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(* Title: ZF/Cardinal_AC.thy
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ID: $Id$
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1478
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1994 University of Cambridge
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Cardinal Arithmetic WITH the Axiom of Choice
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These results help justify infinite-branching datatypes
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*)
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theory Cardinal_AC = CardinalArith + Zorn:
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(*** Strengthened versions of existing theorems about cardinals ***)
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lemma cardinal_eqpoll: "|A| eqpoll A"
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apply (rule AC_well_ord [THEN exE])
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apply (erule well_ord_cardinal_eqpoll)
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done
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lemmas cardinal_idem = cardinal_eqpoll [THEN cardinal_cong, standard]
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lemma cardinal_eqE: "|X| = |Y| ==> X eqpoll Y"
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apply (rule AC_well_ord [THEN exE])
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apply (rule AC_well_ord [THEN exE])
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apply (rule well_ord_cardinal_eqE)
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apply assumption+
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done
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lemma cardinal_eqpoll_iff: "|X| = |Y| <-> X eqpoll Y"
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apply (blast intro: cardinal_cong cardinal_eqE)
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done
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lemma cardinal_disjoint_Un: "[| |A|=|B|; |C|=|D|; A Int C = 0; B Int D = 0 |] ==>
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|A Un C| = |B Un D|"
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apply (simp add: cardinal_eqpoll_iff eqpoll_disjoint_Un)
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done
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lemma lepoll_imp_Card_le: "A lepoll B ==> |A| le |B|"
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apply (rule AC_well_ord [THEN exE])
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apply (erule well_ord_lepoll_imp_Card_le)
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apply assumption
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done
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lemma cadd_assoc: "(i |+| j) |+| k = i |+| (j |+| k)"
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apply (rule AC_well_ord [THEN exE])
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apply (rule AC_well_ord [THEN exE])
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apply (rule AC_well_ord [THEN exE])
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apply (rule well_ord_cadd_assoc)
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apply assumption+
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done
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lemma cmult_assoc: "(i |*| j) |*| k = i |*| (j |*| k)"
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apply (rule AC_well_ord [THEN exE])
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apply (rule AC_well_ord [THEN exE])
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apply (rule AC_well_ord [THEN exE])
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apply (rule well_ord_cmult_assoc)
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apply assumption+
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done
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lemma cadd_cmult_distrib: "(i |+| j) |*| k = (i |*| k) |+| (j |*| k)"
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apply (rule AC_well_ord [THEN exE])
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apply (rule AC_well_ord [THEN exE])
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apply (rule AC_well_ord [THEN exE])
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apply (rule well_ord_cadd_cmult_distrib)
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apply assumption+
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done
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lemma InfCard_square_eq: "InfCard(|A|) ==> A*A eqpoll A"
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apply (rule AC_well_ord [THEN exE])
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apply (erule well_ord_InfCard_square_eq)
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apply assumption
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done
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(*** Other applications of AC ***)
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lemma Card_le_imp_lepoll: "|A| le |B| ==> A lepoll B"
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apply (rule cardinal_eqpoll
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[THEN eqpoll_sym, THEN eqpoll_imp_lepoll, THEN lepoll_trans])
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apply (erule le_imp_subset [THEN subset_imp_lepoll, THEN lepoll_trans])
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apply (rule cardinal_eqpoll [THEN eqpoll_imp_lepoll])
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done
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lemma le_Card_iff: "Card(K) ==> |A| le K <-> A lepoll K"
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apply (erule Card_cardinal_eq [THEN subst], rule iffI,
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erule Card_le_imp_lepoll);
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apply (erule lepoll_imp_Card_le)
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done
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lemma surj_implies_inj: "f: surj(X,Y) ==> EX g. g: inj(Y,X)"
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apply (unfold surj_def)
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apply (erule CollectE)
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apply (rule_tac A1 = "Y" and B1 = "%y. f-``{y}" in AC_Pi [THEN exE])
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apply (fast elim!: apply_Pair)
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apply (blast dest: apply_type Pi_memberD intro: apply_equality Pi_type f_imp_injective)
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done
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(*Kunen's Lemma 10.20*)
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lemma surj_implies_cardinal_le: "f: surj(X,Y) ==> |Y| le |X|"
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apply (rule lepoll_imp_Card_le)
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apply (erule surj_implies_inj [THEN exE])
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apply (unfold lepoll_def)
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apply (erule exI)
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done
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(*Kunen's Lemma 10.21*)
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lemma cardinal_UN_le: "[| InfCard(K); ALL i:K. |X(i)| le K |] ==> |UN i:K. X(i)| le K"
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apply (simp add: InfCard_is_Card le_Card_iff)
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apply (rule lepoll_trans)
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prefer 2
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apply (rule InfCard_square_eq [THEN eqpoll_imp_lepoll])
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apply (simp add: InfCard_is_Card Card_cardinal_eq)
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apply (unfold lepoll_def)
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apply (frule InfCard_is_Card [THEN Card_is_Ord])
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apply (erule AC_ball_Pi [THEN exE])
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apply (rule exI)
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(*Lemma needed in both subgoals, for a fixed z*)
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apply (subgoal_tac "ALL z: (UN i:K. X (i)). z: X (LEAST i. z:X (i)) & (LEAST i. z:X (i)) : K")
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prefer 2
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apply (fast intro!: Least_le [THEN lt_trans1, THEN ltD] ltI
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elim!: LeastI Ord_in_Ord)
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apply (rule_tac c = "%z. <LEAST i. z:X (i) , f ` (LEAST i. z:X (i)) ` z>"
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and d = "%<i,j>. converse (f`i) ` j" in lam_injective)
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(*Instantiate the lemma proved above*)
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apply (blast intro: inj_is_fun [THEN apply_type] dest: apply_type)
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apply (force );
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done
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(*The same again, using csucc*)
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lemma cardinal_UN_lt_csucc:
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"[| InfCard(K); ALL i:K. |X(i)| < csucc(K) |]
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==> |UN i:K. X(i)| < csucc(K)"
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apply (simp add: Card_lt_csucc_iff cardinal_UN_le InfCard_is_Card Card_cardinal)
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done
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(*The same again, for a union of ordinals. In use, j(i) is a bit like rank(i),
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the least ordinal j such that i:Vfrom(A,j). *)
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lemma cardinal_UN_Ord_lt_csucc:
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"[| InfCard(K); ALL i:K. j(i) < csucc(K) |]
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==> (UN i:K. j(i)) < csucc(K)"
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apply (rule cardinal_UN_lt_csucc [THEN Card_lt_imp_lt])
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apply assumption
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apply (blast intro: Ord_cardinal_le [THEN lt_trans1] elim: ltE)
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apply (blast intro!: Ord_UN elim: ltE)
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apply (erule InfCard_is_Card [THEN Card_is_Ord, THEN Card_csucc])
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done
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(** Main result for infinite-branching datatypes. As above, but the index
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set need not be a cardinal. Surprisingly complicated proof!
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**)
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(*Work backwards along the injection from W into K, given that W~=0.*)
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lemma inj_UN_subset:
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"[| f: inj(A,B); a:A |] ==>
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(UN x:A. C(x)) <= (UN y:B. C(if y: range(f) then converse(f)`y else a))"
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apply (rule UN_least)
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apply (rule_tac x1= "f`x" in subset_trans [OF _ UN_upper])
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apply (simp add: inj_is_fun [THEN apply_rangeI])
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apply (blast intro: inj_is_fun [THEN apply_type])
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done
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(*Simpler to require |W|=K; we'd have a bijection; but the theorem would
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be weaker.*)
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lemma le_UN_Ord_lt_csucc:
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"[| InfCard(K); |W| le K; ALL w:W. j(w) < csucc(K) |]
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==> (UN w:W. j(w)) < csucc(K)"
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apply (case_tac "W=0")
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(*solve the easy 0 case*)
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apply (simp add: InfCard_is_Card Card_is_Ord [THEN Card_csucc]
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Card_is_Ord Ord_0_lt_csucc)
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apply (simp add: InfCard_is_Card le_Card_iff lepoll_def)
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apply (safe intro!: equalityI)
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apply (erule swap);
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apply (rule lt_subset_trans [OF inj_UN_subset cardinal_UN_Ord_lt_csucc])
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apply assumption+
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apply (simp add: inj_converse_fun [THEN apply_type])
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apply (blast intro!: Ord_UN elim: ltE)
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done
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end
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