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(* Title: ZF/AC/Hartog.thy
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ID: $Id$
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Author: Krzysztof Grabczewski
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Hartog's function.
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*)
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theory Hartog = AC_Equiv:
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constdefs
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Hartog :: "i => i"
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"Hartog(X) == LEAST i. ~ i \<lesssim> X"
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lemma Ords_in_set: "\<forall>a. Ord(a) --> a \<in> X ==> P"
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apply (rule_tac X1 = "{y \<in> X. Ord (y) }" in ON_class [THEN revcut_rl])
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apply fast
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done
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lemma Ord_lepoll_imp_ex_well_ord:
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"[| Ord(a); a \<lesssim> X |]
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==> \<exists>Y. Y \<subseteq> X & (\<exists>R. well_ord(Y,R) & ordertype(Y,R)=a)"
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apply (unfold lepoll_def)
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apply (erule exE)
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apply (intro exI conjI)
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apply (erule inj_is_fun [THEN fun_is_rel, THEN image_subset])
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apply (rule well_ord_rvimage [OF bij_is_inj well_ord_Memrel])
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apply (erule restrict_bij [THEN bij_converse_bij])
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apply (rule subset_refl, assumption)
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apply (rule trans)
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apply (rule bij_ordertype_vimage)
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apply (erule restrict_bij [THEN bij_converse_bij])
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apply (rule subset_refl)
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apply (erule well_ord_Memrel)
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apply (erule ordertype_Memrel)
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done
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lemma Ord_lepoll_imp_eq_ordertype:
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"[| Ord(a); a \<lesssim> X |] ==> \<exists>Y. Y \<subseteq> X & (\<exists>R. R \<subseteq> X*X & ordertype(Y,R)=a)"
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apply (drule Ord_lepoll_imp_ex_well_ord, assumption, clarify)
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apply (intro exI conjI)
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apply (erule_tac [3] ordertype_Int, auto)
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done
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lemma Ords_lepoll_set_lemma:
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"(\<forall>a. Ord(a) --> a \<lesssim> X) ==>
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\<forall>a. Ord(a) -->
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a \<in> {b. Z \<in> Pow(X)*Pow(X*X), \<exists>Y R. Z=<Y,R> & ordertype(Y,R)=b}"
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apply (intro allI impI)
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apply (elim allE impE, assumption)
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apply (blast dest!: Ord_lepoll_imp_eq_ordertype intro: sym)
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done
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lemma Ords_lepoll_set: "\<forall>a. Ord(a) --> a \<lesssim> X ==> P"
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by (erule Ords_lepoll_set_lemma [THEN Ords_in_set])
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lemma ex_Ord_not_lepoll: "\<exists>a. Ord(a) & ~a \<lesssim> X"
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apply (rule ccontr)
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apply (best intro: Ords_lepoll_set)
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done
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lemma not_Hartog_lepoll_self: "~ Hartog(A) \<lesssim> A"
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apply (unfold Hartog_def)
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apply (rule ex_Ord_not_lepoll [THEN exE])
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apply (rule LeastI, auto)
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done
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lemmas Hartog_lepoll_selfE = not_Hartog_lepoll_self [THEN notE, standard]
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lemma Ord_Hartog: "Ord(Hartog(A))"
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by (unfold Hartog_def, rule Ord_Least)
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lemma less_HartogE1: "[| i < Hartog(A); ~ i \<lesssim> A |] ==> P"
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by (unfold Hartog_def, fast elim: less_LeastE)
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lemma less_HartogE: "[| i < Hartog(A); i \<approx> Hartog(A) |] ==> P"
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by (blast intro: less_HartogE1 eqpoll_sym eqpoll_imp_lepoll
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lepoll_trans [THEN Hartog_lepoll_selfE]);
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lemma Card_Hartog: "Card(Hartog(A))"
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by (fast intro!: CardI Ord_Hartog elim: less_HartogE)
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end
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