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1 (* Title: ZF/AC/WO1_AC.ML |
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2 ID: $Id$ |
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3 Author: Krzysztof Gr`abczewski |
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4 |
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5 The proofs of WO1 ==> AC1 and WO1 ==> AC10(n) for n >= 1 |
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6 |
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7 The latter proof is referred to as clear by the Rubins. |
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8 However it seems to be quite complicated. |
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9 The formal proof presented below is a mechanisation of the proof |
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10 by Lawrence C. Paulson which is the following: |
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11 |
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12 Assume WO1. Let s be a set of infinite sets. |
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13 |
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14 Suppose x:s. Then x is equipollent to |x| (by WO1), an infinite cardinal; |
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15 call it K. Since K = K |+| K = |K+K| (by InfCard_cdouble_eq) there is an |
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16 isomorphism h: bij(K+K, x). (Here + means disjoint sum.) |
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17 |
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18 So there is a partition of x into 2-element sets, namely |
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19 |
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20 {{h(Inl(i)), h(Inr(i))} . i:K} |
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21 |
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22 So for all x:s the desired partition exists. By AC1 (which follows from WO1) |
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23 there exists a function f that chooses a partition for each x:s. Therefore we |
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24 have AC10(2). |
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25 |
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26 *) |
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27 |
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28 open WO1_AC; |
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29 |
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30 (* ********************************************************************** *) |
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31 (* WO1 ==> AC1 *) |
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32 (* ********************************************************************** *) |
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33 |
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34 goalw thy [AC1_def, WO1_def] "!!Z. WO1 ==> AC1"; |
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35 by (fast_tac (AC_cs addSEs [ex_choice_fun]) 1); |
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36 qed "WO1_AC1"; |
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37 |
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38 (* ********************************************************************** *) |
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39 (* WO1 ==> AC10(n) (n >= 1) *) |
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40 (* ********************************************************************** *) |
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41 |
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42 goalw thy [WO1_def] "!!A. [| WO1; ALL B:A. EX C:D(B). P(C,B) |] \ |
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43 \ ==> EX f. ALL B:A. P(f`B,B)"; |
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44 by (eres_inst_tac [("x","Union({{C:D(B). P(C,B)}. B:A})")] allE 1); |
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45 by (eresolve_tac [exE] 1); |
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46 by (dresolve_tac [ex_choice_fun] 1); |
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47 by (fast_tac (AC_cs addEs [RepFunE, sym RS equals0D]) 1); |
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48 by (eresolve_tac [exE] 1); |
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49 by (res_inst_tac [("x","lam x:A. f`{C:D(x). P(C,x)}")] exI 1); |
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50 by (asm_full_simp_tac AC_ss 1); |
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51 by (fast_tac (AC_cs addSDs [RepFunI RSN (2, apply_type)] |
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52 addSEs [CollectD2]) 1); |
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53 val lemma1 = result(); |
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54 |
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55 goalw thy [WO1_def] "!!A. [| ~Finite(B); WO1 |] ==> |B| + |B| eqpoll B"; |
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56 by (resolve_tac [eqpoll_trans] 1); |
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57 by (fast_tac (AC_cs addSEs [well_ord_cardinal_eqpoll]) 2); |
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58 by (resolve_tac [eqpoll_sym RS eqpoll_trans] 1); |
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59 by (fast_tac (AC_cs addSEs [well_ord_cardinal_eqpoll]) 1); |
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60 by (resolve_tac [cadd_def RS def_imp_eq RS subst] 1); |
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61 by (resolve_tac [Card_cardinal RSN (2, Inf_Card_is_InfCard) RS |
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62 InfCard_cdouble_eq RS ssubst] 1); |
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63 by (resolve_tac [eqpoll_refl] 2); |
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64 by (resolve_tac [notI] 1); |
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65 by (eresolve_tac [notE] 1); |
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66 by (resolve_tac [eqpoll_sym RS eqpoll_imp_lepoll RS lepoll_Finite] 1 |
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67 THEN (assume_tac 2)); |
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68 by (fast_tac (AC_cs addSEs [well_ord_cardinal_eqpoll]) 1); |
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69 val lemma2_1 = result(); |
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70 |
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71 goal thy "!!f. f : bij(D+D, B) ==> {{f`Inl(i), f`Inr(i)}. i:D} : Pow(Pow(B))"; |
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72 by (fast_tac (AC_cs addSIs [InlI, InrI] |
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73 addSEs [RepFunE, bij_is_fun RS apply_type]) 1); |
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74 val lemma2_2 = result(); |
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75 |
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76 goal thy "!!f. [| f:inj(A,B); f`a = f`b; a:A; b:A |] ==> a=b"; |
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77 by (resolve_tac [inj_equality] 1); |
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78 by (TRYALL (fast_tac (AC_cs addSEs [inj_is_fun RS apply_Pair] addEs [subst]))); |
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79 val lemma = result(); |
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80 |
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81 goalw thy AC_aux_defs |
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82 "!!f. f : bij(D+D, B) ==> \ |
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83 \ pairwise_disjoint({{f`Inl(i), f`Inr(i)}. i:D})"; |
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84 by (fast_tac (AC_cs addSEs [RepFunE, not_emptyE] |
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85 addDs [bij_is_inj RS lemma, Inl_iff RS iffD1, |
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86 Inr_iff RS iffD1, Inl_Inr_iff RS iffD1 RS FalseE, |
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87 Inr_Inl_iff RS iffD1 RS FalseE] |
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88 addSIs [InlI, InrI]) 1); |
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89 val lemma2_3 = result(); |
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90 |
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91 val [major, minor] = goalw thy AC_aux_defs |
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92 "[| f : bij(D+D, B); 1 le n |] ==> \ |
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93 \ sets_of_size_between({{f`Inl(i), f`Inr(i)}. i:D}, 2, succ(n))"; |
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94 by (rewrite_goals_tac [succ_def]); |
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95 by (fast_tac (AC_cs addSIs [cons_lepoll_cong, minor, lepoll_refl, InlI, InrI] |
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96 addIs [singleton_eqpoll_1 RS eqpoll_imp_lepoll RS lepoll_trans, |
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97 le_imp_subset RS subset_imp_lepoll] |
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98 addDs [major RS bij_is_inj RS lemma, Inl_Inr_iff RS iffD1 RS FalseE] |
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99 addSEs [RepFunE, not_emptyE, mem_irrefl]) 1); |
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100 val lemma2_4 = result(); |
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101 |
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102 goalw thy [bij_def, surj_def] |
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103 "!!f. f : bij(D+D, B) ==> Union({{f`Inl(i), f`Inr(i)}. i:D})=B"; |
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104 by (fast_tac (AC_cs addSEs [inj_is_fun RS apply_type, CollectE, sumE] |
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105 addSIs [InlI, InrI, equalityI]) 1); |
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106 val lemma2_5 = result(); |
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107 |
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108 goal thy "!!A. [| WO1; ~Finite(B); 1 le n |] \ |
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109 \ ==> EX C:Pow(Pow(B)). pairwise_disjoint(C) & \ |
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110 \ sets_of_size_between(C, 2, succ(n)) & \ |
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111 \ Union(C)=B"; |
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112 by (eresolve_tac [lemma2_1 RS (eqpoll_def RS def_imp_iff RS iffD1 RS exE)] 1 |
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113 THEN (assume_tac 1)); |
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114 by (fast_tac (FOL_cs addSIs [bexI] |
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115 addSEs [lemma2_2, lemma2_3, lemma2_4, lemma2_5]) 1); |
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116 val lemma2 = result(); |
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117 |
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118 goalw thy AC_defs "!!n. [| WO1; 1 le n |] ==> AC10(n)"; |
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119 by (fast_tac (AC_cs addSIs [lemma1] addSEs [lemma2]) 1); |
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120 qed "WO1_AC10"; |