author | paulson |
Thu, 10 Sep 1998 17:42:44 +0200 | |
changeset 5470 | 855654b691db |
parent 5265 | 9d1d4c43c76d |
child 6112 | 5e4871c5136b |
permissions | -rw-r--r-- |
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(* Title: ZF/AC/WO1_AC.ML |
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ID: $Id$ |
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Author: Krzysztof Grabczewski |
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The proofs of WO1 ==> AC1 and WO1 ==> AC10(n) for n >= 1 |
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The latter proof is referred to as clear by the Rubins. |
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However it seems to be quite complicated. |
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The formal proof presented below is a mechanisation of the proof |
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by Lawrence C. Paulson which is the following: |
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Assume WO1. Let s be a set of infinite sets. |
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Suppose x:s. Then x is equipollent to |x| (by WO1), an infinite cardinal; |
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call it K. Since K = K |+| K = |K+K| (by InfCard_cdouble_eq) there is an |
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isomorphism h: bij(K+K, x). (Here + means disjoint sum.) |
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So there is a partition of x into 2-element sets, namely |
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{{h(Inl(i)), h(Inr(i))} . i:K} |
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So for all x:s the desired partition exists. By AC1 (which follows from WO1) |
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there exists a function f that chooses a partition for each x:s. Therefore we |
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have AC10(2). |
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*) |
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open WO1_AC; |
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(* ********************************************************************** *) |
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(* WO1 ==> AC1 *) |
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(* ********************************************************************** *) |
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Goalw [AC1_def, WO1_def] "WO1 ==> AC1"; |
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by (fast_tac (claset() addSEs [ex_choice_fun]) 1); |
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qed "WO1_AC1"; |
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(* ********************************************************************** *) |
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(* WO1 ==> AC10(n) (n >= 1) *) |
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(* ********************************************************************** *) |
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Goalw [WO1_def] "[| WO1; ALL B:A. EX C:D(B). P(C,B) |] \ |
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\ ==> EX f. ALL B:A. P(f`B,B)"; |
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by (eres_inst_tac [("x","Union({{C:D(B). P(C,B)}. B:A})")] allE 1); |
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by (etac exE 1); |
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by (dtac ex_choice_fun 1); |
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by (Fast_tac 1); |
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by (etac exE 1); |
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by (res_inst_tac [("x","lam x:A. f`{C:D(x). P(C,x)}")] exI 1); |
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by (Asm_full_simp_tac 1); |
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by (blast_tac (claset() addSDs [RepFunI RSN (2, apply_type)]) 1); |
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val lemma1 = result(); |
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Goalw [WO1_def] "[| ~Finite(B); WO1 |] ==> |B| + |B| eqpoll B"; |
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by (rtac eqpoll_trans 1); |
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by (fast_tac (claset() addSEs [well_ord_cardinal_eqpoll]) 2); |
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by (resolve_tac [eqpoll_sym RS eqpoll_trans] 1); |
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by (fast_tac (claset() addSEs [well_ord_cardinal_eqpoll]) 1); |
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by (resolve_tac [cadd_def RS def_imp_eq RS subst] 1); |
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by (resolve_tac [Card_cardinal RSN (2, Inf_Card_is_InfCard) RS |
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InfCard_cdouble_eq RS ssubst] 1); |
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by (rtac eqpoll_refl 2); |
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by (rtac notI 1); |
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by (etac notE 1); |
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by (resolve_tac [eqpoll_sym RS eqpoll_imp_lepoll RS lepoll_Finite] 1 |
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THEN (assume_tac 2)); |
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by (fast_tac (claset() addSEs [well_ord_cardinal_eqpoll]) 1); |
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val lemma2_1 = result(); |
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Goal "f : bij(D+D, B) ==> {{f`Inl(i), f`Inr(i)}. i:D} : Pow(Pow(B))"; |
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by (fast_tac (claset() addSEs [bij_is_fun RS apply_type]) 1); |
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val lemma2_2 = result(); |
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Goal "[| f:inj(A,B); f`a = f`b; a:A; b:A |] ==> a=b"; |
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by (rtac inj_equality 1); |
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by (TRYALL (fast_tac (claset() addSEs [inj_is_fun RS apply_Pair] addEs [subst]))); |
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val lemma = result(); |
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Goalw AC_aux_defs |
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"f : bij(D+D, B) ==> pairwise_disjoint({{f`Inl(i), f`Inr(i)}. i:D})"; |
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by (blast_tac (claset() addDs [bij_is_inj RS lemma]) 1); |
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val lemma2_3 = result(); |
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val [major, minor] = goalw thy AC_aux_defs |
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"[| f : bij(D+D, B); 1 le n |] ==> \ |
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\ sets_of_size_between({{f`Inl(i), f`Inr(i)}. i:D}, 2, succ(n))"; |
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by (rewtac succ_def); |
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by (fast_tac (claset() |
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addSIs [cons_lepoll_cong, minor, lepoll_refl] |
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addIs [singleton_eqpoll_1 RS eqpoll_imp_lepoll RS lepoll_trans, |
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le_imp_subset RS subset_imp_lepoll] |
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addDs [major RS bij_is_inj RS lemma] |
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addSEs [mem_irrefl]) 1); |
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val lemma2_4 = result(); |
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Goalw [bij_def, surj_def] |
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"f : bij(D+D, B) ==> Union({{f`Inl(i), f`Inr(i)}. i:D})=B"; |
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by (fast_tac (claset() addSEs [inj_is_fun RS apply_type]) 1); |
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val lemma2_5 = result(); |
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Goal "[| WO1; ~Finite(B); 1 le n |] \ |
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\ ==> EX C:Pow(Pow(B)). pairwise_disjoint(C) & \ |
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\ sets_of_size_between(C, 2, succ(n)) & \ |
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\ Union(C)=B"; |
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by (eresolve_tac [lemma2_1 RS (eqpoll_def RS def_imp_iff RS iffD1 RS exE)] 1 |
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THEN (assume_tac 1)); |
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by (fast_tac (FOL_cs addSIs [bexI] |
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addSEs [lemma2_2, lemma2_3, lemma2_4, lemma2_5]) 1); |
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val lemma2 = result(); |
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Goalw AC_defs "[| WO1; 1 le n |] ==> AC10(n)"; |
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by (fast_tac (claset() addSIs [lemma1] addSEs [lemma2]) 1); |
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qed "WO1_AC10"; |