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(* Title: ZF/AC/WO1_AC.ML
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ID: $Id$
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Author: Krzysztof Gr`abczewski
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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 thy [AC1_def, WO1_def] "!!Z. WO1 ==> AC1";
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by (fast_tac (AC_cs 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 thy [WO1_def] "!!A. [| 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 (eresolve_tac [exE] 1);
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by (dresolve_tac [ex_choice_fun] 1);
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by (fast_tac (AC_cs addEs [RepFunE, sym RS equals0D]) 1);
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by (eresolve_tac [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 AC_ss 1);
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by (fast_tac (AC_cs addSDs [RepFunI RSN (2, apply_type)]
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addSEs [CollectD2]) 1);
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val lemma1 = result();
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goalw thy [WO1_def] "!!A. [| ~Finite(B); WO1 |] ==> |B| + |B| eqpoll B";
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by (resolve_tac [eqpoll_trans] 1);
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by (fast_tac (AC_cs 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 (AC_cs 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 (resolve_tac [eqpoll_refl] 2);
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by (resolve_tac [notI] 1);
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by (eresolve_tac [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 (AC_cs addSEs [well_ord_cardinal_eqpoll]) 1);
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val lemma2_1 = result();
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goal thy "!!f. f : bij(D+D, B) ==> {{f`Inl(i), f`Inr(i)}. i:D} : Pow(Pow(B))";
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by (fast_tac (AC_cs addSIs [InlI, InrI]
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addSEs [RepFunE, bij_is_fun RS apply_type]) 1);
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val lemma2_2 = result();
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goal thy "!!f. [| f:inj(A,B); f`a = f`b; a:A; b:A |] ==> a=b";
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by (resolve_tac [inj_equality] 1);
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by (TRYALL (fast_tac (AC_cs addSEs [inj_is_fun RS apply_Pair] addEs [subst])));
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val lemma = result();
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goalw thy AC_aux_defs
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"!!f. f : bij(D+D, B) ==> \
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\ pairwise_disjoint({{f`Inl(i), f`Inr(i)}. i:D})";
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by (fast_tac (AC_cs addSEs [RepFunE, not_emptyE]
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addDs [bij_is_inj RS lemma, Inl_iff RS iffD1,
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Inr_iff RS iffD1, Inl_Inr_iff RS iffD1 RS FalseE,
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Inr_Inl_iff RS iffD1 RS FalseE]
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addSIs [InlI, InrI]) 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 (rewrite_goals_tac [succ_def]);
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by (fast_tac (AC_cs addSIs [cons_lepoll_cong, minor, lepoll_refl, InlI, InrI]
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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, Inl_Inr_iff RS iffD1 RS FalseE]
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addSEs [RepFunE, not_emptyE, mem_irrefl]) 1);
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val lemma2_4 = result();
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goalw thy [bij_def, surj_def]
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"!!f. f : bij(D+D, B) ==> Union({{f`Inl(i), f`Inr(i)}. i:D})=B";
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by (fast_tac (AC_cs addSEs [inj_is_fun RS apply_type, CollectE, sumE]
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addSIs [InlI, InrI, equalityI]) 1);
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val lemma2_5 = result();
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goal thy "!!A. [| 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 thy AC_defs "!!n. [| WO1; 1 le n |] ==> AC10(n)";
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by (fast_tac (AC_cs addSIs [lemma1] addSEs [lemma2]) 1);
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qed "WO1_AC10";
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