(* Title: ZF/AC/WO1_AC.thy
Author: Krzysztof Grabczewski
The proofs of WO1 ==> AC1 and WO1 ==> AC10(n) for n >= 1
The latter proof is referred to as clear by the Rubins.
However it seems to be quite complicated.
The formal proof presented below is a mechanisation of the proof
by Lawrence C. Paulson which is the following:
Assume WO1. Let s be a set of infinite sets.
Suppose x \<in> s. Then x is equipollent to |x| (by WO1), an infinite cardinal
call it K. Since K = K \<oplus> K = |K+K| (by InfCard_cdouble_eq) there is an
isomorphism h \<in> bij(K+K, x). (Here + means disjoint sum.)
So there is a partition of x into 2-element sets, namely
{{h(Inl(i)), h(Inr(i))} . i \<in> K}
So for all x \<in> s the desired partition exists. By AC1 (which follows from WO1)
there exists a function f that chooses a partition for each x \<in> s. Therefore we
have AC10(2).
*)
theory WO1_AC
imports AC_Equiv
begin
(* ********************************************************************** *)
(* WO1 ==> AC1 *)
(* ********************************************************************** *)
theorem WO1_AC1: "WO1 ==> AC1"
by (unfold AC1_def WO1_def, fast elim!: ex_choice_fun)
(* ********************************************************************** *)
(* WO1 ==> AC10(n) (n >= 1) *)
(* ********************************************************************** *)
lemma lemma1: "[| WO1; \B \ A. \C \ D(B). P(C,B) |] ==> \f. \B \ A. P(f`B,B)"
apply (unfold WO1_def)
apply (erule_tac x = "\({{C \ D (B) . P (C,B) }. B \ A}) " in allE)
apply (erule exE, drule ex_choice_fun, fast)
apply (erule exE)
apply (rule_tac x = "\x \ A. f`{C \ D (x) . P (C,x) }" in exI)
apply (simp, blast dest!: apply_type [OF _ RepFunI])
done
lemma lemma2_1: "[| ~Finite(B); WO1 |] ==> |B| + |B| \ B"
apply (unfold WO1_def)
apply (rule eqpoll_trans)
prefer 2 apply (fast elim!: well_ord_cardinal_eqpoll)
apply (rule eqpoll_sym [THEN eqpoll_trans])
apply (fast elim!: well_ord_cardinal_eqpoll)
apply (drule spec [of _ B])
apply (clarify dest!: eqpoll_imp_Finite_iff [OF well_ord_cardinal_eqpoll])
apply (simp add: cadd_def [symmetric]
eqpoll_refl InfCard_cdouble_eq Card_cardinal Inf_Card_is_InfCard)
done
lemma lemma2_2:
"f \ bij(D+D, B) ==> {{f`Inl(i), f`Inr(i)}. i \ D} \ Pow(Pow(B))"
by (fast elim!: bij_is_fun [THEN apply_type])
lemma lemma2_3:
"f \ bij(D+D, B) ==> pairwise_disjoint({{f`Inl(i), f`Inr(i)}. i \ D})"
apply (unfold pairwise_disjoint_def)
apply (blast dest: bij_is_inj [THEN inj_apply_equality])
done
lemma lemma2_4:
"[| f \ bij(D+D, B); 1\n |]
==> sets_of_size_between({{f`Inl(i), f`Inr(i)}. i \<in> D}, 2, succ(n))"
apply (simp (no_asm_simp) add: sets_of_size_between_def succ_def)
apply (blast intro!: cons_lepoll_cong
intro: singleton_eqpoll_1 [THEN eqpoll_imp_lepoll]
le_imp_subset [THEN subset_imp_lepoll] lepoll_trans
dest: bij_is_inj [THEN inj_apply_equality] elim!: mem_irrefl)
done
lemma lemma2_5:
"f \ bij(D+D, B) ==> \({{f`Inl(i), f`Inr(i)}. i \ D})=B"
apply (unfold bij_def surj_def)
apply (fast elim!: inj_is_fun [THEN apply_type])
done
lemma lemma2:
"[| WO1; ~Finite(B); 1\n |]
==> \<exists>C \<in> Pow(Pow(B)). pairwise_disjoint(C) &
sets_of_size_between(C, 2, succ(n)) &
\<Union>(C)=B"
apply (drule lemma2_1 [THEN eqpoll_def [THEN def_imp_iff, THEN iffD1]],
assumption)
apply (blast intro!: lemma2_2 lemma2_3 lemma2_4 lemma2_5)
done
theorem WO1_AC10: "[| WO1; 1\n |] ==> AC10(n)"
apply (unfold AC10_def)
apply (fast intro!: lemma1 elim!: lemma2)
done
end
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