(* Title: ZF/AC/WO1_AC.thy ID: $Id: WO1_AC.thy,v 1.3 2005/06/17 14:15:10 haftmann Exp $ 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 ∈ s. Then x is equipollent to |x| (by WO1), an infinite cardinal call it K. Since K = K |+| K = |K+K| (by InfCard_cdouble_eq) there is an isomorphism h ∈ 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 ∈ K} So for all x ∈ s the desired partition exists. By AC1 (which follows from WO1) there exists a function f that chooses a partition for each x ∈ 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 = "Union ({{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 = "λx ∈ 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 ∈ 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) ==> Union({{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 |] ==> ∃C ∈ 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
theorem WO1_AC1:
WO1 ==> AC1
lemma lemma1:
[| WO1; ∀B∈A. ∃C∈D(B). P(C, B) |] ==> ∃f. ∀B∈A. P(f ` B, B)
lemma lemma2_1:
[| ¬ Finite(B); WO1 |] ==> |B| + |B| ≈ B
lemma lemma2_2:
f ∈ bij(D + D, B) ==> {{f ` Inl(i), f ` Inr(i)} . i ∈ D} ∈ Pow(Pow(B))
lemma lemma2_3:
f ∈ bij(D + D, B) ==> pairwise_disjoint({{f ` Inl(i), f ` Inr(i)} . i ∈ D})
lemma lemma2_4:
[| f ∈ bij(D + D, B); 1 ≤ n |]
==> sets_of_size_between({{f ` Inl(i), f ` Inr(i)} . i ∈ D}, 2, succ(n))
lemma lemma2_5:
f ∈ bij(D + D, B) ==> (\<Union>i∈D. {f ` Inl(i), f ` Inr(i)}) = B
lemma lemma2:
[| WO1; ¬ Finite(B); 1 ≤ n |]
==> ∃C∈Pow(Pow(B)).
pairwise_disjoint(C) ∧
sets_of_size_between(C, 2, succ(n)) ∧ \<Union>C = B
theorem WO1_AC10:
[| WO1; 1 ≤ n |] ==> AC10(n)