# Theory WO1_AC

theory WO1_AC
imports AC_Equiv
```(*  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 ∈ 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 = "⋃({{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])
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) ==> ⋃({{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)) &
⋃(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

```