Theory AC16_lemmas

theory AC16_lemmas
imports Hartog Cardinal_aux
(*  Title:      ZF/AC/AC16_lemmas.thy
Author: Krzysztof Grabczewski

Lemmas used in the proofs concerning AC16
*)


theory AC16_lemmas
imports AC_Equiv Hartog Cardinal_aux
begin

lemma cons_Diff_eq: "a∉A ==> cons(a,A)-{a}=A"
by fast

lemma nat_1_lepoll_iff: "1\<lesssim>X <-> (∃x. x ∈ X)"
apply (unfold lepoll_def)
apply (rule iffI)
apply (fast intro: inj_is_fun [THEN apply_type])
apply (erule exE)
apply (rule_tac x = "λa ∈ 1. x" in exI)
apply (fast intro!: lam_injective)
done

lemma eqpoll_1_iff_singleton: "X≈1 <-> (∃x. X={x})"
apply (rule iffI)
apply (erule eqpollE)
apply (drule nat_1_lepoll_iff [THEN iffD1])
apply (fast intro!: lepoll_1_is_sing)
apply (fast intro!: singleton_eqpoll_1)
done

lemma cons_eqpoll_succ: "[| x≈n; y∉x |] ==> cons(y,x)≈succ(n)"
apply (unfold succ_def)
apply (fast elim!: cons_eqpoll_cong mem_irrefl)
done

lemma subsets_eqpoll_1_eq: "{Y ∈ Pow(X). Y≈1} = {{x}. x ∈ X}"
apply (rule equalityI)
apply (rule subsetI)
apply (erule CollectE)
apply (drule eqpoll_1_iff_singleton [THEN iffD1])
apply (fast intro!: RepFunI)
apply (rule subsetI)
apply (erule RepFunE)
apply (rule CollectI, fast)
apply (fast intro!: singleton_eqpoll_1)
done

lemma eqpoll_RepFun_sing: "X≈{{x}. x ∈ X}"
apply (unfold eqpoll_def bij_def)
apply (rule_tac x = "λx ∈ X. {x}" in exI)
apply (rule IntI)
apply (unfold inj_def surj_def, simp)
apply (fast intro!: lam_type RepFunI intro: singleton_eq_iff [THEN iffD1], simp)
apply (fast intro!: lam_type)
done

lemma subsets_eqpoll_1_eqpoll: "{Y ∈ Pow(X). Y≈1}≈X"
apply (rule subsets_eqpoll_1_eq [THEN ssubst])
apply (rule eqpoll_RepFun_sing [THEN eqpoll_sym])
done

lemma InfCard_Least_in:
"[| InfCard(x); y ⊆ x; y ≈ succ(z) |] ==> (LEAST i. i ∈ y) ∈ y"
apply (erule eqpoll_sym [THEN eqpoll_imp_lepoll,
THEN succ_lepoll_imp_not_empty, THEN not_emptyE])
apply (fast intro: LeastI
dest!: InfCard_is_Card [THEN Card_is_Ord]
elim: Ord_in_Ord)
done

lemma subsets_lepoll_lemma1:
"[| InfCard(x); n ∈ nat |]
==> {y ∈ Pow(x). y≈succ(succ(n))} \<lesssim> x*{y ∈ Pow(x). y≈succ(n)}"

apply (unfold lepoll_def)
apply (rule_tac x = "λy ∈ {y ∈ Pow(x) . y≈succ (succ (n))}.
<LEAST i. i ∈ y, y-{LEAST i. i ∈ y}>"
in exI)
apply (rule_tac d = "%z. cons (fst(z), snd(z))" in lam_injective)
apply (blast intro!: Diff_sing_eqpoll intro: InfCard_Least_in)
apply (simp, blast intro: InfCard_Least_in)
done

lemma set_of_Ord_succ_Union: "(∀y ∈ z. Ord(y)) ==> z ⊆ succ(\<Union>(z))"
apply (rule subsetI)
apply (case_tac "∀y ∈ z. y ⊆ x", blast )
apply (simp, erule bexE)
apply (rule_tac i=y and j=x in Ord_linear_le)
apply (blast dest: le_imp_subset elim: leE ltE)+
done

lemma subset_not_mem: "j ⊆ i ==> i ∉ j"
by (fast elim!: mem_irrefl)

lemma succ_Union_not_mem:
"(!!y. y ∈ z ==> Ord(y)) ==> succ(\<Union>(z)) ∉ z"
apply (rule set_of_Ord_succ_Union [THEN subset_not_mem], blast)
done

lemma Union_cons_eq_succ_Union:
"\<Union>(cons(succ(\<Union>(z)),z)) = succ(\<Union>(z))"
by fast

lemma Un_Ord_disj: "[| Ord(i); Ord(j) |] ==> i ∪ j = i | i ∪ j = j"
by (fast dest!: le_imp_subset elim: Ord_linear_le)

lemma Union_eq_Un: "x ∈ X ==> \<Union>(X) = x ∪ \<Union>(X-{x})"
by fast

lemma Union_in_lemma [rule_format]:
"n ∈ nat ==> ∀z. (∀y ∈ z. Ord(y)) & z≈n & z≠0 --> \<Union>(z) ∈ z"
apply (induct_tac "n")
apply (fast dest!: eqpoll_imp_lepoll [THEN lepoll_0_is_0])
apply (intro allI impI)
apply (erule natE)
apply (fast dest!: eqpoll_1_iff_singleton [THEN iffD1]
intro!: Union_singleton, clarify)
apply (elim not_emptyE)
apply (erule_tac x = "z-{xb}" in allE)
apply (erule impE)
apply (fast elim!: Diff_sing_eqpoll
Diff_sing_eqpoll [THEN eqpoll_succ_imp_not_empty])
apply (subgoal_tac "xb ∪ \<Union>(z - {xb}) ∈ z")
apply (simp add: Union_eq_Un [symmetric])
apply (frule bspec, assumption)
apply (drule bspec)
apply (erule Diff_subset [THEN subsetD])
apply (drule Un_Ord_disj, assumption, auto)
done

lemma Union_in: "[| ∀x ∈ z. Ord(x); z≈n; z≠0; n ∈ nat |] ==> \<Union>(z) ∈ z"
by (blast intro: Union_in_lemma)

lemma succ_Union_in_x:
"[| InfCard(x); z ∈ Pow(x); z≈n; n ∈ nat |] ==> succ(\<Union>(z)) ∈ x"
apply (rule Limit_has_succ [THEN ltE])
prefer 3 apply assumption
apply (erule InfCard_is_Limit)
apply (case_tac "z=0")
apply (simp, fast intro!: InfCard_is_Limit [THEN Limit_has_0])
apply (rule ltI [OF PowD [THEN subsetD] InfCard_is_Card [THEN Card_is_Ord]], assumption)
apply (blast intro: Union_in
InfCard_is_Card [THEN Card_is_Ord, THEN Ord_in_Ord])+
done

lemma succ_lepoll_succ_succ:
"[| InfCard(x); n ∈ nat |]
==> {y ∈ Pow(x). y≈succ(n)} \<lesssim> {y ∈ Pow(x). y≈succ(succ(n))}"

apply (unfold lepoll_def)
apply (rule_tac x = "λz ∈ {y∈Pow(x). y≈succ(n)}. cons(succ(\<Union>(z)), z)"
in exI)
apply (rule_tac d = "%z. z-{\<Union>(z) }" in lam_injective)
apply (blast intro!: succ_Union_in_x succ_Union_not_mem
intro: cons_eqpoll_succ Ord_in_Ord
dest!: InfCard_is_Card [THEN Card_is_Ord])
apply (simp only: Union_cons_eq_succ_Union)
apply (rule cons_Diff_eq)
apply (fast dest!: InfCard_is_Card [THEN Card_is_Ord]
elim: Ord_in_Ord
intro!: succ_Union_not_mem)
done

lemma subsets_eqpoll_X:
"[| InfCard(X); n ∈ nat |] ==> {Y ∈ Pow(X). Y≈succ(n)} ≈ X"
apply (induct_tac "n")
apply (rule subsets_eqpoll_1_eqpoll)
apply (rule eqpollI)
apply (rule subsets_lepoll_lemma1 [THEN lepoll_trans], assumption+)
apply (rule eqpoll_trans [THEN eqpoll_imp_lepoll])
apply (erule eqpoll_refl [THEN prod_eqpoll_cong])
apply (erule InfCard_square_eqpoll)
apply (fast elim: eqpoll_sym [THEN eqpoll_imp_lepoll, THEN lepoll_trans]
intro!: succ_lepoll_succ_succ)
done

lemma image_vimage_eq:
"[| f ∈ surj(A,B); y ⊆ B |] ==> f``(converse(f)``y) = y"
apply (unfold surj_def)
apply (fast dest: apply_equality2 elim: apply_iff [THEN iffD2])
done

lemma vimage_image_eq: "[| f ∈ inj(A,B); y ⊆ A |] ==> converse(f)``(f``y) = y"
by (fast elim!: inj_is_fun [THEN apply_Pair] dest: inj_equality)

lemma subsets_eqpoll:
"A≈B ==> {Y ∈ Pow(A). Y≈n}≈{Y ∈ Pow(B). Y≈n}"
apply (unfold eqpoll_def)
apply (erule exE)
apply (rule_tac x = "λX ∈ {Y ∈ Pow (A) . ∃f. f ∈ bij (Y, n) }. f``X" in exI)
apply (rule_tac d = "%Z. converse (f) ``Z" in lam_bijective)
apply (fast intro!: bij_is_inj [THEN restrict_bij, THEN bij_converse_bij,
THEN comp_bij]
elim!: bij_is_fun [THEN fun_is_rel, THEN image_subset])
apply (blast intro!: bij_is_inj [THEN restrict_bij]
comp_bij bij_converse_bij
bij_is_fun [THEN fun_is_rel, THEN image_subset])
apply (fast elim!: bij_is_inj [THEN vimage_image_eq])
apply (fast elim!: bij_is_surj [THEN image_vimage_eq])
done

lemma WO2_imp_ex_Card: "WO2 ==> ∃a. Card(a) & X≈a"
apply (unfold WO2_def)
apply (drule spec [of _ X])
apply (blast intro: Card_cardinal eqpoll_trans
well_ord_Memrel [THEN well_ord_cardinal_eqpoll, THEN eqpoll_sym])
done

lemma lepoll_infinite: "[| X\<lesssim>Y; ~Finite(X) |] ==> ~Finite(Y)"
by (blast intro: lepoll_Finite)

lemma infinite_Card_is_InfCard: "[| ~Finite(X); Card(X) |] ==> InfCard(X)"
apply (unfold InfCard_def)
apply (fast elim!: Card_is_Ord [THEN nat_le_infinite_Ord])
done

lemma WO2_infinite_subsets_eqpoll_X: "[| WO2; n ∈ nat; ~Finite(X) |]
==> {Y ∈ Pow(X). Y≈succ(n)}≈X"

apply (drule WO2_imp_ex_Card)
apply (elim allE exE conjE)
apply (frule eqpoll_imp_lepoll [THEN lepoll_infinite], assumption)
apply (drule infinite_Card_is_InfCard, assumption)
apply (blast intro: subsets_eqpoll subsets_eqpoll_X eqpoll_sym eqpoll_trans)
done

lemma well_ord_imp_ex_Card: "well_ord(X,R) ==> ∃a. Card(a) & X≈a"
by (fast elim!: well_ord_cardinal_eqpoll [THEN eqpoll_sym]
intro!: Card_cardinal)

lemma well_ord_infinite_subsets_eqpoll_X:
"[| well_ord(X,R); n ∈ nat; ~Finite(X) |] ==> {Y ∈ Pow(X). Y≈succ(n)}≈X"
apply (drule well_ord_imp_ex_Card)
apply (elim allE exE conjE)
apply (frule eqpoll_imp_lepoll [THEN lepoll_infinite], assumption)
apply (drule infinite_Card_is_InfCard, assumption)
apply (blast intro: subsets_eqpoll subsets_eqpoll_X eqpoll_sym eqpoll_trans)
done

end