# Theory Cardinal_AC

```(*  Title:      ZF/Cardinal_AC.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

These results help justify infinite-branching datatypes
*)

section‹Cardinal Arithmetic Using AC›

theory Cardinal_AC imports CardinalArith Zorn begin

subsection‹Strengthened Forms of Existing Theorems on Cardinals›

lemma cardinal_eqpoll: "|A| ≈ A"
apply (rule AC_well_ord [THEN exE])
apply (erule well_ord_cardinal_eqpoll)
done

text‹The theorem \<^term>‹||A|| = |A|››
lemmas cardinal_idem = cardinal_eqpoll [THEN cardinal_cong, simp]

lemma cardinal_eqE: "|X| = |Y| ⟹ X ≈ Y"
apply (rule AC_well_ord [THEN exE])
apply (rule AC_well_ord [THEN exE])
apply (rule well_ord_cardinal_eqE, assumption+)
done

lemma cardinal_eqpoll_iff: "|X| = |Y| ⟷ X ≈ Y"
by (blast intro: cardinal_cong cardinal_eqE)

lemma cardinal_disjoint_Un:
"⟦|A|=|B|;  |C|=|D|;  A ∩ C = 0;  B ∩ D = 0⟧
⟹ |A ∪ C| = |B ∪ D|"

lemma lepoll_imp_cardinal_le: "A ≲ B ⟹ |A| ≤ |B|"
apply (rule AC_well_ord [THEN exE])
apply (erule well_ord_lepoll_imp_cardinal_le, assumption)
done

lemma cadd_assoc: "(i ⊕ j) ⊕ k = i ⊕ (j ⊕ k)"
apply (rule AC_well_ord [THEN exE])
apply (rule AC_well_ord [THEN exE])
apply (rule AC_well_ord [THEN exE])
done

lemma cmult_assoc: "(i ⊗ j) ⊗ k = i ⊗ (j ⊗ k)"
apply (rule AC_well_ord [THEN exE])
apply (rule AC_well_ord [THEN exE])
apply (rule AC_well_ord [THEN exE])
apply (rule well_ord_cmult_assoc, assumption+)
done

lemma cadd_cmult_distrib: "(i ⊕ j) ⊗ k = (i ⊗ k) ⊕ (j ⊗ k)"
apply (rule AC_well_ord [THEN exE])
apply (rule AC_well_ord [THEN exE])
apply (rule AC_well_ord [THEN exE])
done

lemma InfCard_square_eq: "InfCard(|A|) ⟹ A*A ≈ A"
apply (rule AC_well_ord [THEN exE])
apply (erule well_ord_InfCard_square_eq, assumption)
done

subsection ‹The relationship between cardinality and le-pollence›

lemma Card_le_imp_lepoll:
assumes "|A| ≤ |B|" shows "A ≲ B"
proof -
have "A ≈ |A|"
by (rule cardinal_eqpoll [THEN eqpoll_sym])
also have "... ≲ |B|"
by (rule le_imp_subset [THEN subset_imp_lepoll]) (rule assms)
also have "... ≈ B"
by (rule cardinal_eqpoll)
finally show ?thesis .
qed

lemma le_Card_iff: "Card(K) ⟹ |A| ≤ K ⟷ A ≲ K"
apply (erule Card_cardinal_eq [THEN subst], rule iffI,
erule Card_le_imp_lepoll)
apply (erule lepoll_imp_cardinal_le)
done

lemma cardinal_0_iff_0 [simp]: "|A| = 0 ⟷ A = 0"
apply auto
apply (drule cardinal_0 [THEN ssubst])
apply (blast intro: eqpoll_0_iff [THEN iffD1] cardinal_eqpoll_iff [THEN iffD1])
done

lemma cardinal_lt_iff_lesspoll:
assumes i: "Ord(i)" shows "i < |A| ⟷ i ≺ A"
proof
assume "i < |A|"
hence  "i ≺ |A|"
by (blast intro: lt_Card_imp_lesspoll Card_cardinal)
also have "...  ≈ A"
by (rule cardinal_eqpoll)
finally show "i ≺ A" .
next
assume "i ≺ A"
also have "...  ≈ |A|"
by (blast intro: cardinal_eqpoll eqpoll_sym)
finally have "i ≺ |A|" .
thus  "i < |A|" using i
by (force intro: cardinal_lt_imp_lt lesspoll_cardinal_lt)
qed

lemma cardinal_le_imp_lepoll: " i ≤ |A| ⟹ i ≲ A"
by (blast intro: lt_Ord Card_le_imp_lepoll Ord_cardinal_le le_trans)

subsection‹Other Applications of AC›

lemma surj_implies_inj:
assumes f: "f ∈ surj(X,Y)" shows "∃g. g ∈ inj(Y,X)"
proof -
from f AC_Pi [of Y "λy. f-``{y}"]
obtain z where z: "z ∈ (∏y∈Y. f -`` {y})"
by (auto simp add: surj_def) (fast dest: apply_Pair)
show ?thesis
proof
show "z ∈ inj(Y, X)" using z surj_is_fun [OF f]
by (blast dest: apply_type Pi_memberD
intro: apply_equality Pi_type f_imp_injective)
qed
qed

text‹Kunen's Lemma 10.20›
lemma surj_implies_cardinal_le:
assumes f: "f ∈ surj(X,Y)" shows "|Y| ≤ |X|"
proof (rule lepoll_imp_cardinal_le)
from f [THEN surj_implies_inj] obtain g where "g ∈ inj(Y,X)" ..
thus "Y ≲ X"
qed

text‹Kunen's Lemma 10.21›
lemma cardinal_UN_le:
assumes K: "InfCard(K)"
shows "(⋀i. i∈K ⟹ |X(i)| ≤ K) ⟹ |⋃i∈K. X(i)| ≤ K"
proof (simp add: K InfCard_is_Card le_Card_iff)
have [intro]: "Ord(K)" by (blast intro: InfCard_is_Card Card_is_Ord K)
assume "⋀i. i∈K ⟹ X(i) ≲ K"
hence "⋀i. i∈K ⟹ ∃f. f ∈ inj(X(i), K)" by (simp add: lepoll_def)
with AC_Pi obtain f where f: "f ∈ (∏i∈K. inj(X(i), K))"
by force
{ fix z
assume z: "z ∈ (⋃i∈K. X(i))"
then obtain i where i: "i ∈ K" "Ord(i)" "z ∈ X(i)"
by (blast intro: Ord_in_Ord [of K])
hence "(μ i. z ∈ X(i)) ≤ i" by (fast intro: Least_le)
hence "(μ i. z ∈ X(i)) < K" by (best intro: lt_trans1 ltI i)
hence "(μ i. z ∈ X(i)) ∈ K" and "z ∈ X(μ i. z ∈ X(i))"
by (auto intro: LeastI ltD i)
} note mems = this
have "(⋃i∈K. X(i)) ≲ K × K"
proof (unfold lepoll_def)
show "∃f. f ∈ inj(⋃RepFun(K, X), K × K)"
apply (rule exI)
apply (rule_tac c = "λz. ⟨μ i. z ∈ X(i), f ` (μ i. z ∈ X(i)) ` z⟩"
and d = "λ⟨i,j⟩. converse (f`i) ` j" in lam_injective)
apply (force intro: f inj_is_fun mems apply_type Perm.left_inverse)+
done
qed
also have "... ≈ K"
by (simp add: K InfCard_square_eq InfCard_is_Card Card_cardinal_eq)
finally show "(⋃i∈K. X(i)) ≲ K" .
qed

text‹The same again, using \<^term>‹csucc››
lemma cardinal_UN_lt_csucc:
"⟦InfCard(K);  ⋀i. i∈K ⟹ |X(i)| < csucc(K)⟧
⟹ |⋃i∈K. X(i)| < csucc(K)"
by (simp add: Card_lt_csucc_iff cardinal_UN_le InfCard_is_Card Card_cardinal)

text‹The same again, for a union of ordinals.  In use, j(i) is a bit like rank(i),
the least ordinal j such that i:Vfrom(A,j).›
lemma cardinal_UN_Ord_lt_csucc:
"⟦InfCard(K);  ⋀i. i∈K ⟹ j(i) < csucc(K)⟧
⟹ (⋃i∈K. j(i)) < csucc(K)"
apply (rule cardinal_UN_lt_csucc [THEN Card_lt_imp_lt], assumption)
apply (blast intro: Ord_cardinal_le [THEN lt_trans1] elim: ltE)
apply (blast intro!: Ord_UN elim: ltE)
apply (erule InfCard_is_Card [THEN Card_is_Ord, THEN Card_csucc])
done

subsection‹The Main Result for Infinite-Branching Datatypes›

text‹As above, but the index set need not be a cardinal. Work
backwards along the injection from \<^term>‹W› into \<^term>‹K›, given
that \<^term>‹W≠0›.›

lemma inj_UN_subset:
assumes f: "f ∈ inj(A,B)" and a: "a ∈ A"
shows "(⋃x∈A. C(x)) ⊆ (⋃y∈B. C(if y ∈ range(f) then converse(f)`y else a))"
proof (rule UN_least)
fix x
assume x: "x ∈ A"
hence fx: "f ` x ∈ B" by (blast intro: f inj_is_fun [THEN apply_type])
have "C(x) ⊆ C(if f ` x ∈ range(f) then converse(f) ` (f ` x) else a)"
using f x by (simp add: inj_is_fun [THEN apply_rangeI])
also have "... ⊆ (⋃y∈B. C(if y ∈ range(f) then converse(f) ` y else a))"
by (rule UN_upper [OF fx])
finally show "C(x) ⊆ (⋃y∈B. C(if y ∈ range(f) then converse(f)`y else a))" .
qed

theorem le_UN_Ord_lt_csucc:
assumes IK: "InfCard(K)" and WK: "|W| ≤ K" and j: "⋀w. w∈W ⟹ j(w) < csucc(K)"
shows "(⋃w∈W. j(w)) < csucc(K)"
proof -
have CK: "Card(K)"
then obtain f where f: "f ∈ inj(W, K)" using WK
have OU: "Ord(⋃w∈W. j(w))" using j
by (blast elim: ltE)
note lt_subset_trans [OF _ _ OU, trans]
show ?thesis
proof (cases "W=0")
case True  ― ‹solve the easy 0 case›
thus ?thesis by (simp add: CK Card_is_Ord Card_csucc Ord_0_lt_csucc)
next
case False
then obtain x where x: "x ∈ W" by blast
have "(⋃x∈W. j(x)) ⊆ (⋃k∈K. j(if k ∈ range(f) then converse(f) ` k else x))"
by (rule inj_UN_subset [OF f x])
also have "... < csucc(K)" using IK
proof (rule cardinal_UN_Ord_lt_csucc)
fix k
assume "k ∈ K"
thus "j(if k ∈ range(f) then converse(f) ` k else x) < csucc(K)" using f x j
by (simp add: inj_converse_fun [THEN apply_type])
qed
finally show ?thesis .
qed
qed

end
```