Theory ZF

section‹Main ZF Theory: Everything Except AC›

theory ZF imports List IntDiv CardinalArith begin

(*The theory of "iterates" logically belongs to Nat, but can't go there because
  primrec isn't available into after Datatype.*)

subsection‹Iteration of the function \<^term>‹F››

consts  iterates :: "[i⇒i,i,i] ⇒ i"  (‹(‹notation=‹mixfix iterates››_^_ '(_'))› [60,1000,1000] 60)

primrec
    "F^0 (x) = x"
    "F^(succ(n)) (x) = F(F^n (x))"

definition
  iterates_omega :: "[i⇒i,i] ⇒ i" (‹(‹notation=‹mixfix iterates_omega››_^ω '(_'))› [60,1000] 60) where
    "F^ω (x) ≡ ⋃n∈nat. F^n (x)"

lemma iterates_triv:
     "⟦n∈nat;  F(x) = x⟧ ⟹ F^n (x) = x"
by (induct n rule: nat_induct, simp_all)

lemma iterates_type [TC]:
     "⟦n ∈ nat;  a ∈ A; ⋀x. x ∈ A ⟹ F(x) ∈ A⟧
      ⟹ F^n (a) ∈ A"
by (induct n rule: nat_induct, simp_all)

lemma iterates_omega_triv:
    "F(x) = x ⟹ F^ω (x) = x"
by (simp add: iterates_omega_def iterates_triv)

lemma Ord_iterates [simp]:
     "⟦n∈nat;  ⋀i. Ord(i) ⟹ Ord(F(i));  Ord(x)⟧
      ⟹ Ord(F^n (x))"
by (induct n rule: nat_induct, simp_all)

lemma iterates_commute: "n ∈ nat ⟹ F(F^n (x)) = F^n (F(x))"
by (induct_tac n, simp_all)


subsection‹Transfinite Recursion›

text‹Transfinite recursion for definitions based on the
    three cases of ordinals›

definition
  transrec3 :: "[i, i, [i,i]⇒i, [i,i]⇒i] ⇒i" where
    "transrec3(k, a, b, c) ≡
       transrec(k, λx r.
         if x=0 then a
         else if Limit(x) then c(x, λy∈x. r`y)
         else b(Arith.pred(x), r ` Arith.pred(x)))"

lemma transrec3_0 [simp]: "transrec3(0,a,b,c) = a"
by (rule transrec3_def [THEN def_transrec, THEN trans], simp)

lemma transrec3_succ [simp]:
     "transrec3(succ(i),a,b,c) = b(i, transrec3(i,a,b,c))"
by (rule transrec3_def [THEN def_transrec, THEN trans], simp)

lemma transrec3_Limit:
     "Limit(i) ⟹
      transrec3(i,a,b,c) = c(i, λj∈i. transrec3(j,a,b,c))"
by (rule transrec3_def [THEN def_transrec, THEN trans], force)


declaration ‹fn _ =>
  Simplifier.map_ss (Simplifier.set_mksimps (fn ctxt =>
    map mk_eq o Ord_atomize o Variable.gen_all ctxt))
›

end