section‹Main ZF Theory: Everything Except AC›
theory ZF imports List IntDiv CardinalArith begin
subsection‹Iteration of the function \<^term>‹F››
consts iterates :: "[i=>i,i,i] => i" (‹(_^_ '(_'))› [60,1000,1000] 60)
primrec
"F^0 (x) = x"
"F^(succ(n)) (x) = F(F^n (x))"
definition
iterates_omega :: "[i=>i,i] => i" (‹(_^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