src/ZF/ZF.thy
 author wenzelm Mon, 08 Aug 2022 20:27:54 +0200 changeset 75798 8b0dbfbde032 parent 69593 3dda49e08b9d permissions -rw-r--r--
```
section\<open>Main ZF Theory: Everything Except AC\<close>

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\<open>Iteration of the function \<^term>\<open>F\<close>\<close>

consts  iterates :: "[i=>i,i,i] => i"   (\<open>(_^_ '(_'))\<close> [60,1000,1000] 60)

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

definition
iterates_omega :: "[i=>i,i] => i" (\<open>(_^\<omega> '(_'))\<close> [60,1000] 60) where
"F^\<omega> (x) == \<Union>n\<in>nat. F^n (x)"

lemma iterates_triv:
"[| n\<in>nat;  F(x) = x |] ==> F^n (x) = x"
by (induct n rule: nat_induct, simp_all)

lemma iterates_type [TC]:
"[| n \<in> nat;  a \<in> A; !!x. x \<in> A ==> F(x) \<in> A |]
==> F^n (a) \<in> A"
by (induct n rule: nat_induct, simp_all)

lemma iterates_omega_triv:
"F(x) = x ==> F^\<omega> (x) = x"

lemma Ord_iterates [simp]:
"[| n\<in>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 \<in> nat ==> F(F^n (x)) = F^n (F(x))"
by (induct_tac n, simp_all)

subsection\<open>Transfinite Recursion\<close>

text\<open>Transfinite recursion for definitions based on the
three cases of ordinals\<close>

definition
transrec3 :: "[i, i, [i,i]=>i, [i,i]=>i] =>i" where
"transrec3(k, a, b, c) ==
transrec(k, \<lambda>x r.
if x=0 then a
else if Limit(x) then c(x, \<lambda>y\<in>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, \<lambda>j\<in>i. transrec3(j,a,b,c))"
by (rule transrec3_def [THEN def_transrec, THEN trans], force)

declaration \<open>fn _ =>
Simplifier.map_ss (Simplifier.set_mksimps (fn ctxt =>
map mk_eq o Ord_atomize o Variable.gen_all ctxt))
\<close>

end
```