section{*Theory Main: Everything Except AC*}

theory Main_ZF imports List_ZF IntDiv_ZF 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"   ("(_^_ '(_'))" [60,1000,1000] 60)

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

definition
  iterates_omega :: "[i=>i,i] => i"  where
    "iterates_omega(F,x) == \<Union>n\<in>nat. F^n (x)"

notation (xsymbols)
  iterates_omega  ("(_^\<omega> '(_'))" [60,1000] 60)
notation (HTML output)
  iterates_omega  ("(_^\<omega> '(_'))" [60,1000] 60)

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"
by (simp add: iterates_omega_def iterates_triv)

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{* 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, \<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 {* fn _ =>
  Simplifier.map_ss (Simplifier.set_mksimps (K (map mk_eq o Ord_atomize o gen_all)))
*}

end
