Theory ZF

sectionMain 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.*)

subsectionIteration of the function termF

consts  iterates :: "[ii,i,i]  i"   ((_^_ '(_')) [60,1000,1000] 60)

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

definition
  iterates_omega :: "[ii,i]  i" ((_ '(_')) [60,1000] 60) where
    "F (x)  nnat. F^n (x)"

lemma iterates_triv:
     "nnat;  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 (x) = x"
by (simp add: iterates_omega_def iterates_triv)

lemma Ord_iterates [simp]:
     "nnat;  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)


subsectionTransfinite Recursion

textTransfinite 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, λyx. 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, λji. 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