src/ZF/Univ.thy

Univ no longer requires Arith (really it never did)

(*  Title:      ZF/univ.thy
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1992  University of Cambridge

The cumulative hierarchy and a small universe for recursive types

Standard notation for Vset(i) is V(i), but users might want V for a variable

NOTE: univ(A) could be a translation; would simplify many proofs!
  But Ind_Syntax.univ refers to the constant "Univ.univ"
*)

Univ = Epsilon + Sum + Finite + mono +

consts
    Vfrom       :: [i,i]=>i
    Vset        :: i=>i
    Vrec        :: [i, [i,i]=>i] =>i
    Vrecursor   :: [[i,i]=>i, i] =>i
    univ        :: i=>i

translations
    "Vset(x)"   ==      "Vfrom(0,x)"


defs
    Vfrom_def   "Vfrom(A,i) == transrec(i, %x f. A Un (UN y:x. Pow(f`y)))"

    Vrec_def
        "Vrec(a,H) == transrec(rank(a), %x g. lam z: Vset(succ(x)).      
                             H(z, lam w:Vset(x). g`rank(w)`w)) ` a"

    Vrecursor_def
        "Vrecursor(H,a) == transrec(rank(a), %x g. lam z: Vset(succ(x)).      
                                    H(lam w:Vset(x). g`rank(w)`w, z)) ` a"

    univ_def    "univ(A) == Vfrom(A,nat)"

end