const In : set set prop term iIn = In infix iIn 2000 2000 term nIn = \x:set.\y:set.~ x iIn y const Pi : set (set set) set term setexp = \x:set.\y:set.Pi y \z:set.x const ordinal : set prop const omega : set axiom omega_ordinal: ordinal omega const SNoS_ : set set const SNoLev : set set const SNo : set prop const SNo_ : set set prop axiom SNoS_E2: !x:set.ordinal x -> !y:set.y iIn SNoS_ x -> !P:prop.(SNoLev y iIn x -> ordinal (SNoLev y) -> SNo y -> SNo_ (SNoLev y) y -> P) -> P const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < z const nat_p : set prop axiom nat_p_omega: !x:set.nat_p x -> x iIn omega const ap : set set set const Sigma : set (set set) set axiom beta: !x:set.!f:set set.!y:set.y iIn x -> ap (Sigma x f) y = f y const ordsucc : set set axiom ordsuccE: !x:set.!y:set.y iIn ordsucc x -> y iIn x | y = x const add_SNo : set set set term + = add_SNo infix + 2281 2280 const eps_ : set set const nat_primrec : set (set set set) set set const Eps_i : (set prop) set const Empty : set var x:set var y:set hyp !z:set.nat_p z -> nat_primrec (Eps_i \w:set.w iIn SNoS_ omega & w < x & x < w + eps_ Empty) (\w:set.\u:set.Eps_i \v:set.v iIn SNoS_ omega & v < x & x < v + eps_ (ordsucc w) & u < v) (ordsucc z) = Eps_i \w:set.w iIn SNoS_ omega & w < x & x < w + eps_ (ordsucc z) & nat_primrec (Eps_i \u:set.u iIn SNoS_ omega & u < x & x < u + eps_ Empty) (\u:set.\v:set.Eps_i \x2:set.x2 iIn SNoS_ omega & x2 < x & x < x2 + eps_ (ordsucc u) & v < x2) z < w hyp nat_p y hyp nat_primrec (Eps_i \z:set.z iIn SNoS_ omega & z < x & x < z + eps_ Empty) (\z:set.\w:set.Eps_i \u:set.u iIn SNoS_ omega & u < x & x < u + eps_ (ordsucc z) & w < u) y iIn SNoS_ omega & nat_primrec (Eps_i \z:set.z iIn SNoS_ omega & z < x & x < z + eps_ Empty) (\z:set.\w:set.Eps_i \u:set.u iIn SNoS_ omega & u < x & x < u + eps_ (ordsucc z) & w < u) y < x & x < nat_primrec (Eps_i \z:set.z iIn SNoS_ omega & z < x & x < z + eps_ Empty) (\z:set.\w:set.Eps_i \u:set.u iIn SNoS_ omega & u < x & x < u + eps_ (ordsucc z) & w < u) y + eps_ y & !z:set.z iIn y -> SNo (ap (Sigma omega (nat_primrec (Eps_i \w:set.w iIn SNoS_ omega & w < x & x < w + eps_ Empty) \w:set.\u:set.Eps_i \v:set.v iIn SNoS_ omega & v < x & x < v + eps_ (ordsucc w) & u < v)) z) -> ap (Sigma omega (nat_primrec (Eps_i \w:set.w iIn SNoS_ omega & w < x & x < w + eps_ Empty) \w:set.\u:set.Eps_i \v:set.v iIn SNoS_ omega & v < x & x < v + eps_ (ordsucc w) & u < v)) z < nat_primrec (Eps_i \w:set.w iIn SNoS_ omega & w < x & x < w + eps_ Empty) (\w:set.\u:set.Eps_i \v:set.v iIn SNoS_ omega & v < x & x < v + eps_ (ordsucc w) & u < v) y hyp SNo (nat_primrec (Eps_i \z:set.z iIn SNoS_ omega & z < x & x < z + eps_ Empty) (\z:set.\w:set.Eps_i \u:set.u iIn SNoS_ omega & u < x & x < u + eps_ (ordsucc z) & w < u) y) hyp ?z:set.z iIn SNoS_ omega & z < x & x < z + eps_ (ordsucc y) & nat_primrec (Eps_i \w:set.w iIn SNoS_ omega & w < x & x < w + eps_ Empty) (\w:set.\u:set.Eps_i \v:set.v iIn SNoS_ omega & v < x & x < v + eps_ (ordsucc w) & u < v) y < z claim nat_primrec (Eps_i \z:set.z iIn SNoS_ omega & z < x & x < z + eps_ Empty) (\z:set.\w:set.Eps_i \u:set.u iIn SNoS_ omega & u < x & x < u + eps_ (ordsucc z) & w < u) (ordsucc y) iIn SNoS_ omega & nat_primrec (Eps_i \z:set.z iIn SNoS_ omega & z < x & x < z + eps_ Empty) (\z:set.\w:set.Eps_i \u:set.u iIn SNoS_ omega & u < x & x < u + eps_ (ordsucc z) & w < u) (ordsucc y) < x & x < nat_primrec (Eps_i \z:set.z iIn SNoS_ omega & z < x & x < z + eps_ Empty) (\z:set.\w:set.Eps_i \u:set.u iIn SNoS_ omega & u < x & x < u + eps_ (ordsucc z) & w < u) (ordsucc y) + eps_ (ordsucc y) & nat_primrec (Eps_i \z:set.z iIn SNoS_ omega & z < x & x < z + eps_ Empty) (\z:set.\w:set.Eps_i \u:set.u iIn SNoS_ omega & u < x & x < u + eps_ (ordsucc z) & w < u) y < nat_primrec (Eps_i \z:set.z iIn SNoS_ omega & z < x & x < z + eps_ Empty) (\z:set.\w:set.Eps_i \u:set.u iIn SNoS_ omega & u < x & x < u + eps_ (ordsucc z) & w < u) (ordsucc y) -> nat_primrec (Eps_i \z:set.z iIn SNoS_ omega & z < x & x < z + eps_ Empty) (\z:set.\w:set.Eps_i \u:set.u iIn SNoS_ omega & u < x & x < u + eps_ (ordsucc z) & w < u) (ordsucc y) iIn SNoS_ omega & nat_primrec (Eps_i \z:set.z iIn SNoS_ omega & z < x & x < z + eps_ Empty) (\z:set.\w:set.Eps_i \u:set.u iIn SNoS_ omega & u < x & x < u + eps_ (ordsucc z) & w < u) (ordsucc y) < x & x < nat_primrec (Eps_i \z:set.z iIn SNoS_ omega & z < x & x < z + eps_ Empty) (\z:set.\w:set.Eps_i \u:set.u iIn SNoS_ omega & u < x & x < u + eps_ (ordsucc z) & w < u) (ordsucc y) + eps_ (ordsucc y) & !z:set.z iIn ordsucc y -> SNo (ap (Sigma omega (nat_primrec (Eps_i \w:set.w iIn SNoS_ omega & w < x & x < w + eps_ Empty) \w:set.\u:set.Eps_i \v:set.v iIn SNoS_ omega & v < x & x < v + eps_ (ordsucc w) & u < v)) z) -> ap (Sigma omega (nat_primrec (Eps_i \w:set.w iIn SNoS_ omega & w < x & x < w + eps_ Empty) \w:set.\u:set.Eps_i \v:set.v iIn SNoS_ omega & v < x & x < v + eps_ (ordsucc w) & u < v)) z < nat_primrec (Eps_i \w:set.w iIn SNoS_ omega & w < x & x < w + eps_ Empty) (\w:set.\u:set.Eps_i \v:set.v iIn SNoS_ omega & v < x & x < v + eps_ (ordsucc w) & u < v) (ordsucc y)