const In : set set prop term iIn = In infix iIn 2000 2000 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y const SNo : set prop const SNoLt : set set prop term < = SNoLt infix < 2020 2020 term SNoCutP = \x:set.\y:set.(!z:set.z iIn x -> SNo z) & (!z:set.z iIn y -> SNo z) & !z:set.z iIn x -> !w:set.w iIn y -> z < w const ordsucc : set set const Empty : set axiom SNo_1: SNo (ordsucc Empty) const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) const omega : set axiom SNo_omega: SNo omega const SNoS_ : set set axiom omega_SNoS_omega: Subq omega (SNoS_ omega) const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 axiom SNo_mul_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x * y) axiom SNo_0: SNo Empty axiom add_SNo_Lt2: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> y < z -> (x + y) < x + z axiom add_SNo_0R: !x:set.SNo x -> x + Empty = x const nat_p : set prop axiom omega_nat_p: !x:set.x iIn omega -> nat_p x const eps_ : set set axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) axiom SNo_2: SNo (ordsucc (ordsucc Empty)) axiom SNoLt_0_2: Empty < ordsucc (ordsucc Empty) const exp_SNo_nat : set set set axiom exp_SNo_nat_pos: !x:set.SNo x -> Empty < x -> !y:set.nat_p y -> Empty < exp_SNo_nat x y const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom SNoLtLe: !x:set.!y:set.x < y -> x <= y axiom nonneg_mul_SNo_Le': !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> Empty <= z -> x <= y -> x * z <= y * z axiom mul_SNo_eps_power_2: !x:set.nat_p x -> eps_ x * exp_SNo_nat (ordsucc (ordsucc Empty)) x = ordsucc Empty axiom add_SNo_Le1: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x <= z -> (x + y) <= z + y axiom SNoLtLe_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y <= z -> x < z axiom mul_SNo_oneR: !x:set.SNo x -> x * ordsucc Empty = x axiom mul_SNo_distrL: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x * (y + z) = x * y + x * z const Sep : set (set prop) set axiom SepI: !x:set.!p:set prop.!y:set.y iIn x -> p y -> y iIn Sep x p const ordinal : set prop axiom omega_ordinal: ordinal omega axiom ordinal_In_SNoLt: !x:set.ordinal x -> !y:set.y iIn x -> y < x axiom SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < z const SNoCut : set set set var x:set var y:set hyp Empty < x hyp SNo x hyp SNo (SNoCut (Sep (SNoS_ omega) \z:set.x * z < ordsucc Empty) (Sep (SNoS_ omega) \z:set.ordsucc Empty < x * z)) hyp !z:set.z iIn Sep (SNoS_ omega) (\w:set.ordsucc Empty < x * w) -> SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w) < z hyp y iIn omega hyp eps_ y <= x hyp nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) y) hyp SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) y) claim exp_SNo_nat (ordsucc (ordsucc Empty)) y + ordsucc Empty iIn omega -> SNoCut (Sep (SNoS_ omega) \z:set.x * z < ordsucc Empty) (Sep (SNoS_ omega) \z:set.ordsucc Empty < x * z) < omega