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 nat_p : set prop const omega : set axiom nat_p_omega: !x:set.nat_p x -> x iIn omega const ordsucc : set set axiom omega_ordsucc: !x:set.x iIn omega -> ordsucc x iIn omega const add_SNo : set set set term + = add_SNo infix + 2281 2280 const Empty : set axiom add_SNo_1_ordsucc: !x:set.x iIn omega -> x + ordsucc Empty = ordsucc x const SNoCut : set set set const Sep : set (set prop) set const SNoS_ : set set const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 const eps_ : set set const exp_SNo_nat : set set set lemma !x:set.!y:set.Empty < x -> SNo x -> SNo (SNoCut (Sep (SNoS_ omega) \z:set.x * z < ordsucc Empty) (Sep (SNoS_ omega) \z:set.ordsucc Empty < x * z)) -> (!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) -> y iIn omega -> eps_ y <= x -> nat_p (exp_SNo_nat (ordsucc (ordsucc Empty)) y) -> SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) y) -> 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 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) claim SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) y) -> SNoCut (Sep (SNoS_ omega) \z:set.x * z < ordsucc Empty) (Sep (SNoS_ omega) \z:set.ordsucc Empty < x * z) < omega