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 minus_SNo : set set term - = minus_SNo axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x 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 abs_SNo : set set axiom SNo_abs_SNo: !x:set.SNo x -> SNo (abs_SNo x) const omega : set const eps_ : set set axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) 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) const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom SNo_triangle2: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> abs_SNo (x + - z) <= abs_SNo (x + - y) + abs_SNo (y + - z) const nat_p : set prop axiom omega_nat_p: !x:set.x iIn omega -> nat_p x 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 axiom SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < z axiom SNoLt_irref: !x:set.~ x < x axiom FalseE: ~ False axiom SNoLtLe_or: !x:set.!y:set.SNo x -> SNo y -> x < y | y <= x axiom omega_ordsucc: !x:set.x iIn omega -> ordsucc x iIn omega const SNoS_ : set set axiom SNo_eps_SNoS_omega: !x:set.x iIn omega -> eps_ x iIn SNoS_ omega axiom add_SNo_SNoS_omega: !x:set.x iIn SNoS_ omega -> !y:set.y iIn SNoS_ omega -> x + y iIn SNoS_ omega axiom SNoLeLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x <= y -> y < z -> x < z axiom abs_SNo_intvl_bd: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x <= y -> y < x + z -> abs_SNo (y + - x) < z axiom pos_mul_SNo_Lt: !x:set.!y:set.!z:set.SNo x -> Empty < x -> SNo y -> SNo z -> y < z -> x * y < x * z axiom SNoLtLe: !x:set.!y:set.x < y -> x <= y axiom abs_SNo_dist_swap: !x:set.!y:set.SNo x -> SNo y -> abs_SNo (x + - y) = abs_SNo (y + - x) axiom add_SNo_Lt3: !x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> SNo z -> SNo w -> x < z -> y < w -> (x + y) < z + w axiom eps_ordsucc_half_add: !x:set.nat_p x -> eps_ (ordsucc x) + eps_ (ordsucc x) = eps_ x const SNoCut : set set set var x:set var y:set var z:set hyp Empty < x hyp x < ordsucc Empty hyp SNo x hyp !w:set.w iIn Sep (SNoS_ omega) (\u:set.x * u < ordsucc Empty) -> w < SNoCut (Sep (SNoS_ omega) \u:set.x * u < ordsucc Empty) (Sep (SNoS_ omega) \u:set.ordsucc Empty < x * u) hyp !w:set.w iIn Sep (SNoS_ omega) (\u:set.ordsucc Empty < x * u) -> SNoCut (Sep (SNoS_ omega) \u:set.x * u < ordsucc Empty) (Sep (SNoS_ omega) \u:set.ordsucc Empty < x * u) < w hyp SNo (SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w)) hyp SNo (x * SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w)) hyp SNo (x * SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w)) hyp y iIn omega hyp z iIn SNoS_ omega hyp z < SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w) hyp SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w) < z + eps_ (ordsucc y) hyp SNo z hyp SNo (x * z) hyp SNo (eps_ (ordsucc y)) hyp SNo (x * eps_ (ordsucc y)) hyp SNo (z + eps_ (ordsucc y)) hyp SNo (ordsucc Empty + - x * z) hyp SNo (x * z + - x * SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w)) claim x * (z + eps_ (ordsucc y)) < x * z + eps_ (ordsucc y) -> abs_SNo (ordsucc Empty + - x * SNoCut (Sep (SNoS_ omega) \w:set.x * w < ordsucc Empty) (Sep (SNoS_ omega) \w:set.ordsucc Empty < x * w)) < eps_ y