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 term nIn = \x:set.\y:set.~ x iIn y term TransSet = \x:set.!y:set.y iIn x -> Subq y x 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 SNo : set prop const SNoLev : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const SNoL : set set axiom SNoL_I: !x:set.SNo x -> !y:set.SNo y -> SNoLev y iIn SNoLev x -> y < x -> y iIn SNoL x axiom FalseE: ~ False const SNoR : set set axiom SNoR_I: !x:set.SNo x -> !y:set.SNo y -> SNoLev y iIn SNoLev x -> x < y -> y iIn SNoR x axiom SNoLt_trichotomy_or_impred: !x:set.!y:set.SNo x -> SNo y -> !P:prop.(x < y -> P) -> (x = y -> P) -> (y < x -> P) -> P const SNoS_ : set set 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 SNoCutP : set set prop const add_SNo : set set set term + = add_SNo infix + 2281 2280 const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const minus_SNo : set set term - = minus_SNo const SNoCut : set set set axiom mul_SNo_eq_3: !x:set.!y:set.SNo x -> SNo y -> !P:prop.(!z:set.!w:set.SNoCutP z w -> (!u:set.u iIn z -> !Q:prop.(!v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoL y -> u = v * y + x * x2 + - v * x2 -> Q) -> (!v:set.v iIn SNoR x -> !x2:set.x2 iIn SNoR y -> u = v * y + x * x2 + - v * x2 -> Q) -> Q) -> (!u:set.u iIn SNoL x -> !v:set.v iIn SNoL y -> u * y + x * v + - u * v iIn z) -> (!u:set.u iIn SNoR x -> !v:set.v iIn SNoR y -> u * y + x * v + - u * v iIn z) -> (!u:set.u iIn w -> !Q:prop.(!v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoR y -> u = v * y + x * x2 + - v * x2 -> Q) -> (!v:set.v iIn SNoR x -> !x2:set.x2 iIn SNoL y -> u = v * y + x * x2 + - v * x2 -> Q) -> Q) -> (!u:set.u iIn SNoL x -> !v:set.v iIn SNoR y -> u * y + x * v + - u * v iIn w) -> (!u:set.u iIn SNoR x -> !v:set.v iIn SNoL y -> u * y + x * v + - u * v iIn w) -> x * y = SNoCut z w -> P) -> P const Empty : set const real : set const ordsucc : set set const eps_ : set set const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 const abs_SNo : set set lemma !x:set.!y:set.!z:set.!w:set.Empty < x -> Empty < y -> ~ x * y iIn real -> SNo x -> x iIn SNoS_ (ordsucc omega) -> x < omega -> SNo y -> y iIn SNoS_ (ordsucc omega) -> y < omega -> (!u:set.u iIn omega -> !P:prop.(!v:set.v iIn SNoS_ omega -> Empty < v -> v < x -> x < v + eps_ u -> P) -> P) -> (!u:set.u iIn omega -> !P:prop.(!v:set.v iIn SNoS_ omega -> Empty < v -> v < y -> y < v + eps_ u -> P) -> P) -> SNo (x * y) -> SNo - x * y -> (!u:set.SNo u -> SNoLev u iIn omega -> SNoLev u iIn SNoLev (x * y)) -> (!u:set.u iIn SNoL x -> !P:prop.(!v:set.v iIn omega -> eps_ v <= x + - u -> P) -> P) -> (!u:set.u iIn SNoR x -> !P:prop.(!v:set.v iIn omega -> eps_ v <= u + - x -> P) -> P) -> (!u:set.u iIn SNoL y -> !P:prop.(!v:set.v iIn omega -> eps_ v <= y + - u -> P) -> P) -> (!u:set.u iIn SNoR y -> !P:prop.(!v:set.v iIn omega -> eps_ v <= u + - y -> P) -> P) -> SNoCutP z w -> (!u:set.u iIn z -> !P:prop.(!v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoL y -> u = v * y + x * x2 + - v * x2 -> P) -> (!v:set.v iIn SNoR x -> !x2:set.x2 iIn SNoR y -> u = v * y + x * x2 + - v * x2 -> P) -> P) -> (!u:set.u iIn w -> !P:prop.(!v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoR y -> u = v * y + x * x2 + - v * x2 -> P) -> (!v:set.v iIn SNoR x -> !x2:set.x2 iIn SNoL y -> u = v * y + x * x2 + - v * x2 -> P) -> P) -> x * y = SNoCut z w -> ~ !u:set.u iIn SNoS_ omega -> (!v:set.v iIn omega -> abs_SNo (u + - x * y) < eps_ v) -> u = x * y lemma !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo (x * y) -> SNo - x * y -> (!w:set.SNo w -> SNoLev w iIn omega -> SNoLev w iIn SNoLev (x * y)) -> (!w:set.w iIn SNoL x -> !P:prop.(!u:set.u iIn omega -> eps_ u <= x + - w -> P) -> P) -> (!w:set.w iIn SNoR x -> !P:prop.(!u:set.u iIn omega -> eps_ u <= w + - x -> P) -> P) -> (!w:set.w iIn SNoL y -> !P:prop.(!u:set.u iIn omega -> eps_ u <= y + - w -> P) -> P) -> (!w:set.w iIn SNoR y -> !P:prop.(!u:set.u iIn omega -> eps_ u <= w + - y -> P) -> P) -> (!w:set.w iIn omega -> abs_SNo (z + - x * y) < eps_ w) -> SNoLev z iIn omega -> SNo z -> z < x * y -> ~ z iIn SNoL (x * y) lemma !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo (x * y) -> SNo - x * y -> (!w:set.SNo w -> SNoLev w iIn omega -> SNoLev w iIn SNoLev (x * y)) -> (!w:set.w iIn SNoL x -> !P:prop.(!u:set.u iIn omega -> eps_ u <= x + - w -> P) -> P) -> (!w:set.w iIn SNoR x -> !P:prop.(!u:set.u iIn omega -> eps_ u <= w + - x -> P) -> P) -> (!w:set.w iIn SNoL y -> !P:prop.(!u:set.u iIn omega -> eps_ u <= y + - w -> P) -> P) -> (!w:set.w iIn SNoR y -> !P:prop.(!u:set.u iIn omega -> eps_ u <= w + - y -> P) -> P) -> (!w:set.w iIn omega -> abs_SNo (z + - x * y) < eps_ w) -> SNoLev z iIn omega -> SNo z -> x * y < z -> ~ z iIn SNoR (x * y) var x:set var y:set hyp Empty < x hyp Empty < y hyp ~ x * y iIn real hyp SNo x hyp x iIn SNoS_ (ordsucc omega) hyp x < omega hyp SNo y hyp y iIn SNoS_ (ordsucc omega) hyp y < omega hyp !z:set.z iIn SNoS_ omega -> (!w:set.w iIn omega -> abs_SNo (z + - y) < eps_ w) -> z = y hyp !z:set.z iIn omega -> !P:prop.(!w:set.w iIn SNoS_ omega -> Empty < w -> w < x -> x < w + eps_ z -> P) -> P hyp !z:set.z iIn omega -> !P:prop.(!w:set.w iIn SNoS_ omega -> Empty < w -> w < y -> y < w + eps_ z -> P) -> P hyp SNo (x * y) hyp SNo - x * y hyp !z:set.SNo z -> SNoLev z iIn omega -> SNoLev z iIn SNoLev (x * y) hyp Subq (SNoR y) (SNoS_ omega) hyp !z:set.z iIn SNoL x -> !P:prop.(!w:set.w iIn omega -> eps_ w <= x + - z -> P) -> P hyp !z:set.z iIn SNoR x -> !P:prop.(!w:set.w iIn omega -> eps_ w <= z + - x -> P) -> P hyp !z:set.z iIn SNoL y -> !P:prop.(!w:set.w iIn omega -> eps_ w <= y + - z -> P) -> P claim ~ !z:set.z iIn SNoR y -> !P:prop.(!w:set.w iIn omega -> eps_ w <= z + - y -> P) -> P