const SNo : set prop const In : set set prop term iIn = In infix iIn 2000 2000 const SNoL : set set const SNoS_ : set set const SNoLev : set set axiom SNoL_SNoS: !x:set.SNo x -> !y:set.y iIn SNoL x -> y iIn SNoS_ (SNoLev 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 SNoR : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNoR_E: !x:set.SNo x -> !y:set.y iIn SNoR x -> !P:prop.(SNo y -> SNoLev y iIn SNoLev x -> x < y -> P) -> P const add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom add_SNo_SNoR_interpolate: !x:set.!y:set.SNo x -> SNo y -> !z:set.z iIn SNoR (x + y) -> (?w:set.w iIn SNoR x & (w + y) <= z) | ?w:set.w iIn SNoR y & (x + w) <= z axiom add_SNo_assoc: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x + y + z = (x + y) + z const minus_SNo : set set term - = minus_SNo axiom add_SNo_minus_Lt2b3: !x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> SNo z -> SNo w -> (w + z) < x + y -> w < x + y + - z lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.SNo x -> SNo y -> SNo z -> (!x2:set.x2 iIn SNoS_ (SNoLev x) -> (x2 + y) * z = x2 * z + y * z) -> (!x2:set.x2 iIn SNoS_ (SNoLev x) -> !y2:set.y2 iIn SNoS_ (SNoLev z) -> (x2 + y) * y2 = x2 * y2 + y * y2) -> SNo (x * z) -> SNo (y * z) -> u iIn SNoL z -> SNo w -> SNo u -> u < z -> SNo (x * u) -> SNo (y * u) -> SNo (w * z) -> SNo (w * u) -> SNo (w * z + x * u + y * u) -> SNo (x * z + y * z + w * u) -> v iIn SNoR x -> (v + y) <= w -> SNo v -> x < v -> SNo (v * u) -> (x * z + y * z + w * u) < w * z + x * u + y * u lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.SNo x -> SNo y -> SNo z -> (!x2:set.x2 iIn SNoS_ (SNoLev y) -> (x + x2) * z = x * z + x2 * z) -> (!x2:set.x2 iIn SNoS_ (SNoLev y) -> !y2:set.y2 iIn SNoS_ (SNoLev z) -> (x + x2) * y2 = x * y2 + x2 * y2) -> SNo (x * z) -> SNo (y * z) -> u iIn SNoL z -> SNo w -> SNo u -> u < z -> SNo (x * u) -> SNo (y * u) -> SNo (w * z) -> SNo (w * u) -> SNo (w * z + x * u + y * u) -> SNo (x * z + y * z + w * u) -> v iIn SNoR y -> (x + v) <= w -> SNo v -> y < v -> SNo (v * u) -> (x * z + y * z + w * u) < w * z + x * u + y * u var x:set var y:set var z:set var w:set var u:set hyp SNo x hyp SNo y hyp SNo z hyp !v:set.v iIn SNoS_ (SNoLev x) -> (v + y) * z = v * z + y * z hyp !v:set.v iIn SNoS_ (SNoLev y) -> (x + v) * z = x * z + v * z hyp !v:set.v iIn SNoS_ (SNoLev z) -> (x + y) * v = x * v + y * v hyp !v:set.v iIn SNoS_ (SNoLev x) -> !x2:set.x2 iIn SNoS_ (SNoLev z) -> (v + y) * x2 = v * x2 + y * x2 hyp !v:set.v iIn SNoS_ (SNoLev y) -> !x2:set.x2 iIn SNoS_ (SNoLev z) -> (x + v) * x2 = x * x2 + v * x2 hyp SNo (x * z) hyp SNo (y * z) hyp SNo (x * z + y * z) hyp w iIn SNoR (x + y) hyp u iIn SNoL z hyp SNo w hyp SNo u hyp u < z hyp SNo (x * u) hyp SNo (y * u) hyp SNo (w * z) hyp SNo ((x + y) * u) hyp SNo (w * u) hyp SNo (w * z + x * u + y * u) claim SNo (x * z + y * z + w * u) -> (x * z + y * z) < w * z + (x + y) * u + - w * u