const SNo : set prop 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 add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom add_SNo_com: !x:set.!y:set.SNo x -> SNo y -> x + y = y + x const In : set set prop term iIn = In infix iIn 2000 2000 const SNoL : set set const SNoLev : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNoL_E: !x:set.SNo x -> !y:set.y iIn SNoL x -> !P:prop.(SNo y -> SNoLev y iIn SNoLev x -> y < x -> P) -> P const SNoR : set set 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 SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom mul_SNo_SNoL_interpolate_impred: !x:set.!y:set.SNo x -> SNo y -> !z:set.z iIn SNoL (x * y) -> !P:prop.(!w:set.w iIn SNoL x -> !u:set.u iIn SNoL y -> (z + w * u) <= w * y + x * u -> P) -> (!w:set.w iIn SNoR x -> !u:set.u iIn SNoR y -> (z + w * u) <= w * y + x * u -> P) -> P const SNoCutP : set set prop 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 SNoS_ : set set lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.!x2:set.SNo x -> SNo y -> SNo z -> (!y2:set.y2 iIn SNoS_ (SNoLev x) -> y2 * y * z = (y2 * y) * z) -> (!y2:set.y2 iIn SNoS_ (SNoLev y) -> x * y2 * z = (x * y2) * z) -> (!y2:set.y2 iIn SNoS_ (SNoLev z) -> x * y * y2 = (x * y) * y2) -> (!y2:set.y2 iIn SNoS_ (SNoLev x) -> !z2:set.z2 iIn SNoS_ (SNoLev y) -> y2 * z2 * z = (y2 * z2) * z) -> (!y2:set.y2 iIn SNoS_ (SNoLev x) -> !z2:set.z2 iIn SNoS_ (SNoLev z) -> y2 * y * z2 = (y2 * y) * z2) -> (!y2:set.y2 iIn SNoS_ (SNoLev y) -> !z2:set.z2 iIn SNoS_ (SNoLev z) -> x * y2 * z2 = (x * y2) * z2) -> (!y2:set.y2 iIn SNoS_ (SNoLev x) -> !z2:set.z2 iIn SNoS_ (SNoLev y) -> !w2:set.w2 iIn SNoS_ (SNoLev z) -> y2 * z2 * w2 = (y2 * z2) * w2) -> SNo (x * y) -> SNoCutP w u -> (!y2:set.y2 iIn w -> !P:prop.(!z2:set.z2 iIn SNoL x -> !w2:set.w2 iIn SNoL (y * z) -> y2 = z2 * y * z + x * w2 + - z2 * w2 -> P) -> (!z2:set.z2 iIn SNoR x -> !w2:set.w2 iIn SNoR (y * z) -> y2 = z2 * y * z + x * w2 + - z2 * w2 -> P) -> P) -> (!y2:set.y2 iIn u -> !P:prop.(!z2:set.z2 iIn SNoL x -> !w2:set.w2 iIn SNoR (y * z) -> y2 = z2 * y * z + x * w2 + - z2 * w2 -> P) -> (!z2:set.z2 iIn SNoR x -> !w2:set.w2 iIn SNoL (y * z) -> y2 = z2 * y * z + x * w2 + - z2 * w2 -> P) -> P) -> x * y * z = SNoCut w u -> SNoCutP v x2 -> (!y2:set.y2 iIn v -> !P:prop.(!z2:set.z2 iIn SNoL (x * y) -> !w2:set.w2 iIn SNoL z -> y2 = z2 * z + (x * y) * w2 + - z2 * w2 -> P) -> (!z2:set.z2 iIn SNoR (x * y) -> !w2:set.w2 iIn SNoR z -> y2 = z2 * z + (x * y) * w2 + - z2 * w2 -> P) -> P) -> (!y2:set.y2 iIn x2 -> !P:prop.(!z2:set.z2 iIn SNoL (x * y) -> !w2:set.w2 iIn SNoR z -> y2 = z2 * z + (x * y) * w2 + - z2 * w2 -> P) -> (!z2:set.z2 iIn SNoR (x * y) -> !w2:set.w2 iIn SNoL z -> y2 = z2 * z + (x * y) * w2 + - z2 * w2 -> P) -> P) -> (x * y) * z = SNoCut v x2 -> (!y2:set.!z2:set.SNo y2 -> SNo z2 -> !w2:set.w2 iIn SNoL (z2 * y2) -> !P:prop.(!u2:set.u2 iIn SNoL y2 -> !v2:set.v2 iIn SNoL z2 -> (w2 + v2 * u2) <= z2 * u2 + v2 * y2 -> P) -> (!u2:set.u2 iIn SNoR y2 -> !v2:set.v2 iIn SNoR z2 -> (w2 + v2 * u2) <= z2 * u2 + v2 * y2 -> P) -> P) -> SNoCut w u = SNoCut v x2 var x:set var y:set var z:set hyp SNo x hyp SNo y hyp SNo z hyp !w:set.w iIn SNoS_ (SNoLev x) -> w * y * z = (w * y) * z hyp !w:set.w iIn SNoS_ (SNoLev y) -> x * w * z = (x * w) * z hyp !w:set.w iIn SNoS_ (SNoLev z) -> x * y * w = (x * y) * w hyp !w:set.w iIn SNoS_ (SNoLev x) -> !u:set.u iIn SNoS_ (SNoLev y) -> w * u * z = (w * u) * z hyp !w:set.w iIn SNoS_ (SNoLev x) -> !u:set.u iIn SNoS_ (SNoLev z) -> w * y * u = (w * y) * u hyp !w:set.w iIn SNoS_ (SNoLev y) -> !u:set.u iIn SNoS_ (SNoLev z) -> x * w * u = (x * w) * u hyp !w:set.w iIn SNoS_ (SNoLev x) -> !u:set.u iIn SNoS_ (SNoLev y) -> !v:set.v iIn SNoS_ (SNoLev z) -> w * u * v = (w * u) * v hyp SNo (x * y) hyp SNo (y * z) claim SNo ((x * y) * z) -> x * y * z = (x * y) * z