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 mul_SNo_distrL: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x * (y + z) = x * y + x * z axiom mul_SNo_distrR: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> (x + y) * z = x * z + y * z const In : set set prop term iIn = In infix iIn 2000 2000 const SNoL : set set const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 const SNoR : set set 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 axiom mul_SNo_SNoR_interpolate_impred: !x:set.!y:set.SNo x -> SNo y -> !z:set.z iIn SNoR (x * y) -> !P:prop.(!w:set.w iIn SNoL x -> !u:set.u iIn SNoR y -> (w * y + x * u) <= z + w * u -> P) -> (!w:set.w iIn SNoR x -> !u:set.u iIn SNoL y -> (w * y + x * u) <= z + w * u -> P) -> P const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom mul_SNo_Lt: !x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> SNo z -> SNo w -> z < x -> w < y -> (z * y + x * w) < x * y + z * w axiom mul_SNo_Le: !x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> SNo z -> SNo w -> z <= x -> w <= y -> (z * y + x * w) <= x * y + z * w const SNoS_ : set set const SNoLev : set set const minus_SNo : set set term - = minus_SNo axiom mul_SNo_assoc_lem1: !g:set set set.(!x:set.!y:set.SNo x -> SNo y -> SNo (g x y)) -> (!x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> g x (y + z) = g x y + g x z) -> (!x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> g (x + y) z = g x z + g y z) -> (!x:set.!y:set.SNo x -> SNo y -> !z:set.z iIn SNoL (g x y) -> !P:prop.(!w:set.w iIn SNoL x -> !u:set.u iIn SNoL y -> (z + g w u) <= g w y + g x u -> P) -> (!w:set.w iIn SNoR x -> !u:set.u iIn SNoR y -> (z + g w u) <= g w y + g x u -> P) -> P) -> (!x:set.!y:set.SNo x -> SNo y -> !z:set.z iIn SNoR (g x y) -> !P:prop.(!w:set.w iIn SNoL x -> !u:set.u iIn SNoR y -> (g w y + g x u) <= z + g w u -> P) -> (!w:set.w iIn SNoR x -> !u:set.u iIn SNoL y -> (g w y + g x u) <= z + g w u -> P) -> P) -> (!x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> SNo z -> SNo w -> z < x -> w < y -> (g z y + g x w) < g x y + g z w) -> (!x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> SNo z -> SNo w -> z <= x -> w <= y -> (g z y + g x w) <= g x y + g z w) -> !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> (!w:set.w iIn SNoS_ (SNoLev x) -> g w (g y z) = g (g w y) z) -> (!w:set.w iIn SNoS_ (SNoLev y) -> g x (g w z) = g (g x w) z) -> (!w:set.w iIn SNoS_ (SNoLev z) -> g x (g y w) = g (g x y) w) -> (!w:set.w iIn SNoS_ (SNoLev x) -> !u:set.u iIn SNoS_ (SNoLev y) -> g w (g u z) = g (g w u) z) -> (!w:set.w iIn SNoS_ (SNoLev x) -> !u:set.u iIn SNoS_ (SNoLev z) -> g w (g y u) = g (g w y) u) -> (!w:set.w iIn SNoS_ (SNoLev y) -> !u:set.u iIn SNoS_ (SNoLev z) -> g x (g w u) = g (g x w) u) -> (!w:set.w iIn SNoS_ (SNoLev x) -> !u:set.u iIn SNoS_ (SNoLev y) -> !v:set.v iIn SNoS_ (SNoLev z) -> g w (g u v) = g (g w u) v) -> !w:set.(!u:set.u iIn w -> !P:prop.(!v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoL (g y z) -> u = g v (g y z) + g x x2 + - g v x2 -> P) -> (!v:set.v iIn SNoR x -> !x2:set.x2 iIn SNoR (g y z) -> u = g v (g y z) + g x x2 + - g v x2 -> P) -> P) -> !u:set.u iIn w -> u < g (g x y) z axiom mul_SNo_assoc_lem2: !g:set set set.(!x:set.!y:set.SNo x -> SNo y -> SNo (g x y)) -> (!x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> g x (y + z) = g x y + g x z) -> (!x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> g (x + y) z = g x z + g y z) -> (!x:set.!y:set.SNo x -> SNo y -> !z:set.z iIn SNoL (g x y) -> !P:prop.(!w:set.w iIn SNoL x -> !u:set.u iIn SNoL y -> (z + g w u) <= g w y + g x u -> P) -> (!w:set.w iIn SNoR x -> !u:set.u iIn SNoR y -> (z + g w u) <= g w y + g x u -> P) -> P) -> (!x:set.!y:set.SNo x -> SNo y -> !z:set.z iIn SNoR (g x y) -> !P:prop.(!w:set.w iIn SNoL x -> !u:set.u iIn SNoR y -> (g w y + g x u) <= z + g w u -> P) -> (!w:set.w iIn SNoR x -> !u:set.u iIn SNoL y -> (g w y + g x u) <= z + g w u -> P) -> P) -> (!x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> SNo z -> SNo w -> z < x -> w < y -> (g z y + g x w) < g x y + g z w) -> (!x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> SNo z -> SNo w -> z <= x -> w <= y -> (g z y + g x w) <= g x y + g z w) -> !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> (!w:set.w iIn SNoS_ (SNoLev x) -> g w (g y z) = g (g w y) z) -> (!w:set.w iIn SNoS_ (SNoLev y) -> g x (g w z) = g (g x w) z) -> (!w:set.w iIn SNoS_ (SNoLev z) -> g x (g y w) = g (g x y) w) -> (!w:set.w iIn SNoS_ (SNoLev x) -> !u:set.u iIn SNoS_ (SNoLev y) -> g w (g u z) = g (g w u) z) -> (!w:set.w iIn SNoS_ (SNoLev x) -> !u:set.u iIn SNoS_ (SNoLev z) -> g w (g y u) = g (g w y) u) -> (!w:set.w iIn SNoS_ (SNoLev y) -> !u:set.u iIn SNoS_ (SNoLev z) -> g x (g w u) = g (g x w) u) -> (!w:set.w iIn SNoS_ (SNoLev x) -> !u:set.u iIn SNoS_ (SNoLev y) -> !v:set.v iIn SNoS_ (SNoLev z) -> g w (g u v) = g (g w u) v) -> !w:set.(!u:set.u iIn w -> !P:prop.(!v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoR (g y z) -> u = g v (g y z) + g x x2 + - g v x2 -> P) -> (!v:set.v iIn SNoR x -> !x2:set.x2 iIn SNoL (g y z) -> u = g v (g y z) + g x x2 + - g v x2 -> P) -> P) -> !u:set.u iIn w -> g (g x y) z < u axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x axiom add_SNo_com_3_0_1: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x + y + z = y + x + z 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 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 SNoCutP : set set prop const SNoCut : set set set axiom SNoCut_ext: !x:set.!y:set.!z:set.!w:set.SNoCutP x y -> SNoCutP z w -> (!u:set.u iIn x -> u < SNoCut z w) -> (!u:set.u iIn y -> SNoCut z w < u) -> (!u:set.u iIn z -> u < SNoCut x y) -> (!u:set.u iIn w -> SNoCut x y < u) -> SNoCut x y = SNoCut z w var x:set var y:set var z:set var w:set var u:set var v:set var x2:set hyp SNo x hyp SNo y hyp SNo z hyp !y2:set.y2 iIn SNoS_ (SNoLev x) -> y2 * y * z = (y2 * y) * z hyp !y2:set.y2 iIn SNoS_ (SNoLev y) -> x * y2 * z = (x * y2) * z hyp !y2:set.y2 iIn SNoS_ (SNoLev z) -> x * y * y2 = (x * y) * y2 hyp !y2:set.y2 iIn SNoS_ (SNoLev x) -> !z2:set.z2 iIn SNoS_ (SNoLev y) -> y2 * z2 * z = (y2 * z2) * z hyp !y2:set.y2 iIn SNoS_ (SNoLev x) -> !z2:set.z2 iIn SNoS_ (SNoLev z) -> y2 * y * z2 = (y2 * y) * z2 hyp !y2:set.y2 iIn SNoS_ (SNoLev y) -> !z2:set.z2 iIn SNoS_ (SNoLev z) -> x * y2 * z2 = (x * y2) * z2 hyp !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 hyp SNo (x * y) hyp SNoCutP w u hyp !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 hyp !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 hyp x * y * z = SNoCut w u hyp SNoCutP v x2 hyp !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 hyp !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 hyp (x * y) * z = SNoCut v x2 hyp !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 hyp !y2:set.!z2:set.SNo y2 -> SNo z2 -> !w2:set.w2 iIn SNoR (z2 * y2) -> !P:prop.(!u2:set.u2 iIn SNoL y2 -> !v2:set.v2 iIn SNoR z2 -> (z2 * u2 + v2 * y2) <= w2 + v2 * u2 -> P) -> (!u2:set.u2 iIn SNoR y2 -> !v2:set.v2 iIn SNoL z2 -> (z2 * u2 + v2 * y2) <= w2 + v2 * u2 -> P) -> P hyp !y2:set.!z2:set.!w2:set.!u2:set.SNo y2 -> SNo z2 -> SNo w2 -> SNo u2 -> w2 < y2 -> u2 < z2 -> (z2 * w2 + u2 * y2) < z2 * y2 + u2 * w2 hyp !y2:set.!z2:set.!w2:set.!u2:set.SNo y2 -> SNo z2 -> SNo w2 -> SNo u2 -> w2 <= y2 -> u2 <= z2 -> (z2 * w2 + u2 * y2) <= z2 * y2 + u2 * w2 hyp !y2:set.y2 iIn SNoS_ (SNoLev x) -> (y2 * y) * z = y2 * y * z hyp !y2:set.y2 iIn SNoS_ (SNoLev y) -> (x * y2) * z = x * y2 * z hyp !y2:set.y2 iIn SNoS_ (SNoLev z) -> (x * y) * y2 = x * y * y2 hyp !y2:set.y2 iIn SNoS_ (SNoLev y) -> !z2:set.z2 iIn SNoS_ (SNoLev x) -> (z2 * y2) * z = z2 * y2 * z hyp !y2:set.y2 iIn SNoS_ (SNoLev z) -> !z2:set.z2 iIn SNoS_ (SNoLev x) -> (z2 * y) * y2 = z2 * y * y2 hyp !y2:set.y2 iIn SNoS_ (SNoLev z) -> !z2:set.z2 iIn SNoS_ (SNoLev y) -> (x * z2) * y2 = x * z2 * y2 claim (!y2:set.y2 iIn SNoS_ (SNoLev z) -> !z2:set.z2 iIn SNoS_ (SNoLev y) -> !w2:set.w2 iIn SNoS_ (SNoLev x) -> (w2 * z2) * y2 = w2 * z2 * y2) -> SNoCut w u = SNoCut v x2