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 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 add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom add_SNo_SNoL_interpolate: !x:set.!y:set.SNo x -> SNo y -> !z:set.z iIn SNoL (x + y) -> (?w:set.w iIn SNoL x & z <= w + y) | ?w:set.w iIn SNoL y & z <= x + w 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_Lt1b3: !x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> SNo z -> SNo w -> (x + y) < w + z -> (x + y + - z) < w 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 SNoL x -> w <= v + y -> SNo v -> v < x -> SNo (v * u) -> (w * z + x * u + y * u) < x * z + y * z + w * 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 SNoL y -> w <= x + v -> SNo v -> v < y -> SNo (v * u) -> (w * z + x * u + y * u) < x * z + y * z + w * 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 SNoL (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) -> (w * z + (x + y) * u + - w * u) < x * z + y * z