const SNo : set prop const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom SNo_add_SNo_3: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> SNo (x + y + z) axiom SNo_add_SNo_4: !x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> SNo z -> SNo w -> SNo (x + y + z + w) const In : set set prop term iIn = In infix iIn 2000 2000 const SNoR : set set const SNoS_ : set set const SNoLev : set set axiom SNoR_SNoS: !x:set.SNo x -> !y:set.y iIn SNoR x -> y iIn SNoS_ (SNoLev x) const SNoL : 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 add_SNo_Lt2: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> y < z -> (x + y) < x + z 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 add_SNo_com: !x:set.!y:set.SNo x -> SNo y -> x + y = y + x axiom add_SNo_assoc: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x + y + z = (x + y) + z axiom SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom add_SNo_Le1: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x <= z -> (x + y) <= z + y axiom add_SNo_com_4_inner_mid: !x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> SNo z -> SNo w -> (x + y) + z + w = (x + z) + y + w axiom SNoLtLe_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y <= z -> x < z axiom add_SNo_Lt1_cancel: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> (x + y) < z + y -> x < z var x:set var y:set var z:set var w:set var u:set var v:set hyp SNo x hyp SNo y hyp SNo z hyp !x2:set.x2 iIn SNoS_ (SNoLev y) -> (x + x2) * z = x * z + x2 * z hyp !x2:set.x2 iIn SNoS_ (SNoLev z) -> (x + y) * x2 = x * x2 + y * x2 hyp !x2:set.x2 iIn SNoS_ (SNoLev y) -> !y2:set.y2 iIn SNoS_ (SNoLev z) -> (x + x2) * y2 = x * y2 + x2 * y2 hyp SNo (x + y) hyp SNo ((x + y) * z) hyp SNo (x * z) hyp SNo w hyp u iIn SNoL y hyp v iIn SNoR z hyp (u * z + y * v) <= w + u * v hyp SNo u hyp u < y hyp SNo v hyp z < v hyp SNo (u * v) hyp SNo (x + u) hyp SNo (w + u * v) hyp SNo ((x + y) * v) hyp SNo (u * z) hyp SNo (x * v) hyp SNo (y * v) hyp SNo (w + x * z) claim SNo (u * v + x * v) -> (x + y) * z < w + x * z