const SNo : set prop const add_SNo : set set set term + = add_SNo infix + 2281 2280 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) axiom SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) axiom SNo_add_SNo_3: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> SNo (x + y + z) const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 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_Lt2: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> y < z -> (x + y) < x + z 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 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 SNoR : set set axiom SNoR_SNoS: !x:set.SNo x -> !y:set.y iIn SNoR x -> y iIn SNoS_ (SNoLev x) axiom SNo_mul_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x * y) const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom SNoLtLe: !x:set.!y:set.x < y -> x <= y 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 axiom add_SNo_assoc: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x + y + z = (x + y) + z axiom add_SNo_Le2: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> y <= z -> (x + y) <= x + z axiom add_SNo_rotate_4_1: !x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> SNo z -> SNo w -> x + y + z + w = w + x + y + z axiom SNoLtLe_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y <= z -> x < z axiom add_SNo_assoc_4: !x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> SNo z -> SNo w -> x + y + z + w = (x + y + z) + w 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 x) -> (x2 + y) * z = x2 * z + y * z hyp !x2:set.x2 iIn SNoS_ (SNoLev x) -> !y2:set.y2 iIn SNoS_ (SNoLev z) -> (x2 + y) * y2 = x2 * y2 + y * y2 hyp SNo (x * z) hyp SNo (y * z) hyp u iIn SNoR z hyp SNo w hyp SNo u hyp z < u hyp SNo (x * u) hyp SNo (y * u) hyp SNo (w * z) hyp SNo (w * u) hyp SNo (w * z + x * u + y * u) hyp SNo (x * z + y * z + w * u) hyp v iIn SNoL x hyp w <= v + y hyp SNo v hyp v < x hyp SNo (v * u) hyp SNo (v * z) claim SNo (v + y) -> (x * z + y * z + w * u) < w * z + x * u + y * u