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