const SNo : set prop 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 SNoLt : set set prop term < = SNoLt infix < 2020 2020 const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom SNoLtLe: !x:set.!y:set.x < y -> x <= y const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom add_SNo_com: !x:set.!y:set.SNo x -> SNo y -> x + y = y + x axiom SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) 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.!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) -> SNo w -> x < w -> SNo u -> y < u -> SNo v -> v < z -> SNo (g w y) -> SNo (g w u) -> SNo (g x u) -> SNo (g w y + g x u) -> (g (g w y + g x u) z + g (g x y + g w u) v) < g (g x y + g w u) z + g (g w y + g x u) v const minus_SNo : set set term - = minus_SNo var g:set set set var x:set var y:set var z:set var w:set var u:set var v:set var x2:set var y2:set hyp !z2:set.!w2:set.SNo z2 -> SNo w2 -> SNo (g z2 w2) hyp !z2:set.!w2:set.!u2:set.!v2:set.SNo z2 -> SNo w2 -> SNo u2 -> SNo v2 -> u2 < z2 -> v2 < w2 -> (g u2 w2 + g z2 v2) < g z2 w2 + g u2 v2 hyp !z2:set.!w2:set.!u2:set.!v2:set.SNo z2 -> SNo w2 -> SNo u2 -> SNo v2 -> u2 <= z2 -> v2 <= w2 -> (g u2 w2 + g z2 v2) <= g z2 w2 + g u2 v2 hyp SNo x hyp SNo y hyp SNo z hyp SNo (g x y) hyp !z2:set.z2 iIn SNoS_ (SNoLev x) -> !w2:set.SNo w2 -> !u2:set.u2 iIn SNoS_ (SNoLev y) -> !v2:set.v2 iIn SNoS_ (SNoLev z) -> w = g z2 (g y z) + g x w2 + - g z2 w2 -> (g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2)) <= g x (g u2 z + g y v2) + g z2 (w2 + g u2 v2) -> (g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2) < g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2 -> w < g (g x y) z hyp u iIn SNoR x hyp w = g u (g y z) + g x v + - g u v hyp SNo u hyp x < u hyp SNo v hyp x2 iIn SNoR y hyp y2 iIn SNoL z hyp (g x2 z + g y y2) <= v + g x2 y2 hyp SNo x2 hyp y < x2 hyp SNo y2 hyp y2 < z hyp SNo (g u y) hyp SNo (g u x2) hyp SNo (g x x2) hyp SNo (g x2 z + g y y2) hyp SNo (v + g x2 y2) hyp SNo (g x (v + g x2 y2)) hyp SNo (g u (v + g x2 y2)) hyp SNo (g u (g x2 z + g y y2)) hyp SNo (g x (g x2 z + g y y2)) hyp SNo (g x y + g u x2) claim SNo (g (g x y + g u x2) y2) -> w < g (g x y) z