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