const SNo : set prop const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) 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 add_SNo_assoc: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x + y + z = (x + y) + 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 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 add_SNo_rotate_5_2: !x:set.!y:set.!z:set.!w:set.!u:set.SNo x -> SNo y -> SNo z -> SNo w -> SNo u -> x + y + z + w + u = w + u + x + y + z axiom add_SNo_com: !x:set.!y:set.SNo x -> SNo y -> x + y = y + x 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 SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom add_SNo_Le2: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> y <= z -> (x + y) <= x + 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_Lt1_cancel: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> (x + y) < z + y -> x < z const minus_SNo : set set term - = minus_SNo axiom add_SNo_minus_Lt2b3: !x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> SNo z -> SNo w -> (w + z) < x + y -> w < x + y + - z const In : set set prop term iIn = In infix iIn 2000 2000 const SNoS_ : set set const SNoLev : set set 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.!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 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 x (g x2 z + g y y2) + g u (v + g x2 y2)) <= g u (g x2 z + g y y2) + g x (v + g x2 y2) hyp (g (g x y + g u x2) z + g (g u y + g x x2) y2) < g (g u y + g x x2) z + g (g x y + g u x2) y2 hyp SNo u 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) hyp SNo (g x2 z) hyp SNo (g y y2) hyp SNo (g u (g x2 z)) hyp SNo (g u (g y y2)) hyp SNo (g x2 y2) hyp SNo (g x (g x2 y2)) hyp SNo (g x (g x2 z)) hyp SNo (g x (g y y2)) hyp SNo (g u (g y z) + g x v) hyp SNo (g (g x y) z + g u v) hyp SNo (g u (g x2 y2)) hyp SNo (g u (v + g x2 y2)) hyp SNo (g u (g x2 z + g y y2)) hyp SNo (g u (g y z) + g x (g x2 z) + g x (g y y2) + g u v + g u (g x2 y2)) claim SNo (g u (g x2 z) + g u (g y y2) + g x (g x2 y2)) -> g (g x y) z < w