const SNo : set prop const In : set set prop term iIn = In infix iIn 2000 2000 const SNoR : set set const SNoLev : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNoR_E: !x:set.SNo x -> !y:set.y iIn SNoR x -> !P:prop.(SNo y -> SNoLev y iIn SNoLev x -> x < y -> P) -> P const SNoL : set set axiom SNoL_E: !x:set.SNo x -> !y:set.y iIn SNoL x -> !P:prop.(SNo y -> SNoLev y iIn SNoLev x -> y < x -> P) -> P const add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 const SNoS_ : 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.(!z2:set.!w2:set.SNo z2 -> SNo w2 -> SNo (g z2 w2)) -> (!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) -> (!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) -> SNo x -> SNo y -> SNo z -> SNo (g x y) -> (!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 x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) <= g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) -> (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 -> g (g x y) z < w) -> u iIn SNoL x -> 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) -> x2 iIn SNoL y -> y2 iIn SNoR z -> (g x2 z + g y y2) <= v + g x2 y2 -> SNo x2 -> x2 < y -> SNo y2 -> z < y2 -> SNo (g x x2) -> g (g x y) z < w 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.!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) -> (!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) -> SNo x -> SNo y -> SNo z -> SNo (g x y) -> (!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 x (g u2 z + g y v2) + g z2 (w2 + g u2 v2)) <= g z2 (g u2 z + g y v2) + g x (w2 + g u2 v2) -> (g (g x y + g z2 u2) z + g (g z2 y + g x u2) v2) < g (g z2 y + g x u2) z + g (g x y + g z2 u2) v2 -> g (g x y) z < w) -> u iIn SNoL x -> w = g u (g y z) + g x v + - g u v -> SNo u -> u < x -> SNo v -> x2 iIn SNoR y -> y2 iIn SNoL z -> (g x2 z + g y y2) <= v + g x2 y2 -> SNo x2 -> y < x2 -> SNo y2 -> y2 < z -> 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 hyp SNo (g u v) claim SNo (g u y) -> g (g x y) z < w