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