const In : set set prop term iIn = In infix iIn 2000 2000 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y term TransSet = \x:set.!y:set.y iIn x -> Subq y x const SNo : set prop const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 term SNo_max_of = \x:set.\y:set.y iIn x & SNo y & !z:set.z iIn x -> SNo z -> z <= y 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) const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNoLeLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x <= y -> y < z -> x < z axiom SNoLtLe: !x:set.!y:set.x < y -> x <= y axiom FalseE: ~ False const SNoR : set set const SNoLev : 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 axiom add_SNo_SNoR_interpolate: !x:set.!y:set.SNo x -> SNo y -> !z:set.z iIn SNoR (x + y) -> (?w:set.w iIn SNoR x & (w + y) <= z) | ?w:set.w iIn SNoR y & (x + w) <= z 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 axiom add_SNo_com: !x:set.!y:set.SNo x -> SNo y -> x + y = y + x axiom add_SNo_SNoL_interpolate: !x:set.!y:set.SNo x -> SNo y -> !z:set.z iIn SNoL (x + y) -> (?w:set.w iIn SNoL x & z <= w + y) | ?w:set.w iIn SNoL y & z <= x + w axiom SNoLe_antisym: !x:set.!y:set.SNo x -> SNo y -> x <= y -> y <= x -> x = y axiom SNoLtLe_or: !x:set.!y:set.SNo x -> SNo y -> x < y | y <= x axiom SNoLt_SNoL_or_SNoR_impred: !x:set.!y:set.SNo x -> SNo y -> x < y -> !P:prop.(!z:set.z iIn SNoL y -> z iIn SNoR x -> P) -> (x iIn SNoL y -> P) -> (y iIn SNoR x -> P) -> P const SNoS_ : set set var x:set var y:set var z:set hyp SNo x hyp SNo y hyp SNo z hyp !w:set.w iIn SNoS_ (SNoLev z) -> x < w -> (y + w) < x + x -> ?u:set.u iIn SNoR w & y + u = x + x hyp x < z hyp (y + z) < x + x hyp SNo (x + x) hyp SNo (y + z) hyp !w:set.w iIn SNoR y -> ~ (w + z) <= x + x hyp !w:set.w iIn SNoL x -> ~ (y + z) <= w + x claim (!w:set.w iIn SNoR z -> (y + w) < x + x -> ?u:set.u iIn SNoR z & y + u = x + x) -> ?w:set.w iIn SNoR z & y + w = x + x