const In : set set prop term iIn = In infix iIn 2000 2000 const SNo : set prop const SNoLt : set set prop term < = SNoLt infix < 2020 2020 term SNoCutP = \x:set.\y:set.(!z:set.z iIn x -> SNo z) & (!z:set.z iIn y -> SNo z) & !z:set.z iIn x -> !w:set.w iIn y -> z < w const Pi : set (set set) set term setexp = \x:set.\y:set.Pi y \z:set.x 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 Repl : set (set set) set axiom ReplI: !x:set.!f:set set.!y:set.y iIn x -> f y iIn Repl x f const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom SNoLtLe_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y <= z -> x < z const SNoS_ : set set const omega : set const ap : set set set const Sigma : set (set set) set const ordsucc : set set const SNoCut : set set set const minus_SNo : set set term - = minus_SNo const eps_ : set set const abs_SNo : set set var x:set var y:set var z:set var w:set var u:set var v:set var x2:set hyp SNo x hyp SNo y hyp SNo (x + y) hyp !y2:set.y2 iIn SNoS_ omega -> (!z2:set.z2 iIn omega -> abs_SNo (y2 + - x) < eps_ z2) -> y2 = x hyp !y2:set.y2 iIn omega -> SNo (ap (Sigma omega \z2:set.ap w (ordsucc z2) + ap v (ordsucc z2)) y2) hyp !y2:set.y2 iIn omega -> (ap (Sigma omega \z2:set.ap w (ordsucc z2) + ap v (ordsucc z2)) y2 + - eps_ y2) < x + y hyp SNo (SNoCut (Repl omega (ap (Sigma omega \y2:set.ap z (ordsucc y2) + ap u (ordsucc y2)))) (Repl omega (ap (Sigma omega \y2:set.ap w (ordsucc y2) + ap v (ordsucc y2))))) hyp !y2:set.y2 iIn Repl omega (ap (Sigma omega \z2:set.ap w (ordsucc z2) + ap v (ordsucc z2))) -> SNoCut (Repl omega (ap (Sigma omega \z2:set.ap z (ordsucc z2) + ap u (ordsucc z2)))) (Repl omega (ap (Sigma omega \z2:set.ap w (ordsucc z2) + ap v (ordsucc z2)))) < y2 hyp SNo x2 hyp x < x2 hyp x2 iIn SNoS_ omega claim (?y2:set.y2 iIn omega & ap (Sigma omega \z2:set.ap w (ordsucc z2) + ap v (ordsucc z2)) y2 <= x2 + y) -> SNoCut (Repl omega (ap (Sigma omega \y2:set.ap z (ordsucc y2) + ap u (ordsucc y2)))) (Repl omega (ap (Sigma omega \y2:set.ap w (ordsucc y2) + ap v (ordsucc y2)))) < x2 + y