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 Empty : set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const minus_SNo : set set term - = minus_SNo axiom SNoLt_minus_pos: !x:set.!y:set.SNo x -> SNo y -> x < y -> Empty < y + - x axiom dneg: !P:prop.~ ~ P -> P const SNoS_ : set set const omega : set const abs_SNo : set set const eps_ : set set const ap : set set set const Sigma : set (set set) set const ordsucc : set set const SNoCut : set set set const Repl : set (set set) set const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.!x2:set.SNo x -> SNo y -> SNo (x + y) -> (!y2:set.y2 iIn SNoS_ omega -> (!z2:set.z2 iIn omega -> abs_SNo (y2 + - y) < eps_ z2) -> y2 = y) -> (!y2:set.y2 iIn omega -> SNo (ap (Sigma omega \z2:set.ap w (ordsucc z2) + ap v (ordsucc z2)) y2)) -> (!y2:set.y2 iIn omega -> (ap (Sigma omega \z2:set.ap w (ordsucc z2) + ap v (ordsucc z2)) y2 + - eps_ y2) < x + y) -> 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))))) -> (!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) -> SNo x2 -> y < x2 -> x2 iIn SNoS_ omega -> (?y2:set.y2 iIn omega & ap (Sigma omega \z2:set.ap w (ordsucc z2) + ap v (ordsucc z2)) y2 <= x + x2) -> 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)))) < x + x2 lemma !x:set.!y:set.!z:set.!w:set.!u:set.SNo x -> SNo y -> SNo (x + y) -> (!v:set.v iIn SNoS_ omega -> (!x2:set.x2 iIn omega -> abs_SNo (v + - y) < eps_ x2) -> v = y) -> (!v:set.v iIn omega -> SNo (ap (Sigma omega \x2:set.ap z (ordsucc x2) + ap w (ordsucc x2)) v)) -> (!v:set.v iIn omega -> (ap (Sigma omega \x2:set.ap z (ordsucc x2) + ap w (ordsucc x2)) v + - eps_ v) < x + y) -> SNo u -> y < u -> u iIn SNoS_ omega -> ~ (?v:set.v iIn omega & ap (Sigma omega \x2:set.ap z (ordsucc x2) + ap w (ordsucc x2)) v <= x + u) -> ~ Empty < u + - y const SNoLev : set set const real : set var x:set var y:set var z:set var w:set var u:set var v:set var x2:set hyp y iIn real 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 + - y) < eps_ z2) -> y2 = y 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 SNoLev x2 iIn SNoLev y hyp y < x2 claim x2 iIn SNoS_ omega -> 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)))) < x + x2