const Pi : set (set set) set term setexp = \x:set.\y:set.Pi y \z:set.x const SNo : set prop const minus_SNo : set set term - = minus_SNo axiom SNo_minus_SNo: !x:set.SNo x -> SNo - 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 In : set set prop term iIn = In infix iIn 2000 2000 const omega : set const eps_ : set set axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom add_SNo_Le1: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x <= z -> (x + y) <= z + y axiom add_SNo_com: !x:set.!y:set.SNo x -> SNo y -> x + y = y + x axiom add_SNo_minus_R2': !x:set.!y:set.SNo x -> SNo y -> (x + - y) + y = x axiom FalseE: ~ False const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNoLtLe_or: !x:set.!y:set.SNo x -> SNo y -> x < y | y <= x const Empty : set const abs_SNo : set set axiom pos_abs_SNo: !x:set.Empty < x -> abs_SNo x = x axiom dneg: !P:prop.~ ~ P -> P const SNoS_ : set set const ap : set set set const SNoCut : set set set const Repl : set (set set) set lemma !x:set.!y:set.!z:set.!w:set.SNo x -> (!u:set.u iIn SNoS_ omega -> (!v:set.v iIn omega -> abs_SNo (u + - x) < eps_ v) -> u = x) -> (!u:set.u iIn omega -> SNo (ap z u)) -> (!u:set.u iIn omega -> (ap z u + - eps_ u) < x) -> SNo (SNoCut (Repl omega (ap y)) (Repl omega (ap z))) -> (!u:set.u iIn omega -> SNoCut (Repl omega (ap y)) (Repl omega (ap z)) < ap z u) -> SNo w -> x < w -> w iIn SNoS_ omega -> Empty < w + - x -> (?u:set.u iIn omega & (x + eps_ u) <= w) -> SNoCut (Repl omega (ap y)) (Repl omega (ap z)) < w lemma !x:set.!y:set.SNo x -> (!z:set.z iIn SNoS_ omega -> (!w:set.w iIn omega -> abs_SNo (z + - x) < eps_ w) -> z = x) -> SNo y -> x < y -> y iIn SNoS_ omega -> Empty < y + - x -> ~ (?z:set.z iIn omega & (x + eps_ z) <= y) -> y != x var x:set var y:set var z:set var w:set hyp SNo x hyp !u:set.u iIn SNoS_ omega -> (!v:set.v iIn omega -> abs_SNo (u + - x) < eps_ v) -> u = x hyp !u:set.u iIn omega -> SNo (ap z u) hyp !u:set.u iIn omega -> (ap z u + - eps_ u) < x hyp SNo (SNoCut (Repl omega (ap y)) (Repl omega (ap z))) hyp !u:set.u iIn omega -> SNoCut (Repl omega (ap y)) (Repl omega (ap z)) < ap z u hyp SNo w hyp x < w hyp w iIn SNoS_ omega claim Empty < w + - x -> SNoCut (Repl omega (ap y)) (Repl omega (ap z)) < w