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 nIn = \x:set.\y:set.~ x iIn y term TransSet = \x:set.!y:set.y iIn x -> Subq y x const Pi : set (set set) set term setexp = \x:set.\y:set.Pi y \z:set.x const SNo : set prop const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 axiom SNo_mul_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x * y) const SNoR : set set const SNoLev : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 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 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 const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom mul_SNo_SNoR_interpolate_impred: !x:set.!y:set.SNo x -> SNo y -> !z:set.z iIn SNoR (x * y) -> !P:prop.(!w:set.w iIn SNoL x -> !u:set.u iIn SNoR y -> (w * y + x * u) <= z + w * u -> P) -> (!w:set.w iIn SNoR x -> !u:set.u iIn SNoL y -> (w * y + x * u) <= z + w * u -> P) -> P const minus_SNo : set set term - = minus_SNo const omega : set const eps_ : set set const abs_SNo : set set lemma !x:set.!y:set.!z:set.!w:set.!u:set.SNo x -> SNo y -> SNo (x * y) -> SNo - x * y -> (!v:set.v iIn SNoL x -> !P:prop.(!x2:set.x2 iIn omega -> eps_ x2 <= x + - v -> P) -> P) -> (!v:set.v iIn SNoR y -> !P:prop.(!x2:set.x2 iIn omega -> eps_ x2 <= v + - y -> P) -> P) -> (!v:set.v iIn omega -> abs_SNo (z + - x * y) < eps_ v) -> SNo z -> x * y < z -> w iIn SNoL x -> u iIn SNoR y -> SNo w -> SNo u -> (w * y + x * u) <= z + w * u -> ~ SNo (w * y) lemma !x:set.!y:set.!z:set.!w:set.!u:set.SNo x -> SNo y -> SNo (x * y) -> SNo - x * y -> (!v:set.v iIn SNoR x -> !P:prop.(!x2:set.x2 iIn omega -> eps_ x2 <= v + - x -> P) -> P) -> (!v:set.v iIn SNoL y -> !P:prop.(!x2:set.x2 iIn omega -> eps_ x2 <= y + - v -> P) -> P) -> (!v:set.v iIn omega -> abs_SNo (z + - x * y) < eps_ v) -> SNo z -> x * y < z -> w iIn SNoR x -> u iIn SNoL y -> SNo w -> SNo u -> (w * y + x * u) <= z + w * u -> ~ SNo (w * y) var x:set var y:set var z:set hyp SNo x hyp SNo y hyp SNo (x * y) hyp SNo - x * y hyp !w:set.SNo w -> SNoLev w iIn omega -> SNoLev w iIn SNoLev (x * y) hyp !w:set.w iIn SNoL x -> !P:prop.(!u:set.u iIn omega -> eps_ u <= x + - w -> P) -> P hyp !w:set.w iIn SNoR x -> !P:prop.(!u:set.u iIn omega -> eps_ u <= w + - x -> P) -> P hyp !w:set.w iIn SNoL y -> !P:prop.(!u:set.u iIn omega -> eps_ u <= y + - w -> P) -> P hyp !w:set.w iIn SNoR y -> !P:prop.(!u:set.u iIn omega -> eps_ u <= w + - y -> P) -> P hyp !w:set.w iIn omega -> abs_SNo (z + - x * y) < eps_ w hyp SNoLev z iIn omega hyp SNo z hyp x * y < z claim ~ z iIn SNoR (x * y)