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 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 SNoS_ : set set const SNoLev : set set const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const SNoL : set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoR : set set lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.SNo x -> (!x2:set.x2 iIn SNoS_ (SNoLev x) -> !y2:set.SNo y2 -> !P:prop.(SNo (x2 * y2) -> (!z2:set.z2 iIn SNoL x2 -> !w2:set.w2 iIn SNoL y2 -> (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2) -> (!z2:set.z2 iIn SNoR x2 -> !w2:set.w2 iIn SNoR y2 -> (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2) -> (!z2:set.z2 iIn SNoL x2 -> !w2:set.w2 iIn SNoR y2 -> (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2) -> (!z2:set.z2 iIn SNoR x2 -> !w2:set.w2 iIn SNoL y2 -> (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2) -> P) -> P) -> SNo y -> (!x2:set.x2 iIn SNoS_ (SNoLev y) -> !P:prop.(SNo (x * x2) -> (!y2:set.y2 iIn SNoL x -> !z2:set.z2 iIn SNoL x2 -> (y2 * x2 + x * z2) < x * x2 + y2 * z2) -> (!y2:set.y2 iIn SNoR x -> !z2:set.z2 iIn SNoR x2 -> (y2 * x2 + x * z2) < x * x2 + y2 * z2) -> (!y2:set.y2 iIn SNoL x -> !z2:set.z2 iIn SNoR x2 -> (x * x2 + y2 * z2) < y2 * x2 + x * z2) -> (!y2:set.y2 iIn SNoR x -> !z2:set.z2 iIn SNoL x2 -> (x * x2 + y2 * z2) < y2 * x2 + x * z2) -> P) -> P) -> (!x2:set.x2 iIn SNoR x -> !y2:set.SNo y2 -> SNo (x2 * y2)) -> (!x2:set.x2 iIn SNoR x -> SNo (x2 * y)) -> z iIn SNoR x -> w iIn SNoR y -> SNo z -> SNoLev z iIn SNoLev x -> x < z -> SNo w -> SNoLev w iIn SNoLev y -> u iIn SNoR x -> v iIn SNoL y -> SNo u -> SNoLev u iIn SNoLev x -> x < u -> SNo v -> SNo (z * y) -> SNo (x * w) -> SNo (z * w) -> SNo (u * y) -> SNo (x * v) -> SNo (u * v) -> SNo (z * v) -> SNo (u * w) -> v < w -> (!x2:set.x2 iIn SNoR x -> (x * w + x2 * v) < x2 * w + x * v) -> (z * y + x * w + u * v) < u * y + x * v + z * w lemma !x:set.!y:set.!z:set.!w:set.!u:set.SNo y -> (!v:set.v iIn SNoS_ (SNoLev y) -> !P:prop.(SNo (x * v) -> (!x2:set.x2 iIn SNoL x -> !y2:set.y2 iIn SNoL v -> (x2 * v + x * y2) < x * v + x2 * y2) -> (!x2:set.x2 iIn SNoR x -> !y2:set.y2 iIn SNoR v -> (x2 * v + x * y2) < x * v + x2 * y2) -> (!x2:set.x2 iIn SNoL x -> !y2:set.y2 iIn SNoR v -> (x * v + x2 * y2) < x2 * v + x * y2) -> (!x2:set.x2 iIn SNoR x -> !y2:set.y2 iIn SNoL v -> (x * v + x2 * y2) < x2 * v + x * y2) -> P) -> P) -> (!v:set.v iIn SNoR x -> !x2:set.SNo x2 -> SNo (v * x2)) -> z iIn SNoR y -> SNo z -> SNoLev z iIn SNoLev y -> w iIn SNoL y -> SNo w -> SNo (x * z) -> SNo (x * w) -> w < z -> u iIn SNoR x -> SNo (u * z) -> (x * z + u * w) < u * z + x * w var x:set var y:set var z:set var w:set var u:set var v:set hyp SNo x hyp !x2:set.x2 iIn SNoS_ (SNoLev x) -> !y2:set.SNo y2 -> !P:prop.(SNo (x2 * y2) -> (!z2:set.z2 iIn SNoL x2 -> !w2:set.w2 iIn SNoL y2 -> (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2) -> (!z2:set.z2 iIn SNoR x2 -> !w2:set.w2 iIn SNoR y2 -> (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2) -> (!z2:set.z2 iIn SNoL x2 -> !w2:set.w2 iIn SNoR y2 -> (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2) -> (!z2:set.z2 iIn SNoR x2 -> !w2:set.w2 iIn SNoL y2 -> (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2) -> P) -> P hyp SNo y hyp !x2:set.x2 iIn SNoS_ (SNoLev y) -> !P:prop.(SNo (x * x2) -> (!y2:set.y2 iIn SNoL x -> !z2:set.z2 iIn SNoL x2 -> (y2 * x2 + x * z2) < x * x2 + y2 * z2) -> (!y2:set.y2 iIn SNoR x -> !z2:set.z2 iIn SNoR x2 -> (y2 * x2 + x * z2) < x * x2 + y2 * z2) -> (!y2:set.y2 iIn SNoL x -> !z2:set.z2 iIn SNoR x2 -> (x * x2 + y2 * z2) < y2 * x2 + x * z2) -> (!y2:set.y2 iIn SNoR x -> !z2:set.z2 iIn SNoL x2 -> (x * x2 + y2 * z2) < y2 * x2 + x * z2) -> P) -> P hyp !x2:set.x2 iIn SNoR x -> !y2:set.SNo y2 -> SNo (x2 * y2) hyp !x2:set.x2 iIn SNoR x -> SNo (x2 * y) hyp z iIn SNoR x hyp w iIn SNoR y hyp SNo z hyp SNoLev z iIn SNoLev x hyp x < z hyp SNo w hyp SNoLev w iIn SNoLev y hyp y < w hyp u iIn SNoR x hyp v iIn SNoL y hyp SNo u hyp SNoLev u iIn SNoLev x hyp x < u hyp SNo v hyp v < y hyp SNo (z * y) hyp SNo (x * w) hyp SNo (z * w) hyp SNo (u * y) hyp SNo (x * v) hyp SNo (u * v) hyp SNo (z * v) hyp SNo (u * w) claim v < w -> (z * y + x * w + u * v) < u * y + x * v + z * w