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 TransSet = \x:set.!y:set.y iIn x -> Subq y x const SNo : set prop 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 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 SNoLt : set set prop term < = SNoLt infix < 2020 2020 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 SNoR : set set const SNoL : set set lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.SNo x -> SNo y -> SNo z -> (!x2:set.x2 iIn SNoR x -> !y2:set.y2 iIn SNoL y -> (z + x2 * y2) < x2 * y + x * y2) -> SNo w -> w < z -> u iIn SNoR x -> v iIn SNoL y -> (u * y + x * v) <= w + u * v -> SNo u -> SNo v -> SNo (u * v) -> (z + u * v) < w + u * v -> w < x * y var x:set var y:set var z:set var w:set var u:set var v:set hyp SNo x hyp SNo y hyp SNo z hyp !x2:set.x2 iIn SNoR x -> !y2:set.y2 iIn SNoL y -> (z + x2 * y2) < x2 * y + x * y2 hyp SNo w hyp w < z hyp u iIn SNoR x hyp v iIn SNoL y hyp (u * y + x * v) <= w + u * v hyp SNo u hyp SNo v claim SNo (u * v) -> w < x * y