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 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) axiom add_SNo_Lt2: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> y < z -> (x + y) < x + z axiom add_SNo_com_3_0_1: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x + y + z = y + x + z const ordinal : set prop const SNoLev : set set axiom SNoLev_ordinal: !x:set.SNo x -> ordinal (SNoLev x) axiom ordinal_TransSet: !x:set.ordinal x -> TransSet x axiom SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < z const SNoR : set set axiom SNoR_I: !x:set.SNo x -> !y:set.SNo y -> SNoLev y iIn SNoLev x -> x < y -> y iIn SNoR x 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 SNoS_ : set set axiom SNoR_SNoS: !x:set.SNo x -> !y:set.y iIn SNoR x -> y iIn SNoS_ (SNoLev x) axiom add_SNo_com: !x:set.!y:set.SNo x -> SNo y -> x + y = y + x const SNoL : set set axiom SNoL_or_SNoR_impred: !x:set.!y:set.SNo x -> SNo y -> !P:prop.(x = y -> P) -> (!z:set.z iIn SNoL y -> z iIn SNoR x -> P) -> (x iIn SNoL y -> P) -> (y iIn SNoR x -> P) -> (!z:set.z iIn SNoR y -> z iIn SNoL x -> P) -> (x iIn SNoR y -> P) -> (y iIn SNoL x -> P) -> P const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.!x2:set.SNo y -> (!y2:set.y2 iIn SNoS_ (SNoLev y) -> !P:prop.(SNo (x * y2) -> (!z2:set.z2 iIn SNoL x -> !w2:set.w2 iIn SNoL y2 -> (z2 * y2 + x * w2) < x * y2 + z2 * w2) -> (!z2:set.z2 iIn SNoR x -> !w2:set.w2 iIn SNoR y2 -> (z2 * y2 + x * w2) < x * y2 + z2 * w2) -> (!z2:set.z2 iIn SNoL x -> !w2:set.w2 iIn SNoR y2 -> (x * y2 + z2 * w2) < z2 * y2 + x * w2) -> (!z2:set.z2 iIn SNoR x -> !w2:set.w2 iIn SNoL y2 -> (x * y2 + z2 * w2) < z2 * y2 + x * w2) -> P) -> P) -> (!y2:set.y2 iIn SNoR x -> !z2:set.SNo z2 -> SNo (y2 * z2)) -> (!y2:set.y2 iIn SNoR y -> SNo (x * y2)) -> z iIn SNoR x -> w iIn SNoR y -> v iIn SNoR y -> SNo v -> SNoLev v iIn SNoLev y -> y < v -> SNo (z * y) -> SNo (x * w) -> SNo (z * w) -> SNo (u * y) -> SNo (x * v) -> SNo (u * v) -> SNo (z * v) -> (z * y + u * v) < u * y + z * v -> x2 iIn SNoL w -> x2 iIn SNoR v -> SNo x2 -> SNoLev x2 iIn SNoLev v -> v < x2 -> x2 iIn SNoR y -> (z * y + x * w + u * v) < u * y + x * v + z * w lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.!x2:set.SNo y -> (!y2:set.y2 iIn SNoS_ (SNoLev y) -> !P:prop.(SNo (x * y2) -> (!z2:set.z2 iIn SNoL x -> !w2:set.w2 iIn SNoL y2 -> (z2 * y2 + x * w2) < x * y2 + z2 * w2) -> (!z2:set.z2 iIn SNoR x -> !w2:set.w2 iIn SNoR y2 -> (z2 * y2 + x * w2) < x * y2 + z2 * w2) -> (!z2:set.z2 iIn SNoL x -> !w2:set.w2 iIn SNoR y2 -> (x * y2 + z2 * w2) < z2 * y2 + x * w2) -> (!z2:set.z2 iIn SNoR x -> !w2:set.w2 iIn SNoL y2 -> (x * y2 + z2 * w2) < z2 * y2 + x * w2) -> P) -> P) -> (!y2:set.y2 iIn SNoL x -> !z2:set.SNo z2 -> SNo (y2 * z2)) -> (!y2:set.y2 iIn SNoR y -> SNo (x * y2)) -> w iIn SNoR y -> SNo w -> SNoLev w iIn SNoLev y -> y < w -> u iIn SNoL x -> v iIn SNoR y -> SNo (z * y) -> SNo (x * w) -> SNo (z * w) -> SNo (u * y) -> SNo (x * v) -> SNo (u * v) -> SNo (u * w) -> (z * y + u * w) < u * y + z * w -> x2 iIn SNoR w -> x2 iIn SNoL v -> SNo x2 -> SNoLev x2 iIn SNoLev w -> w < x2 -> x2 iIn SNoR y -> (z * y + x * w + u * v) < u * y + x * v + z * w var x:set var y:set var z:set var w:set var u:set var v:set 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 SNoL x -> !y2:set.SNo y2 -> SNo (x2 * y2) hyp !x2:set.x2 iIn SNoR x -> !y2:set.SNo y2 -> SNo (x2 * y2) hyp !x2:set.x2 iIn SNoR y -> SNo (x * x2) hyp z iIn SNoR x hyp w iIn SNoR y hyp SNo w hyp SNoLev w iIn SNoLev y hyp y < w hyp u iIn SNoL x hyp v iIn SNoR y hyp SNo v hyp SNoLev v iIn SNoLev y hyp y < v 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) hyp (z * y + u * w) < u * y + z * w hyp (z * y + u * v) < u * y + z * v hyp (x * w + u * v) < u * w + x * v -> (z * y + x * w + u * v) < u * y + x * v + z * w claim ((x * w + z * v) < x * v + z * w -> (z * y + x * w + u * v) < u * y + x * v + z * w) -> (z * y + x * w + u * v) < u * y + x * v + z * w