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 const SNo : set prop const minus_SNo : set set term - = minus_SNo axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x const SNoL : set set const SNoLev : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 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 SNoR : set set 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 add_SNo : set set set term + = add_SNo infix + 2281 2280 const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 axiom mul_SNo_Subq_lem: !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.!x2:set.!y2:set.(!z2:set.z2 iIn x2 -> !P:prop.(!w2:set.w2 iIn z -> !u2:set.u2 iIn w -> z2 = w2 * y + x * u2 + - w2 * u2 -> P) -> (!w2:set.w2 iIn u -> !u2:set.u2 iIn v -> z2 = w2 * y + x * u2 + - w2 * u2 -> P) -> P) -> (!z2:set.z2 iIn z -> !w2:set.w2 iIn w -> z2 * y + x * w2 + - z2 * w2 iIn y2) -> (!z2:set.z2 iIn u -> !w2:set.w2 iIn v -> z2 * y + x * w2 + - z2 * w2 iIn y2) -> Subq x2 y2 const Repl : set (set set) set axiom ReplE_impred: !x:set.!f:set set.!y:set.y iIn Repl x f -> !P:prop.(!z:set.z iIn x -> y = f z -> P) -> P axiom set_ext: !x:set.!y:set.Subq x y -> Subq y x -> x = y const SNoS_ : set set lemma !x:set.!y:set.!z:set.!w:set.!u:set.SNo x -> SNo y -> (!v:set.v iIn SNoS_ (SNoLev x) -> (- v) * y = - v * y) -> (!v:set.v iIn SNoS_ (SNoLev y) -> (- x) * v = - x * v) -> (!v:set.v iIn SNoS_ (SNoLev x) -> !x2:set.x2 iIn SNoS_ (SNoLev y) -> (- v) * x2 = - v * x2) -> (!v:set.v iIn SNoR x -> !x2:set.x2 iIn SNoL y -> v * y + x * x2 + - v * x2 iIn z) -> u iIn SNoL y -> SNo w -> SNoLev w iIn SNoLev - x -> w < - x -> SNo u -> SNo - w -> w * y + (- x) * u + - w * u iIn Repl z minus_SNo lemma !x:set.!y:set.!z:set.!w:set.!u:set.SNo x -> SNo y -> (!v:set.v iIn SNoS_ (SNoLev x) -> (- v) * y = - v * y) -> (!v:set.v iIn SNoS_ (SNoLev y) -> (- x) * v = - x * v) -> (!v:set.v iIn SNoS_ (SNoLev x) -> !x2:set.x2 iIn SNoS_ (SNoLev y) -> (- v) * x2 = - v * x2) -> (!v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoR y -> v * y + x * x2 + - v * x2 iIn z) -> u iIn SNoR y -> SNo w -> SNoLev w iIn SNoLev - x -> - x < w -> SNo u -> SNo - w -> w * y + (- x) * u + - w * u iIn Repl z minus_SNo lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.SNo x -> SNo y -> (!x2:set.x2 iIn SNoS_ (SNoLev x) -> (- x2) * y = - x2 * y) -> (!x2:set.x2 iIn SNoS_ (SNoLev y) -> (- x) * x2 = - x * x2) -> (!x2:set.x2 iIn SNoS_ (SNoLev x) -> !y2:set.y2 iIn SNoS_ (SNoLev y) -> (- x2) * y2 = - x2 * y2) -> (!x2:set.x2 iIn SNoR - x -> !y2:set.y2 iIn SNoR y -> x2 * y + (- x) * y2 + - x2 * y2 iIn z) -> u iIn SNoL x -> v iIn SNoR y -> w = u * y + x * v + - u * v -> SNo u -> SNoLev u iIn SNoLev x -> u < x -> SNo v -> SNo - u -> - w iIn z lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.SNo x -> SNo y -> (!x2:set.x2 iIn SNoS_ (SNoLev x) -> (- x2) * y = - x2 * y) -> (!x2:set.x2 iIn SNoS_ (SNoLev y) -> (- x) * x2 = - x * x2) -> (!x2:set.x2 iIn SNoS_ (SNoLev x) -> !y2:set.y2 iIn SNoS_ (SNoLev y) -> (- x2) * y2 = - x2 * y2) -> (!x2:set.x2 iIn SNoL - x -> !y2:set.y2 iIn SNoL y -> x2 * y + (- x) * y2 + - x2 * y2 iIn z) -> u iIn SNoR x -> v iIn SNoL y -> w = u * y + x * v + - u * v -> SNo u -> SNoLev u iIn SNoLev x -> x < u -> SNo v -> SNo - u -> - w iIn z lemma !x:set.!y:set.!z:set.!w:set.!u:set.SNo x -> SNo y -> (!v:set.v iIn SNoS_ (SNoLev x) -> (- v) * y = - v * y) -> (!v:set.v iIn SNoS_ (SNoLev y) -> (- x) * v = - x * v) -> (!v:set.v iIn SNoS_ (SNoLev x) -> !x2:set.x2 iIn SNoS_ (SNoLev y) -> (- v) * x2 = - v * x2) -> (!v:set.v iIn SNoR x -> !x2:set.x2 iIn SNoR y -> v * y + x * x2 + - v * x2 iIn z) -> u iIn SNoR y -> SNo w -> SNoLev w iIn SNoLev - x -> w < - x -> SNo u -> SNo - w -> w * y + (- x) * u + - w * u iIn Repl z minus_SNo lemma !x:set.!y:set.!z:set.!w:set.!u:set.SNo x -> SNo y -> (!v:set.v iIn SNoS_ (SNoLev x) -> (- v) * y = - v * y) -> (!v:set.v iIn SNoS_ (SNoLev y) -> (- x) * v = - x * v) -> (!v:set.v iIn SNoS_ (SNoLev x) -> !x2:set.x2 iIn SNoS_ (SNoLev y) -> (- v) * x2 = - v * x2) -> (!v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoL y -> v * y + x * x2 + - v * x2 iIn z) -> u iIn SNoL y -> SNo w -> SNoLev w iIn SNoLev - x -> - x < w -> SNo u -> SNo - w -> w * y + (- x) * u + - w * u iIn Repl z minus_SNo lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.SNo x -> SNo y -> (!x2:set.x2 iIn SNoS_ (SNoLev x) -> (- x2) * y = - x2 * y) -> (!x2:set.x2 iIn SNoS_ (SNoLev y) -> (- x) * x2 = - x * x2) -> (!x2:set.x2 iIn SNoS_ (SNoLev x) -> !y2:set.y2 iIn SNoS_ (SNoLev y) -> (- x2) * y2 = - x2 * y2) -> (!x2:set.x2 iIn SNoR - x -> !y2:set.y2 iIn SNoL y -> x2 * y + (- x) * y2 + - x2 * y2 iIn z) -> u iIn SNoL x -> v iIn SNoL y -> w = u * y + x * v + - u * v -> SNo u -> SNoLev u iIn SNoLev x -> u < x -> SNo v -> SNo - u -> - w iIn z lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.SNo x -> SNo y -> (!x2:set.x2 iIn SNoS_ (SNoLev x) -> (- x2) * y = - x2 * y) -> (!x2:set.x2 iIn SNoS_ (SNoLev y) -> (- x) * x2 = - x * x2) -> (!x2:set.x2 iIn SNoS_ (SNoLev x) -> !y2:set.y2 iIn SNoS_ (SNoLev y) -> (- x2) * y2 = - x2 * y2) -> (!x2:set.x2 iIn SNoL - x -> !y2:set.y2 iIn SNoR y -> x2 * y + (- x) * y2 + - x2 * y2 iIn z) -> u iIn SNoR x -> v iIn SNoR y -> w = u * y + x * v + - u * v -> SNo u -> SNoLev u iIn SNoLev x -> x < u -> SNo v -> SNo - u -> - w iIn z const SNoCut : set set set const SNoCutP : set set prop 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 !x2:set.x2 iIn SNoS_ (SNoLev x) -> (- x2) * y = - x2 * y hyp !x2:set.x2 iIn SNoS_ (SNoLev y) -> (- x) * x2 = - x * x2 hyp !x2:set.x2 iIn SNoS_ (SNoLev x) -> !y2:set.y2 iIn SNoS_ (SNoLev y) -> (- x2) * y2 = - x2 * y2 hyp SNo - x hyp SNoCutP z w hyp !x2:set.x2 iIn z -> !P:prop.(!y2:set.y2 iIn SNoL x -> !z2:set.z2 iIn SNoL y -> x2 = y2 * y + x * z2 + - y2 * z2 -> P) -> (!y2:set.y2 iIn SNoR x -> !z2:set.z2 iIn SNoR y -> x2 = y2 * y + x * z2 + - y2 * z2 -> P) -> P hyp !x2:set.x2 iIn SNoL x -> !y2:set.y2 iIn SNoL y -> x2 * y + x * y2 + - x2 * y2 iIn z hyp !x2:set.x2 iIn SNoR x -> !y2:set.y2 iIn SNoR y -> x2 * y + x * y2 + - x2 * y2 iIn z hyp !x2:set.x2 iIn w -> !P:prop.(!y2:set.y2 iIn SNoL x -> !z2:set.z2 iIn SNoR y -> x2 = y2 * y + x * z2 + - y2 * z2 -> P) -> (!y2:set.y2 iIn SNoR x -> !z2:set.z2 iIn SNoL y -> x2 = y2 * y + x * z2 + - y2 * z2 -> P) -> P hyp !x2:set.x2 iIn SNoL x -> !y2:set.y2 iIn SNoR y -> x2 * y + x * y2 + - x2 * y2 iIn w hyp !x2:set.x2 iIn SNoR x -> !y2:set.y2 iIn SNoL y -> x2 * y + x * y2 + - x2 * y2 iIn w hyp x * y = SNoCut z w hyp !x2:set.x2 iIn u -> !P:prop.(!y2:set.y2 iIn SNoL - x -> !z2:set.z2 iIn SNoL y -> x2 = y2 * y + (- x) * z2 + - y2 * z2 -> P) -> (!y2:set.y2 iIn SNoR - x -> !z2:set.z2 iIn SNoR y -> x2 = y2 * y + (- x) * z2 + - y2 * z2 -> P) -> P hyp !x2:set.x2 iIn SNoL - x -> !y2:set.y2 iIn SNoL y -> x2 * y + (- x) * y2 + - x2 * y2 iIn u hyp !x2:set.x2 iIn SNoR - x -> !y2:set.y2 iIn SNoR y -> x2 * y + (- x) * y2 + - x2 * y2 iIn u hyp !x2:set.x2 iIn v -> !P:prop.(!y2:set.y2 iIn SNoL - x -> !z2:set.z2 iIn SNoR y -> x2 = y2 * y + (- x) * z2 + - y2 * z2 -> P) -> (!y2:set.y2 iIn SNoR - x -> !z2:set.z2 iIn SNoL y -> x2 = y2 * y + (- x) * z2 + - y2 * z2 -> P) -> P hyp !x2:set.x2 iIn SNoL - x -> !y2:set.y2 iIn SNoR y -> x2 * y + (- x) * y2 + - x2 * y2 iIn v hyp !x2:set.x2 iIn SNoR - x -> !y2:set.y2 iIn SNoL y -> x2 * y + (- x) * y2 + - x2 * y2 iIn v hyp (- x) * y = SNoCut u v claim - x * y = SNoCut (Repl w minus_SNo) (Repl z minus_SNo) -> (- x) * y = - x * y