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 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 const binunion : set set set axiom binunionE: !x:set.!y:set.!z:set.z iIn binunion x y -> z iIn x | z iIn y const SNoLev : set set const SNoS_ : set set axiom SNoS_I2: !x:set.!y:set.SNo x -> SNo y -> SNoLev x iIn SNoLev y -> x iIn SNoS_ (SNoLev y) 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 add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoR : set set lemma !x:set.!y:set.(!z:set.!w:set.SNo (z + w) & (!u:set.u iIn SNoL z -> (u + w) < z + w) & (!u:set.u iIn SNoR z -> (z + w) < u + w) & (!u:set.u iIn SNoL w -> (z + u) < z + w) & (!u:set.u iIn SNoR w -> (z + w) < z + u) & SNoCutP (binunion (Repl (SNoL z) \u:set.u + w) (Repl (SNoL w) (add_SNo z))) (binunion (Repl (SNoR z) \u:set.u + w) (Repl (SNoR w) (add_SNo z))) -> !P:prop.(SNo (z + w) -> (!u:set.u iIn SNoL z -> (u + w) < z + w) -> (!u:set.u iIn SNoR z -> (z + w) < u + w) -> (!u:set.u iIn SNoL w -> (z + u) < z + w) -> (!u:set.u iIn SNoR w -> (z + w) < z + u) -> P) -> P) -> SNo x -> SNo y -> (!z:set.z iIn SNoS_ (SNoLev x) -> SNo (z + y) & (!w:set.w iIn SNoL z -> (w + y) < z + y) & (!w:set.w iIn SNoR z -> (z + y) < w + y) & (!w:set.w iIn SNoL y -> (z + w) < z + y) & (!w:set.w iIn SNoR y -> (z + y) < z + w) & SNoCutP (binunion (Repl (SNoL z) \w:set.w + y) (Repl (SNoL y) (add_SNo z))) (binunion (Repl (SNoR z) \w:set.w + y) (Repl (SNoR y) (add_SNo z)))) -> (!z:set.z iIn SNoS_ (SNoLev y) -> SNo (x + z) & (!w:set.w iIn SNoL x -> (w + z) < x + z) & (!w:set.w iIn SNoR x -> (x + z) < w + z) & (!w:set.w iIn SNoL z -> (x + w) < x + z) & (!w:set.w iIn SNoR z -> (x + z) < x + w) & SNoCutP (binunion (Repl (SNoL x) \w:set.w + z) (Repl (SNoL z) (add_SNo x))) (binunion (Repl (SNoR x) \w:set.w + z) (Repl (SNoR z) (add_SNo x)))) -> (!z:set.z iIn SNoS_ (SNoLev x) -> !w:set.w iIn SNoS_ (SNoLev y) -> SNo (z + w) & (!u:set.u iIn SNoL z -> (u + w) < z + w) & (!u:set.u iIn SNoR z -> (z + w) < u + w) & (!u:set.u iIn SNoL w -> (z + u) < z + w) & (!u:set.u iIn SNoR w -> (z + w) < z + u) & SNoCutP (binunion (Repl (SNoL z) \u:set.u + w) (Repl (SNoL w) (add_SNo z))) (binunion (Repl (SNoR z) \u:set.u + w) (Repl (SNoR w) (add_SNo z)))) -> TransSet (SNoLev x) -> TransSet (SNoLev y) -> (!z:set.z iIn SNoL x -> SNo (z + y) & (!w:set.w iIn SNoL z -> (w + y) < z + y) & (!w:set.w iIn SNoR z -> (z + y) < w + y) & (!w:set.w iIn SNoL y -> (z + w) < z + y) & (!w:set.w iIn SNoR y -> (z + y) < z + w) & SNoCutP (binunion (Repl (SNoL z) \w:set.w + y) (Repl (SNoL y) (add_SNo z))) (binunion (Repl (SNoR z) \w:set.w + y) (Repl (SNoR y) (add_SNo z)))) -> (!z:set.z iIn SNoR x -> SNo (z + y) & (!w:set.w iIn SNoL z -> (w + y) < z + y) & (!w:set.w iIn SNoR z -> (z + y) < w + y) & (!w:set.w iIn SNoL y -> (z + w) < z + y) & (!w:set.w iIn SNoR y -> (z + y) < z + w) & SNoCutP (binunion (Repl (SNoL z) \w:set.w + y) (Repl (SNoL y) (add_SNo z))) (binunion (Repl (SNoR z) \w:set.w + y) (Repl (SNoR y) (add_SNo z)))) -> (!z:set.z iIn SNoL y -> SNo (x + z) & (!w:set.w iIn SNoL x -> (w + z) < x + z) & (!w:set.w iIn SNoR x -> (x + z) < w + z) & (!w:set.w iIn SNoL z -> (x + w) < x + z) & (!w:set.w iIn SNoR z -> (x + z) < x + w) & SNoCutP (binunion (Repl (SNoL x) \w:set.w + z) (Repl (SNoL z) (add_SNo x))) (binunion (Repl (SNoR x) \w:set.w + z) (Repl (SNoR z) (add_SNo x)))) -> (!z:set.z iIn SNoR y -> SNo (x + z) & (!w:set.w iIn SNoL x -> (w + z) < x + z) & (!w:set.w iIn SNoR x -> (x + z) < w + z) & (!w:set.w iIn SNoL z -> (x + w) < x + z) & (!w:set.w iIn SNoR z -> (x + z) < x + w) & SNoCutP (binunion (Repl (SNoL x) \w:set.w + z) (Repl (SNoL z) (add_SNo x))) (binunion (Repl (SNoR x) \w:set.w + z) (Repl (SNoR z) (add_SNo x)))) -> SNoCutP (binunion (Repl (SNoL x) \z:set.z + y) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) \z:set.z + y) (Repl (SNoR y) (add_SNo x))) -> SNo (x + y) & (!z:set.z iIn SNoL x -> (z + y) < x + y) & (!z:set.z iIn SNoR x -> (x + y) < z + y) & (!z:set.z iIn SNoL y -> (x + z) < x + y) & (!z:set.z iIn SNoR y -> (x + y) < x + z) & SNoCutP (binunion (Repl (SNoL x) \z:set.z + y) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) \z:set.z + y) (Repl (SNoR y) (add_SNo x))) lemma !x:set.!y:set.!z:set.!w:set.!u:set.(!v:set.!x2:set.SNo (v + x2) & (!y2:set.y2 iIn SNoL v -> (y2 + x2) < v + x2) & (!y2:set.y2 iIn SNoR v -> (v + x2) < y2 + x2) & (!y2:set.y2 iIn SNoL x2 -> (v + y2) < v + x2) & (!y2:set.y2 iIn SNoR x2 -> (v + x2) < v + y2) & SNoCutP (binunion (Repl (SNoL v) \y2:set.y2 + x2) (Repl (SNoL x2) (add_SNo v))) (binunion (Repl (SNoR v) \y2:set.y2 + x2) (Repl (SNoR x2) (add_SNo v))) -> !P:prop.(SNo (v + x2) -> (!y2:set.y2 iIn SNoL v -> (y2 + x2) < v + x2) -> (!y2:set.y2 iIn SNoR v -> (v + x2) < y2 + x2) -> (!y2:set.y2 iIn SNoL x2 -> (v + y2) < v + x2) -> (!y2:set.y2 iIn SNoR x2 -> (v + x2) < v + y2) -> P) -> P) -> SNo x -> SNo y -> (!v:set.v iIn SNoS_ (SNoLev x) -> SNo (v + y) & (!x2:set.x2 iIn SNoL v -> (x2 + y) < v + y) & (!x2:set.x2 iIn SNoR v -> (v + y) < x2 + y) & (!x2:set.x2 iIn SNoL y -> (v + x2) < v + y) & (!x2:set.x2 iIn SNoR y -> (v + y) < v + x2) & SNoCutP (binunion (Repl (SNoL v) \x2:set.x2 + y) (Repl (SNoL y) (add_SNo v))) (binunion (Repl (SNoR v) \x2:set.x2 + y) (Repl (SNoR y) (add_SNo v)))) -> (!v:set.v iIn SNoS_ (SNoLev x) -> !x2:set.x2 iIn SNoS_ (SNoLev y) -> SNo (v + x2) & (!y2:set.y2 iIn SNoL v -> (y2 + x2) < v + x2) & (!y2:set.y2 iIn SNoR v -> (v + x2) < y2 + x2) & (!y2:set.y2 iIn SNoL x2 -> (v + y2) < v + x2) & (!y2:set.y2 iIn SNoR x2 -> (v + x2) < v + y2) & SNoCutP (binunion (Repl (SNoL v) \y2:set.y2 + x2) (Repl (SNoL x2) (add_SNo v))) (binunion (Repl (SNoR v) \y2:set.y2 + x2) (Repl (SNoR x2) (add_SNo v)))) -> TransSet (SNoLev x) -> (!v:set.v iIn SNoL x -> SNo (v + y) & (!x2:set.x2 iIn SNoL v -> (x2 + y) < v + y) & (!x2:set.x2 iIn SNoR v -> (v + y) < x2 + y) & (!x2:set.x2 iIn SNoL y -> (v + x2) < v + y) & (!x2:set.x2 iIn SNoR y -> (v + y) < v + x2) & SNoCutP (binunion (Repl (SNoL v) \x2:set.x2 + y) (Repl (SNoL y) (add_SNo v))) (binunion (Repl (SNoR v) \x2:set.x2 + y) (Repl (SNoR y) (add_SNo v)))) -> (!v:set.v iIn SNoR x -> SNo (v + y) & (!x2:set.x2 iIn SNoL v -> (x2 + y) < v + y) & (!x2:set.x2 iIn SNoR v -> (v + y) < x2 + y) & (!x2:set.x2 iIn SNoL y -> (v + x2) < v + y) & (!x2:set.x2 iIn SNoR y -> (v + y) < v + x2) & SNoCutP (binunion (Repl (SNoL v) \x2:set.x2 + y) (Repl (SNoL y) (add_SNo v))) (binunion (Repl (SNoR v) \x2:set.x2 + y) (Repl (SNoR y) (add_SNo v)))) -> (!v:set.v iIn SNoR y -> SNo (x + v) & (!x2:set.x2 iIn SNoL x -> (x2 + v) < x + v) & (!x2:set.x2 iIn SNoR x -> (x + v) < x2 + v) & (!x2:set.x2 iIn SNoL v -> (x + x2) < x + v) & (!x2:set.x2 iIn SNoR v -> (x + v) < x + x2) & SNoCutP (binunion (Repl (SNoL x) \x2:set.x2 + v) (Repl (SNoL v) (add_SNo x))) (binunion (Repl (SNoR x) \x2:set.x2 + v) (Repl (SNoR v) (add_SNo x)))) -> w iIn binunion (Repl (SNoR x) \v:set.v + y) (Repl (SNoR y) (add_SNo x)) -> u iIn SNoL x -> z = u + y -> SNo u -> SNoLev u iIn SNoLev x -> u < x -> u iIn SNoS_ (SNoLev x) -> z < w lemma !x:set.!y:set.!z:set.!w:set.!u:set.(!v:set.!x2:set.SNo (v + x2) & (!y2:set.y2 iIn SNoL v -> (y2 + x2) < v + x2) & (!y2:set.y2 iIn SNoR v -> (v + x2) < y2 + x2) & (!y2:set.y2 iIn SNoL x2 -> (v + y2) < v + x2) & (!y2:set.y2 iIn SNoR x2 -> (v + x2) < v + y2) & SNoCutP (binunion (Repl (SNoL v) \y2:set.y2 + x2) (Repl (SNoL x2) (add_SNo v))) (binunion (Repl (SNoR v) \y2:set.y2 + x2) (Repl (SNoR x2) (add_SNo v))) -> !P:prop.(SNo (v + x2) -> (!y2:set.y2 iIn SNoL v -> (y2 + x2) < v + x2) -> (!y2:set.y2 iIn SNoR v -> (v + x2) < y2 + x2) -> (!y2:set.y2 iIn SNoL x2 -> (v + y2) < v + x2) -> (!y2:set.y2 iIn SNoR x2 -> (v + x2) < v + y2) -> P) -> P) -> SNo x -> SNo y -> (!v:set.v iIn SNoS_ (SNoLev y) -> SNo (x + v) & (!x2:set.x2 iIn SNoL x -> (x2 + v) < x + v) & (!x2:set.x2 iIn SNoR x -> (x + v) < x2 + v) & (!x2:set.x2 iIn SNoL v -> (x + x2) < x + v) & (!x2:set.x2 iIn SNoR v -> (x + v) < x + x2) & SNoCutP (binunion (Repl (SNoL x) \x2:set.x2 + v) (Repl (SNoL v) (add_SNo x))) (binunion (Repl (SNoR x) \x2:set.x2 + v) (Repl (SNoR v) (add_SNo x)))) -> (!v:set.v iIn SNoS_ (SNoLev x) -> !x2:set.x2 iIn SNoS_ (SNoLev y) -> SNo (v + x2) & (!y2:set.y2 iIn SNoL v -> (y2 + x2) < v + x2) & (!y2:set.y2 iIn SNoR v -> (v + x2) < y2 + x2) & (!y2:set.y2 iIn SNoL x2 -> (v + y2) < v + x2) & (!y2:set.y2 iIn SNoR x2 -> (v + x2) < v + y2) & SNoCutP (binunion (Repl (SNoL v) \y2:set.y2 + x2) (Repl (SNoL x2) (add_SNo v))) (binunion (Repl (SNoR v) \y2:set.y2 + x2) (Repl (SNoR x2) (add_SNo v)))) -> TransSet (SNoLev y) -> (!v:set.v iIn SNoR x -> SNo (v + y) & (!x2:set.x2 iIn SNoL v -> (x2 + y) < v + y) & (!x2:set.x2 iIn SNoR v -> (v + y) < x2 + y) & (!x2:set.x2 iIn SNoL y -> (v + x2) < v + y) & (!x2:set.x2 iIn SNoR y -> (v + y) < v + x2) & SNoCutP (binunion (Repl (SNoL v) \x2:set.x2 + y) (Repl (SNoL y) (add_SNo v))) (binunion (Repl (SNoR v) \x2:set.x2 + y) (Repl (SNoR y) (add_SNo v)))) -> (!v:set.v iIn SNoL y -> SNo (x + v) & (!x2:set.x2 iIn SNoL x -> (x2 + v) < x + v) & (!x2:set.x2 iIn SNoR x -> (x + v) < x2 + v) & (!x2:set.x2 iIn SNoL v -> (x + x2) < x + v) & (!x2:set.x2 iIn SNoR v -> (x + v) < x + x2) & SNoCutP (binunion (Repl (SNoL x) \x2:set.x2 + v) (Repl (SNoL v) (add_SNo x))) (binunion (Repl (SNoR x) \x2:set.x2 + v) (Repl (SNoR v) (add_SNo x)))) -> (!v:set.v iIn SNoR y -> SNo (x + v) & (!x2:set.x2 iIn SNoL x -> (x2 + v) < x + v) & (!x2:set.x2 iIn SNoR x -> (x + v) < x2 + v) & (!x2:set.x2 iIn SNoL v -> (x + x2) < x + v) & (!x2:set.x2 iIn SNoR v -> (x + v) < x + x2) & SNoCutP (binunion (Repl (SNoL x) \x2:set.x2 + v) (Repl (SNoL v) (add_SNo x))) (binunion (Repl (SNoR x) \x2:set.x2 + v) (Repl (SNoR v) (add_SNo x)))) -> w iIn binunion (Repl (SNoR x) \v:set.v + y) (Repl (SNoR y) (add_SNo x)) -> u iIn SNoL y -> z = x + u -> SNo u -> SNoLev u iIn SNoLev y -> u < y -> u iIn SNoS_ (SNoLev y) -> z < w var x:set var y:set hyp !z:set.!w:set.SNo (z + w) & (!u:set.u iIn SNoL z -> (u + w) < z + w) & (!u:set.u iIn SNoR z -> (z + w) < u + w) & (!u:set.u iIn SNoL w -> (z + u) < z + w) & (!u:set.u iIn SNoR w -> (z + w) < z + u) & SNoCutP (binunion (Repl (SNoL z) \u:set.u + w) (Repl (SNoL w) (add_SNo z))) (binunion (Repl (SNoR z) \u:set.u + w) (Repl (SNoR w) (add_SNo z))) -> !P:prop.(SNo (z + w) -> (!u:set.u iIn SNoL z -> (u + w) < z + w) -> (!u:set.u iIn SNoR z -> (z + w) < u + w) -> (!u:set.u iIn SNoL w -> (z + u) < z + w) -> (!u:set.u iIn SNoR w -> (z + w) < z + u) -> P) -> P hyp SNo x hyp SNo y hyp !z:set.z iIn SNoS_ (SNoLev x) -> SNo (z + y) & (!w:set.w iIn SNoL z -> (w + y) < z + y) & (!w:set.w iIn SNoR z -> (z + y) < w + y) & (!w:set.w iIn SNoL y -> (z + w) < z + y) & (!w:set.w iIn SNoR y -> (z + y) < z + w) & SNoCutP (binunion (Repl (SNoL z) \w:set.w + y) (Repl (SNoL y) (add_SNo z))) (binunion (Repl (SNoR z) \w:set.w + y) (Repl (SNoR y) (add_SNo z))) hyp !z:set.z iIn SNoS_ (SNoLev y) -> SNo (x + z) & (!w:set.w iIn SNoL x -> (w + z) < x + z) & (!w:set.w iIn SNoR x -> (x + z) < w + z) & (!w:set.w iIn SNoL z -> (x + w) < x + z) & (!w:set.w iIn SNoR z -> (x + z) < x + w) & SNoCutP (binunion (Repl (SNoL x) \w:set.w + z) (Repl (SNoL z) (add_SNo x))) (binunion (Repl (SNoR x) \w:set.w + z) (Repl (SNoR z) (add_SNo x))) hyp !z:set.z iIn SNoS_ (SNoLev x) -> !w:set.w iIn SNoS_ (SNoLev y) -> SNo (z + w) & (!u:set.u iIn SNoL z -> (u + w) < z + w) & (!u:set.u iIn SNoR z -> (z + w) < u + w) & (!u:set.u iIn SNoL w -> (z + u) < z + w) & (!u:set.u iIn SNoR w -> (z + w) < z + u) & SNoCutP (binunion (Repl (SNoL z) \u:set.u + w) (Repl (SNoL w) (add_SNo z))) (binunion (Repl (SNoR z) \u:set.u + w) (Repl (SNoR w) (add_SNo z))) hyp TransSet (SNoLev x) hyp TransSet (SNoLev y) hyp !z:set.z iIn SNoL x -> SNo (z + y) & (!w:set.w iIn SNoL z -> (w + y) < z + y) & (!w:set.w iIn SNoR z -> (z + y) < w + y) & (!w:set.w iIn SNoL y -> (z + w) < z + y) & (!w:set.w iIn SNoR y -> (z + y) < z + w) & SNoCutP (binunion (Repl (SNoL z) \w:set.w + y) (Repl (SNoL y) (add_SNo z))) (binunion (Repl (SNoR z) \w:set.w + y) (Repl (SNoR y) (add_SNo z))) hyp !z:set.z iIn SNoR x -> SNo (z + y) & (!w:set.w iIn SNoL z -> (w + y) < z + y) & (!w:set.w iIn SNoR z -> (z + y) < w + y) & (!w:set.w iIn SNoL y -> (z + w) < z + y) & (!w:set.w iIn SNoR y -> (z + y) < z + w) & SNoCutP (binunion (Repl (SNoL z) \w:set.w + y) (Repl (SNoL y) (add_SNo z))) (binunion (Repl (SNoR z) \w:set.w + y) (Repl (SNoR y) (add_SNo z))) hyp !z:set.z iIn SNoL y -> SNo (x + z) & (!w:set.w iIn SNoL x -> (w + z) < x + z) & (!w:set.w iIn SNoR x -> (x + z) < w + z) & (!w:set.w iIn SNoL z -> (x + w) < x + z) & (!w:set.w iIn SNoR z -> (x + z) < x + w) & SNoCutP (binunion (Repl (SNoL x) \w:set.w + z) (Repl (SNoL z) (add_SNo x))) (binunion (Repl (SNoR x) \w:set.w + z) (Repl (SNoR z) (add_SNo x))) claim (!z:set.z iIn SNoR y -> SNo (x + z) & (!w:set.w iIn SNoL x -> (w + z) < x + z) & (!w:set.w iIn SNoR x -> (x + z) < w + z) & (!w:set.w iIn SNoL z -> (x + w) < x + z) & (!w:set.w iIn SNoR z -> (x + z) < x + w) & SNoCutP (binunion (Repl (SNoL x) \w:set.w + z) (Repl (SNoL z) (add_SNo x))) (binunion (Repl (SNoR x) \w:set.w + z) (Repl (SNoR z) (add_SNo x)))) -> SNo (x + y) & (!z:set.z iIn SNoL x -> (z + y) < x + y) & (!z:set.z iIn SNoR x -> (x + y) < z + y) & (!z:set.z iIn SNoL y -> (x + z) < x + y) & (!z:set.z iIn SNoR y -> (x + y) < x + z) & SNoCutP (binunion (Repl (SNoL x) \z:set.z + y) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) \z:set.z + y) (Repl (SNoR y) (add_SNo x)))