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