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 term ordinal = \x:set.TransSet x & !y:set.y iIn x -> TransSet y const SNo : set prop const SNoCutP : set set prop const binunion : set set set const Repl : set (set set) set const SNoL : set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoR : set set axiom add_SNo_SNoCutP: !x:set.!y:set.SNo x -> SNo y -> 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))) axiom SNoCutP_SNoL_SNoR: !x:set.SNo x -> SNoCutP (SNoL x) (SNoR x) axiom SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) 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 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 binunionE: !x:set.!y:set.!z:set.z iIn binunion x y -> z iIn x | z iIn y const SNoCut : set set set axiom SNo_eta: !x:set.SNo x -> x = SNoCut (SNoL x) (SNoR x) axiom In_no2cycle: !x:set.!y:set.x iIn y -> ~ y iIn x axiom FalseE: ~ False const SNoS_ : set set axiom SNoR_SNoS_: !x:set.Subq (SNoR x) (SNoS_ (SNoLev x)) axiom SNoLt_trichotomy_or: !x:set.!y:set.SNo x -> SNo y -> x < y | x = y | y < 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 axiom add_SNo_eq: !x:set.SNo x -> !y:set.SNo y -> x + y = SNoCut (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))) const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom SNoCut_Le: !x:set.!y:set.!z:set.!w:set.SNoCutP x y -> SNoCutP z w -> (!u:set.u iIn x -> u < SNoCut z w) -> (!u:set.u iIn w -> SNoCut x y < u) -> SNoCut x y <= SNoCut z w axiom SNoLtLe_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y <= z -> x < z axiom SNoLt_irref: !x:set.~ x < x axiom dneg: !P:prop.~ ~ P -> P axiom SNoLev_ind: !p:set prop.(!x:set.SNo x -> (!y:set.y iIn SNoS_ (SNoLev x) -> p y) -> p x) -> !x:set.SNo x -> p x lemma !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) -> (!z:set.SNo z -> SNoLev z iIn SNoLev (x + y) -> z < x + y -> (?w:set.w iIn SNoL x & z <= w + y) | ?w:set.w iIn SNoL y & z <= x + w) -> !z:set.z iIn SNoL (x + y) -> (?w:set.w iIn SNoL x & z <= w + y) | ?w:set.w iIn SNoL y & z <= x + w lemma !x:set.!y:set.!z:set.!w:set.SNo y -> SNo z -> ~ ((?u:set.u iIn SNoL x & z <= u + y) | ?u:set.u iIn SNoL y & z <= x + u) -> w iIn SNoL x -> SNo w -> SNo (w + y) -> (w + y) < z lemma !x:set.!y:set.!z:set.!w:set.SNo x -> SNo z -> ~ ((?u:set.u iIn SNoL x & z <= u + y) | ?u:set.u iIn SNoL y & z <= x + u) -> w iIn SNoL y -> SNo w -> SNo (x + w) -> (x + w) < z lemma !x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> SNo (x + y) -> SNo z -> (!u:set.u iIn SNoS_ (SNoLev z) -> SNoLev u iIn SNoLev (x + y) -> u < x + y -> (?v:set.v iIn SNoL x & u <= v + y) | ?v:set.v iIn SNoL y & u <= x + v) -> SNoLev z iIn SNoLev (x + y) -> ~ ((?u:set.u iIn SNoL x & z <= u + y) | ?u:set.u iIn SNoL y & z <= x + u) -> w iIn SNoR z -> SNo w -> SNoLev w iIn SNoLev z -> z < w -> w < x + y -> ~ w iIn SNoS_ (SNoLev z) var x:set var y:set hyp SNo x hyp SNo y claim SNo (x + y) -> !z:set.z iIn SNoL (x + y) -> (?w:set.w iIn SNoL x & z <= w + y) | ?w:set.w iIn SNoL y & z <= x + w