%------------------------------------------------------------------------------ % File : Duper---1.0 % Problem : NUM409+1 : TPTP v9.2.0. Released v3.2.0. % Transfm : none % Format : tptp:raw % Command : duper %s % Computer : n028.cluster.edu % Model : x86_64 x86_64 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz % Memory : 8042.1875MB % OS : Linux 3.10.0-693.el7.x86_64 % CPULimit : 300s % WCLimit : 300s % DateTime : Fri Oct 3 07:56:26 PM UTC 2025 % Result : Theorem 5.22s 5.44s % Output : Proof 5.30s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.11/0.12 % Problem : NUM409+1 : TPTP v9.2.0. Released v3.2.0. % 0.11/0.13 % Command : duper %s % 0.12/0.34 % Computer : n028.cluster.edu % 0.12/0.34 % Model : x86_64 x86_64 % 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.12/0.34 % Memory : 8042.1875MB % 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64 % 0.12/0.34 % CPULimit : 300 % 0.12/0.34 % WCLimit : 300 % 0.12/0.34 % DateTime : Fri Oct 3 07:22:08 EDT 2025 % 0.12/0.34 % CPUTime : % 5.22/5.44 SZS status Theorem for theBenchmark.p % 5.22/5.44 SZS output start Proof for theBenchmark.p % 5.22/5.44 Clause #1 (by assumption #[]): Eq (∀ (A : Iota), empty A → function A) True % 5.22/5.44 Clause #3 (by assumption #[]): Eq (∀ (A : Iota), empty A → relation A) True % 5.22/5.44 Clause #6 (by assumption #[]): Eq (∀ (A : Iota), empty A → And (And (epsilon_transitive A) (epsilon_connected A)) (ordinal A)) True % 5.22/5.44 Clause #10 (by assumption #[]): Eq (∀ (A : Iota), And (relation A) (function A) → Iff (transfinite_sequence A) (ordinal (relation_dom A))) True % 5.22/5.44 Clause #11 (by assumption #[]): Eq % 5.22/5.44 (∀ (A B : Iota), % 5.22/5.44 And (And (relation B) (function B)) (transfinite_sequence B) → % 5.22/5.44 Iff (transfinite_sequence_of B A) (subset (relation_rng B) A)) % 5.22/5.44 True % 5.22/5.44 Clause #16 (by assumption #[]): Eq (empty empty_set) True % 5.22/5.44 Clause #23 (by assumption #[]): Eq (∀ (A : Iota), empty A → And (empty (relation_dom A)) (relation (relation_dom A))) True % 5.22/5.44 Clause #24 (by assumption #[]): Eq (∀ (A : Iota), empty A → And (empty (relation_rng A)) (relation (relation_rng A))) True % 5.22/5.44 Clause #42 (by assumption #[]): Eq (∀ (A : Iota), subset empty_set A) True % 5.22/5.44 Clause #44 (by assumption #[]): Eq (Not (∀ (A : Iota), transfinite_sequence_of empty_set A)) True % 5.22/5.44 Clause #48 (by assumption #[]): Eq (∀ (A : Iota), empty A → Eq A empty_set) True % 5.22/5.44 Clause #54 (by clausification #[3]): ∀ (a : Iota), Eq (empty a → relation a) True % 5.22/5.44 Clause #55 (by clausification #[54]): ∀ (a : Iota), Or (Eq (empty a) False) (Eq (relation a) True) % 5.22/5.44 Clause #56 (by superposition #[55, 16]): Or (Eq (relation empty_set) True) (Eq False True) % 5.22/5.44 Clause #57 (by clausification #[56]): Eq (relation empty_set) True % 5.22/5.44 Clause #58 (by clausification #[1]): ∀ (a : Iota), Eq (empty a → function a) True % 5.22/5.44 Clause #59 (by clausification #[58]): ∀ (a : Iota), Or (Eq (empty a) False) (Eq (function a) True) % 5.22/5.44 Clause #60 (by superposition #[59, 16]): Or (Eq (function empty_set) True) (Eq False True) % 5.22/5.44 Clause #65 (by clausification #[60]): Eq (function empty_set) True % 5.22/5.44 Clause #66 (by clausification #[42]): ∀ (a : Iota), Eq (subset empty_set a) True % 5.22/5.44 Clause #67 (by clausification #[44]): Eq (∀ (A : Iota), transfinite_sequence_of empty_set A) False % 5.22/5.44 Clause #68 (by clausification #[67]): ∀ (a : Iota), Eq (Not (transfinite_sequence_of empty_set (skS.0 0 a))) True % 5.22/5.44 Clause #69 (by clausification #[68]): ∀ (a : Iota), Eq (transfinite_sequence_of empty_set (skS.0 0 a)) False % 5.22/5.44 Clause #85 (by clausification #[48]): ∀ (a : Iota), Eq (empty a → Eq a empty_set) True % 5.22/5.44 Clause #86 (by clausification #[85]): ∀ (a : Iota), Or (Eq (empty a) False) (Eq (Eq a empty_set) True) % 5.22/5.44 Clause #87 (by clausification #[86]): ∀ (a : Iota), Or (Eq (empty a) False) (Eq a empty_set) % 5.22/5.44 Clause #120 (by clausification #[6]): ∀ (a : Iota), Eq (empty a → And (And (epsilon_transitive a) (epsilon_connected a)) (ordinal a)) True % 5.22/5.44 Clause #121 (by clausification #[120]): ∀ (a : Iota), Or (Eq (empty a) False) (Eq (And (And (epsilon_transitive a) (epsilon_connected a)) (ordinal a)) True) % 5.22/5.44 Clause #122 (by clausification #[121]): ∀ (a : Iota), Or (Eq (empty a) False) (Eq (ordinal a) True) % 5.22/5.44 Clause #124 (by superposition #[122, 16]): Or (Eq (ordinal empty_set) True) (Eq False True) % 5.22/5.44 Clause #126 (by clausification #[124]): Eq (ordinal empty_set) True % 5.22/5.44 Clause #175 (by clausification #[10]): ∀ (a : Iota), Eq (And (relation a) (function a) → Iff (transfinite_sequence a) (ordinal (relation_dom a))) True % 5.22/5.44 Clause #176 (by clausification #[175]): ∀ (a : Iota), % 5.22/5.44 Or (Eq (And (relation a) (function a)) False) (Eq (Iff (transfinite_sequence a) (ordinal (relation_dom a))) True) % 5.22/5.44 Clause #177 (by clausification #[176]): ∀ (a : Iota), % 5.22/5.44 Or (Eq (Iff (transfinite_sequence a) (ordinal (relation_dom a))) True) % 5.22/5.44 (Or (Eq (relation a) False) (Eq (function a) False)) % 5.22/5.44 Clause #178 (by clausification #[177]): ∀ (a : Iota), % 5.22/5.44 Or (Eq (relation a) False) % 5.22/5.44 (Or (Eq (function a) False) (Or (Eq (transfinite_sequence a) True) (Eq (ordinal (relation_dom a)) False))) % 5.22/5.44 Clause #180 (by superposition #[178, 57]): Or (Eq (function empty_set) False) % 5.22/5.44 (Or (Eq (transfinite_sequence empty_set) True) (Or (Eq (ordinal (relation_dom empty_set)) False) (Eq False True))) % 5.30/5.46 Clause #200 (by clausification #[11]): ∀ (a : Iota), % 5.30/5.46 Eq % 5.30/5.46 (∀ (B : Iota), % 5.30/5.46 And (And (relation B) (function B)) (transfinite_sequence B) → % 5.30/5.46 Iff (transfinite_sequence_of B a) (subset (relation_rng B) a)) % 5.30/5.46 True % 5.30/5.46 Clause #201 (by clausification #[200]): ∀ (a a_1 : Iota), % 5.30/5.46 Eq % 5.30/5.46 (And (And (relation a) (function a)) (transfinite_sequence a) → % 5.30/5.46 Iff (transfinite_sequence_of a a_1) (subset (relation_rng a) a_1)) % 5.30/5.46 True % 5.30/5.46 Clause #202 (by clausification #[201]): ∀ (a a_1 : Iota), % 5.30/5.46 Or (Eq (And (And (relation a) (function a)) (transfinite_sequence a)) False) % 5.30/5.46 (Eq (Iff (transfinite_sequence_of a a_1) (subset (relation_rng a) a_1)) True) % 5.30/5.46 Clause #203 (by clausification #[202]): ∀ (a a_1 : Iota), % 5.30/5.46 Or (Eq (Iff (transfinite_sequence_of a a_1) (subset (relation_rng a) a_1)) True) % 5.30/5.46 (Or (Eq (And (relation a) (function a)) False) (Eq (transfinite_sequence a) False)) % 5.30/5.46 Clause #204 (by clausification #[203]): ∀ (a a_1 : Iota), % 5.30/5.46 Or (Eq (And (relation a) (function a)) False) % 5.30/5.46 (Or (Eq (transfinite_sequence a) False) % 5.30/5.46 (Or (Eq (transfinite_sequence_of a a_1) True) (Eq (subset (relation_rng a) a_1) False))) % 5.30/5.46 Clause #206 (by clausification #[204]): ∀ (a a_1 : Iota), % 5.30/5.46 Or (Eq (transfinite_sequence a) False) % 5.30/5.46 (Or (Eq (transfinite_sequence_of a a_1) True) % 5.30/5.46 (Or (Eq (subset (relation_rng a) a_1) False) (Or (Eq (relation a) False) (Eq (function a) False)))) % 5.30/5.46 Clause #243 (by clausification #[24]): ∀ (a : Iota), Eq (empty a → And (empty (relation_rng a)) (relation (relation_rng a))) True % 5.30/5.46 Clause #244 (by clausification #[243]): ∀ (a : Iota), Or (Eq (empty a) False) (Eq (And (empty (relation_rng a)) (relation (relation_rng a))) True) % 5.30/5.46 Clause #246 (by clausification #[244]): ∀ (a : Iota), Or (Eq (empty a) False) (Eq (empty (relation_rng a)) True) % 5.30/5.46 Clause #248 (by clausification #[23]): ∀ (a : Iota), Eq (empty a → And (empty (relation_dom a)) (relation (relation_dom a))) True % 5.30/5.46 Clause #249 (by clausification #[248]): ∀ (a : Iota), Or (Eq (empty a) False) (Eq (And (empty (relation_dom a)) (relation (relation_dom a))) True) % 5.30/5.46 Clause #251 (by clausification #[249]): ∀ (a : Iota), Or (Eq (empty a) False) (Eq (empty (relation_dom a)) True) % 5.30/5.46 Clause #263 (by superposition #[251, 16]): Or (Eq (empty (relation_dom empty_set)) True) (Eq False True) % 5.30/5.46 Clause #264 (by clausification #[263]): Eq (empty (relation_dom empty_set)) True % 5.30/5.46 Clause #267 (by superposition #[264, 87]): Or (Eq True False) (Eq (relation_dom empty_set) empty_set) % 5.30/5.46 Clause #274 (by clausification #[267]): Eq (relation_dom empty_set) empty_set % 5.30/5.46 Clause #292 (by superposition #[246, 16]): Or (Eq (empty (relation_rng empty_set)) True) (Eq False True) % 5.30/5.46 Clause #293 (by clausification #[292]): Eq (empty (relation_rng empty_set)) True % 5.30/5.46 Clause #296 (by superposition #[293, 87]): Or (Eq True False) (Eq (relation_rng empty_set) empty_set) % 5.30/5.46 Clause #309 (by clausification #[296]): Eq (relation_rng empty_set) empty_set % 5.30/5.46 Clause #451 (by clausification #[180]): Or (Eq (function empty_set) False) % 5.30/5.46 (Or (Eq (transfinite_sequence empty_set) True) (Eq (ordinal (relation_dom empty_set)) False)) % 5.30/5.46 Clause #452 (by forward demodulation #[451, 65]): Or (Eq True False) (Or (Eq (transfinite_sequence empty_set) True) (Eq (ordinal (relation_dom empty_set)) False)) % 5.30/5.46 Clause #453 (by clausification #[452]): Or (Eq (transfinite_sequence empty_set) True) (Eq (ordinal (relation_dom empty_set)) False) % 5.30/5.46 Clause #454 (by forward demodulation #[453, 274]): Or (Eq (transfinite_sequence empty_set) True) (Eq (ordinal empty_set) False) % 5.30/5.46 Clause #455 (by forward demodulation #[454, 126]): Or (Eq (transfinite_sequence empty_set) True) (Eq True False) % 5.30/5.46 Clause #456 (by clausification #[455]): Eq (transfinite_sequence empty_set) True % 5.30/5.46 Clause #457 (by superposition #[456, 206]): ∀ (a : Iota), % 5.30/5.46 Or (Eq True False) % 5.30/5.46 (Or (Eq (transfinite_sequence_of empty_set a) True) % 5.30/5.46 (Or (Eq (subset (relation_rng empty_set) a) False) % 5.30/5.46 (Or (Eq (relation empty_set) False) (Eq (function empty_set) False)))) % 5.30/5.46 Clause #715 (by clausification #[457]): ∀ (a : Iota), % 5.30/5.47 Or (Eq (transfinite_sequence_of empty_set a) True) % 5.30/5.47 (Or (Eq (subset (relation_rng empty_set) a) False) % 5.30/5.47 (Or (Eq (relation empty_set) False) (Eq (function empty_set) False))) % 5.30/5.47 Clause #716 (by forward demodulation #[715, 309]): ∀ (a : Iota), % 5.30/5.47 Or (Eq (transfinite_sequence_of empty_set a) True) % 5.30/5.47 (Or (Eq (subset empty_set a) False) (Or (Eq (relation empty_set) False) (Eq (function empty_set) False))) % 5.30/5.47 Clause #717 (by forward demodulation #[716, 66]): ∀ (a : Iota), % 5.30/5.47 Or (Eq (transfinite_sequence_of empty_set a) True) % 5.30/5.47 (Or (Eq True False) (Or (Eq (relation empty_set) False) (Eq (function empty_set) False))) % 5.30/5.47 Clause #718 (by clausification #[717]): ∀ (a : Iota), % 5.30/5.47 Or (Eq (transfinite_sequence_of empty_set a) True) % 5.30/5.47 (Or (Eq (relation empty_set) False) (Eq (function empty_set) False)) % 5.30/5.47 Clause #719 (by forward demodulation #[718, 57]): ∀ (a : Iota), Or (Eq (transfinite_sequence_of empty_set a) True) (Or (Eq True False) (Eq (function empty_set) False)) % 5.30/5.47 Clause #720 (by clausification #[719]): ∀ (a : Iota), Or (Eq (transfinite_sequence_of empty_set a) True) (Eq (function empty_set) False) % 5.30/5.47 Clause #721 (by forward demodulation #[720, 65]): ∀ (a : Iota), Or (Eq (transfinite_sequence_of empty_set a) True) (Eq True False) % 5.30/5.47 Clause #722 (by clausification #[721]): ∀ (a : Iota), Eq (transfinite_sequence_of empty_set a) True % 5.30/5.47 Clause #723 (by superposition #[722, 69]): Eq True False % 5.30/5.47 Clause #728 (by clausification #[723]): False % 5.30/5.47 SZS output end Proof for theBenchmark.p %------------------------------------------------------------------------------