%------------------------------------------------------------------------------ % File : Duper---1.0 % Problem : SET902+1 : TPTP v9.2.0. Released v3.2.0. % Transfm : none % Format : tptp:raw % Command : duper %s % Computer : n023.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 08:00:39 PM UTC 2025 % Result : Theorem 4.24s 4.46s % Output : Proof 4.35s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.06/0.12 % Problem : SET902+1 : TPTP v9.2.0. Released v3.2.0. % 0.06/0.14 % Command : duper %s % 0.14/0.35 % Computer : n023.cluster.edu % 0.14/0.35 % Model : x86_64 x86_64 % 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.14/0.35 % Memory : 8042.1875MB % 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64 % 0.14/0.35 % CPULimit : 300 % 0.14/0.35 % WCLimit : 300 % 0.14/0.35 % DateTime : Thu Oct 2 13:39:38 EDT 2025 % 0.14/0.35 % CPUTime : % 4.24/4.46 SZS status Theorem for theBenchmark.p % 4.24/4.46 SZS output start Proof for theBenchmark.p % 4.24/4.46 Clause #0 (by assumption #[]): Eq (∀ (A B : Iota), Eq (set_union2 A B) (set_union2 B A)) True % 4.24/4.46 Clause #4 (by assumption #[]): Eq (∀ (A : Iota), Iota → Eq (set_union2 A A) A) True % 4.24/4.46 Clause #5 (by assumption #[]): Eq (∀ (A : Iota), Ne (singleton A) empty_set) True % 4.24/4.46 Clause #6 (by assumption #[]): Eq (∀ (A B : Iota), Iff (subset A (singleton B)) (Or (Eq A empty_set) (Eq A (singleton B)))) True % 4.24/4.46 Clause #10 (by assumption #[]): Eq % 4.24/4.46 (Not % 4.24/4.46 (∀ (A B C : Iota), % 4.24/4.46 Not % 4.24/4.46 (And % 4.24/4.46 (And (And (Eq (singleton A) (set_union2 B C)) (Not (And (Eq B (singleton A)) (Eq C (singleton A))))) % 4.24/4.46 (Not (And (Eq B empty_set) (Eq C (singleton A))))) % 4.24/4.46 (Not (And (Eq B (singleton A)) (Eq C empty_set)))))) % 4.24/4.46 True % 4.24/4.46 Clause #11 (by assumption #[]): Eq (∀ (A B : Iota), subset A (set_union2 A B)) True % 4.24/4.46 Clause #18 (by clausification #[5]): ∀ (a : Iota), Eq (Ne (singleton a) empty_set) True % 4.24/4.46 Clause #19 (by clausification #[18]): ∀ (a : Iota), Ne (singleton a) empty_set % 4.24/4.46 Clause #20 (by clausification #[0]): ∀ (a : Iota), Eq (∀ (B : Iota), Eq (set_union2 a B) (set_union2 B a)) True % 4.24/4.46 Clause #21 (by clausification #[20]): ∀ (a a_1 : Iota), Eq (Eq (set_union2 a a_1) (set_union2 a_1 a)) True % 4.24/4.46 Clause #22 (by clausification #[21]): ∀ (a a_1 : Iota), Eq (set_union2 a a_1) (set_union2 a_1 a) % 4.24/4.46 Clause #23 (by clausification #[11]): ∀ (a : Iota), Eq (∀ (B : Iota), subset a (set_union2 a B)) True % 4.24/4.46 Clause #24 (by clausification #[23]): ∀ (a a_1 : Iota), Eq (subset a (set_union2 a a_1)) True % 4.24/4.46 Clause #25 (by superposition #[24, 22]): ∀ (a a_1 : Iota), Eq (subset a (set_union2 a_1 a)) True % 4.24/4.46 Clause #26 (by clausification #[4]): ∀ (a : Iota), Eq (Iota → Eq (set_union2 a a) a) True % 4.24/4.46 Clause #27 (by clausification #[26]): ∀ (a : Iota), Iota → Eq (Eq (set_union2 a a) a) True % 4.24/4.46 Clause #28 (by clausification #[27]): ∀ (a : Iota), Eq (set_union2 a a) a % 4.24/4.46 Clause #41 (by clausification #[6]): ∀ (a : Iota), Eq (∀ (B : Iota), Iff (subset a (singleton B)) (Or (Eq a empty_set) (Eq a (singleton B)))) True % 4.24/4.46 Clause #42 (by clausification #[41]): ∀ (a a_1 : Iota), Eq (Iff (subset a (singleton a_1)) (Or (Eq a empty_set) (Eq a (singleton a_1)))) True % 4.24/4.46 Clause #44 (by clausification #[42]): ∀ (a a_1 : Iota), Or (Eq (subset a (singleton a_1)) False) (Eq (Or (Eq a empty_set) (Eq a (singleton a_1))) True) % 4.24/4.46 Clause #51 (by clausification #[44]): ∀ (a a_1 : Iota), % 4.24/4.46 Or (Eq (subset a (singleton a_1)) False) (Or (Eq (Eq a empty_set) True) (Eq (Eq a (singleton a_1)) True)) % 4.24/4.46 Clause #52 (by clausification #[51]): ∀ (a a_1 : Iota), Or (Eq (subset a (singleton a_1)) False) (Or (Eq (Eq a (singleton a_1)) True) (Eq a empty_set)) % 4.24/4.46 Clause #53 (by clausification #[52]): ∀ (a a_1 : Iota), Or (Eq (subset a (singleton a_1)) False) (Or (Eq a empty_set) (Eq a (singleton a_1))) % 4.24/4.46 Clause #56 (by clausification #[10]): Eq % 4.24/4.46 (∀ (A B C : Iota), % 4.24/4.46 Not % 4.24/4.46 (And % 4.24/4.46 (And (And (Eq (singleton A) (set_union2 B C)) (Not (And (Eq B (singleton A)) (Eq C (singleton A))))) % 4.24/4.46 (Not (And (Eq B empty_set) (Eq C (singleton A))))) % 4.24/4.46 (Not (And (Eq B (singleton A)) (Eq C empty_set))))) % 4.24/4.46 False % 4.24/4.46 Clause #57 (by clausification #[56]): ∀ (a : Iota), % 4.24/4.46 Eq % 4.24/4.46 (Not % 4.24/4.46 (∀ (B C : Iota), % 4.24/4.46 Not % 4.24/4.46 (And % 4.24/4.46 (And % 4.24/4.46 (And (Eq (singleton (skS.0 2 a)) (set_union2 B C)) % 4.24/4.46 (Not (And (Eq B (singleton (skS.0 2 a))) (Eq C (singleton (skS.0 2 a)))))) % 4.24/4.46 (Not (And (Eq B empty_set) (Eq C (singleton (skS.0 2 a)))))) % 4.24/4.46 (Not (And (Eq B (singleton (skS.0 2 a))) (Eq C empty_set)))))) % 4.24/4.46 True % 4.24/4.46 Clause #58 (by clausification #[57]): ∀ (a : Iota), % 4.24/4.46 Eq % 4.24/4.46 (∀ (B C : Iota), % 4.24/4.46 Not % 4.24/4.46 (And % 4.24/4.46 (And % 4.24/4.46 (And (Eq (singleton (skS.0 2 a)) (set_union2 B C)) % 4.24/4.46 (Not (And (Eq B (singleton (skS.0 2 a))) (Eq C (singleton (skS.0 2 a)))))) % 4.24/4.46 (Not (And (Eq B empty_set) (Eq C (singleton (skS.0 2 a)))))) % 4.24/4.46 (Not (And (Eq B (singleton (skS.0 2 a))) (Eq C empty_set))))) % 4.24/4.46 False % 4.24/4.46 Clause #59 (by clausification #[58]): ∀ (a a_1 : Iota), % 4.24/4.48 Eq % 4.24/4.48 (Not % 4.24/4.48 (∀ (C : Iota), % 4.24/4.48 Not % 4.24/4.48 (And % 4.24/4.48 (And % 4.24/4.48 (And (Eq (singleton (skS.0 2 a)) (set_union2 (skS.0 3 a a_1) C)) % 4.24/4.48 (Not (And (Eq (skS.0 3 a a_1) (singleton (skS.0 2 a))) (Eq C (singleton (skS.0 2 a)))))) % 4.24/4.48 (Not (And (Eq (skS.0 3 a a_1) empty_set) (Eq C (singleton (skS.0 2 a)))))) % 4.24/4.48 (Not (And (Eq (skS.0 3 a a_1) (singleton (skS.0 2 a))) (Eq C empty_set)))))) % 4.24/4.48 True % 4.24/4.48 Clause #60 (by clausification #[59]): ∀ (a a_1 : Iota), % 4.24/4.48 Eq % 4.24/4.48 (∀ (C : Iota), % 4.24/4.48 Not % 4.24/4.48 (And % 4.24/4.48 (And % 4.24/4.48 (And (Eq (singleton (skS.0 2 a)) (set_union2 (skS.0 3 a a_1) C)) % 4.24/4.48 (Not (And (Eq (skS.0 3 a a_1) (singleton (skS.0 2 a))) (Eq C (singleton (skS.0 2 a)))))) % 4.24/4.48 (Not (And (Eq (skS.0 3 a a_1) empty_set) (Eq C (singleton (skS.0 2 a)))))) % 4.24/4.48 (Not (And (Eq (skS.0 3 a a_1) (singleton (skS.0 2 a))) (Eq C empty_set))))) % 4.24/4.48 False % 4.24/4.48 Clause #61 (by clausification #[60]): ∀ (a a_1 a_2 : Iota), % 4.24/4.48 Eq % 4.24/4.48 (Not % 4.24/4.48 (Not % 4.24/4.48 (And % 4.24/4.48 (And % 4.24/4.48 (And (Eq (singleton (skS.0 2 a)) (set_union2 (skS.0 3 a a_1) (skS.0 4 a a_1 a_2))) % 4.24/4.48 (Not (And (Eq (skS.0 3 a a_1) (singleton (skS.0 2 a))) (Eq (skS.0 4 a a_1 a_2) (singleton (skS.0 2 a)))))) % 4.24/4.48 (Not (And (Eq (skS.0 3 a a_1) empty_set) (Eq (skS.0 4 a a_1 a_2) (singleton (skS.0 2 a)))))) % 4.24/4.48 (Not (And (Eq (skS.0 3 a a_1) (singleton (skS.0 2 a))) (Eq (skS.0 4 a a_1 a_2) empty_set)))))) % 4.24/4.48 True % 4.24/4.48 Clause #62 (by clausification #[61]): ∀ (a a_1 a_2 : Iota), % 4.24/4.48 Eq % 4.24/4.48 (Not % 4.24/4.48 (And % 4.24/4.48 (And % 4.24/4.48 (And (Eq (singleton (skS.0 2 a)) (set_union2 (skS.0 3 a a_1) (skS.0 4 a a_1 a_2))) % 4.24/4.48 (Not (And (Eq (skS.0 3 a a_1) (singleton (skS.0 2 a))) (Eq (skS.0 4 a a_1 a_2) (singleton (skS.0 2 a)))))) % 4.24/4.48 (Not (And (Eq (skS.0 3 a a_1) empty_set) (Eq (skS.0 4 a a_1 a_2) (singleton (skS.0 2 a)))))) % 4.24/4.48 (Not (And (Eq (skS.0 3 a a_1) (singleton (skS.0 2 a))) (Eq (skS.0 4 a a_1 a_2) empty_set))))) % 4.24/4.48 False % 4.24/4.48 Clause #63 (by clausification #[62]): ∀ (a a_1 a_2 : Iota), % 4.24/4.48 Eq % 4.24/4.48 (And % 4.24/4.48 (And % 4.24/4.48 (And (Eq (singleton (skS.0 2 a)) (set_union2 (skS.0 3 a a_1) (skS.0 4 a a_1 a_2))) % 4.24/4.48 (Not (And (Eq (skS.0 3 a a_1) (singleton (skS.0 2 a))) (Eq (skS.0 4 a a_1 a_2) (singleton (skS.0 2 a)))))) % 4.24/4.48 (Not (And (Eq (skS.0 3 a a_1) empty_set) (Eq (skS.0 4 a a_1 a_2) (singleton (skS.0 2 a)))))) % 4.24/4.48 (Not (And (Eq (skS.0 3 a a_1) (singleton (skS.0 2 a))) (Eq (skS.0 4 a a_1 a_2) empty_set)))) % 4.24/4.48 True % 4.24/4.48 Clause #64 (by clausification #[63]): ∀ (a a_1 a_2 : Iota), % 4.24/4.48 Eq (Not (And (Eq (skS.0 3 a a_1) (singleton (skS.0 2 a))) (Eq (skS.0 4 a a_1 a_2) empty_set))) True % 4.24/4.48 Clause #65 (by clausification #[63]): ∀ (a a_1 a_2 : Iota), % 4.24/4.48 Eq % 4.24/4.48 (And % 4.24/4.48 (And (Eq (singleton (skS.0 2 a)) (set_union2 (skS.0 3 a a_1) (skS.0 4 a a_1 a_2))) % 4.24/4.48 (Not (And (Eq (skS.0 3 a a_1) (singleton (skS.0 2 a))) (Eq (skS.0 4 a a_1 a_2) (singleton (skS.0 2 a)))))) % 4.24/4.48 (Not (And (Eq (skS.0 3 a a_1) empty_set) (Eq (skS.0 4 a a_1 a_2) (singleton (skS.0 2 a)))))) % 4.24/4.48 True % 4.24/4.48 Clause #66 (by clausification #[64]): ∀ (a a_1 a_2 : Iota), Eq (And (Eq (skS.0 3 a a_1) (singleton (skS.0 2 a))) (Eq (skS.0 4 a a_1 a_2) empty_set)) False % 4.24/4.48 Clause #67 (by clausification #[66]): ∀ (a a_1 a_2 : Iota), % 4.24/4.48 Or (Eq (Eq (skS.0 3 a a_1) (singleton (skS.0 2 a))) False) (Eq (Eq (skS.0 4 a a_1 a_2) empty_set) False) % 4.24/4.48 Clause #68 (by clausification #[67]): ∀ (a a_1 a_2 : Iota), Or (Eq (Eq (skS.0 4 a a_1 a_2) empty_set) False) (Ne (skS.0 3 a a_1) (singleton (skS.0 2 a))) % 4.24/4.48 Clause #69 (by clausification #[68]): ∀ (a a_1 a_2 : Iota), Or (Ne (skS.0 3 a a_1) (singleton (skS.0 2 a))) (Ne (skS.0 4 a a_1 a_2) empty_set) % 4.24/4.48 Clause #72 (by clausification #[65]): ∀ (a a_1 a_2 : Iota), % 4.24/4.48 Eq (Not (And (Eq (skS.0 3 a a_1) empty_set) (Eq (skS.0 4 a a_1 a_2) (singleton (skS.0 2 a))))) True % 4.24/4.48 Clause #73 (by clausification #[65]): ∀ (a a_1 a_2 : Iota), % 4.24/4.48 Eq % 4.24/4.48 (And (Eq (singleton (skS.0 2 a)) (set_union2 (skS.0 3 a a_1) (skS.0 4 a a_1 a_2))) % 4.24/4.48 (Not (And (Eq (skS.0 3 a a_1) (singleton (skS.0 2 a))) (Eq (skS.0 4 a a_1 a_2) (singleton (skS.0 2 a)))))) % 4.35/4.51 True % 4.35/4.51 Clause #74 (by clausification #[72]): ∀ (a a_1 a_2 : Iota), Eq (And (Eq (skS.0 3 a a_1) empty_set) (Eq (skS.0 4 a a_1 a_2) (singleton (skS.0 2 a)))) False % 4.35/4.51 Clause #75 (by clausification #[74]): ∀ (a a_1 a_2 : Iota), % 4.35/4.51 Or (Eq (Eq (skS.0 3 a a_1) empty_set) False) (Eq (Eq (skS.0 4 a a_1 a_2) (singleton (skS.0 2 a))) False) % 4.35/4.51 Clause #76 (by clausification #[75]): ∀ (a a_1 a_2 : Iota), Or (Eq (Eq (skS.0 4 a a_1 a_2) (singleton (skS.0 2 a))) False) (Ne (skS.0 3 a a_1) empty_set) % 4.35/4.51 Clause #77 (by clausification #[76]): ∀ (a a_1 a_2 : Iota), Or (Ne (skS.0 3 a a_1) empty_set) (Ne (skS.0 4 a a_1 a_2) (singleton (skS.0 2 a))) % 4.35/4.51 Clause #78 (by clausification #[73]): ∀ (a a_1 a_2 : Iota), % 4.35/4.51 Eq (Not (And (Eq (skS.0 3 a a_1) (singleton (skS.0 2 a))) (Eq (skS.0 4 a a_1 a_2) (singleton (skS.0 2 a))))) True % 4.35/4.51 Clause #79 (by clausification #[73]): ∀ (a a_1 a_2 : Iota), Eq (Eq (singleton (skS.0 2 a)) (set_union2 (skS.0 3 a a_1) (skS.0 4 a a_1 a_2))) True % 4.35/4.51 Clause #80 (by clausification #[78]): ∀ (a a_1 a_2 : Iota), % 4.35/4.51 Eq (And (Eq (skS.0 3 a a_1) (singleton (skS.0 2 a))) (Eq (skS.0 4 a a_1 a_2) (singleton (skS.0 2 a)))) False % 4.35/4.51 Clause #81 (by clausification #[80]): ∀ (a a_1 a_2 : Iota), % 4.35/4.51 Or (Eq (Eq (skS.0 3 a a_1) (singleton (skS.0 2 a))) False) (Eq (Eq (skS.0 4 a a_1 a_2) (singleton (skS.0 2 a))) False) % 4.35/4.51 Clause #82 (by clausification #[81]): ∀ (a a_1 a_2 : Iota), % 4.35/4.51 Or (Eq (Eq (skS.0 4 a a_1 a_2) (singleton (skS.0 2 a))) False) (Ne (skS.0 3 a a_1) (singleton (skS.0 2 a))) % 4.35/4.51 Clause #83 (by clausification #[82]): ∀ (a a_1 a_2 : Iota), Or (Ne (skS.0 3 a a_1) (singleton (skS.0 2 a))) (Ne (skS.0 4 a a_1 a_2) (singleton (skS.0 2 a))) % 4.35/4.51 Clause #84 (by clausification #[79]): ∀ (a a_1 a_2 : Iota), Eq (singleton (skS.0 2 a)) (set_union2 (skS.0 3 a a_1) (skS.0 4 a a_1 a_2)) % 4.35/4.51 Clause #85 (by superposition #[84, 24]): ∀ (a a_1 : Iota), Eq (subset (skS.0 3 a a_1) (singleton (skS.0 2 a))) True % 4.35/4.51 Clause #86 (by superposition #[84, 25]): ∀ (a a_1 a_2 : Iota), Eq (subset (skS.0 4 a a_1 a_2) (singleton (skS.0 2 a))) True % 4.35/4.51 Clause #89 (by superposition #[85, 53]): ∀ (a a_1 : Iota), Or (Eq True False) (Or (Eq (skS.0 3 a a_1) empty_set) (Eq (skS.0 3 a a_1) (singleton (skS.0 2 a)))) % 4.35/4.51 Clause #90 (by superposition #[86, 53]): ∀ (a a_1 a_2 : Iota), % 4.35/4.51 Or (Eq True False) (Or (Eq (skS.0 4 a a_1 a_2) empty_set) (Eq (skS.0 4 a a_1 a_2) (singleton (skS.0 2 a)))) % 4.35/4.51 Clause #91 (by clausification #[89]): ∀ (a a_1 : Iota), Or (Eq (skS.0 3 a a_1) empty_set) (Eq (skS.0 3 a a_1) (singleton (skS.0 2 a))) % 4.35/4.51 Clause #92 (by superposition #[91, 69]): ∀ (a a_1 a_2 : Iota), % 4.35/4.51 Or (Eq (skS.0 3 a a_1) empty_set) % 4.35/4.51 (Or (Ne (singleton (skS.0 2 a)) (singleton (skS.0 2 a))) (Ne (skS.0 4 a a_1 a_2) empty_set)) % 4.35/4.51 Clause #94 (by superposition #[91, 83]): ∀ (a a_1 a_2 : Iota), % 4.35/4.51 Or (Eq (skS.0 3 a a_1) empty_set) % 4.35/4.51 (Or (Ne (singleton (skS.0 2 a)) (singleton (skS.0 2 a))) (Ne (skS.0 4 a a_1 a_2) (singleton (skS.0 2 a)))) % 4.35/4.51 Clause #98 (by clausification #[90]): ∀ (a a_1 a_2 : Iota), Or (Eq (skS.0 4 a a_1 a_2) empty_set) (Eq (skS.0 4 a a_1 a_2) (singleton (skS.0 2 a))) % 4.35/4.51 Clause #102 (by eliminate resolved literals #[92]): ∀ (a a_1 a_2 : Iota), Or (Eq (skS.0 3 a a_1) empty_set) (Ne (skS.0 4 a a_1 a_2) empty_set) % 4.35/4.51 Clause #118 (by eliminate resolved literals #[94]): ∀ (a a_1 a_2 : Iota), Or (Eq (skS.0 3 a a_1) empty_set) (Ne (skS.0 4 a a_1 a_2) (singleton (skS.0 2 a))) % 4.35/4.51 Clause #119 (by forward contextual literal cutting #[118, 77]): ∀ (a a_1 a_2 : Iota), Ne (skS.0 4 a a_1 a_2) (singleton (skS.0 2 a)) % 4.35/4.51 Clause #120 (by backward contextual literal cutting #[119, 98]): ∀ (a a_1 a_2 : Iota), Eq (skS.0 4 a a_1 a_2) empty_set % 4.35/4.51 Clause #123 (by backward demodulation #[120, 84]): ∀ (a a_1 : Iota), Eq (singleton (skS.0 2 a)) (set_union2 (skS.0 3 a a_1) empty_set) % 4.35/4.51 Clause #129 (by backward contextual literal cutting #[120, 102]): ∀ (a a_1 : Iota), Eq (skS.0 3 a a_1) empty_set % 4.35/4.51 Clause #134 (by forward demodulation #[123, 129]): ∀ (a : Iota), Eq (singleton (skS.0 2 a)) (set_union2 empty_set empty_set) % 4.35/4.51 Clause #135 (by forward demodulation #[134, 28]): ∀ (a : Iota), Eq (singleton (skS.0 2 a)) empty_set % 4.35/4.51 Clause #136 (by forward contextual literal cutting #[135, 19]): False % 4.35/4.51 SZS output end Proof for theBenchmark.p %------------------------------------------------------------------------------