%------------------------------------------------------------------------------ % File : Duper---1.0 % Problem : SEU579^2 : TPTP v9.2.0. Released v3.7.0. % Transfm : none % Format : tptp:raw % Command : duper %s % Computer : n008.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 08:01:57 PM UTC 2025 % Result : Theorem 10.33s 10.48s % Output : Proof 10.39s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.00/0.12 % Problem : SEU579^2 : TPTP v9.2.0. Released v3.7.0. % 0.13/0.13 % Command : duper %s % 0.14/0.34 % Computer : n008.cluster.edu % 0.14/0.34 % Model : x86_64 x86_64 % 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.14/0.34 % Memory : 8042.1875MB % 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64 % 0.14/0.34 % CPULimit : 300 % 0.14/0.34 % WCLimit : 300 % 0.14/0.34 % DateTime : Fri Oct 3 04:31:08 EDT 2025 % 0.20/0.34 % CPUTime : % 10.33/10.48 SZS status Theorem for theBenchmark.p % 10.33/10.48 SZS output start Proof for theBenchmark.p % 10.33/10.48 Clause #0 (by assumption #[]): Eq (Eq powersetI (∀ (A B : Iota), (∀ (Xx : Iota), in Xx B → in Xx A) → in B (powerset A))) True % 10.33/10.48 Clause #1 (by assumption #[]): Eq (Eq powersetE (∀ (A B Xx : Iota), in B (powerset A) → in Xx B → in Xx A)) True % 10.33/10.48 Clause #2 (by assumption #[]): Eq (Eq subsetI2 (∀ (A B : Iota), (∀ (Xx : Iota), in Xx A → in Xx B) → subset A B)) True % 10.33/10.48 Clause #3 (by assumption #[]): Eq (Eq subsetE (∀ (A B Xx : Iota), subset A B → in Xx A → in Xx B)) True % 10.33/10.48 Clause #4 (by assumption #[]): Eq (Not (powersetI → powersetE → subsetI2 → subsetE → ∀ (A B : Iota), subset A B → subset (powerset A) (powerset B))) % 10.33/10.48 True % 10.33/10.48 Clause #5 (by clausification #[2]): Eq subsetI2 (∀ (A B : Iota), (∀ (Xx : Iota), in Xx A → in Xx B) → subset A B) % 10.33/10.48 Clause #7 (by clausify Prop equality #[5]): Or (Eq subsetI2 False) (Eq (∀ (A B : Iota), (∀ (Xx : Iota), in Xx A → in Xx B) → subset A B) True) % 10.33/10.48 Clause #9 (by clausification #[7]): ∀ (a : Iota), Or (Eq subsetI2 False) (Eq (∀ (B : Iota), (∀ (Xx : Iota), in Xx a → in Xx B) → subset a B) True) % 10.33/10.48 Clause #10 (by clausification #[9]): ∀ (a a_1 : Iota), Or (Eq subsetI2 False) (Eq ((∀ (Xx : Iota), in Xx a → in Xx a_1) → subset a a_1) True) % 10.33/10.48 Clause #11 (by clausification #[10]): ∀ (a a_1 : Iota), Or (Eq subsetI2 False) (Or (Eq (∀ (Xx : Iota), in Xx a → in Xx a_1) False) (Eq (subset a a_1) True)) % 10.33/10.48 Clause #12 (by clausification #[11]): ∀ (a a_1 a_2 : Iota), % 10.33/10.48 Or (Eq subsetI2 False) % 10.33/10.48 (Or (Eq (subset a a_1) True) (Eq (Not (in (skS.0 0 a a_1 a_2) a → in (skS.0 0 a a_1 a_2) a_1)) True)) % 10.33/10.48 Clause #13 (by clausification #[12]): ∀ (a a_1 a_2 : Iota), % 10.33/10.48 Or (Eq subsetI2 False) % 10.33/10.48 (Or (Eq (subset a a_1) True) (Eq (in (skS.0 0 a a_1 a_2) a → in (skS.0 0 a a_1 a_2) a_1) False)) % 10.33/10.48 Clause #14 (by clausification #[13]): ∀ (a a_1 a_2 : Iota), Or (Eq subsetI2 False) (Or (Eq (subset a a_1) True) (Eq (in (skS.0 0 a a_1 a_2) a) True)) % 10.33/10.48 Clause #15 (by clausification #[13]): ∀ (a a_1 a_2 : Iota), Or (Eq subsetI2 False) (Or (Eq (subset a a_1) True) (Eq (in (skS.0 0 a a_1 a_2) a_1) False)) % 10.33/10.48 Clause #24 (by clausification #[0]): Eq powersetI (∀ (A B : Iota), (∀ (Xx : Iota), in Xx B → in Xx A) → in B (powerset A)) % 10.33/10.48 Clause #28 (by clausification #[1]): Eq powersetE (∀ (A B Xx : Iota), in B (powerset A) → in Xx B → in Xx A) % 10.33/10.48 Clause #60 (by clausification #[3]): Eq subsetE (∀ (A B Xx : Iota), subset A B → in Xx A → in Xx B) % 10.33/10.48 Clause #77 (by clausification #[4]): Eq (powersetI → powersetE → subsetI2 → subsetE → ∀ (A B : Iota), subset A B → subset (powerset A) (powerset B)) False % 10.33/10.48 Clause #78 (by clausification #[77]): Eq powersetI True % 10.33/10.48 Clause #79 (by clausification #[77]): Eq (powersetE → subsetI2 → subsetE → ∀ (A B : Iota), subset A B → subset (powerset A) (powerset B)) False % 10.33/10.48 Clause #80 (by backward demodulation #[78, 24]): Eq True (∀ (A B : Iota), (∀ (Xx : Iota), in Xx B → in Xx A) → in B (powerset A)) % 10.33/10.48 Clause #83 (by clausification #[80]): ∀ (a : Iota), Eq (∀ (B : Iota), (∀ (Xx : Iota), in Xx B → in Xx a) → in B (powerset a)) True % 10.33/10.48 Clause #84 (by clausification #[83]): ∀ (a a_1 : Iota), Eq ((∀ (Xx : Iota), in Xx a → in Xx a_1) → in a (powerset a_1)) True % 10.33/10.48 Clause #85 (by clausification #[84]): ∀ (a a_1 : Iota), Or (Eq (∀ (Xx : Iota), in Xx a → in Xx a_1) False) (Eq (in a (powerset a_1)) True) % 10.33/10.48 Clause #86 (by clausification #[85]): ∀ (a a_1 a_2 : Iota), % 10.33/10.48 Or (Eq (in a (powerset a_1)) True) (Eq (Not (in (skS.0 12 a a_1 a_2) a → in (skS.0 12 a a_1 a_2) a_1)) True) % 10.33/10.48 Clause #87 (by clausification #[86]): ∀ (a a_1 a_2 : Iota), % 10.33/10.48 Or (Eq (in a (powerset a_1)) True) (Eq (in (skS.0 12 a a_1 a_2) a → in (skS.0 12 a a_1 a_2) a_1) False) % 10.33/10.48 Clause #88 (by clausification #[87]): ∀ (a a_1 a_2 : Iota), Or (Eq (in a (powerset a_1)) True) (Eq (in (skS.0 12 a a_1 a_2) a) True) % 10.33/10.48 Clause #89 (by clausification #[87]): ∀ (a a_1 a_2 : Iota), Or (Eq (in a (powerset a_1)) True) (Eq (in (skS.0 12 a a_1 a_2) a_1) False) % 10.33/10.48 Clause #91 (by clausification #[79]): Eq powersetE True % 10.33/10.50 Clause #92 (by clausification #[79]): Eq (subsetI2 → subsetE → ∀ (A B : Iota), subset A B → subset (powerset A) (powerset B)) False % 10.33/10.50 Clause #93 (by backward demodulation #[91, 28]): Eq True (∀ (A B Xx : Iota), in B (powerset A) → in Xx B → in Xx A) % 10.33/10.50 Clause #96 (by clausification #[93]): ∀ (a : Iota), Eq (∀ (B Xx : Iota), in B (powerset a) → in Xx B → in Xx a) True % 10.33/10.50 Clause #97 (by clausification #[96]): ∀ (a a_1 : Iota), Eq (∀ (Xx : Iota), in a (powerset a_1) → in Xx a → in Xx a_1) True % 10.33/10.50 Clause #98 (by clausification #[97]): ∀ (a a_1 a_2 : Iota), Eq (in a (powerset a_1) → in a_2 a → in a_2 a_1) True % 10.33/10.50 Clause #99 (by clausification #[98]): ∀ (a a_1 a_2 : Iota), Or (Eq (in a (powerset a_1)) False) (Eq (in a_2 a → in a_2 a_1) True) % 10.33/10.50 Clause #100 (by clausification #[99]): ∀ (a a_1 a_2 : Iota), Or (Eq (in a (powerset a_1)) False) (Or (Eq (in a_2 a) False) (Eq (in a_2 a_1) True)) % 10.33/10.50 Clause #102 (by clausification #[92]): Eq subsetI2 True % 10.33/10.50 Clause #103 (by clausification #[92]): Eq (subsetE → ∀ (A B : Iota), subset A B → subset (powerset A) (powerset B)) False % 10.33/10.51 Clause #104 (by backward demodulation #[102, 5]): Eq True (∀ (A B : Iota), (∀ (Xx : Iota), in Xx A → in Xx B) → subset A B) % 10.33/10.51 Clause #105 (by backward demodulation #[102, 14]): ∀ (a a_1 a_2 : Iota), Or (Eq True False) (Or (Eq (subset a a_1) True) (Eq (in (skS.0 0 a a_1 a_2) a) True)) % 10.33/10.51 Clause #107 (by backward demodulation #[102, 15]): ∀ (a a_1 a_2 : Iota), Or (Eq True False) (Or (Eq (subset a a_1) True) (Eq (in (skS.0 0 a a_1 a_2) a_1) False)) % 10.33/10.51 Clause #108 (by clausification #[104]): ∀ (a : Iota), Eq (∀ (B : Iota), (∀ (Xx : Iota), in Xx a → in Xx B) → subset a B) True % 10.33/10.51 Clause #109 (by clausification #[108]): ∀ (a a_1 : Iota), Eq ((∀ (Xx : Iota), in Xx a → in Xx a_1) → subset a a_1) True % 10.33/10.51 Clause #110 (by clausification #[109]): ∀ (a a_1 : Iota), Or (Eq (∀ (Xx : Iota), in Xx a → in Xx a_1) False) (Eq (subset a a_1) True) % 10.33/10.51 Clause #111 (by clausification #[110]): ∀ (a a_1 a_2 : Iota), % 10.33/10.51 Or (Eq (subset a a_1) True) (Eq (Not (in (skS.0 13 a a_1 a_2) a → in (skS.0 13 a a_1 a_2) a_1)) True) % 10.33/10.51 Clause #112 (by clausification #[111]): ∀ (a a_1 a_2 : Iota), Or (Eq (subset a a_1) True) (Eq (in (skS.0 13 a a_1 a_2) a → in (skS.0 13 a a_1 a_2) a_1) False) % 10.33/10.51 Clause #113 (by clausification #[112]): ∀ (a a_1 a_2 : Iota), Or (Eq (subset a a_1) True) (Eq (in (skS.0 13 a a_1 a_2) a) True) % 10.33/10.51 Clause #114 (by clausification #[112]): ∀ (a a_1 a_2 : Iota), Or (Eq (subset a a_1) True) (Eq (in (skS.0 13 a a_1 a_2) a_1) False) % 10.33/10.51 Clause #115 (by superposition #[113, 100]): ∀ (a a_1 a_2 a_3 : Iota), % 10.33/10.51 Or (Eq (subset (powerset a) a_1) True) % 10.33/10.51 (Or (Eq True False) (Or (Eq (in a_2 (skS.0 13 (powerset a) a_1 a_3)) False) (Eq (in a_2 a) True))) % 10.33/10.51 Clause #117 (by clausification #[103]): Eq subsetE True % 10.33/10.51 Clause #118 (by clausification #[103]): Eq (∀ (A B : Iota), subset A B → subset (powerset A) (powerset B)) False % 10.33/10.51 Clause #119 (by backward demodulation #[117, 60]): Eq True (∀ (A B Xx : Iota), subset A B → in Xx A → in Xx B) % 10.33/10.51 Clause #122 (by clausification #[119]): ∀ (a : Iota), Eq (∀ (B Xx : Iota), subset a B → in Xx a → in Xx B) True % 10.33/10.51 Clause #123 (by clausification #[122]): ∀ (a a_1 : Iota), Eq (∀ (Xx : Iota), subset a a_1 → in Xx a → in Xx a_1) True % 10.33/10.51 Clause #124 (by clausification #[123]): ∀ (a a_1 a_2 : Iota), Eq (subset a a_1 → in a_2 a → in a_2 a_1) True % 10.33/10.51 Clause #125 (by clausification #[124]): ∀ (a a_1 a_2 : Iota), Or (Eq (subset a a_1) False) (Eq (in a_2 a → in a_2 a_1) True) % 10.33/10.51 Clause #126 (by clausification #[125]): ∀ (a a_1 a_2 : Iota), Or (Eq (subset a a_1) False) (Or (Eq (in a_2 a) False) (Eq (in a_2 a_1) True)) % 10.33/10.51 Clause #127 (by clausification #[118]): ∀ (a : Iota), Eq (Not (∀ (B : Iota), subset (skS.0 14 a) B → subset (powerset (skS.0 14 a)) (powerset B))) True % 10.33/10.51 Clause #128 (by clausification #[127]): ∀ (a : Iota), Eq (∀ (B : Iota), subset (skS.0 14 a) B → subset (powerset (skS.0 14 a)) (powerset B)) False % 10.33/10.51 Clause #129 (by clausification #[128]): ∀ (a a_1 : Iota), % 10.37/10.53 Eq (Not (subset (skS.0 14 a) (skS.0 15 a a_1) → subset (powerset (skS.0 14 a)) (powerset (skS.0 15 a a_1)))) True % 10.37/10.53 Clause #130 (by clausification #[129]): ∀ (a a_1 : Iota), % 10.37/10.53 Eq (subset (skS.0 14 a) (skS.0 15 a a_1) → subset (powerset (skS.0 14 a)) (powerset (skS.0 15 a a_1))) False % 10.37/10.53 Clause #131 (by clausification #[130]): ∀ (a a_1 : Iota), Eq (subset (skS.0 14 a) (skS.0 15 a a_1)) True % 10.37/10.53 Clause #132 (by clausification #[130]): ∀ (a a_1 : Iota), Eq (subset (powerset (skS.0 14 a)) (powerset (skS.0 15 a a_1))) False % 10.37/10.53 Clause #133 (by superposition #[131, 126]): ∀ (a a_1 a_2 : Iota), Or (Eq True False) (Or (Eq (in a (skS.0 14 a_1)) False) (Eq (in a (skS.0 15 a_1 a_2)) True)) % 10.37/10.53 Clause #145 (by clausification #[105]): ∀ (a a_1 a_2 : Iota), Or (Eq (subset a a_1) True) (Eq (in (skS.0 0 a a_1 a_2) a) True) % 10.37/10.53 Clause #147 (by clausification #[107]): ∀ (a a_1 a_2 : Iota), Or (Eq (subset a a_1) True) (Eq (in (skS.0 0 a a_1 a_2) a_1) False) % 10.37/10.53 Clause #155 (by clausification #[133]): ∀ (a a_1 a_2 : Iota), Or (Eq (in a (skS.0 14 a_1)) False) (Eq (in a (skS.0 15 a_1 a_2)) True) % 10.37/10.53 Clause #164 (by clausification #[115]): ∀ (a a_1 a_2 a_3 : Iota), % 10.37/10.53 Or (Eq (subset (powerset a) a_1) True) (Or (Eq (in a_2 (skS.0 13 (powerset a) a_1 a_3)) False) (Eq (in a_2 a) True)) % 10.37/10.53 Clause #167 (by superposition #[164, 145]): ∀ (a a_1 a_2 a_3 a_4 : Iota), % 10.37/10.53 Or (Eq (subset (powerset a) a_1) True) % 10.37/10.53 (Or (Eq (in (skS.0 0 (skS.0 13 (powerset a) a_1 a_2) a_3 a_4) a) True) % 10.37/10.53 (Or (Eq (subset (skS.0 13 (powerset a) a_1 a_2) a_3) True) (Eq False True))) % 10.37/10.53 Clause #195 (by clausification #[167]): ∀ (a a_1 a_2 a_3 a_4 : Iota), % 10.37/10.53 Or (Eq (subset (powerset a) a_1) True) % 10.37/10.53 (Or (Eq (in (skS.0 0 (skS.0 13 (powerset a) a_1 a_2) a_3 a_4) a) True) % 10.37/10.53 (Eq (subset (skS.0 13 (powerset a) a_1 a_2) a_3) True)) % 10.37/10.53 Clause #197 (by superposition #[195, 155]): ∀ (a a_1 a_2 a_3 a_4 a_5 : Iota), % 10.37/10.53 Or (Eq (subset (powerset (skS.0 14 a)) a_1) True) % 10.37/10.53 (Or (Eq (subset (skS.0 13 (powerset (skS.0 14 a)) a_1 a_2) a_3) True) % 10.37/10.53 (Or (Eq True False) (Eq (in (skS.0 0 (skS.0 13 (powerset (skS.0 14 a)) a_1 a_2) a_3 a_4) (skS.0 15 a a_5)) True))) % 10.37/10.53 Clause #396 (by clausification #[197]): ∀ (a a_1 a_2 a_3 a_4 a_5 : Iota), % 10.37/10.53 Or (Eq (subset (powerset (skS.0 14 a)) a_1) True) % 10.37/10.53 (Or (Eq (subset (skS.0 13 (powerset (skS.0 14 a)) a_1 a_2) a_3) True) % 10.37/10.53 (Eq (in (skS.0 0 (skS.0 13 (powerset (skS.0 14 a)) a_1 a_2) a_3 a_4) (skS.0 15 a a_5)) True)) % 10.37/10.53 Clause #397 (by superposition #[396, 147]): ∀ (a a_1 a_2 a_3 : Iota), % 10.37/10.53 Or (Eq (subset (powerset (skS.0 14 a)) a_1) True) % 10.37/10.53 (Or (Eq (subset (skS.0 13 (powerset (skS.0 14 a)) a_1 a_2) (skS.0 15 a a_3)) True) % 10.37/10.53 (Or (Eq (subset (skS.0 13 (powerset (skS.0 14 a)) a_1 a_2) (skS.0 15 a a_3)) True) (Eq True False))) % 10.37/10.53 Clause #719 (by clausification #[397]): ∀ (a a_1 a_2 a_3 : Iota), % 10.37/10.53 Or (Eq (subset (powerset (skS.0 14 a)) a_1) True) % 10.37/10.53 (Or (Eq (subset (skS.0 13 (powerset (skS.0 14 a)) a_1 a_2) (skS.0 15 a a_3)) True) % 10.37/10.53 (Eq (subset (skS.0 13 (powerset (skS.0 14 a)) a_1 a_2) (skS.0 15 a a_3)) True)) % 10.37/10.53 Clause #720 (by eliminate duplicate literals #[719]): ∀ (a a_1 a_2 a_3 : Iota), % 10.37/10.53 Or (Eq (subset (powerset (skS.0 14 a)) a_1) True) % 10.37/10.53 (Eq (subset (skS.0 13 (powerset (skS.0 14 a)) a_1 a_2) (skS.0 15 a a_3)) True) % 10.37/10.53 Clause #721 (by superposition #[720, 126]): ∀ (a a_1 a_2 a_3 a_4 : Iota), % 10.37/10.53 Or (Eq (subset (powerset (skS.0 14 a)) a_1) True) % 10.37/10.53 (Or (Eq True False) % 10.37/10.53 (Or (Eq (in a_2 (skS.0 13 (powerset (skS.0 14 a)) a_1 a_3)) False) (Eq (in a_2 (skS.0 15 a a_4)) True))) % 10.37/10.53 Clause #722 (by clausification #[721]): ∀ (a a_1 a_2 a_3 a_4 : Iota), % 10.37/10.53 Or (Eq (subset (powerset (skS.0 14 a)) a_1) True) % 10.37/10.53 (Or (Eq (in a_2 (skS.0 13 (powerset (skS.0 14 a)) a_1 a_3)) False) (Eq (in a_2 (skS.0 15 a a_4)) True)) % 10.37/10.53 Clause #723 (by superposition #[722, 88]): ∀ (a a_1 a_2 a_3 a_4 a_5 : Iota), % 10.37/10.53 Or (Eq (subset (powerset (skS.0 14 a)) a_1) True) % 10.37/10.53 (Or (Eq (in (skS.0 12 (skS.0 13 (powerset (skS.0 14 a)) a_1 a_2) a_3 a_4) (skS.0 15 a a_5)) True) % 10.37/10.53 (Or (Eq (in (skS.0 13 (powerset (skS.0 14 a)) a_1 a_2) (powerset a_3)) True) (Eq False True))) % 10.39/10.55 Clause #889 (by clausification #[723]): ∀ (a a_1 a_2 a_3 a_4 a_5 : Iota), % 10.39/10.55 Or (Eq (subset (powerset (skS.0 14 a)) a_1) True) % 10.39/10.55 (Or (Eq (in (skS.0 12 (skS.0 13 (powerset (skS.0 14 a)) a_1 a_2) a_3 a_4) (skS.0 15 a a_5)) True) % 10.39/10.55 (Eq (in (skS.0 13 (powerset (skS.0 14 a)) a_1 a_2) (powerset a_3)) True)) % 10.39/10.55 Clause #890 (by superposition #[889, 89]): ∀ (a a_1 a_2 a_3 : Iota), % 10.39/10.55 Or (Eq (subset (powerset (skS.0 14 a)) a_1) True) % 10.39/10.55 (Or (Eq (in (skS.0 13 (powerset (skS.0 14 a)) a_1 a_2) (powerset (skS.0 15 a a_3))) True) % 10.39/10.55 (Or (Eq (in (skS.0 13 (powerset (skS.0 14 a)) a_1 a_2) (powerset (skS.0 15 a a_3))) True) (Eq True False))) % 10.39/10.55 Clause #917 (by clausification #[890]): ∀ (a a_1 a_2 a_3 : Iota), % 10.39/10.55 Or (Eq (subset (powerset (skS.0 14 a)) a_1) True) % 10.39/10.55 (Or (Eq (in (skS.0 13 (powerset (skS.0 14 a)) a_1 a_2) (powerset (skS.0 15 a a_3))) True) % 10.39/10.55 (Eq (in (skS.0 13 (powerset (skS.0 14 a)) a_1 a_2) (powerset (skS.0 15 a a_3))) True)) % 10.39/10.55 Clause #918 (by eliminate duplicate literals #[917]): ∀ (a a_1 a_2 a_3 : Iota), % 10.39/10.55 Or (Eq (subset (powerset (skS.0 14 a)) a_1) True) % 10.39/10.55 (Eq (in (skS.0 13 (powerset (skS.0 14 a)) a_1 a_2) (powerset (skS.0 15 a a_3))) True) % 10.39/10.55 Clause #919 (by superposition #[918, 114]): ∀ (a a_1 : Iota), % 10.39/10.55 Or (Eq (subset (powerset (skS.0 14 a)) (powerset (skS.0 15 a a_1))) True) % 10.39/10.55 (Or (Eq (subset (powerset (skS.0 14 a)) (powerset (skS.0 15 a a_1))) True) (Eq True False)) % 10.39/10.55 Clause #920 (by clausification #[919]): ∀ (a a_1 : Iota), % 10.39/10.55 Or (Eq (subset (powerset (skS.0 14 a)) (powerset (skS.0 15 a a_1))) True) % 10.39/10.55 (Eq (subset (powerset (skS.0 14 a)) (powerset (skS.0 15 a a_1))) True) % 10.39/10.55 Clause #921 (by eliminate duplicate literals #[920]): ∀ (a a_1 : Iota), Eq (subset (powerset (skS.0 14 a)) (powerset (skS.0 15 a a_1))) True % 10.39/10.55 Clause #922 (by superposition #[921, 132]): Eq True False % 10.39/10.55 Clause #924 (by clausification #[922]): False % 10.39/10.55 SZS output end Proof for theBenchmark.p %------------------------------------------------------------------------------