%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET968+1 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n005.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 : Sat Jun 21 05:26:44 AM UTC 2025
% Result : Theorem 0.20s 0.37s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SET968+1 : TPTP v9.0.0. Released v3.2.0.
% 0.03/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n005.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Jun 20 05:53:20 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.20/0.37 % SZS status Theorem
% 0.20/0.37 % SZS output start Proof
% 0.20/0.37 tff(set_union2_type, type, (
% 0.20/0.37 set_union2: ( $i * $i ) > $i)).
% 0.20/0.37 tff(cartesian_product2_type, type, (
% 0.20/0.37 cartesian_product2: ( $i * $i ) > $i)).
% 0.20/0.37 tff(tptp_fun_D_2_type, type, (
% 0.20/0.37 tptp_fun_D_2: $i)).
% 0.20/0.37 tff(tptp_fun_B_4_type, type, (
% 0.20/0.37 tptp_fun_B_4: $i)).
% 0.20/0.37 tff(tptp_fun_C_3_type, type, (
% 0.20/0.37 tptp_fun_C_3: $i)).
% 0.20/0.37 tff(tptp_fun_A_5_type, type, (
% 0.20/0.37 tptp_fun_A_5: $i)).
% 0.20/0.37 tff(1,plain,
% 0.20/0.37 (^[A: $i, B: $i, C: $i] : refl((set_union2(set_union2(A, B), C) = set_union2(A, set_union2(B, C))) <=> (set_union2(set_union2(A, B), C) = set_union2(A, set_union2(B, C))))),
% 0.20/0.37 inference(bind,[status(th)],[])).
% 0.20/0.37 tff(2,plain,
% 0.20/0.37 (![A: $i, B: $i, C: $i] : (set_union2(set_union2(A, B), C) = set_union2(A, set_union2(B, C))) <=> ![A: $i, B: $i, C: $i] : (set_union2(set_union2(A, B), C) = set_union2(A, set_union2(B, C)))),
% 0.20/0.37 inference(quant_intro,[status(thm)],[1])).
% 0.20/0.37 tff(3,plain,
% 0.20/0.37 (![A: $i, B: $i, C: $i] : (set_union2(set_union2(A, B), C) = set_union2(A, set_union2(B, C))) <=> ![A: $i, B: $i, C: $i] : (set_union2(set_union2(A, B), C) = set_union2(A, set_union2(B, C)))),
% 0.20/0.37 inference(rewrite,[status(thm)],[])).
% 0.20/0.37 tff(4,axiom,(![A: $i, B: $i, C: $i] : (set_union2(set_union2(A, B), C) = set_union2(A, set_union2(B, C)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t4_xboole_1')).
% 0.20/0.37 tff(5,plain,
% 0.20/0.37 (![A: $i, B: $i, C: $i] : (set_union2(set_union2(A, B), C) = set_union2(A, set_union2(B, C)))),
% 0.20/0.37 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.20/0.37 tff(6,plain,(
% 0.20/0.37 ![A: $i, B: $i, C: $i] : (set_union2(set_union2(A, B), C) = set_union2(A, set_union2(B, C)))),
% 0.20/0.37 inference(skolemize,[status(sab)],[5])).
% 0.20/0.37 tff(7,plain,
% 0.20/0.37 (![A: $i, B: $i, C: $i] : (set_union2(set_union2(A, B), C) = set_union2(A, set_union2(B, C)))),
% 0.20/0.37 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.20/0.37 tff(8,plain,
% 0.20/0.37 ((~![A: $i, B: $i, C: $i] : (set_union2(set_union2(A, B), C) = set_union2(A, set_union2(B, C)))) | (set_union2(set_union2(set_union2(cartesian_product2(A!5, C!3), cartesian_product2(A!5, D!2)), cartesian_product2(B!4, C!3)), cartesian_product2(B!4, D!2)) = set_union2(set_union2(cartesian_product2(A!5, C!3), cartesian_product2(A!5, D!2)), set_union2(cartesian_product2(B!4, C!3), cartesian_product2(B!4, D!2))))),
% 0.20/0.37 inference(quant_inst,[status(thm)],[])).
% 0.20/0.37 tff(9,plain,
% 0.20/0.37 (set_union2(set_union2(set_union2(cartesian_product2(A!5, C!3), cartesian_product2(A!5, D!2)), cartesian_product2(B!4, C!3)), cartesian_product2(B!4, D!2)) = set_union2(set_union2(cartesian_product2(A!5, C!3), cartesian_product2(A!5, D!2)), set_union2(cartesian_product2(B!4, C!3), cartesian_product2(B!4, D!2)))),
% 0.20/0.37 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.20/0.37 tff(10,plain,
% 0.20/0.37 (set_union2(set_union2(cartesian_product2(A!5, C!3), cartesian_product2(A!5, D!2)), set_union2(cartesian_product2(B!4, C!3), cartesian_product2(B!4, D!2))) = set_union2(set_union2(set_union2(cartesian_product2(A!5, C!3), cartesian_product2(A!5, D!2)), cartesian_product2(B!4, C!3)), cartesian_product2(B!4, D!2))),
% 0.20/0.37 inference(symmetry,[status(thm)],[9])).
% 0.20/0.37 tff(11,plain,
% 0.20/0.37 (^[A: $i, B: $i, C: $i] : refl((~((~(cartesian_product2(set_union2(A, B), C) = set_union2(cartesian_product2(A, C), cartesian_product2(B, C)))) | (~(cartesian_product2(C, set_union2(A, B)) = set_union2(cartesian_product2(C, A), cartesian_product2(C, B)))))) <=> (~((~(cartesian_product2(set_union2(A, B), C) = set_union2(cartesian_product2(A, C), cartesian_product2(B, C)))) | (~(cartesian_product2(C, set_union2(A, B)) = set_union2(cartesian_product2(C, A), cartesian_product2(C, B)))))))),
% 0.20/0.37 inference(bind,[status(th)],[])).
% 0.20/0.37 tff(12,plain,
% 0.20/0.37 (![A: $i, B: $i, C: $i] : (~((~(cartesian_product2(set_union2(A, B), C) = set_union2(cartesian_product2(A, C), cartesian_product2(B, C)))) | (~(cartesian_product2(C, set_union2(A, B)) = set_union2(cartesian_product2(C, A), cartesian_product2(C, B)))))) <=> ![A: $i, B: $i, C: $i] : (~((~(cartesian_product2(set_union2(A, B), C) = set_union2(cartesian_product2(A, C), cartesian_product2(B, C)))) | (~(cartesian_product2(C, set_union2(A, B)) = set_union2(cartesian_product2(C, A), cartesian_product2(C, B))))))),
% 0.20/0.37 inference(quant_intro,[status(thm)],[11])).
% 0.20/0.37 tff(13,plain,
% 0.20/0.37 (^[A: $i, B: $i, C: $i] : rewrite(((cartesian_product2(set_union2(A, B), C) = set_union2(cartesian_product2(A, C), cartesian_product2(B, C))) & (cartesian_product2(C, set_union2(A, B)) = set_union2(cartesian_product2(C, A), cartesian_product2(C, B)))) <=> (~((~(cartesian_product2(set_union2(A, B), C) = set_union2(cartesian_product2(A, C), cartesian_product2(B, C)))) | (~(cartesian_product2(C, set_union2(A, B)) = set_union2(cartesian_product2(C, A), cartesian_product2(C, B)))))))),
% 0.20/0.37 inference(bind,[status(th)],[])).
% 0.20/0.37 tff(14,plain,
% 0.20/0.37 (![A: $i, B: $i, C: $i] : ((cartesian_product2(set_union2(A, B), C) = set_union2(cartesian_product2(A, C), cartesian_product2(B, C))) & (cartesian_product2(C, set_union2(A, B)) = set_union2(cartesian_product2(C, A), cartesian_product2(C, B)))) <=> ![A: $i, B: $i, C: $i] : (~((~(cartesian_product2(set_union2(A, B), C) = set_union2(cartesian_product2(A, C), cartesian_product2(B, C)))) | (~(cartesian_product2(C, set_union2(A, B)) = set_union2(cartesian_product2(C, A), cartesian_product2(C, B))))))),
% 0.20/0.37 inference(quant_intro,[status(thm)],[13])).
% 0.20/0.37 tff(15,plain,
% 0.20/0.37 (![A: $i, B: $i, C: $i] : ((cartesian_product2(set_union2(A, B), C) = set_union2(cartesian_product2(A, C), cartesian_product2(B, C))) & (cartesian_product2(C, set_union2(A, B)) = set_union2(cartesian_product2(C, A), cartesian_product2(C, B)))) <=> ![A: $i, B: $i, C: $i] : ((cartesian_product2(set_union2(A, B), C) = set_union2(cartesian_product2(A, C), cartesian_product2(B, C))) & (cartesian_product2(C, set_union2(A, B)) = set_union2(cartesian_product2(C, A), cartesian_product2(C, B))))),
% 0.20/0.37 inference(rewrite,[status(thm)],[])).
% 0.20/0.37 tff(16,axiom,(![A: $i, B: $i, C: $i] : ((cartesian_product2(set_union2(A, B), C) = set_union2(cartesian_product2(A, C), cartesian_product2(B, C))) & (cartesian_product2(C, set_union2(A, B)) = set_union2(cartesian_product2(C, A), cartesian_product2(C, B))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t120_zfmisc_1')).
% 0.20/0.37 tff(17,plain,
% 0.20/0.37 (![A: $i, B: $i, C: $i] : ((cartesian_product2(set_union2(A, B), C) = set_union2(cartesian_product2(A, C), cartesian_product2(B, C))) & (cartesian_product2(C, set_union2(A, B)) = set_union2(cartesian_product2(C, A), cartesian_product2(C, B))))),
% 0.20/0.37 inference(modus_ponens,[status(thm)],[16, 15])).
% 0.20/0.37 tff(18,plain,(
% 0.20/0.37 ![A: $i, B: $i, C: $i] : ((cartesian_product2(set_union2(A, B), C) = set_union2(cartesian_product2(A, C), cartesian_product2(B, C))) & (cartesian_product2(C, set_union2(A, B)) = set_union2(cartesian_product2(C, A), cartesian_product2(C, B))))),
% 0.20/0.37 inference(skolemize,[status(sab)],[17])).
% 0.20/0.37 tff(19,plain,
% 0.20/0.37 (![A: $i, B: $i, C: $i] : (~((~(cartesian_product2(set_union2(A, B), C) = set_union2(cartesian_product2(A, C), cartesian_product2(B, C)))) | (~(cartesian_product2(C, set_union2(A, B)) = set_union2(cartesian_product2(C, A), cartesian_product2(C, B))))))),
% 0.20/0.37 inference(modus_ponens,[status(thm)],[18, 14])).
% 0.20/0.37 tff(20,plain,
% 0.20/0.37 (![A: $i, B: $i, C: $i] : (~((~(cartesian_product2(set_union2(A, B), C) = set_union2(cartesian_product2(A, C), cartesian_product2(B, C)))) | (~(cartesian_product2(C, set_union2(A, B)) = set_union2(cartesian_product2(C, A), cartesian_product2(C, B))))))),
% 0.20/0.37 inference(modus_ponens,[status(thm)],[19, 12])).
% 0.20/0.37 tff(21,plain,
% 0.20/0.37 ((~![A: $i, B: $i, C: $i] : (~((~(cartesian_product2(set_union2(A, B), C) = set_union2(cartesian_product2(A, C), cartesian_product2(B, C)))) | (~(cartesian_product2(C, set_union2(A, B)) = set_union2(cartesian_product2(C, A), cartesian_product2(C, B))))))) | (~((~(cartesian_product2(set_union2(C!3, D!2), B!4) = set_union2(cartesian_product2(C!3, B!4), cartesian_product2(D!2, B!4)))) | (~(cartesian_product2(B!4, set_union2(C!3, D!2)) = set_union2(cartesian_product2(B!4, C!3), cartesian_product2(B!4, D!2))))))),
% 0.20/0.37 inference(quant_inst,[status(thm)],[])).
% 0.20/0.37 tff(22,plain,
% 0.20/0.37 (~((~(cartesian_product2(set_union2(C!3, D!2), B!4) = set_union2(cartesian_product2(C!3, B!4), cartesian_product2(D!2, B!4)))) | (~(cartesian_product2(B!4, set_union2(C!3, D!2)) = set_union2(cartesian_product2(B!4, C!3), cartesian_product2(B!4, D!2)))))),
% 0.20/0.37 inference(unit_resolution,[status(thm)],[21, 20])).
% 0.20/0.37 tff(23,plain,
% 0.20/0.37 (((~(cartesian_product2(set_union2(C!3, D!2), B!4) = set_union2(cartesian_product2(C!3, B!4), cartesian_product2(D!2, B!4)))) | (~(cartesian_product2(B!4, set_union2(C!3, D!2)) = set_union2(cartesian_product2(B!4, C!3), cartesian_product2(B!4, D!2))))) | (cartesian_product2(B!4, set_union2(C!3, D!2)) = set_union2(cartesian_product2(B!4, C!3), cartesian_product2(B!4, D!2)))),
% 0.20/0.37 inference(tautology,[status(thm)],[])).
% 0.20/0.37 tff(24,plain,
% 0.20/0.37 (cartesian_product2(B!4, set_union2(C!3, D!2)) = set_union2(cartesian_product2(B!4, C!3), cartesian_product2(B!4, D!2))),
% 0.20/0.37 inference(unit_resolution,[status(thm)],[23, 22])).
% 0.20/0.37 tff(25,plain,
% 0.20/0.37 ((~![A: $i, B: $i, C: $i] : (~((~(cartesian_product2(set_union2(A, B), C) = set_union2(cartesian_product2(A, C), cartesian_product2(B, C)))) | (~(cartesian_product2(C, set_union2(A, B)) = set_union2(cartesian_product2(C, A), cartesian_product2(C, B))))))) | (~((~(cartesian_product2(set_union2(C!3, D!2), A!5) = set_union2(cartesian_product2(C!3, A!5), cartesian_product2(D!2, A!5)))) | (~(cartesian_product2(A!5, set_union2(C!3, D!2)) = set_union2(cartesian_product2(A!5, C!3), cartesian_product2(A!5, D!2))))))),
% 0.20/0.37 inference(quant_inst,[status(thm)],[])).
% 0.20/0.37 tff(26,plain,
% 0.20/0.37 (~((~(cartesian_product2(set_union2(C!3, D!2), A!5) = set_union2(cartesian_product2(C!3, A!5), cartesian_product2(D!2, A!5)))) | (~(cartesian_product2(A!5, set_union2(C!3, D!2)) = set_union2(cartesian_product2(A!5, C!3), cartesian_product2(A!5, D!2)))))),
% 0.20/0.37 inference(unit_resolution,[status(thm)],[25, 20])).
% 0.20/0.37 tff(27,plain,
% 0.20/0.37 (((~(cartesian_product2(set_union2(C!3, D!2), A!5) = set_union2(cartesian_product2(C!3, A!5), cartesian_product2(D!2, A!5)))) | (~(cartesian_product2(A!5, set_union2(C!3, D!2)) = set_union2(cartesian_product2(A!5, C!3), cartesian_product2(A!5, D!2))))) | (cartesian_product2(A!5, set_union2(C!3, D!2)) = set_union2(cartesian_product2(A!5, C!3), cartesian_product2(A!5, D!2)))),
% 0.20/0.37 inference(tautology,[status(thm)],[])).
% 0.20/0.37 tff(28,plain,
% 0.20/0.37 (cartesian_product2(A!5, set_union2(C!3, D!2)) = set_union2(cartesian_product2(A!5, C!3), cartesian_product2(A!5, D!2))),
% 0.20/0.37 inference(unit_resolution,[status(thm)],[27, 26])).
% 0.20/0.37 tff(29,plain,
% 0.20/0.37 (set_union2(cartesian_product2(A!5, set_union2(C!3, D!2)), cartesian_product2(B!4, set_union2(C!3, D!2))) = set_union2(set_union2(cartesian_product2(A!5, C!3), cartesian_product2(A!5, D!2)), set_union2(cartesian_product2(B!4, C!3), cartesian_product2(B!4, D!2)))),
% 0.20/0.37 inference(monotonicity,[status(thm)],[28, 24])).
% 0.20/0.37 tff(30,plain,
% 0.20/0.37 ((~![A: $i, B: $i, C: $i] : (~((~(cartesian_product2(set_union2(A, B), C) = set_union2(cartesian_product2(A, C), cartesian_product2(B, C)))) | (~(cartesian_product2(C, set_union2(A, B)) = set_union2(cartesian_product2(C, A), cartesian_product2(C, B))))))) | (~((~(cartesian_product2(set_union2(A!5, B!4), set_union2(C!3, D!2)) = set_union2(cartesian_product2(A!5, set_union2(C!3, D!2)), cartesian_product2(B!4, set_union2(C!3, D!2))))) | (~(cartesian_product2(set_union2(C!3, D!2), set_union2(A!5, B!4)) = set_union2(cartesian_product2(set_union2(C!3, D!2), A!5), cartesian_product2(set_union2(C!3, D!2), B!4))))))),
% 0.20/0.37 inference(quant_inst,[status(thm)],[])).
% 0.20/0.37 tff(31,plain,
% 0.20/0.37 (~((~(cartesian_product2(set_union2(A!5, B!4), set_union2(C!3, D!2)) = set_union2(cartesian_product2(A!5, set_union2(C!3, D!2)), cartesian_product2(B!4, set_union2(C!3, D!2))))) | (~(cartesian_product2(set_union2(C!3, D!2), set_union2(A!5, B!4)) = set_union2(cartesian_product2(set_union2(C!3, D!2), A!5), cartesian_product2(set_union2(C!3, D!2), B!4)))))),
% 0.20/0.37 inference(unit_resolution,[status(thm)],[30, 20])).
% 0.20/0.37 tff(32,plain,
% 0.20/0.37 (((~(cartesian_product2(set_union2(A!5, B!4), set_union2(C!3, D!2)) = set_union2(cartesian_product2(A!5, set_union2(C!3, D!2)), cartesian_product2(B!4, set_union2(C!3, D!2))))) | (~(cartesian_product2(set_union2(C!3, D!2), set_union2(A!5, B!4)) = set_union2(cartesian_product2(set_union2(C!3, D!2), A!5), cartesian_product2(set_union2(C!3, D!2), B!4))))) | (cartesian_product2(set_union2(A!5, B!4), set_union2(C!3, D!2)) = set_union2(cartesian_product2(A!5, set_union2(C!3, D!2)), cartesian_product2(B!4, set_union2(C!3, D!2))))),
% 0.20/0.38 inference(tautology,[status(thm)],[])).
% 0.20/0.38 tff(33,plain,
% 0.20/0.38 (cartesian_product2(set_union2(A!5, B!4), set_union2(C!3, D!2)) = set_union2(cartesian_product2(A!5, set_union2(C!3, D!2)), cartesian_product2(B!4, set_union2(C!3, D!2)))),
% 0.20/0.38 inference(unit_resolution,[status(thm)],[32, 31])).
% 0.20/0.38 tff(34,plain,
% 0.20/0.38 (cartesian_product2(set_union2(A!5, B!4), set_union2(C!3, D!2)) = set_union2(set_union2(set_union2(cartesian_product2(A!5, C!3), cartesian_product2(A!5, D!2)), cartesian_product2(B!4, C!3)), cartesian_product2(B!4, D!2))),
% 0.20/0.38 inference(transitivity,[status(thm)],[33, 29, 10])).
% 0.20/0.38 tff(35,plain,
% 0.20/0.38 ((~![A: $i, B: $i, C: $i, D: $i] : (cartesian_product2(set_union2(A, B), set_union2(C, D)) = set_union2(set_union2(set_union2(cartesian_product2(A, C), cartesian_product2(A, D)), cartesian_product2(B, C)), cartesian_product2(B, D)))) <=> (~![A: $i, B: $i, C: $i, D: $i] : (cartesian_product2(set_union2(A, B), set_union2(C, D)) = set_union2(set_union2(set_union2(cartesian_product2(A, C), cartesian_product2(A, D)), cartesian_product2(B, C)), cartesian_product2(B, D))))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(36,axiom,(~![A: $i, B: $i, C: $i, D: $i] : (cartesian_product2(set_union2(A, B), set_union2(C, D)) = set_union2(set_union2(set_union2(cartesian_product2(A, C), cartesian_product2(A, D)), cartesian_product2(B, C)), cartesian_product2(B, D)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t121_zfmisc_1')).
% 0.20/0.38 tff(37,plain,
% 0.20/0.38 (~![A: $i, B: $i, C: $i, D: $i] : (cartesian_product2(set_union2(A, B), set_union2(C, D)) = set_union2(set_union2(set_union2(cartesian_product2(A, C), cartesian_product2(A, D)), cartesian_product2(B, C)), cartesian_product2(B, D)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[36, 35])).
% 0.20/0.38 tff(38,plain,
% 0.20/0.38 (~![A: $i, B: $i, C: $i, D: $i] : (cartesian_product2(set_union2(A, B), set_union2(C, D)) = set_union2(set_union2(set_union2(cartesian_product2(A, C), cartesian_product2(A, D)), cartesian_product2(B, C)), cartesian_product2(B, D)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[37, 35])).
% 0.20/0.38 tff(39,plain,
% 0.20/0.38 (~![A: $i, B: $i, C: $i, D: $i] : (cartesian_product2(set_union2(A, B), set_union2(C, D)) = set_union2(set_union2(set_union2(cartesian_product2(A, C), cartesian_product2(A, D)), cartesian_product2(B, C)), cartesian_product2(B, D)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[38, 35])).
% 0.20/0.38 tff(40,plain,
% 0.20/0.38 (~![A: $i, B: $i, C: $i, D: $i] : (cartesian_product2(set_union2(A, B), set_union2(C, D)) = set_union2(set_union2(set_union2(cartesian_product2(A, C), cartesian_product2(A, D)), cartesian_product2(B, C)), cartesian_product2(B, D)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[39, 35])).
% 0.20/0.38 tff(41,plain,(
% 0.20/0.38 ~(cartesian_product2(set_union2(A!5, B!4), set_union2(C!3, D!2)) = set_union2(set_union2(set_union2(cartesian_product2(A!5, C!3), cartesian_product2(A!5, D!2)), cartesian_product2(B!4, C!3)), cartesian_product2(B!4, D!2)))),
% 0.20/0.38 inference(skolemize,[status(sab)],[40])).
% 0.20/0.38 tff(42,plain,
% 0.20/0.38 ($false),
% 0.20/0.38 inference(unit_resolution,[status(thm)],[41, 34])).
% 0.20/0.38 % SZS output end Proof
% 0.20/0.38 % E exiting
%------------------------------------------------------------------------------