%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET903+1 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n003.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 05:26:38 AM UTC 2025
% Result : Theorem 0.14s 0.40s
% Output : Proof 0.14s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.13/0.14 % Problem : SET903+1 : TPTP v9.0.0. Released v3.2.0.
% 0.13/0.14 % Command : run_E %s %d THM
% 0.14/0.36 % Computer : n003.cluster.edu
% 0.14/0.36 % Model : x86_64 x86_64
% 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.36 % Memory : 8042.1875MB
% 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.36 % CPULimit : 300
% 0.14/0.36 % WCLimit : 300
% 0.14/0.36 % DateTime : Fri Jun 20 05:50:24 EDT 2025
% 0.14/0.36 % CPUTime :
% 0.14/0.40 % SZS status Theorem
% 0.14/0.40 % SZS output start Proof
% 0.14/0.40 tff(tptp_fun_C_2_type, type, (
% 0.14/0.40 tptp_fun_C_2: $i)).
% 0.14/0.40 tff(tptp_fun_B_3_type, type, (
% 0.14/0.40 tptp_fun_B_3: $i)).
% 0.14/0.40 tff(singleton_type, type, (
% 0.14/0.40 singleton: $i > $i)).
% 0.14/0.40 tff(tptp_fun_A_4_type, type, (
% 0.14/0.40 tptp_fun_A_4: $i)).
% 0.14/0.40 tff(empty_set_type, type, (
% 0.14/0.40 empty_set: $i)).
% 0.14/0.40 tff(set_union2_type, type, (
% 0.14/0.40 set_union2: ( $i * $i ) > $i)).
% 0.14/0.40 tff(1,plain,
% 0.14/0.40 ((singleton(A!4) = C!2) <=> (C!2 = singleton(A!4))),
% 0.14/0.40 inference(commutativity,[status(thm)],[])).
% 0.14/0.40 tff(2,plain,
% 0.14/0.40 ((~![A: $i, B: $i, C: $i] : (~((singleton(A) = set_union2(B, C)) & (~(B = C)) & (~(B = empty_set)) & (~(C = empty_set))))) <=> (~![A: $i, B: $i, C: $i] : (~((singleton(A) = set_union2(B, C)) & (~(B = C)) & (~(B = empty_set)) & (~(C = empty_set)))))),
% 0.14/0.40 inference(rewrite,[status(thm)],[])).
% 0.14/0.40 tff(3,plain,
% 0.14/0.40 ((~![A: $i, B: $i, C: $i] : (~((((singleton(A) = set_union2(B, C)) & (~(B = C))) & (~(B = empty_set))) & (~(C = empty_set))))) <=> (~![A: $i, B: $i, C: $i] : (~((singleton(A) = set_union2(B, C)) & (~(B = C)) & (~(B = empty_set)) & (~(C = empty_set)))))),
% 0.14/0.40 inference(rewrite,[status(thm)],[])).
% 0.14/0.40 tff(4,axiom,(~![A: $i, B: $i, C: $i] : (~((((singleton(A) = set_union2(B, C)) & (~(B = C))) & (~(B = empty_set))) & (~(C = empty_set))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t44_zfmisc_1')).
% 0.14/0.40 tff(5,plain,
% 0.14/0.40 (~![A: $i, B: $i, C: $i] : (~((singleton(A) = set_union2(B, C)) & (~(B = C)) & (~(B = empty_set)) & (~(C = empty_set))))),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.14/0.40 tff(6,plain,
% 0.14/0.40 (~![A: $i, B: $i, C: $i] : (~((singleton(A) = set_union2(B, C)) & (~(B = C)) & (~(B = empty_set)) & (~(C = empty_set))))),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[5, 2])).
% 0.14/0.40 tff(7,plain,
% 0.14/0.40 (~![A: $i, B: $i, C: $i] : (~((singleton(A) = set_union2(B, C)) & (~(B = C)) & (~(B = empty_set)) & (~(C = empty_set))))),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.14/0.40 tff(8,plain,
% 0.14/0.40 (~![A: $i, B: $i, C: $i] : (~((singleton(A) = set_union2(B, C)) & (~(B = C)) & (~(B = empty_set)) & (~(C = empty_set))))),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[7, 2])).
% 0.14/0.40 tff(9,plain,(
% 0.14/0.40 (singleton(A!4) = set_union2(B!3, C!2)) & (~(B!3 = C!2)) & (~(B!3 = empty_set)) & (~(C!2 = empty_set))),
% 0.14/0.40 inference(skolemize,[status(sab)],[8])).
% 0.14/0.40 tff(10,plain,
% 0.14/0.40 (~(C!2 = empty_set)),
% 0.14/0.40 inference(and_elim,[status(thm)],[9])).
% 0.14/0.40 tff(11,plain,
% 0.14/0.40 (((~(C!2 = empty_set)) | (~(B!3 = singleton(A!4)))) | (C!2 = empty_set)),
% 0.14/0.40 inference(tautology,[status(thm)],[])).
% 0.14/0.40 tff(12,plain,
% 0.14/0.40 ((~(C!2 = empty_set)) | (~(B!3 = singleton(A!4)))),
% 0.14/0.40 inference(unit_resolution,[status(thm)],[11, 10])).
% 0.14/0.40 tff(13,plain,
% 0.14/0.40 (~(B!3 = empty_set)),
% 0.14/0.40 inference(and_elim,[status(thm)],[9])).
% 0.14/0.40 tff(14,plain,
% 0.14/0.40 (((~(B!3 = empty_set)) | (~(C!2 = singleton(A!4)))) | (B!3 = empty_set)),
% 0.14/0.40 inference(tautology,[status(thm)],[])).
% 0.14/0.40 tff(15,plain,
% 0.14/0.40 ((~(B!3 = empty_set)) | (~(C!2 = singleton(A!4)))),
% 0.14/0.40 inference(unit_resolution,[status(thm)],[14, 13])).
% 0.14/0.40 tff(16,plain,
% 0.14/0.40 (singleton(A!4) = set_union2(B!3, C!2)),
% 0.14/0.40 inference(and_elim,[status(thm)],[9])).
% 0.14/0.40 tff(17,plain,
% 0.14/0.40 (^[A: $i, B: $i, C: $i] : refl(((~((~(B = singleton(A))) | (~(C = singleton(A))))) | (~((~(B = empty_set)) | (~(C = singleton(A))))) | (~((~(C = empty_set)) | (~(B = singleton(A))))) | (~(singleton(A) = set_union2(B, C)))) <=> ((~((~(B = singleton(A))) | (~(C = singleton(A))))) | (~((~(B = empty_set)) | (~(C = singleton(A))))) | (~((~(C = empty_set)) | (~(B = singleton(A))))) | (~(singleton(A) = set_union2(B, C)))))),
% 0.14/0.40 inference(bind,[status(th)],[])).
% 0.14/0.40 tff(18,plain,
% 0.14/0.40 (![A: $i, B: $i, C: $i] : ((~((~(B = singleton(A))) | (~(C = singleton(A))))) | (~((~(B = empty_set)) | (~(C = singleton(A))))) | (~((~(C = empty_set)) | (~(B = singleton(A))))) | (~(singleton(A) = set_union2(B, C)))) <=> ![A: $i, B: $i, C: $i] : ((~((~(B = singleton(A))) | (~(C = singleton(A))))) | (~((~(B = empty_set)) | (~(C = singleton(A))))) | (~((~(C = empty_set)) | (~(B = singleton(A))))) | (~(singleton(A) = set_union2(B, C))))),
% 0.14/0.40 inference(quant_intro,[status(thm)],[17])).
% 0.14/0.40 tff(19,plain,
% 0.14/0.40 (^[A: $i, B: $i, C: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(rewrite(((B = singleton(A)) & (C = singleton(A))) <=> (~((~(B = singleton(A))) | (~(C = singleton(A)))))), ((~((B = singleton(A)) & (C = singleton(A)))) <=> (~(~((~(B = singleton(A))) | (~(C = singleton(A)))))))), rewrite((~(~((~(B = singleton(A))) | (~(C = singleton(A)))))) <=> ((~(B = singleton(A))) | (~(C = singleton(A))))), ((~((B = singleton(A)) & (C = singleton(A)))) <=> ((~(B = singleton(A))) | (~(C = singleton(A)))))), trans(monotonicity(rewrite(((B = empty_set) & (C = singleton(A))) <=> (~((~(B = empty_set)) | (~(C = singleton(A)))))), ((~((B = empty_set) & (C = singleton(A)))) <=> (~(~((~(B = empty_set)) | (~(C = singleton(A)))))))), rewrite((~(~((~(B = empty_set)) | (~(C = singleton(A)))))) <=> ((~(B = empty_set)) | (~(C = singleton(A))))), ((~((B = empty_set) & (C = singleton(A)))) <=> ((~(B = empty_set)) | (~(C = singleton(A)))))), trans(monotonicity(rewrite(((B = singleton(A)) & (C = empty_set)) <=> (~((~(C = empty_set)) | (~(B = singleton(A)))))), ((~((B = singleton(A)) & (C = empty_set))) <=> (~(~((~(C = empty_set)) | (~(B = singleton(A)))))))), rewrite((~(~((~(C = empty_set)) | (~(B = singleton(A)))))) <=> ((~(C = empty_set)) | (~(B = singleton(A))))), ((~((B = singleton(A)) & (C = empty_set))) <=> ((~(C = empty_set)) | (~(B = singleton(A)))))), (((singleton(A) = set_union2(B, C)) & (~((B = singleton(A)) & (C = singleton(A)))) & (~((B = empty_set) & (C = singleton(A)))) & (~((B = singleton(A)) & (C = empty_set)))) <=> ((singleton(A) = set_union2(B, C)) & ((~(B = singleton(A))) | (~(C = singleton(A)))) & ((~(B = empty_set)) | (~(C = singleton(A)))) & ((~(C = empty_set)) | (~(B = singleton(A))))))), rewrite(((singleton(A) = set_union2(B, C)) & ((~(B = singleton(A))) | (~(C = singleton(A)))) & ((~(B = empty_set)) | (~(C = singleton(A)))) & ((~(C = empty_set)) | (~(B = singleton(A))))) <=> (~((~((~(B = singleton(A))) | (~(C = singleton(A))))) | (~((~(B = empty_set)) | (~(C = singleton(A))))) | (~((~(C = empty_set)) | (~(B = singleton(A))))) | (~(singleton(A) = set_union2(B, C)))))), (((singleton(A) = set_union2(B, C)) & (~((B = singleton(A)) & (C = singleton(A)))) & (~((B = empty_set) & (C = singleton(A)))) & (~((B = singleton(A)) & (C = empty_set)))) <=> (~((~((~(B = singleton(A))) | (~(C = singleton(A))))) | (~((~(B = empty_set)) | (~(C = singleton(A))))) | (~((~(C = empty_set)) | (~(B = singleton(A))))) | (~(singleton(A) = set_union2(B, C))))))), ((~((singleton(A) = set_union2(B, C)) & (~((B = singleton(A)) & (C = singleton(A)))) & (~((B = empty_set) & (C = singleton(A)))) & (~((B = singleton(A)) & (C = empty_set))))) <=> (~(~((~((~(B = singleton(A))) | (~(C = singleton(A))))) | (~((~(B = empty_set)) | (~(C = singleton(A))))) | (~((~(C = empty_set)) | (~(B = singleton(A))))) | (~(singleton(A) = set_union2(B, C)))))))), rewrite((~(~((~((~(B = singleton(A))) | (~(C = singleton(A))))) | (~((~(B = empty_set)) | (~(C = singleton(A))))) | (~((~(C = empty_set)) | (~(B = singleton(A))))) | (~(singleton(A) = set_union2(B, C)))))) <=> ((~((~(B = singleton(A))) | (~(C = singleton(A))))) | (~((~(B = empty_set)) | (~(C = singleton(A))))) | (~((~(C = empty_set)) | (~(B = singleton(A))))) | (~(singleton(A) = set_union2(B, C))))), ((~((singleton(A) = set_union2(B, C)) & (~((B = singleton(A)) & (C = singleton(A)))) & (~((B = empty_set) & (C = singleton(A)))) & (~((B = singleton(A)) & (C = empty_set))))) <=> ((~((~(B = singleton(A))) | (~(C = singleton(A))))) | (~((~(B = empty_set)) | (~(C = singleton(A))))) | (~((~(C = empty_set)) | (~(B = singleton(A))))) | (~(singleton(A) = set_union2(B, C))))))),
% 0.14/0.40 inference(bind,[status(th)],[])).
% 0.14/0.40 tff(20,plain,
% 0.14/0.40 (![A: $i, B: $i, C: $i] : (~((singleton(A) = set_union2(B, C)) & (~((B = singleton(A)) & (C = singleton(A)))) & (~((B = empty_set) & (C = singleton(A)))) & (~((B = singleton(A)) & (C = empty_set))))) <=> ![A: $i, B: $i, C: $i] : ((~((~(B = singleton(A))) | (~(C = singleton(A))))) | (~((~(B = empty_set)) | (~(C = singleton(A))))) | (~((~(C = empty_set)) | (~(B = singleton(A))))) | (~(singleton(A) = set_union2(B, C))))),
% 0.14/0.40 inference(quant_intro,[status(thm)],[19])).
% 0.14/0.40 tff(21,plain,
% 0.14/0.40 (![A: $i, B: $i, C: $i] : (~((singleton(A) = set_union2(B, C)) & (~((B = singleton(A)) & (C = singleton(A)))) & (~((B = empty_set) & (C = singleton(A)))) & (~((B = singleton(A)) & (C = empty_set))))) <=> ![A: $i, B: $i, C: $i] : (~((singleton(A) = set_union2(B, C)) & (~((B = singleton(A)) & (C = singleton(A)))) & (~((B = empty_set) & (C = singleton(A)))) & (~((B = singleton(A)) & (C = empty_set)))))),
% 0.14/0.40 inference(rewrite,[status(thm)],[])).
% 0.14/0.40 tff(22,plain,
% 0.14/0.40 (^[A: $i, B: $i, C: $i] : rewrite((~((((singleton(A) = set_union2(B, C)) & (~((B = singleton(A)) & (C = singleton(A))))) & (~((B = empty_set) & (C = singleton(A))))) & (~((B = singleton(A)) & (C = empty_set))))) <=> (~((singleton(A) = set_union2(B, C)) & (~((B = singleton(A)) & (C = singleton(A)))) & (~((B = empty_set) & (C = singleton(A)))) & (~((B = singleton(A)) & (C = empty_set))))))),
% 0.14/0.40 inference(bind,[status(th)],[])).
% 0.14/0.40 tff(23,plain,
% 0.14/0.40 (![A: $i, B: $i, C: $i] : (~((((singleton(A) = set_union2(B, C)) & (~((B = singleton(A)) & (C = singleton(A))))) & (~((B = empty_set) & (C = singleton(A))))) & (~((B = singleton(A)) & (C = empty_set))))) <=> ![A: $i, B: $i, C: $i] : (~((singleton(A) = set_union2(B, C)) & (~((B = singleton(A)) & (C = singleton(A)))) & (~((B = empty_set) & (C = singleton(A)))) & (~((B = singleton(A)) & (C = empty_set)))))),
% 0.14/0.40 inference(quant_intro,[status(thm)],[22])).
% 0.14/0.40 tff(24,axiom,(![A: $i, B: $i, C: $i] : (~((((singleton(A) = set_union2(B, C)) & (~((B = singleton(A)) & (C = singleton(A))))) & (~((B = empty_set) & (C = singleton(A))))) & (~((B = singleton(A)) & (C = empty_set)))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t43_zfmisc_1')).
% 0.14/0.40 tff(25,plain,
% 0.14/0.40 (![A: $i, B: $i, C: $i] : (~((singleton(A) = set_union2(B, C)) & (~((B = singleton(A)) & (C = singleton(A)))) & (~((B = empty_set) & (C = singleton(A)))) & (~((B = singleton(A)) & (C = empty_set)))))),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[24, 23])).
% 0.14/0.40 tff(26,plain,
% 0.14/0.40 (![A: $i, B: $i, C: $i] : (~((singleton(A) = set_union2(B, C)) & (~((B = singleton(A)) & (C = singleton(A)))) & (~((B = empty_set) & (C = singleton(A)))) & (~((B = singleton(A)) & (C = empty_set)))))),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[25, 21])).
% 0.14/0.40 tff(27,plain,(
% 0.14/0.40 ![A: $i, B: $i, C: $i] : (~((singleton(A) = set_union2(B, C)) & (~((B = singleton(A)) & (C = singleton(A)))) & (~((B = empty_set) & (C = singleton(A)))) & (~((B = singleton(A)) & (C = empty_set)))))),
% 0.14/0.40 inference(skolemize,[status(sab)],[26])).
% 0.14/0.40 tff(28,plain,
% 0.14/0.40 (![A: $i, B: $i, C: $i] : ((~((~(B = singleton(A))) | (~(C = singleton(A))))) | (~((~(B = empty_set)) | (~(C = singleton(A))))) | (~((~(C = empty_set)) | (~(B = singleton(A))))) | (~(singleton(A) = set_union2(B, C))))),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[27, 20])).
% 0.14/0.40 tff(29,plain,
% 0.14/0.40 (![A: $i, B: $i, C: $i] : ((~((~(B = singleton(A))) | (~(C = singleton(A))))) | (~((~(B = empty_set)) | (~(C = singleton(A))))) | (~((~(C = empty_set)) | (~(B = singleton(A))))) | (~(singleton(A) = set_union2(B, C))))),
% 0.14/0.40 inference(modus_ponens,[status(thm)],[28, 18])).
% 0.14/0.40 tff(30,plain,
% 0.14/0.40 (((~![A: $i, B: $i, C: $i] : ((~((~(B = singleton(A))) | (~(C = singleton(A))))) | (~((~(B = empty_set)) | (~(C = singleton(A))))) | (~((~(C = empty_set)) | (~(B = singleton(A))))) | (~(singleton(A) = set_union2(B, C))))) | ((~((~(B!3 = singleton(A!4))) | (~(C!2 = singleton(A!4))))) | (~((~(B!3 = empty_set)) | (~(C!2 = singleton(A!4))))) | (~((~(C!2 = empty_set)) | (~(B!3 = singleton(A!4))))) | (~(singleton(A!4) = set_union2(B!3, C!2))))) <=> ((~![A: $i, B: $i, C: $i] : ((~((~(B = singleton(A))) | (~(C = singleton(A))))) | (~((~(B = empty_set)) | (~(C = singleton(A))))) | (~((~(C = empty_set)) | (~(B = singleton(A))))) | (~(singleton(A) = set_union2(B, C))))) | (~((~(B!3 = singleton(A!4))) | (~(C!2 = singleton(A!4))))) | (~((~(B!3 = empty_set)) | (~(C!2 = singleton(A!4))))) | (~((~(C!2 = empty_set)) | (~(B!3 = singleton(A!4))))) | (~(singleton(A!4) = set_union2(B!3, C!2))))),
% 0.14/0.40 inference(rewrite,[status(thm)],[])).
% 0.14/0.40 tff(31,plain,
% 0.14/0.40 ((~![A: $i, B: $i, C: $i] : ((~((~(B = singleton(A))) | (~(C = singleton(A))))) | (~((~(B = empty_set)) | (~(C = singleton(A))))) | (~((~(C = empty_set)) | (~(B = singleton(A))))) | (~(singleton(A) = set_union2(B, C))))) | ((~((~(B!3 = singleton(A!4))) | (~(C!2 = singleton(A!4))))) | (~((~(B!3 = empty_set)) | (~(C!2 = singleton(A!4))))) | (~((~(C!2 = empty_set)) | (~(B!3 = singleton(A!4))))) | (~(singleton(A!4) = set_union2(B!3, C!2))))),
% 0.14/0.41 inference(quant_inst,[status(thm)],[])).
% 0.14/0.41 tff(32,plain,
% 0.14/0.41 ((~![A: $i, B: $i, C: $i] : ((~((~(B = singleton(A))) | (~(C = singleton(A))))) | (~((~(B = empty_set)) | (~(C = singleton(A))))) | (~((~(C = empty_set)) | (~(B = singleton(A))))) | (~(singleton(A) = set_union2(B, C))))) | (~((~(B!3 = singleton(A!4))) | (~(C!2 = singleton(A!4))))) | (~((~(B!3 = empty_set)) | (~(C!2 = singleton(A!4))))) | (~((~(C!2 = empty_set)) | (~(B!3 = singleton(A!4))))) | (~(singleton(A!4) = set_union2(B!3, C!2)))),
% 0.14/0.41 inference(modus_ponens,[status(thm)],[31, 30])).
% 0.14/0.41 tff(33,plain,
% 0.14/0.41 (~((~(B!3 = singleton(A!4))) | (~(C!2 = singleton(A!4))))),
% 0.14/0.41 inference(unit_resolution,[status(thm)],[32, 29, 16, 15, 12])).
% 0.14/0.41 tff(34,plain,
% 0.14/0.41 (((~(B!3 = singleton(A!4))) | (~(C!2 = singleton(A!4)))) | (B!3 = singleton(A!4))),
% 0.14/0.41 inference(tautology,[status(thm)],[])).
% 0.14/0.41 tff(35,plain,
% 0.14/0.41 (B!3 = singleton(A!4)),
% 0.14/0.41 inference(unit_resolution,[status(thm)],[34, 33])).
% 0.14/0.41 tff(36,plain,
% 0.14/0.41 ((B!3 = C!2) <=> (singleton(A!4) = C!2)),
% 0.14/0.41 inference(monotonicity,[status(thm)],[35])).
% 0.14/0.41 tff(37,plain,
% 0.14/0.41 ((B!3 = C!2) <=> (C!2 = singleton(A!4))),
% 0.14/0.41 inference(transitivity,[status(thm)],[36, 1])).
% 0.14/0.41 tff(38,plain,
% 0.14/0.41 ((C!2 = singleton(A!4)) <=> (B!3 = C!2)),
% 0.14/0.41 inference(symmetry,[status(thm)],[37])).
% 0.14/0.41 tff(39,plain,
% 0.14/0.41 (((~(B!3 = singleton(A!4))) | (~(C!2 = singleton(A!4)))) | (C!2 = singleton(A!4))),
% 0.14/0.41 inference(tautology,[status(thm)],[])).
% 0.14/0.41 tff(40,plain,
% 0.14/0.41 (C!2 = singleton(A!4)),
% 0.14/0.41 inference(unit_resolution,[status(thm)],[39, 33])).
% 0.14/0.41 tff(41,plain,
% 0.14/0.41 (B!3 = C!2),
% 0.14/0.41 inference(modus_ponens,[status(thm)],[40, 38])).
% 0.14/0.41 tff(42,plain,
% 0.14/0.41 (~(B!3 = C!2)),
% 0.14/0.41 inference(and_elim,[status(thm)],[9])).
% 0.14/0.41 tff(43,plain,
% 0.14/0.41 ($false),
% 0.14/0.41 inference(unit_resolution,[status(thm)],[42, 41])).
% 0.14/0.41 % SZS output end Proof
% 0.14/0.41 % E exiting
%------------------------------------------------------------------------------