%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET162+3 : TPTP v9.0.0. Released v2.2.0.
% 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 05:25:23 AM UTC 2025
% Result : Theorem 0.20s 0.40s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : SET162+3 : TPTP v9.0.0. Released v2.2.0.
% 0.07/0.13 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n003.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 Jun 20 04:28:39 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.20/0.40 % SZS status Theorem
% 0.20/0.40 % SZS output start Proof
% 0.20/0.40 tff(member_type, type, (
% 0.20/0.40 member: ( $i * $i ) > $o)).
% 0.20/0.40 tff(empty_set_type, type, (
% 0.20/0.40 empty_set: $i)).
% 0.20/0.40 tff(tptp_fun_D_2_type, type, (
% 0.20/0.40 tptp_fun_D_2: ( $i * $i ) > $i)).
% 0.20/0.40 tff(tptp_fun_B_3_type, type, (
% 0.20/0.40 tptp_fun_B_3: $i)).
% 0.20/0.40 tff(union_type, type, (
% 0.20/0.40 union: ( $i * $i ) > $i)).
% 0.20/0.40 tff(1,assumption,(~member(tptp_fun_D_2(union(empty_set, B!3), B!3), union(empty_set, B!3))), introduced(assumption)).
% 0.20/0.40 tff(2,plain,
% 0.20/0.40 (^[B: $i, C: $i, D: $i] : refl((~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))) <=> (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(3,plain,
% 0.20/0.40 (![B: $i, C: $i, D: $i] : (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))) <=> ![B: $i, C: $i, D: $i] : (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[2])).
% 0.20/0.40 tff(4,plain,
% 0.20/0.40 (![B: $i, C: $i] : ![D: $i] : (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))) <=> ![B: $i, C: $i, D: $i] : (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))),
% 0.20/0.40 inference(pull_quant,[status(thm)],[])).
% 0.20/0.40 tff(5,plain,
% 0.20/0.40 (^[B: $i, C: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(pull_quant(((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C))) <=> ![D: $i] : ((~(B = C)) | (member(D, B) <=> member(D, C)))), ((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) <=> (~![D: $i] : ((~(B = C)) | (member(D, B) <=> member(D, C)))))), pull_quant((~![D: $i] : ((~(B = C)) | (member(D, B) <=> member(D, C)))) <=> ?[D: $i] : (~((~(B = C)) | (member(D, B) <=> member(D, C))))), ((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) <=> ?[D: $i] : (~((~(B = C)) | (member(D, B) <=> member(D, C)))))), (((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))) <=> (?[D: $i] : (~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))), pull_quant((?[D: $i] : (~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))) <=> ?[D: $i] : ((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))), (((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))) <=> ?[D: $i] : ((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))), ((~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))) <=> (~?[D: $i] : ((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))))), pull_quant((~?[D: $i] : ((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))) <=> ![D: $i] : (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))), ((~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))) <=> ![D: $i] : (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(6,plain,
% 0.20/0.40 (![B: $i, C: $i] : (~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))) <=> ![B: $i, C: $i] : ![D: $i] : (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[5])).
% 0.20/0.40 tff(7,plain,
% 0.20/0.40 (![B: $i, C: $i] : (~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))) <=> ![B: $i, C: $i, D: $i] : (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))),
% 0.20/0.40 inference(transitivity,[status(thm)],[6, 4])).
% 0.20/0.40 tff(8,plain,
% 0.20/0.40 (![B: $i, C: $i] : (~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))) <=> ![B: $i, C: $i, D: $i] : (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))),
% 0.20/0.40 inference(transitivity,[status(thm)],[7, 3])).
% 0.20/0.40 tff(9,plain,
% 0.20/0.40 (^[B: $i, C: $i] : rewrite((~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))) <=> (~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(10,plain,
% 0.20/0.40 (![B: $i, C: $i] : (~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))) <=> ![B: $i, C: $i] : (~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[9])).
% 0.20/0.40 tff(11,plain,
% 0.20/0.40 (![B: $i, C: $i] : (~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))) <=> ![B: $i, C: $i, D: $i] : (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))),
% 0.20/0.40 inference(transitivity,[status(thm)],[10, 8])).
% 0.20/0.40 tff(12,plain,
% 0.20/0.40 (^[B: $i, C: $i] : trans(monotonicity(rewrite(((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C))) <=> ((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))), trans(monotonicity(rewrite((~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))) <=> (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))), (((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))) <=> ((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))), rewrite(((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))) <=> ((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))), (((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))) <=> ((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))), ((((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C))) & ((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))) <=> (((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C))) & ((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))), rewrite((((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C))) & ((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))) <=> (~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))), ((((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C))) & ((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))) <=> (~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(13,plain,
% 0.20/0.41 (![B: $i, C: $i] : (((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C))) & ((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))) <=> ![B: $i, C: $i] : (~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[12])).
% 0.20/0.41 tff(14,plain,
% 0.20/0.41 (![B: $i, C: $i] : ((B = C) <=> ![D: $i] : (member(D, B) <=> member(D, C))) <=> ![B: $i, C: $i] : ((B = C) <=> ![D: $i] : (member(D, B) <=> member(D, C)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(15,axiom,(![B: $i, C: $i] : ((B = C) <=> ![D: $i] : (member(D, B) <=> member(D, C)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','equal_member_defn')).
% 0.20/0.41 tff(16,plain,
% 0.20/0.41 (![B: $i, C: $i] : ((B = C) <=> ![D: $i] : (member(D, B) <=> member(D, C)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[15, 14])).
% 0.20/0.41 tff(17,plain,(
% 0.20/0.41 ![B: $i, C: $i] : (((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C))) & ((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))),
% 0.20/0.41 inference(skolemize,[status(sab)],[16])).
% 0.20/0.41 tff(18,plain,
% 0.20/0.41 (![B: $i, C: $i] : (~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[17, 13])).
% 0.20/0.41 tff(19,plain,
% 0.20/0.41 (![B: $i, C: $i, D: $i] : (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[18, 11])).
% 0.20/0.41 tff(20,plain,
% 0.20/0.41 ((~![B: $i, C: $i, D: $i] : (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))) | (~((~((~(B!3 = union(empty_set, B!3))) | (member(tptp_fun_D_2(empty_set, B!3), B!3) <=> member(tptp_fun_D_2(empty_set, B!3), union(empty_set, B!3))))) | (~((B!3 = union(empty_set, B!3)) | (~(member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3) <=> member(tptp_fun_D_2(union(empty_set, B!3), B!3), union(empty_set, B!3))))))))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(21,plain,
% 0.20/0.41 (~((~((~(B!3 = union(empty_set, B!3))) | (member(tptp_fun_D_2(empty_set, B!3), B!3) <=> member(tptp_fun_D_2(empty_set, B!3), union(empty_set, B!3))))) | (~((B!3 = union(empty_set, B!3)) | (~(member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3) <=> member(tptp_fun_D_2(union(empty_set, B!3), B!3), union(empty_set, B!3)))))))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[20, 19])).
% 0.20/0.41 tff(22,plain,
% 0.20/0.41 (((~((~(B!3 = union(empty_set, B!3))) | (member(tptp_fun_D_2(empty_set, B!3), B!3) <=> member(tptp_fun_D_2(empty_set, B!3), union(empty_set, B!3))))) | (~((B!3 = union(empty_set, B!3)) | (~(member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3) <=> member(tptp_fun_D_2(union(empty_set, B!3), B!3), union(empty_set, B!3))))))) | ((B!3 = union(empty_set, B!3)) | (~(member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3) <=> member(tptp_fun_D_2(union(empty_set, B!3), B!3), union(empty_set, B!3)))))),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(23,plain,
% 0.20/0.41 ((B!3 = union(empty_set, B!3)) | (~(member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3) <=> member(tptp_fun_D_2(union(empty_set, B!3), B!3), union(empty_set, B!3))))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[22, 21])).
% 0.20/0.41 tff(24,plain,
% 0.20/0.41 ((B!3 = union(B!3, empty_set)) <=> (union(B!3, empty_set) = B!3)),
% 0.20/0.41 inference(commutativity,[status(thm)],[])).
% 0.20/0.41 tff(25,plain,
% 0.20/0.41 (^[B: $i, C: $i] : refl((union(B, C) = union(C, B)) <=> (union(B, C) = union(C, B)))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(26,plain,
% 0.20/0.41 (![B: $i, C: $i] : (union(B, C) = union(C, B)) <=> ![B: $i, C: $i] : (union(B, C) = union(C, B))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[25])).
% 0.20/0.41 tff(27,plain,
% 0.20/0.41 (![B: $i, C: $i] : (union(B, C) = union(C, B)) <=> ![B: $i, C: $i] : (union(B, C) = union(C, B))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(28,axiom,(![B: $i, C: $i] : (union(B, C) = union(C, B))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','commutativity_of_union')).
% 0.20/0.41 tff(29,plain,
% 0.20/0.41 (![B: $i, C: $i] : (union(B, C) = union(C, B))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[28, 27])).
% 0.20/0.41 tff(30,plain,(
% 0.20/0.41 ![B: $i, C: $i] : (union(B, C) = union(C, B))),
% 0.20/0.41 inference(skolemize,[status(sab)],[29])).
% 0.20/0.41 tff(31,plain,
% 0.20/0.41 (![B: $i, C: $i] : (union(B, C) = union(C, B))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[30, 26])).
% 0.20/0.41 tff(32,plain,
% 0.20/0.41 ((~![B: $i, C: $i] : (union(B, C) = union(C, B))) | (union(B!3, empty_set) = union(empty_set, B!3))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(33,plain,
% 0.20/0.41 (union(B!3, empty_set) = union(empty_set, B!3)),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[32, 31])).
% 0.20/0.41 tff(34,plain,
% 0.20/0.41 (union(empty_set, B!3) = union(B!3, empty_set)),
% 0.20/0.41 inference(symmetry,[status(thm)],[33])).
% 0.20/0.41 tff(35,plain,
% 0.20/0.41 ((B!3 = union(empty_set, B!3)) <=> (B!3 = union(B!3, empty_set))),
% 0.20/0.41 inference(monotonicity,[status(thm)],[34])).
% 0.20/0.41 tff(36,plain,
% 0.20/0.41 ((B!3 = union(empty_set, B!3)) <=> (union(B!3, empty_set) = B!3)),
% 0.20/0.41 inference(transitivity,[status(thm)],[35, 24])).
% 0.20/0.41 tff(37,plain,
% 0.20/0.41 ((union(B!3, empty_set) = B!3) <=> (B!3 = union(empty_set, B!3))),
% 0.20/0.41 inference(symmetry,[status(thm)],[36])).
% 0.20/0.41 tff(38,plain,
% 0.20/0.41 ((~(union(B!3, empty_set) = B!3)) <=> (~(B!3 = union(empty_set, B!3)))),
% 0.20/0.41 inference(monotonicity,[status(thm)],[37])).
% 0.20/0.41 tff(39,plain,
% 0.20/0.41 ((~![B: $i] : (union(B, empty_set) = B)) <=> (~![B: $i] : (union(B, empty_set) = B))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(40,axiom,(~![B: $i] : (union(B, empty_set) = B)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_union_empty_set')).
% 0.20/0.41 tff(41,plain,
% 0.20/0.41 (~![B: $i] : (union(B, empty_set) = B)),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[40, 39])).
% 0.20/0.41 tff(42,plain,
% 0.20/0.41 (~![B: $i] : (union(B, empty_set) = B)),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[41, 39])).
% 0.20/0.41 tff(43,plain,
% 0.20/0.41 (~![B: $i] : (union(B, empty_set) = B)),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[42, 39])).
% 0.20/0.41 tff(44,plain,
% 0.20/0.41 (~![B: $i] : (union(B, empty_set) = B)),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[43, 39])).
% 0.20/0.41 tff(45,plain,(
% 0.20/0.41 ~(union(B!3, empty_set) = B!3)),
% 0.20/0.41 inference(skolemize,[status(sab)],[44])).
% 0.20/0.41 tff(46,plain,
% 0.20/0.41 (~(B!3 = union(empty_set, B!3))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[45, 38])).
% 0.20/0.41 tff(47,plain,
% 0.20/0.41 ((~((B!3 = union(empty_set, B!3)) | (~(member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3) <=> member(tptp_fun_D_2(union(empty_set, B!3), B!3), union(empty_set, B!3)))))) | (B!3 = union(empty_set, B!3)) | (~(member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3) <=> member(tptp_fun_D_2(union(empty_set, B!3), B!3), union(empty_set, B!3))))),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(48,plain,
% 0.20/0.41 (~(member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3) <=> member(tptp_fun_D_2(union(empty_set, B!3), B!3), union(empty_set, B!3)))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[47, 46, 23])).
% 0.20/0.41 tff(49,plain,
% 0.20/0.41 ((member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3) <=> member(tptp_fun_D_2(union(empty_set, B!3), B!3), union(empty_set, B!3))) | member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3) | member(tptp_fun_D_2(union(empty_set, B!3), B!3), union(empty_set, B!3))),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(50,plain,
% 0.20/0.41 (member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3) | member(tptp_fun_D_2(union(empty_set, B!3), B!3), union(empty_set, B!3))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[49, 48])).
% 0.20/0.41 tff(51,plain,
% 0.20/0.41 (member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3)),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[50, 1])).
% 0.20/0.41 tff(52,plain,
% 0.20/0.41 (^[B: $i, C: $i, D: $i] : refl((member(D, union(B, C)) <=> (member(D, C) | member(D, B))) <=> (member(D, union(B, C)) <=> (member(D, C) | member(D, B))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(53,plain,
% 0.20/0.41 (![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B))) <=> ![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[52])).
% 0.20/0.41 tff(54,plain,
% 0.20/0.41 (![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B))) <=> ![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(55,plain,
% 0.20/0.41 (^[B: $i, C: $i, D: $i] : rewrite((member(D, union(B, C)) <=> (member(D, B) | member(D, C))) <=> (member(D, union(B, C)) <=> (member(D, C) | member(D, B))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(56,plain,
% 0.20/0.41 (![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, B) | member(D, C))) <=> ![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[55])).
% 0.20/0.41 tff(57,axiom,(![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, B) | member(D, C)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','union_defn')).
% 0.20/0.41 tff(58,plain,
% 0.20/0.41 (![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[57, 56])).
% 0.20/0.41 tff(59,plain,
% 0.20/0.41 (![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[58, 54])).
% 0.20/0.41 tff(60,plain,(
% 0.20/0.41 ![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))),
% 0.20/0.41 inference(skolemize,[status(sab)],[59])).
% 0.20/0.41 tff(61,plain,
% 0.20/0.41 (![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[60, 53])).
% 0.20/0.41 tff(62,plain,
% 0.20/0.41 ((~![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))) | (member(tptp_fun_D_2(union(empty_set, B!3), B!3), union(empty_set, B!3)) <=> (member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3) | member(tptp_fun_D_2(union(empty_set, B!3), B!3), empty_set)))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(63,plain,
% 0.20/0.41 (member(tptp_fun_D_2(union(empty_set, B!3), B!3), union(empty_set, B!3)) <=> (member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3) | member(tptp_fun_D_2(union(empty_set, B!3), B!3), empty_set))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[62, 61])).
% 0.20/0.41 tff(64,plain,
% 0.20/0.41 ((~(member(tptp_fun_D_2(union(empty_set, B!3), B!3), union(empty_set, B!3)) <=> (member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3) | member(tptp_fun_D_2(union(empty_set, B!3), B!3), empty_set)))) | member(tptp_fun_D_2(union(empty_set, B!3), B!3), union(empty_set, B!3)) | (~(member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3) | member(tptp_fun_D_2(union(empty_set, B!3), B!3), empty_set)))),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(65,plain,
% 0.20/0.41 (member(tptp_fun_D_2(union(empty_set, B!3), B!3), union(empty_set, B!3)) | (~(member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3) | member(tptp_fun_D_2(union(empty_set, B!3), B!3), empty_set)))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[64, 63])).
% 0.20/0.41 tff(66,plain,
% 0.20/0.41 (~(member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3) | member(tptp_fun_D_2(union(empty_set, B!3), B!3), empty_set))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[65, 1])).
% 0.20/0.41 tff(67,plain,
% 0.20/0.41 ((member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3) | member(tptp_fun_D_2(union(empty_set, B!3), B!3), empty_set)) | (~member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3))),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(68,plain,
% 0.20/0.41 ($false),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[67, 66, 51])).
% 0.20/0.41 tff(69,plain,(member(tptp_fun_D_2(union(empty_set, B!3), B!3), union(empty_set, B!3))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.41 tff(70,plain,
% 0.20/0.41 ((~(member(tptp_fun_D_2(union(empty_set, B!3), B!3), union(empty_set, B!3)) <=> (member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3) | member(tptp_fun_D_2(union(empty_set, B!3), B!3), empty_set)))) | (~member(tptp_fun_D_2(union(empty_set, B!3), B!3), union(empty_set, B!3))) | (member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3) | member(tptp_fun_D_2(union(empty_set, B!3), B!3), empty_set))),
% 0.20/0.42 inference(tautology,[status(thm)],[])).
% 0.20/0.42 tff(71,plain,
% 0.20/0.42 ((~member(tptp_fun_D_2(union(empty_set, B!3), B!3), union(empty_set, B!3))) | (member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3) | member(tptp_fun_D_2(union(empty_set, B!3), B!3), empty_set))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[70, 63])).
% 0.20/0.42 tff(72,plain,
% 0.20/0.42 (member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3) | member(tptp_fun_D_2(union(empty_set, B!3), B!3), empty_set)),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[71, 69])).
% 0.20/0.42 tff(73,plain,
% 0.20/0.42 ((member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3) <=> member(tptp_fun_D_2(union(empty_set, B!3), B!3), union(empty_set, B!3))) | (~member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3)) | (~member(tptp_fun_D_2(union(empty_set, B!3), B!3), union(empty_set, B!3)))),
% 0.20/0.42 inference(tautology,[status(thm)],[])).
% 0.20/0.42 tff(74,plain,
% 0.20/0.42 ((~member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3)) | (~member(tptp_fun_D_2(union(empty_set, B!3), B!3), union(empty_set, B!3)))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[73, 48])).
% 0.20/0.42 tff(75,plain,
% 0.20/0.42 (~member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3)),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[74, 69])).
% 0.20/0.42 tff(76,plain,
% 0.20/0.42 ((~(member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3) | member(tptp_fun_D_2(union(empty_set, B!3), B!3), empty_set))) | member(tptp_fun_D_2(union(empty_set, B!3), B!3), B!3) | member(tptp_fun_D_2(union(empty_set, B!3), B!3), empty_set)),
% 0.20/0.42 inference(tautology,[status(thm)],[])).
% 0.20/0.42 tff(77,plain,
% 0.20/0.42 (member(tptp_fun_D_2(union(empty_set, B!3), B!3), empty_set)),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[76, 75, 72])).
% 0.20/0.42 tff(78,plain,
% 0.20/0.42 (^[B: $i] : refl((~member(B, empty_set)) <=> (~member(B, empty_set)))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(79,plain,
% 0.20/0.42 (![B: $i] : (~member(B, empty_set)) <=> ![B: $i] : (~member(B, empty_set))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[78])).
% 0.20/0.42 tff(80,plain,
% 0.20/0.42 (![B: $i] : (~member(B, empty_set)) <=> ![B: $i] : (~member(B, empty_set))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(81,axiom,(![B: $i] : (~member(B, empty_set))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','empty_set_defn')).
% 0.20/0.42 tff(82,plain,
% 0.20/0.42 (![B: $i] : (~member(B, empty_set))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[81, 80])).
% 0.20/0.42 tff(83,plain,(
% 0.20/0.42 ![B: $i] : (~member(B, empty_set))),
% 0.20/0.42 inference(skolemize,[status(sab)],[82])).
% 0.20/0.42 tff(84,plain,
% 0.20/0.42 (![B: $i] : (~member(B, empty_set))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[83, 79])).
% 0.20/0.42 tff(85,plain,
% 0.20/0.42 ((~![B: $i] : (~member(B, empty_set))) | (~member(tptp_fun_D_2(union(empty_set, B!3), B!3), empty_set))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(86,plain,
% 0.20/0.42 ($false),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[85, 84, 77])).
% 0.20/0.42 % SZS output end Proof
% 0.20/0.42 % E exiting
%------------------------------------------------------------------------------