%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET592+3 : TPTP v9.0.0. Released v2.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n013.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 05:25:56 AM UTC 2025
% Result : Theorem 0.21s 0.41s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : SET592+3 : TPTP v9.0.0. Released v2.2.0.
% 0.07/0.13 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n013.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Jun 20 05:30:08 EDT 2025
% 0.13/0.35 % CPUTime :
% 0.21/0.41 % SZS status Theorem
% 0.21/0.41 % SZS output start Proof
% 0.21/0.41 tff(member_type, type, (
% 0.21/0.41 member: ( $i * $i ) > $o)).
% 0.21/0.41 tff(empty_set_type, type, (
% 0.21/0.41 empty_set: $i)).
% 0.21/0.41 tff(tptp_fun_D_2_type, type, (
% 0.21/0.41 tptp_fun_D_2: ( $i * $i ) > $i)).
% 0.21/0.41 tff(tptp_fun_B_5_type, type, (
% 0.21/0.41 tptp_fun_B_5: $i)).
% 0.21/0.41 tff(tptp_fun_D_3_type, type, (
% 0.21/0.41 tptp_fun_D_3: $i)).
% 0.21/0.41 tff(subset_type, type, (
% 0.21/0.41 subset: ( $i * $i ) > $o)).
% 0.21/0.41 tff(tptp_fun_D_0_type, type, (
% 0.21/0.41 tptp_fun_D_0: ( $i * $i ) > $i)).
% 0.21/0.41 tff(intersection_type, type, (
% 0.21/0.41 intersection: ( $i * $i ) > $i)).
% 0.21/0.41 tff(tptp_fun_C_4_type, type, (
% 0.21/0.41 tptp_fun_C_4: $i)).
% 0.21/0.41 tff(1,assumption,(~(member(tptp_fun_D_2(empty_set, B!5), B!5) <=> member(tptp_fun_D_2(empty_set, B!5), empty_set))), introduced(assumption)).
% 0.21/0.41 tff(2,plain,
% 0.21/0.41 (^[B: $i] : refl((~member(B, empty_set)) <=> (~member(B, empty_set)))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(3,plain,
% 0.21/0.41 (![B: $i] : (~member(B, empty_set)) <=> ![B: $i] : (~member(B, empty_set))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[2])).
% 0.21/0.41 tff(4,plain,
% 0.21/0.41 (![B: $i] : (~member(B, empty_set)) <=> ![B: $i] : (~member(B, empty_set))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(5,axiom,(![B: $i] : (~member(B, empty_set))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','empty_set_defn')).
% 0.21/0.41 tff(6,plain,
% 0.21/0.41 (![B: $i] : (~member(B, empty_set))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.21/0.41 tff(7,plain,(
% 0.21/0.41 ![B: $i] : (~member(B, empty_set))),
% 0.21/0.41 inference(skolemize,[status(sab)],[6])).
% 0.21/0.41 tff(8,plain,
% 0.21/0.41 (![B: $i] : (~member(B, empty_set))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.21/0.41 tff(9,plain,
% 0.21/0.41 ((~![B: $i] : (~member(B, empty_set))) | (~member(tptp_fun_D_2(empty_set, B!5), empty_set))),
% 0.21/0.41 inference(quant_inst,[status(thm)],[])).
% 0.21/0.41 tff(10,plain,
% 0.21/0.41 (~member(tptp_fun_D_2(empty_set, B!5), empty_set)),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[9, 8])).
% 0.21/0.41 tff(11,plain,
% 0.21/0.41 ((member(tptp_fun_D_2(empty_set, B!5), B!5) <=> member(tptp_fun_D_2(empty_set, B!5), empty_set)) | member(tptp_fun_D_2(empty_set, B!5), B!5) | member(tptp_fun_D_2(empty_set, B!5), empty_set)),
% 0.21/0.41 inference(tautology,[status(thm)],[])).
% 0.21/0.41 tff(12,plain,
% 0.21/0.41 (member(tptp_fun_D_2(empty_set, B!5), B!5)),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[11, 10, 1])).
% 0.21/0.41 tff(13,plain,
% 0.21/0.41 (^[B: $i, C: $i] : refl((~((~((~subset(B, C)) | ![D: $i] : ((~member(D, B)) | member(D, C)))) | (~(subset(B, C) | (~((~member(tptp_fun_D_0(C, B), B)) | member(tptp_fun_D_0(C, B), C))))))) <=> (~((~((~subset(B, C)) | ![D: $i] : ((~member(D, B)) | member(D, C)))) | (~(subset(B, C) | (~((~member(tptp_fun_D_0(C, B), B)) | member(tptp_fun_D_0(C, B), C))))))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(14,plain,
% 0.21/0.41 (![B: $i, C: $i] : (~((~((~subset(B, C)) | ![D: $i] : ((~member(D, B)) | member(D, C)))) | (~(subset(B, C) | (~((~member(tptp_fun_D_0(C, B), B)) | member(tptp_fun_D_0(C, B), C))))))) <=> ![B: $i, C: $i] : (~((~((~subset(B, C)) | ![D: $i] : ((~member(D, B)) | member(D, C)))) | (~(subset(B, C) | (~((~member(tptp_fun_D_0(C, B), B)) | member(tptp_fun_D_0(C, B), C)))))))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[13])).
% 0.21/0.41 tff(15,plain,
% 0.21/0.41 (^[B: $i, C: $i] : rewrite((~((~((~subset(B, C)) | ![D: $i] : ((~member(D, B)) | member(D, C)))) | (~(subset(B, C) | (~((~member(tptp_fun_D_0(C, B), B)) | member(tptp_fun_D_0(C, B), C))))))) <=> (~((~((~subset(B, C)) | ![D: $i] : ((~member(D, B)) | member(D, C)))) | (~(subset(B, C) | (~((~member(tptp_fun_D_0(C, B), B)) | member(tptp_fun_D_0(C, B), C))))))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(16,plain,
% 0.21/0.41 (![B: $i, C: $i] : (~((~((~subset(B, C)) | ![D: $i] : ((~member(D, B)) | member(D, C)))) | (~(subset(B, C) | (~((~member(tptp_fun_D_0(C, B), B)) | member(tptp_fun_D_0(C, B), C))))))) <=> ![B: $i, C: $i] : (~((~((~subset(B, C)) | ![D: $i] : ((~member(D, B)) | member(D, C)))) | (~(subset(B, C) | (~((~member(tptp_fun_D_0(C, B), B)) | member(tptp_fun_D_0(C, B), C)))))))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[15])).
% 0.21/0.41 tff(17,plain,
% 0.21/0.41 (![B: $i, C: $i] : (~((~((~subset(B, C)) | ![D: $i] : ((~member(D, B)) | member(D, C)))) | (~(subset(B, C) | (~((~member(tptp_fun_D_0(C, B), B)) | member(tptp_fun_D_0(C, B), C))))))) <=> ![B: $i, C: $i] : (~((~((~subset(B, C)) | ![D: $i] : ((~member(D, B)) | member(D, C)))) | (~(subset(B, C) | (~((~member(tptp_fun_D_0(C, B), B)) | member(tptp_fun_D_0(C, B), C)))))))),
% 0.21/0.42 inference(transitivity,[status(thm)],[16, 14])).
% 0.21/0.42 tff(18,plain,
% 0.21/0.42 (^[B: $i, C: $i] : trans(monotonicity(rewrite(((~subset(B, C)) | ![D: $i] : ((~member(D, B)) | member(D, C))) <=> ((~subset(B, C)) | ![D: $i] : ((~member(D, B)) | member(D, C)))), rewrite((subset(B, C) | (~((~member(tptp_fun_D_0(C, B), B)) | member(tptp_fun_D_0(C, B), C)))) <=> (subset(B, C) | (~((~member(tptp_fun_D_0(C, B), B)) | member(tptp_fun_D_0(C, B), C))))), ((((~subset(B, C)) | ![D: $i] : ((~member(D, B)) | member(D, C))) & (subset(B, C) | (~((~member(tptp_fun_D_0(C, B), B)) | member(tptp_fun_D_0(C, B), C))))) <=> (((~subset(B, C)) | ![D: $i] : ((~member(D, B)) | member(D, C))) & (subset(B, C) | (~((~member(tptp_fun_D_0(C, B), B)) | member(tptp_fun_D_0(C, B), C))))))), rewrite((((~subset(B, C)) | ![D: $i] : ((~member(D, B)) | member(D, C))) & (subset(B, C) | (~((~member(tptp_fun_D_0(C, B), B)) | member(tptp_fun_D_0(C, B), C))))) <=> (~((~((~subset(B, C)) | ![D: $i] : ((~member(D, B)) | member(D, C)))) | (~(subset(B, C) | (~((~member(tptp_fun_D_0(C, B), B)) | member(tptp_fun_D_0(C, B), C)))))))), ((((~subset(B, C)) | ![D: $i] : ((~member(D, B)) | member(D, C))) & (subset(B, C) | (~((~member(tptp_fun_D_0(C, B), B)) | member(tptp_fun_D_0(C, B), C))))) <=> (~((~((~subset(B, C)) | ![D: $i] : ((~member(D, B)) | member(D, C)))) | (~(subset(B, C) | (~((~member(tptp_fun_D_0(C, B), B)) | member(tptp_fun_D_0(C, B), C)))))))))),
% 0.21/0.42 inference(bind,[status(th)],[])).
% 0.21/0.42 tff(19,plain,
% 0.21/0.42 (![B: $i, C: $i] : (((~subset(B, C)) | ![D: $i] : ((~member(D, B)) | member(D, C))) & (subset(B, C) | (~((~member(tptp_fun_D_0(C, B), B)) | member(tptp_fun_D_0(C, B), C))))) <=> ![B: $i, C: $i] : (~((~((~subset(B, C)) | ![D: $i] : ((~member(D, B)) | member(D, C)))) | (~(subset(B, C) | (~((~member(tptp_fun_D_0(C, B), B)) | member(tptp_fun_D_0(C, B), C)))))))),
% 0.21/0.42 inference(quant_intro,[status(thm)],[18])).
% 0.21/0.42 tff(20,plain,
% 0.21/0.42 (![B: $i, C: $i] : (subset(B, C) <=> ![D: $i] : ((~member(D, B)) | member(D, C))) <=> ![B: $i, C: $i] : (subset(B, C) <=> ![D: $i] : ((~member(D, B)) | member(D, C)))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(21,plain,
% 0.21/0.42 (^[B: $i, C: $i] : rewrite((subset(B, C) <=> ![D: $i] : (member(D, B) => member(D, C))) <=> (subset(B, C) <=> ![D: $i] : ((~member(D, B)) | member(D, C))))),
% 0.21/0.42 inference(bind,[status(th)],[])).
% 0.21/0.42 tff(22,plain,
% 0.21/0.42 (![B: $i, C: $i] : (subset(B, C) <=> ![D: $i] : (member(D, B) => member(D, C))) <=> ![B: $i, C: $i] : (subset(B, C) <=> ![D: $i] : ((~member(D, B)) | member(D, C)))),
% 0.21/0.42 inference(quant_intro,[status(thm)],[21])).
% 0.21/0.42 tff(23,axiom,(![B: $i, C: $i] : (subset(B, C) <=> ![D: $i] : (member(D, B) => member(D, C)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','subset_defn')).
% 0.21/0.42 tff(24,plain,
% 0.21/0.42 (![B: $i, C: $i] : (subset(B, C) <=> ![D: $i] : ((~member(D, B)) | member(D, C)))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[23, 22])).
% 0.21/0.42 tff(25,plain,
% 0.21/0.42 (![B: $i, C: $i] : (subset(B, C) <=> ![D: $i] : ((~member(D, B)) | member(D, C)))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[24, 20])).
% 0.21/0.42 tff(26,plain,(
% 0.21/0.42 ![B: $i, C: $i] : (((~subset(B, C)) | ![D: $i] : ((~member(D, B)) | member(D, C))) & (subset(B, C) | (~((~member(tptp_fun_D_0(C, B), B)) | member(tptp_fun_D_0(C, B), C)))))),
% 0.21/0.42 inference(skolemize,[status(sab)],[25])).
% 0.21/0.42 tff(27,plain,
% 0.21/0.42 (![B: $i, C: $i] : (~((~((~subset(B, C)) | ![D: $i] : ((~member(D, B)) | member(D, C)))) | (~(subset(B, C) | (~((~member(tptp_fun_D_0(C, B), B)) | member(tptp_fun_D_0(C, B), C)))))))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[26, 19])).
% 0.21/0.42 tff(28,plain,
% 0.21/0.42 (![B: $i, C: $i] : (~((~((~subset(B, C)) | ![D: $i] : ((~member(D, B)) | member(D, C)))) | (~(subset(B, C) | (~((~member(tptp_fun_D_0(C, B), B)) | member(tptp_fun_D_0(C, B), C)))))))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[27, 17])).
% 0.21/0.42 tff(29,plain,
% 0.21/0.42 ((~![B: $i, C: $i] : (~((~((~subset(B, C)) | ![D: $i] : ((~member(D, B)) | member(D, C)))) | (~(subset(B, C) | (~((~member(tptp_fun_D_0(C, B), B)) | member(tptp_fun_D_0(C, B), C)))))))) | (~((~((~subset(B!5, D!3)) | ![D: $i] : ((~member(D, B!5)) | member(D, D!3)))) | (~(subset(B!5, D!3) | (~((~member(tptp_fun_D_0(D!3, B!5), B!5)) | member(tptp_fun_D_0(D!3, B!5), D!3)))))))),
% 0.21/0.42 inference(quant_inst,[status(thm)],[])).
% 0.21/0.42 tff(30,plain,
% 0.21/0.42 (~((~((~subset(B!5, D!3)) | ![D: $i] : ((~member(D, B!5)) | member(D, D!3)))) | (~(subset(B!5, D!3) | (~((~member(tptp_fun_D_0(D!3, B!5), B!5)) | member(tptp_fun_D_0(D!3, B!5), D!3))))))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[29, 28])).
% 0.21/0.42 tff(31,plain,
% 0.21/0.42 (((~((~subset(B!5, D!3)) | ![D: $i] : ((~member(D, B!5)) | member(D, D!3)))) | (~(subset(B!5, D!3) | (~((~member(tptp_fun_D_0(D!3, B!5), B!5)) | member(tptp_fun_D_0(D!3, B!5), D!3)))))) | ((~subset(B!5, D!3)) | ![D: $i] : ((~member(D, B!5)) | member(D, D!3)))),
% 0.21/0.42 inference(tautology,[status(thm)],[])).
% 0.21/0.42 tff(32,plain,
% 0.21/0.42 ((~subset(B!5, D!3)) | ![D: $i] : ((~member(D, B!5)) | member(D, D!3))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[31, 30])).
% 0.21/0.42 tff(33,plain,
% 0.21/0.42 ((~![B: $i, C: $i, D: $i] : ((~(subset(B, C) & subset(B, D) & (intersection(C, D) = empty_set))) | (B = empty_set))) <=> (~![B: $i, C: $i, D: $i] : ((~(subset(B, C) & subset(B, D) & (intersection(C, D) = empty_set))) | (B = empty_set)))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(34,plain,
% 0.21/0.42 ((~![B: $i, C: $i, D: $i] : (((subset(B, C) & subset(B, D)) & (intersection(C, D) = empty_set)) => (B = empty_set))) <=> (~![B: $i, C: $i, D: $i] : ((~(subset(B, C) & subset(B, D) & (intersection(C, D) = empty_set))) | (B = empty_set)))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(35,axiom,(~![B: $i, C: $i, D: $i] : (((subset(B, C) & subset(B, D)) & (intersection(C, D) = empty_set)) => (B = empty_set))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_th51')).
% 0.21/0.42 tff(36,plain,
% 0.21/0.42 (~![B: $i, C: $i, D: $i] : ((~(subset(B, C) & subset(B, D) & (intersection(C, D) = empty_set))) | (B = empty_set))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.21/0.42 tff(37,plain,
% 0.21/0.42 (~![B: $i, C: $i, D: $i] : ((~(subset(B, C) & subset(B, D) & (intersection(C, D) = empty_set))) | (B = empty_set))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[36, 33])).
% 0.21/0.42 tff(38,plain,
% 0.21/0.42 (~![B: $i, C: $i, D: $i] : ((~(subset(B, C) & subset(B, D) & (intersection(C, D) = empty_set))) | (B = empty_set))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[37, 33])).
% 0.21/0.42 tff(39,plain,
% 0.21/0.42 (~![B: $i, C: $i, D: $i] : ((~(subset(B, C) & subset(B, D) & (intersection(C, D) = empty_set))) | (B = empty_set))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[38, 33])).
% 0.21/0.42 tff(40,plain,(
% 0.21/0.42 ~((~(subset(B!5, C!4) & subset(B!5, D!3) & (intersection(C!4, D!3) = empty_set))) | (B!5 = empty_set))),
% 0.21/0.42 inference(skolemize,[status(sab)],[39])).
% 0.21/0.42 tff(41,plain,
% 0.21/0.42 (subset(B!5, C!4) & subset(B!5, D!3) & (intersection(C!4, D!3) = empty_set)),
% 0.21/0.42 inference(or_elim,[status(thm)],[40])).
% 0.21/0.42 tff(42,plain,
% 0.21/0.42 (subset(B!5, D!3)),
% 0.21/0.42 inference(and_elim,[status(thm)],[41])).
% 0.21/0.42 tff(43,plain,
% 0.21/0.42 ((~((~subset(B!5, D!3)) | ![D: $i] : ((~member(D, B!5)) | member(D, D!3)))) | (~subset(B!5, D!3)) | ![D: $i] : ((~member(D, B!5)) | member(D, D!3))),
% 0.21/0.42 inference(tautology,[status(thm)],[])).
% 0.21/0.42 tff(44,plain,
% 0.21/0.42 ((~((~subset(B!5, D!3)) | ![D: $i] : ((~member(D, B!5)) | member(D, D!3)))) | ![D: $i] : ((~member(D, B!5)) | member(D, D!3))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[43, 42])).
% 0.21/0.42 tff(45,plain,
% 0.21/0.42 (![D: $i] : ((~member(D, B!5)) | member(D, D!3))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[44, 32])).
% 0.21/0.42 tff(46,plain,
% 0.21/0.42 (((~![D: $i] : ((~member(D, B!5)) | member(D, D!3))) | ((~member(tptp_fun_D_2(empty_set, B!5), B!5)) | member(tptp_fun_D_2(empty_set, B!5), D!3))) <=> ((~![D: $i] : ((~member(D, B!5)) | member(D, D!3))) | (~member(tptp_fun_D_2(empty_set, B!5), B!5)) | member(tptp_fun_D_2(empty_set, B!5), D!3))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(47,plain,
% 0.21/0.42 ((~![D: $i] : ((~member(D, B!5)) | member(D, D!3))) | ((~member(tptp_fun_D_2(empty_set, B!5), B!5)) | member(tptp_fun_D_2(empty_set, B!5), D!3))),
% 0.21/0.42 inference(quant_inst,[status(thm)],[])).
% 0.21/0.42 tff(48,plain,
% 0.21/0.42 ((~![D: $i] : ((~member(D, B!5)) | member(D, D!3))) | (~member(tptp_fun_D_2(empty_set, B!5), B!5)) | member(tptp_fun_D_2(empty_set, B!5), D!3)),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[47, 46])).
% 0.21/0.42 tff(49,plain,
% 0.21/0.42 ((~member(tptp_fun_D_2(empty_set, B!5), B!5)) | member(tptp_fun_D_2(empty_set, B!5), D!3)),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[48, 45])).
% 0.21/0.42 tff(50,plain,
% 0.21/0.42 (member(tptp_fun_D_2(empty_set, B!5), D!3)),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[49, 12])).
% 0.21/0.42 tff(51,assumption,(~(member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4)))), introduced(assumption)).
% 0.21/0.42 tff(52,plain,
% 0.21/0.42 (^[B: $i, C: $i, D: $i] : refl((~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C)))) <=> (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C)))))),
% 0.21/0.42 inference(bind,[status(th)],[])).
% 0.21/0.42 tff(53,plain,
% 0.21/0.42 (![B: $i, C: $i, D: $i] : (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C)))) <=> ![B: $i, C: $i, D: $i] : (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C))))),
% 0.21/0.42 inference(quant_intro,[status(thm)],[52])).
% 0.21/0.42 tff(54,plain,
% 0.21/0.42 (^[B: $i, C: $i, D: $i] : trans(trans(monotonicity(rewrite((member(D, B) & member(D, C)) <=> (~((~member(D, C)) | (~member(D, B))))), ((member(D, intersection(B, C)) <=> (member(D, B) & member(D, C))) <=> (member(D, intersection(B, C)) <=> (~((~member(D, C)) | (~member(D, B))))))), rewrite((member(D, intersection(B, C)) <=> (~((~member(D, C)) | (~member(D, B))))) <=> (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C))))), ((member(D, intersection(B, C)) <=> (member(D, B) & member(D, C))) <=> (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C)))))), rewrite((~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C)))) <=> (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C))))), ((member(D, intersection(B, C)) <=> (member(D, B) & member(D, C))) <=> (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C))))))),
% 0.21/0.42 inference(bind,[status(th)],[])).
% 0.21/0.42 tff(55,plain,
% 0.21/0.42 (![B: $i, C: $i, D: $i] : (member(D, intersection(B, C)) <=> (member(D, B) & member(D, C))) <=> ![B: $i, C: $i, D: $i] : (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C))))),
% 0.21/0.42 inference(quant_intro,[status(thm)],[54])).
% 0.21/0.42 tff(56,plain,
% 0.21/0.42 (![B: $i, C: $i, D: $i] : (member(D, intersection(B, C)) <=> (member(D, B) & member(D, C))) <=> ![B: $i, C: $i, D: $i] : (member(D, intersection(B, C)) <=> (member(D, B) & member(D, C)))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(57,axiom,(![B: $i, C: $i, D: $i] : (member(D, intersection(B, C)) <=> (member(D, B) & member(D, C)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','intersection_defn')).
% 0.21/0.42 tff(58,plain,
% 0.21/0.42 (![B: $i, C: $i, D: $i] : (member(D, intersection(B, C)) <=> (member(D, B) & member(D, C)))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[57, 56])).
% 0.21/0.42 tff(59,plain,(
% 0.21/0.42 ![B: $i, C: $i, D: $i] : (member(D, intersection(B, C)) <=> (member(D, B) & member(D, C)))),
% 0.21/0.42 inference(skolemize,[status(sab)],[58])).
% 0.21/0.42 tff(60,plain,
% 0.21/0.42 (![B: $i, C: $i, D: $i] : (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C))))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[59, 55])).
% 0.21/0.42 tff(61,plain,
% 0.21/0.42 (![B: $i, C: $i, D: $i] : (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C))))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[60, 53])).
% 0.21/0.42 tff(62,plain,
% 0.21/0.42 (((~![B: $i, C: $i, D: $i] : (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C))))) | (member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4)))) <=> ((~![B: $i, C: $i, D: $i] : (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C))))) | (member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4))))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(63,plain,
% 0.21/0.43 ((~(~(member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4))))) <=> (member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4)))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(64,plain,
% 0.21/0.43 ((~(member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4)))) <=> (~(member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(65,plain,
% 0.21/0.43 (((~member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4)) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4))) <=> (~(member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(66,plain,
% 0.21/0.43 (((~member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4)) | (~member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4))) <=> (~member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(67,plain,
% 0.21/0.43 ((((~member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4)) | (~member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4))) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4))) <=> ((~member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4)) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4)))),
% 0.21/0.43 inference(monotonicity,[status(thm)],[66])).
% 0.21/0.43 tff(68,plain,
% 0.21/0.43 ((((~member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4)) | (~member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4))) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4))) <=> (~(member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4))))),
% 0.21/0.43 inference(transitivity,[status(thm)],[67, 65])).
% 0.21/0.43 tff(69,plain,
% 0.21/0.43 ((((~member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4)) | (~member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4))) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4))) <=> (~(member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4))))),
% 0.21/0.43 inference(transitivity,[status(thm)],[68, 64])).
% 0.21/0.43 tff(70,plain,
% 0.21/0.43 ((~(((~member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4)) | (~member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4))) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4)))) <=> (~(~(member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4)))))),
% 0.21/0.43 inference(monotonicity,[status(thm)],[69])).
% 0.21/0.43 tff(71,plain,
% 0.21/0.43 ((~(((~member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4)) | (~member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4))) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4)))) <=> (member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4)))),
% 0.21/0.43 inference(transitivity,[status(thm)],[70, 63])).
% 0.21/0.43 tff(72,plain,
% 0.21/0.43 (((~![B: $i, C: $i, D: $i] : (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C))))) | (~(((~member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4)) | (~member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4))) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4))))) <=> ((~![B: $i, C: $i, D: $i] : (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C))))) | (member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4))))),
% 0.21/0.43 inference(monotonicity,[status(thm)],[71])).
% 0.21/0.43 tff(73,plain,
% 0.21/0.43 (((~![B: $i, C: $i, D: $i] : (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C))))) | (~(((~member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4)) | (~member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4))) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4))))) <=> ((~![B: $i, C: $i, D: $i] : (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C))))) | (member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4))))),
% 0.21/0.43 inference(transitivity,[status(thm)],[72, 62])).
% 0.21/0.43 tff(74,plain,
% 0.21/0.43 ((~![B: $i, C: $i, D: $i] : (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C))))) | (~(((~member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4)) | (~member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4))) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4))))),
% 0.21/0.43 inference(quant_inst,[status(thm)],[])).
% 0.21/0.43 tff(75,plain,
% 0.21/0.43 ((~![B: $i, C: $i, D: $i] : (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C))))) | (member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4)))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[74, 73])).
% 0.21/0.43 tff(76,plain,
% 0.21/0.43 ($false),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[75, 61, 51])).
% 0.21/0.43 tff(77,plain,(member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4))), inference(lemma,lemma(discharge,[]))).
% 0.21/0.43 tff(78,plain,
% 0.21/0.43 (^[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))))))))),
% 0.21/0.43 inference(bind,[status(th)],[])).
% 0.21/0.43 tff(79,plain,
% 0.21/0.43 (![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)))))))),
% 0.21/0.43 inference(quant_intro,[status(thm)],[78])).
% 0.21/0.43 tff(80,plain,
% 0.21/0.43 (![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)))))))),
% 0.21/0.43 inference(pull_quant,[status(thm)],[])).
% 0.21/0.43 tff(81,plain,
% 0.21/0.43 (^[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)))))))))),
% 0.21/0.43 inference(bind,[status(th)],[])).
% 0.21/0.43 tff(82,plain,
% 0.21/0.43 (![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)))))))),
% 0.21/0.43 inference(quant_intro,[status(thm)],[81])).
% 0.21/0.43 tff(83,plain,
% 0.21/0.43 (![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)))))))),
% 0.21/0.43 inference(transitivity,[status(thm)],[82, 80])).
% 0.21/0.43 tff(84,plain,
% 0.21/0.43 (![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)))))))),
% 0.21/0.43 inference(transitivity,[status(thm)],[83, 79])).
% 0.21/0.43 tff(85,plain,
% 0.21/0.43 (^[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))))))))),
% 0.21/0.43 inference(bind,[status(th)],[])).
% 0.21/0.43 tff(86,plain,
% 0.21/0.43 (![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)))))))),
% 0.21/0.43 inference(quant_intro,[status(thm)],[85])).
% 0.21/0.43 tff(87,plain,
% 0.21/0.43 (![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)))))))),
% 0.21/0.43 inference(transitivity,[status(thm)],[86, 84])).
% 0.21/0.43 tff(88,plain,
% 0.21/0.43 (^[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)))))))))),
% 0.21/0.43 inference(bind,[status(th)],[])).
% 0.21/0.43 tff(89,plain,
% 0.21/0.43 (![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)))))))),
% 0.21/0.43 inference(quant_intro,[status(thm)],[88])).
% 0.21/0.43 tff(90,plain,
% 0.21/0.43 (![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)))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(91,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')).
% 0.21/0.43 tff(92,plain,
% 0.21/0.43 (![B: $i, C: $i] : ((B = C) <=> ![D: $i] : (member(D, B) <=> member(D, C)))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[91, 90])).
% 0.21/0.43 tff(93,plain,(
% 0.21/0.43 ![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)))))),
% 0.21/0.43 inference(skolemize,[status(sab)],[92])).
% 0.21/0.43 tff(94,plain,
% 0.21/0.43 (![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)))))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[93, 89])).
% 0.21/0.43 tff(95,plain,
% 0.21/0.43 (![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)))))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[94, 87])).
% 0.21/0.43 tff(96,plain,
% 0.21/0.43 ((~![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)))))))) | (~((~((~(C!4 = intersection(C!4, C!4))) | (member(tptp_fun_D_2(B!5, C!4), C!4) <=> member(tptp_fun_D_2(B!5, C!4), intersection(C!4, C!4))))) | (~((C!4 = intersection(C!4, C!4)) | (~(member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4))))))))),
% 0.21/0.43 inference(quant_inst,[status(thm)],[])).
% 0.21/0.43 tff(97,plain,
% 0.21/0.43 (~((~((~(C!4 = intersection(C!4, C!4))) | (member(tptp_fun_D_2(B!5, C!4), C!4) <=> member(tptp_fun_D_2(B!5, C!4), intersection(C!4, C!4))))) | (~((C!4 = intersection(C!4, C!4)) | (~(member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4)))))))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[96, 95])).
% 0.21/0.43 tff(98,plain,
% 0.21/0.43 (((~((~(C!4 = intersection(C!4, C!4))) | (member(tptp_fun_D_2(B!5, C!4), C!4) <=> member(tptp_fun_D_2(B!5, C!4), intersection(C!4, C!4))))) | (~((C!4 = intersection(C!4, C!4)) | (~(member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4))))))) | ((C!4 = intersection(C!4, C!4)) | (~(member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4)))))),
% 0.21/0.44 inference(tautology,[status(thm)],[])).
% 0.21/0.44 tff(99,plain,
% 0.21/0.44 ((C!4 = intersection(C!4, C!4)) | (~(member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4))))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[98, 97])).
% 0.21/0.44 tff(100,plain,
% 0.21/0.44 ((~((C!4 = intersection(C!4, C!4)) | (~(member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4)))))) | (C!4 = intersection(C!4, C!4)) | (~(member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4))))),
% 0.21/0.44 inference(tautology,[status(thm)],[])).
% 0.21/0.44 tff(101,plain,
% 0.21/0.44 ((C!4 = intersection(C!4, C!4)) | (~(member(tptp_fun_D_2(intersection(C!4, C!4), C!4), C!4) <=> member(tptp_fun_D_2(intersection(C!4, C!4), C!4), intersection(C!4, C!4))))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[100, 99])).
% 0.21/0.44 tff(102,plain,
% 0.21/0.44 (C!4 = intersection(C!4, C!4)),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[101, 77])).
% 0.21/0.44 tff(103,plain,
% 0.21/0.44 (member(tptp_fun_D_2(empty_set, B!5), C!4) <=> member(tptp_fun_D_2(empty_set, B!5), intersection(C!4, C!4))),
% 0.21/0.44 inference(monotonicity,[status(thm)],[102])).
% 0.21/0.44 tff(104,plain,
% 0.21/0.44 ((~![B: $i, C: $i] : (~((~((~subset(B, C)) | ![D: $i] : ((~member(D, B)) | member(D, C)))) | (~(subset(B, C) | (~((~member(tptp_fun_D_0(C, B), B)) | member(tptp_fun_D_0(C, B), C)))))))) | (~((~((~subset(B!5, C!4)) | ![D: $i] : ((~member(D, B!5)) | member(D, C!4)))) | (~(subset(B!5, C!4) | (~((~member(tptp_fun_D_0(C!4, B!5), B!5)) | member(tptp_fun_D_0(C!4, B!5), C!4)))))))),
% 0.21/0.44 inference(quant_inst,[status(thm)],[])).
% 0.21/0.44 tff(105,plain,
% 0.21/0.44 (~((~((~subset(B!5, C!4)) | ![D: $i] : ((~member(D, B!5)) | member(D, C!4)))) | (~(subset(B!5, C!4) | (~((~member(tptp_fun_D_0(C!4, B!5), B!5)) | member(tptp_fun_D_0(C!4, B!5), C!4))))))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[104, 28])).
% 0.21/0.44 tff(106,plain,
% 0.21/0.44 (((~((~subset(B!5, C!4)) | ![D: $i] : ((~member(D, B!5)) | member(D, C!4)))) | (~(subset(B!5, C!4) | (~((~member(tptp_fun_D_0(C!4, B!5), B!5)) | member(tptp_fun_D_0(C!4, B!5), C!4)))))) | ((~subset(B!5, C!4)) | ![D: $i] : ((~member(D, B!5)) | member(D, C!4)))),
% 0.21/0.44 inference(tautology,[status(thm)],[])).
% 0.21/0.44 tff(107,plain,
% 0.21/0.44 ((~subset(B!5, C!4)) | ![D: $i] : ((~member(D, B!5)) | member(D, C!4))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[106, 105])).
% 0.21/0.44 tff(108,plain,
% 0.21/0.44 (subset(B!5, C!4)),
% 0.21/0.44 inference(and_elim,[status(thm)],[41])).
% 0.21/0.44 tff(109,plain,
% 0.21/0.44 ((~((~subset(B!5, C!4)) | ![D: $i] : ((~member(D, B!5)) | member(D, C!4)))) | (~subset(B!5, C!4)) | ![D: $i] : ((~member(D, B!5)) | member(D, C!4))),
% 0.21/0.44 inference(tautology,[status(thm)],[])).
% 0.21/0.44 tff(110,plain,
% 0.21/0.44 ((~((~subset(B!5, C!4)) | ![D: $i] : ((~member(D, B!5)) | member(D, C!4)))) | ![D: $i] : ((~member(D, B!5)) | member(D, C!4))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[109, 108])).
% 0.21/0.44 tff(111,plain,
% 0.21/0.44 (![D: $i] : ((~member(D, B!5)) | member(D, C!4))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[110, 107])).
% 0.21/0.44 tff(112,plain,
% 0.21/0.44 (((~![D: $i] : ((~member(D, B!5)) | member(D, C!4))) | ((~member(tptp_fun_D_2(empty_set, B!5), B!5)) | member(tptp_fun_D_2(empty_set, B!5), C!4))) <=> ((~![D: $i] : ((~member(D, B!5)) | member(D, C!4))) | (~member(tptp_fun_D_2(empty_set, B!5), B!5)) | member(tptp_fun_D_2(empty_set, B!5), C!4))),
% 0.21/0.44 inference(rewrite,[status(thm)],[])).
% 0.21/0.44 tff(113,plain,
% 0.21/0.44 ((~![D: $i] : ((~member(D, B!5)) | member(D, C!4))) | ((~member(tptp_fun_D_2(empty_set, B!5), B!5)) | member(tptp_fun_D_2(empty_set, B!5), C!4))),
% 0.21/0.44 inference(quant_inst,[status(thm)],[])).
% 0.21/0.44 tff(114,plain,
% 0.21/0.44 ((~![D: $i] : ((~member(D, B!5)) | member(D, C!4))) | (~member(tptp_fun_D_2(empty_set, B!5), B!5)) | member(tptp_fun_D_2(empty_set, B!5), C!4)),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[113, 112])).
% 0.21/0.44 tff(115,plain,
% 0.21/0.44 ((~member(tptp_fun_D_2(empty_set, B!5), B!5)) | member(tptp_fun_D_2(empty_set, B!5), C!4)),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[114, 111])).
% 0.21/0.44 tff(116,plain,
% 0.21/0.44 (member(tptp_fun_D_2(empty_set, B!5), C!4)),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[115, 12])).
% 0.21/0.44 tff(117,plain,
% 0.21/0.44 (member(tptp_fun_D_2(empty_set, B!5), intersection(C!4, C!4))),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[116, 103])).
% 0.21/0.44 tff(118,plain,
% 0.21/0.44 ((~![B: $i, C: $i, D: $i] : (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C))))) | (~(((~member(tptp_fun_D_2(empty_set, B!5), intersection(C!4, C!4))) | (~member(tptp_fun_D_2(empty_set, B!5), D!3))) <=> member(tptp_fun_D_2(empty_set, B!5), intersection(D!3, intersection(C!4, C!4)))))),
% 0.21/0.44 inference(quant_inst,[status(thm)],[])).
% 0.21/0.44 tff(119,plain,
% 0.21/0.44 (~(((~member(tptp_fun_D_2(empty_set, B!5), intersection(C!4, C!4))) | (~member(tptp_fun_D_2(empty_set, B!5), D!3))) <=> member(tptp_fun_D_2(empty_set, B!5), intersection(D!3, intersection(C!4, C!4))))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[118, 61])).
% 0.21/0.44 tff(120,plain,
% 0.21/0.44 (intersection(C!4, D!3) = empty_set),
% 0.21/0.44 inference(and_elim,[status(thm)],[41])).
% 0.21/0.44 tff(121,plain,
% 0.21/0.44 (^[B: $i, C: $i] : refl((intersection(B, C) = intersection(C, B)) <=> (intersection(B, C) = intersection(C, B)))),
% 0.21/0.44 inference(bind,[status(th)],[])).
% 0.21/0.44 tff(122,plain,
% 0.21/0.44 (![B: $i, C: $i] : (intersection(B, C) = intersection(C, B)) <=> ![B: $i, C: $i] : (intersection(B, C) = intersection(C, B))),
% 0.21/0.44 inference(quant_intro,[status(thm)],[121])).
% 0.21/0.44 tff(123,plain,
% 0.21/0.44 (![B: $i, C: $i] : (intersection(B, C) = intersection(C, B)) <=> ![B: $i, C: $i] : (intersection(B, C) = intersection(C, B))),
% 0.21/0.44 inference(rewrite,[status(thm)],[])).
% 0.21/0.44 tff(124,axiom,(![B: $i, C: $i] : (intersection(B, C) = intersection(C, B))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','commutativity_of_intersection')).
% 0.21/0.44 tff(125,plain,
% 0.21/0.44 (![B: $i, C: $i] : (intersection(B, C) = intersection(C, B))),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[124, 123])).
% 0.21/0.44 tff(126,plain,(
% 0.21/0.44 ![B: $i, C: $i] : (intersection(B, C) = intersection(C, B))),
% 0.21/0.44 inference(skolemize,[status(sab)],[125])).
% 0.21/0.44 tff(127,plain,
% 0.21/0.44 (![B: $i, C: $i] : (intersection(B, C) = intersection(C, B))),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[126, 122])).
% 0.21/0.44 tff(128,plain,
% 0.21/0.44 ((~![B: $i, C: $i] : (intersection(B, C) = intersection(C, B))) | (intersection(C!4, D!3) = intersection(D!3, C!4))),
% 0.21/0.44 inference(quant_inst,[status(thm)],[])).
% 0.21/0.44 tff(129,plain,
% 0.21/0.44 (intersection(C!4, D!3) = intersection(D!3, C!4)),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[128, 127])).
% 0.21/0.44 tff(130,plain,
% 0.21/0.44 (intersection(D!3, C!4) = intersection(C!4, D!3)),
% 0.21/0.44 inference(symmetry,[status(thm)],[129])).
% 0.21/0.44 tff(131,plain,
% 0.21/0.44 (intersection(C!4, C!4) = C!4),
% 0.21/0.44 inference(symmetry,[status(thm)],[102])).
% 0.21/0.44 tff(132,plain,
% 0.21/0.44 (intersection(D!3, intersection(C!4, C!4)) = intersection(D!3, C!4)),
% 0.21/0.44 inference(monotonicity,[status(thm)],[131])).
% 0.21/0.44 tff(133,plain,
% 0.21/0.44 (intersection(D!3, intersection(C!4, C!4)) = empty_set),
% 0.21/0.44 inference(transitivity,[status(thm)],[132, 130, 120])).
% 0.21/0.44 tff(134,plain,
% 0.21/0.44 (member(tptp_fun_D_2(empty_set, B!5), intersection(D!3, intersection(C!4, C!4))) <=> member(tptp_fun_D_2(empty_set, B!5), empty_set)),
% 0.21/0.44 inference(monotonicity,[status(thm)],[133])).
% 0.21/0.44 tff(135,plain,
% 0.21/0.44 (member(tptp_fun_D_2(empty_set, B!5), empty_set) <=> member(tptp_fun_D_2(empty_set, B!5), intersection(D!3, intersection(C!4, C!4)))),
% 0.21/0.44 inference(symmetry,[status(thm)],[134])).
% 0.21/0.44 tff(136,plain,
% 0.21/0.44 ((~member(tptp_fun_D_2(empty_set, B!5), empty_set)) <=> (~member(tptp_fun_D_2(empty_set, B!5), intersection(D!3, intersection(C!4, C!4))))),
% 0.21/0.44 inference(monotonicity,[status(thm)],[135])).
% 0.21/0.44 tff(137,plain,
% 0.21/0.44 (~member(tptp_fun_D_2(empty_set, B!5), intersection(D!3, intersection(C!4, C!4)))),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[10, 136])).
% 0.21/0.44 tff(138,plain,
% 0.21/0.44 ((((~member(tptp_fun_D_2(empty_set, B!5), intersection(C!4, C!4))) | (~member(tptp_fun_D_2(empty_set, B!5), D!3))) <=> member(tptp_fun_D_2(empty_set, B!5), intersection(D!3, intersection(C!4, C!4)))) | ((~member(tptp_fun_D_2(empty_set, B!5), intersection(C!4, C!4))) | (~member(tptp_fun_D_2(empty_set, B!5), D!3))) | member(tptp_fun_D_2(empty_set, B!5), intersection(D!3, intersection(C!4, C!4)))),
% 0.21/0.44 inference(tautology,[status(thm)],[])).
% 0.21/0.44 tff(139,plain,
% 0.21/0.44 ((~member(tptp_fun_D_2(empty_set, B!5), intersection(C!4, C!4))) | (~member(tptp_fun_D_2(empty_set, B!5), D!3))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[138, 137, 119])).
% 0.21/0.44 tff(140,plain,
% 0.21/0.44 ((~((~member(tptp_fun_D_2(empty_set, B!5), intersection(C!4, C!4))) | (~member(tptp_fun_D_2(empty_set, B!5), D!3)))) | (~member(tptp_fun_D_2(empty_set, B!5), intersection(C!4, C!4))) | (~member(tptp_fun_D_2(empty_set, B!5), D!3))),
% 0.21/0.44 inference(tautology,[status(thm)],[])).
% 0.21/0.44 tff(141,plain,
% 0.21/0.44 ($false),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[140, 139, 117, 50])).
% 0.21/0.44 tff(142,plain,(member(tptp_fun_D_2(empty_set, B!5), B!5) <=> member(tptp_fun_D_2(empty_set, B!5), empty_set)), inference(lemma,lemma(discharge,[]))).
% 0.21/0.44 tff(143,plain,
% 0.21/0.44 (~(B!5 = empty_set)),
% 0.21/0.44 inference(or_elim,[status(thm)],[40])).
% 0.21/0.44 tff(144,plain,
% 0.21/0.44 ((~((B!5 = empty_set) | (~(member(tptp_fun_D_2(empty_set, B!5), B!5) <=> member(tptp_fun_D_2(empty_set, B!5), empty_set))))) | (B!5 = empty_set) | (~(member(tptp_fun_D_2(empty_set, B!5), B!5) <=> member(tptp_fun_D_2(empty_set, B!5), empty_set)))),
% 0.21/0.44 inference(tautology,[status(thm)],[])).
% 0.21/0.44 tff(145,plain,
% 0.21/0.44 (~((B!5 = empty_set) | (~(member(tptp_fun_D_2(empty_set, B!5), B!5) <=> member(tptp_fun_D_2(empty_set, B!5), empty_set))))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[144, 143, 142])).
% 0.21/0.44 tff(146,plain,
% 0.21/0.44 (((~((~(B!5 = empty_set)) | (member(tptp_fun_D_2(empty_set, B!5), B!5) <=> member(tptp_fun_D_2(empty_set, B!5), empty_set)))) | (~((B!5 = empty_set) | (~(member(tptp_fun_D_2(empty_set, B!5), B!5) <=> member(tptp_fun_D_2(empty_set, B!5), empty_set)))))) | ((B!5 = empty_set) | (~(member(tptp_fun_D_2(empty_set, B!5), B!5) <=> member(tptp_fun_D_2(empty_set, B!5), empty_set))))),
% 0.21/0.44 inference(tautology,[status(thm)],[])).
% 0.21/0.44 tff(147,plain,
% 0.21/0.44 ((~((~(B!5 = empty_set)) | (member(tptp_fun_D_2(empty_set, B!5), B!5) <=> member(tptp_fun_D_2(empty_set, B!5), empty_set)))) | (~((B!5 = empty_set) | (~(member(tptp_fun_D_2(empty_set, B!5), B!5) <=> member(tptp_fun_D_2(empty_set, B!5), empty_set)))))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[146, 145])).
% 0.21/0.44 tff(148,plain,
% 0.21/0.44 ((~![B: $i, C: $i, D: $i] : (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))) | (~((~((~(B!5 = empty_set)) | (member(tptp_fun_D_2(empty_set, B!5), B!5) <=> member(tptp_fun_D_2(empty_set, B!5), empty_set)))) | (~((B!5 = empty_set) | (~(member(tptp_fun_D_2(empty_set, B!5), B!5) <=> member(tptp_fun_D_2(empty_set, B!5), empty_set)))))))),
% 0.21/0.44 inference(quant_inst,[status(thm)],[])).
% 0.21/0.44 tff(149,plain,
% 0.21/0.44 ($false),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[148, 95, 147])).
% 0.21/0.44 % SZS output end Proof
% 0.21/0.44 % E exiting
%------------------------------------------------------------------------------