%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET638+3 : TPTP v9.0.0. Released v2.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n010.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 05:26:05 AM UTC 2025
% Result : Theorem 0.21s 0.37s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SET638+3 : TPTP v9.0.0. Released v2.2.0.
% 0.03/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n010.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Jun 20 05:34:11 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.21/0.37 % SZS status Theorem
% 0.21/0.37 % SZS output start Proof
% 0.21/0.37 tff(member_type, type, (
% 0.21/0.37 member: ( $i * $i ) > $o)).
% 0.21/0.37 tff(intersection_type, type, (
% 0.21/0.37 intersection: ( $i * $i ) > $i)).
% 0.21/0.37 tff(tptp_fun_C_4_type, type, (
% 0.21/0.37 tptp_fun_C_4: $i)).
% 0.21/0.37 tff(tptp_fun_B_5_type, type, (
% 0.21/0.37 tptp_fun_B_5: $i)).
% 0.21/0.37 tff(tptp_fun_D_0_type, type, (
% 0.21/0.37 tptp_fun_D_0: ( $i * $i ) > $i)).
% 0.21/0.37 tff(subset_type, type, (
% 0.21/0.37 subset: ( $i * $i ) > $o)).
% 0.21/0.37 tff(empty_set_type, type, (
% 0.21/0.37 empty_set: $i)).
% 0.21/0.37 tff(tptp_fun_D_3_type, type, (
% 0.21/0.37 tptp_fun_D_3: $i)).
% 0.21/0.37 tff(union_type, type, (
% 0.21/0.37 union: ( $i * $i ) > $i)).
% 0.21/0.37 tff(1,plain,
% 0.21/0.37 (^[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))))))))),
% 0.21/0.37 inference(bind,[status(th)],[])).
% 0.21/0.37 tff(2,plain,
% 0.21/0.37 (![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)))))))),
% 0.21/0.37 inference(quant_intro,[status(thm)],[1])).
% 0.21/0.37 tff(3,plain,
% 0.21/0.37 (^[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))))))))),
% 0.21/0.37 inference(bind,[status(th)],[])).
% 0.21/0.37 tff(4,plain,
% 0.21/0.37 (![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)))))))),
% 0.21/0.37 inference(quant_intro,[status(thm)],[3])).
% 0.21/0.37 tff(5,plain,
% 0.21/0.37 (![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)))))))),
% 0.21/0.37 inference(transitivity,[status(thm)],[4, 2])).
% 0.21/0.37 tff(6,plain,
% 0.21/0.37 (^[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)))))))))),
% 0.21/0.37 inference(bind,[status(th)],[])).
% 0.21/0.37 tff(7,plain,
% 0.21/0.37 (![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)))))))),
% 0.21/0.37 inference(quant_intro,[status(thm)],[6])).
% 0.21/0.37 tff(8,plain,
% 0.21/0.37 (![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)))),
% 0.21/0.37 inference(rewrite,[status(thm)],[])).
% 0.21/0.37 tff(9,plain,
% 0.21/0.37 (^[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))))),
% 0.21/0.37 inference(bind,[status(th)],[])).
% 0.21/0.37 tff(10,plain,
% 0.21/0.37 (![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)))),
% 0.21/0.37 inference(quant_intro,[status(thm)],[9])).
% 0.21/0.37 tff(11,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')).
% 0.21/0.37 tff(12,plain,
% 0.21/0.37 (![B: $i, C: $i] : (subset(B, C) <=> ![D: $i] : ((~member(D, B)) | member(D, C)))),
% 0.21/0.37 inference(modus_ponens,[status(thm)],[11, 10])).
% 0.21/0.37 tff(13,plain,
% 0.21/0.37 (![B: $i, C: $i] : (subset(B, C) <=> ![D: $i] : ((~member(D, B)) | member(D, C)))),
% 0.21/0.37 inference(modus_ponens,[status(thm)],[12, 8])).
% 0.21/0.37 tff(14,plain,(
% 0.21/0.37 ![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)))))),
% 0.21/0.37 inference(skolemize,[status(sab)],[13])).
% 0.21/0.37 tff(15,plain,
% 0.21/0.37 (![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)))))))),
% 0.21/0.37 inference(modus_ponens,[status(thm)],[14, 7])).
% 0.21/0.37 tff(16,plain,
% 0.21/0.37 (![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)))))))),
% 0.21/0.37 inference(modus_ponens,[status(thm)],[15, 5])).
% 0.21/0.37 tff(17,plain,
% 0.21/0.37 ((~![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)))))))),
% 0.21/0.37 inference(quant_inst,[status(thm)],[])).
% 0.21/0.37 tff(18,plain,
% 0.21/0.37 (~((~((~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))))))),
% 0.21/0.37 inference(unit_resolution,[status(thm)],[17, 16])).
% 0.21/0.37 tff(19,plain,
% 0.21/0.37 (((~((~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) | (~((~member(tptp_fun_D_0(C!4, B!5), B!5)) | member(tptp_fun_D_0(C!4, B!5), C!4))))),
% 0.21/0.37 inference(tautology,[status(thm)],[])).
% 0.21/0.37 tff(20,plain,
% 0.21/0.37 (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)))),
% 0.21/0.37 inference(unit_resolution,[status(thm)],[19, 18])).
% 0.21/0.37 tff(21,plain,
% 0.21/0.37 ((~![B: $i, C: $i, D: $i] : ((~(subset(B, union(C, D)) & (intersection(B, D) = empty_set))) | subset(B, C))) <=> (~![B: $i, C: $i, D: $i] : ((~(subset(B, union(C, D)) & (intersection(B, D) = empty_set))) | subset(B, C)))),
% 0.21/0.37 inference(rewrite,[status(thm)],[])).
% 0.21/0.37 tff(22,plain,
% 0.21/0.37 ((~![B: $i, C: $i, D: $i] : ((subset(B, union(C, D)) & (intersection(B, D) = empty_set)) => subset(B, C))) <=> (~![B: $i, C: $i, D: $i] : ((~(subset(B, union(C, D)) & (intersection(B, D) = empty_set))) | subset(B, C)))),
% 0.21/0.37 inference(rewrite,[status(thm)],[])).
% 0.21/0.37 tff(23,axiom,(~![B: $i, C: $i, D: $i] : ((subset(B, union(C, D)) & (intersection(B, D) = empty_set)) => subset(B, C))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_th120')).
% 0.21/0.37 tff(24,plain,
% 0.21/0.37 (~![B: $i, C: $i, D: $i] : ((~(subset(B, union(C, D)) & (intersection(B, D) = empty_set))) | subset(B, C))),
% 0.21/0.37 inference(modus_ponens,[status(thm)],[23, 22])).
% 0.21/0.37 tff(25,plain,
% 0.21/0.37 (~![B: $i, C: $i, D: $i] : ((~(subset(B, union(C, D)) & (intersection(B, D) = empty_set))) | subset(B, C))),
% 0.21/0.37 inference(modus_ponens,[status(thm)],[24, 21])).
% 0.21/0.37 tff(26,plain,
% 0.21/0.37 (~![B: $i, C: $i, D: $i] : ((~(subset(B, union(C, D)) & (intersection(B, D) = empty_set))) | subset(B, C))),
% 0.21/0.37 inference(modus_ponens,[status(thm)],[25, 21])).
% 0.21/0.37 tff(27,plain,
% 0.21/0.37 (~![B: $i, C: $i, D: $i] : ((~(subset(B, union(C, D)) & (intersection(B, D) = empty_set))) | subset(B, C))),
% 0.21/0.37 inference(modus_ponens,[status(thm)],[26, 21])).
% 0.21/0.37 tff(28,plain,(
% 0.21/0.37 ~((~(subset(B!5, union(C!4, D!3)) & (intersection(B!5, D!3) = empty_set))) | subset(B!5, C!4))),
% 0.21/0.37 inference(skolemize,[status(sab)],[27])).
% 0.21/0.37 tff(29,plain,
% 0.21/0.37 (~subset(B!5, C!4)),
% 0.21/0.37 inference(or_elim,[status(thm)],[28])).
% 0.21/0.37 tff(30,plain,
% 0.21/0.37 ((~(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) | (~((~member(tptp_fun_D_0(C!4, B!5), B!5)) | member(tptp_fun_D_0(C!4, B!5), C!4)))),
% 0.21/0.37 inference(tautology,[status(thm)],[])).
% 0.21/0.37 tff(31,plain,
% 0.21/0.37 ((~(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))))) | (~((~member(tptp_fun_D_0(C!4, B!5), B!5)) | member(tptp_fun_D_0(C!4, B!5), C!4)))),
% 0.21/0.37 inference(unit_resolution,[status(thm)],[30, 29])).
% 0.21/0.37 tff(32,plain,
% 0.21/0.37 (~((~member(tptp_fun_D_0(C!4, B!5), B!5)) | member(tptp_fun_D_0(C!4, B!5), C!4))),
% 0.21/0.37 inference(unit_resolution,[status(thm)],[31, 20])).
% 0.21/0.37 tff(33,plain,
% 0.21/0.37 (((~member(tptp_fun_D_0(C!4, B!5), B!5)) | member(tptp_fun_D_0(C!4, B!5), C!4)) | (~member(tptp_fun_D_0(C!4, B!5), C!4))),
% 0.21/0.37 inference(tautology,[status(thm)],[])).
% 0.21/0.37 tff(34,plain,
% 0.21/0.37 (~member(tptp_fun_D_0(C!4, B!5), C!4)),
% 0.21/0.37 inference(unit_resolution,[status(thm)],[33, 32])).
% 0.21/0.37 tff(35,plain,
% 0.21/0.37 (((~member(tptp_fun_D_0(C!4, B!5), B!5)) | (~member(tptp_fun_D_0(C!4, B!5), C!4))) | member(tptp_fun_D_0(C!4, B!5), C!4)),
% 0.21/0.37 inference(tautology,[status(thm)],[])).
% 0.21/0.37 tff(36,plain,
% 0.21/0.37 ((~member(tptp_fun_D_0(C!4, B!5), B!5)) | (~member(tptp_fun_D_0(C!4, B!5), C!4))),
% 0.21/0.37 inference(unit_resolution,[status(thm)],[35, 34])).
% 0.21/0.37 tff(37,plain,
% 0.21/0.37 (subset(B!5, union(C!4, D!3)) & (intersection(B!5, D!3) = empty_set)),
% 0.21/0.37 inference(or_elim,[status(thm)],[28])).
% 0.21/0.37 tff(38,plain,
% 0.21/0.37 (subset(B!5, union(C!4, D!3))),
% 0.21/0.37 inference(and_elim,[status(thm)],[37])).
% 0.21/0.37 tff(39,plain,
% 0.21/0.37 (^[B: $i, C: $i] : refl(((~subset(B, C)) | (intersection(B, C) = B)) <=> ((~subset(B, C)) | (intersection(B, C) = B)))),
% 0.21/0.37 inference(bind,[status(th)],[])).
% 0.21/0.37 tff(40,plain,
% 0.21/0.37 (![B: $i, C: $i] : ((~subset(B, C)) | (intersection(B, C) = B)) <=> ![B: $i, C: $i] : ((~subset(B, C)) | (intersection(B, C) = B))),
% 0.21/0.37 inference(quant_intro,[status(thm)],[39])).
% 0.21/0.37 tff(41,plain,
% 0.21/0.37 (![B: $i, C: $i] : ((~subset(B, C)) | (intersection(B, C) = B)) <=> ![B: $i, C: $i] : ((~subset(B, C)) | (intersection(B, C) = B))),
% 0.21/0.37 inference(rewrite,[status(thm)],[])).
% 0.21/0.37 tff(42,plain,
% 0.21/0.37 (^[B: $i, C: $i] : rewrite((subset(B, C) => (intersection(B, C) = B)) <=> ((~subset(B, C)) | (intersection(B, C) = B)))),
% 0.21/0.37 inference(bind,[status(th)],[])).
% 0.21/0.37 tff(43,plain,
% 0.21/0.37 (![B: $i, C: $i] : (subset(B, C) => (intersection(B, C) = B)) <=> ![B: $i, C: $i] : ((~subset(B, C)) | (intersection(B, C) = B))),
% 0.21/0.37 inference(quant_intro,[status(thm)],[42])).
% 0.21/0.37 tff(44,axiom,(![B: $i, C: $i] : (subset(B, C) => (intersection(B, C) = B))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','subset_intersection')).
% 0.21/0.37 tff(45,plain,
% 0.21/0.37 (![B: $i, C: $i] : ((~subset(B, C)) | (intersection(B, C) = B))),
% 0.21/0.37 inference(modus_ponens,[status(thm)],[44, 43])).
% 0.21/0.37 tff(46,plain,
% 0.21/0.37 (![B: $i, C: $i] : ((~subset(B, C)) | (intersection(B, C) = B))),
% 0.21/0.37 inference(modus_ponens,[status(thm)],[45, 41])).
% 0.21/0.37 tff(47,plain,(
% 0.21/0.37 ![B: $i, C: $i] : ((~subset(B, C)) | (intersection(B, C) = B))),
% 0.21/0.37 inference(skolemize,[status(sab)],[46])).
% 0.21/0.37 tff(48,plain,
% 0.21/0.37 (![B: $i, C: $i] : ((~subset(B, C)) | (intersection(B, C) = B))),
% 0.21/0.37 inference(modus_ponens,[status(thm)],[47, 40])).
% 0.21/0.37 tff(49,plain,
% 0.21/0.37 (((~![B: $i, C: $i] : ((~subset(B, C)) | (intersection(B, C) = B))) | ((~subset(B!5, union(C!4, D!3))) | (intersection(B!5, union(C!4, D!3)) = B!5))) <=> ((~![B: $i, C: $i] : ((~subset(B, C)) | (intersection(B, C) = B))) | (~subset(B!5, union(C!4, D!3))) | (intersection(B!5, union(C!4, D!3)) = B!5))),
% 0.21/0.37 inference(rewrite,[status(thm)],[])).
% 0.21/0.37 tff(50,plain,
% 0.21/0.37 ((~![B: $i, C: $i] : ((~subset(B, C)) | (intersection(B, C) = B))) | ((~subset(B!5, union(C!4, D!3))) | (intersection(B!5, union(C!4, D!3)) = B!5))),
% 0.21/0.37 inference(quant_inst,[status(thm)],[])).
% 0.21/0.37 tff(51,plain,
% 0.21/0.37 ((~![B: $i, C: $i] : ((~subset(B, C)) | (intersection(B, C) = B))) | (~subset(B!5, union(C!4, D!3))) | (intersection(B!5, union(C!4, D!3)) = B!5)),
% 0.21/0.37 inference(modus_ponens,[status(thm)],[50, 49])).
% 0.21/0.37 tff(52,plain,
% 0.21/0.37 (intersection(B!5, union(C!4, D!3)) = B!5),
% 0.21/0.37 inference(unit_resolution,[status(thm)],[51, 48, 38])).
% 0.21/0.37 tff(53,plain,
% 0.21/0.37 (^[B: $i, C: $i, D: $i] : refl((intersection(B, union(C, D)) = union(intersection(B, C), intersection(B, D))) <=> (intersection(B, union(C, D)) = union(intersection(B, C), intersection(B, D))))),
% 0.21/0.37 inference(bind,[status(th)],[])).
% 0.21/0.37 tff(54,plain,
% 0.21/0.37 (![B: $i, C: $i, D: $i] : (intersection(B, union(C, D)) = union(intersection(B, C), intersection(B, D))) <=> ![B: $i, C: $i, D: $i] : (intersection(B, union(C, D)) = union(intersection(B, C), intersection(B, D)))),
% 0.21/0.37 inference(quant_intro,[status(thm)],[53])).
% 0.21/0.37 tff(55,plain,
% 0.21/0.37 (![B: $i, C: $i, D: $i] : (intersection(B, union(C, D)) = union(intersection(B, C), intersection(B, D))) <=> ![B: $i, C: $i, D: $i] : (intersection(B, union(C, D)) = union(intersection(B, C), intersection(B, D)))),
% 0.21/0.37 inference(rewrite,[status(thm)],[])).
% 0.21/0.37 tff(56,axiom,(![B: $i, C: $i, D: $i] : (intersection(B, union(C, D)) = union(intersection(B, C), intersection(B, D)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','intersection_distributes_over_union')).
% 0.21/0.37 tff(57,plain,
% 0.21/0.37 (![B: $i, C: $i, D: $i] : (intersection(B, union(C, D)) = union(intersection(B, C), intersection(B, D)))),
% 0.21/0.37 inference(modus_ponens,[status(thm)],[56, 55])).
% 0.21/0.37 tff(58,plain,(
% 0.21/0.37 ![B: $i, C: $i, D: $i] : (intersection(B, union(C, D)) = union(intersection(B, C), intersection(B, D)))),
% 0.21/0.37 inference(skolemize,[status(sab)],[57])).
% 0.21/0.37 tff(59,plain,
% 0.21/0.37 (![B: $i, C: $i, D: $i] : (intersection(B, union(C, D)) = union(intersection(B, C), intersection(B, D)))),
% 0.21/0.37 inference(modus_ponens,[status(thm)],[58, 54])).
% 0.21/0.37 tff(60,plain,
% 0.21/0.37 ((~![B: $i, C: $i, D: $i] : (intersection(B, union(C, D)) = union(intersection(B, C), intersection(B, D)))) | (intersection(B!5, union(C!4, D!3)) = union(intersection(B!5, C!4), intersection(B!5, D!3)))),
% 0.21/0.37 inference(quant_inst,[status(thm)],[])).
% 0.21/0.37 tff(61,plain,
% 0.21/0.37 (intersection(B!5, union(C!4, D!3)) = union(intersection(B!5, C!4), intersection(B!5, D!3))),
% 0.21/0.37 inference(unit_resolution,[status(thm)],[60, 59])).
% 0.21/0.37 tff(62,plain,
% 0.21/0.37 (union(intersection(B!5, C!4), intersection(B!5, D!3)) = intersection(B!5, union(C!4, D!3))),
% 0.21/0.37 inference(symmetry,[status(thm)],[61])).
% 0.21/0.37 tff(63,plain,
% 0.21/0.37 (intersection(B!5, D!3) = empty_set),
% 0.21/0.37 inference(and_elim,[status(thm)],[37])).
% 0.21/0.37 tff(64,plain,
% 0.21/0.37 (empty_set = intersection(B!5, D!3)),
% 0.21/0.37 inference(symmetry,[status(thm)],[63])).
% 0.21/0.37 tff(65,plain,
% 0.21/0.37 (union(intersection(B!5, C!4), empty_set) = union(intersection(B!5, C!4), intersection(B!5, D!3))),
% 0.21/0.37 inference(monotonicity,[status(thm)],[64])).
% 0.21/0.37 tff(66,plain,
% 0.21/0.37 (^[B: $i] : refl((union(B, empty_set) = B) <=> (union(B, empty_set) = B))),
% 0.21/0.37 inference(bind,[status(th)],[])).
% 0.21/0.37 tff(67,plain,
% 0.21/0.37 (![B: $i] : (union(B, empty_set) = B) <=> ![B: $i] : (union(B, empty_set) = B)),
% 0.21/0.37 inference(quant_intro,[status(thm)],[66])).
% 0.21/0.37 tff(68,plain,
% 0.21/0.37 (![B: $i] : (union(B, empty_set) = B) <=> ![B: $i] : (union(B, empty_set) = B)),
% 0.21/0.37 inference(rewrite,[status(thm)],[])).
% 0.21/0.37 tff(69,axiom,(![B: $i] : (union(B, empty_set) = B)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','union_empty_set')).
% 0.21/0.37 tff(70,plain,
% 0.21/0.37 (![B: $i] : (union(B, empty_set) = B)),
% 0.21/0.37 inference(modus_ponens,[status(thm)],[69, 68])).
% 0.21/0.37 tff(71,plain,(
% 0.21/0.37 ![B: $i] : (union(B, empty_set) = B)),
% 0.21/0.37 inference(skolemize,[status(sab)],[70])).
% 0.21/0.37 tff(72,plain,
% 0.21/0.37 (![B: $i] : (union(B, empty_set) = B)),
% 0.21/0.37 inference(modus_ponens,[status(thm)],[71, 67])).
% 0.21/0.37 tff(73,plain,
% 0.21/0.37 ((~![B: $i] : (union(B, empty_set) = B)) | (union(intersection(B!5, C!4), empty_set) = intersection(B!5, C!4))),
% 0.21/0.37 inference(quant_inst,[status(thm)],[])).
% 0.21/0.37 tff(74,plain,
% 0.21/0.37 (union(intersection(B!5, C!4), empty_set) = intersection(B!5, C!4)),
% 0.21/0.37 inference(unit_resolution,[status(thm)],[73, 72])).
% 0.21/0.37 tff(75,plain,
% 0.21/0.37 (intersection(B!5, C!4) = union(intersection(B!5, C!4), empty_set)),
% 0.21/0.37 inference(symmetry,[status(thm)],[74])).
% 0.21/0.37 tff(76,plain,
% 0.21/0.37 (intersection(B!5, C!4) = B!5),
% 0.21/0.37 inference(transitivity,[status(thm)],[75, 65, 62, 52])).
% 0.21/0.37 tff(77,plain,
% 0.21/0.37 (member(tptp_fun_D_0(C!4, B!5), intersection(B!5, C!4)) <=> member(tptp_fun_D_0(C!4, B!5), B!5)),
% 0.21/0.37 inference(monotonicity,[status(thm)],[76])).
% 0.21/0.37 tff(78,plain,
% 0.21/0.37 (member(tptp_fun_D_0(C!4, B!5), B!5) <=> member(tptp_fun_D_0(C!4, B!5), intersection(B!5, C!4))),
% 0.21/0.37 inference(symmetry,[status(thm)],[77])).
% 0.21/0.37 tff(79,plain,
% 0.21/0.37 (((~member(tptp_fun_D_0(C!4, B!5), B!5)) | member(tptp_fun_D_0(C!4, B!5), C!4)) | member(tptp_fun_D_0(C!4, B!5), B!5)),
% 0.21/0.37 inference(tautology,[status(thm)],[])).
% 0.21/0.37 tff(80,plain,
% 0.21/0.37 (member(tptp_fun_D_0(C!4, B!5), B!5)),
% 0.21/0.37 inference(unit_resolution,[status(thm)],[79, 32])).
% 0.21/0.37 tff(81,plain,
% 0.21/0.37 (member(tptp_fun_D_0(C!4, B!5), intersection(B!5, C!4))),
% 0.21/0.37 inference(modus_ponens,[status(thm)],[80, 78])).
% 0.21/0.37 tff(82,plain,
% 0.21/0.37 ((((~member(tptp_fun_D_0(C!4, B!5), B!5)) | (~member(tptp_fun_D_0(C!4, B!5), C!4))) <=> member(tptp_fun_D_0(C!4, B!5), intersection(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)))) | (~member(tptp_fun_D_0(C!4, B!5), intersection(B!5, C!4)))),
% 0.21/0.37 inference(tautology,[status(thm)],[])).
% 0.21/0.37 tff(83,plain,
% 0.21/0.37 (((~member(tptp_fun_D_0(C!4, B!5), B!5)) | (~member(tptp_fun_D_0(C!4, B!5), C!4))) <=> member(tptp_fun_D_0(C!4, B!5), intersection(B!5, C!4))),
% 0.21/0.37 inference(unit_resolution,[status(thm)],[82, 81, 36])).
% 0.21/0.37 tff(84,plain,
% 0.21/0.37 (^[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)))))),
% 0.21/0.37 inference(bind,[status(th)],[])).
% 0.21/0.37 tff(85,plain,
% 0.21/0.37 (![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))))),
% 0.21/0.37 inference(quant_intro,[status(thm)],[84])).
% 0.21/0.37 tff(86,plain,
% 0.21/0.37 (^[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))))))),
% 0.21/0.37 inference(bind,[status(th)],[])).
% 0.21/0.37 tff(87,plain,
% 0.21/0.37 (![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))))),
% 0.21/0.37 inference(quant_intro,[status(thm)],[86])).
% 0.21/0.37 tff(88,plain,
% 0.21/0.37 (![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)))),
% 0.21/0.37 inference(rewrite,[status(thm)],[])).
% 0.21/0.37 tff(89,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')).
% 0.21/0.37 tff(90,plain,
% 0.21/0.37 (![B: $i, C: $i, D: $i] : (member(D, intersection(B, C)) <=> (member(D, B) & member(D, C)))),
% 0.21/0.37 inference(modus_ponens,[status(thm)],[89, 88])).
% 0.21/0.37 tff(91,plain,(
% 0.21/0.37 ![B: $i, C: $i, D: $i] : (member(D, intersection(B, C)) <=> (member(D, B) & member(D, C)))),
% 0.21/0.37 inference(skolemize,[status(sab)],[90])).
% 0.21/0.37 tff(92,plain,
% 0.21/0.37 (![B: $i, C: $i, D: $i] : (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C))))),
% 0.21/0.37 inference(modus_ponens,[status(thm)],[91, 87])).
% 0.21/0.37 tff(93,plain,
% 0.21/0.37 (![B: $i, C: $i, D: $i] : (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C))))),
% 0.21/0.37 inference(modus_ponens,[status(thm)],[92, 85])).
% 0.21/0.37 tff(94,plain,
% 0.21/0.37 (((~![B: $i, C: $i, D: $i] : (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C))))) | (~(((~member(tptp_fun_D_0(C!4, B!5), B!5)) | (~member(tptp_fun_D_0(C!4, B!5), C!4))) <=> member(tptp_fun_D_0(C!4, B!5), intersection(B!5, C!4))))) <=> ((~![B: $i, C: $i, D: $i] : (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C))))) | (~(((~member(tptp_fun_D_0(C!4, B!5), B!5)) | (~member(tptp_fun_D_0(C!4, B!5), C!4))) <=> member(tptp_fun_D_0(C!4, B!5), intersection(B!5, C!4)))))),
% 0.21/0.37 inference(rewrite,[status(thm)],[])).
% 0.21/0.37 tff(95,plain,
% 0.21/0.37 ((~(((~member(tptp_fun_D_0(C!4, B!5), B!5)) | (~member(tptp_fun_D_0(C!4, B!5), C!4))) <=> member(tptp_fun_D_0(C!4, B!5), intersection(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))) <=> member(tptp_fun_D_0(C!4, B!5), intersection(B!5, C!4))))),
% 0.21/0.37 inference(rewrite,[status(thm)],[])).
% 0.21/0.37 tff(96,plain,
% 0.21/0.37 ((((~member(tptp_fun_D_0(C!4, B!5), C!4)) | (~member(tptp_fun_D_0(C!4, B!5), B!5))) <=> member(tptp_fun_D_0(C!4, B!5), intersection(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))) <=> member(tptp_fun_D_0(C!4, B!5), intersection(B!5, C!4)))),
% 0.21/0.37 inference(rewrite,[status(thm)],[])).
% 0.21/0.37 tff(97,plain,
% 0.21/0.37 ((~(((~member(tptp_fun_D_0(C!4, B!5), C!4)) | (~member(tptp_fun_D_0(C!4, B!5), B!5))) <=> member(tptp_fun_D_0(C!4, B!5), intersection(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))) <=> member(tptp_fun_D_0(C!4, B!5), intersection(B!5, C!4))))),
% 0.21/0.37 inference(monotonicity,[status(thm)],[96])).
% 0.21/0.37 tff(98,plain,
% 0.21/0.37 ((~(((~member(tptp_fun_D_0(C!4, B!5), C!4)) | (~member(tptp_fun_D_0(C!4, B!5), B!5))) <=> member(tptp_fun_D_0(C!4, B!5), intersection(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))) <=> member(tptp_fun_D_0(C!4, B!5), intersection(B!5, C!4))))),
% 0.21/0.37 inference(transitivity,[status(thm)],[97, 95])).
% 0.21/0.37 tff(99,plain,
% 0.21/0.37 (((~![B: $i, C: $i, D: $i] : (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C))))) | (~(((~member(tptp_fun_D_0(C!4, B!5), C!4)) | (~member(tptp_fun_D_0(C!4, B!5), B!5))) <=> member(tptp_fun_D_0(C!4, B!5), intersection(B!5, C!4))))) <=> ((~![B: $i, C: $i, D: $i] : (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C))))) | (~(((~member(tptp_fun_D_0(C!4, B!5), B!5)) | (~member(tptp_fun_D_0(C!4, B!5), C!4))) <=> member(tptp_fun_D_0(C!4, B!5), intersection(B!5, C!4)))))),
% 0.21/0.37 inference(monotonicity,[status(thm)],[98])).
% 0.21/0.37 tff(100,plain,
% 0.21/0.37 (((~![B: $i, C: $i, D: $i] : (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C))))) | (~(((~member(tptp_fun_D_0(C!4, B!5), C!4)) | (~member(tptp_fun_D_0(C!4, B!5), B!5))) <=> member(tptp_fun_D_0(C!4, B!5), intersection(B!5, C!4))))) <=> ((~![B: $i, C: $i, D: $i] : (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C))))) | (~(((~member(tptp_fun_D_0(C!4, B!5), B!5)) | (~member(tptp_fun_D_0(C!4, B!5), C!4))) <=> member(tptp_fun_D_0(C!4, B!5), intersection(B!5, C!4)))))),
% 0.21/0.38 inference(transitivity,[status(thm)],[99, 94])).
% 0.21/0.38 tff(101,plain,
% 0.21/0.38 ((~![B: $i, C: $i, D: $i] : (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C))))) | (~(((~member(tptp_fun_D_0(C!4, B!5), C!4)) | (~member(tptp_fun_D_0(C!4, B!5), B!5))) <=> member(tptp_fun_D_0(C!4, B!5), intersection(B!5, C!4))))),
% 0.21/0.38 inference(quant_inst,[status(thm)],[])).
% 0.21/0.38 tff(102,plain,
% 0.21/0.38 ((~![B: $i, C: $i, D: $i] : (~(((~member(D, C)) | (~member(D, B))) <=> member(D, intersection(B, C))))) | (~(((~member(tptp_fun_D_0(C!4, B!5), B!5)) | (~member(tptp_fun_D_0(C!4, B!5), C!4))) <=> member(tptp_fun_D_0(C!4, B!5), intersection(B!5, C!4))))),
% 0.21/0.38 inference(modus_ponens,[status(thm)],[101, 100])).
% 0.21/0.38 tff(103,plain,
% 0.21/0.38 ($false),
% 0.21/0.38 inference(unit_resolution,[status(thm)],[102, 93, 83])).
% 0.21/0.38 % SZS output end Proof
% 0.21/0.38 % E exiting
%------------------------------------------------------------------------------