%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET798+4 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n012.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 05:26:22 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
% 0.12/0.13 % Problem : SET798+4 : TPTP v9.0.0. Released v3.2.0.
% 0.12/0.13 % Command : run_E %s %d THM
% 0.13/0.35 % Computer : n012.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.20/0.35 % CPULimit : 300
% 0.20/0.35 % WCLimit : 300
% 0.20/0.35 % DateTime : Fri Jun 20 05:42:40 EDT 2025
% 0.20/0.35 % 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(tptp_fun_Y_19_type, type, (
% 0.20/0.40 tptp_fun_Y_19: $i)).
% 0.20/0.40 tff(tptp_fun_X_12_type, type, (
% 0.20/0.40 tptp_fun_X_12: ( $i * $i * $i ) > $i)).
% 0.20/0.40 tff(tptp_fun_R_18_type, type, (
% 0.20/0.40 tptp_fun_R_18: $i)).
% 0.20/0.40 tff(tptp_fun_X_20_type, type, (
% 0.20/0.40 tptp_fun_X_20: $i)).
% 0.20/0.40 tff(tptp_fun_M_21_type, type, (
% 0.20/0.40 tptp_fun_M_21: $i)).
% 0.20/0.40 tff(apply_type, type, (
% 0.20/0.40 apply: ( $i * $i * $i ) > $o)).
% 0.20/0.40 tff(lower_bound_type, type, (
% 0.20/0.40 lower_bound: ( $i * $i * $i ) > $o)).
% 0.20/0.40 tff(subset_type, type, (
% 0.20/0.40 subset: ( $i * $i ) > $o)).
% 0.20/0.40 tff(tptp_fun_E_17_type, type, (
% 0.20/0.40 tptp_fun_E_17: $i)).
% 0.20/0.40 tff(order_type, type, (
% 0.20/0.40 order: ( $i * $i ) > $o)).
% 0.20/0.40 tff(tptp_fun_X_0_type, type, (
% 0.20/0.40 tptp_fun_X_0: ( $i * $i ) > $i)).
% 0.20/0.40 tff(1,plain,
% 0.20/0.40 (^[R: $i, E: $i, M: $i] : refl((~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))))) <=> (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(2,plain,
% 0.20/0.40 (![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))))) <=> ![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[1])).
% 0.20/0.40 tff(3,plain,
% 0.20/0.40 (^[R: $i, E: $i, M: $i] : rewrite((~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))))) <=> (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(4,plain,
% 0.20/0.40 (![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))))) <=> ![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[3])).
% 0.20/0.40 tff(5,plain,
% 0.20/0.40 (![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))))) <=> ![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))),
% 0.20/0.40 inference(transitivity,[status(thm)],[4, 2])).
% 0.20/0.40 tff(6,plain,
% 0.20/0.40 (^[R: $i, E: $i, M: $i] : trans(monotonicity(rewrite(((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X))) <=> ((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))), rewrite((lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))) <=> (lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))), ((((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X))) & (lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))) <=> (((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X))) & (lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))))), rewrite((((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X))) & (lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))) <=> (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))), ((((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X))) & (lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))) <=> (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(7,plain,
% 0.20/0.40 (![R: $i, E: $i, M: $i] : (((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X))) & (lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))) <=> ![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[6])).
% 0.20/0.40 tff(8,plain,
% 0.20/0.40 (![R: $i, E: $i, M: $i] : (lower_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, M, X))) <=> ![R: $i, E: $i, M: $i] : (lower_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, M, X)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(9,plain,
% 0.20/0.40 (^[R: $i, E: $i, M: $i] : rewrite((lower_bound(M, R, E) <=> ![X: $i] : (member(X, E) => apply(R, M, X))) <=> (lower_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, M, X))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(10,plain,
% 0.20/0.40 (![R: $i, E: $i, M: $i] : (lower_bound(M, R, E) <=> ![X: $i] : (member(X, E) => apply(R, M, X))) <=> ![R: $i, E: $i, M: $i] : (lower_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, M, X)))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[9])).
% 0.20/0.40 tff(11,axiom,(![R: $i, E: $i, M: $i] : (lower_bound(M, R, E) <=> ![X: $i] : (member(X, E) => apply(R, M, X)))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+3.ax','lower_bound')).
% 0.20/0.40 tff(12,plain,
% 0.20/0.40 (![R: $i, E: $i, M: $i] : (lower_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, M, X)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[11, 10])).
% 0.20/0.40 tff(13,plain,
% 0.20/0.40 (![R: $i, E: $i, M: $i] : (lower_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, M, X)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[12, 8])).
% 0.20/0.40 tff(14,plain,(
% 0.20/0.40 ![R: $i, E: $i, M: $i] : (((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X))) & (lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))),
% 0.20/0.40 inference(skolemize,[status(sab)],[13])).
% 0.20/0.40 tff(15,plain,
% 0.20/0.40 (![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[14, 7])).
% 0.20/0.40 tff(16,plain,
% 0.20/0.40 (![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[15, 5])).
% 0.20/0.40 tff(17,plain,
% 0.20/0.40 ((~![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))) | (~((~((~lower_bound(M!21, R!18, X!20)) | ![X: $i] : ((~member(X, X!20)) | apply(R!18, M!21, X)))) | (~(lower_bound(M!21, R!18, X!20) | (~((~member(tptp_fun_X_12(M!21, X!20, R!18), X!20)) | apply(R!18, M!21, tptp_fun_X_12(M!21, X!20, R!18))))))))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(18,plain,
% 0.20/0.40 (~((~((~lower_bound(M!21, R!18, X!20)) | ![X: $i] : ((~member(X, X!20)) | apply(R!18, M!21, X)))) | (~(lower_bound(M!21, R!18, X!20) | (~((~member(tptp_fun_X_12(M!21, X!20, R!18), X!20)) | apply(R!18, M!21, tptp_fun_X_12(M!21, X!20, R!18)))))))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[17, 16])).
% 0.20/0.40 tff(19,plain,
% 0.20/0.40 (((~((~lower_bound(M!21, R!18, X!20)) | ![X: $i] : ((~member(X, X!20)) | apply(R!18, M!21, X)))) | (~(lower_bound(M!21, R!18, X!20) | (~((~member(tptp_fun_X_12(M!21, X!20, R!18), X!20)) | apply(R!18, M!21, tptp_fun_X_12(M!21, X!20, R!18))))))) | (lower_bound(M!21, R!18, X!20) | (~((~member(tptp_fun_X_12(M!21, X!20, R!18), X!20)) | apply(R!18, M!21, tptp_fun_X_12(M!21, X!20, R!18)))))),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(20,plain,
% 0.20/0.40 (lower_bound(M!21, R!18, X!20) | (~((~member(tptp_fun_X_12(M!21, X!20, R!18), X!20)) | apply(R!18, M!21, tptp_fun_X_12(M!21, X!20, R!18))))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[19, 18])).
% 0.20/0.40 tff(21,plain,
% 0.20/0.40 ((order(R!18, E!17) & (subset(X!20, E!17) & subset(Y!19, E!17) & subset(X!20, Y!19) & (~((~lower_bound(M!21, R!18, Y!19)) | lower_bound(M!21, R!18, X!20))))) <=> (order(R!18, E!17) & subset(X!20, E!17) & subset(Y!19, E!17) & subset(X!20, Y!19) & (~((~lower_bound(M!21, R!18, Y!19)) | lower_bound(M!21, R!18, X!20))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(22,plain,
% 0.20/0.40 (((subset(X!20, E!17) & subset(Y!19, E!17) & subset(X!20, Y!19)) & (~((~lower_bound(M!21, R!18, Y!19)) | lower_bound(M!21, R!18, X!20)))) <=> (subset(X!20, E!17) & subset(Y!19, E!17) & subset(X!20, Y!19) & (~((~lower_bound(M!21, R!18, Y!19)) | lower_bound(M!21, R!18, X!20))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(23,plain,
% 0.20/0.40 ((order(R!18, E!17) & ((subset(X!20, E!17) & subset(Y!19, E!17) & subset(X!20, Y!19)) & (~((~lower_bound(M!21, R!18, Y!19)) | lower_bound(M!21, R!18, X!20))))) <=> (order(R!18, E!17) & (subset(X!20, E!17) & subset(Y!19, E!17) & subset(X!20, Y!19) & (~((~lower_bound(M!21, R!18, Y!19)) | lower_bound(M!21, R!18, X!20)))))),
% 0.20/0.40 inference(monotonicity,[status(thm)],[22])).
% 0.20/0.40 tff(24,plain,
% 0.20/0.40 ((order(R!18, E!17) & ((subset(X!20, E!17) & subset(Y!19, E!17) & subset(X!20, Y!19)) & (~((~lower_bound(M!21, R!18, Y!19)) | lower_bound(M!21, R!18, X!20))))) <=> (order(R!18, E!17) & subset(X!20, E!17) & subset(Y!19, E!17) & subset(X!20, Y!19) & (~((~lower_bound(M!21, R!18, Y!19)) | lower_bound(M!21, R!18, X!20))))),
% 0.20/0.40 inference(transitivity,[status(thm)],[23, 21])).
% 0.20/0.40 tff(25,plain,
% 0.20/0.40 ((~![R: $i, E: $i] : ((~order(R, E)) | ![X: $i, Y: $i] : ((~(subset(X, E) & subset(Y, E) & subset(X, Y))) | ![M: $i] : ((~lower_bound(M, R, Y)) | lower_bound(M, R, X))))) <=> (~![R: $i, E: $i] : ((~order(R, E)) | ![X: $i, Y: $i] : ((~(subset(X, E) & subset(Y, E) & subset(X, Y))) | ![M: $i] : ((~lower_bound(M, R, Y)) | lower_bound(M, R, X)))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(26,plain,
% 0.20/0.40 ((~![R: $i, E: $i] : (order(R, E) => ![X: $i, Y: $i] : (((subset(X, E) & subset(Y, E)) & subset(X, Y)) => ![M: $i] : (lower_bound(M, R, Y) => lower_bound(M, R, X))))) <=> (~![R: $i, E: $i] : ((~order(R, E)) | ![X: $i, Y: $i] : ((~(subset(X, E) & subset(Y, E) & subset(X, Y))) | ![M: $i] : ((~lower_bound(M, R, Y)) | lower_bound(M, R, X)))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(27,axiom,(~![R: $i, E: $i] : (order(R, E) => ![X: $i, Y: $i] : (((subset(X, E) & subset(Y, E)) & subset(X, Y)) => ![M: $i] : (lower_bound(M, R, Y) => lower_bound(M, R, X))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','thIV10')).
% 0.20/0.40 tff(28,plain,
% 0.20/0.40 (~![R: $i, E: $i] : ((~order(R, E)) | ![X: $i, Y: $i] : ((~(subset(X, E) & subset(Y, E) & subset(X, Y))) | ![M: $i] : ((~lower_bound(M, R, Y)) | lower_bound(M, R, X))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[27, 26])).
% 0.20/0.40 tff(29,plain,
% 0.20/0.40 (~![R: $i, E: $i] : ((~order(R, E)) | ![X: $i, Y: $i] : ((~(subset(X, E) & subset(Y, E) & subset(X, Y))) | ![M: $i] : ((~lower_bound(M, R, Y)) | lower_bound(M, R, X))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[28, 25])).
% 0.20/0.40 tff(30,plain,
% 0.20/0.40 (~![R: $i, E: $i] : ((~order(R, E)) | ![X: $i, Y: $i] : ((~(subset(X, E) & subset(Y, E) & subset(X, Y))) | ![M: $i] : ((~lower_bound(M, R, Y)) | lower_bound(M, R, X))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[29, 25])).
% 0.20/0.41 tff(31,plain,
% 0.20/0.41 (~![R: $i, E: $i] : ((~order(R, E)) | ![X: $i, Y: $i] : ((~(subset(X, E) & subset(Y, E) & subset(X, Y))) | ![M: $i] : ((~lower_bound(M, R, Y)) | lower_bound(M, R, X))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[30, 25])).
% 0.20/0.41 tff(32,plain,
% 0.20/0.41 (order(R!18, E!17) & subset(X!20, E!17) & subset(Y!19, E!17) & subset(X!20, Y!19) & (~((~lower_bound(M!21, R!18, Y!19)) | lower_bound(M!21, R!18, X!20)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[31, 24])).
% 0.20/0.41 tff(33,plain,
% 0.20/0.41 (~((~lower_bound(M!21, R!18, Y!19)) | lower_bound(M!21, R!18, X!20))),
% 0.20/0.41 inference(and_elim,[status(thm)],[32])).
% 0.20/0.41 tff(34,plain,
% 0.20/0.41 (~lower_bound(M!21, R!18, X!20)),
% 0.20/0.41 inference(or_elim,[status(thm)],[33])).
% 0.20/0.41 tff(35,plain,
% 0.20/0.41 ((~(lower_bound(M!21, R!18, X!20) | (~((~member(tptp_fun_X_12(M!21, X!20, R!18), X!20)) | apply(R!18, M!21, tptp_fun_X_12(M!21, X!20, R!18)))))) | lower_bound(M!21, R!18, X!20) | (~((~member(tptp_fun_X_12(M!21, X!20, R!18), X!20)) | apply(R!18, M!21, tptp_fun_X_12(M!21, X!20, R!18))))),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(36,plain,
% 0.20/0.41 ((~(lower_bound(M!21, R!18, X!20) | (~((~member(tptp_fun_X_12(M!21, X!20, R!18), X!20)) | apply(R!18, M!21, tptp_fun_X_12(M!21, X!20, R!18)))))) | (~((~member(tptp_fun_X_12(M!21, X!20, R!18), X!20)) | apply(R!18, M!21, tptp_fun_X_12(M!21, X!20, R!18))))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[35, 34])).
% 0.20/0.41 tff(37,plain,
% 0.20/0.41 (~((~member(tptp_fun_X_12(M!21, X!20, R!18), X!20)) | apply(R!18, M!21, tptp_fun_X_12(M!21, X!20, R!18)))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[36, 20])).
% 0.20/0.41 tff(38,plain,
% 0.20/0.41 (((~member(tptp_fun_X_12(M!21, X!20, R!18), X!20)) | apply(R!18, M!21, tptp_fun_X_12(M!21, X!20, R!18))) | member(tptp_fun_X_12(M!21, X!20, R!18), X!20)),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(39,plain,
% 0.20/0.41 (member(tptp_fun_X_12(M!21, X!20, R!18), X!20)),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[38, 37])).
% 0.20/0.41 tff(40,plain,
% 0.20/0.41 (^[A: $i, B: $i] : refl((~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))) <=> (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(41,plain,
% 0.20/0.41 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[40])).
% 0.20/0.41 tff(42,plain,
% 0.20/0.41 (^[A: $i, B: $i] : rewrite((~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))) <=> (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(43,plain,
% 0.20/0.41 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[42])).
% 0.20/0.41 tff(44,plain,
% 0.20/0.41 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))),
% 0.20/0.41 inference(transitivity,[status(thm)],[43, 41])).
% 0.20/0.41 tff(45,plain,
% 0.20/0.41 (^[A: $i, B: $i] : trans(monotonicity(rewrite(((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) <=> ((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))), rewrite((subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))) <=> (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))), ((((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) & (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))) <=> (((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) & (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))), rewrite((((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) & (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))) <=> (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))), ((((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) & (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))) <=> (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(46,plain,
% 0.20/0.41 (![A: $i, B: $i] : (((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) & (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[45])).
% 0.20/0.41 tff(47,plain,
% 0.20/0.41 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B))) <=> ![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(48,plain,
% 0.20/0.41 (^[A: $i, B: $i] : rewrite((subset(A, B) <=> ![X: $i] : (member(X, A) => member(X, B))) <=> (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(49,plain,
% 0.20/0.41 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : (member(X, A) => member(X, B))) <=> ![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[48])).
% 0.20/0.41 tff(50,axiom,(![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : (member(X, A) => member(X, B)))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+0.ax','subset')).
% 0.20/0.41 tff(51,plain,
% 0.20/0.41 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[50, 49])).
% 0.20/0.41 tff(52,plain,
% 0.20/0.41 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[51, 47])).
% 0.20/0.41 tff(53,plain,(
% 0.20/0.41 ![A: $i, B: $i] : (((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) & (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))),
% 0.20/0.41 inference(skolemize,[status(sab)],[52])).
% 0.20/0.41 tff(54,plain,
% 0.20/0.41 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[53, 46])).
% 0.20/0.41 tff(55,plain,
% 0.20/0.41 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[54, 44])).
% 0.20/0.41 tff(56,plain,
% 0.20/0.41 ((~![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))) | (~((~((~subset(X!20, Y!19)) | ![X: $i] : ((~member(X, X!20)) | member(X, Y!19)))) | (~(subset(X!20, Y!19) | (~((~member(tptp_fun_X_0(Y!19, X!20), X!20)) | member(tptp_fun_X_0(Y!19, X!20), Y!19)))))))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(57,plain,
% 0.20/0.41 (~((~((~subset(X!20, Y!19)) | ![X: $i] : ((~member(X, X!20)) | member(X, Y!19)))) | (~(subset(X!20, Y!19) | (~((~member(tptp_fun_X_0(Y!19, X!20), X!20)) | member(tptp_fun_X_0(Y!19, X!20), Y!19))))))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[56, 55])).
% 0.20/0.41 tff(58,plain,
% 0.20/0.41 (((~((~subset(X!20, Y!19)) | ![X: $i] : ((~member(X, X!20)) | member(X, Y!19)))) | (~(subset(X!20, Y!19) | (~((~member(tptp_fun_X_0(Y!19, X!20), X!20)) | member(tptp_fun_X_0(Y!19, X!20), Y!19)))))) | ((~subset(X!20, Y!19)) | ![X: $i] : ((~member(X, X!20)) | member(X, Y!19)))),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(59,plain,
% 0.20/0.41 ((~subset(X!20, Y!19)) | ![X: $i] : ((~member(X, X!20)) | member(X, Y!19))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[58, 57])).
% 0.20/0.41 tff(60,plain,
% 0.20/0.41 (subset(X!20, Y!19)),
% 0.20/0.41 inference(and_elim,[status(thm)],[32])).
% 0.20/0.41 tff(61,plain,
% 0.20/0.41 ((~((~subset(X!20, Y!19)) | ![X: $i] : ((~member(X, X!20)) | member(X, Y!19)))) | (~subset(X!20, Y!19)) | ![X: $i] : ((~member(X, X!20)) | member(X, Y!19))),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(62,plain,
% 0.20/0.41 ((~((~subset(X!20, Y!19)) | ![X: $i] : ((~member(X, X!20)) | member(X, Y!19)))) | ![X: $i] : ((~member(X, X!20)) | member(X, Y!19))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[61, 60])).
% 0.20/0.41 tff(63,plain,
% 0.20/0.41 (![X: $i] : ((~member(X, X!20)) | member(X, Y!19))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[62, 59])).
% 0.20/0.41 tff(64,plain,
% 0.20/0.41 (((~![X: $i] : ((~member(X, X!20)) | member(X, Y!19))) | ((~member(tptp_fun_X_12(M!21, X!20, R!18), X!20)) | member(tptp_fun_X_12(M!21, X!20, R!18), Y!19))) <=> ((~![X: $i] : ((~member(X, X!20)) | member(X, Y!19))) | (~member(tptp_fun_X_12(M!21, X!20, R!18), X!20)) | member(tptp_fun_X_12(M!21, X!20, R!18), Y!19))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(65,plain,
% 0.20/0.41 ((~![X: $i] : ((~member(X, X!20)) | member(X, Y!19))) | ((~member(tptp_fun_X_12(M!21, X!20, R!18), X!20)) | member(tptp_fun_X_12(M!21, X!20, R!18), Y!19))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(66,plain,
% 0.20/0.41 ((~![X: $i] : ((~member(X, X!20)) | member(X, Y!19))) | (~member(tptp_fun_X_12(M!21, X!20, R!18), X!20)) | member(tptp_fun_X_12(M!21, X!20, R!18), Y!19)),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[65, 64])).
% 0.20/0.41 tff(67,plain,
% 0.20/0.41 (member(tptp_fun_X_12(M!21, X!20, R!18), Y!19)),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[66, 63, 39])).
% 0.20/0.41 tff(68,plain,
% 0.20/0.41 (((~member(tptp_fun_X_12(M!21, X!20, R!18), X!20)) | apply(R!18, M!21, tptp_fun_X_12(M!21, X!20, R!18))) | (~apply(R!18, M!21, tptp_fun_X_12(M!21, X!20, R!18)))),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(69,plain,
% 0.20/0.41 (~apply(R!18, M!21, tptp_fun_X_12(M!21, X!20, R!18))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[68, 37])).
% 0.20/0.41 tff(70,plain,
% 0.20/0.41 ((~![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))) | (~((~((~lower_bound(M!21, R!18, Y!19)) | ![X: $i] : ((~member(X, Y!19)) | apply(R!18, M!21, X)))) | (~(lower_bound(M!21, R!18, Y!19) | (~((~member(tptp_fun_X_12(M!21, Y!19, R!18), Y!19)) | apply(R!18, M!21, tptp_fun_X_12(M!21, Y!19, R!18))))))))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(71,plain,
% 0.20/0.41 (~((~((~lower_bound(M!21, R!18, Y!19)) | ![X: $i] : ((~member(X, Y!19)) | apply(R!18, M!21, X)))) | (~(lower_bound(M!21, R!18, Y!19) | (~((~member(tptp_fun_X_12(M!21, Y!19, R!18), Y!19)) | apply(R!18, M!21, tptp_fun_X_12(M!21, Y!19, R!18)))))))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[70, 16])).
% 0.20/0.41 tff(72,plain,
% 0.20/0.41 (((~((~lower_bound(M!21, R!18, Y!19)) | ![X: $i] : ((~member(X, Y!19)) | apply(R!18, M!21, X)))) | (~(lower_bound(M!21, R!18, Y!19) | (~((~member(tptp_fun_X_12(M!21, Y!19, R!18), Y!19)) | apply(R!18, M!21, tptp_fun_X_12(M!21, Y!19, R!18))))))) | ((~lower_bound(M!21, R!18, Y!19)) | ![X: $i] : ((~member(X, Y!19)) | apply(R!18, M!21, X)))),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(73,plain,
% 0.20/0.41 ((~lower_bound(M!21, R!18, Y!19)) | ![X: $i] : ((~member(X, Y!19)) | apply(R!18, M!21, X))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[72, 71])).
% 0.20/0.42 tff(74,plain,
% 0.20/0.42 (lower_bound(M!21, R!18, Y!19)),
% 0.20/0.42 inference(or_elim,[status(thm)],[33])).
% 0.20/0.42 tff(75,plain,
% 0.20/0.42 ((~((~lower_bound(M!21, R!18, Y!19)) | ![X: $i] : ((~member(X, Y!19)) | apply(R!18, M!21, X)))) | (~lower_bound(M!21, R!18, Y!19)) | ![X: $i] : ((~member(X, Y!19)) | apply(R!18, M!21, X))),
% 0.20/0.42 inference(tautology,[status(thm)],[])).
% 0.20/0.42 tff(76,plain,
% 0.20/0.42 ((~((~lower_bound(M!21, R!18, Y!19)) | ![X: $i] : ((~member(X, Y!19)) | apply(R!18, M!21, X)))) | ![X: $i] : ((~member(X, Y!19)) | apply(R!18, M!21, X))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[75, 74])).
% 0.20/0.42 tff(77,plain,
% 0.20/0.42 (![X: $i] : ((~member(X, Y!19)) | apply(R!18, M!21, X))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[76, 73])).
% 0.20/0.42 tff(78,plain,
% 0.20/0.42 (((~![X: $i] : ((~member(X, Y!19)) | apply(R!18, M!21, X))) | ((~member(tptp_fun_X_12(M!21, X!20, R!18), Y!19)) | apply(R!18, M!21, tptp_fun_X_12(M!21, X!20, R!18)))) <=> ((~![X: $i] : ((~member(X, Y!19)) | apply(R!18, M!21, X))) | (~member(tptp_fun_X_12(M!21, X!20, R!18), Y!19)) | apply(R!18, M!21, tptp_fun_X_12(M!21, X!20, R!18)))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(79,plain,
% 0.20/0.42 ((~![X: $i] : ((~member(X, Y!19)) | apply(R!18, M!21, X))) | ((~member(tptp_fun_X_12(M!21, X!20, R!18), Y!19)) | apply(R!18, M!21, tptp_fun_X_12(M!21, X!20, R!18)))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(80,plain,
% 0.20/0.42 ((~![X: $i] : ((~member(X, Y!19)) | apply(R!18, M!21, X))) | (~member(tptp_fun_X_12(M!21, X!20, R!18), Y!19)) | apply(R!18, M!21, tptp_fun_X_12(M!21, X!20, R!18))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[79, 78])).
% 0.20/0.42 tff(81,plain,
% 0.20/0.42 ($false),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[80, 77, 69, 67])).
% 0.20/0.42 % SZS output end Proof
% 0.20/0.42 % E exiting
%------------------------------------------------------------------------------