%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET800+4 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n005.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 0.19s 0.38s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SET800+4 : TPTP v9.0.0. Released v3.2.0.
% 0.03/0.12 % Command : run_E %s %d THM
% 0.11/0.33 % Computer : n005.cluster.edu
% 0.11/0.33 % Model : x86_64 x86_64
% 0.11/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.33 % Memory : 8042.1875MB
% 0.11/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.33 % CPULimit : 300
% 0.11/0.33 % WCLimit : 300
% 0.11/0.33 % DateTime : Fri Jun 20 05:43:20 EDT 2025
% 0.11/0.33 % CPUTime :
% 0.19/0.38 % SZS status Theorem
% 0.19/0.38 % SZS output start Proof
% 0.19/0.38 tff(apply_type, type, (
% 0.19/0.38 apply: ( $i * $i * $i ) > $o)).
% 0.19/0.38 tff(tptp_fun_M2_21_type, type, (
% 0.19/0.38 tptp_fun_M2_21: $i)).
% 0.19/0.38 tff(tptp_fun_R_18_type, type, (
% 0.19/0.38 tptp_fun_R_18: $i)).
% 0.19/0.38 tff(member_type, type, (
% 0.19/0.38 member: ( $i * $i ) > $o)).
% 0.19/0.38 tff(tptp_fun_X2_19_type, type, (
% 0.19/0.38 tptp_fun_X2_19: $i)).
% 0.19/0.38 tff(lower_bound_type, type, (
% 0.19/0.38 lower_bound: ( $i * $i * $i ) > $o)).
% 0.19/0.38 tff(tptp_fun_X_12_type, type, (
% 0.19/0.38 tptp_fun_X_12: ( $i * $i * $i ) > $i)).
% 0.19/0.38 tff(tptp_fun_E_17_type, type, (
% 0.19/0.38 tptp_fun_E_17: $i)).
% 0.19/0.38 tff(greatest_lower_bound_type, type, (
% 0.19/0.38 greatest_lower_bound: ( $i * $i * $i * $i ) > $o)).
% 0.19/0.38 tff(tptp_fun_M_16_type, type, (
% 0.19/0.38 tptp_fun_M_16: ( $i * $i * $i * $i ) > $i)).
% 0.19/0.38 tff(tptp_fun_X1_20_type, type, (
% 0.19/0.38 tptp_fun_X1_20: $i)).
% 0.19/0.38 tff(tptp_fun_M1_22_type, type, (
% 0.19/0.38 tptp_fun_M1_22: $i)).
% 0.19/0.38 tff(subset_type, type, (
% 0.19/0.38 subset: ( $i * $i ) > $o)).
% 0.19/0.38 tff(order_type, type, (
% 0.19/0.38 order: ( $i * $i ) > $o)).
% 0.19/0.38 tff(tptp_fun_X_0_type, type, (
% 0.19/0.38 tptp_fun_X_0: ( $i * $i ) > $i)).
% 0.19/0.38 tff(1,plain,
% 0.19/0.38 (^[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)))))))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(2,plain,
% 0.19/0.38 (![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))))))))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[1])).
% 0.19/0.38 tff(3,plain,
% 0.19/0.38 (^[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)))))))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(4,plain,
% 0.19/0.38 (![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))))))))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[3])).
% 0.19/0.38 tff(5,plain,
% 0.19/0.38 (![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))))))))),
% 0.19/0.38 inference(transitivity,[status(thm)],[4, 2])).
% 0.19/0.38 tff(6,plain,
% 0.19/0.38 (^[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))))))))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(7,plain,
% 0.19/0.38 (![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))))))))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[6])).
% 0.19/0.38 tff(8,plain,
% 0.19/0.38 (![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)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(9,plain,
% 0.19/0.38 (^[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))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(10,plain,
% 0.19/0.38 (![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)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[9])).
% 0.19/0.38 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/sandbox/benchmark/Axioms/SET006+3.ax','lower_bound')).
% 0.19/0.38 tff(12,plain,
% 0.19/0.38 (![R: $i, E: $i, M: $i] : (lower_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, M, X)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[11, 10])).
% 0.19/0.38 tff(13,plain,
% 0.19/0.38 (![R: $i, E: $i, M: $i] : (lower_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, M, X)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[12, 8])).
% 0.19/0.38 tff(14,plain,(
% 0.19/0.38 ![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))))))),
% 0.19/0.38 inference(skolemize,[status(sab)],[13])).
% 0.19/0.38 tff(15,plain,
% 0.19/0.38 (![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))))))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[14, 7])).
% 0.19/0.38 tff(16,plain,
% 0.19/0.38 (![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))))))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[15, 5])).
% 0.19/0.38 tff(17,plain,
% 0.19/0.38 ((~![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(M2!21, R!18, X2!19)) | ![X: $i] : ((~member(X, X2!19)) | apply(R!18, M2!21, X)))) | (~(lower_bound(M2!21, R!18, X2!19) | (~((~member(tptp_fun_X_12(M2!21, X2!19, R!18), X2!19)) | apply(R!18, M2!21, tptp_fun_X_12(M2!21, X2!19, R!18))))))))),
% 0.19/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.38 tff(18,plain,
% 0.19/0.38 (~((~((~lower_bound(M2!21, R!18, X2!19)) | ![X: $i] : ((~member(X, X2!19)) | apply(R!18, M2!21, X)))) | (~(lower_bound(M2!21, R!18, X2!19) | (~((~member(tptp_fun_X_12(M2!21, X2!19, R!18), X2!19)) | apply(R!18, M2!21, tptp_fun_X_12(M2!21, X2!19, R!18)))))))),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[17, 16])).
% 0.19/0.38 tff(19,plain,
% 0.19/0.38 (((~((~lower_bound(M2!21, R!18, X2!19)) | ![X: $i] : ((~member(X, X2!19)) | apply(R!18, M2!21, X)))) | (~(lower_bound(M2!21, R!18, X2!19) | (~((~member(tptp_fun_X_12(M2!21, X2!19, R!18), X2!19)) | apply(R!18, M2!21, tptp_fun_X_12(M2!21, X2!19, R!18))))))) | ((~lower_bound(M2!21, R!18, X2!19)) | ![X: $i] : ((~member(X, X2!19)) | apply(R!18, M2!21, X)))),
% 0.19/0.38 inference(tautology,[status(thm)],[])).
% 0.19/0.38 tff(20,plain,
% 0.19/0.38 ((~lower_bound(M2!21, R!18, X2!19)) | ![X: $i] : ((~member(X, X2!19)) | apply(R!18, M2!21, X))),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[19, 18])).
% 0.19/0.38 tff(21,plain,
% 0.19/0.38 (^[A: $i, X: $i, R: $i, E: $i] : rewrite((~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X)))))))) | (~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))))) <=> (~((~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))) | (~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X)))))))))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(22,plain,
% 0.19/0.38 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X)))))))) | (~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))))) <=> ![A: $i, X: $i, R: $i, E: $i] : (~((~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))) | (~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X))))))))))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[21])).
% 0.19/0.38 tff(23,plain,
% 0.19/0.38 (^[A: $i, X: $i, R: $i, E: $i] : refl((~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X)))))))) | (~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))))) <=> (~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X)))))))) | (~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(24,plain,
% 0.19/0.38 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X)))))))) | (~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))))) <=> ![A: $i, X: $i, R: $i, E: $i] : (~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X)))))))) | (~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X))))))))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[23])).
% 0.19/0.38 tff(25,plain,
% 0.19/0.38 (^[A: $i, X: $i, R: $i, E: $i] : rewrite((~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X)))))))) | (~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))))) <=> (~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X)))))))) | (~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(26,plain,
% 0.19/0.38 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X)))))))) | (~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))))) <=> ![A: $i, X: $i, R: $i, E: $i] : (~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X)))))))) | (~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X))))))))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[25])).
% 0.19/0.38 tff(27,plain,
% 0.19/0.38 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X)))))))) | (~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))))) <=> ![A: $i, X: $i, R: $i, E: $i] : (~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X)))))))) | (~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X))))))))),
% 0.19/0.38 inference(transitivity,[status(thm)],[26, 24])).
% 0.19/0.38 tff(28,plain,
% 0.19/0.38 (^[A: $i, X: $i, R: $i, E: $i] : trans(monotonicity(rewrite(((~greatest_lower_bound(A, X, R, E)) | (member(A, X) & lower_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & lower_bound(M, R, X))) | apply(R, M, A)))) <=> ((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X)))))))), trans(monotonicity(rewrite((~((~(member(tptp_fun_M_16(E, R, X, A), E) & lower_bound(tptp_fun_M_16(E, R, X, A), R, X))) | apply(R, tptp_fun_M_16(E, R, X, A), A))) <=> (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X))))), ((greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~((~(member(tptp_fun_M_16(E, R, X, A), E) & lower_bound(tptp_fun_M_16(E, R, X, A), R, X))) | apply(R, tptp_fun_M_16(E, R, X, A), A)))) <=> (greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X))))))), rewrite((greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X))))) <=> (greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))), ((greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~((~(member(tptp_fun_M_16(E, R, X, A), E) & lower_bound(tptp_fun_M_16(E, R, X, A), R, X))) | apply(R, tptp_fun_M_16(E, R, X, A), A)))) <=> (greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X))))))), ((((~greatest_lower_bound(A, X, R, E)) | (member(A, X) & lower_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & lower_bound(M, R, X))) | apply(R, M, A)))) & (greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~((~(member(tptp_fun_M_16(E, R, X, A), E) & lower_bound(tptp_fun_M_16(E, R, X, A), R, X))) | apply(R, tptp_fun_M_16(E, R, X, A), A))))) <=> (((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X))))))) & (greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))))), rewrite((((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X))))))) & (greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))) <=> (~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X)))))))) | (~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X))))))))), ((((~greatest_lower_bound(A, X, R, E)) | (member(A, X) & lower_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & lower_bound(M, R, X))) | apply(R, M, A)))) & (greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~((~(member(tptp_fun_M_16(E, R, X, A), E) & lower_bound(tptp_fun_M_16(E, R, X, A), R, X))) | apply(R, tptp_fun_M_16(E, R, X, A), A))))) <=> (~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X)))))))) | (~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X))))))))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(29,plain,
% 0.19/0.39 (![A: $i, X: $i, R: $i, E: $i] : (((~greatest_lower_bound(A, X, R, E)) | (member(A, X) & lower_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & lower_bound(M, R, X))) | apply(R, M, A)))) & (greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~((~(member(tptp_fun_M_16(E, R, X, A), E) & lower_bound(tptp_fun_M_16(E, R, X, A), R, X))) | apply(R, tptp_fun_M_16(E, R, X, A), A))))) <=> ![A: $i, X: $i, R: $i, E: $i] : (~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X)))))))) | (~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X))))))))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[28])).
% 0.19/0.39 tff(30,plain,
% 0.19/0.39 (^[A: $i, X: $i, R: $i, E: $i] : rewrite((((~greatest_lower_bound(A, X, R, E)) | (member(A, X) & lower_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & lower_bound(M, R, X))) | apply(R, M, A)))) & (greatest_lower_bound(A, X, R, E) | ((~member(A, X)) | (~lower_bound(A, R, X)) | (~((~(member(tptp_fun_M_16(E, R, X, A), E) & lower_bound(tptp_fun_M_16(E, R, X, A), R, X))) | apply(R, tptp_fun_M_16(E, R, X, A), A)))))) <=> (((~greatest_lower_bound(A, X, R, E)) | (member(A, X) & lower_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & lower_bound(M, R, X))) | apply(R, M, A)))) & (greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~((~(member(tptp_fun_M_16(E, R, X, A), E) & lower_bound(tptp_fun_M_16(E, R, X, A), R, X))) | apply(R, tptp_fun_M_16(E, R, X, A), A))))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(31,plain,
% 0.19/0.39 (![A: $i, X: $i, R: $i, E: $i] : (((~greatest_lower_bound(A, X, R, E)) | (member(A, X) & lower_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & lower_bound(M, R, X))) | apply(R, M, A)))) & (greatest_lower_bound(A, X, R, E) | ((~member(A, X)) | (~lower_bound(A, R, X)) | (~((~(member(tptp_fun_M_16(E, R, X, A), E) & lower_bound(tptp_fun_M_16(E, R, X, A), R, X))) | apply(R, tptp_fun_M_16(E, R, X, A), A)))))) <=> ![A: $i, X: $i, R: $i, E: $i] : (((~greatest_lower_bound(A, X, R, E)) | (member(A, X) & lower_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & lower_bound(M, R, X))) | apply(R, M, A)))) & (greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~((~(member(tptp_fun_M_16(E, R, X, A), E) & lower_bound(tptp_fun_M_16(E, R, X, A), R, X))) | apply(R, tptp_fun_M_16(E, R, X, A), A)))))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[30])).
% 0.19/0.39 tff(32,plain,
% 0.19/0.39 (![A: $i, X: $i, R: $i, E: $i] : (greatest_lower_bound(A, X, R, E) <=> (member(A, X) & lower_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & lower_bound(M, R, X))) | apply(R, M, A)))) <=> ![A: $i, X: $i, R: $i, E: $i] : (greatest_lower_bound(A, X, R, E) <=> (member(A, X) & lower_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & lower_bound(M, R, X))) | apply(R, M, A))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(33,plain,
% 0.19/0.39 (^[A: $i, X: $i, R: $i, E: $i] : rewrite((greatest_lower_bound(A, X, R, E) <=> ((member(A, X) & lower_bound(A, R, X)) & ![M: $i] : ((member(M, E) & lower_bound(M, R, X)) => apply(R, M, A)))) <=> (greatest_lower_bound(A, X, R, E) <=> (member(A, X) & lower_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & lower_bound(M, R, X))) | apply(R, M, A)))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(34,plain,
% 0.19/0.39 (![A: $i, X: $i, R: $i, E: $i] : (greatest_lower_bound(A, X, R, E) <=> ((member(A, X) & lower_bound(A, R, X)) & ![M: $i] : ((member(M, E) & lower_bound(M, R, X)) => apply(R, M, A)))) <=> ![A: $i, X: $i, R: $i, E: $i] : (greatest_lower_bound(A, X, R, E) <=> (member(A, X) & lower_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & lower_bound(M, R, X))) | apply(R, M, A))))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[33])).
% 0.19/0.39 tff(35,axiom,(![A: $i, X: $i, R: $i, E: $i] : (greatest_lower_bound(A, X, R, E) <=> ((member(A, X) & lower_bound(A, R, X)) & ![M: $i] : ((member(M, E) & lower_bound(M, R, X)) => apply(R, M, A))))), file('/export/starexec/sandbox/benchmark/Axioms/SET006+3.ax','greatest_lower_bound')).
% 0.19/0.39 tff(36,plain,
% 0.19/0.39 (![A: $i, X: $i, R: $i, E: $i] : (greatest_lower_bound(A, X, R, E) <=> (member(A, X) & lower_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & lower_bound(M, R, X))) | apply(R, M, A))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.19/0.39 tff(37,plain,
% 0.19/0.39 (![A: $i, X: $i, R: $i, E: $i] : (greatest_lower_bound(A, X, R, E) <=> (member(A, X) & lower_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & lower_bound(M, R, X))) | apply(R, M, A))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[36, 32])).
% 0.19/0.39 tff(38,plain,(
% 0.19/0.39 ![A: $i, X: $i, R: $i, E: $i] : (((~greatest_lower_bound(A, X, R, E)) | (member(A, X) & lower_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & lower_bound(M, R, X))) | apply(R, M, A)))) & (greatest_lower_bound(A, X, R, E) | ((~member(A, X)) | (~lower_bound(A, R, X)) | (~((~(member(tptp_fun_M_16(E, R, X, A), E) & lower_bound(tptp_fun_M_16(E, R, X, A), R, X))) | apply(R, tptp_fun_M_16(E, R, X, A), A))))))),
% 0.19/0.39 inference(skolemize,[status(sab)],[37])).
% 0.19/0.39 tff(39,plain,
% 0.19/0.39 (![A: $i, X: $i, R: $i, E: $i] : (((~greatest_lower_bound(A, X, R, E)) | (member(A, X) & lower_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & lower_bound(M, R, X))) | apply(R, M, A)))) & (greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~((~(member(tptp_fun_M_16(E, R, X, A), E) & lower_bound(tptp_fun_M_16(E, R, X, A), R, X))) | apply(R, tptp_fun_M_16(E, R, X, A), A)))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[38, 31])).
% 0.19/0.39 tff(40,plain,
% 0.19/0.39 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X)))))))) | (~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X))))))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[39, 29])).
% 0.19/0.39 tff(41,plain,
% 0.19/0.39 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X)))))))) | (~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X))))))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[40, 27])).
% 0.19/0.39 tff(42,plain,
% 0.19/0.39 (![A: $i, X: $i, R: $i, E: $i] : (~((~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))) | (~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X))))))))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[41, 22])).
% 0.19/0.39 tff(43,plain,
% 0.19/0.39 (((~![A: $i, X: $i, R: $i, E: $i] : (~((~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))) | (~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X))))))))))) | (~((~(greatest_lower_bound(M2!21, X2!19, R!18, E!17) | (~lower_bound(M2!21, R!18, X2!19)) | (~member(M2!21, X2!19)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X2!19, M2!21), R!18, X2!19)) | (~member(tptp_fun_M_16(E!17, R!18, X2!19, M2!21), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X2!19, M2!21), M2!21))))) | (~((~greatest_lower_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~lower_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M2!21) | (~lower_bound(M, R!18, X2!19))))))))))) <=> ((~![A: $i, X: $i, R: $i, E: $i] : (~((~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))) | (~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X))))))))))) | (~((~(greatest_lower_bound(M2!21, X2!19, R!18, E!17) | (~lower_bound(M2!21, R!18, X2!19)) | (~member(M2!21, X2!19)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X2!19, M2!21), R!18, X2!19)) | (~member(tptp_fun_M_16(E!17, R!18, X2!19, M2!21), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X2!19, M2!21), M2!21))))) | (~((~greatest_lower_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~lower_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M2!21) | (~lower_bound(M, R!18, X2!19)))))))))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(44,plain,
% 0.19/0.39 ((~((~(greatest_lower_bound(M2!21, X2!19, R!18, E!17) | (~member(M2!21, X2!19)) | (~lower_bound(M2!21, R!18, X2!19)) | (~(apply(R!18, tptp_fun_M_16(E!17, R!18, X2!19, M2!21), M2!21) | (~member(tptp_fun_M_16(E!17, R!18, X2!19, M2!21), E!17)) | (~lower_bound(tptp_fun_M_16(E!17, R!18, X2!19, M2!21), R!18, X2!19)))))) | (~((~greatest_lower_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~lower_bound(M2!21, R!18, X2!19)) | (~![M: $i] : (apply(R!18, M, M2!21) | (~member(M, E!17)) | (~lower_bound(M, R!18, X2!19)))))))))) <=> (~((~(greatest_lower_bound(M2!21, X2!19, R!18, E!17) | (~lower_bound(M2!21, R!18, X2!19)) | (~member(M2!21, X2!19)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X2!19, M2!21), R!18, X2!19)) | (~member(tptp_fun_M_16(E!17, R!18, X2!19, M2!21), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X2!19, M2!21), M2!21))))) | (~((~greatest_lower_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~lower_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M2!21) | (~lower_bound(M, R!18, X2!19))))))))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(45,plain,
% 0.19/0.39 (((~![A: $i, X: $i, R: $i, E: $i] : (~((~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))) | (~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X))))))))))) | (~((~(greatest_lower_bound(M2!21, X2!19, R!18, E!17) | (~member(M2!21, X2!19)) | (~lower_bound(M2!21, R!18, X2!19)) | (~(apply(R!18, tptp_fun_M_16(E!17, R!18, X2!19, M2!21), M2!21) | (~member(tptp_fun_M_16(E!17, R!18, X2!19, M2!21), E!17)) | (~lower_bound(tptp_fun_M_16(E!17, R!18, X2!19, M2!21), R!18, X2!19)))))) | (~((~greatest_lower_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~lower_bound(M2!21, R!18, X2!19)) | (~![M: $i] : (apply(R!18, M, M2!21) | (~member(M, E!17)) | (~lower_bound(M, R!18, X2!19))))))))))) <=> ((~![A: $i, X: $i, R: $i, E: $i] : (~((~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))) | (~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X))))))))))) | (~((~(greatest_lower_bound(M2!21, X2!19, R!18, E!17) | (~lower_bound(M2!21, R!18, X2!19)) | (~member(M2!21, X2!19)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X2!19, M2!21), R!18, X2!19)) | (~member(tptp_fun_M_16(E!17, R!18, X2!19, M2!21), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X2!19, M2!21), M2!21))))) | (~((~greatest_lower_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~lower_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M2!21) | (~lower_bound(M, R!18, X2!19)))))))))))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[44])).
% 0.19/0.39 tff(46,plain,
% 0.19/0.39 (((~![A: $i, X: $i, R: $i, E: $i] : (~((~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))) | (~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X))))))))))) | (~((~(greatest_lower_bound(M2!21, X2!19, R!18, E!17) | (~member(M2!21, X2!19)) | (~lower_bound(M2!21, R!18, X2!19)) | (~(apply(R!18, tptp_fun_M_16(E!17, R!18, X2!19, M2!21), M2!21) | (~member(tptp_fun_M_16(E!17, R!18, X2!19, M2!21), E!17)) | (~lower_bound(tptp_fun_M_16(E!17, R!18, X2!19, M2!21), R!18, X2!19)))))) | (~((~greatest_lower_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~lower_bound(M2!21, R!18, X2!19)) | (~![M: $i] : (apply(R!18, M, M2!21) | (~member(M, E!17)) | (~lower_bound(M, R!18, X2!19))))))))))) <=> ((~![A: $i, X: $i, R: $i, E: $i] : (~((~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))) | (~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X))))))))))) | (~((~(greatest_lower_bound(M2!21, X2!19, R!18, E!17) | (~lower_bound(M2!21, R!18, X2!19)) | (~member(M2!21, X2!19)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X2!19, M2!21), R!18, X2!19)) | (~member(tptp_fun_M_16(E!17, R!18, X2!19, M2!21), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X2!19, M2!21), M2!21))))) | (~((~greatest_lower_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~lower_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M2!21) | (~lower_bound(M, R!18, X2!19)))))))))))),
% 0.19/0.40 inference(transitivity,[status(thm)],[45, 43])).
% 0.19/0.40 tff(47,plain,
% 0.19/0.40 ((~![A: $i, X: $i, R: $i, E: $i] : (~((~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))) | (~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X))))))))))) | (~((~(greatest_lower_bound(M2!21, X2!19, R!18, E!17) | (~member(M2!21, X2!19)) | (~lower_bound(M2!21, R!18, X2!19)) | (~(apply(R!18, tptp_fun_M_16(E!17, R!18, X2!19, M2!21), M2!21) | (~member(tptp_fun_M_16(E!17, R!18, X2!19, M2!21), E!17)) | (~lower_bound(tptp_fun_M_16(E!17, R!18, X2!19, M2!21), R!18, X2!19)))))) | (~((~greatest_lower_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~lower_bound(M2!21, R!18, X2!19)) | (~![M: $i] : (apply(R!18, M, M2!21) | (~member(M, E!17)) | (~lower_bound(M, R!18, X2!19))))))))))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(48,plain,
% 0.19/0.40 ((~![A: $i, X: $i, R: $i, E: $i] : (~((~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))) | (~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X))))))))))) | (~((~(greatest_lower_bound(M2!21, X2!19, R!18, E!17) | (~lower_bound(M2!21, R!18, X2!19)) | (~member(M2!21, X2!19)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X2!19, M2!21), R!18, X2!19)) | (~member(tptp_fun_M_16(E!17, R!18, X2!19, M2!21), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X2!19, M2!21), M2!21))))) | (~((~greatest_lower_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~lower_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M2!21) | (~lower_bound(M, R!18, X2!19))))))))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[47, 46])).
% 0.19/0.40 tff(49,plain,
% 0.19/0.40 (~((~(greatest_lower_bound(M2!21, X2!19, R!18, E!17) | (~lower_bound(M2!21, R!18, X2!19)) | (~member(M2!21, X2!19)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X2!19, M2!21), R!18, X2!19)) | (~member(tptp_fun_M_16(E!17, R!18, X2!19, M2!21), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X2!19, M2!21), M2!21))))) | (~((~greatest_lower_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~lower_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M2!21) | (~lower_bound(M, R!18, X2!19)))))))))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[48, 42])).
% 0.19/0.40 tff(50,plain,
% 0.19/0.40 (((~(greatest_lower_bound(M2!21, X2!19, R!18, E!17) | (~lower_bound(M2!21, R!18, X2!19)) | (~member(M2!21, X2!19)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X2!19, M2!21), R!18, X2!19)) | (~member(tptp_fun_M_16(E!17, R!18, X2!19, M2!21), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X2!19, M2!21), M2!21))))) | (~((~greatest_lower_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~lower_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M2!21) | (~lower_bound(M, R!18, X2!19))))))))) | ((~greatest_lower_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~lower_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M2!21) | (~lower_bound(M, R!18, X2!19)))))))),
% 0.19/0.40 inference(tautology,[status(thm)],[])).
% 0.19/0.40 tff(51,plain,
% 0.19/0.40 ((~greatest_lower_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~lower_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M2!21) | (~lower_bound(M, R!18, X2!19))))))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[50, 49])).
% 0.19/0.40 tff(52,plain,
% 0.19/0.40 ((order(R!18, E!17) & (subset(X1!20, E!17) & subset(X2!19, E!17) & subset(X1!20, X2!19) & (~((~(greatest_lower_bound(M1!22, X1!20, R!18, E!17) & greatest_lower_bound(M2!21, X2!19, R!18, E!17))) | apply(R!18, M2!21, M1!22))))) <=> (order(R!18, E!17) & subset(X1!20, E!17) & subset(X2!19, E!17) & subset(X1!20, X2!19) & (~((~(greatest_lower_bound(M1!22, X1!20, R!18, E!17) & greatest_lower_bound(M2!21, X2!19, R!18, E!17))) | apply(R!18, M2!21, M1!22))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(53,plain,
% 0.19/0.40 (((subset(X1!20, E!17) & subset(X2!19, E!17) & subset(X1!20, X2!19)) & (~((~(greatest_lower_bound(M1!22, X1!20, R!18, E!17) & greatest_lower_bound(M2!21, X2!19, R!18, E!17))) | apply(R!18, M2!21, M1!22)))) <=> (subset(X1!20, E!17) & subset(X2!19, E!17) & subset(X1!20, X2!19) & (~((~(greatest_lower_bound(M1!22, X1!20, R!18, E!17) & greatest_lower_bound(M2!21, X2!19, R!18, E!17))) | apply(R!18, M2!21, M1!22))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(54,plain,
% 0.19/0.40 ((order(R!18, E!17) & ((subset(X1!20, E!17) & subset(X2!19, E!17) & subset(X1!20, X2!19)) & (~((~(greatest_lower_bound(M1!22, X1!20, R!18, E!17) & greatest_lower_bound(M2!21, X2!19, R!18, E!17))) | apply(R!18, M2!21, M1!22))))) <=> (order(R!18, E!17) & (subset(X1!20, E!17) & subset(X2!19, E!17) & subset(X1!20, X2!19) & (~((~(greatest_lower_bound(M1!22, X1!20, R!18, E!17) & greatest_lower_bound(M2!21, X2!19, R!18, E!17))) | apply(R!18, M2!21, M1!22)))))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[53])).
% 0.19/0.40 tff(55,plain,
% 0.19/0.40 ((order(R!18, E!17) & ((subset(X1!20, E!17) & subset(X2!19, E!17) & subset(X1!20, X2!19)) & (~((~(greatest_lower_bound(M1!22, X1!20, R!18, E!17) & greatest_lower_bound(M2!21, X2!19, R!18, E!17))) | apply(R!18, M2!21, M1!22))))) <=> (order(R!18, E!17) & subset(X1!20, E!17) & subset(X2!19, E!17) & subset(X1!20, X2!19) & (~((~(greatest_lower_bound(M1!22, X1!20, R!18, E!17) & greatest_lower_bound(M2!21, X2!19, R!18, E!17))) | apply(R!18, M2!21, M1!22))))),
% 0.19/0.40 inference(transitivity,[status(thm)],[54, 52])).
% 0.19/0.40 tff(56,plain,
% 0.19/0.40 ((~![R: $i, E: $i] : ((~order(R, E)) | ![X1: $i, X2: $i] : ((~(subset(X1, E) & subset(X2, E) & subset(X1, X2))) | ![M1: $i, M2: $i] : ((~(greatest_lower_bound(M1, X1, R, E) & greatest_lower_bound(M2, X2, R, E))) | apply(R, M2, M1))))) <=> (~![R: $i, E: $i] : ((~order(R, E)) | ![X1: $i, X2: $i] : ((~(subset(X1, E) & subset(X2, E) & subset(X1, X2))) | ![M1: $i, M2: $i] : ((~(greatest_lower_bound(M1, X1, R, E) & greatest_lower_bound(M2, X2, R, E))) | apply(R, M2, M1)))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(57,plain,
% 0.19/0.40 ((~![R: $i, E: $i] : (order(R, E) => ![X1: $i, X2: $i] : (((subset(X1, E) & subset(X2, E)) & subset(X1, X2)) => ![M1: $i, M2: $i] : ((greatest_lower_bound(M1, X1, R, E) & greatest_lower_bound(M2, X2, R, E)) => apply(R, M2, M1))))) <=> (~![R: $i, E: $i] : ((~order(R, E)) | ![X1: $i, X2: $i] : ((~(subset(X1, E) & subset(X2, E) & subset(X1, X2))) | ![M1: $i, M2: $i] : ((~(greatest_lower_bound(M1, X1, R, E) & greatest_lower_bound(M2, X2, R, E))) | apply(R, M2, M1)))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(58,axiom,(~![R: $i, E: $i] : (order(R, E) => ![X1: $i, X2: $i] : (((subset(X1, E) & subset(X2, E)) & subset(X1, X2)) => ![M1: $i, M2: $i] : ((greatest_lower_bound(M1, X1, R, E) & greatest_lower_bound(M2, X2, R, E)) => apply(R, M2, M1))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','thIV12')).
% 0.19/0.41 tff(59,plain,
% 0.19/0.41 (~![R: $i, E: $i] : ((~order(R, E)) | ![X1: $i, X2: $i] : ((~(subset(X1, E) & subset(X2, E) & subset(X1, X2))) | ![M1: $i, M2: $i] : ((~(greatest_lower_bound(M1, X1, R, E) & greatest_lower_bound(M2, X2, R, E))) | apply(R, M2, M1))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[58, 57])).
% 0.19/0.41 tff(60,plain,
% 0.19/0.41 (~![R: $i, E: $i] : ((~order(R, E)) | ![X1: $i, X2: $i] : ((~(subset(X1, E) & subset(X2, E) & subset(X1, X2))) | ![M1: $i, M2: $i] : ((~(greatest_lower_bound(M1, X1, R, E) & greatest_lower_bound(M2, X2, R, E))) | apply(R, M2, M1))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[59, 56])).
% 0.19/0.41 tff(61,plain,
% 0.19/0.41 (~![R: $i, E: $i] : ((~order(R, E)) | ![X1: $i, X2: $i] : ((~(subset(X1, E) & subset(X2, E) & subset(X1, X2))) | ![M1: $i, M2: $i] : ((~(greatest_lower_bound(M1, X1, R, E) & greatest_lower_bound(M2, X2, R, E))) | apply(R, M2, M1))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[60, 56])).
% 0.19/0.41 tff(62,plain,
% 0.19/0.41 (~![R: $i, E: $i] : ((~order(R, E)) | ![X1: $i, X2: $i] : ((~(subset(X1, E) & subset(X2, E) & subset(X1, X2))) | ![M1: $i, M2: $i] : ((~(greatest_lower_bound(M1, X1, R, E) & greatest_lower_bound(M2, X2, R, E))) | apply(R, M2, M1))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[61, 56])).
% 0.19/0.41 tff(63,plain,
% 0.19/0.41 (order(R!18, E!17) & subset(X1!20, E!17) & subset(X2!19, E!17) & subset(X1!20, X2!19) & (~((~(greatest_lower_bound(M1!22, X1!20, R!18, E!17) & greatest_lower_bound(M2!21, X2!19, R!18, E!17))) | apply(R!18, M2!21, M1!22)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[62, 55])).
% 0.19/0.41 tff(64,plain,
% 0.19/0.41 (~((~(greatest_lower_bound(M1!22, X1!20, R!18, E!17) & greatest_lower_bound(M2!21, X2!19, R!18, E!17))) | apply(R!18, M2!21, M1!22))),
% 0.19/0.41 inference(and_elim,[status(thm)],[63])).
% 0.19/0.41 tff(65,plain,
% 0.19/0.41 (greatest_lower_bound(M1!22, X1!20, R!18, E!17) & greatest_lower_bound(M2!21, X2!19, R!18, E!17)),
% 0.19/0.41 inference(or_elim,[status(thm)],[64])).
% 0.19/0.41 tff(66,plain,
% 0.19/0.41 (greatest_lower_bound(M2!21, X2!19, R!18, E!17)),
% 0.19/0.41 inference(and_elim,[status(thm)],[65])).
% 0.19/0.41 tff(67,plain,
% 0.19/0.41 ((~((~greatest_lower_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~lower_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M2!21) | (~lower_bound(M, R!18, X2!19)))))))) | (~greatest_lower_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~lower_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M2!21) | (~lower_bound(M, R!18, X2!19))))))),
% 0.19/0.41 inference(tautology,[status(thm)],[])).
% 0.19/0.41 tff(68,plain,
% 0.19/0.41 ((~((~greatest_lower_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~lower_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M2!21) | (~lower_bound(M, R!18, X2!19)))))))) | (~((~member(M2!21, X2!19)) | (~lower_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M2!21) | (~lower_bound(M, R!18, X2!19))))))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[67, 66])).
% 0.19/0.41 tff(69,plain,
% 0.19/0.41 (~((~member(M2!21, X2!19)) | (~lower_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M2!21) | (~lower_bound(M, R!18, X2!19)))))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[68, 51])).
% 0.19/0.41 tff(70,plain,
% 0.19/0.41 (((~member(M2!21, X2!19)) | (~lower_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M2!21) | (~lower_bound(M, R!18, X2!19))))) | lower_bound(M2!21, R!18, X2!19)),
% 0.19/0.41 inference(tautology,[status(thm)],[])).
% 0.19/0.41 tff(71,plain,
% 0.19/0.41 (lower_bound(M2!21, R!18, X2!19)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[70, 69])).
% 0.19/0.41 tff(72,plain,
% 0.19/0.41 ((~((~lower_bound(M2!21, R!18, X2!19)) | ![X: $i] : ((~member(X, X2!19)) | apply(R!18, M2!21, X)))) | (~lower_bound(M2!21, R!18, X2!19)) | ![X: $i] : ((~member(X, X2!19)) | apply(R!18, M2!21, X))),
% 0.19/0.41 inference(tautology,[status(thm)],[])).
% 0.19/0.41 tff(73,plain,
% 0.19/0.41 ((~((~lower_bound(M2!21, R!18, X2!19)) | ![X: $i] : ((~member(X, X2!19)) | apply(R!18, M2!21, X)))) | ![X: $i] : ((~member(X, X2!19)) | apply(R!18, M2!21, X))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[72, 71])).
% 0.19/0.41 tff(74,plain,
% 0.19/0.41 (![X: $i] : ((~member(X, X2!19)) | apply(R!18, M2!21, X))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[73, 20])).
% 0.19/0.41 tff(75,plain,
% 0.19/0.41 (((~![A: $i, X: $i, R: $i, E: $i] : (~((~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))) | (~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X))))))))))) | (~((~(greatest_lower_bound(M1!22, X1!20, R!18, E!17) | (~lower_bound(M1!22, R!18, X1!20)) | (~member(M1!22, X1!20)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X1!20, M1!22), R!18, X1!20)) | (~member(tptp_fun_M_16(E!17, R!18, X1!20, M1!22), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X1!20, M1!22), M1!22))))) | (~((~greatest_lower_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~lower_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M1!22) | (~lower_bound(M, R!18, X1!20))))))))))) <=> ((~![A: $i, X: $i, R: $i, E: $i] : (~((~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))) | (~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X))))))))))) | (~((~(greatest_lower_bound(M1!22, X1!20, R!18, E!17) | (~lower_bound(M1!22, R!18, X1!20)) | (~member(M1!22, X1!20)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X1!20, M1!22), R!18, X1!20)) | (~member(tptp_fun_M_16(E!17, R!18, X1!20, M1!22), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X1!20, M1!22), M1!22))))) | (~((~greatest_lower_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~lower_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M1!22) | (~lower_bound(M, R!18, X1!20)))))))))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(76,plain,
% 0.19/0.41 ((~((~(greatest_lower_bound(M1!22, X1!20, R!18, E!17) | (~member(M1!22, X1!20)) | (~lower_bound(M1!22, R!18, X1!20)) | (~(apply(R!18, tptp_fun_M_16(E!17, R!18, X1!20, M1!22), M1!22) | (~member(tptp_fun_M_16(E!17, R!18, X1!20, M1!22), E!17)) | (~lower_bound(tptp_fun_M_16(E!17, R!18, X1!20, M1!22), R!18, X1!20)))))) | (~((~greatest_lower_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~lower_bound(M1!22, R!18, X1!20)) | (~![M: $i] : (apply(R!18, M, M1!22) | (~member(M, E!17)) | (~lower_bound(M, R!18, X1!20)))))))))) <=> (~((~(greatest_lower_bound(M1!22, X1!20, R!18, E!17) | (~lower_bound(M1!22, R!18, X1!20)) | (~member(M1!22, X1!20)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X1!20, M1!22), R!18, X1!20)) | (~member(tptp_fun_M_16(E!17, R!18, X1!20, M1!22), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X1!20, M1!22), M1!22))))) | (~((~greatest_lower_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~lower_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M1!22) | (~lower_bound(M, R!18, X1!20))))))))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(77,plain,
% 0.19/0.41 (((~![A: $i, X: $i, R: $i, E: $i] : (~((~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))) | (~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X))))))))))) | (~((~(greatest_lower_bound(M1!22, X1!20, R!18, E!17) | (~member(M1!22, X1!20)) | (~lower_bound(M1!22, R!18, X1!20)) | (~(apply(R!18, tptp_fun_M_16(E!17, R!18, X1!20, M1!22), M1!22) | (~member(tptp_fun_M_16(E!17, R!18, X1!20, M1!22), E!17)) | (~lower_bound(tptp_fun_M_16(E!17, R!18, X1!20, M1!22), R!18, X1!20)))))) | (~((~greatest_lower_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~lower_bound(M1!22, R!18, X1!20)) | (~![M: $i] : (apply(R!18, M, M1!22) | (~member(M, E!17)) | (~lower_bound(M, R!18, X1!20))))))))))) <=> ((~![A: $i, X: $i, R: $i, E: $i] : (~((~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))) | (~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X))))))))))) | (~((~(greatest_lower_bound(M1!22, X1!20, R!18, E!17) | (~lower_bound(M1!22, R!18, X1!20)) | (~member(M1!22, X1!20)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X1!20, M1!22), R!18, X1!20)) | (~member(tptp_fun_M_16(E!17, R!18, X1!20, M1!22), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X1!20, M1!22), M1!22))))) | (~((~greatest_lower_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~lower_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M1!22) | (~lower_bound(M, R!18, X1!20)))))))))))),
% 0.19/0.41 inference(monotonicity,[status(thm)],[76])).
% 0.19/0.41 tff(78,plain,
% 0.19/0.41 (((~![A: $i, X: $i, R: $i, E: $i] : (~((~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))) | (~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X))))))))))) | (~((~(greatest_lower_bound(M1!22, X1!20, R!18, E!17) | (~member(M1!22, X1!20)) | (~lower_bound(M1!22, R!18, X1!20)) | (~(apply(R!18, tptp_fun_M_16(E!17, R!18, X1!20, M1!22), M1!22) | (~member(tptp_fun_M_16(E!17, R!18, X1!20, M1!22), E!17)) | (~lower_bound(tptp_fun_M_16(E!17, R!18, X1!20, M1!22), R!18, X1!20)))))) | (~((~greatest_lower_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~lower_bound(M1!22, R!18, X1!20)) | (~![M: $i] : (apply(R!18, M, M1!22) | (~member(M, E!17)) | (~lower_bound(M, R!18, X1!20))))))))))) <=> ((~![A: $i, X: $i, R: $i, E: $i] : (~((~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))) | (~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X))))))))))) | (~((~(greatest_lower_bound(M1!22, X1!20, R!18, E!17) | (~lower_bound(M1!22, R!18, X1!20)) | (~member(M1!22, X1!20)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X1!20, M1!22), R!18, X1!20)) | (~member(tptp_fun_M_16(E!17, R!18, X1!20, M1!22), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X1!20, M1!22), M1!22))))) | (~((~greatest_lower_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~lower_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M1!22) | (~lower_bound(M, R!18, X1!20)))))))))))),
% 0.19/0.41 inference(transitivity,[status(thm)],[77, 75])).
% 0.19/0.41 tff(79,plain,
% 0.19/0.41 ((~![A: $i, X: $i, R: $i, E: $i] : (~((~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))) | (~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X))))))))))) | (~((~(greatest_lower_bound(M1!22, X1!20, R!18, E!17) | (~member(M1!22, X1!20)) | (~lower_bound(M1!22, R!18, X1!20)) | (~(apply(R!18, tptp_fun_M_16(E!17, R!18, X1!20, M1!22), M1!22) | (~member(tptp_fun_M_16(E!17, R!18, X1!20, M1!22), E!17)) | (~lower_bound(tptp_fun_M_16(E!17, R!18, X1!20, M1!22), R!18, X1!20)))))) | (~((~greatest_lower_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~lower_bound(M1!22, R!18, X1!20)) | (~![M: $i] : (apply(R!18, M, M1!22) | (~member(M, E!17)) | (~lower_bound(M, R!18, X1!20))))))))))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(80,plain,
% 0.19/0.42 ((~![A: $i, X: $i, R: $i, E: $i] : (~((~(greatest_lower_bound(A, X, R, E) | (~member(A, X)) | (~lower_bound(A, R, X)) | (~(apply(R, tptp_fun_M_16(E, R, X, A), A) | (~member(tptp_fun_M_16(E, R, X, A), E)) | (~lower_bound(tptp_fun_M_16(E, R, X, A), R, X)))))) | (~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![M: $i] : (apply(R, M, A) | (~member(M, E)) | (~lower_bound(M, R, X))))))))))) | (~((~(greatest_lower_bound(M1!22, X1!20, R!18, E!17) | (~lower_bound(M1!22, R!18, X1!20)) | (~member(M1!22, X1!20)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X1!20, M1!22), R!18, X1!20)) | (~member(tptp_fun_M_16(E!17, R!18, X1!20, M1!22), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X1!20, M1!22), M1!22))))) | (~((~greatest_lower_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~lower_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M1!22) | (~lower_bound(M, R!18, X1!20))))))))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[79, 78])).
% 0.19/0.42 tff(81,plain,
% 0.19/0.42 (~((~(greatest_lower_bound(M1!22, X1!20, R!18, E!17) | (~lower_bound(M1!22, R!18, X1!20)) | (~member(M1!22, X1!20)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X1!20, M1!22), R!18, X1!20)) | (~member(tptp_fun_M_16(E!17, R!18, X1!20, M1!22), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X1!20, M1!22), M1!22))))) | (~((~greatest_lower_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~lower_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M1!22) | (~lower_bound(M, R!18, X1!20)))))))))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[80, 42])).
% 0.19/0.42 tff(82,plain,
% 0.19/0.42 (((~(greatest_lower_bound(M1!22, X1!20, R!18, E!17) | (~lower_bound(M1!22, R!18, X1!20)) | (~member(M1!22, X1!20)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X1!20, M1!22), R!18, X1!20)) | (~member(tptp_fun_M_16(E!17, R!18, X1!20, M1!22), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X1!20, M1!22), M1!22))))) | (~((~greatest_lower_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~lower_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M1!22) | (~lower_bound(M, R!18, X1!20))))))))) | ((~greatest_lower_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~lower_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M1!22) | (~lower_bound(M, R!18, X1!20)))))))),
% 0.19/0.42 inference(tautology,[status(thm)],[])).
% 0.19/0.42 tff(83,plain,
% 0.19/0.42 ((~greatest_lower_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~lower_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M1!22) | (~lower_bound(M, R!18, X1!20))))))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[82, 81])).
% 0.19/0.42 tff(84,plain,
% 0.19/0.42 (greatest_lower_bound(M1!22, X1!20, R!18, E!17)),
% 0.19/0.42 inference(and_elim,[status(thm)],[65])).
% 0.19/0.42 tff(85,plain,
% 0.19/0.42 ((~((~greatest_lower_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~lower_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M1!22) | (~lower_bound(M, R!18, X1!20)))))))) | (~greatest_lower_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~lower_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M1!22) | (~lower_bound(M, R!18, X1!20))))))),
% 0.19/0.42 inference(tautology,[status(thm)],[])).
% 0.19/0.42 tff(86,plain,
% 0.19/0.42 ((~((~greatest_lower_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~lower_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M1!22) | (~lower_bound(M, R!18, X1!20)))))))) | (~((~member(M1!22, X1!20)) | (~lower_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M1!22) | (~lower_bound(M, R!18, X1!20))))))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[85, 84])).
% 0.19/0.42 tff(87,plain,
% 0.19/0.42 (~((~member(M1!22, X1!20)) | (~lower_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M1!22) | (~lower_bound(M, R!18, X1!20)))))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[86, 83])).
% 0.19/0.42 tff(88,plain,
% 0.19/0.42 (((~member(M1!22, X1!20)) | (~lower_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M1!22) | (~lower_bound(M, R!18, X1!20))))) | member(M1!22, X1!20)),
% 0.19/0.42 inference(tautology,[status(thm)],[])).
% 0.19/0.42 tff(89,plain,
% 0.19/0.42 (member(M1!22, X1!20)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[88, 87])).
% 0.19/0.42 tff(90,plain,
% 0.19/0.42 (^[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))))))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(91,plain,
% 0.19/0.42 (![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)))))))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[90])).
% 0.19/0.42 tff(92,plain,
% 0.19/0.42 (^[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))))))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(93,plain,
% 0.19/0.42 (![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)))))))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[92])).
% 0.19/0.42 tff(94,plain,
% 0.19/0.42 (![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)))))))),
% 0.19/0.42 inference(transitivity,[status(thm)],[93, 91])).
% 0.19/0.42 tff(95,plain,
% 0.19/0.42 (^[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)))))))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(96,plain,
% 0.19/0.42 (![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)))))))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[95])).
% 0.19/0.42 tff(97,plain,
% 0.19/0.42 (![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)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(98,plain,
% 0.19/0.42 (^[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))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(99,plain,
% 0.19/0.42 (![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)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[98])).
% 0.19/0.42 tff(100,axiom,(![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : (member(X, A) => member(X, B)))), file('/export/starexec/sandbox/benchmark/Axioms/SET006+0.ax','subset')).
% 0.19/0.42 tff(101,plain,
% 0.19/0.42 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[100, 99])).
% 0.19/0.42 tff(102,plain,
% 0.19/0.42 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[101, 97])).
% 0.19/0.42 tff(103,plain,(
% 0.19/0.42 ![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)))))),
% 0.19/0.42 inference(skolemize,[status(sab)],[102])).
% 0.19/0.42 tff(104,plain,
% 0.19/0.42 (![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)))))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[103, 96])).
% 0.19/0.42 tff(105,plain,
% 0.19/0.42 (![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)))))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[104, 94])).
% 0.19/0.42 tff(106,plain,
% 0.19/0.42 ((~![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(X1!20, X2!19)) | ![X: $i] : ((~member(X, X1!20)) | member(X, X2!19)))) | (~(subset(X1!20, X2!19) | (~((~member(tptp_fun_X_0(X2!19, X1!20), X1!20)) | member(tptp_fun_X_0(X2!19, X1!20), X2!19)))))))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(107,plain,
% 0.19/0.42 (~((~((~subset(X1!20, X2!19)) | ![X: $i] : ((~member(X, X1!20)) | member(X, X2!19)))) | (~(subset(X1!20, X2!19) | (~((~member(tptp_fun_X_0(X2!19, X1!20), X1!20)) | member(tptp_fun_X_0(X2!19, X1!20), X2!19))))))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[106, 105])).
% 0.19/0.42 tff(108,plain,
% 0.19/0.42 (((~((~subset(X1!20, X2!19)) | ![X: $i] : ((~member(X, X1!20)) | member(X, X2!19)))) | (~(subset(X1!20, X2!19) | (~((~member(tptp_fun_X_0(X2!19, X1!20), X1!20)) | member(tptp_fun_X_0(X2!19, X1!20), X2!19)))))) | ((~subset(X1!20, X2!19)) | ![X: $i] : ((~member(X, X1!20)) | member(X, X2!19)))),
% 0.19/0.42 inference(tautology,[status(thm)],[])).
% 0.19/0.42 tff(109,plain,
% 0.19/0.42 ((~subset(X1!20, X2!19)) | ![X: $i] : ((~member(X, X1!20)) | member(X, X2!19))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[108, 107])).
% 0.19/0.42 tff(110,plain,
% 0.19/0.42 (subset(X1!20, X2!19)),
% 0.19/0.42 inference(and_elim,[status(thm)],[63])).
% 0.19/0.43 tff(111,plain,
% 0.19/0.43 ((~((~subset(X1!20, X2!19)) | ![X: $i] : ((~member(X, X1!20)) | member(X, X2!19)))) | (~subset(X1!20, X2!19)) | ![X: $i] : ((~member(X, X1!20)) | member(X, X2!19))),
% 0.19/0.43 inference(tautology,[status(thm)],[])).
% 0.19/0.43 tff(112,plain,
% 0.19/0.43 ((~((~subset(X1!20, X2!19)) | ![X: $i] : ((~member(X, X1!20)) | member(X, X2!19)))) | ![X: $i] : ((~member(X, X1!20)) | member(X, X2!19))),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[111, 110])).
% 0.19/0.43 tff(113,plain,
% 0.19/0.43 (![X: $i] : ((~member(X, X1!20)) | member(X, X2!19))),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[112, 109])).
% 0.19/0.43 tff(114,plain,
% 0.19/0.43 (((~![X: $i] : ((~member(X, X1!20)) | member(X, X2!19))) | ((~member(M1!22, X1!20)) | member(M1!22, X2!19))) <=> ((~![X: $i] : ((~member(X, X1!20)) | member(X, X2!19))) | (~member(M1!22, X1!20)) | member(M1!22, X2!19))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(115,plain,
% 0.19/0.43 ((~![X: $i] : ((~member(X, X1!20)) | member(X, X2!19))) | ((~member(M1!22, X1!20)) | member(M1!22, X2!19))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(116,plain,
% 0.19/0.43 ((~![X: $i] : ((~member(X, X1!20)) | member(X, X2!19))) | (~member(M1!22, X1!20)) | member(M1!22, X2!19)),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[115, 114])).
% 0.19/0.43 tff(117,plain,
% 0.19/0.43 (member(M1!22, X2!19)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[116, 113, 89])).
% 0.19/0.43 tff(118,plain,
% 0.19/0.43 (~apply(R!18, M2!21, M1!22)),
% 0.19/0.43 inference(or_elim,[status(thm)],[64])).
% 0.19/0.43 tff(119,plain,
% 0.19/0.43 (((~![X: $i] : ((~member(X, X2!19)) | apply(R!18, M2!21, X))) | ((~member(M1!22, X2!19)) | apply(R!18, M2!21, M1!22))) <=> ((~![X: $i] : ((~member(X, X2!19)) | apply(R!18, M2!21, X))) | (~member(M1!22, X2!19)) | apply(R!18, M2!21, M1!22))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(120,plain,
% 0.19/0.43 ((~![X: $i] : ((~member(X, X2!19)) | apply(R!18, M2!21, X))) | ((~member(M1!22, X2!19)) | apply(R!18, M2!21, M1!22))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(121,plain,
% 0.19/0.43 ((~![X: $i] : ((~member(X, X2!19)) | apply(R!18, M2!21, X))) | (~member(M1!22, X2!19)) | apply(R!18, M2!21, M1!22)),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[120, 119])).
% 0.19/0.43 tff(122,plain,
% 0.19/0.43 ($false),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[121, 118, 117, 74])).
% 0.19/0.43 % SZS output end Proof
% 0.19/0.43 % E exiting
%------------------------------------------------------------------------------