%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET802+4 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n015.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 0.21s 0.39s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.13 % Problem : SET802+4 : TPTP v9.0.0. Released v3.2.0.
% 0.03/0.13 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n015.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Jun 20 05:43:26 EDT 2025
% 0.13/0.34 % CPUTime :
% 0.21/0.39 % SZS status Theorem
% 0.21/0.39 % SZS output start Proof
% 0.21/0.39 tff(apply_type, type, (
% 0.21/0.39 apply: ( $i * $i * $i ) > $o)).
% 0.21/0.39 tff(tptp_fun_M_16_type, type, (
% 0.21/0.39 tptp_fun_M_16: ( $i * $i * $i * $i ) > $i)).
% 0.21/0.39 tff(tptp_fun_M_20_type, type, (
% 0.21/0.39 tptp_fun_M_20: $i)).
% 0.21/0.39 tff(tptp_fun_X_19_type, type, (
% 0.21/0.39 tptp_fun_X_19: $i)).
% 0.21/0.39 tff(tptp_fun_R_18_type, type, (
% 0.21/0.39 tptp_fun_R_18: $i)).
% 0.21/0.39 tff(tptp_fun_E_17_type, type, (
% 0.21/0.39 tptp_fun_E_17: $i)).
% 0.21/0.39 tff(member_type, type, (
% 0.21/0.39 member: ( $i * $i ) > $o)).
% 0.21/0.39 tff(lower_bound_type, type, (
% 0.21/0.39 lower_bound: ( $i * $i * $i ) > $o)).
% 0.21/0.39 tff(tptp_fun_X_12_type, type, (
% 0.21/0.39 tptp_fun_X_12: ( $i * $i * $i ) > $i)).
% 0.21/0.39 tff(greatest_lower_bound_type, type, (
% 0.21/0.39 greatest_lower_bound: ( $i * $i * $i * $i ) > $o)).
% 0.21/0.39 tff(least_type, type, (
% 0.21/0.39 least: ( $i * $i * $i ) > $o)).
% 0.21/0.39 tff(subset_type, type, (
% 0.21/0.39 subset: ( $i * $i ) > $o)).
% 0.21/0.39 tff(order_type, type, (
% 0.21/0.39 order: ( $i * $i ) > $o)).
% 0.21/0.39 tff(1,plain,
% 0.21/0.39 (^[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)))))))))),
% 0.21/0.39 inference(bind,[status(th)],[])).
% 0.21/0.39 tff(2,plain,
% 0.21/0.39 (![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))))))))),
% 0.21/0.39 inference(quant_intro,[status(thm)],[1])).
% 0.21/0.39 tff(3,plain,
% 0.21/0.39 (^[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)))))))))),
% 0.21/0.39 inference(bind,[status(th)],[])).
% 0.21/0.39 tff(4,plain,
% 0.21/0.39 (![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))))))))),
% 0.21/0.39 inference(quant_intro,[status(thm)],[3])).
% 0.21/0.39 tff(5,plain,
% 0.21/0.39 (![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))))))))),
% 0.21/0.39 inference(transitivity,[status(thm)],[4, 2])).
% 0.21/0.39 tff(6,plain,
% 0.21/0.39 (^[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))))))))))),
% 0.21/0.39 inference(bind,[status(th)],[])).
% 0.21/0.39 tff(7,plain,
% 0.21/0.39 (![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))))))))),
% 0.21/0.39 inference(quant_intro,[status(thm)],[6])).
% 0.21/0.39 tff(8,plain,
% 0.21/0.39 (![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)))),
% 0.21/0.39 inference(rewrite,[status(thm)],[])).
% 0.21/0.39 tff(9,plain,
% 0.21/0.39 (^[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))))),
% 0.21/0.39 inference(bind,[status(th)],[])).
% 0.21/0.39 tff(10,plain,
% 0.21/0.39 (![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)))),
% 0.21/0.39 inference(quant_intro,[status(thm)],[9])).
% 0.21/0.39 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')).
% 0.21/0.39 tff(12,plain,
% 0.21/0.39 (![R: $i, E: $i, M: $i] : (lower_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, M, X)))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[11, 10])).
% 0.21/0.39 tff(13,plain,
% 0.21/0.39 (![R: $i, E: $i, M: $i] : (lower_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, M, X)))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[12, 8])).
% 0.21/0.39 tff(14,plain,(
% 0.21/0.39 ![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))))))),
% 0.21/0.39 inference(skolemize,[status(sab)],[13])).
% 0.21/0.39 tff(15,plain,
% 0.21/0.39 (![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))))))))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[14, 7])).
% 0.21/0.39 tff(16,plain,
% 0.21/0.39 (![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))))))))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[15, 5])).
% 0.21/0.39 tff(17,plain,
% 0.21/0.39 ((~![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(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), X)))) | (~(lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19) | (~((~member(tptp_fun_X_12(tptp_fun_M_16(E!17, R!18, X!19, M!20), X!19, R!18), X!19)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), tptp_fun_X_12(tptp_fun_M_16(E!17, R!18, X!19, M!20), X!19, R!18))))))))),
% 0.21/0.40 inference(quant_inst,[status(thm)],[])).
% 0.21/0.40 tff(18,plain,
% 0.21/0.40 (~((~((~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), X)))) | (~(lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19) | (~((~member(tptp_fun_X_12(tptp_fun_M_16(E!17, R!18, X!19, M!20), X!19, R!18), X!19)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), tptp_fun_X_12(tptp_fun_M_16(E!17, R!18, X!19, M!20), X!19, R!18)))))))),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[17, 16])).
% 0.21/0.40 tff(19,plain,
% 0.21/0.40 (((~((~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), X)))) | (~(lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19) | (~((~member(tptp_fun_X_12(tptp_fun_M_16(E!17, R!18, X!19, M!20), X!19, R!18), X!19)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), tptp_fun_X_12(tptp_fun_M_16(E!17, R!18, X!19, M!20), X!19, R!18))))))) | ((~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), X)))),
% 0.21/0.40 inference(tautology,[status(thm)],[])).
% 0.21/0.40 tff(20,plain,
% 0.21/0.40 ((~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), X))),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[19, 18])).
% 0.21/0.40 tff(21,plain,
% 0.21/0.40 (^[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)))))))))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(22,plain,
% 0.21/0.40 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![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))))))))))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[21])).
% 0.21/0.40 tff(23,plain,
% 0.21/0.40 (^[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)))))))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(24,plain,
% 0.21/0.40 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![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))))))))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[23])).
% 0.21/0.40 tff(25,plain,
% 0.21/0.40 (^[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)))))))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(26,plain,
% 0.21/0.40 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![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))))))))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[25])).
% 0.21/0.40 tff(27,plain,
% 0.21/0.40 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![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))))))))),
% 0.21/0.40 inference(transitivity,[status(thm)],[26, 24])).
% 0.21/0.40 tff(28,plain,
% 0.21/0.40 (^[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))))))))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(29,plain,
% 0.21/0.40 (![A: $i, X: $i, R: $i, E: $i] : (((~greatest_lower_bound(A, X, R, E)) | (member(A, X) & lower_bound(A, R, X) & ![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))))))))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[28])).
% 0.21/0.40 tff(30,plain,
% 0.21/0.40 (^[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))))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(31,plain,
% 0.21/0.40 (![A: $i, X: $i, R: $i, E: $i] : (((~greatest_lower_bound(A, X, R, E)) | (member(A, X) & lower_bound(A, R, X) & ![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)))))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[30])).
% 0.21/0.40 tff(32,plain,
% 0.21/0.40 (![A: $i, X: $i, R: $i, E: $i] : (greatest_lower_bound(A, X, R, E) <=> (member(A, X) & lower_bound(A, R, X) & ![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))))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(33,plain,
% 0.21/0.40 (^[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)))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(34,plain,
% 0.21/0.40 (![A: $i, X: $i, R: $i, E: $i] : (greatest_lower_bound(A, X, R, E) <=> ((member(A, X) & lower_bound(A, R, X)) & ![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))))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[33])).
% 0.21/0.40 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/sandbox2/benchmark/Axioms/SET006+3.ax','greatest_lower_bound')).
% 0.21/0.40 tff(36,plain,
% 0.21/0.40 (![A: $i, X: $i, R: $i, E: $i] : (greatest_lower_bound(A, X, R, E) <=> (member(A, X) & lower_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & lower_bound(M, R, X))) | apply(R, M, A))))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.21/0.40 tff(37,plain,
% 0.21/0.40 (![A: $i, X: $i, R: $i, E: $i] : (greatest_lower_bound(A, X, R, E) <=> (member(A, X) & lower_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & lower_bound(M, R, X))) | apply(R, M, A))))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[36, 32])).
% 0.21/0.40 tff(38,plain,(
% 0.21/0.40 ![A: $i, X: $i, R: $i, E: $i] : (((~greatest_lower_bound(A, X, R, E)) | (member(A, X) & lower_bound(A, R, X) & ![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))))))),
% 0.21/0.40 inference(skolemize,[status(sab)],[37])).
% 0.21/0.40 tff(39,plain,
% 0.21/0.40 (![A: $i, X: $i, R: $i, E: $i] : (((~greatest_lower_bound(A, X, R, E)) | (member(A, X) & lower_bound(A, R, X) & ![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)))))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[38, 31])).
% 0.21/0.40 tff(40,plain,
% 0.21/0.40 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![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))))))))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[39, 29])).
% 0.21/0.40 tff(41,plain,
% 0.21/0.40 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![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))))))))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[40, 27])).
% 0.21/0.40 tff(42,plain,
% 0.21/0.40 (![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))))))))))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[41, 22])).
% 0.21/0.40 tff(43,plain,
% 0.21/0.40 (((~![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(M!20, X!19, R!18, E!17) | (~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)) | (~member(tptp_fun_M_16(E!17, R!18, X!19, M!20), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20))))) | (~((~greatest_lower_bound(M!20, X!19, R!18, E!17)) | (~((~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M!20) | (~lower_bound(M, R!18, X!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(M!20, X!19, R!18, E!17) | (~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)) | (~member(tptp_fun_M_16(E!17, R!18, X!19, M!20), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20))))) | (~((~greatest_lower_bound(M!20, X!19, R!18, E!17)) | (~((~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M!20) | (~lower_bound(M, R!18, X!19)))))))))))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(44,plain,
% 0.21/0.41 ((~((~(greatest_lower_bound(M!20, X!19, R!18, E!17) | (~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~(apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20) | (~member(tptp_fun_M_16(E!17, R!18, X!19, M!20), E!17)) | (~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)))))) | (~((~greatest_lower_bound(M!20, X!19, R!18, E!17)) | (~((~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~![M: $i] : (apply(R!18, M, M!20) | (~member(M, E!17)) | (~lower_bound(M, R!18, X!19)))))))))) <=> (~((~(greatest_lower_bound(M!20, X!19, R!18, E!17) | (~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)) | (~member(tptp_fun_M_16(E!17, R!18, X!19, M!20), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20))))) | (~((~greatest_lower_bound(M!20, X!19, R!18, E!17)) | (~((~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M!20) | (~lower_bound(M, R!18, X!19))))))))))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(45,plain,
% 0.21/0.41 (((~![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(M!20, X!19, R!18, E!17) | (~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~(apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20) | (~member(tptp_fun_M_16(E!17, R!18, X!19, M!20), E!17)) | (~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)))))) | (~((~greatest_lower_bound(M!20, X!19, R!18, E!17)) | (~((~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~![M: $i] : (apply(R!18, M, M!20) | (~member(M, E!17)) | (~lower_bound(M, R!18, X!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(M!20, X!19, R!18, E!17) | (~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)) | (~member(tptp_fun_M_16(E!17, R!18, X!19, M!20), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20))))) | (~((~greatest_lower_bound(M!20, X!19, R!18, E!17)) | (~((~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M!20) | (~lower_bound(M, R!18, X!19)))))))))))),
% 0.21/0.41 inference(monotonicity,[status(thm)],[44])).
% 0.21/0.41 tff(46,plain,
% 0.21/0.41 (((~![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(M!20, X!19, R!18, E!17) | (~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~(apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20) | (~member(tptp_fun_M_16(E!17, R!18, X!19, M!20), E!17)) | (~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)))))) | (~((~greatest_lower_bound(M!20, X!19, R!18, E!17)) | (~((~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~![M: $i] : (apply(R!18, M, M!20) | (~member(M, E!17)) | (~lower_bound(M, R!18, X!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(M!20, X!19, R!18, E!17) | (~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)) | (~member(tptp_fun_M_16(E!17, R!18, X!19, M!20), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20))))) | (~((~greatest_lower_bound(M!20, X!19, R!18, E!17)) | (~((~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M!20) | (~lower_bound(M, R!18, X!19)))))))))))),
% 0.21/0.41 inference(transitivity,[status(thm)],[45, 43])).
% 0.21/0.41 tff(47,plain,
% 0.21/0.41 ((~![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(M!20, X!19, R!18, E!17) | (~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~(apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20) | (~member(tptp_fun_M_16(E!17, R!18, X!19, M!20), E!17)) | (~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)))))) | (~((~greatest_lower_bound(M!20, X!19, R!18, E!17)) | (~((~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~![M: $i] : (apply(R!18, M, M!20) | (~member(M, E!17)) | (~lower_bound(M, R!18, X!19))))))))))),
% 0.21/0.41 inference(quant_inst,[status(thm)],[])).
% 0.21/0.41 tff(48,plain,
% 0.21/0.41 ((~![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(M!20, X!19, R!18, E!17) | (~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)) | (~member(tptp_fun_M_16(E!17, R!18, X!19, M!20), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20))))) | (~((~greatest_lower_bound(M!20, X!19, R!18, E!17)) | (~((~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M!20) | (~lower_bound(M, R!18, X!19))))))))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[47, 46])).
% 0.21/0.41 tff(49,plain,
% 0.21/0.41 (~((~(greatest_lower_bound(M!20, X!19, R!18, E!17) | (~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)) | (~member(tptp_fun_M_16(E!17, R!18, X!19, M!20), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20))))) | (~((~greatest_lower_bound(M!20, X!19, R!18, E!17)) | (~((~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M!20) | (~lower_bound(M, R!18, X!19)))))))))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[48, 42])).
% 0.21/0.42 tff(50,plain,
% 0.21/0.42 (((~(greatest_lower_bound(M!20, X!19, R!18, E!17) | (~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)) | (~member(tptp_fun_M_16(E!17, R!18, X!19, M!20), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20))))) | (~((~greatest_lower_bound(M!20, X!19, R!18, E!17)) | (~((~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M!20) | (~lower_bound(M, R!18, X!19))))))))) | (greatest_lower_bound(M!20, X!19, R!18, E!17) | (~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)) | (~member(tptp_fun_M_16(E!17, R!18, X!19, M!20), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20))))),
% 0.21/0.42 inference(tautology,[status(thm)],[])).
% 0.21/0.42 tff(51,plain,
% 0.21/0.42 (greatest_lower_bound(M!20, X!19, R!18, E!17) | (~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)) | (~member(tptp_fun_M_16(E!17, R!18, X!19, M!20), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20)))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[50, 49])).
% 0.21/0.42 tff(52,assumption,((~((~lower_bound(M!20, R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X)))) | (~((~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))) | lower_bound(M!20, R!18, X!19)))), introduced(assumption)).
% 0.21/0.42 tff(53,plain,
% 0.21/0.42 (((~![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!20, R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X)))) | (~((~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))) | lower_bound(M!20, R!18, X!19)))))) <=> ((~![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!20, R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X)))) | (~((~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))) | lower_bound(M!20, R!18, X!19))))))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(54,plain,
% 0.21/0.42 ((~((~((~lower_bound(M!20, R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X)))) | (~(lower_bound(M!20, R!18, X!19) | (~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))))))) <=> (~((~((~lower_bound(M!20, R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X)))) | (~((~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))) | lower_bound(M!20, R!18, X!19)))))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(55,plain,
% 0.21/0.42 (((~![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!20, R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X)))) | (~(lower_bound(M!20, R!18, X!19) | (~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18))))))))) <=> ((~![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!20, R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X)))) | (~((~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))) | lower_bound(M!20, R!18, X!19))))))),
% 0.21/0.42 inference(monotonicity,[status(thm)],[54])).
% 0.21/0.42 tff(56,plain,
% 0.21/0.42 (((~![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!20, R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X)))) | (~(lower_bound(M!20, R!18, X!19) | (~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18))))))))) <=> ((~![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!20, R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X)))) | (~((~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))) | lower_bound(M!20, R!18, X!19))))))),
% 0.21/0.42 inference(transitivity,[status(thm)],[55, 53])).
% 0.21/0.42 tff(57,plain,
% 0.21/0.42 ((~![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!20, R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X)))) | (~(lower_bound(M!20, R!18, X!19) | (~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18))))))))),
% 0.21/0.42 inference(quant_inst,[status(thm)],[])).
% 0.21/0.42 tff(58,plain,
% 0.21/0.42 ((~![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!20, R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X)))) | (~((~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))) | lower_bound(M!20, R!18, X!19)))))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[57, 56])).
% 0.21/0.42 tff(59,plain,
% 0.21/0.42 ($false),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[58, 16, 52])).
% 0.21/0.42 tff(60,plain,(~((~((~lower_bound(M!20, R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X)))) | (~((~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))) | lower_bound(M!20, R!18, X!19))))), inference(lemma,lemma(discharge,[]))).
% 0.21/0.42 tff(61,plain,
% 0.21/0.42 (((~((~lower_bound(M!20, R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X)))) | (~((~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))) | lower_bound(M!20, R!18, X!19)))) | ((~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))) | lower_bound(M!20, R!18, X!19))),
% 0.21/0.42 inference(tautology,[status(thm)],[])).
% 0.21/0.42 tff(62,plain,
% 0.21/0.42 ((~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))) | lower_bound(M!20, R!18, X!19)),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[61, 60])).
% 0.21/0.42 tff(63,plain,
% 0.21/0.42 (~((~((~lower_bound(M!20, R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X)))) | (~((~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))) | lower_bound(M!20, R!18, X!19))))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[58, 16])).
% 0.21/0.42 tff(64,plain,
% 0.21/0.42 (((~((~lower_bound(M!20, R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X)))) | (~((~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))) | lower_bound(M!20, R!18, X!19)))) | ((~lower_bound(M!20, R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X)))),
% 0.21/0.42 inference(tautology,[status(thm)],[])).
% 0.21/0.42 tff(65,plain,
% 0.21/0.42 ((~lower_bound(M!20, R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[64, 63])).
% 0.21/0.42 tff(66,plain,
% 0.21/0.42 (((~(greatest_lower_bound(M!20, X!19, R!18, E!17) | (~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)) | (~member(tptp_fun_M_16(E!17, R!18, X!19, M!20), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20))))) | (~((~greatest_lower_bound(M!20, X!19, R!18, E!17)) | (~((~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M!20) | (~lower_bound(M, R!18, X!19))))))))) | ((~greatest_lower_bound(M!20, X!19, R!18, E!17)) | (~((~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M!20) | (~lower_bound(M, R!18, X!19)))))))),
% 0.21/0.42 inference(tautology,[status(thm)],[])).
% 0.21/0.42 tff(67,plain,
% 0.21/0.42 ((~greatest_lower_bound(M!20, X!19, R!18, E!17)) | (~((~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M!20) | (~lower_bound(M, R!18, X!19))))))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[66, 49])).
% 0.21/0.42 tff(68,assumption,(~least(M!20, R!18, X!19)), introduced(assumption)).
% 0.21/0.42 tff(69,plain,
% 0.21/0.42 ((~(~(((~member(M!20, X!19)) | (~greatest_lower_bound(M!20, X!19, R!18, E!17))) <=> least(M!20, R!18, X!19)))) <=> (((~member(M!20, X!19)) | (~greatest_lower_bound(M!20, X!19, R!18, E!17))) <=> least(M!20, R!18, X!19))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(70,plain,
% 0.21/0.42 ((~(((~member(M!20, X!19)) | (~greatest_lower_bound(M!20, X!19, R!18, E!17))) <=> least(M!20, R!18, X!19))) <=> (~(((~member(M!20, X!19)) | (~greatest_lower_bound(M!20, X!19, R!18, E!17))) <=> least(M!20, R!18, X!19)))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(71,plain,
% 0.21/0.42 ((least(M!20, R!18, X!19) <=> (~((~member(M!20, X!19)) | (~greatest_lower_bound(M!20, X!19, R!18, E!17))))) <=> (~(((~member(M!20, X!19)) | (~greatest_lower_bound(M!20, X!19, R!18, E!17))) <=> least(M!20, R!18, X!19)))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(72,plain,
% 0.21/0.42 ((member(M!20, X!19) & greatest_lower_bound(M!20, X!19, R!18, E!17)) <=> (~((~member(M!20, X!19)) | (~greatest_lower_bound(M!20, X!19, R!18, E!17))))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(73,plain,
% 0.21/0.42 ((least(M!20, R!18, X!19) <=> (member(M!20, X!19) & greatest_lower_bound(M!20, X!19, R!18, E!17))) <=> (least(M!20, R!18, X!19) <=> (~((~member(M!20, X!19)) | (~greatest_lower_bound(M!20, X!19, R!18, E!17)))))),
% 0.21/0.42 inference(monotonicity,[status(thm)],[72])).
% 0.21/0.42 tff(74,plain,
% 0.21/0.42 ((least(M!20, R!18, X!19) <=> (member(M!20, X!19) & greatest_lower_bound(M!20, X!19, R!18, E!17))) <=> (~(((~member(M!20, X!19)) | (~greatest_lower_bound(M!20, X!19, R!18, E!17))) <=> least(M!20, R!18, X!19)))),
% 0.21/0.42 inference(transitivity,[status(thm)],[73, 71])).
% 0.21/0.42 tff(75,plain,
% 0.21/0.42 ((least(M!20, R!18, X!19) <=> (member(M!20, X!19) & greatest_lower_bound(M!20, X!19, R!18, E!17))) <=> (~(((~member(M!20, X!19)) | (~greatest_lower_bound(M!20, X!19, R!18, E!17))) <=> least(M!20, R!18, X!19)))),
% 0.21/0.42 inference(transitivity,[status(thm)],[74, 70])).
% 0.21/0.42 tff(76,plain,
% 0.21/0.42 ((~(least(M!20, R!18, X!19) <=> (member(M!20, X!19) & greatest_lower_bound(M!20, X!19, R!18, E!17)))) <=> (~(~(((~member(M!20, X!19)) | (~greatest_lower_bound(M!20, X!19, R!18, E!17))) <=> least(M!20, R!18, X!19))))),
% 0.21/0.42 inference(monotonicity,[status(thm)],[75])).
% 0.21/0.42 tff(77,plain,
% 0.21/0.42 ((~(least(M!20, R!18, X!19) <=> (member(M!20, X!19) & greatest_lower_bound(M!20, X!19, R!18, E!17)))) <=> (((~member(M!20, X!19)) | (~greatest_lower_bound(M!20, X!19, R!18, E!17))) <=> least(M!20, R!18, X!19))),
% 0.21/0.42 inference(transitivity,[status(thm)],[76, 69])).
% 0.21/0.42 tff(78,plain,
% 0.21/0.42 ((order(R!18, E!17) & (subset(X!19, E!17) & (~(least(M!20, R!18, X!19) <=> (member(M!20, X!19) & greatest_lower_bound(M!20, X!19, R!18, E!17)))))) <=> (order(R!18, E!17) & subset(X!19, E!17) & (~(least(M!20, R!18, X!19) <=> (member(M!20, X!19) & greatest_lower_bound(M!20, X!19, R!18, E!17)))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(79,plain,
% 0.21/0.43 ((subset(X!19, E!17) & (~(least(M!20, R!18, X!19) <=> (member(M!20, X!19) & greatest_lower_bound(M!20, X!19, R!18, E!17))))) <=> (subset(X!19, E!17) & (~(least(M!20, R!18, X!19) <=> (member(M!20, X!19) & greatest_lower_bound(M!20, X!19, R!18, E!17)))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(80,plain,
% 0.21/0.43 ((order(R!18, E!17) & (subset(X!19, E!17) & (~(least(M!20, R!18, X!19) <=> (member(M!20, X!19) & greatest_lower_bound(M!20, X!19, R!18, E!17)))))) <=> (order(R!18, E!17) & (subset(X!19, E!17) & (~(least(M!20, R!18, X!19) <=> (member(M!20, X!19) & greatest_lower_bound(M!20, X!19, R!18, E!17))))))),
% 0.21/0.43 inference(monotonicity,[status(thm)],[79])).
% 0.21/0.43 tff(81,plain,
% 0.21/0.43 ((order(R!18, E!17) & (subset(X!19, E!17) & (~(least(M!20, R!18, X!19) <=> (member(M!20, X!19) & greatest_lower_bound(M!20, X!19, R!18, E!17)))))) <=> (order(R!18, E!17) & subset(X!19, E!17) & (~(least(M!20, R!18, X!19) <=> (member(M!20, X!19) & greatest_lower_bound(M!20, X!19, R!18, E!17)))))),
% 0.21/0.43 inference(transitivity,[status(thm)],[80, 78])).
% 0.21/0.43 tff(82,plain,
% 0.21/0.43 ((~![R: $i, E: $i] : ((~order(R, E)) | ![X: $i] : ((~subset(X, E)) | ![M: $i] : (least(M, R, X) <=> (member(M, X) & greatest_lower_bound(M, X, R, E)))))) <=> (~![R: $i, E: $i] : ((~order(R, E)) | ![X: $i] : ((~subset(X, E)) | ![M: $i] : (least(M, R, X) <=> (member(M, X) & greatest_lower_bound(M, X, R, E))))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(83,plain,
% 0.21/0.43 ((~![R: $i, E: $i] : (order(R, E) => ![X: $i] : (subset(X, E) => ![M: $i] : (least(M, R, X) <=> (member(M, X) & greatest_lower_bound(M, X, R, E)))))) <=> (~![R: $i, E: $i] : ((~order(R, E)) | ![X: $i] : ((~subset(X, E)) | ![M: $i] : (least(M, R, X) <=> (member(M, X) & greatest_lower_bound(M, X, R, E))))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(84,axiom,(~![R: $i, E: $i] : (order(R, E) => ![X: $i] : (subset(X, E) => ![M: $i] : (least(M, R, X) <=> (member(M, X) & greatest_lower_bound(M, X, R, E)))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','thIV14')).
% 0.21/0.43 tff(85,plain,
% 0.21/0.43 (~![R: $i, E: $i] : ((~order(R, E)) | ![X: $i] : ((~subset(X, E)) | ![M: $i] : (least(M, R, X) <=> (member(M, X) & greatest_lower_bound(M, X, R, E)))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[84, 83])).
% 0.21/0.43 tff(86,plain,
% 0.21/0.43 (~![R: $i, E: $i] : ((~order(R, E)) | ![X: $i] : ((~subset(X, E)) | ![M: $i] : (least(M, R, X) <=> (member(M, X) & greatest_lower_bound(M, X, R, E)))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[85, 82])).
% 0.21/0.43 tff(87,plain,
% 0.21/0.43 (~![R: $i, E: $i] : ((~order(R, E)) | ![X: $i] : ((~subset(X, E)) | ![M: $i] : (least(M, R, X) <=> (member(M, X) & greatest_lower_bound(M, X, R, E)))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[86, 82])).
% 0.21/0.43 tff(88,plain,
% 0.21/0.43 (~![R: $i, E: $i] : ((~order(R, E)) | ![X: $i] : ((~subset(X, E)) | ![M: $i] : (least(M, R, X) <=> (member(M, X) & greatest_lower_bound(M, X, R, E)))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[87, 82])).
% 0.21/0.43 tff(89,plain,
% 0.21/0.43 (order(R!18, E!17) & subset(X!19, E!17) & (~(least(M!20, R!18, X!19) <=> (member(M!20, X!19) & greatest_lower_bound(M!20, X!19, R!18, E!17))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[88, 81])).
% 0.21/0.43 tff(90,plain,
% 0.21/0.43 (~(least(M!20, R!18, X!19) <=> (member(M!20, X!19) & greatest_lower_bound(M!20, X!19, R!18, E!17)))),
% 0.21/0.43 inference(and_elim,[status(thm)],[89])).
% 0.21/0.43 tff(91,plain,
% 0.21/0.43 (((~member(M!20, X!19)) | (~greatest_lower_bound(M!20, X!19, R!18, E!17))) <=> least(M!20, R!18, X!19)),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[90, 77])).
% 0.21/0.43 tff(92,plain,
% 0.21/0.43 ((~((~member(M!20, X!19)) | (~greatest_lower_bound(M!20, X!19, R!18, E!17)))) | least(M!20, R!18, X!19) | (~(((~member(M!20, X!19)) | (~greatest_lower_bound(M!20, X!19, R!18, E!17))) <=> least(M!20, R!18, X!19)))),
% 0.21/0.43 inference(tautology,[status(thm)],[])).
% 0.21/0.43 tff(93,plain,
% 0.21/0.43 ((~((~member(M!20, X!19)) | (~greatest_lower_bound(M!20, X!19, R!18, E!17)))) | least(M!20, R!18, X!19)),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[92, 91])).
% 0.21/0.43 tff(94,plain,
% 0.21/0.43 (~((~member(M!20, X!19)) | (~greatest_lower_bound(M!20, X!19, R!18, E!17)))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[93, 68])).
% 0.21/0.43 tff(95,plain,
% 0.21/0.43 (((~member(M!20, X!19)) | (~greatest_lower_bound(M!20, X!19, R!18, E!17))) | greatest_lower_bound(M!20, X!19, R!18, E!17)),
% 0.21/0.43 inference(tautology,[status(thm)],[])).
% 0.21/0.43 tff(96,plain,
% 0.21/0.43 (greatest_lower_bound(M!20, X!19, R!18, E!17)),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[95, 94])).
% 0.21/0.43 tff(97,plain,
% 0.21/0.43 ((~((~greatest_lower_bound(M!20, X!19, R!18, E!17)) | (~((~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M!20) | (~lower_bound(M, R!18, X!19)))))))) | (~greatest_lower_bound(M!20, X!19, R!18, E!17)) | (~((~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M!20) | (~lower_bound(M, R!18, X!19))))))),
% 0.21/0.43 inference(tautology,[status(thm)],[])).
% 0.21/0.43 tff(98,plain,
% 0.21/0.43 ((~((~greatest_lower_bound(M!20, X!19, R!18, E!17)) | (~((~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M!20) | (~lower_bound(M, R!18, X!19)))))))) | (~((~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M!20) | (~lower_bound(M, R!18, X!19))))))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[97, 96])).
% 0.21/0.43 tff(99,plain,
% 0.21/0.43 (~((~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M!20) | (~lower_bound(M, R!18, X!19)))))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[98, 67])).
% 0.21/0.43 tff(100,plain,
% 0.21/0.43 (((~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M, M!20) | (~lower_bound(M, R!18, X!19))))) | lower_bound(M!20, R!18, X!19)),
% 0.21/0.43 inference(tautology,[status(thm)],[])).
% 0.21/0.43 tff(101,plain,
% 0.21/0.43 (lower_bound(M!20, R!18, X!19)),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[100, 99])).
% 0.21/0.43 tff(102,plain,
% 0.21/0.43 ((~((~lower_bound(M!20, R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X)))) | (~lower_bound(M!20, R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X))),
% 0.21/0.43 inference(tautology,[status(thm)],[])).
% 0.21/0.43 tff(103,plain,
% 0.21/0.43 ((~((~lower_bound(M!20, R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X)))) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[102, 101])).
% 0.21/0.43 tff(104,plain,
% 0.21/0.43 (![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[103, 65])).
% 0.21/0.43 tff(105,plain,
% 0.21/0.43 (((~member(M!20, X!19)) | (~greatest_lower_bound(M!20, X!19, R!18, E!17))) | member(M!20, X!19)),
% 0.21/0.43 inference(tautology,[status(thm)],[])).
% 0.21/0.43 tff(106,plain,
% 0.21/0.43 (member(M!20, X!19)),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[105, 94])).
% 0.21/0.43 tff(107,plain,
% 0.21/0.43 (^[R: $i, E: $i, M: $i] : rewrite((~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))))) <=> (~((~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))) | (~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))))))),
% 0.21/0.43 inference(bind,[status(th)],[])).
% 0.21/0.43 tff(108,plain,
% 0.21/0.43 (![R: $i, E: $i, M: $i] : (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, 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] : (~((~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))) | (~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X)))))))))),
% 0.21/0.43 inference(quant_intro,[status(thm)],[107])).
% 0.21/0.43 tff(109,plain,
% 0.21/0.43 (^[R: $i, E: $i, M: $i] : refl((~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))))) <=> (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))))))),
% 0.21/0.43 inference(bind,[status(th)],[])).
% 0.21/0.43 tff(110,plain,
% 0.21/0.43 (![R: $i, E: $i, M: $i] : (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, 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] : (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))),
% 0.21/0.43 inference(quant_intro,[status(thm)],[109])).
% 0.21/0.43 tff(111,plain,
% 0.21/0.43 (^[R: $i, E: $i, M: $i] : rewrite((~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))))) <=> (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))))))),
% 0.21/0.43 inference(bind,[status(th)],[])).
% 0.21/0.43 tff(112,plain,
% 0.21/0.43 (![R: $i, E: $i, M: $i] : (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, 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] : (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))),
% 0.21/0.43 inference(quant_intro,[status(thm)],[111])).
% 0.21/0.43 tff(113,plain,
% 0.21/0.43 (![R: $i, E: $i, M: $i] : (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, 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] : (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))),
% 0.21/0.43 inference(transitivity,[status(thm)],[112, 110])).
% 0.21/0.43 tff(114,plain,
% 0.21/0.43 (^[R: $i, E: $i, M: $i] : trans(monotonicity(rewrite(((~least(M, R, E)) | (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) <=> ((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))), rewrite((least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))) <=> (least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))), ((((~least(M, R, E)) | (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) & (least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))) <=> (((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X)))))) & (least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))))), rewrite((((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X)))))) & (least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))) <=> (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))), ((((~least(M, R, E)) | (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) & (least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))) <=> (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))))),
% 0.21/0.43 inference(bind,[status(th)],[])).
% 0.21/0.43 tff(115,plain,
% 0.21/0.43 (![R: $i, E: $i, M: $i] : (((~least(M, R, E)) | (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) & (least(M, R, E) | (~member(M, 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] : (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))),
% 0.21/0.43 inference(quant_intro,[status(thm)],[114])).
% 0.21/0.43 tff(116,plain,
% 0.21/0.43 (^[R: $i, E: $i, M: $i] : rewrite((((~least(M, R, E)) | (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) & (least(M, R, E) | ((~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))) <=> (((~least(M, R, E)) | (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) & (least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))))),
% 0.21/0.43 inference(bind,[status(th)],[])).
% 0.21/0.43 tff(117,plain,
% 0.21/0.43 (![R: $i, E: $i, M: $i] : (((~least(M, R, E)) | (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) & (least(M, R, E) | ((~member(M, 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] : (((~least(M, R, E)) | (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) & (least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))),
% 0.21/0.43 inference(quant_intro,[status(thm)],[116])).
% 0.21/0.43 tff(118,plain,
% 0.21/0.43 (![R: $i, E: $i, M: $i] : (least(M, R, E) <=> (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) <=> ![R: $i, E: $i, M: $i] : (least(M, R, E) <=> (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(119,plain,
% 0.21/0.43 (^[R: $i, E: $i, M: $i] : rewrite((least(M, R, E) <=> (member(M, E) & ![X: $i] : (member(X, E) => apply(R, M, X)))) <=> (least(M, R, E) <=> (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))))),
% 0.21/0.43 inference(bind,[status(th)],[])).
% 0.21/0.43 tff(120,plain,
% 0.21/0.43 (![R: $i, E: $i, M: $i] : (least(M, R, E) <=> (member(M, E) & ![X: $i] : (member(X, E) => apply(R, M, X)))) <=> ![R: $i, E: $i, M: $i] : (least(M, R, E) <=> (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X))))),
% 0.21/0.43 inference(quant_intro,[status(thm)],[119])).
% 0.21/0.43 tff(121,axiom,(![R: $i, E: $i, M: $i] : (least(M, R, E) <=> (member(M, E) & ![X: $i] : (member(X, E) => apply(R, M, X))))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+3.ax','least')).
% 0.21/0.43 tff(122,plain,
% 0.21/0.43 (![R: $i, E: $i, M: $i] : (least(M, R, E) <=> (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[121, 120])).
% 0.21/0.43 tff(123,plain,
% 0.21/0.43 (![R: $i, E: $i, M: $i] : (least(M, R, E) <=> (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[122, 118])).
% 0.21/0.43 tff(124,plain,(
% 0.21/0.43 ![R: $i, E: $i, M: $i] : (((~least(M, R, E)) | (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) & (least(M, R, E) | ((~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))))),
% 0.21/0.43 inference(skolemize,[status(sab)],[123])).
% 0.21/0.43 tff(125,plain,
% 0.21/0.43 (![R: $i, E: $i, M: $i] : (((~least(M, R, E)) | (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) & (least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[124, 117])).
% 0.21/0.44 tff(126,plain,
% 0.21/0.44 (![R: $i, E: $i, M: $i] : (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[125, 115])).
% 0.21/0.44 tff(127,plain,
% 0.21/0.44 (![R: $i, E: $i, M: $i] : (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[126, 113])).
% 0.21/0.44 tff(128,plain,
% 0.21/0.44 (![R: $i, E: $i, M: $i] : (~((~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))) | (~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X)))))))))),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[127, 108])).
% 0.21/0.44 tff(129,plain,
% 0.21/0.44 ((~![R: $i, E: $i, M: $i] : (~((~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))) | (~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X)))))))))) | (~((~(least(M!20, R!18, X!19) | (~member(M!20, X!19)) | (~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))))) | (~((~least(M!20, R!18, X!19)) | (~((~member(M!20, X!19)) | (~![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X)))))))))),
% 0.21/0.44 inference(quant_inst,[status(thm)],[])).
% 0.21/0.44 tff(130,plain,
% 0.21/0.44 (~((~(least(M!20, R!18, X!19) | (~member(M!20, X!19)) | (~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))))) | (~((~least(M!20, R!18, X!19)) | (~((~member(M!20, X!19)) | (~![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X))))))))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[129, 128])).
% 0.21/0.44 tff(131,plain,
% 0.21/0.44 (((~(least(M!20, R!18, X!19) | (~member(M!20, X!19)) | (~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))))) | (~((~least(M!20, R!18, X!19)) | (~((~member(M!20, X!19)) | (~![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X)))))))) | (least(M!20, R!18, X!19) | (~member(M!20, X!19)) | (~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))))),
% 0.21/0.44 inference(tautology,[status(thm)],[])).
% 0.21/0.44 tff(132,plain,
% 0.21/0.44 (least(M!20, R!18, X!19) | (~member(M!20, X!19)) | (~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18))))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[131, 130])).
% 0.21/0.44 tff(133,plain,
% 0.21/0.44 ((~(least(M!20, R!18, X!19) | (~member(M!20, X!19)) | (~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))))) | least(M!20, R!18, X!19) | (~member(M!20, X!19)) | (~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18))))),
% 0.21/0.44 inference(tautology,[status(thm)],[])).
% 0.21/0.44 tff(134,plain,
% 0.21/0.44 (least(M!20, R!18, X!19) | (~member(M!20, X!19)) | (~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18))))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[133, 132])).
% 0.21/0.44 tff(135,plain,
% 0.21/0.44 (~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[134, 106, 68])).
% 0.21/0.44 tff(136,plain,
% 0.21/0.44 (((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18))) | (~apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))),
% 0.21/0.44 inference(tautology,[status(thm)],[])).
% 0.21/0.44 tff(137,plain,
% 0.21/0.44 (~apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[136, 135])).
% 0.21/0.44 tff(138,plain,
% 0.21/0.44 (((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18))) | member(tptp_fun_X_12(M!20, X!19, R!18), X!19)),
% 0.21/0.44 inference(tautology,[status(thm)],[])).
% 0.21/0.44 tff(139,plain,
% 0.21/0.44 (member(tptp_fun_X_12(M!20, X!19, R!18), X!19)),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[138, 135])).
% 0.21/0.44 tff(140,plain,
% 0.21/0.44 (((~![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X))) | ((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))) <=> ((~![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X))) | (~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))),
% 0.21/0.44 inference(rewrite,[status(thm)],[])).
% 0.21/0.44 tff(141,plain,
% 0.21/0.44 ((~![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X))) | ((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))),
% 0.21/0.44 inference(quant_inst,[status(thm)],[])).
% 0.21/0.44 tff(142,plain,
% 0.21/0.44 ((~![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X))) | (~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18))),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[141, 140])).
% 0.21/0.44 tff(143,plain,
% 0.21/0.44 ($false),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[142, 139, 137, 104])).
% 0.21/0.44 tff(144,plain,(least(M!20, R!18, X!19)), inference(lemma,lemma(discharge,[]))).
% 0.21/0.44 tff(145,plain,
% 0.21/0.44 (((~(least(M!20, R!18, X!19) | (~member(M!20, X!19)) | (~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))))) | (~((~least(M!20, R!18, X!19)) | (~((~member(M!20, X!19)) | (~![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X)))))))) | ((~least(M!20, R!18, X!19)) | (~((~member(M!20, X!19)) | (~![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X))))))),
% 0.21/0.44 inference(tautology,[status(thm)],[])).
% 0.21/0.44 tff(146,plain,
% 0.21/0.44 ((~least(M!20, R!18, X!19)) | (~((~member(M!20, X!19)) | (~![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X)))))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[145, 130])).
% 0.21/0.44 tff(147,plain,
% 0.21/0.44 ((~((~least(M!20, R!18, X!19)) | (~((~member(M!20, X!19)) | (~![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X))))))) | (~least(M!20, R!18, X!19)) | (~((~member(M!20, X!19)) | (~![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X)))))),
% 0.21/0.44 inference(tautology,[status(thm)],[])).
% 0.21/0.44 tff(148,plain,
% 0.21/0.44 ((~least(M!20, R!18, X!19)) | (~((~member(M!20, X!19)) | (~![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X)))))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[147, 146])).
% 0.21/0.44 tff(149,plain,
% 0.21/0.44 (~((~member(M!20, X!19)) | (~![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X))))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[148, 144])).
% 0.21/0.44 tff(150,plain,
% 0.21/0.44 (((~member(M!20, X!19)) | (~![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X)))) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X))),
% 0.21/0.44 inference(tautology,[status(thm)],[])).
% 0.21/0.44 tff(151,plain,
% 0.21/0.44 (![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[150, 149])).
% 0.21/0.44 tff(152,assumption,(~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))), introduced(assumption)).
% 0.21/0.44 tff(153,plain,
% 0.21/0.44 (~apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[136, 152])).
% 0.21/0.44 tff(154,plain,
% 0.21/0.44 (member(tptp_fun_X_12(M!20, X!19, R!18), X!19)),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[138, 152])).
% 0.21/0.44 tff(155,plain,
% 0.21/0.44 ($false),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[142, 154, 153, 151])).
% 0.21/0.44 tff(156,plain,((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18))), inference(lemma,lemma(discharge,[]))).
% 0.21/0.44 tff(157,plain,
% 0.21/0.44 ((~((~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))) | lower_bound(M!20, R!18, X!19))) | (~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))) | lower_bound(M!20, R!18, X!19)),
% 0.21/0.44 inference(tautology,[status(thm)],[])).
% 0.21/0.44 tff(158,plain,
% 0.21/0.44 ((~((~((~member(tptp_fun_X_12(M!20, X!19, R!18), X!19)) | apply(R!18, M!20, tptp_fun_X_12(M!20, X!19, R!18)))) | lower_bound(M!20, R!18, X!19))) | lower_bound(M!20, R!18, X!19)),
% 0.21/0.45 inference(unit_resolution,[status(thm)],[157, 156])).
% 0.21/0.45 tff(159,plain,
% 0.21/0.45 (lower_bound(M!20, R!18, X!19)),
% 0.21/0.45 inference(unit_resolution,[status(thm)],[158, 62])).
% 0.21/0.45 tff(160,plain,
% 0.21/0.45 (((~member(M!20, X!19)) | (~greatest_lower_bound(M!20, X!19, R!18, E!17))) | (~least(M!20, R!18, X!19)) | (~(((~member(M!20, X!19)) | (~greatest_lower_bound(M!20, X!19, R!18, E!17))) <=> least(M!20, R!18, X!19)))),
% 0.21/0.45 inference(tautology,[status(thm)],[])).
% 0.21/0.45 tff(161,plain,
% 0.21/0.45 (((~member(M!20, X!19)) | (~greatest_lower_bound(M!20, X!19, R!18, E!17))) | (~least(M!20, R!18, X!19))),
% 0.21/0.45 inference(unit_resolution,[status(thm)],[160, 91])).
% 0.21/0.45 tff(162,plain,
% 0.21/0.45 ((~member(M!20, X!19)) | (~greatest_lower_bound(M!20, X!19, R!18, E!17))),
% 0.21/0.45 inference(unit_resolution,[status(thm)],[161, 144])).
% 0.21/0.45 tff(163,plain,
% 0.21/0.45 (((~member(M!20, X!19)) | (~![X: $i] : ((~member(X, X!19)) | apply(R!18, M!20, X)))) | member(M!20, X!19)),
% 0.21/0.45 inference(tautology,[status(thm)],[])).
% 0.21/0.45 tff(164,plain,
% 0.21/0.45 (member(M!20, X!19)),
% 0.21/0.45 inference(unit_resolution,[status(thm)],[163, 149])).
% 0.21/0.45 tff(165,plain,
% 0.21/0.45 ((~((~member(M!20, X!19)) | (~greatest_lower_bound(M!20, X!19, R!18, E!17)))) | (~member(M!20, X!19)) | (~greatest_lower_bound(M!20, X!19, R!18, E!17))),
% 0.21/0.45 inference(tautology,[status(thm)],[])).
% 0.21/0.45 tff(166,plain,
% 0.21/0.45 (~greatest_lower_bound(M!20, X!19, R!18, E!17)),
% 0.21/0.45 inference(unit_resolution,[status(thm)],[165, 164, 162])).
% 0.21/0.45 tff(167,plain,
% 0.21/0.45 ((~(greatest_lower_bound(M!20, X!19, R!18, E!17) | (~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)) | (~member(tptp_fun_M_16(E!17, R!18, X!19, M!20), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20))))) | greatest_lower_bound(M!20, X!19, R!18, E!17) | (~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)) | (~member(tptp_fun_M_16(E!17, R!18, X!19, M!20), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20)))),
% 0.21/0.45 inference(tautology,[status(thm)],[])).
% 0.21/0.45 tff(168,plain,
% 0.21/0.45 ((~(greatest_lower_bound(M!20, X!19, R!18, E!17) | (~member(M!20, X!19)) | (~lower_bound(M!20, R!18, X!19)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)) | (~member(tptp_fun_M_16(E!17, R!18, X!19, M!20), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20))))) | (~lower_bound(M!20, R!18, X!19)) | (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)) | (~member(tptp_fun_M_16(E!17, R!18, X!19, M!20), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20)))),
% 0.21/0.45 inference(unit_resolution,[status(thm)],[167, 164, 166])).
% 0.21/0.45 tff(169,plain,
% 0.21/0.45 (~((~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)) | (~member(tptp_fun_M_16(E!17, R!18, X!19, M!20), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20))),
% 0.21/0.45 inference(unit_resolution,[status(thm)],[168, 159, 51])).
% 0.21/0.45 tff(170,plain,
% 0.21/0.45 (((~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)) | (~member(tptp_fun_M_16(E!17, R!18, X!19, M!20), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20)) | lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)),
% 0.21/0.45 inference(tautology,[status(thm)],[])).
% 0.21/0.45 tff(171,plain,
% 0.21/0.45 (lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)),
% 0.21/0.45 inference(unit_resolution,[status(thm)],[170, 169])).
% 0.21/0.45 tff(172,plain,
% 0.21/0.45 ((~((~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), X)))) | (~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), X))),
% 0.21/0.45 inference(tautology,[status(thm)],[])).
% 0.21/0.45 tff(173,plain,
% 0.21/0.45 ((~((~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), X)))) | ![X: $i] : ((~member(X, X!19)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), X))),
% 0.21/0.45 inference(unit_resolution,[status(thm)],[172, 171])).
% 0.21/0.45 tff(174,plain,
% 0.21/0.45 (![X: $i] : ((~member(X, X!19)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), X))),
% 0.21/0.45 inference(unit_resolution,[status(thm)],[173, 20])).
% 0.21/0.45 tff(175,plain,
% 0.21/0.45 (((~lower_bound(tptp_fun_M_16(E!17, R!18, X!19, M!20), R!18, X!19)) | (~member(tptp_fun_M_16(E!17, R!18, X!19, M!20), E!17)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20)) | (~apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20))),
% 0.21/0.45 inference(tautology,[status(thm)],[])).
% 0.21/0.45 tff(176,plain,
% 0.21/0.45 (~apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20)),
% 0.21/0.45 inference(unit_resolution,[status(thm)],[175, 169])).
% 0.21/0.45 tff(177,plain,
% 0.21/0.45 (((~![X: $i] : ((~member(X, X!19)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), X))) | ((~member(M!20, X!19)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20))) <=> ((~![X: $i] : ((~member(X, X!19)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), X))) | (~member(M!20, X!19)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20))),
% 0.21/0.45 inference(rewrite,[status(thm)],[])).
% 0.21/0.45 tff(178,plain,
% 0.21/0.45 ((~![X: $i] : ((~member(X, X!19)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), X))) | ((~member(M!20, X!19)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20))),
% 0.21/0.45 inference(quant_inst,[status(thm)],[])).
% 0.21/0.45 tff(179,plain,
% 0.21/0.45 ((~![X: $i] : ((~member(X, X!19)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), X))) | (~member(M!20, X!19)) | apply(R!18, tptp_fun_M_16(E!17, R!18, X!19, M!20), M!20)),
% 0.21/0.45 inference(modus_ponens,[status(thm)],[178, 177])).
% 0.21/0.45 tff(180,plain,
% 0.21/0.45 ($false),
% 0.21/0.45 inference(unit_resolution,[status(thm)],[179, 164, 176, 174])).
% 0.21/0.45 % SZS output end Proof
% 0.21/0.45 % E exiting
%------------------------------------------------------------------------------