TSTP Solution File: SET800+4 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SET800+4 : TPTP v8.1.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n022.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 : Tue Sep 20 05:08:04 EDT 2022
% Result : Theorem 0.21s 0.41s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : SET800+4 : TPTP v8.1.0. Released v3.2.0.
% 0.11/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.34 % Computer : n022.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 : Sat Sep 3 08:02:25 EDT 2022
% 0.13/0.34 % CPUTime :
% 0.13/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.13/0.34 Usage: tptp [options] [-file:]file
% 0.13/0.34 -h, -? prints this message.
% 0.13/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.13/0.34 -m, -model generate model.
% 0.13/0.34 -p, -proof generate proof.
% 0.13/0.34 -c, -core generate unsat core of named formulas.
% 0.13/0.34 -st, -statistics display statistics.
% 0.13/0.34 -t:timeout set timeout (in second).
% 0.13/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.13/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.13/0.34 -<param>:<value> configuration parameter and value.
% 0.13/0.34 -o:<output-file> file to place output in.
% 0.21/0.41 % SZS status Theorem
% 0.21/0.41 % SZS output start Proof
% 0.21/0.41 tff(apply_type, type, (
% 0.21/0.41 apply: ( $i * $i * $i ) > $o)).
% 0.21/0.41 tff(tptp_fun_M2_21_type, type, (
% 0.21/0.41 tptp_fun_M2_21: $i)).
% 0.21/0.41 tff(tptp_fun_R_18_type, type, (
% 0.21/0.41 tptp_fun_R_18: $i)).
% 0.21/0.41 tff(member_type, type, (
% 0.21/0.41 member: ( $i * $i ) > $o)).
% 0.21/0.41 tff(tptp_fun_X2_19_type, type, (
% 0.21/0.41 tptp_fun_X2_19: $i)).
% 0.21/0.41 tff(lower_bound_type, type, (
% 0.21/0.41 lower_bound: ( $i * $i * $i ) > $o)).
% 0.21/0.41 tff(tptp_fun_X_12_type, type, (
% 0.21/0.41 tptp_fun_X_12: ( $i * $i * $i ) > $i)).
% 0.21/0.41 tff(tptp_fun_E_17_type, type, (
% 0.21/0.41 tptp_fun_E_17: $i)).
% 0.21/0.41 tff(greatest_lower_bound_type, type, (
% 0.21/0.41 greatest_lower_bound: ( $i * $i * $i * $i ) > $o)).
% 0.21/0.41 tff(tptp_fun_M_16_type, type, (
% 0.21/0.41 tptp_fun_M_16: ( $i * $i * $i * $i ) > $i)).
% 0.21/0.41 tff(tptp_fun_X1_20_type, type, (
% 0.21/0.41 tptp_fun_X1_20: $i)).
% 0.21/0.41 tff(tptp_fun_M1_22_type, type, (
% 0.21/0.41 tptp_fun_M1_22: $i)).
% 0.21/0.41 tff(subset_type, type, (
% 0.21/0.41 subset: ( $i * $i ) > $o)).
% 0.21/0.41 tff(order_type, type, (
% 0.21/0.41 order: ( $i * $i ) > $o)).
% 0.21/0.41 tff(tptp_fun_X_0_type, type, (
% 0.21/0.41 tptp_fun_X_0: ( $i * $i ) > $i)).
% 0.21/0.41 tff(1,plain,
% 0.21/0.41 (^[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)))))))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(2,plain,
% 0.21/0.41 (![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))))) <=> ![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))))))))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[1])).
% 0.21/0.41 tff(3,plain,
% 0.21/0.41 (^[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)))))))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(4,plain,
% 0.21/0.41 (![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))))) <=> ![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))))))))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[3])).
% 0.21/0.41 tff(5,plain,
% 0.21/0.41 (![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))))) <=> ![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))))))))),
% 0.21/0.41 inference(transitivity,[status(thm)],[4, 2])).
% 0.21/0.41 tff(6,plain,
% 0.21/0.41 (^[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))))))))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(7,plain,
% 0.21/0.41 (![R: $i, E: $i, M: $i] : (((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X))) & (lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))) <=> ![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))))))))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[6])).
% 0.21/0.41 tff(8,plain,
% 0.21/0.41 (![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)))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(9,plain,
% 0.21/0.41 (^[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))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(10,plain,
% 0.21/0.41 (![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)))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[9])).
% 0.21/0.41 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')).
% 0.21/0.41 tff(12,plain,
% 0.21/0.41 (![R: $i, E: $i, M: $i] : (lower_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, M, X)))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[11, 10])).
% 0.21/0.41 tff(13,plain,
% 0.21/0.41 (![R: $i, E: $i, M: $i] : (lower_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, M, X)))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[12, 8])).
% 0.21/0.41 tff(14,plain,(
% 0.21/0.41 ![R: $i, E: $i, M: $i] : (((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X))) & (lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))),
% 0.21/0.41 inference(skolemize,[status(sab)],[13])).
% 0.21/0.41 tff(15,plain,
% 0.21/0.41 (![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[14, 7])).
% 0.21/0.41 tff(16,plain,
% 0.21/0.41 (![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[15, 5])).
% 0.21/0.41 tff(17,plain,
% 0.21/0.41 ((~![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))) | (~((~((~lower_bound(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))))))))),
% 0.21/0.41 inference(quant_inst,[status(thm)],[])).
% 0.21/0.41 tff(18,plain,
% 0.21/0.41 (~((~((~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) | (~((~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)))))))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[17, 16])).
% 0.21/0.41 tff(19,plain,
% 0.21/0.41 (((~((~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) | (~((~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)))),
% 0.21/0.41 inference(tautology,[status(thm)],[])).
% 0.21/0.41 tff(20,plain,
% 0.21/0.41 ((~lower_bound(M2!21, R!18, X2!19)) | ![X: $i] : ((~member(X, X2!19)) | apply(R!18, M2!21, X))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[19, 18])).
% 0.21/0.41 tff(21,plain,
% 0.21/0.41 (^[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)))))))))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(22,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)) | (~![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))))))))))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[21])).
% 0.21/0.41 tff(23,plain,
% 0.21/0.41 (^[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)))))))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(24,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)) | (~![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))))))))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[23])).
% 0.21/0.41 tff(25,plain,
% 0.21/0.41 (^[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)))))))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(26,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)) | (~![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))))))))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[25])).
% 0.21/0.41 tff(27,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)) | (~![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))))))))),
% 0.21/0.41 inference(transitivity,[status(thm)],[26, 24])).
% 0.21/0.41 tff(28,plain,
% 0.21/0.41 (^[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))))))))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(29,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) & ![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))))))))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[28])).
% 0.21/0.41 tff(30,plain,
% 0.21/0.41 (^[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))))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(31,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) & ![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)))))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[30])).
% 0.21/0.41 tff(32,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) & ![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))))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(33,plain,
% 0.21/0.41 (^[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)))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(34,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)) & ![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))))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[33])).
% 0.21/0.41 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')).
% 0.21/0.41 tff(36,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) & ![M: $i] : ((~(member(M, E) & lower_bound(M, R, X))) | apply(R, M, A))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.21/0.41 tff(37,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) & ![M: $i] : ((~(member(M, E) & lower_bound(M, R, X))) | apply(R, M, A))))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[36, 32])).
% 0.21/0.42 tff(38,plain,(
% 0.21/0.42 ![A: $i, X: $i, R: $i, E: $i] : (((~greatest_lower_bound(A, X, R, E)) | (member(A, X) & lower_bound(A, R, X) & ![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))))))),
% 0.21/0.42 inference(skolemize,[status(sab)],[37])).
% 0.21/0.42 tff(39,plain,
% 0.21/0.42 (![A: $i, X: $i, R: $i, E: $i] : (((~greatest_lower_bound(A, X, R, E)) | (member(A, X) & lower_bound(A, R, X) & ![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)))))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[38, 31])).
% 0.21/0.42 tff(40,plain,
% 0.21/0.42 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![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))))))))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[39, 29])).
% 0.21/0.42 tff(41,plain,
% 0.21/0.42 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![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))))))))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[40, 27])).
% 0.21/0.42 tff(42,plain,
% 0.21/0.42 (![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))))))))))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[41, 22])).
% 0.21/0.42 tff(43,plain,
% 0.21/0.42 (((~![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(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] : ((~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) | (~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] : ((~member(M, E!17)) | apply(R!18, M, M2!21) | (~lower_bound(M, R!18, X2!19)))))))))))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(44,plain,
% 0.21/0.42 ((~((~(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) | (~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] : ((~member(M, E!17)) | apply(R!18, M, M2!21) | (~lower_bound(M, R!18, X2!19))))))))))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(45,plain,
% 0.21/0.42 (((~![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(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) | (~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] : ((~member(M, E!17)) | apply(R!18, M, M2!21) | (~lower_bound(M, R!18, X2!19)))))))))))),
% 0.21/0.42 inference(monotonicity,[status(thm)],[44])).
% 0.21/0.42 tff(46,plain,
% 0.21/0.42 (((~![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(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) | (~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] : ((~member(M, E!17)) | apply(R!18, M, M2!21) | (~lower_bound(M, R!18, X2!19)))))))))))),
% 0.21/0.43 inference(transitivity,[status(thm)],[45, 43])).
% 0.21/0.43 tff(47,plain,
% 0.21/0.43 ((~![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))))))))))),
% 0.21/0.43 inference(quant_inst,[status(thm)],[])).
% 0.21/0.43 tff(48,plain,
% 0.21/0.43 ((~![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] : ((~member(M, E!17)) | apply(R!18, M, M2!21) | (~lower_bound(M, R!18, X2!19))))))))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[47, 46])).
% 0.21/0.43 tff(49,plain,
% 0.21/0.43 (~((~(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] : ((~member(M, E!17)) | apply(R!18, M, M2!21) | (~lower_bound(M, R!18, X2!19)))))))))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[48, 42])).
% 0.21/0.43 tff(50,plain,
% 0.21/0.43 (((~(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] : ((~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)))))))),
% 0.21/0.43 inference(tautology,[status(thm)],[])).
% 0.21/0.43 tff(51,plain,
% 0.21/0.43 ((~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))))))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[50, 49])).
% 0.21/0.43 tff(52,plain,
% 0.21/0.43 ((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))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(53,plain,
% 0.21/0.43 (((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))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(54,plain,
% 0.21/0.43 ((~(~(subset(X1!20, E!17) & subset(X2!19, E!17) & subset(X1!20, X2!19)))) <=> (subset(X1!20, E!17) & subset(X2!19, E!17) & subset(X1!20, X2!19))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(55,plain,
% 0.21/0.43 (((~(~(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))))),
% 0.21/0.43 inference(monotonicity,[status(thm)],[54])).
% 0.21/0.43 tff(56,plain,
% 0.21/0.43 (((~(~(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))))),
% 0.21/0.43 inference(transitivity,[status(thm)],[55, 53])).
% 0.21/0.43 tff(57,plain,
% 0.21/0.43 ((~(~order(R!18, E!17))) <=> order(R!18, E!17)),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(58,plain,
% 0.21/0.43 (((~(~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)))))),
% 0.21/0.43 inference(monotonicity,[status(thm)],[57, 56])).
% 0.21/0.43 tff(59,plain,
% 0.21/0.43 (((~(~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))))),
% 0.21/0.43 inference(transitivity,[status(thm)],[58, 52])).
% 0.21/0.43 tff(60,plain,
% 0.21/0.43 ((~![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)))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(61,plain,
% 0.21/0.43 ((~![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)))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(62,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/sandbox2/benchmark/theBenchmark.p','thIV12')).
% 0.21/0.43 tff(63,plain,
% 0.21/0.43 (~![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))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[62, 61])).
% 0.21/0.43 tff(64,plain,
% 0.21/0.43 (~![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))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[63, 60])).
% 0.21/0.43 tff(65,plain,
% 0.21/0.43 (~![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))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[64, 60])).
% 0.21/0.43 tff(66,plain,
% 0.21/0.43 (~![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))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[65, 60])).
% 0.21/0.43 tff(67,plain,
% 0.21/0.43 (~![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))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[66, 60])).
% 0.21/0.43 tff(68,plain,
% 0.21/0.43 (~![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))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[67, 60])).
% 0.21/0.43 tff(69,plain,
% 0.21/0.43 (~![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))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[68, 60])).
% 0.21/0.43 tff(70,plain,
% 0.21/0.43 (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)))),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[69, 59])).
% 0.21/0.44 tff(71,plain,
% 0.21/0.44 (~((~(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))),
% 0.21/0.44 inference(and_elim,[status(thm)],[70])).
% 0.21/0.44 tff(72,plain,
% 0.21/0.44 (greatest_lower_bound(M1!22, X1!20, R!18, E!17) & greatest_lower_bound(M2!21, X2!19, R!18, E!17)),
% 0.21/0.44 inference(or_elim,[status(thm)],[71])).
% 0.21/0.44 tff(73,plain,
% 0.21/0.44 (greatest_lower_bound(M2!21, X2!19, R!18, E!17)),
% 0.21/0.44 inference(and_elim,[status(thm)],[72])).
% 0.21/0.44 tff(74,plain,
% 0.21/0.44 ((~((~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))))))),
% 0.21/0.44 inference(tautology,[status(thm)],[])).
% 0.21/0.44 tff(75,plain,
% 0.21/0.44 ((~((~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))))))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[74, 73])).
% 0.21/0.44 tff(76,plain,
% 0.21/0.44 (~((~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)))))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[75, 51])).
% 0.21/0.44 tff(77,plain,
% 0.21/0.44 (((~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)),
% 0.21/0.44 inference(tautology,[status(thm)],[])).
% 0.21/0.44 tff(78,plain,
% 0.21/0.44 (lower_bound(M2!21, R!18, X2!19)),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[77, 76])).
% 0.21/0.44 tff(79,plain,
% 0.21/0.44 ((~((~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))),
% 0.21/0.44 inference(tautology,[status(thm)],[])).
% 0.21/0.44 tff(80,plain,
% 0.21/0.44 ((~((~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))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[79, 78])).
% 0.21/0.44 tff(81,plain,
% 0.21/0.44 (![X: $i] : ((~member(X, X2!19)) | apply(R!18, M2!21, X))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[80, 20])).
% 0.21/0.44 tff(82,plain,
% 0.21/0.44 (((~![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] : ((~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) | (~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] : ((~member(M, E!17)) | apply(R!18, M, M1!22) | (~lower_bound(M, R!18, X1!20)))))))))))),
% 0.21/0.44 inference(rewrite,[status(thm)],[])).
% 0.21/0.44 tff(83,plain,
% 0.21/0.44 ((~((~(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) | (~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] : ((~member(M, E!17)) | apply(R!18, M, M1!22) | (~lower_bound(M, R!18, X1!20))))))))))),
% 0.21/0.44 inference(rewrite,[status(thm)],[])).
% 0.21/0.44 tff(84,plain,
% 0.21/0.44 (((~![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) | (~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] : ((~member(M, E!17)) | apply(R!18, M, M1!22) | (~lower_bound(M, R!18, X1!20)))))))))))),
% 0.21/0.44 inference(monotonicity,[status(thm)],[83])).
% 0.21/0.44 tff(85,plain,
% 0.21/0.44 (((~![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) | (~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] : ((~member(M, E!17)) | apply(R!18, M, M1!22) | (~lower_bound(M, R!18, X1!20)))))))))))),
% 0.21/0.45 inference(transitivity,[status(thm)],[84, 82])).
% 0.21/0.45 tff(86,plain,
% 0.21/0.45 ((~![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))))))))))),
% 0.21/0.45 inference(quant_inst,[status(thm)],[])).
% 0.21/0.45 tff(87,plain,
% 0.21/0.45 ((~![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] : ((~member(M, E!17)) | apply(R!18, M, M1!22) | (~lower_bound(M, R!18, X1!20))))))))))),
% 0.21/0.45 inference(modus_ponens,[status(thm)],[86, 85])).
% 0.21/0.45 tff(88,plain,
% 0.21/0.45 (~((~(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] : ((~member(M, E!17)) | apply(R!18, M, M1!22) | (~lower_bound(M, R!18, X1!20)))))))))),
% 0.21/0.45 inference(unit_resolution,[status(thm)],[87, 42])).
% 0.21/0.45 tff(89,plain,
% 0.21/0.45 (((~(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] : ((~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)))))))),
% 0.21/0.45 inference(tautology,[status(thm)],[])).
% 0.21/0.45 tff(90,plain,
% 0.21/0.45 ((~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))))))),
% 0.21/0.45 inference(unit_resolution,[status(thm)],[89, 88])).
% 0.21/0.45 tff(91,plain,
% 0.21/0.45 (greatest_lower_bound(M1!22, X1!20, R!18, E!17)),
% 0.21/0.45 inference(and_elim,[status(thm)],[72])).
% 0.21/0.45 tff(92,plain,
% 0.21/0.45 ((~((~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))))))),
% 0.21/0.45 inference(tautology,[status(thm)],[])).
% 0.21/0.45 tff(93,plain,
% 0.21/0.45 ((~((~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))))))),
% 0.21/0.45 inference(unit_resolution,[status(thm)],[92, 91])).
% 0.21/0.45 tff(94,plain,
% 0.21/0.45 (~((~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)))))),
% 0.21/0.45 inference(unit_resolution,[status(thm)],[93, 90])).
% 0.21/0.45 tff(95,plain,
% 0.21/0.45 (((~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)),
% 0.21/0.45 inference(tautology,[status(thm)],[])).
% 0.21/0.45 tff(96,plain,
% 0.21/0.45 (member(M1!22, X1!20)),
% 0.21/0.45 inference(unit_resolution,[status(thm)],[95, 94])).
% 0.21/0.45 tff(97,plain,
% 0.21/0.45 (^[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))))))))),
% 0.21/0.45 inference(bind,[status(th)],[])).
% 0.21/0.45 tff(98,plain,
% 0.21/0.45 (![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)))))))),
% 0.21/0.45 inference(quant_intro,[status(thm)],[97])).
% 0.21/0.45 tff(99,plain,
% 0.21/0.45 (^[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))))))))),
% 0.21/0.45 inference(bind,[status(th)],[])).
% 0.21/0.45 tff(100,plain,
% 0.21/0.45 (![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)))))))),
% 0.21/0.45 inference(quant_intro,[status(thm)],[99])).
% 0.21/0.45 tff(101,plain,
% 0.21/0.45 (![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)))))))),
% 0.21/0.45 inference(transitivity,[status(thm)],[100, 98])).
% 0.21/0.45 tff(102,plain,
% 0.21/0.45 (^[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)))))))))),
% 0.21/0.45 inference(bind,[status(th)],[])).
% 0.21/0.45 tff(103,plain,
% 0.21/0.45 (![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)))))))),
% 0.21/0.45 inference(quant_intro,[status(thm)],[102])).
% 0.21/0.45 tff(104,plain,
% 0.21/0.45 (![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)))),
% 0.21/0.45 inference(rewrite,[status(thm)],[])).
% 0.21/0.45 tff(105,plain,
% 0.21/0.45 (^[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))))),
% 0.21/0.45 inference(bind,[status(th)],[])).
% 0.21/0.45 tff(106,plain,
% 0.21/0.45 (![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)))),
% 0.21/0.45 inference(quant_intro,[status(thm)],[105])).
% 0.21/0.45 tff(107,axiom,(![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : (member(X, A) => member(X, B)))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+0.ax','subset')).
% 0.21/0.45 tff(108,plain,
% 0.21/0.45 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.21/0.45 inference(modus_ponens,[status(thm)],[107, 106])).
% 0.21/0.46 tff(109,plain,
% 0.21/0.46 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.21/0.46 inference(modus_ponens,[status(thm)],[108, 104])).
% 0.21/0.46 tff(110,plain,(
% 0.21/0.46 ![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)))))),
% 0.21/0.46 inference(skolemize,[status(sab)],[109])).
% 0.21/0.46 tff(111,plain,
% 0.21/0.46 (![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)))))))),
% 0.21/0.46 inference(modus_ponens,[status(thm)],[110, 103])).
% 0.21/0.46 tff(112,plain,
% 0.21/0.46 (![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)))))))),
% 0.21/0.46 inference(modus_ponens,[status(thm)],[111, 101])).
% 0.21/0.46 tff(113,plain,
% 0.21/0.46 ((~![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)))))))),
% 0.21/0.46 inference(quant_inst,[status(thm)],[])).
% 0.21/0.46 tff(114,plain,
% 0.21/0.46 (~((~((~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))))))),
% 0.21/0.46 inference(unit_resolution,[status(thm)],[113, 112])).
% 0.21/0.46 tff(115,plain,
% 0.21/0.46 (((~((~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)))),
% 0.21/0.46 inference(tautology,[status(thm)],[])).
% 0.21/0.46 tff(116,plain,
% 0.21/0.46 ((~subset(X1!20, X2!19)) | ![X: $i] : ((~member(X, X1!20)) | member(X, X2!19))),
% 0.21/0.46 inference(unit_resolution,[status(thm)],[115, 114])).
% 0.21/0.46 tff(117,plain,
% 0.21/0.46 (subset(X1!20, X2!19)),
% 0.21/0.46 inference(and_elim,[status(thm)],[70])).
% 0.21/0.46 tff(118,plain,
% 0.21/0.46 ((~((~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))),
% 0.21/0.46 inference(tautology,[status(thm)],[])).
% 0.21/0.46 tff(119,plain,
% 0.21/0.46 ((~((~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))),
% 0.21/0.46 inference(unit_resolution,[status(thm)],[118, 117])).
% 0.21/0.46 tff(120,plain,
% 0.21/0.46 (![X: $i] : ((~member(X, X1!20)) | member(X, X2!19))),
% 0.21/0.46 inference(unit_resolution,[status(thm)],[119, 116])).
% 0.21/0.46 tff(121,plain,
% 0.21/0.46 (((~![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))),
% 0.21/0.46 inference(rewrite,[status(thm)],[])).
% 0.21/0.46 tff(122,plain,
% 0.21/0.46 ((~![X: $i] : ((~member(X, X1!20)) | member(X, X2!19))) | ((~member(M1!22, X1!20)) | member(M1!22, X2!19))),
% 0.21/0.46 inference(quant_inst,[status(thm)],[])).
% 0.21/0.46 tff(123,plain,
% 0.21/0.46 ((~![X: $i] : ((~member(X, X1!20)) | member(X, X2!19))) | (~member(M1!22, X1!20)) | member(M1!22, X2!19)),
% 0.21/0.46 inference(modus_ponens,[status(thm)],[122, 121])).
% 0.21/0.46 tff(124,plain,
% 0.21/0.46 (member(M1!22, X2!19)),
% 0.21/0.46 inference(unit_resolution,[status(thm)],[123, 120, 96])).
% 0.21/0.46 tff(125,plain,
% 0.21/0.46 (~apply(R!18, M2!21, M1!22)),
% 0.21/0.46 inference(or_elim,[status(thm)],[71])).
% 0.21/0.46 tff(126,plain,
% 0.21/0.46 (((~![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))),
% 0.21/0.46 inference(rewrite,[status(thm)],[])).
% 0.21/0.46 tff(127,plain,
% 0.21/0.46 ((~![X: $i] : ((~member(X, X2!19)) | apply(R!18, M2!21, X))) | ((~member(M1!22, X2!19)) | apply(R!18, M2!21, M1!22))),
% 0.21/0.46 inference(quant_inst,[status(thm)],[])).
% 0.21/0.46 tff(128,plain,
% 0.21/0.46 ((~![X: $i] : ((~member(X, X2!19)) | apply(R!18, M2!21, X))) | (~member(M1!22, X2!19)) | apply(R!18, M2!21, M1!22)),
% 0.21/0.46 inference(modus_ponens,[status(thm)],[127, 126])).
% 0.21/0.46 tff(129,plain,
% 0.21/0.46 ($false),
% 0.21/0.46 inference(unit_resolution,[status(thm)],[128, 125, 124, 81])).
% 0.21/0.46 % SZS output end Proof
%------------------------------------------------------------------------------