TSTP Solution File: SET799+4 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SET799+4 : TPTP v8.1.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n014.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 0.16s 0.37s
% Output : Proof 0.16s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.02/0.10 % Problem : SET799+4 : TPTP v8.1.0. Released v3.2.0.
% 0.02/0.10 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.10/0.30 % Computer : n014.cluster.edu
% 0.10/0.30 % Model : x86_64 x86_64
% 0.10/0.30 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.30 % Memory : 8042.1875MB
% 0.10/0.30 % OS : Linux 3.10.0-693.el7.x86_64
% 0.10/0.30 % CPULimit : 300
% 0.10/0.30 % WCLimit : 300
% 0.10/0.30 % DateTime : Sat Sep 3 08:01:45 EDT 2022
% 0.10/0.31 % CPUTime :
% 0.10/0.31 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.10/0.31 Usage: tptp [options] [-file:]file
% 0.10/0.31 -h, -? prints this message.
% 0.10/0.31 -smt2 print SMT-LIB2 benchmark.
% 0.10/0.31 -m, -model generate model.
% 0.10/0.31 -p, -proof generate proof.
% 0.10/0.31 -c, -core generate unsat core of named formulas.
% 0.10/0.31 -st, -statistics display statistics.
% 0.10/0.31 -t:timeout set timeout (in second).
% 0.10/0.31 -smt2status display status in smt2 format instead of SZS.
% 0.10/0.31 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.10/0.31 -<param>:<value> configuration parameter and value.
% 0.10/0.31 -o:<output-file> file to place output in.
% 0.16/0.37 % SZS status Theorem
% 0.16/0.37 % SZS output start Proof
% 0.16/0.37 tff(apply_type, type, (
% 0.16/0.37 apply: ( $i * $i * $i ) > $o)).
% 0.16/0.37 tff(tptp_fun_M2_21_type, type, (
% 0.16/0.37 tptp_fun_M2_21: $i)).
% 0.16/0.37 tff(tptp_fun_R_18_type, type, (
% 0.16/0.37 tptp_fun_R_18: $i)).
% 0.16/0.37 tff(member_type, type, (
% 0.16/0.37 member: ( $i * $i ) > $o)).
% 0.16/0.37 tff(tptp_fun_X2_19_type, type, (
% 0.16/0.37 tptp_fun_X2_19: $i)).
% 0.16/0.37 tff(upper_bound_type, type, (
% 0.16/0.37 upper_bound: ( $i * $i * $i ) > $o)).
% 0.16/0.37 tff(tptp_fun_X_11_type, type, (
% 0.16/0.37 tptp_fun_X_11: ( $i * $i * $i ) > $i)).
% 0.16/0.37 tff(tptp_fun_E_17_type, type, (
% 0.16/0.37 tptp_fun_E_17: $i)).
% 0.16/0.37 tff(least_upper_bound_type, type, (
% 0.16/0.37 least_upper_bound: ( $i * $i * $i * $i ) > $o)).
% 0.16/0.37 tff(tptp_fun_M_15_type, type, (
% 0.16/0.37 tptp_fun_M_15: ( $i * $i * $i * $i ) > $i)).
% 0.16/0.37 tff(tptp_fun_X1_20_type, type, (
% 0.16/0.37 tptp_fun_X1_20: $i)).
% 0.16/0.37 tff(tptp_fun_M1_22_type, type, (
% 0.16/0.37 tptp_fun_M1_22: $i)).
% 0.16/0.37 tff(subset_type, type, (
% 0.16/0.37 subset: ( $i * $i ) > $o)).
% 0.16/0.37 tff(order_type, type, (
% 0.16/0.37 order: ( $i * $i ) > $o)).
% 0.16/0.37 tff(tptp_fun_X_0_type, type, (
% 0.16/0.37 tptp_fun_X_0: ( $i * $i ) > $i)).
% 0.16/0.37 tff(1,plain,
% 0.16/0.37 (^[R: $i, E: $i, M: $i] : refl((~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M))))))) <=> (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M))))))))),
% 0.16/0.37 inference(bind,[status(th)],[])).
% 0.16/0.37 tff(2,plain,
% 0.16/0.37 (![R: $i, E: $i, M: $i] : (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M))))))) <=> ![R: $i, E: $i, M: $i] : (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M)))))))),
% 0.16/0.37 inference(quant_intro,[status(thm)],[1])).
% 0.16/0.37 tff(3,plain,
% 0.16/0.37 (^[R: $i, E: $i, M: $i] : rewrite((~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M))))))) <=> (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M))))))))),
% 0.16/0.37 inference(bind,[status(th)],[])).
% 0.16/0.37 tff(4,plain,
% 0.16/0.37 (![R: $i, E: $i, M: $i] : (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M))))))) <=> ![R: $i, E: $i, M: $i] : (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M)))))))),
% 0.16/0.37 inference(quant_intro,[status(thm)],[3])).
% 0.16/0.37 tff(5,plain,
% 0.16/0.37 (![R: $i, E: $i, M: $i] : (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M))))))) <=> ![R: $i, E: $i, M: $i] : (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M)))))))),
% 0.16/0.37 inference(transitivity,[status(thm)],[4, 2])).
% 0.16/0.37 tff(6,plain,
% 0.16/0.37 (^[R: $i, E: $i, M: $i] : trans(monotonicity(rewrite(((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M))) <=> ((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))), rewrite((upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M)))) <=> (upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M))))), ((((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M))) & (upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M))))) <=> (((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M))) & (upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M))))))), rewrite((((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M))) & (upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M))))) <=> (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M)))))))), ((((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M))) & (upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M))))) <=> (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M)))))))))),
% 0.16/0.37 inference(bind,[status(th)],[])).
% 0.16/0.37 tff(7,plain,
% 0.16/0.37 (![R: $i, E: $i, M: $i] : (((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M))) & (upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M))))) <=> ![R: $i, E: $i, M: $i] : (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M)))))))),
% 0.16/0.37 inference(quant_intro,[status(thm)],[6])).
% 0.16/0.37 tff(8,plain,
% 0.16/0.37 (![R: $i, E: $i, M: $i] : (upper_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, X, M))) <=> ![R: $i, E: $i, M: $i] : (upper_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, X, M)))),
% 0.16/0.37 inference(rewrite,[status(thm)],[])).
% 0.16/0.37 tff(9,plain,
% 0.16/0.37 (^[R: $i, E: $i, M: $i] : rewrite((upper_bound(M, R, E) <=> ![X: $i] : (member(X, E) => apply(R, X, M))) <=> (upper_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, X, M))))),
% 0.16/0.37 inference(bind,[status(th)],[])).
% 0.16/0.37 tff(10,plain,
% 0.16/0.37 (![R: $i, E: $i, M: $i] : (upper_bound(M, R, E) <=> ![X: $i] : (member(X, E) => apply(R, X, M))) <=> ![R: $i, E: $i, M: $i] : (upper_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, X, M)))),
% 0.16/0.37 inference(quant_intro,[status(thm)],[9])).
% 0.16/0.37 tff(11,axiom,(![R: $i, E: $i, M: $i] : (upper_bound(M, R, E) <=> ![X: $i] : (member(X, E) => apply(R, X, M)))), file('/export/starexec/sandbox/benchmark/Axioms/SET006+3.ax','upper_bound')).
% 0.16/0.37 tff(12,plain,
% 0.16/0.37 (![R: $i, E: $i, M: $i] : (upper_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, X, M)))),
% 0.16/0.37 inference(modus_ponens,[status(thm)],[11, 10])).
% 0.16/0.37 tff(13,plain,
% 0.16/0.37 (![R: $i, E: $i, M: $i] : (upper_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, X, M)))),
% 0.16/0.37 inference(modus_ponens,[status(thm)],[12, 8])).
% 0.16/0.37 tff(14,plain,(
% 0.16/0.37 ![R: $i, E: $i, M: $i] : (((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M))) & (upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M)))))),
% 0.16/0.37 inference(skolemize,[status(sab)],[13])).
% 0.16/0.37 tff(15,plain,
% 0.16/0.37 (![R: $i, E: $i, M: $i] : (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M)))))))),
% 0.16/0.37 inference(modus_ponens,[status(thm)],[14, 7])).
% 0.16/0.37 tff(16,plain,
% 0.16/0.37 (![R: $i, E: $i, M: $i] : (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M)))))))),
% 0.16/0.37 inference(modus_ponens,[status(thm)],[15, 5])).
% 0.16/0.37 tff(17,plain,
% 0.16/0.37 ((~![R: $i, E: $i, M: $i] : (~((~((~upper_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, X, M)))) | (~(upper_bound(M, R, E) | (~((~member(tptp_fun_X_11(M, E, R), E)) | apply(R, tptp_fun_X_11(M, E, R), M)))))))) | (~((~((~upper_bound(M2!21, R!18, X2!19)) | ![X: $i] : ((~member(X, X2!19)) | apply(R!18, X, M2!21)))) | (~(upper_bound(M2!21, R!18, X2!19) | (~((~member(tptp_fun_X_11(M2!21, X2!19, R!18), X2!19)) | apply(R!18, tptp_fun_X_11(M2!21, X2!19, R!18), M2!21)))))))),
% 0.16/0.37 inference(quant_inst,[status(thm)],[])).
% 0.16/0.37 tff(18,plain,
% 0.16/0.37 (~((~((~upper_bound(M2!21, R!18, X2!19)) | ![X: $i] : ((~member(X, X2!19)) | apply(R!18, X, M2!21)))) | (~(upper_bound(M2!21, R!18, X2!19) | (~((~member(tptp_fun_X_11(M2!21, X2!19, R!18), X2!19)) | apply(R!18, tptp_fun_X_11(M2!21, X2!19, R!18), M2!21))))))),
% 0.16/0.37 inference(unit_resolution,[status(thm)],[17, 16])).
% 0.16/0.37 tff(19,plain,
% 0.16/0.37 (((~((~upper_bound(M2!21, R!18, X2!19)) | ![X: $i] : ((~member(X, X2!19)) | apply(R!18, X, M2!21)))) | (~(upper_bound(M2!21, R!18, X2!19) | (~((~member(tptp_fun_X_11(M2!21, X2!19, R!18), X2!19)) | apply(R!18, tptp_fun_X_11(M2!21, X2!19, R!18), M2!21)))))) | ((~upper_bound(M2!21, R!18, X2!19)) | ![X: $i] : ((~member(X, X2!19)) | apply(R!18, X, M2!21)))),
% 0.16/0.37 inference(tautology,[status(thm)],[])).
% 0.16/0.37 tff(20,plain,
% 0.16/0.37 ((~upper_bound(M2!21, R!18, X2!19)) | ![X: $i] : ((~member(X, X2!19)) | apply(R!18, X, M2!21))),
% 0.16/0.37 inference(unit_resolution,[status(thm)],[19, 18])).
% 0.16/0.37 tff(21,plain,
% 0.16/0.37 (^[A: $i, X: $i, R: $i, E: $i] : rewrite((~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))))) <=> (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))))))),
% 0.16/0.37 inference(bind,[status(th)],[])).
% 0.16/0.37 tff(22,plain,
% 0.16/0.37 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))))) <=> ![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))),
% 0.16/0.37 inference(quant_intro,[status(thm)],[21])).
% 0.16/0.37 tff(23,plain,
% 0.16/0.37 (^[A: $i, X: $i, R: $i, E: $i] : refl((~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))))) <=> (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))))))),
% 0.16/0.37 inference(bind,[status(th)],[])).
% 0.16/0.37 tff(24,plain,
% 0.16/0.37 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))))) <=> ![A: $i, X: $i, R: $i, E: $i] : (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X))))))))),
% 0.16/0.37 inference(quant_intro,[status(thm)],[23])).
% 0.16/0.37 tff(25,plain,
% 0.16/0.37 (^[A: $i, X: $i, R: $i, E: $i] : rewrite((~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))))) <=> (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))))))),
% 0.16/0.37 inference(bind,[status(th)],[])).
% 0.16/0.37 tff(26,plain,
% 0.16/0.37 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))))) <=> ![A: $i, X: $i, R: $i, E: $i] : (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X))))))))),
% 0.16/0.37 inference(quant_intro,[status(thm)],[25])).
% 0.16/0.37 tff(27,plain,
% 0.16/0.37 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))))) <=> ![A: $i, X: $i, R: $i, E: $i] : (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X))))))))),
% 0.16/0.37 inference(transitivity,[status(thm)],[26, 24])).
% 0.16/0.37 tff(28,plain,
% 0.16/0.37 (^[A: $i, X: $i, R: $i, E: $i] : trans(monotonicity(rewrite(((~least_upper_bound(A, X, R, E)) | (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M)))) <=> ((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))), trans(monotonicity(rewrite((~((~(member(tptp_fun_M_15(E, R, X, A), E) & upper_bound(tptp_fun_M_15(E, R, X, A), R, X))) | apply(R, A, tptp_fun_M_15(E, R, X, A)))) <=> (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X))))), ((least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~((~(member(tptp_fun_M_15(E, R, X, A), E) & upper_bound(tptp_fun_M_15(E, R, X, A), R, X))) | apply(R, A, tptp_fun_M_15(E, R, X, A))))) <=> (least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X))))))), rewrite((least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X))))) <=> (least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))), ((least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~((~(member(tptp_fun_M_15(E, R, X, A), E) & upper_bound(tptp_fun_M_15(E, R, X, A), R, X))) | apply(R, A, tptp_fun_M_15(E, R, X, A))))) <=> (least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X))))))), ((((~least_upper_bound(A, X, R, E)) | (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M)))) & (least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~((~(member(tptp_fun_M_15(E, R, X, A), E) & upper_bound(tptp_fun_M_15(E, R, X, A), R, X))) | apply(R, A, tptp_fun_M_15(E, R, X, A)))))) <=> (((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))) & (least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))))), rewrite((((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))) & (least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) <=> (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X))))))))), ((((~least_upper_bound(A, X, R, E)) | (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M)))) & (least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~((~(member(tptp_fun_M_15(E, R, X, A), E) & upper_bound(tptp_fun_M_15(E, R, X, A), R, X))) | apply(R, A, tptp_fun_M_15(E, R, X, A)))))) <=> (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X))))))))))),
% 0.16/0.37 inference(bind,[status(th)],[])).
% 0.16/0.37 tff(29,plain,
% 0.16/0.37 (![A: $i, X: $i, R: $i, E: $i] : (((~least_upper_bound(A, X, R, E)) | (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M)))) & (least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~((~(member(tptp_fun_M_15(E, R, X, A), E) & upper_bound(tptp_fun_M_15(E, R, X, A), R, X))) | apply(R, A, tptp_fun_M_15(E, R, X, A)))))) <=> ![A: $i, X: $i, R: $i, E: $i] : (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X))))))))),
% 0.16/0.38 inference(quant_intro,[status(thm)],[28])).
% 0.16/0.38 tff(30,plain,
% 0.16/0.38 (^[A: $i, X: $i, R: $i, E: $i] : rewrite((((~least_upper_bound(A, X, R, E)) | (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M)))) & (least_upper_bound(A, X, R, E) | ((~member(A, X)) | (~upper_bound(A, R, X)) | (~((~(member(tptp_fun_M_15(E, R, X, A), E) & upper_bound(tptp_fun_M_15(E, R, X, A), R, X))) | apply(R, A, tptp_fun_M_15(E, R, X, A))))))) <=> (((~least_upper_bound(A, X, R, E)) | (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M)))) & (least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~((~(member(tptp_fun_M_15(E, R, X, A), E) & upper_bound(tptp_fun_M_15(E, R, X, A), R, X))) | apply(R, A, tptp_fun_M_15(E, R, X, A)))))))),
% 0.16/0.38 inference(bind,[status(th)],[])).
% 0.16/0.38 tff(31,plain,
% 0.16/0.38 (![A: $i, X: $i, R: $i, E: $i] : (((~least_upper_bound(A, X, R, E)) | (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M)))) & (least_upper_bound(A, X, R, E) | ((~member(A, X)) | (~upper_bound(A, R, X)) | (~((~(member(tptp_fun_M_15(E, R, X, A), E) & upper_bound(tptp_fun_M_15(E, R, X, A), R, X))) | apply(R, A, tptp_fun_M_15(E, R, X, A))))))) <=> ![A: $i, X: $i, R: $i, E: $i] : (((~least_upper_bound(A, X, R, E)) | (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M)))) & (least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~((~(member(tptp_fun_M_15(E, R, X, A), E) & upper_bound(tptp_fun_M_15(E, R, X, A), R, X))) | apply(R, A, tptp_fun_M_15(E, R, X, A))))))),
% 0.16/0.38 inference(quant_intro,[status(thm)],[30])).
% 0.16/0.38 tff(32,plain,
% 0.16/0.38 (![A: $i, X: $i, R: $i, E: $i] : (least_upper_bound(A, X, R, E) <=> (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M)))) <=> ![A: $i, X: $i, R: $i, E: $i] : (least_upper_bound(A, X, R, E) <=> (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M))))),
% 0.16/0.38 inference(rewrite,[status(thm)],[])).
% 0.16/0.38 tff(33,plain,
% 0.16/0.38 (^[A: $i, X: $i, R: $i, E: $i] : rewrite((least_upper_bound(A, X, R, E) <=> ((member(A, X) & upper_bound(A, R, X)) & ![M: $i] : ((member(M, E) & upper_bound(M, R, X)) => apply(R, A, M)))) <=> (least_upper_bound(A, X, R, E) <=> (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M)))))),
% 0.16/0.38 inference(bind,[status(th)],[])).
% 0.16/0.38 tff(34,plain,
% 0.16/0.38 (![A: $i, X: $i, R: $i, E: $i] : (least_upper_bound(A, X, R, E) <=> ((member(A, X) & upper_bound(A, R, X)) & ![M: $i] : ((member(M, E) & upper_bound(M, R, X)) => apply(R, A, M)))) <=> ![A: $i, X: $i, R: $i, E: $i] : (least_upper_bound(A, X, R, E) <=> (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M))))),
% 0.16/0.38 inference(quant_intro,[status(thm)],[33])).
% 0.16/0.38 tff(35,axiom,(![A: $i, X: $i, R: $i, E: $i] : (least_upper_bound(A, X, R, E) <=> ((member(A, X) & upper_bound(A, R, X)) & ![M: $i] : ((member(M, E) & upper_bound(M, R, X)) => apply(R, A, M))))), file('/export/starexec/sandbox/benchmark/Axioms/SET006+3.ax','least_upper_bound')).
% 0.16/0.38 tff(36,plain,
% 0.16/0.38 (![A: $i, X: $i, R: $i, E: $i] : (least_upper_bound(A, X, R, E) <=> (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M))))),
% 0.16/0.38 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.16/0.38 tff(37,plain,
% 0.16/0.38 (![A: $i, X: $i, R: $i, E: $i] : (least_upper_bound(A, X, R, E) <=> (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M))))),
% 0.16/0.38 inference(modus_ponens,[status(thm)],[36, 32])).
% 0.16/0.38 tff(38,plain,(
% 0.16/0.38 ![A: $i, X: $i, R: $i, E: $i] : (((~least_upper_bound(A, X, R, E)) | (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M)))) & (least_upper_bound(A, X, R, E) | ((~member(A, X)) | (~upper_bound(A, R, X)) | (~((~(member(tptp_fun_M_15(E, R, X, A), E) & upper_bound(tptp_fun_M_15(E, R, X, A), R, X))) | apply(R, A, tptp_fun_M_15(E, R, X, A)))))))),
% 0.16/0.38 inference(skolemize,[status(sab)],[37])).
% 0.16/0.38 tff(39,plain,
% 0.16/0.38 (![A: $i, X: $i, R: $i, E: $i] : (((~least_upper_bound(A, X, R, E)) | (member(A, X) & upper_bound(A, R, X) & ![M: $i] : ((~(member(M, E) & upper_bound(M, R, X))) | apply(R, A, M)))) & (least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~((~(member(tptp_fun_M_15(E, R, X, A), E) & upper_bound(tptp_fun_M_15(E, R, X, A), R, X))) | apply(R, A, tptp_fun_M_15(E, R, X, A))))))),
% 0.16/0.38 inference(modus_ponens,[status(thm)],[38, 31])).
% 0.16/0.38 tff(40,plain,
% 0.16/0.38 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X))))))))),
% 0.16/0.38 inference(modus_ponens,[status(thm)],[39, 29])).
% 0.16/0.38 tff(41,plain,
% 0.16/0.38 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X)))))))) | (~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X))))))))),
% 0.16/0.38 inference(modus_ponens,[status(thm)],[40, 27])).
% 0.16/0.38 tff(42,plain,
% 0.16/0.38 (![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))),
% 0.16/0.38 inference(modus_ponens,[status(thm)],[41, 22])).
% 0.16/0.38 tff(43,plain,
% 0.16/0.38 (((~![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))) | (~((~(least_upper_bound(M2!21, X2!19, R!18, E!17) | (~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~(apply(R!18, M2!21, tptp_fun_M_15(E!17, R!18, X2!19, M2!21)) | (~member(tptp_fun_M_15(E!17, R!18, X2!19, M2!21), E!17)) | (~upper_bound(tptp_fun_M_15(E!17, R!18, X2!19, M2!21), R!18, X2!19)))))) | (~((~least_upper_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M2!21, M) | (~upper_bound(M, R!18, X2!19))))))))))) <=> ((~![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))) | (~((~(least_upper_bound(M2!21, X2!19, R!18, E!17) | (~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~(apply(R!18, M2!21, tptp_fun_M_15(E!17, R!18, X2!19, M2!21)) | (~member(tptp_fun_M_15(E!17, R!18, X2!19, M2!21), E!17)) | (~upper_bound(tptp_fun_M_15(E!17, R!18, X2!19, M2!21), R!18, X2!19)))))) | (~((~least_upper_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M2!21, M) | (~upper_bound(M, R!18, X2!19)))))))))))),
% 0.16/0.38 inference(rewrite,[status(thm)],[])).
% 0.16/0.38 tff(44,plain,
% 0.16/0.38 ((~((~(least_upper_bound(M2!21, X2!19, R!18, E!17) | (~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~(apply(R!18, M2!21, tptp_fun_M_15(E!17, R!18, X2!19, M2!21)) | (~member(tptp_fun_M_15(E!17, R!18, X2!19, M2!21), E!17)) | (~upper_bound(tptp_fun_M_15(E!17, R!18, X2!19, M2!21), R!18, X2!19)))))) | (~((~least_upper_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~![M: $i] : (apply(R!18, M2!21, M) | (~member(M, E!17)) | (~upper_bound(M, R!18, X2!19)))))))))) <=> (~((~(least_upper_bound(M2!21, X2!19, R!18, E!17) | (~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~(apply(R!18, M2!21, tptp_fun_M_15(E!17, R!18, X2!19, M2!21)) | (~member(tptp_fun_M_15(E!17, R!18, X2!19, M2!21), E!17)) | (~upper_bound(tptp_fun_M_15(E!17, R!18, X2!19, M2!21), R!18, X2!19)))))) | (~((~least_upper_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M2!21, M) | (~upper_bound(M, R!18, X2!19))))))))))),
% 0.16/0.38 inference(rewrite,[status(thm)],[])).
% 0.16/0.38 tff(45,plain,
% 0.16/0.38 (((~![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))) | (~((~(least_upper_bound(M2!21, X2!19, R!18, E!17) | (~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~(apply(R!18, M2!21, tptp_fun_M_15(E!17, R!18, X2!19, M2!21)) | (~member(tptp_fun_M_15(E!17, R!18, X2!19, M2!21), E!17)) | (~upper_bound(tptp_fun_M_15(E!17, R!18, X2!19, M2!21), R!18, X2!19)))))) | (~((~least_upper_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~![M: $i] : (apply(R!18, M2!21, M) | (~member(M, E!17)) | (~upper_bound(M, R!18, X2!19))))))))))) <=> ((~![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))) | (~((~(least_upper_bound(M2!21, X2!19, R!18, E!17) | (~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~(apply(R!18, M2!21, tptp_fun_M_15(E!17, R!18, X2!19, M2!21)) | (~member(tptp_fun_M_15(E!17, R!18, X2!19, M2!21), E!17)) | (~upper_bound(tptp_fun_M_15(E!17, R!18, X2!19, M2!21), R!18, X2!19)))))) | (~((~least_upper_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M2!21, M) | (~upper_bound(M, R!18, X2!19)))))))))))),
% 0.16/0.38 inference(monotonicity,[status(thm)],[44])).
% 0.16/0.38 tff(46,plain,
% 0.16/0.38 (((~![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))) | (~((~(least_upper_bound(M2!21, X2!19, R!18, E!17) | (~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~(apply(R!18, M2!21, tptp_fun_M_15(E!17, R!18, X2!19, M2!21)) | (~member(tptp_fun_M_15(E!17, R!18, X2!19, M2!21), E!17)) | (~upper_bound(tptp_fun_M_15(E!17, R!18, X2!19, M2!21), R!18, X2!19)))))) | (~((~least_upper_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~![M: $i] : (apply(R!18, M2!21, M) | (~member(M, E!17)) | (~upper_bound(M, R!18, X2!19))))))))))) <=> ((~![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))) | (~((~(least_upper_bound(M2!21, X2!19, R!18, E!17) | (~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~(apply(R!18, M2!21, tptp_fun_M_15(E!17, R!18, X2!19, M2!21)) | (~member(tptp_fun_M_15(E!17, R!18, X2!19, M2!21), E!17)) | (~upper_bound(tptp_fun_M_15(E!17, R!18, X2!19, M2!21), R!18, X2!19)))))) | (~((~least_upper_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M2!21, M) | (~upper_bound(M, R!18, X2!19)))))))))))),
% 0.16/0.39 inference(transitivity,[status(thm)],[45, 43])).
% 0.16/0.39 tff(47,plain,
% 0.16/0.39 ((~![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))) | (~((~(least_upper_bound(M2!21, X2!19, R!18, E!17) | (~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~(apply(R!18, M2!21, tptp_fun_M_15(E!17, R!18, X2!19, M2!21)) | (~member(tptp_fun_M_15(E!17, R!18, X2!19, M2!21), E!17)) | (~upper_bound(tptp_fun_M_15(E!17, R!18, X2!19, M2!21), R!18, X2!19)))))) | (~((~least_upper_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~![M: $i] : (apply(R!18, M2!21, M) | (~member(M, E!17)) | (~upper_bound(M, R!18, X2!19))))))))))),
% 0.16/0.39 inference(quant_inst,[status(thm)],[])).
% 0.16/0.39 tff(48,plain,
% 0.16/0.39 ((~![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))) | (~((~(least_upper_bound(M2!21, X2!19, R!18, E!17) | (~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~(apply(R!18, M2!21, tptp_fun_M_15(E!17, R!18, X2!19, M2!21)) | (~member(tptp_fun_M_15(E!17, R!18, X2!19, M2!21), E!17)) | (~upper_bound(tptp_fun_M_15(E!17, R!18, X2!19, M2!21), R!18, X2!19)))))) | (~((~least_upper_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M2!21, M) | (~upper_bound(M, R!18, X2!19))))))))))),
% 0.16/0.39 inference(modus_ponens,[status(thm)],[47, 46])).
% 0.16/0.39 tff(49,plain,
% 0.16/0.39 (~((~(least_upper_bound(M2!21, X2!19, R!18, E!17) | (~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~(apply(R!18, M2!21, tptp_fun_M_15(E!17, R!18, X2!19, M2!21)) | (~member(tptp_fun_M_15(E!17, R!18, X2!19, M2!21), E!17)) | (~upper_bound(tptp_fun_M_15(E!17, R!18, X2!19, M2!21), R!18, X2!19)))))) | (~((~least_upper_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M2!21, M) | (~upper_bound(M, R!18, X2!19)))))))))),
% 0.16/0.39 inference(unit_resolution,[status(thm)],[48, 42])).
% 0.16/0.39 tff(50,plain,
% 0.16/0.39 (((~(least_upper_bound(M2!21, X2!19, R!18, E!17) | (~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~(apply(R!18, M2!21, tptp_fun_M_15(E!17, R!18, X2!19, M2!21)) | (~member(tptp_fun_M_15(E!17, R!18, X2!19, M2!21), E!17)) | (~upper_bound(tptp_fun_M_15(E!17, R!18, X2!19, M2!21), R!18, X2!19)))))) | (~((~least_upper_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M2!21, M) | (~upper_bound(M, R!18, X2!19))))))))) | ((~least_upper_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M2!21, M) | (~upper_bound(M, R!18, X2!19)))))))),
% 0.16/0.39 inference(tautology,[status(thm)],[])).
% 0.16/0.39 tff(51,plain,
% 0.16/0.39 ((~least_upper_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M2!21, M) | (~upper_bound(M, R!18, X2!19))))))),
% 0.16/0.39 inference(unit_resolution,[status(thm)],[50, 49])).
% 0.16/0.39 tff(52,plain,
% 0.16/0.39 ((order(R!18, E!17) & (subset(X1!20, E!17) & subset(X2!19, E!17) & subset(X1!20, X2!19) & (~((~(least_upper_bound(M1!22, X1!20, R!18, E!17) & least_upper_bound(M2!21, X2!19, R!18, E!17))) | apply(R!18, M1!22, M2!21))))) <=> (order(R!18, E!17) & subset(X1!20, E!17) & subset(X2!19, E!17) & subset(X1!20, X2!19) & (~((~(least_upper_bound(M1!22, X1!20, R!18, E!17) & least_upper_bound(M2!21, X2!19, R!18, E!17))) | apply(R!18, M1!22, M2!21))))),
% 0.16/0.39 inference(rewrite,[status(thm)],[])).
% 0.16/0.39 tff(53,plain,
% 0.16/0.39 (((subset(X1!20, E!17) & subset(X2!19, E!17) & subset(X1!20, X2!19)) & (~((~(least_upper_bound(M1!22, X1!20, R!18, E!17) & least_upper_bound(M2!21, X2!19, R!18, E!17))) | apply(R!18, M1!22, M2!21)))) <=> (subset(X1!20, E!17) & subset(X2!19, E!17) & subset(X1!20, X2!19) & (~((~(least_upper_bound(M1!22, X1!20, R!18, E!17) & least_upper_bound(M2!21, X2!19, R!18, E!17))) | apply(R!18, M1!22, M2!21))))),
% 0.16/0.39 inference(rewrite,[status(thm)],[])).
% 0.16/0.39 tff(54,plain,
% 0.16/0.39 ((~(~(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))),
% 0.16/0.39 inference(rewrite,[status(thm)],[])).
% 0.16/0.39 tff(55,plain,
% 0.16/0.39 (((~(~(subset(X1!20, E!17) & subset(X2!19, E!17) & subset(X1!20, X2!19)))) & (~((~(least_upper_bound(M1!22, X1!20, R!18, E!17) & least_upper_bound(M2!21, X2!19, R!18, E!17))) | apply(R!18, M1!22, M2!21)))) <=> ((subset(X1!20, E!17) & subset(X2!19, E!17) & subset(X1!20, X2!19)) & (~((~(least_upper_bound(M1!22, X1!20, R!18, E!17) & least_upper_bound(M2!21, X2!19, R!18, E!17))) | apply(R!18, M1!22, M2!21))))),
% 0.16/0.39 inference(monotonicity,[status(thm)],[54])).
% 0.16/0.39 tff(56,plain,
% 0.16/0.39 (((~(~(subset(X1!20, E!17) & subset(X2!19, E!17) & subset(X1!20, X2!19)))) & (~((~(least_upper_bound(M1!22, X1!20, R!18, E!17) & least_upper_bound(M2!21, X2!19, R!18, E!17))) | apply(R!18, M1!22, M2!21)))) <=> (subset(X1!20, E!17) & subset(X2!19, E!17) & subset(X1!20, X2!19) & (~((~(least_upper_bound(M1!22, X1!20, R!18, E!17) & least_upper_bound(M2!21, X2!19, R!18, E!17))) | apply(R!18, M1!22, M2!21))))),
% 0.16/0.39 inference(transitivity,[status(thm)],[55, 53])).
% 0.16/0.39 tff(57,plain,
% 0.16/0.39 ((~(~order(R!18, E!17))) <=> order(R!18, E!17)),
% 0.16/0.39 inference(rewrite,[status(thm)],[])).
% 0.16/0.39 tff(58,plain,
% 0.16/0.39 (((~(~order(R!18, E!17))) & ((~(~(subset(X1!20, E!17) & subset(X2!19, E!17) & subset(X1!20, X2!19)))) & (~((~(least_upper_bound(M1!22, X1!20, R!18, E!17) & least_upper_bound(M2!21, X2!19, R!18, E!17))) | apply(R!18, M1!22, M2!21))))) <=> (order(R!18, E!17) & (subset(X1!20, E!17) & subset(X2!19, E!17) & subset(X1!20, X2!19) & (~((~(least_upper_bound(M1!22, X1!20, R!18, E!17) & least_upper_bound(M2!21, X2!19, R!18, E!17))) | apply(R!18, M1!22, M2!21)))))),
% 0.16/0.39 inference(monotonicity,[status(thm)],[57, 56])).
% 0.16/0.39 tff(59,plain,
% 0.16/0.39 (((~(~order(R!18, E!17))) & ((~(~(subset(X1!20, E!17) & subset(X2!19, E!17) & subset(X1!20, X2!19)))) & (~((~(least_upper_bound(M1!22, X1!20, R!18, E!17) & least_upper_bound(M2!21, X2!19, R!18, E!17))) | apply(R!18, M1!22, M2!21))))) <=> (order(R!18, E!17) & subset(X1!20, E!17) & subset(X2!19, E!17) & subset(X1!20, X2!19) & (~((~(least_upper_bound(M1!22, X1!20, R!18, E!17) & least_upper_bound(M2!21, X2!19, R!18, E!17))) | apply(R!18, M1!22, M2!21))))),
% 0.16/0.39 inference(transitivity,[status(thm)],[58, 52])).
% 0.16/0.39 tff(60,plain,
% 0.16/0.39 ((~![R: $i, E: $i] : ((~order(R, E)) | ![X1: $i, X2: $i] : ((~(subset(X1, E) & subset(X2, E) & subset(X1, X2))) | ![M1: $i, M2: $i] : ((~(least_upper_bound(M1, X1, R, E) & least_upper_bound(M2, X2, R, E))) | apply(R, M1, M2))))) <=> (~![R: $i, E: $i] : ((~order(R, E)) | ![X1: $i, X2: $i] : ((~(subset(X1, E) & subset(X2, E) & subset(X1, X2))) | ![M1: $i, M2: $i] : ((~(least_upper_bound(M1, X1, R, E) & least_upper_bound(M2, X2, R, E))) | apply(R, M1, M2)))))),
% 0.16/0.39 inference(rewrite,[status(thm)],[])).
% 0.16/0.39 tff(61,plain,
% 0.16/0.39 ((~![R: $i, E: $i] : (order(R, E) => ![X1: $i, X2: $i] : (((subset(X1, E) & subset(X2, E)) & subset(X1, X2)) => ![M1: $i, M2: $i] : ((least_upper_bound(M1, X1, R, E) & least_upper_bound(M2, X2, R, E)) => apply(R, M1, M2))))) <=> (~![R: $i, E: $i] : ((~order(R, E)) | ![X1: $i, X2: $i] : ((~(subset(X1, E) & subset(X2, E) & subset(X1, X2))) | ![M1: $i, M2: $i] : ((~(least_upper_bound(M1, X1, R, E) & least_upper_bound(M2, X2, R, E))) | apply(R, M1, M2)))))),
% 0.16/0.39 inference(rewrite,[status(thm)],[])).
% 0.16/0.39 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] : ((least_upper_bound(M1, X1, R, E) & least_upper_bound(M2, X2, R, E)) => apply(R, M1, M2))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','thIV11')).
% 0.16/0.39 tff(63,plain,
% 0.16/0.39 (~![R: $i, E: $i] : ((~order(R, E)) | ![X1: $i, X2: $i] : ((~(subset(X1, E) & subset(X2, E) & subset(X1, X2))) | ![M1: $i, M2: $i] : ((~(least_upper_bound(M1, X1, R, E) & least_upper_bound(M2, X2, R, E))) | apply(R, M1, M2))))),
% 0.16/0.39 inference(modus_ponens,[status(thm)],[62, 61])).
% 0.16/0.39 tff(64,plain,
% 0.16/0.39 (~![R: $i, E: $i] : ((~order(R, E)) | ![X1: $i, X2: $i] : ((~(subset(X1, E) & subset(X2, E) & subset(X1, X2))) | ![M1: $i, M2: $i] : ((~(least_upper_bound(M1, X1, R, E) & least_upper_bound(M2, X2, R, E))) | apply(R, M1, M2))))),
% 0.16/0.39 inference(modus_ponens,[status(thm)],[63, 60])).
% 0.16/0.39 tff(65,plain,
% 0.16/0.39 (~![R: $i, E: $i] : ((~order(R, E)) | ![X1: $i, X2: $i] : ((~(subset(X1, E) & subset(X2, E) & subset(X1, X2))) | ![M1: $i, M2: $i] : ((~(least_upper_bound(M1, X1, R, E) & least_upper_bound(M2, X2, R, E))) | apply(R, M1, M2))))),
% 0.16/0.39 inference(modus_ponens,[status(thm)],[64, 60])).
% 0.16/0.39 tff(66,plain,
% 0.16/0.39 (~![R: $i, E: $i] : ((~order(R, E)) | ![X1: $i, X2: $i] : ((~(subset(X1, E) & subset(X2, E) & subset(X1, X2))) | ![M1: $i, M2: $i] : ((~(least_upper_bound(M1, X1, R, E) & least_upper_bound(M2, X2, R, E))) | apply(R, M1, M2))))),
% 0.16/0.39 inference(modus_ponens,[status(thm)],[65, 60])).
% 0.16/0.39 tff(67,plain,
% 0.16/0.39 (~![R: $i, E: $i] : ((~order(R, E)) | ![X1: $i, X2: $i] : ((~(subset(X1, E) & subset(X2, E) & subset(X1, X2))) | ![M1: $i, M2: $i] : ((~(least_upper_bound(M1, X1, R, E) & least_upper_bound(M2, X2, R, E))) | apply(R, M1, M2))))),
% 0.16/0.39 inference(modus_ponens,[status(thm)],[66, 60])).
% 0.16/0.39 tff(68,plain,
% 0.16/0.39 (~![R: $i, E: $i] : ((~order(R, E)) | ![X1: $i, X2: $i] : ((~(subset(X1, E) & subset(X2, E) & subset(X1, X2))) | ![M1: $i, M2: $i] : ((~(least_upper_bound(M1, X1, R, E) & least_upper_bound(M2, X2, R, E))) | apply(R, M1, M2))))),
% 0.16/0.39 inference(modus_ponens,[status(thm)],[67, 60])).
% 0.16/0.39 tff(69,plain,
% 0.16/0.39 (~![R: $i, E: $i] : ((~order(R, E)) | ![X1: $i, X2: $i] : ((~(subset(X1, E) & subset(X2, E) & subset(X1, X2))) | ![M1: $i, M2: $i] : ((~(least_upper_bound(M1, X1, R, E) & least_upper_bound(M2, X2, R, E))) | apply(R, M1, M2))))),
% 0.16/0.39 inference(modus_ponens,[status(thm)],[68, 60])).
% 0.16/0.39 tff(70,plain,
% 0.16/0.39 (order(R!18, E!17) & subset(X1!20, E!17) & subset(X2!19, E!17) & subset(X1!20, X2!19) & (~((~(least_upper_bound(M1!22, X1!20, R!18, E!17) & least_upper_bound(M2!21, X2!19, R!18, E!17))) | apply(R!18, M1!22, M2!21)))),
% 0.16/0.39 inference(modus_ponens,[status(thm)],[69, 59])).
% 0.16/0.39 tff(71,plain,
% 0.16/0.39 (~((~(least_upper_bound(M1!22, X1!20, R!18, E!17) & least_upper_bound(M2!21, X2!19, R!18, E!17))) | apply(R!18, M1!22, M2!21))),
% 0.16/0.39 inference(and_elim,[status(thm)],[70])).
% 0.16/0.39 tff(72,plain,
% 0.16/0.39 (least_upper_bound(M1!22, X1!20, R!18, E!17) & least_upper_bound(M2!21, X2!19, R!18, E!17)),
% 0.16/0.40 inference(or_elim,[status(thm)],[71])).
% 0.16/0.40 tff(73,plain,
% 0.16/0.40 (least_upper_bound(M2!21, X2!19, R!18, E!17)),
% 0.16/0.40 inference(and_elim,[status(thm)],[72])).
% 0.16/0.40 tff(74,plain,
% 0.16/0.40 ((~((~least_upper_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M2!21, M) | (~upper_bound(M, R!18, X2!19)))))))) | (~least_upper_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M2!21, M) | (~upper_bound(M, R!18, X2!19))))))),
% 0.16/0.40 inference(tautology,[status(thm)],[])).
% 0.16/0.40 tff(75,plain,
% 0.16/0.40 ((~((~least_upper_bound(M2!21, X2!19, R!18, E!17)) | (~((~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M2!21, M) | (~upper_bound(M, R!18, X2!19)))))))) | (~((~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M2!21, M) | (~upper_bound(M, R!18, X2!19))))))),
% 0.16/0.40 inference(unit_resolution,[status(thm)],[74, 73])).
% 0.16/0.40 tff(76,plain,
% 0.16/0.40 (~((~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M2!21, M) | (~upper_bound(M, R!18, X2!19)))))),
% 0.16/0.40 inference(unit_resolution,[status(thm)],[75, 51])).
% 0.16/0.40 tff(77,plain,
% 0.16/0.40 (((~member(M2!21, X2!19)) | (~upper_bound(M2!21, R!18, X2!19)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M2!21, M) | (~upper_bound(M, R!18, X2!19))))) | upper_bound(M2!21, R!18, X2!19)),
% 0.16/0.40 inference(tautology,[status(thm)],[])).
% 0.16/0.40 tff(78,plain,
% 0.16/0.40 (upper_bound(M2!21, R!18, X2!19)),
% 0.16/0.40 inference(unit_resolution,[status(thm)],[77, 76])).
% 0.16/0.40 tff(79,plain,
% 0.16/0.40 ((~((~upper_bound(M2!21, R!18, X2!19)) | ![X: $i] : ((~member(X, X2!19)) | apply(R!18, X, M2!21)))) | (~upper_bound(M2!21, R!18, X2!19)) | ![X: $i] : ((~member(X, X2!19)) | apply(R!18, X, M2!21))),
% 0.16/0.40 inference(tautology,[status(thm)],[])).
% 0.16/0.40 tff(80,plain,
% 0.16/0.40 ((~((~upper_bound(M2!21, R!18, X2!19)) | ![X: $i] : ((~member(X, X2!19)) | apply(R!18, X, M2!21)))) | ![X: $i] : ((~member(X, X2!19)) | apply(R!18, X, M2!21))),
% 0.16/0.40 inference(unit_resolution,[status(thm)],[79, 78])).
% 0.16/0.40 tff(81,plain,
% 0.16/0.40 (![X: $i] : ((~member(X, X2!19)) | apply(R!18, X, M2!21))),
% 0.16/0.40 inference(unit_resolution,[status(thm)],[80, 20])).
% 0.16/0.40 tff(82,plain,
% 0.16/0.40 (((~![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))) | (~((~(least_upper_bound(M1!22, X1!20, R!18, E!17) | (~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~(apply(R!18, M1!22, tptp_fun_M_15(E!17, R!18, X1!20, M1!22)) | (~member(tptp_fun_M_15(E!17, R!18, X1!20, M1!22), E!17)) | (~upper_bound(tptp_fun_M_15(E!17, R!18, X1!20, M1!22), R!18, X1!20)))))) | (~((~least_upper_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M1!22, M) | (~upper_bound(M, R!18, X1!20))))))))))) <=> ((~![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))) | (~((~(least_upper_bound(M1!22, X1!20, R!18, E!17) | (~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~(apply(R!18, M1!22, tptp_fun_M_15(E!17, R!18, X1!20, M1!22)) | (~member(tptp_fun_M_15(E!17, R!18, X1!20, M1!22), E!17)) | (~upper_bound(tptp_fun_M_15(E!17, R!18, X1!20, M1!22), R!18, X1!20)))))) | (~((~least_upper_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M1!22, M) | (~upper_bound(M, R!18, X1!20)))))))))))),
% 0.16/0.40 inference(rewrite,[status(thm)],[])).
% 0.16/0.40 tff(83,plain,
% 0.16/0.40 ((~((~(least_upper_bound(M1!22, X1!20, R!18, E!17) | (~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~(apply(R!18, M1!22, tptp_fun_M_15(E!17, R!18, X1!20, M1!22)) | (~member(tptp_fun_M_15(E!17, R!18, X1!20, M1!22), E!17)) | (~upper_bound(tptp_fun_M_15(E!17, R!18, X1!20, M1!22), R!18, X1!20)))))) | (~((~least_upper_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~![M: $i] : (apply(R!18, M1!22, M) | (~member(M, E!17)) | (~upper_bound(M, R!18, X1!20)))))))))) <=> (~((~(least_upper_bound(M1!22, X1!20, R!18, E!17) | (~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~(apply(R!18, M1!22, tptp_fun_M_15(E!17, R!18, X1!20, M1!22)) | (~member(tptp_fun_M_15(E!17, R!18, X1!20, M1!22), E!17)) | (~upper_bound(tptp_fun_M_15(E!17, R!18, X1!20, M1!22), R!18, X1!20)))))) | (~((~least_upper_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M1!22, M) | (~upper_bound(M, R!18, X1!20))))))))))),
% 0.16/0.40 inference(rewrite,[status(thm)],[])).
% 0.16/0.40 tff(84,plain,
% 0.16/0.40 (((~![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))) | (~((~(least_upper_bound(M1!22, X1!20, R!18, E!17) | (~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~(apply(R!18, M1!22, tptp_fun_M_15(E!17, R!18, X1!20, M1!22)) | (~member(tptp_fun_M_15(E!17, R!18, X1!20, M1!22), E!17)) | (~upper_bound(tptp_fun_M_15(E!17, R!18, X1!20, M1!22), R!18, X1!20)))))) | (~((~least_upper_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~![M: $i] : (apply(R!18, M1!22, M) | (~member(M, E!17)) | (~upper_bound(M, R!18, X1!20))))))))))) <=> ((~![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))) | (~((~(least_upper_bound(M1!22, X1!20, R!18, E!17) | (~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~(apply(R!18, M1!22, tptp_fun_M_15(E!17, R!18, X1!20, M1!22)) | (~member(tptp_fun_M_15(E!17, R!18, X1!20, M1!22), E!17)) | (~upper_bound(tptp_fun_M_15(E!17, R!18, X1!20, M1!22), R!18, X1!20)))))) | (~((~least_upper_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M1!22, M) | (~upper_bound(M, R!18, X1!20)))))))))))),
% 0.16/0.40 inference(monotonicity,[status(thm)],[83])).
% 0.16/0.40 tff(85,plain,
% 0.16/0.40 (((~![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))) | (~((~(least_upper_bound(M1!22, X1!20, R!18, E!17) | (~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~(apply(R!18, M1!22, tptp_fun_M_15(E!17, R!18, X1!20, M1!22)) | (~member(tptp_fun_M_15(E!17, R!18, X1!20, M1!22), E!17)) | (~upper_bound(tptp_fun_M_15(E!17, R!18, X1!20, M1!22), R!18, X1!20)))))) | (~((~least_upper_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~![M: $i] : (apply(R!18, M1!22, M) | (~member(M, E!17)) | (~upper_bound(M, R!18, X1!20))))))))))) <=> ((~![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))) | (~((~(least_upper_bound(M1!22, X1!20, R!18, E!17) | (~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~(apply(R!18, M1!22, tptp_fun_M_15(E!17, R!18, X1!20, M1!22)) | (~member(tptp_fun_M_15(E!17, R!18, X1!20, M1!22), E!17)) | (~upper_bound(tptp_fun_M_15(E!17, R!18, X1!20, M1!22), R!18, X1!20)))))) | (~((~least_upper_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M1!22, M) | (~upper_bound(M, R!18, X1!20)))))))))))),
% 0.16/0.41 inference(transitivity,[status(thm)],[84, 82])).
% 0.16/0.41 tff(86,plain,
% 0.16/0.41 ((~![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))) | (~((~(least_upper_bound(M1!22, X1!20, R!18, E!17) | (~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~(apply(R!18, M1!22, tptp_fun_M_15(E!17, R!18, X1!20, M1!22)) | (~member(tptp_fun_M_15(E!17, R!18, X1!20, M1!22), E!17)) | (~upper_bound(tptp_fun_M_15(E!17, R!18, X1!20, M1!22), R!18, X1!20)))))) | (~((~least_upper_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~![M: $i] : (apply(R!18, M1!22, M) | (~member(M, E!17)) | (~upper_bound(M, R!18, X1!20))))))))))),
% 0.16/0.41 inference(quant_inst,[status(thm)],[])).
% 0.16/0.41 tff(87,plain,
% 0.16/0.41 ((~![A: $i, X: $i, R: $i, E: $i] : (~((~(least_upper_bound(A, X, R, E) | (~member(A, X)) | (~upper_bound(A, R, X)) | (~(apply(R, A, tptp_fun_M_15(E, R, X, A)) | (~member(tptp_fun_M_15(E, R, X, A), E)) | (~upper_bound(tptp_fun_M_15(E, R, X, A), R, X)))))) | (~((~least_upper_bound(A, X, R, E)) | (~((~member(A, X)) | (~upper_bound(A, R, X)) | (~![M: $i] : (apply(R, A, M) | (~member(M, E)) | (~upper_bound(M, R, X))))))))))) | (~((~(least_upper_bound(M1!22, X1!20, R!18, E!17) | (~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~(apply(R!18, M1!22, tptp_fun_M_15(E!17, R!18, X1!20, M1!22)) | (~member(tptp_fun_M_15(E!17, R!18, X1!20, M1!22), E!17)) | (~upper_bound(tptp_fun_M_15(E!17, R!18, X1!20, M1!22), R!18, X1!20)))))) | (~((~least_upper_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M1!22, M) | (~upper_bound(M, R!18, X1!20))))))))))),
% 0.16/0.41 inference(modus_ponens,[status(thm)],[86, 85])).
% 0.16/0.41 tff(88,plain,
% 0.16/0.41 (~((~(least_upper_bound(M1!22, X1!20, R!18, E!17) | (~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~(apply(R!18, M1!22, tptp_fun_M_15(E!17, R!18, X1!20, M1!22)) | (~member(tptp_fun_M_15(E!17, R!18, X1!20, M1!22), E!17)) | (~upper_bound(tptp_fun_M_15(E!17, R!18, X1!20, M1!22), R!18, X1!20)))))) | (~((~least_upper_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M1!22, M) | (~upper_bound(M, R!18, X1!20)))))))))),
% 0.16/0.41 inference(unit_resolution,[status(thm)],[87, 42])).
% 0.16/0.41 tff(89,plain,
% 0.16/0.41 (((~(least_upper_bound(M1!22, X1!20, R!18, E!17) | (~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~(apply(R!18, M1!22, tptp_fun_M_15(E!17, R!18, X1!20, M1!22)) | (~member(tptp_fun_M_15(E!17, R!18, X1!20, M1!22), E!17)) | (~upper_bound(tptp_fun_M_15(E!17, R!18, X1!20, M1!22), R!18, X1!20)))))) | (~((~least_upper_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M1!22, M) | (~upper_bound(M, R!18, X1!20))))))))) | ((~least_upper_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M1!22, M) | (~upper_bound(M, R!18, X1!20)))))))),
% 0.16/0.41 inference(tautology,[status(thm)],[])).
% 0.16/0.41 tff(90,plain,
% 0.16/0.41 ((~least_upper_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M1!22, M) | (~upper_bound(M, R!18, X1!20))))))),
% 0.16/0.41 inference(unit_resolution,[status(thm)],[89, 88])).
% 0.16/0.41 tff(91,plain,
% 0.16/0.41 (least_upper_bound(M1!22, X1!20, R!18, E!17)),
% 0.16/0.41 inference(and_elim,[status(thm)],[72])).
% 0.16/0.41 tff(92,plain,
% 0.16/0.41 ((~((~least_upper_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M1!22, M) | (~upper_bound(M, R!18, X1!20)))))))) | (~least_upper_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M1!22, M) | (~upper_bound(M, R!18, X1!20))))))),
% 0.16/0.41 inference(tautology,[status(thm)],[])).
% 0.16/0.41 tff(93,plain,
% 0.16/0.41 ((~((~least_upper_bound(M1!22, X1!20, R!18, E!17)) | (~((~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M1!22, M) | (~upper_bound(M, R!18, X1!20)))))))) | (~((~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M1!22, M) | (~upper_bound(M, R!18, X1!20))))))),
% 0.16/0.41 inference(unit_resolution,[status(thm)],[92, 91])).
% 0.16/0.41 tff(94,plain,
% 0.16/0.41 (~((~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M1!22, M) | (~upper_bound(M, R!18, X1!20)))))),
% 0.16/0.41 inference(unit_resolution,[status(thm)],[93, 90])).
% 0.16/0.41 tff(95,plain,
% 0.16/0.41 (((~member(M1!22, X1!20)) | (~upper_bound(M1!22, R!18, X1!20)) | (~![M: $i] : ((~member(M, E!17)) | apply(R!18, M1!22, M) | (~upper_bound(M, R!18, X1!20))))) | member(M1!22, X1!20)),
% 0.16/0.41 inference(tautology,[status(thm)],[])).
% 0.16/0.41 tff(96,plain,
% 0.16/0.41 (member(M1!22, X1!20)),
% 0.16/0.41 inference(unit_resolution,[status(thm)],[95, 94])).
% 0.16/0.41 tff(97,plain,
% 0.16/0.41 (^[A: $i, B: $i] : refl((~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))) <=> (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))))),
% 0.16/0.41 inference(bind,[status(th)],[])).
% 0.16/0.41 tff(98,plain,
% 0.16/0.41 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))),
% 0.16/0.41 inference(quant_intro,[status(thm)],[97])).
% 0.16/0.41 tff(99,plain,
% 0.16/0.41 (^[A: $i, B: $i] : rewrite((~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))) <=> (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))))),
% 0.16/0.41 inference(bind,[status(th)],[])).
% 0.16/0.41 tff(100,plain,
% 0.16/0.41 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))),
% 0.16/0.41 inference(quant_intro,[status(thm)],[99])).
% 0.16/0.41 tff(101,plain,
% 0.16/0.41 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))),
% 0.16/0.41 inference(transitivity,[status(thm)],[100, 98])).
% 0.16/0.41 tff(102,plain,
% 0.16/0.41 (^[A: $i, B: $i] : trans(monotonicity(rewrite(((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) <=> ((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))), rewrite((subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))) <=> (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))), ((((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) & (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))) <=> (((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) & (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))))), rewrite((((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) & (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))) <=> (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))), ((((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) & (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))) <=> (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))))),
% 0.16/0.41 inference(bind,[status(th)],[])).
% 0.16/0.41 tff(103,plain,
% 0.16/0.41 (![A: $i, B: $i] : (((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) & (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))),
% 0.16/0.41 inference(quant_intro,[status(thm)],[102])).
% 0.16/0.41 tff(104,plain,
% 0.16/0.41 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B))) <=> ![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.16/0.41 inference(rewrite,[status(thm)],[])).
% 0.16/0.41 tff(105,plain,
% 0.16/0.41 (^[A: $i, B: $i] : rewrite((subset(A, B) <=> ![X: $i] : (member(X, A) => member(X, B))) <=> (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B))))),
% 0.16/0.41 inference(bind,[status(th)],[])).
% 0.16/0.41 tff(106,plain,
% 0.16/0.41 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : (member(X, A) => member(X, B))) <=> ![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.16/0.41 inference(quant_intro,[status(thm)],[105])).
% 0.16/0.41 tff(107,axiom,(![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : (member(X, A) => member(X, B)))), file('/export/starexec/sandbox/benchmark/Axioms/SET006+0.ax','subset')).
% 0.16/0.41 tff(108,plain,
% 0.16/0.41 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.16/0.41 inference(modus_ponens,[status(thm)],[107, 106])).
% 0.16/0.41 tff(109,plain,
% 0.16/0.41 (![A: $i, B: $i] : (subset(A, B) <=> ![X: $i] : ((~member(X, A)) | member(X, B)))),
% 0.16/0.41 inference(modus_ponens,[status(thm)],[108, 104])).
% 0.16/0.41 tff(110,plain,(
% 0.16/0.41 ![A: $i, B: $i] : (((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B))) & (subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))),
% 0.16/0.41 inference(skolemize,[status(sab)],[109])).
% 0.16/0.41 tff(111,plain,
% 0.16/0.41 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))),
% 0.16/0.41 inference(modus_ponens,[status(thm)],[110, 103])).
% 0.16/0.41 tff(112,plain,
% 0.16/0.42 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))),
% 0.16/0.42 inference(modus_ponens,[status(thm)],[111, 101])).
% 0.16/0.42 tff(113,plain,
% 0.16/0.42 ((~![A: $i, B: $i] : (~((~((~subset(A, B)) | ![X: $i] : ((~member(X, A)) | member(X, B)))) | (~(subset(A, B) | (~((~member(tptp_fun_X_0(B, A), A)) | member(tptp_fun_X_0(B, A), B)))))))) | (~((~((~subset(X1!20, X2!19)) | ![X: $i] : ((~member(X, X1!20)) | member(X, X2!19)))) | (~(subset(X1!20, X2!19) | (~((~member(tptp_fun_X_0(X2!19, X1!20), X1!20)) | member(tptp_fun_X_0(X2!19, X1!20), X2!19)))))))),
% 0.16/0.42 inference(quant_inst,[status(thm)],[])).
% 0.16/0.42 tff(114,plain,
% 0.16/0.42 (~((~((~subset(X1!20, X2!19)) | ![X: $i] : ((~member(X, X1!20)) | member(X, X2!19)))) | (~(subset(X1!20, X2!19) | (~((~member(tptp_fun_X_0(X2!19, X1!20), X1!20)) | member(tptp_fun_X_0(X2!19, X1!20), X2!19))))))),
% 0.16/0.42 inference(unit_resolution,[status(thm)],[113, 112])).
% 0.16/0.42 tff(115,plain,
% 0.16/0.42 (((~((~subset(X1!20, X2!19)) | ![X: $i] : ((~member(X, X1!20)) | member(X, X2!19)))) | (~(subset(X1!20, X2!19) | (~((~member(tptp_fun_X_0(X2!19, X1!20), X1!20)) | member(tptp_fun_X_0(X2!19, X1!20), X2!19)))))) | ((~subset(X1!20, X2!19)) | ![X: $i] : ((~member(X, X1!20)) | member(X, X2!19)))),
% 0.16/0.42 inference(tautology,[status(thm)],[])).
% 0.16/0.42 tff(116,plain,
% 0.16/0.42 ((~subset(X1!20, X2!19)) | ![X: $i] : ((~member(X, X1!20)) | member(X, X2!19))),
% 0.16/0.42 inference(unit_resolution,[status(thm)],[115, 114])).
% 0.16/0.42 tff(117,plain,
% 0.16/0.42 (subset(X1!20, X2!19)),
% 0.16/0.42 inference(and_elim,[status(thm)],[70])).
% 0.16/0.42 tff(118,plain,
% 0.16/0.42 ((~((~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))),
% 0.16/0.42 inference(tautology,[status(thm)],[])).
% 0.16/0.42 tff(119,plain,
% 0.16/0.42 ((~((~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))),
% 0.16/0.42 inference(unit_resolution,[status(thm)],[118, 117])).
% 0.16/0.42 tff(120,plain,
% 0.16/0.42 (![X: $i] : ((~member(X, X1!20)) | member(X, X2!19))),
% 0.16/0.42 inference(unit_resolution,[status(thm)],[119, 116])).
% 0.16/0.42 tff(121,plain,
% 0.16/0.42 (((~![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))),
% 0.16/0.42 inference(rewrite,[status(thm)],[])).
% 0.16/0.42 tff(122,plain,
% 0.16/0.42 ((~![X: $i] : ((~member(X, X1!20)) | member(X, X2!19))) | ((~member(M1!22, X1!20)) | member(M1!22, X2!19))),
% 0.16/0.42 inference(quant_inst,[status(thm)],[])).
% 0.16/0.42 tff(123,plain,
% 0.16/0.42 ((~![X: $i] : ((~member(X, X1!20)) | member(X, X2!19))) | (~member(M1!22, X1!20)) | member(M1!22, X2!19)),
% 0.16/0.42 inference(modus_ponens,[status(thm)],[122, 121])).
% 0.16/0.42 tff(124,plain,
% 0.16/0.42 (member(M1!22, X2!19)),
% 0.16/0.42 inference(unit_resolution,[status(thm)],[123, 120, 96])).
% 0.16/0.42 tff(125,plain,
% 0.16/0.42 (~apply(R!18, M1!22, M2!21)),
% 0.16/0.42 inference(or_elim,[status(thm)],[71])).
% 0.16/0.42 tff(126,plain,
% 0.16/0.42 (((~![X: $i] : ((~member(X, X2!19)) | apply(R!18, X, M2!21))) | ((~member(M1!22, X2!19)) | apply(R!18, M1!22, M2!21))) <=> ((~![X: $i] : ((~member(X, X2!19)) | apply(R!18, X, M2!21))) | (~member(M1!22, X2!19)) | apply(R!18, M1!22, M2!21))),
% 0.16/0.42 inference(rewrite,[status(thm)],[])).
% 0.16/0.42 tff(127,plain,
% 0.16/0.42 ((~![X: $i] : ((~member(X, X2!19)) | apply(R!18, X, M2!21))) | ((~member(M1!22, X2!19)) | apply(R!18, M1!22, M2!21))),
% 0.16/0.42 inference(quant_inst,[status(thm)],[])).
% 0.16/0.42 tff(128,plain,
% 0.16/0.42 ((~![X: $i] : ((~member(X, X2!19)) | apply(R!18, X, M2!21))) | (~member(M1!22, X2!19)) | apply(R!18, M1!22, M2!21)),
% 0.16/0.42 inference(modus_ponens,[status(thm)],[127, 126])).
% 0.16/0.42 tff(129,plain,
% 0.16/0.42 ($false),
% 0.16/0.42 inference(unit_resolution,[status(thm)],[128, 125, 124, 81])).
% 0.16/0.42 % SZS output end Proof
%------------------------------------------------------------------------------