%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET796+4 : TPTP v9.0.0. Bugfixed v4.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n015.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:26:22 AM UTC 2025
% Result : Theorem 0.12s 0.39s
% Output : Proof 0.22s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : SET796+4 : TPTP v9.0.0. Bugfixed v4.0.0.
% 0.07/0.13 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n015.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Fri Jun 20 05:43:41 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.12/0.39 % SZS status Theorem
% 0.12/0.39 % SZS output start Proof
% 0.12/0.39 tff(apply_type, type, (
% 0.12/0.39 apply: ( $i * $i * $i ) > $o)).
% 0.12/0.39 tff(tptp_fun_M_16_type, type, (
% 0.12/0.39 tptp_fun_M_16: ( $i * $i * $i * $i ) > $i)).
% 0.12/0.39 tff(tptp_fun_A_18_type, type, (
% 0.12/0.39 tptp_fun_A_18: $i)).
% 0.12/0.39 tff(unordered_pair_type, type, (
% 0.12/0.39 unordered_pair: ( $i * $i ) > $i)).
% 0.12/0.39 tff(tptp_fun_B_17_type, type, (
% 0.12/0.39 tptp_fun_B_17: $i)).
% 0.12/0.39 tff(tptp_fun_R_20_type, type, (
% 0.12/0.39 tptp_fun_R_20: $i)).
% 0.12/0.39 tff(tptp_fun_E_19_type, type, (
% 0.12/0.39 tptp_fun_E_19: $i)).
% 0.12/0.39 tff(member_type, type, (
% 0.12/0.39 member: ( $i * $i ) > $o)).
% 0.12/0.39 tff(lower_bound_type, type, (
% 0.12/0.39 lower_bound: ( $i * $i * $i ) > $o)).
% 0.12/0.39 tff(tptp_fun_X_12_type, type, (
% 0.12/0.39 tptp_fun_X_12: ( $i * $i * $i ) > $i)).
% 0.12/0.39 tff(order_type, type, (
% 0.12/0.39 order: ( $i * $i ) > $o)).
% 0.12/0.39 tff(greatest_lower_bound_type, type, (
% 0.12/0.39 greatest_lower_bound: ( $i * $i * $i * $i ) > $o)).
% 0.12/0.39 tff(tptp_fun_Z_6_type, type, (
% 0.12/0.39 tptp_fun_Z_6: ( $i * $i ) > $i)).
% 0.12/0.39 tff(tptp_fun_X_8_type, type, (
% 0.12/0.39 tptp_fun_X_8: ( $i * $i ) > $i)).
% 0.12/0.39 tff(tptp_fun_Y_7_type, type, (
% 0.12/0.39 tptp_fun_Y_7: ( $i * $i ) > $i)).
% 0.12/0.39 tff(tptp_fun_Y_4_type, type, (
% 0.12/0.39 tptp_fun_Y_4: ( $i * $i ) > $i)).
% 0.12/0.39 tff(tptp_fun_X_5_type, type, (
% 0.12/0.39 tptp_fun_X_5: ( $i * $i ) > $i)).
% 0.12/0.39 tff(tptp_fun_X_3_type, type, (
% 0.12/0.39 tptp_fun_X_3: ( $i * $i ) > $i)).
% 0.12/0.39 tff(1,plain,
% 0.12/0.39 (^[R: $i, E: $i, M: $i] : refl((~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))))) <=> (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(2,plain,
% 0.12/0.39 (![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))))) <=> ![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[1])).
% 0.12/0.39 tff(3,plain,
% 0.12/0.39 (^[R: $i, E: $i, M: $i] : rewrite((~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))))) <=> (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(4,plain,
% 0.12/0.39 (![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))))) <=> ![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[3])).
% 0.12/0.39 tff(5,plain,
% 0.12/0.39 (![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))))) <=> ![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))),
% 0.12/0.39 inference(transitivity,[status(thm)],[4, 2])).
% 0.12/0.39 tff(6,plain,
% 0.12/0.39 (^[R: $i, E: $i, M: $i] : trans(monotonicity(rewrite(((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X))) <=> ((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))), rewrite((lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))) <=> (lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))), ((((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X))) & (lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))) <=> (((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X))) & (lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))))), rewrite((((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X))) & (lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))) <=> (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))), ((((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X))) & (lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))) <=> (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))))),
% 0.22/0.40 inference(bind,[status(th)],[])).
% 0.22/0.40 tff(7,plain,
% 0.22/0.40 (![R: $i, E: $i, M: $i] : (((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X))) & (lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R)))))) <=> ![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))),
% 0.22/0.40 inference(quant_intro,[status(thm)],[6])).
% 0.22/0.40 tff(8,plain,
% 0.22/0.40 (![R: $i, E: $i, M: $i] : (lower_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, M, X))) <=> ![R: $i, E: $i, M: $i] : (lower_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, M, X)))),
% 0.22/0.40 inference(rewrite,[status(thm)],[])).
% 0.22/0.40 tff(9,plain,
% 0.22/0.40 (^[R: $i, E: $i, M: $i] : rewrite((lower_bound(M, R, E) <=> ![X: $i] : (member(X, E) => apply(R, M, X))) <=> (lower_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, M, X))))),
% 0.22/0.40 inference(bind,[status(th)],[])).
% 0.22/0.40 tff(10,plain,
% 0.22/0.40 (![R: $i, E: $i, M: $i] : (lower_bound(M, R, E) <=> ![X: $i] : (member(X, E) => apply(R, M, X))) <=> ![R: $i, E: $i, M: $i] : (lower_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, M, X)))),
% 0.22/0.40 inference(quant_intro,[status(thm)],[9])).
% 0.22/0.40 tff(11,axiom,(![R: $i, E: $i, M: $i] : (lower_bound(M, R, E) <=> ![X: $i] : (member(X, E) => apply(R, M, X)))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+3.ax','lower_bound')).
% 0.22/0.40 tff(12,plain,
% 0.22/0.40 (![R: $i, E: $i, M: $i] : (lower_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, M, X)))),
% 0.22/0.40 inference(modus_ponens,[status(thm)],[11, 10])).
% 0.22/0.40 tff(13,plain,
% 0.22/0.40 (![R: $i, E: $i, M: $i] : (lower_bound(M, R, E) <=> ![X: $i] : ((~member(X, E)) | apply(R, M, X)))),
% 0.22/0.40 inference(modus_ponens,[status(thm)],[12, 8])).
% 0.22/0.40 tff(14,plain,(
% 0.22/0.40 ![R: $i, E: $i, M: $i] : (((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X))) & (lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))),
% 0.22/0.40 inference(skolemize,[status(sab)],[13])).
% 0.22/0.40 tff(15,plain,
% 0.22/0.40 (![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))),
% 0.22/0.40 inference(modus_ponens,[status(thm)],[14, 7])).
% 0.22/0.40 tff(16,plain,
% 0.22/0.40 (![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))),
% 0.22/0.40 inference(modus_ponens,[status(thm)],[15, 5])).
% 0.22/0.40 tff(17,plain,
% 0.22/0.40 ((~![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))) | (~((~((~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))) | ![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), X)))) | (~(lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17)) | (~((~member(tptp_fun_X_12(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), tptp_fun_X_12(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), unordered_pair(A!18, B!17), R!20))))))))),
% 0.22/0.40 inference(quant_inst,[status(thm)],[])).
% 0.22/0.40 tff(18,plain,
% 0.22/0.40 (~((~((~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))) | ![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), X)))) | (~(lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17)) | (~((~member(tptp_fun_X_12(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), tptp_fun_X_12(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), unordered_pair(A!18, B!17), R!20)))))))),
% 0.22/0.40 inference(unit_resolution,[status(thm)],[17, 16])).
% 0.22/0.40 tff(19,plain,
% 0.22/0.40 (((~((~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))) | ![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), X)))) | (~(lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17)) | (~((~member(tptp_fun_X_12(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), tptp_fun_X_12(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), unordered_pair(A!18, B!17), R!20))))))) | ((~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))) | ![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), X)))),
% 0.22/0.40 inference(tautology,[status(thm)],[])).
% 0.22/0.40 tff(20,plain,
% 0.22/0.40 ((~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))) | ![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), X))),
% 0.22/0.40 inference(unit_resolution,[status(thm)],[19, 18])).
% 0.22/0.40 tff(21,assumption,(~member(A!18, unordered_pair(A!18, B!17))), introduced(assumption)).
% 0.22/0.40 tff(22,plain,
% 0.22/0.40 (^[X: $i, A: $i, B: $i] : refl((member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A))) <=> (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A))))),
% 0.22/0.40 inference(bind,[status(th)],[])).
% 0.22/0.40 tff(23,plain,
% 0.22/0.40 (![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A))) <=> ![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))),
% 0.22/0.40 inference(quant_intro,[status(thm)],[22])).
% 0.22/0.40 tff(24,plain,
% 0.22/0.40 (![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A))) <=> ![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))),
% 0.22/0.40 inference(rewrite,[status(thm)],[])).
% 0.22/0.40 tff(25,plain,
% 0.22/0.40 (^[X: $i, A: $i, B: $i] : rewrite((member(X, unordered_pair(A, B)) <=> ((X = A) | (X = B))) <=> (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A))))),
% 0.22/0.40 inference(bind,[status(th)],[])).
% 0.22/0.40 tff(26,plain,
% 0.22/0.40 (![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = A) | (X = B))) <=> ![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))),
% 0.22/0.40 inference(quant_intro,[status(thm)],[25])).
% 0.22/0.40 tff(27,axiom,(![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = A) | (X = B)))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+0.ax','unordered_pair')).
% 0.22/0.40 tff(28,plain,
% 0.22/0.40 (![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))),
% 0.22/0.40 inference(modus_ponens,[status(thm)],[27, 26])).
% 0.22/0.40 tff(29,plain,
% 0.22/0.40 (![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))),
% 0.22/0.40 inference(modus_ponens,[status(thm)],[28, 24])).
% 0.22/0.40 tff(30,plain,(
% 0.22/0.40 ![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))),
% 0.22/0.40 inference(skolemize,[status(sab)],[29])).
% 0.22/0.40 tff(31,plain,
% 0.22/0.40 (![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))),
% 0.22/0.40 inference(modus_ponens,[status(thm)],[30, 23])).
% 0.22/0.40 tff(32,plain,
% 0.22/0.40 (((~![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))) | member(A!18, unordered_pair(A!18, B!17))) <=> ((~![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))) | member(A!18, unordered_pair(A!18, B!17)))),
% 0.22/0.40 inference(rewrite,[status(thm)],[])).
% 0.22/0.40 tff(33,plain,
% 0.22/0.40 ((member(A!18, unordered_pair(A!18, B!17)) <=> $true) <=> member(A!18, unordered_pair(A!18, B!17))),
% 0.22/0.40 inference(rewrite,[status(thm)],[])).
% 0.22/0.40 tff(34,plain,
% 0.22/0.40 (((A!18 = B!17) | $true) <=> $true),
% 0.22/0.40 inference(rewrite,[status(thm)],[])).
% 0.22/0.40 tff(35,plain,
% 0.22/0.40 ((A!18 = A!18) <=> $true),
% 0.22/0.40 inference(rewrite,[status(thm)],[])).
% 0.22/0.40 tff(36,plain,
% 0.22/0.40 (((A!18 = B!17) | (A!18 = A!18)) <=> ((A!18 = B!17) | $true)),
% 0.22/0.40 inference(monotonicity,[status(thm)],[35])).
% 0.22/0.40 tff(37,plain,
% 0.22/0.40 (((A!18 = B!17) | (A!18 = A!18)) <=> $true),
% 0.22/0.40 inference(transitivity,[status(thm)],[36, 34])).
% 0.22/0.40 tff(38,plain,
% 0.22/0.40 ((member(A!18, unordered_pair(A!18, B!17)) <=> ((A!18 = B!17) | (A!18 = A!18))) <=> (member(A!18, unordered_pair(A!18, B!17)) <=> $true)),
% 0.22/0.40 inference(monotonicity,[status(thm)],[37])).
% 0.22/0.40 tff(39,plain,
% 0.22/0.40 ((member(A!18, unordered_pair(A!18, B!17)) <=> ((A!18 = B!17) | (A!18 = A!18))) <=> member(A!18, unordered_pair(A!18, B!17))),
% 0.22/0.40 inference(transitivity,[status(thm)],[38, 33])).
% 0.22/0.40 tff(40,plain,
% 0.22/0.40 (((~![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))) | (member(A!18, unordered_pair(A!18, B!17)) <=> ((A!18 = B!17) | (A!18 = A!18)))) <=> ((~![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))) | member(A!18, unordered_pair(A!18, B!17)))),
% 0.22/0.40 inference(monotonicity,[status(thm)],[39])).
% 0.22/0.40 tff(41,plain,
% 0.22/0.40 (((~![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))) | (member(A!18, unordered_pair(A!18, B!17)) <=> ((A!18 = B!17) | (A!18 = A!18)))) <=> ((~![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))) | member(A!18, unordered_pair(A!18, B!17)))),
% 0.22/0.40 inference(transitivity,[status(thm)],[40, 32])).
% 0.22/0.40 tff(42,plain,
% 0.22/0.40 ((~![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))) | (member(A!18, unordered_pair(A!18, B!17)) <=> ((A!18 = B!17) | (A!18 = A!18)))),
% 0.22/0.40 inference(quant_inst,[status(thm)],[])).
% 0.22/0.40 tff(43,plain,
% 0.22/0.40 ((~![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))) | member(A!18, unordered_pair(A!18, B!17))),
% 0.22/0.40 inference(modus_ponens,[status(thm)],[42, 41])).
% 0.22/0.40 tff(44,plain,
% 0.22/0.40 ($false),
% 0.22/0.40 inference(unit_resolution,[status(thm)],[43, 31, 21])).
% 0.22/0.40 tff(45,plain,(member(A!18, unordered_pair(A!18, B!17))), inference(lemma,lemma(discharge,[]))).
% 0.22/0.40 tff(46,assumption,((~((~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | ![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, A!18, X)))) | (~(lower_bound(A!18, R!20, unordered_pair(A!18, B!17)) | (~((~member(tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, A!18, tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20))))))), introduced(assumption)).
% 0.22/0.40 tff(47,plain,
% 0.22/0.40 ((~![R: $i, E: $i, M: $i] : (~((~((~lower_bound(M, R, E)) | ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) | (~(lower_bound(M, R, E) | (~((~member(tptp_fun_X_12(M, E, R), E)) | apply(R, M, tptp_fun_X_12(M, E, R))))))))) | (~((~((~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | ![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, A!18, X)))) | (~(lower_bound(A!18, R!20, unordered_pair(A!18, B!17)) | (~((~member(tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, A!18, tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20))))))))),
% 0.22/0.41 inference(quant_inst,[status(thm)],[])).
% 0.22/0.41 tff(48,plain,
% 0.22/0.41 ($false),
% 0.22/0.41 inference(unit_resolution,[status(thm)],[47, 16, 46])).
% 0.22/0.41 tff(49,plain,(~((~((~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | ![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, A!18, X)))) | (~(lower_bound(A!18, R!20, unordered_pair(A!18, B!17)) | (~((~member(tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, A!18, tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20)))))))), inference(lemma,lemma(discharge,[]))).
% 0.22/0.41 tff(50,plain,
% 0.22/0.41 (((~((~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | ![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, A!18, X)))) | (~(lower_bound(A!18, R!20, unordered_pair(A!18, B!17)) | (~((~member(tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, A!18, tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20))))))) | (lower_bound(A!18, R!20, unordered_pair(A!18, B!17)) | (~((~member(tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, A!18, tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20)))))),
% 0.22/0.41 inference(tautology,[status(thm)],[])).
% 0.22/0.41 tff(51,plain,
% 0.22/0.41 (lower_bound(A!18, R!20, unordered_pair(A!18, B!17)) | (~((~member(tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, A!18, tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20))))),
% 0.22/0.41 inference(unit_resolution,[status(thm)],[50, 49])).
% 0.22/0.41 tff(52,plain,
% 0.22/0.41 ((~![X: $i, A: $i, B: $i] : (member(X, unordered_pair(A, B)) <=> ((X = B) | (X = A)))) | (member(tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17)) <=> ((tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20) = B!17) | (tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20) = A!18)))),
% 0.22/0.41 inference(quant_inst,[status(thm)],[])).
% 0.22/0.41 tff(53,plain,
% 0.22/0.41 (member(tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17)) <=> ((tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20) = B!17) | (tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20) = A!18))),
% 0.22/0.41 inference(unit_resolution,[status(thm)],[52, 31])).
% 0.22/0.41 tff(54,assumption,(~((~member(tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, A!18, tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20)))), introduced(assumption)).
% 0.22/0.41 tff(55,plain,
% 0.22/0.41 (((~member(tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, A!18, tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20))) | member(tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))),
% 0.22/0.41 inference(tautology,[status(thm)],[])).
% 0.22/0.41 tff(56,plain,
% 0.22/0.41 (member(tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))),
% 0.22/0.41 inference(unit_resolution,[status(thm)],[55, 54])).
% 0.22/0.41 tff(57,plain,
% 0.22/0.41 ((~(member(tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17)) <=> ((tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20) = B!17) | (tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20) = A!18)))) | (~member(tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | ((tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20) = B!17) | (tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20) = A!18))),
% 0.22/0.41 inference(tautology,[status(thm)],[])).
% 0.22/0.41 tff(58,plain,
% 0.22/0.41 ((~(member(tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17)) <=> ((tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20) = B!17) | (tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20) = A!18)))) | ((tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20) = B!17) | (tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20) = A!18))),
% 0.22/0.41 inference(unit_resolution,[status(thm)],[57, 56])).
% 0.22/0.41 tff(59,plain,
% 0.22/0.41 ((tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20) = B!17) | (tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20) = A!18)),
% 0.22/0.41 inference(unit_resolution,[status(thm)],[58, 53])).
% 0.22/0.41 tff(60,plain,
% 0.22/0.41 (((~member(tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, A!18, tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20))) | (~apply(R!20, A!18, tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20)))),
% 0.22/0.41 inference(tautology,[status(thm)],[])).
% 0.22/0.41 tff(61,plain,
% 0.22/0.41 (~apply(R!20, A!18, tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20))),
% 0.22/0.41 inference(unit_resolution,[status(thm)],[60, 54])).
% 0.22/0.41 tff(62,assumption,(tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20) = B!17), introduced(assumption)).
% 0.22/0.41 tff(63,plain,
% 0.22/0.41 (apply(R!20, A!18, tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20)) <=> apply(R!20, A!18, B!17)),
% 0.22/0.41 inference(monotonicity,[status(thm)],[62])).
% 0.22/0.41 tff(64,plain,
% 0.22/0.41 (apply(R!20, A!18, B!17) <=> apply(R!20, A!18, tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20))),
% 0.22/0.41 inference(symmetry,[status(thm)],[63])).
% 0.22/0.41 tff(65,plain,
% 0.22/0.41 ((~![R: $i, E: $i, A: $i, B: $i] : ((~(order(R, E) & member(A, E) & member(B, E) & apply(R, A, B))) | greatest_lower_bound(A, unordered_pair(A, B), R, E))) <=> (~![R: $i, E: $i, A: $i, B: $i] : ((~(order(R, E) & member(A, E) & member(B, E) & apply(R, A, B))) | greatest_lower_bound(A, unordered_pair(A, B), R, E)))),
% 0.22/0.41 inference(rewrite,[status(thm)],[])).
% 0.22/0.41 tff(66,plain,
% 0.22/0.41 ((~![R: $i, E: $i, A: $i, B: $i] : ((((order(R, E) & member(A, E)) & member(B, E)) & apply(R, A, B)) => greatest_lower_bound(A, unordered_pair(A, B), R, E))) <=> (~![R: $i, E: $i, A: $i, B: $i] : ((~(order(R, E) & member(A, E) & member(B, E) & apply(R, A, B))) | greatest_lower_bound(A, unordered_pair(A, B), R, E)))),
% 0.22/0.41 inference(rewrite,[status(thm)],[])).
% 0.22/0.41 tff(67,axiom,(~![R: $i, E: $i, A: $i, B: $i] : ((((order(R, E) & member(A, E)) & member(B, E)) & apply(R, A, B)) => greatest_lower_bound(A, unordered_pair(A, B), R, E))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','thIV8')).
% 0.22/0.41 tff(68,plain,
% 0.22/0.41 (~![R: $i, E: $i, A: $i, B: $i] : ((~(order(R, E) & member(A, E) & member(B, E) & apply(R, A, B))) | greatest_lower_bound(A, unordered_pair(A, B), R, E))),
% 0.22/0.41 inference(modus_ponens,[status(thm)],[67, 66])).
% 0.22/0.41 tff(69,plain,
% 0.22/0.41 (~![R: $i, E: $i, A: $i, B: $i] : ((~(order(R, E) & member(A, E) & member(B, E) & apply(R, A, B))) | greatest_lower_bound(A, unordered_pair(A, B), R, E))),
% 0.22/0.41 inference(modus_ponens,[status(thm)],[68, 65])).
% 0.22/0.41 tff(70,plain,
% 0.22/0.41 (~![R: $i, E: $i, A: $i, B: $i] : ((~(order(R, E) & member(A, E) & member(B, E) & apply(R, A, B))) | greatest_lower_bound(A, unordered_pair(A, B), R, E))),
% 0.22/0.41 inference(modus_ponens,[status(thm)],[69, 65])).
% 0.22/0.41 tff(71,plain,
% 0.22/0.41 (~![R: $i, E: $i, A: $i, B: $i] : ((~(order(R, E) & member(A, E) & member(B, E) & apply(R, A, B))) | greatest_lower_bound(A, unordered_pair(A, B), R, E))),
% 0.22/0.41 inference(modus_ponens,[status(thm)],[70, 65])).
% 0.22/0.41 tff(72,plain,(
% 0.22/0.41 ~((~(order(R!20, E!19) & member(A!18, E!19) & member(B!17, E!19) & apply(R!20, A!18, B!17))) | greatest_lower_bound(A!18, unordered_pair(A!18, B!17), R!20, E!19))),
% 0.22/0.41 inference(skolemize,[status(sab)],[71])).
% 0.22/0.41 tff(73,plain,
% 0.22/0.41 (order(R!20, E!19) & member(A!18, E!19) & member(B!17, E!19) & apply(R!20, A!18, B!17)),
% 0.22/0.41 inference(or_elim,[status(thm)],[72])).
% 0.22/0.41 tff(74,plain,
% 0.22/0.41 (apply(R!20, A!18, B!17)),
% 0.22/0.41 inference(and_elim,[status(thm)],[73])).
% 0.22/0.41 tff(75,plain,
% 0.22/0.41 (apply(R!20, A!18, tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20))),
% 0.22/0.41 inference(modus_ponens,[status(thm)],[74, 64])).
% 0.22/0.41 tff(76,assumption,(~apply(R!20, A!18, tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20))), introduced(assumption)).
% 0.22/0.41 tff(77,plain,
% 0.22/0.41 ($false),
% 0.22/0.41 inference(unit_resolution,[status(thm)],[76, 75])).
% 0.22/0.41 tff(78,plain,((~(tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20) = B!17)) | apply(R!20, A!18, tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20))), inference(lemma,lemma(discharge,[]))).
% 0.22/0.41 tff(79,plain,
% 0.22/0.41 (~(tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20) = B!17)),
% 0.22/0.41 inference(unit_resolution,[status(thm)],[78, 61])).
% 0.22/0.41 tff(80,plain,
% 0.22/0.41 ((~((tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20) = B!17) | (tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20) = A!18))) | (tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20) = B!17) | (tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20) = A!18)),
% 0.22/0.41 inference(tautology,[status(thm)],[])).
% 0.22/0.41 tff(81,plain,
% 0.22/0.41 (tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20) = A!18),
% 0.22/0.41 inference(unit_resolution,[status(thm)],[80, 79, 59])).
% 0.22/0.41 tff(82,plain,
% 0.22/0.41 (apply(R!20, A!18, tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20)) <=> apply(R!20, A!18, A!18)),
% 0.22/0.41 inference(monotonicity,[status(thm)],[81])).
% 0.22/0.41 tff(83,plain,
% 0.22/0.41 (apply(R!20, A!18, A!18) <=> apply(R!20, A!18, tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20))),
% 0.22/0.41 inference(symmetry,[status(thm)],[82])).
% 0.22/0.41 tff(84,plain,
% 0.22/0.41 (^[R: $i, E: $i] : rewrite((~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))))) <=> (~((~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))) | (~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))))))),
% 0.22/0.41 inference(bind,[status(th)],[])).
% 0.22/0.41 tff(85,plain,
% 0.22/0.41 (![R: $i, E: $i] : (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))))) <=> ![R: $i, E: $i] : (~((~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))) | (~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))),
% 0.22/0.41 inference(quant_intro,[status(thm)],[84])).
% 0.22/0.41 tff(86,plain,
% 0.22/0.41 (^[R: $i, E: $i] : refl((~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))))) <=> (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))))))),
% 0.22/0.41 inference(bind,[status(th)],[])).
% 0.22/0.41 tff(87,plain,
% 0.22/0.41 (![R: $i, E: $i] : (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))))) <=> ![R: $i, E: $i] : (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E))))))))),
% 0.22/0.41 inference(quant_intro,[status(thm)],[86])).
% 0.22/0.41 tff(88,plain,
% 0.22/0.41 (^[R: $i, E: $i] : rewrite((~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))))) <=> (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))))))),
% 0.22/0.41 inference(bind,[status(th)],[])).
% 0.22/0.41 tff(89,plain,
% 0.22/0.41 (![R: $i, E: $i] : (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))))) <=> ![R: $i, E: $i] : (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E))))))))),
% 0.22/0.42 inference(quant_intro,[status(thm)],[88])).
% 0.22/0.42 tff(90,plain,
% 0.22/0.42 (![R: $i, E: $i] : (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))))) <=> ![R: $i, E: $i] : (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E))))))))),
% 0.22/0.42 inference(transitivity,[status(thm)],[89, 87])).
% 0.22/0.42 tff(91,plain,
% 0.22/0.42 (^[R: $i, E: $i] : trans(monotonicity(rewrite(((~order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) <=> ((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))), trans(monotonicity(rewrite((~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~(apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R)) & apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R)))) | (~(member(tptp_fun_X_5(E, R), E) & member(tptp_fun_Y_4(E, R), E))))) <=> (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E))))), rewrite((~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R)) & apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R)))) | (~(member(tptp_fun_X_8(E, R), E) & member(tptp_fun_Y_7(E, R), E) & member(tptp_fun_Z_6(E, R), E))))) <=> (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E))))), ((order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~(apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R)) & apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R)))) | (~(member(tptp_fun_X_5(E, R), E) & member(tptp_fun_Y_4(E, R), E))))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R)) & apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R)))) | (~(member(tptp_fun_X_8(E, R), E) & member(tptp_fun_Y_7(E, R), E) & member(tptp_fun_Z_6(E, R), E)))))) <=> (order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E))))))), rewrite((order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E))))) <=> (order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))), ((order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~(apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R)) & apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R)))) | (~(member(tptp_fun_X_5(E, R), E) & member(tptp_fun_Y_4(E, R), E))))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R)) & apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R)))) | (~(member(tptp_fun_X_8(E, R), E) & member(tptp_fun_Y_7(E, R), E) & member(tptp_fun_Z_6(E, R), E)))))) <=> (order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E))))))), ((((~order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~(apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R)) & apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R)))) | (~(member(tptp_fun_X_5(E, R), E) & member(tptp_fun_Y_4(E, R), E))))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R)) & apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R)))) | (~(member(tptp_fun_X_8(E, R), E) & member(tptp_fun_Y_7(E, R), E) & member(tptp_fun_Z_6(E, R), E))))))) <=> (((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))) & (order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))))), rewrite((((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))) & (order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))) <=> (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E))))))))), ((((~order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~(apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R)) & apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R)))) | (~(member(tptp_fun_X_5(E, R), E) & member(tptp_fun_Y_4(E, R), E))))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R)) & apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R)))) | (~(member(tptp_fun_X_8(E, R), E) & member(tptp_fun_Y_7(E, R), E) & member(tptp_fun_Z_6(E, R), E))))))) <=> (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E))))))))))),
% 0.22/0.42 inference(bind,[status(th)],[])).
% 0.22/0.42 tff(92,plain,
% 0.22/0.42 (![R: $i, E: $i] : (((~order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~(apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R)) & apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R)))) | (~(member(tptp_fun_X_5(E, R), E) & member(tptp_fun_Y_4(E, R), E))))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R)) & apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R)))) | (~(member(tptp_fun_X_8(E, R), E) & member(tptp_fun_Y_7(E, R), E) & member(tptp_fun_Z_6(E, R), E))))))) <=> ![R: $i, E: $i] : (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E))))))))),
% 0.22/0.42 inference(quant_intro,[status(thm)],[91])).
% 0.22/0.42 tff(93,plain,
% 0.22/0.42 (^[R: $i, E: $i] : rewrite((((~order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (order(R, E) | ((~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~(apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R)) & apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R)))) | (~(member(tptp_fun_X_5(E, R), E) & member(tptp_fun_Y_4(E, R), E))))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R)) & apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R)))) | (~(member(tptp_fun_X_8(E, R), E) & member(tptp_fun_Y_7(E, R), E) & member(tptp_fun_Z_6(E, R), E)))))))) <=> (((~order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~(apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R)) & apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R)))) | (~(member(tptp_fun_X_5(E, R), E) & member(tptp_fun_Y_4(E, R), E))))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R)) & apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R)))) | (~(member(tptp_fun_X_8(E, R), E) & member(tptp_fun_Y_7(E, R), E) & member(tptp_fun_Z_6(E, R), E))))))))),
% 0.22/0.42 inference(bind,[status(th)],[])).
% 0.22/0.42 tff(94,plain,
% 0.22/0.42 (![R: $i, E: $i] : (((~order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (order(R, E) | ((~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~(apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R)) & apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R)))) | (~(member(tptp_fun_X_5(E, R), E) & member(tptp_fun_Y_4(E, R), E))))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R)) & apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R)))) | (~(member(tptp_fun_X_8(E, R), E) & member(tptp_fun_Y_7(E, R), E) & member(tptp_fun_Z_6(E, R), E)))))))) <=> ![R: $i, E: $i] : (((~order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~(apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R)) & apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R)))) | (~(member(tptp_fun_X_5(E, R), E) & member(tptp_fun_Y_4(E, R), E))))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R)) & apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R)))) | (~(member(tptp_fun_X_8(E, R), E) & member(tptp_fun_Y_7(E, R), E) & member(tptp_fun_Z_6(E, R), E)))))))),
% 0.22/0.42 inference(quant_intro,[status(thm)],[93])).
% 0.22/0.42 tff(95,plain,
% 0.22/0.42 (![R: $i, E: $i] : (order(R, E) <=> (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) <=> ![R: $i, E: $i] : (order(R, E) <=> (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E))))))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(96,plain,
% 0.22/0.42 (^[R: $i, E: $i] : rewrite((order(R, E) <=> ((![X: $i] : (member(X, E) => apply(R, X, X)) & ![X: $i, Y: $i] : ((member(X, E) & member(Y, E)) => ((apply(R, X, Y) & apply(R, Y, X)) => (X = Y)))) & ![X: $i, Y: $i, Z: $i] : (((member(X, E) & member(Y, E)) & member(Z, E)) => ((apply(R, X, Y) & apply(R, Y, Z)) => apply(R, X, Z))))) <=> (order(R, E) <=> (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))))),
% 0.22/0.42 inference(bind,[status(th)],[])).
% 0.22/0.42 tff(97,plain,
% 0.22/0.42 (![R: $i, E: $i] : (order(R, E) <=> ((![X: $i] : (member(X, E) => apply(R, X, X)) & ![X: $i, Y: $i] : ((member(X, E) & member(Y, E)) => ((apply(R, X, Y) & apply(R, Y, X)) => (X = Y)))) & ![X: $i, Y: $i, Z: $i] : (((member(X, E) & member(Y, E)) & member(Z, E)) => ((apply(R, X, Y) & apply(R, Y, Z)) => apply(R, X, Z))))) <=> ![R: $i, E: $i] : (order(R, E) <=> (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E))))))),
% 0.22/0.42 inference(quant_intro,[status(thm)],[96])).
% 0.22/0.42 tff(98,axiom,(![R: $i, E: $i] : (order(R, E) <=> ((![X: $i] : (member(X, E) => apply(R, X, X)) & ![X: $i, Y: $i] : ((member(X, E) & member(Y, E)) => ((apply(R, X, Y) & apply(R, Y, X)) => (X = Y)))) & ![X: $i, Y: $i, Z: $i] : (((member(X, E) & member(Y, E)) & member(Z, E)) => ((apply(R, X, Y) & apply(R, Y, Z)) => apply(R, X, Z)))))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+3.ax','order')).
% 0.22/0.42 tff(99,plain,
% 0.22/0.42 (![R: $i, E: $i] : (order(R, E) <=> (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E))))))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[98, 97])).
% 0.22/0.42 tff(100,plain,
% 0.22/0.42 (![R: $i, E: $i] : (order(R, E) <=> (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E))))))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[99, 95])).
% 0.22/0.42 tff(101,plain,(
% 0.22/0.42 ![R: $i, E: $i] : (((~order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (order(R, E) | ((~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~(apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R)) & apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R)))) | (~(member(tptp_fun_X_5(E, R), E) & member(tptp_fun_Y_4(E, R), E))))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R)) & apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R)))) | (~(member(tptp_fun_X_8(E, R), E) & member(tptp_fun_Y_7(E, R), E) & member(tptp_fun_Z_6(E, R), E))))))))),
% 0.22/0.42 inference(skolemize,[status(sab)],[100])).
% 0.22/0.42 tff(102,plain,
% 0.22/0.42 (![R: $i, E: $i] : (((~order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i] : ((X = Y) | (~(apply(R, X, Y) & apply(R, Y, X))) | (~(member(X, E) & member(Y, E)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~(apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R)) & apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R)))) | (~(member(tptp_fun_X_5(E, R), E) & member(tptp_fun_Y_4(E, R), E))))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R)) & apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R)))) | (~(member(tptp_fun_X_8(E, R), E) & member(tptp_fun_Y_7(E, R), E) & member(tptp_fun_Z_6(E, R), E)))))))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[101, 94])).
% 0.22/0.43 tff(103,plain,
% 0.22/0.43 (![R: $i, E: $i] : (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E))))))))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[102, 92])).
% 0.22/0.43 tff(104,plain,
% 0.22/0.43 (![R: $i, E: $i] : (~((~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E))))))))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[103, 90])).
% 0.22/0.43 tff(105,plain,
% 0.22/0.43 (![R: $i, E: $i] : (~((~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))) | (~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[104, 85])).
% 0.22/0.43 tff(106,plain,
% 0.22/0.43 (((~![R: $i, E: $i] : (~((~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))) | (~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))) | (~((~(order(R!20, E!19) | (~((~member(tptp_fun_X_3(E!19, R!20), E!19)) | apply(R!20, tptp_fun_X_3(E!19, R!20), tptp_fun_X_3(E!19, R!20)))) | (~((~member(tptp_fun_Y_4(E!19, R!20), E!19)) | (~member(tptp_fun_X_5(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_4(E!19, R!20), tptp_fun_X_5(E!19, R!20))) | (~apply(R!20, tptp_fun_X_5(E!19, R!20), tptp_fun_Y_4(E!19, R!20))) | (tptp_fun_X_5(E!19, R!20) = tptp_fun_Y_4(E!19, R!20)))) | (~((~member(tptp_fun_Z_6(E!19, R!20), E!19)) | (~member(tptp_fun_Y_7(E!19, R!20), E!19)) | (~member(tptp_fun_X_8(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_7(E!19, R!20), tptp_fun_Z_6(E!19, R!20))) | (~apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Y_7(E!19, R!20))) | apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Z_6(E!19, R!20)))))) | (~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19))))))))))) <=> ((~![R: $i, E: $i] : (~((~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))) | (~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))) | (~((~(order(R!20, E!19) | (~((~member(tptp_fun_X_3(E!19, R!20), E!19)) | apply(R!20, tptp_fun_X_3(E!19, R!20), tptp_fun_X_3(E!19, R!20)))) | (~((~member(tptp_fun_Y_4(E!19, R!20), E!19)) | (~member(tptp_fun_X_5(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_4(E!19, R!20), tptp_fun_X_5(E!19, R!20))) | (~apply(R!20, tptp_fun_X_5(E!19, R!20), tptp_fun_Y_4(E!19, R!20))) | (tptp_fun_X_5(E!19, R!20) = tptp_fun_Y_4(E!19, R!20)))) | (~((~member(tptp_fun_Z_6(E!19, R!20), E!19)) | (~member(tptp_fun_Y_7(E!19, R!20), E!19)) | (~member(tptp_fun_X_8(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_7(E!19, R!20), tptp_fun_Z_6(E!19, R!20))) | (~apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Y_7(E!19, R!20))) | apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Z_6(E!19, R!20)))))) | (~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19)))))))))))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(107,plain,
% 0.22/0.43 ((~((~(order(R!20, E!19) | (~((~member(tptp_fun_X_3(E!19, R!20), E!19)) | apply(R!20, tptp_fun_X_3(E!19, R!20), tptp_fun_X_3(E!19, R!20)))) | (~((tptp_fun_X_5(E!19, R!20) = tptp_fun_Y_4(E!19, R!20)) | (~apply(R!20, tptp_fun_X_5(E!19, R!20), tptp_fun_Y_4(E!19, R!20))) | (~apply(R!20, tptp_fun_Y_4(E!19, R!20), tptp_fun_X_5(E!19, R!20))) | (~member(tptp_fun_X_5(E!19, R!20), E!19)) | (~member(tptp_fun_Y_4(E!19, R!20), E!19)))) | (~(apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Z_6(E!19, R!20)) | (~apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Y_7(E!19, R!20))) | (~apply(R!20, tptp_fun_Y_7(E!19, R!20), tptp_fun_Z_6(E!19, R!20))) | (~member(tptp_fun_X_8(E!19, R!20), E!19)) | (~member(tptp_fun_Y_7(E!19, R!20), E!19)) | (~member(tptp_fun_Z_6(E!19, R!20), E!19)))))) | (~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E!19)) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, X)) | (~member(X, E!19)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19)))))))))) <=> (~((~(order(R!20, E!19) | (~((~member(tptp_fun_X_3(E!19, R!20), E!19)) | apply(R!20, tptp_fun_X_3(E!19, R!20), tptp_fun_X_3(E!19, R!20)))) | (~((~member(tptp_fun_Y_4(E!19, R!20), E!19)) | (~member(tptp_fun_X_5(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_4(E!19, R!20), tptp_fun_X_5(E!19, R!20))) | (~apply(R!20, tptp_fun_X_5(E!19, R!20), tptp_fun_Y_4(E!19, R!20))) | (tptp_fun_X_5(E!19, R!20) = tptp_fun_Y_4(E!19, R!20)))) | (~((~member(tptp_fun_Z_6(E!19, R!20), E!19)) | (~member(tptp_fun_Y_7(E!19, R!20), E!19)) | (~member(tptp_fun_X_8(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_7(E!19, R!20), tptp_fun_Z_6(E!19, R!20))) | (~apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Y_7(E!19, R!20))) | apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Z_6(E!19, R!20)))))) | (~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19))))))))))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(108,plain,
% 0.22/0.43 (((~![R: $i, E: $i] : (~((~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))) | (~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))) | (~((~(order(R!20, E!19) | (~((~member(tptp_fun_X_3(E!19, R!20), E!19)) | apply(R!20, tptp_fun_X_3(E!19, R!20), tptp_fun_X_3(E!19, R!20)))) | (~((tptp_fun_X_5(E!19, R!20) = tptp_fun_Y_4(E!19, R!20)) | (~apply(R!20, tptp_fun_X_5(E!19, R!20), tptp_fun_Y_4(E!19, R!20))) | (~apply(R!20, tptp_fun_Y_4(E!19, R!20), tptp_fun_X_5(E!19, R!20))) | (~member(tptp_fun_X_5(E!19, R!20), E!19)) | (~member(tptp_fun_Y_4(E!19, R!20), E!19)))) | (~(apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Z_6(E!19, R!20)) | (~apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Y_7(E!19, R!20))) | (~apply(R!20, tptp_fun_Y_7(E!19, R!20), tptp_fun_Z_6(E!19, R!20))) | (~member(tptp_fun_X_8(E!19, R!20), E!19)) | (~member(tptp_fun_Y_7(E!19, R!20), E!19)) | (~member(tptp_fun_Z_6(E!19, R!20), E!19)))))) | (~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E!19)) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, X)) | (~member(X, E!19)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19))))))))))) <=> ((~![R: $i, E: $i] : (~((~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))) | (~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))) | (~((~(order(R!20, E!19) | (~((~member(tptp_fun_X_3(E!19, R!20), E!19)) | apply(R!20, tptp_fun_X_3(E!19, R!20), tptp_fun_X_3(E!19, R!20)))) | (~((~member(tptp_fun_Y_4(E!19, R!20), E!19)) | (~member(tptp_fun_X_5(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_4(E!19, R!20), tptp_fun_X_5(E!19, R!20))) | (~apply(R!20, tptp_fun_X_5(E!19, R!20), tptp_fun_Y_4(E!19, R!20))) | (tptp_fun_X_5(E!19, R!20) = tptp_fun_Y_4(E!19, R!20)))) | (~((~member(tptp_fun_Z_6(E!19, R!20), E!19)) | (~member(tptp_fun_Y_7(E!19, R!20), E!19)) | (~member(tptp_fun_X_8(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_7(E!19, R!20), tptp_fun_Z_6(E!19, R!20))) | (~apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Y_7(E!19, R!20))) | apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Z_6(E!19, R!20)))))) | (~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19)))))))))))),
% 0.22/0.44 inference(monotonicity,[status(thm)],[107])).
% 0.22/0.44 tff(109,plain,
% 0.22/0.44 (((~![R: $i, E: $i] : (~((~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))) | (~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))) | (~((~(order(R!20, E!19) | (~((~member(tptp_fun_X_3(E!19, R!20), E!19)) | apply(R!20, tptp_fun_X_3(E!19, R!20), tptp_fun_X_3(E!19, R!20)))) | (~((tptp_fun_X_5(E!19, R!20) = tptp_fun_Y_4(E!19, R!20)) | (~apply(R!20, tptp_fun_X_5(E!19, R!20), tptp_fun_Y_4(E!19, R!20))) | (~apply(R!20, tptp_fun_Y_4(E!19, R!20), tptp_fun_X_5(E!19, R!20))) | (~member(tptp_fun_X_5(E!19, R!20), E!19)) | (~member(tptp_fun_Y_4(E!19, R!20), E!19)))) | (~(apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Z_6(E!19, R!20)) | (~apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Y_7(E!19, R!20))) | (~apply(R!20, tptp_fun_Y_7(E!19, R!20), tptp_fun_Z_6(E!19, R!20))) | (~member(tptp_fun_X_8(E!19, R!20), E!19)) | (~member(tptp_fun_Y_7(E!19, R!20), E!19)) | (~member(tptp_fun_Z_6(E!19, R!20), E!19)))))) | (~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E!19)) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, X)) | (~member(X, E!19)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19))))))))))) <=> ((~![R: $i, E: $i] : (~((~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))) | (~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))) | (~((~(order(R!20, E!19) | (~((~member(tptp_fun_X_3(E!19, R!20), E!19)) | apply(R!20, tptp_fun_X_3(E!19, R!20), tptp_fun_X_3(E!19, R!20)))) | (~((~member(tptp_fun_Y_4(E!19, R!20), E!19)) | (~member(tptp_fun_X_5(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_4(E!19, R!20), tptp_fun_X_5(E!19, R!20))) | (~apply(R!20, tptp_fun_X_5(E!19, R!20), tptp_fun_Y_4(E!19, R!20))) | (tptp_fun_X_5(E!19, R!20) = tptp_fun_Y_4(E!19, R!20)))) | (~((~member(tptp_fun_Z_6(E!19, R!20), E!19)) | (~member(tptp_fun_Y_7(E!19, R!20), E!19)) | (~member(tptp_fun_X_8(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_7(E!19, R!20), tptp_fun_Z_6(E!19, R!20))) | (~apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Y_7(E!19, R!20))) | apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Z_6(E!19, R!20)))))) | (~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19)))))))))))),
% 0.22/0.44 inference(transitivity,[status(thm)],[108, 106])).
% 0.22/0.44 tff(110,plain,
% 0.22/0.44 ((~![R: $i, E: $i] : (~((~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))) | (~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))) | (~((~(order(R!20, E!19) | (~((~member(tptp_fun_X_3(E!19, R!20), E!19)) | apply(R!20, tptp_fun_X_3(E!19, R!20), tptp_fun_X_3(E!19, R!20)))) | (~((tptp_fun_X_5(E!19, R!20) = tptp_fun_Y_4(E!19, R!20)) | (~apply(R!20, tptp_fun_X_5(E!19, R!20), tptp_fun_Y_4(E!19, R!20))) | (~apply(R!20, tptp_fun_Y_4(E!19, R!20), tptp_fun_X_5(E!19, R!20))) | (~member(tptp_fun_X_5(E!19, R!20), E!19)) | (~member(tptp_fun_Y_4(E!19, R!20), E!19)))) | (~(apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Z_6(E!19, R!20)) | (~apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Y_7(E!19, R!20))) | (~apply(R!20, tptp_fun_Y_7(E!19, R!20), tptp_fun_Z_6(E!19, R!20))) | (~member(tptp_fun_X_8(E!19, R!20), E!19)) | (~member(tptp_fun_Y_7(E!19, R!20), E!19)) | (~member(tptp_fun_Z_6(E!19, R!20), E!19)))))) | (~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E!19)) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, X)) | (~member(X, E!19)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19))))))))))),
% 0.22/0.44 inference(quant_inst,[status(thm)],[])).
% 0.22/0.44 tff(111,plain,
% 0.22/0.44 ((~![R: $i, E: $i] : (~((~(order(R, E) | (~((~member(tptp_fun_X_3(E, R), E)) | apply(R, tptp_fun_X_3(E, R), tptp_fun_X_3(E, R)))) | (~((tptp_fun_X_5(E, R) = tptp_fun_Y_4(E, R)) | (~apply(R, tptp_fun_X_5(E, R), tptp_fun_Y_4(E, R))) | (~apply(R, tptp_fun_Y_4(E, R), tptp_fun_X_5(E, R))) | (~member(tptp_fun_X_5(E, R), E)) | (~member(tptp_fun_Y_4(E, R), E)))) | (~(apply(R, tptp_fun_X_8(E, R), tptp_fun_Z_6(E, R)) | (~apply(R, tptp_fun_X_8(E, R), tptp_fun_Y_7(E, R))) | (~apply(R, tptp_fun_Y_7(E, R), tptp_fun_Z_6(E, R))) | (~member(tptp_fun_X_8(E, R), E)) | (~member(tptp_fun_Y_7(E, R), E)) | (~member(tptp_fun_Z_6(E, R), E)))))) | (~((~order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~member(Y, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)) | (~member(X, E)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))) | (~((~(order(R!20, E!19) | (~((~member(tptp_fun_X_3(E!19, R!20), E!19)) | apply(R!20, tptp_fun_X_3(E!19, R!20), tptp_fun_X_3(E!19, R!20)))) | (~((~member(tptp_fun_Y_4(E!19, R!20), E!19)) | (~member(tptp_fun_X_5(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_4(E!19, R!20), tptp_fun_X_5(E!19, R!20))) | (~apply(R!20, tptp_fun_X_5(E!19, R!20), tptp_fun_Y_4(E!19, R!20))) | (tptp_fun_X_5(E!19, R!20) = tptp_fun_Y_4(E!19, R!20)))) | (~((~member(tptp_fun_Z_6(E!19, R!20), E!19)) | (~member(tptp_fun_Y_7(E!19, R!20), E!19)) | (~member(tptp_fun_X_8(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_7(E!19, R!20), tptp_fun_Z_6(E!19, R!20))) | (~apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Y_7(E!19, R!20))) | apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Z_6(E!19, R!20)))))) | (~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19))))))))))),
% 0.22/0.44 inference(modus_ponens,[status(thm)],[110, 109])).
% 0.22/0.44 tff(112,plain,
% 0.22/0.44 (~((~(order(R!20, E!19) | (~((~member(tptp_fun_X_3(E!19, R!20), E!19)) | apply(R!20, tptp_fun_X_3(E!19, R!20), tptp_fun_X_3(E!19, R!20)))) | (~((~member(tptp_fun_Y_4(E!19, R!20), E!19)) | (~member(tptp_fun_X_5(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_4(E!19, R!20), tptp_fun_X_5(E!19, R!20))) | (~apply(R!20, tptp_fun_X_5(E!19, R!20), tptp_fun_Y_4(E!19, R!20))) | (tptp_fun_X_5(E!19, R!20) = tptp_fun_Y_4(E!19, R!20)))) | (~((~member(tptp_fun_Z_6(E!19, R!20), E!19)) | (~member(tptp_fun_Y_7(E!19, R!20), E!19)) | (~member(tptp_fun_X_8(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_7(E!19, R!20), tptp_fun_Z_6(E!19, R!20))) | (~apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Y_7(E!19, R!20))) | apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Z_6(E!19, R!20)))))) | (~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19)))))))))),
% 0.22/0.44 inference(unit_resolution,[status(thm)],[111, 105])).
% 0.22/0.44 tff(113,plain,
% 0.22/0.44 (((~(order(R!20, E!19) | (~((~member(tptp_fun_X_3(E!19, R!20), E!19)) | apply(R!20, tptp_fun_X_3(E!19, R!20), tptp_fun_X_3(E!19, R!20)))) | (~((~member(tptp_fun_Y_4(E!19, R!20), E!19)) | (~member(tptp_fun_X_5(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_4(E!19, R!20), tptp_fun_X_5(E!19, R!20))) | (~apply(R!20, tptp_fun_X_5(E!19, R!20), tptp_fun_Y_4(E!19, R!20))) | (tptp_fun_X_5(E!19, R!20) = tptp_fun_Y_4(E!19, R!20)))) | (~((~member(tptp_fun_Z_6(E!19, R!20), E!19)) | (~member(tptp_fun_Y_7(E!19, R!20), E!19)) | (~member(tptp_fun_X_8(E!19, R!20), E!19)) | (~apply(R!20, tptp_fun_Y_7(E!19, R!20), tptp_fun_Z_6(E!19, R!20))) | (~apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Y_7(E!19, R!20))) | apply(R!20, tptp_fun_X_8(E!19, R!20), tptp_fun_Z_6(E!19, R!20)))))) | (~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19))))))))) | ((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19)))))))),
% 0.22/0.45 inference(tautology,[status(thm)],[])).
% 0.22/0.45 tff(114,plain,
% 0.22/0.45 ((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19))))))),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[113, 112])).
% 0.22/0.45 tff(115,plain,
% 0.22/0.45 (order(R!20, E!19)),
% 0.22/0.45 inference(and_elim,[status(thm)],[73])).
% 0.22/0.45 tff(116,plain,
% 0.22/0.45 ((~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19)))))))) | (~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19))))))),
% 0.22/0.45 inference(tautology,[status(thm)],[])).
% 0.22/0.45 tff(117,plain,
% 0.22/0.45 ((~((~order(R!20, E!19)) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19)))))))) | (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19))))))),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[116, 115])).
% 0.22/0.45 tff(118,plain,
% 0.22/0.45 (~((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19)))))),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[117, 114])).
% 0.22/0.45 tff(119,plain,
% 0.22/0.45 (((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~![X: $i, Y: $i] : ((X = Y) | (~apply(R!20, X, Y)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~apply(R!20, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R!20, X, Z) | (~apply(R!20, X, Y)) | (~apply(R!20, Y, Z)) | (~member(X, E!19)) | (~member(Y, E!19)) | (~member(Z, E!19))))) | ![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))),
% 0.22/0.45 inference(tautology,[status(thm)],[])).
% 0.22/0.45 tff(120,plain,
% 0.22/0.45 (![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[119, 118])).
% 0.22/0.45 tff(121,plain,
% 0.22/0.45 (member(A!18, E!19)),
% 0.22/0.45 inference(and_elim,[status(thm)],[73])).
% 0.22/0.45 tff(122,plain,
% 0.22/0.45 (((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | ((~member(A!18, E!19)) | apply(R!20, A!18, A!18))) <=> ((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~member(A!18, E!19)) | apply(R!20, A!18, A!18))),
% 0.22/0.45 inference(rewrite,[status(thm)],[])).
% 0.22/0.45 tff(123,plain,
% 0.22/0.45 ((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | ((~member(A!18, E!19)) | apply(R!20, A!18, A!18))),
% 0.22/0.45 inference(quant_inst,[status(thm)],[])).
% 0.22/0.45 tff(124,plain,
% 0.22/0.45 ((~![X: $i] : ((~member(X, E!19)) | apply(R!20, X, X))) | (~member(A!18, E!19)) | apply(R!20, A!18, A!18)),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[123, 122])).
% 0.22/0.45 tff(125,plain,
% 0.22/0.45 (apply(R!20, A!18, A!18)),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[124, 121, 120])).
% 0.22/0.45 tff(126,plain,
% 0.22/0.45 (apply(R!20, A!18, tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20))),
% 0.22/0.45 inference(modus_ponens,[status(thm)],[125, 83])).
% 0.22/0.45 tff(127,plain,
% 0.22/0.45 ($false),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[61, 126])).
% 0.22/0.45 tff(128,plain,((~member(tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, A!18, tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20))), inference(lemma,lemma(discharge,[]))).
% 0.22/0.45 tff(129,plain,
% 0.22/0.45 ((~(lower_bound(A!18, R!20, unordered_pair(A!18, B!17)) | (~((~member(tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, A!18, tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20)))))) | lower_bound(A!18, R!20, unordered_pair(A!18, B!17)) | (~((~member(tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, A!18, tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20))))),
% 0.22/0.45 inference(tautology,[status(thm)],[])).
% 0.22/0.45 tff(130,plain,
% 0.22/0.45 ((~(lower_bound(A!18, R!20, unordered_pair(A!18, B!17)) | (~((~member(tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20), unordered_pair(A!18, B!17))) | apply(R!20, A!18, tptp_fun_X_12(A!18, unordered_pair(A!18, B!17), R!20)))))) | lower_bound(A!18, R!20, unordered_pair(A!18, B!17))),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[129, 128])).
% 0.22/0.45 tff(131,plain,
% 0.22/0.45 (lower_bound(A!18, R!20, unordered_pair(A!18, B!17))),
% 0.22/0.45 inference(unit_resolution,[status(thm)],[130, 51])).
% 0.22/0.45 tff(132,plain,
% 0.22/0.45 (^[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)))))))))))),
% 0.22/0.45 inference(bind,[status(th)],[])).
% 0.22/0.45 tff(133,plain,
% 0.22/0.45 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![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))))))))))),
% 0.22/0.45 inference(quant_intro,[status(thm)],[132])).
% 0.22/0.45 tff(134,plain,
% 0.22/0.45 (^[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)))))))))),
% 0.22/0.45 inference(bind,[status(th)],[])).
% 0.22/0.45 tff(135,plain,
% 0.22/0.45 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![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))))))))),
% 0.22/0.45 inference(quant_intro,[status(thm)],[134])).
% 0.22/0.45 tff(136,plain,
% 0.22/0.45 (^[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)))))))))),
% 0.22/0.45 inference(bind,[status(th)],[])).
% 0.22/0.45 tff(137,plain,
% 0.22/0.45 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![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))))))))),
% 0.22/0.45 inference(quant_intro,[status(thm)],[136])).
% 0.22/0.45 tff(138,plain,
% 0.22/0.45 (![A: $i, X: $i, R: $i, E: $i] : (~((~((~greatest_lower_bound(A, X, R, E)) | (~((~member(A, X)) | (~lower_bound(A, R, X)) | (~![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))))))))),
% 0.22/0.45 inference(transitivity,[status(thm)],[137, 135])).
% 0.22/0.45 tff(139,plain,
% 0.22/0.45 (^[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))))))))))),
% 0.22/0.45 inference(bind,[status(th)],[])).
% 0.22/0.45 tff(140,plain,
% 0.22/0.45 (![A: $i, X: $i, R: $i, E: $i] : (((~greatest_lower_bound(A, X, R, E)) | (member(A, X) & lower_bound(A, R, X) & ![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))))))))),
% 0.22/0.45 inference(quant_intro,[status(thm)],[139])).
% 0.22/0.45 tff(141,plain,
% 0.22/0.45 (^[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))))))),
% 0.22/0.45 inference(bind,[status(th)],[])).
% 0.22/0.45 tff(142,plain,
% 0.22/0.45 (![A: $i, X: $i, R: $i, E: $i] : (((~greatest_lower_bound(A, X, R, E)) | (member(A, X) & lower_bound(A, R, X) & ![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)))))),
% 0.22/0.45 inference(quant_intro,[status(thm)],[141])).
% 0.22/0.45 tff(143,plain,
% 0.22/0.45 (![A: $i, X: $i, R: $i, E: $i] : (greatest_lower_bound(A, X, R, E) <=> (member(A, X) & lower_bound(A, R, X) & ![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))))),
% 0.22/0.45 inference(rewrite,[status(thm)],[])).
% 0.22/0.45 tff(144,plain,
% 0.22/0.45 (^[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)))))),
% 0.22/0.46 inference(bind,[status(th)],[])).
% 0.22/0.46 tff(145,plain,
% 0.22/0.46 (![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))))),
% 0.22/0.46 inference(quant_intro,[status(thm)],[144])).
% 0.22/0.46 tff(146,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')).
% 0.22/0.46 tff(147,plain,
% 0.22/0.46 (![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))))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[146, 145])).
% 0.22/0.46 tff(148,plain,
% 0.22/0.46 (![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))))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[147, 143])).
% 0.22/0.46 tff(149,plain,(
% 0.22/0.46 ![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))))))),
% 0.22/0.46 inference(skolemize,[status(sab)],[148])).
% 0.22/0.46 tff(150,plain,
% 0.22/0.46 (![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)))))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[149, 142])).
% 0.22/0.46 tff(151,plain,
% 0.22/0.46 (![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))))))))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[150, 140])).
% 0.22/0.46 tff(152,plain,
% 0.22/0.46 (![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))))))))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[151, 138])).
% 0.22/0.46 tff(153,plain,
% 0.22/0.46 (![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))))))))))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[152, 133])).
% 0.22/0.46 tff(154,plain,
% 0.22/0.46 (((~![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(A!18, unordered_pair(A!18, B!17), R!20, E!19) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~member(A!18, unordered_pair(A!18, B!17))) | (~((~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), E!19)) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18))))) | (~((~greatest_lower_bound(A!18, unordered_pair(A!18, B!17), R!20, E!19)) | (~((~member(A!18, unordered_pair(A!18, B!17))) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~![M: $i] : ((~member(M, E!19)) | apply(R!20, M, A!18) | (~lower_bound(M, R!20, unordered_pair(A!18, B!17)))))))))))) <=> ((~![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(A!18, unordered_pair(A!18, B!17), R!20, E!19) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~member(A!18, unordered_pair(A!18, B!17))) | (~((~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), E!19)) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18))))) | (~((~greatest_lower_bound(A!18, unordered_pair(A!18, B!17), R!20, E!19)) | (~((~member(A!18, unordered_pair(A!18, B!17))) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~![M: $i] : ((~member(M, E!19)) | apply(R!20, M, A!18) | (~lower_bound(M, R!20, unordered_pair(A!18, B!17))))))))))))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(155,plain,
% 0.22/0.46 ((~((~(greatest_lower_bound(A!18, unordered_pair(A!18, B!17), R!20, E!19) | (~member(A!18, unordered_pair(A!18, B!17))) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~(apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18) | (~member(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), E!19)) | (~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))))))) | (~((~greatest_lower_bound(A!18, unordered_pair(A!18, B!17), R!20, E!19)) | (~((~member(A!18, unordered_pair(A!18, B!17))) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~![M: $i] : (apply(R!20, M, A!18) | (~member(M, E!19)) | (~lower_bound(M, R!20, unordered_pair(A!18, B!17))))))))))) <=> (~((~(greatest_lower_bound(A!18, unordered_pair(A!18, B!17), R!20, E!19) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~member(A!18, unordered_pair(A!18, B!17))) | (~((~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), E!19)) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18))))) | (~((~greatest_lower_bound(A!18, unordered_pair(A!18, B!17), R!20, E!19)) | (~((~member(A!18, unordered_pair(A!18, B!17))) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~![M: $i] : ((~member(M, E!19)) | apply(R!20, M, A!18) | (~lower_bound(M, R!20, unordered_pair(A!18, B!17)))))))))))),
% 0.22/0.46 inference(rewrite,[status(thm)],[])).
% 0.22/0.46 tff(156,plain,
% 0.22/0.46 (((~![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(A!18, unordered_pair(A!18, B!17), R!20, E!19) | (~member(A!18, unordered_pair(A!18, B!17))) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~(apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18) | (~member(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), E!19)) | (~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))))))) | (~((~greatest_lower_bound(A!18, unordered_pair(A!18, B!17), R!20, E!19)) | (~((~member(A!18, unordered_pair(A!18, B!17))) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~![M: $i] : (apply(R!20, M, A!18) | (~member(M, E!19)) | (~lower_bound(M, R!20, unordered_pair(A!18, B!17)))))))))))) <=> ((~![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(A!18, unordered_pair(A!18, B!17), R!20, E!19) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~member(A!18, unordered_pair(A!18, B!17))) | (~((~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), E!19)) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18))))) | (~((~greatest_lower_bound(A!18, unordered_pair(A!18, B!17), R!20, E!19)) | (~((~member(A!18, unordered_pair(A!18, B!17))) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~![M: $i] : ((~member(M, E!19)) | apply(R!20, M, A!18) | (~lower_bound(M, R!20, unordered_pair(A!18, B!17))))))))))))),
% 0.22/0.46 inference(monotonicity,[status(thm)],[155])).
% 0.22/0.46 tff(157,plain,
% 0.22/0.46 (((~![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(A!18, unordered_pair(A!18, B!17), R!20, E!19) | (~member(A!18, unordered_pair(A!18, B!17))) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~(apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18) | (~member(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), E!19)) | (~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))))))) | (~((~greatest_lower_bound(A!18, unordered_pair(A!18, B!17), R!20, E!19)) | (~((~member(A!18, unordered_pair(A!18, B!17))) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~![M: $i] : (apply(R!20, M, A!18) | (~member(M, E!19)) | (~lower_bound(M, R!20, unordered_pair(A!18, B!17)))))))))))) <=> ((~![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(A!18, unordered_pair(A!18, B!17), R!20, E!19) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~member(A!18, unordered_pair(A!18, B!17))) | (~((~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), E!19)) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18))))) | (~((~greatest_lower_bound(A!18, unordered_pair(A!18, B!17), R!20, E!19)) | (~((~member(A!18, unordered_pair(A!18, B!17))) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~![M: $i] : ((~member(M, E!19)) | apply(R!20, M, A!18) | (~lower_bound(M, R!20, unordered_pair(A!18, B!17))))))))))))),
% 0.22/0.46 inference(transitivity,[status(thm)],[156, 154])).
% 0.22/0.46 tff(158,plain,
% 0.22/0.46 ((~![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(A!18, unordered_pair(A!18, B!17), R!20, E!19) | (~member(A!18, unordered_pair(A!18, B!17))) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~(apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18) | (~member(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), E!19)) | (~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))))))) | (~((~greatest_lower_bound(A!18, unordered_pair(A!18, B!17), R!20, E!19)) | (~((~member(A!18, unordered_pair(A!18, B!17))) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~![M: $i] : (apply(R!20, M, A!18) | (~member(M, E!19)) | (~lower_bound(M, R!20, unordered_pair(A!18, B!17)))))))))))),
% 0.22/0.46 inference(quant_inst,[status(thm)],[])).
% 0.22/0.46 tff(159,plain,
% 0.22/0.46 ((~![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(A!18, unordered_pair(A!18, B!17), R!20, E!19) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~member(A!18, unordered_pair(A!18, B!17))) | (~((~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), E!19)) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18))))) | (~((~greatest_lower_bound(A!18, unordered_pair(A!18, B!17), R!20, E!19)) | (~((~member(A!18, unordered_pair(A!18, B!17))) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~![M: $i] : ((~member(M, E!19)) | apply(R!20, M, A!18) | (~lower_bound(M, R!20, unordered_pair(A!18, B!17)))))))))))),
% 0.22/0.46 inference(modus_ponens,[status(thm)],[158, 157])).
% 0.22/0.46 tff(160,plain,
% 0.22/0.46 (~((~(greatest_lower_bound(A!18, unordered_pair(A!18, B!17), R!20, E!19) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~member(A!18, unordered_pair(A!18, B!17))) | (~((~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), E!19)) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18))))) | (~((~greatest_lower_bound(A!18, unordered_pair(A!18, B!17), R!20, E!19)) | (~((~member(A!18, unordered_pair(A!18, B!17))) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~![M: $i] : ((~member(M, E!19)) | apply(R!20, M, A!18) | (~lower_bound(M, R!20, unordered_pair(A!18, B!17))))))))))),
% 0.22/0.46 inference(unit_resolution,[status(thm)],[159, 153])).
% 0.22/0.46 tff(161,plain,
% 0.22/0.46 (((~(greatest_lower_bound(A!18, unordered_pair(A!18, B!17), R!20, E!19) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~member(A!18, unordered_pair(A!18, B!17))) | (~((~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), E!19)) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18))))) | (~((~greatest_lower_bound(A!18, unordered_pair(A!18, B!17), R!20, E!19)) | (~((~member(A!18, unordered_pair(A!18, B!17))) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~![M: $i] : ((~member(M, E!19)) | apply(R!20, M, A!18) | (~lower_bound(M, R!20, unordered_pair(A!18, B!17)))))))))) | (greatest_lower_bound(A!18, unordered_pair(A!18, B!17), R!20, E!19) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~member(A!18, unordered_pair(A!18, B!17))) | (~((~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), E!19)) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18))))),
% 0.22/0.47 inference(tautology,[status(thm)],[])).
% 0.22/0.47 tff(162,plain,
% 0.22/0.47 (greatest_lower_bound(A!18, unordered_pair(A!18, B!17), R!20, E!19) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~member(A!18, unordered_pair(A!18, B!17))) | (~((~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), E!19)) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18)))),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[161, 160])).
% 0.22/0.47 tff(163,plain,
% 0.22/0.47 (~greatest_lower_bound(A!18, unordered_pair(A!18, B!17), R!20, E!19)),
% 0.22/0.47 inference(or_elim,[status(thm)],[72])).
% 0.22/0.47 tff(164,plain,
% 0.22/0.47 ((~(greatest_lower_bound(A!18, unordered_pair(A!18, B!17), R!20, E!19) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~member(A!18, unordered_pair(A!18, B!17))) | (~((~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), E!19)) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18))))) | greatest_lower_bound(A!18, unordered_pair(A!18, B!17), R!20, E!19) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~member(A!18, unordered_pair(A!18, B!17))) | (~((~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), E!19)) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18)))),
% 0.22/0.47 inference(tautology,[status(thm)],[])).
% 0.22/0.47 tff(165,plain,
% 0.22/0.47 ((~(greatest_lower_bound(A!18, unordered_pair(A!18, B!17), R!20, E!19) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~member(A!18, unordered_pair(A!18, B!17))) | (~((~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), E!19)) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18))))) | (~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~member(A!18, unordered_pair(A!18, B!17))) | (~((~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), E!19)) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18)))),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[164, 163])).
% 0.22/0.47 tff(166,plain,
% 0.22/0.47 ((~lower_bound(A!18, R!20, unordered_pair(A!18, B!17))) | (~member(A!18, unordered_pair(A!18, B!17))) | (~((~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), E!19)) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18)))),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[165, 162])).
% 0.22/0.47 tff(167,plain,
% 0.22/0.47 (~((~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), E!19)) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18))),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[166, 131, 45])).
% 0.22/0.47 tff(168,plain,
% 0.22/0.47 (((~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), E!19)) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18)) | lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))),
% 0.22/0.47 inference(tautology,[status(thm)],[])).
% 0.22/0.47 tff(169,plain,
% 0.22/0.47 (lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[168, 167])).
% 0.22/0.47 tff(170,plain,
% 0.22/0.47 ((~((~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))) | ![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), X)))) | (~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))) | ![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), X))),
% 0.22/0.47 inference(tautology,[status(thm)],[])).
% 0.22/0.47 tff(171,plain,
% 0.22/0.47 ((~((~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))) | ![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), X)))) | ![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), X))),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[170, 169])).
% 0.22/0.47 tff(172,plain,
% 0.22/0.47 (![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), X))),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[171, 20])).
% 0.22/0.47 tff(173,plain,
% 0.22/0.47 (((~lower_bound(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), R!20, unordered_pair(A!18, B!17))) | (~member(tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), E!19)) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18)) | (~apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18))),
% 0.22/0.47 inference(tautology,[status(thm)],[])).
% 0.22/0.47 tff(174,plain,
% 0.22/0.47 (~apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18)),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[173, 167])).
% 0.22/0.47 tff(175,plain,
% 0.22/0.47 (((~![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), X))) | ((~member(A!18, unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18))) <=> ((~![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), X))) | (~member(A!18, unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18))),
% 0.22/0.47 inference(rewrite,[status(thm)],[])).
% 0.22/0.47 tff(176,plain,
% 0.22/0.47 ((~![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), X))) | ((~member(A!18, unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18))),
% 0.22/0.47 inference(quant_inst,[status(thm)],[])).
% 0.22/0.47 tff(177,plain,
% 0.22/0.47 ((~![X: $i] : ((~member(X, unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), X))) | (~member(A!18, unordered_pair(A!18, B!17))) | apply(R!20, tptp_fun_M_16(E!19, R!20, unordered_pair(A!18, B!17), A!18), A!18)),
% 0.22/0.47 inference(modus_ponens,[status(thm)],[176, 175])).
% 0.22/0.47 tff(178,plain,
% 0.22/0.47 ($false),
% 0.22/0.47 inference(unit_resolution,[status(thm)],[177, 45, 174, 172])).
% 0.22/0.47 % SZS output end Proof
% 0.22/0.47 % E exiting
%------------------------------------------------------------------------------