%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET804+4 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n012.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 05:26:23 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 : SET804+4 : TPTP v9.0.0. Released v3.2.0.
% 0.07/0.13 % Command : run_E %s %d THM
% 0.12/0.35 % Computer : n012.cluster.edu
% 0.12/0.35 % Model : x86_64 x86_64
% 0.12/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.35 % Memory : 8042.1875MB
% 0.12/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.35 % CPULimit : 300
% 0.12/0.35 % WCLimit : 300
% 0.12/0.35 % DateTime : Fri Jun 20 05:42:25 EDT 2025
% 0.12/0.35 % CPUTime :
% 0.12/0.39 % SZS status Theorem
% 0.12/0.39 % SZS output start Proof
% 0.12/0.39 tff(tptp_fun_M_18_type, type, (
% 0.12/0.39 tptp_fun_M_18: $i)).
% 0.12/0.39 tff(tptp_fun_M2_16_type, type, (
% 0.12/0.39 tptp_fun_M2_16: $i)).
% 0.12/0.39 tff(tptp_fun_M1_17_type, type, (
% 0.12/0.39 tptp_fun_M1_17: $i)).
% 0.12/0.39 tff(apply_type, type, (
% 0.12/0.39 apply: ( $i * $i * $i ) > $o)).
% 0.12/0.39 tff(tptp_fun_R_15_type, type, (
% 0.12/0.39 tptp_fun_R_15: $i)).
% 0.12/0.39 tff(member_type, type, (
% 0.12/0.39 member: ( $i * $i ) > $o)).
% 0.12/0.39 tff(tptp_fun_E_14_type, type, (
% 0.12/0.39 tptp_fun_E_14: $i)).
% 0.12/0.39 tff(min_type, type, (
% 0.12/0.39 min: ( $i * $i * $i ) > $o)).
% 0.12/0.39 tff(tptp_fun_X_11_type, type, (
% 0.12/0.39 tptp_fun_X_11: ( $i * $i * $i ) > $i)).
% 0.12/0.39 tff(least_type, type, (
% 0.12/0.39 least: ( $i * $i * $i ) > $o)).
% 0.12/0.39 tff(order_type, type, (
% 0.12/0.39 order: ( $i * $i ) > $o)).
% 0.12/0.39 tff(tptp_fun_X_9_type, type, (
% 0.12/0.39 tptp_fun_X_9: ( $i * $i * $i ) > $i)).
% 0.12/0.39 tff(1,plain,
% 0.12/0.39 ((M2!16 = M1!17) <=> (M1!17 = M2!16)),
% 0.12/0.39 inference(commutativity,[status(thm)],[])).
% 0.12/0.39 tff(2,plain,
% 0.12/0.39 (^[R: $i, E: $i, M: $i] : rewrite((~((~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M)))))))) | (~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))))) <=> (~((~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))) | (~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M)))))))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(3,plain,
% 0.12/0.39 (![R: $i, E: $i, M: $i] : (~((~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M)))))))) | (~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))))) <=> ![R: $i, E: $i, M: $i] : (~((~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))) | (~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M))))))))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[2])).
% 0.12/0.39 tff(4,plain,
% 0.12/0.39 (^[R: $i, E: $i, M: $i] : refl((~((~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M)))))))) | (~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))))) <=> (~((~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M)))))))) | (~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(5,plain,
% 0.12/0.39 (![R: $i, E: $i, M: $i] : (~((~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M)))))))) | (~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))))) <=> ![R: $i, E: $i, M: $i] : (~((~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M)))))))) | (~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M))))))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[4])).
% 0.12/0.39 tff(6,plain,
% 0.12/0.39 (^[R: $i, E: $i, M: $i] : rewrite((~((~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M)))))))) | (~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))))) <=> (~((~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M)))))))) | (~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(7,plain,
% 0.12/0.39 (![R: $i, E: $i, M: $i] : (~((~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M)))))))) | (~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))))) <=> ![R: $i, E: $i, M: $i] : (~((~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M)))))))) | (~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M))))))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[6])).
% 0.12/0.39 tff(8,plain,
% 0.12/0.39 (![R: $i, E: $i, M: $i] : (~((~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M)))))))) | (~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))))) <=> ![R: $i, E: $i, M: $i] : (~((~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M)))))))) | (~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M))))))))),
% 0.12/0.39 inference(transitivity,[status(thm)],[7, 5])).
% 0.12/0.39 tff(9,plain,
% 0.12/0.39 (^[R: $i, E: $i, M: $i] : trans(monotonicity(rewrite(((~min(M, R, E)) | (member(M, E) & ![X: $i] : ((~(member(X, E) & apply(R, X, M))) | (M = X)))) <=> ((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M)))))))), trans(monotonicity(rewrite((~((~(member(tptp_fun_X_11(M, E, R), E) & apply(R, tptp_fun_X_11(M, E, R), M))) | (M = tptp_fun_X_11(M, E, R)))) <=> (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M))))), ((min(M, R, E) | (~member(M, E)) | (~((~(member(tptp_fun_X_11(M, E, R), E) & apply(R, tptp_fun_X_11(M, E, R), M))) | (M = tptp_fun_X_11(M, E, R))))) <=> (min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M))))))), rewrite((min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M))))) <=> (min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))), ((min(M, R, E) | (~member(M, E)) | (~((~(member(tptp_fun_X_11(M, E, R), E) & apply(R, tptp_fun_X_11(M, E, R), M))) | (M = tptp_fun_X_11(M, E, R))))) <=> (min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M))))))), ((((~min(M, R, E)) | (member(M, E) & ![X: $i] : ((~(member(X, E) & apply(R, X, M))) | (M = X)))) & (min(M, R, E) | (~member(M, E)) | (~((~(member(tptp_fun_X_11(M, E, R), E) & apply(R, tptp_fun_X_11(M, E, R), M))) | (M = tptp_fun_X_11(M, E, R)))))) <=> (((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M))))))) & (min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))))), rewrite((((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M))))))) & (min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))) <=> (~((~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M)))))))) | (~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M))))))))), ((((~min(M, R, E)) | (member(M, E) & ![X: $i] : ((~(member(X, E) & apply(R, X, M))) | (M = X)))) & (min(M, R, E) | (~member(M, E)) | (~((~(member(tptp_fun_X_11(M, E, R), E) & apply(R, tptp_fun_X_11(M, E, R), M))) | (M = tptp_fun_X_11(M, E, R)))))) <=> (~((~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M)))))))) | (~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M))))))))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(10,plain,
% 0.12/0.40 (![R: $i, E: $i, M: $i] : (((~min(M, R, E)) | (member(M, E) & ![X: $i] : ((~(member(X, E) & apply(R, X, M))) | (M = X)))) & (min(M, R, E) | (~member(M, E)) | (~((~(member(tptp_fun_X_11(M, E, R), E) & apply(R, tptp_fun_X_11(M, E, R), M))) | (M = tptp_fun_X_11(M, E, R)))))) <=> ![R: $i, E: $i, M: $i] : (~((~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M)))))))) | (~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M))))))))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[9])).
% 0.12/0.40 tff(11,plain,
% 0.12/0.40 (^[R: $i, E: $i, M: $i] : rewrite((((~min(M, R, E)) | (member(M, E) & ![X: $i] : ((~(member(X, E) & apply(R, X, M))) | (M = X)))) & (min(M, R, E) | ((~member(M, E)) | (~((~(member(tptp_fun_X_11(M, E, R), E) & apply(R, tptp_fun_X_11(M, E, R), M))) | (M = tptp_fun_X_11(M, E, R))))))) <=> (((~min(M, R, E)) | (member(M, E) & ![X: $i] : ((~(member(X, E) & apply(R, X, M))) | (M = X)))) & (min(M, R, E) | (~member(M, E)) | (~((~(member(tptp_fun_X_11(M, E, R), E) & apply(R, tptp_fun_X_11(M, E, R), M))) | (M = tptp_fun_X_11(M, E, R)))))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(12,plain,
% 0.12/0.40 (![R: $i, E: $i, M: $i] : (((~min(M, R, E)) | (member(M, E) & ![X: $i] : ((~(member(X, E) & apply(R, X, M))) | (M = X)))) & (min(M, R, E) | ((~member(M, E)) | (~((~(member(tptp_fun_X_11(M, E, R), E) & apply(R, tptp_fun_X_11(M, E, R), M))) | (M = tptp_fun_X_11(M, E, R))))))) <=> ![R: $i, E: $i, M: $i] : (((~min(M, R, E)) | (member(M, E) & ![X: $i] : ((~(member(X, E) & apply(R, X, M))) | (M = X)))) & (min(M, R, E) | (~member(M, E)) | (~((~(member(tptp_fun_X_11(M, E, R), E) & apply(R, tptp_fun_X_11(M, E, R), M))) | (M = tptp_fun_X_11(M, E, R))))))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[11])).
% 0.12/0.40 tff(13,plain,
% 0.12/0.40 (![R: $i, E: $i, M: $i] : (min(M, R, E) <=> (member(M, E) & ![X: $i] : ((~(member(X, E) & apply(R, X, M))) | (M = X)))) <=> ![R: $i, E: $i, M: $i] : (min(M, R, E) <=> (member(M, E) & ![X: $i] : ((~(member(X, E) & apply(R, X, M))) | (M = X))))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(14,plain,
% 0.12/0.40 (^[R: $i, E: $i, M: $i] : rewrite((min(M, R, E) <=> (member(M, E) & ![X: $i] : ((member(X, E) & apply(R, X, M)) => (M = X)))) <=> (min(M, R, E) <=> (member(M, E) & ![X: $i] : ((~(member(X, E) & apply(R, X, M))) | (M = X)))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(15,plain,
% 0.12/0.40 (![R: $i, E: $i, M: $i] : (min(M, R, E) <=> (member(M, E) & ![X: $i] : ((member(X, E) & apply(R, X, M)) => (M = X)))) <=> ![R: $i, E: $i, M: $i] : (min(M, R, E) <=> (member(M, E) & ![X: $i] : ((~(member(X, E) & apply(R, X, M))) | (M = X))))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[14])).
% 0.12/0.40 tff(16,axiom,(![R: $i, E: $i, M: $i] : (min(M, R, E) <=> (member(M, E) & ![X: $i] : ((member(X, E) & apply(R, X, M)) => (M = X))))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+3.ax','min')).
% 0.12/0.40 tff(17,plain,
% 0.12/0.40 (![R: $i, E: $i, M: $i] : (min(M, R, E) <=> (member(M, E) & ![X: $i] : ((~(member(X, E) & apply(R, X, M))) | (M = X))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[16, 15])).
% 0.12/0.40 tff(18,plain,
% 0.12/0.40 (![R: $i, E: $i, M: $i] : (min(M, R, E) <=> (member(M, E) & ![X: $i] : ((~(member(X, E) & apply(R, X, M))) | (M = X))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[17, 13])).
% 0.12/0.40 tff(19,plain,(
% 0.12/0.40 ![R: $i, E: $i, M: $i] : (((~min(M, R, E)) | (member(M, E) & ![X: $i] : ((~(member(X, E) & apply(R, X, M))) | (M = X)))) & (min(M, R, E) | ((~member(M, E)) | (~((~(member(tptp_fun_X_11(M, E, R), E) & apply(R, tptp_fun_X_11(M, E, R), M))) | (M = tptp_fun_X_11(M, E, R)))))))),
% 0.12/0.40 inference(skolemize,[status(sab)],[18])).
% 0.12/0.40 tff(20,plain,
% 0.12/0.40 (![R: $i, E: $i, M: $i] : (((~min(M, R, E)) | (member(M, E) & ![X: $i] : ((~(member(X, E) & apply(R, X, M))) | (M = X)))) & (min(M, R, E) | (~member(M, E)) | (~((~(member(tptp_fun_X_11(M, E, R), E) & apply(R, tptp_fun_X_11(M, E, R), M))) | (M = tptp_fun_X_11(M, E, R))))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[19, 12])).
% 0.12/0.40 tff(21,plain,
% 0.12/0.40 (![R: $i, E: $i, M: $i] : (~((~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M)))))))) | (~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M))))))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[20, 10])).
% 0.12/0.40 tff(22,plain,
% 0.12/0.40 (![R: $i, E: $i, M: $i] : (~((~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M)))))))) | (~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M))))))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[21, 8])).
% 0.12/0.40 tff(23,plain,
% 0.12/0.40 (![R: $i, E: $i, M: $i] : (~((~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))) | (~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M))))))))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[22, 3])).
% 0.12/0.40 tff(24,plain,
% 0.12/0.40 (((~![R: $i, E: $i, M: $i] : (~((~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))) | (~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M))))))))))) | (~((~(min(M1!17, R!15, E!14) | (~member(M1!17, E!14)) | (~((~apply(R!15, tptp_fun_X_11(M1!17, E!14, R!15), M1!17)) | (~member(tptp_fun_X_11(M1!17, E!14, R!15), E!14)) | (M1!17 = tptp_fun_X_11(M1!17, E!14, R!15)))))) | (~((~min(M1!17, R!15, E!14)) | (~((~member(M1!17, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17))))))))))) <=> ((~![R: $i, E: $i, M: $i] : (~((~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))) | (~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M))))))))))) | (~((~(min(M1!17, R!15, E!14) | (~member(M1!17, E!14)) | (~((~apply(R!15, tptp_fun_X_11(M1!17, E!14, R!15), M1!17)) | (~member(tptp_fun_X_11(M1!17, E!14, R!15), E!14)) | (M1!17 = tptp_fun_X_11(M1!17, E!14, R!15)))))) | (~((~min(M1!17, R!15, E!14)) | (~((~member(M1!17, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17)))))))))))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(25,plain,
% 0.12/0.40 ((~((~(min(M1!17, R!15, E!14) | (~member(M1!17, E!14)) | (~((M1!17 = tptp_fun_X_11(M1!17, E!14, R!15)) | (~member(tptp_fun_X_11(M1!17, E!14, R!15), E!14)) | (~apply(R!15, tptp_fun_X_11(M1!17, E!14, R!15), M1!17)))))) | (~((~min(M1!17, R!15, E!14)) | (~((~member(M1!17, E!14)) | (~![X: $i] : ((M1!17 = X) | (~member(X, E!14)) | (~apply(R!15, X, M1!17)))))))))) <=> (~((~(min(M1!17, R!15, E!14) | (~member(M1!17, E!14)) | (~((~apply(R!15, tptp_fun_X_11(M1!17, E!14, R!15), M1!17)) | (~member(tptp_fun_X_11(M1!17, E!14, R!15), E!14)) | (M1!17 = tptp_fun_X_11(M1!17, E!14, R!15)))))) | (~((~min(M1!17, R!15, E!14)) | (~((~member(M1!17, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17))))))))))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(26,plain,
% 0.12/0.40 (((~![R: $i, E: $i, M: $i] : (~((~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))) | (~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M))))))))))) | (~((~(min(M1!17, R!15, E!14) | (~member(M1!17, E!14)) | (~((M1!17 = tptp_fun_X_11(M1!17, E!14, R!15)) | (~member(tptp_fun_X_11(M1!17, E!14, R!15), E!14)) | (~apply(R!15, tptp_fun_X_11(M1!17, E!14, R!15), M1!17)))))) | (~((~min(M1!17, R!15, E!14)) | (~((~member(M1!17, E!14)) | (~![X: $i] : ((M1!17 = X) | (~member(X, E!14)) | (~apply(R!15, X, M1!17))))))))))) <=> ((~![R: $i, E: $i, M: $i] : (~((~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))) | (~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M))))))))))) | (~((~(min(M1!17, R!15, E!14) | (~member(M1!17, E!14)) | (~((~apply(R!15, tptp_fun_X_11(M1!17, E!14, R!15), M1!17)) | (~member(tptp_fun_X_11(M1!17, E!14, R!15), E!14)) | (M1!17 = tptp_fun_X_11(M1!17, E!14, R!15)))))) | (~((~min(M1!17, R!15, E!14)) | (~((~member(M1!17, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17)))))))))))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[25])).
% 0.12/0.40 tff(27,plain,
% 0.12/0.40 (((~![R: $i, E: $i, M: $i] : (~((~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))) | (~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M))))))))))) | (~((~(min(M1!17, R!15, E!14) | (~member(M1!17, E!14)) | (~((M1!17 = tptp_fun_X_11(M1!17, E!14, R!15)) | (~member(tptp_fun_X_11(M1!17, E!14, R!15), E!14)) | (~apply(R!15, tptp_fun_X_11(M1!17, E!14, R!15), M1!17)))))) | (~((~min(M1!17, R!15, E!14)) | (~((~member(M1!17, E!14)) | (~![X: $i] : ((M1!17 = X) | (~member(X, E!14)) | (~apply(R!15, X, M1!17))))))))))) <=> ((~![R: $i, E: $i, M: $i] : (~((~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))) | (~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M))))))))))) | (~((~(min(M1!17, R!15, E!14) | (~member(M1!17, E!14)) | (~((~apply(R!15, tptp_fun_X_11(M1!17, E!14, R!15), M1!17)) | (~member(tptp_fun_X_11(M1!17, E!14, R!15), E!14)) | (M1!17 = tptp_fun_X_11(M1!17, E!14, R!15)))))) | (~((~min(M1!17, R!15, E!14)) | (~((~member(M1!17, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17)))))))))))),
% 0.12/0.40 inference(transitivity,[status(thm)],[26, 24])).
% 0.12/0.40 tff(28,plain,
% 0.12/0.40 ((~![R: $i, E: $i, M: $i] : (~((~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))) | (~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M))))))))))) | (~((~(min(M1!17, R!15, E!14) | (~member(M1!17, E!14)) | (~((M1!17 = tptp_fun_X_11(M1!17, E!14, R!15)) | (~member(tptp_fun_X_11(M1!17, E!14, R!15), E!14)) | (~apply(R!15, tptp_fun_X_11(M1!17, E!14, R!15), M1!17)))))) | (~((~min(M1!17, R!15, E!14)) | (~((~member(M1!17, E!14)) | (~![X: $i] : ((M1!17 = X) | (~member(X, E!14)) | (~apply(R!15, X, M1!17))))))))))),
% 0.12/0.40 inference(quant_inst,[status(thm)],[])).
% 0.12/0.40 tff(29,plain,
% 0.12/0.40 ((~![R: $i, E: $i, M: $i] : (~((~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))) | (~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M))))))))))) | (~((~(min(M1!17, R!15, E!14) | (~member(M1!17, E!14)) | (~((~apply(R!15, tptp_fun_X_11(M1!17, E!14, R!15), M1!17)) | (~member(tptp_fun_X_11(M1!17, E!14, R!15), E!14)) | (M1!17 = tptp_fun_X_11(M1!17, E!14, R!15)))))) | (~((~min(M1!17, R!15, E!14)) | (~((~member(M1!17, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17))))))))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[28, 27])).
% 0.12/0.40 tff(30,plain,
% 0.12/0.40 (~((~(min(M1!17, R!15, E!14) | (~member(M1!17, E!14)) | (~((~apply(R!15, tptp_fun_X_11(M1!17, E!14, R!15), M1!17)) | (~member(tptp_fun_X_11(M1!17, E!14, R!15), E!14)) | (M1!17 = tptp_fun_X_11(M1!17, E!14, R!15)))))) | (~((~min(M1!17, R!15, E!14)) | (~((~member(M1!17, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17)))))))))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[29, 23])).
% 0.12/0.40 tff(31,plain,
% 0.12/0.40 (((~(min(M1!17, R!15, E!14) | (~member(M1!17, E!14)) | (~((~apply(R!15, tptp_fun_X_11(M1!17, E!14, R!15), M1!17)) | (~member(tptp_fun_X_11(M1!17, E!14, R!15), E!14)) | (M1!17 = tptp_fun_X_11(M1!17, E!14, R!15)))))) | (~((~min(M1!17, R!15, E!14)) | (~((~member(M1!17, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17))))))))) | ((~min(M1!17, R!15, E!14)) | (~((~member(M1!17, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17)))))))),
% 0.12/0.41 inference(tautology,[status(thm)],[])).
% 0.12/0.41 tff(32,plain,
% 0.12/0.41 ((~min(M1!17, R!15, E!14)) | (~((~member(M1!17, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17))))))),
% 0.12/0.41 inference(unit_resolution,[status(thm)],[31, 30])).
% 0.12/0.41 tff(33,plain,
% 0.12/0.41 ((order(R!15, E!14) & (least(M!18, R!15, E!14) & min(M1!17, R!15, E!14) & min(M2!16, R!15, E!14) & (~(M1!17 = M2!16)))) <=> (order(R!15, E!14) & least(M!18, R!15, E!14) & min(M1!17, R!15, E!14) & min(M2!16, R!15, E!14) & (~(M1!17 = M2!16)))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(34,plain,
% 0.12/0.41 ((least(M!18, R!15, E!14) & (min(M1!17, R!15, E!14) & min(M2!16, R!15, E!14) & (~(M1!17 = M2!16)))) <=> (least(M!18, R!15, E!14) & min(M1!17, R!15, E!14) & min(M2!16, R!15, E!14) & (~(M1!17 = M2!16)))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(35,plain,
% 0.12/0.41 ((order(R!15, E!14) & (least(M!18, R!15, E!14) & (min(M1!17, R!15, E!14) & min(M2!16, R!15, E!14) & (~(M1!17 = M2!16))))) <=> (order(R!15, E!14) & (least(M!18, R!15, E!14) & min(M1!17, R!15, E!14) & min(M2!16, R!15, E!14) & (~(M1!17 = M2!16))))),
% 0.12/0.41 inference(monotonicity,[status(thm)],[34])).
% 0.12/0.41 tff(36,plain,
% 0.12/0.41 ((order(R!15, E!14) & (least(M!18, R!15, E!14) & (min(M1!17, R!15, E!14) & min(M2!16, R!15, E!14) & (~(M1!17 = M2!16))))) <=> (order(R!15, E!14) & least(M!18, R!15, E!14) & min(M1!17, R!15, E!14) & min(M2!16, R!15, E!14) & (~(M1!17 = M2!16)))),
% 0.12/0.41 inference(transitivity,[status(thm)],[35, 33])).
% 0.12/0.41 tff(37,plain,
% 0.12/0.41 ((~![R: $i, E: $i] : ((~order(R, E)) | ![M1: $i, M2: $i] : ((~?[M: $i] : least(M, R, E)) | (~(min(M1, R, E) & min(M2, R, E) & (~(M1 = M2))))))) <=> (~![R: $i, E: $i] : ((~order(R, E)) | ![M1: $i, M2: $i] : ((~?[M: $i] : least(M, R, E)) | (~(min(M1, R, E) & min(M2, R, E) & (~(M1 = M2)))))))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(38,plain,
% 0.12/0.41 ((~![R: $i, E: $i] : (order(R, E) => ![M1: $i, M2: $i] : (((min(M1, R, E) & min(M2, R, E)) & (~(M1 = M2))) => (~?[M: $i] : least(M, R, E))))) <=> (~![R: $i, E: $i] : ((~order(R, E)) | ![M1: $i, M2: $i] : ((~?[M: $i] : least(M, R, E)) | (~(min(M1, R, E) & min(M2, R, E) & (~(M1 = M2)))))))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(39,axiom,(~![R: $i, E: $i] : (order(R, E) => ![M1: $i, M2: $i] : (((min(M1, R, E) & min(M2, R, E)) & (~(M1 = M2))) => (~?[M: $i] : least(M, R, E))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','thIV16')).
% 0.12/0.41 tff(40,plain,
% 0.12/0.41 (~![R: $i, E: $i] : ((~order(R, E)) | ![M1: $i, M2: $i] : ((~?[M: $i] : least(M, R, E)) | (~(min(M1, R, E) & min(M2, R, E) & (~(M1 = M2))))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[39, 38])).
% 0.12/0.41 tff(41,plain,
% 0.12/0.41 (~![R: $i, E: $i] : ((~order(R, E)) | ![M1: $i, M2: $i] : ((~?[M: $i] : least(M, R, E)) | (~(min(M1, R, E) & min(M2, R, E) & (~(M1 = M2))))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[40, 37])).
% 0.12/0.41 tff(42,plain,
% 0.12/0.41 (~![R: $i, E: $i] : ((~order(R, E)) | ![M1: $i, M2: $i] : ((~?[M: $i] : least(M, R, E)) | (~(min(M1, R, E) & min(M2, R, E) & (~(M1 = M2))))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[41, 37])).
% 0.12/0.41 tff(43,plain,
% 0.12/0.41 (~![R: $i, E: $i] : ((~order(R, E)) | ![M1: $i, M2: $i] : ((~?[M: $i] : least(M, R, E)) | (~(min(M1, R, E) & min(M2, R, E) & (~(M1 = M2))))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[42, 37])).
% 0.12/0.41 tff(44,plain,
% 0.12/0.41 (order(R!15, E!14) & least(M!18, R!15, E!14) & min(M1!17, R!15, E!14) & min(M2!16, R!15, E!14) & (~(M1!17 = M2!16))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[43, 36])).
% 0.12/0.41 tff(45,plain,
% 0.12/0.41 (min(M1!17, R!15, E!14)),
% 0.12/0.41 inference(and_elim,[status(thm)],[44])).
% 0.12/0.41 tff(46,plain,
% 0.12/0.41 ((~((~min(M1!17, R!15, E!14)) | (~((~member(M1!17, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17)))))))) | (~min(M1!17, R!15, E!14)) | (~((~member(M1!17, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17))))))),
% 0.22/0.41 inference(tautology,[status(thm)],[])).
% 0.22/0.41 tff(47,plain,
% 0.22/0.41 ((~((~min(M1!17, R!15, E!14)) | (~((~member(M1!17, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17)))))))) | (~((~member(M1!17, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17))))))),
% 0.22/0.41 inference(unit_resolution,[status(thm)],[46, 45])).
% 0.22/0.41 tff(48,plain,
% 0.22/0.41 (~((~member(M1!17, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17)))))),
% 0.22/0.41 inference(unit_resolution,[status(thm)],[47, 32])).
% 0.22/0.41 tff(49,plain,
% 0.22/0.41 (((~member(M1!17, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17))))) | member(M1!17, E!14)),
% 0.22/0.41 inference(tautology,[status(thm)],[])).
% 0.22/0.41 tff(50,plain,
% 0.22/0.41 (member(M1!17, E!14)),
% 0.22/0.41 inference(unit_resolution,[status(thm)],[49, 48])).
% 0.22/0.41 tff(51,plain,
% 0.22/0.41 (^[R: $i, E: $i, M: $i] : rewrite((~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R)))))))) <=> (~((~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R)))))) | (~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))))))),
% 0.22/0.41 inference(bind,[status(th)],[])).
% 0.22/0.41 tff(52,plain,
% 0.22/0.41 (![R: $i, E: $i, M: $i] : (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R)))))))) <=> ![R: $i, E: $i, M: $i] : (~((~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R)))))) | (~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X)))))))))),
% 0.22/0.41 inference(quant_intro,[status(thm)],[51])).
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% 0.22/0.41 (^[R: $i, E: $i, M: $i] : refl((~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R)))))))) <=> (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R)))))))))),
% 0.22/0.41 inference(bind,[status(th)],[])).
% 0.22/0.41 tff(54,plain,
% 0.22/0.41 (![R: $i, E: $i, M: $i] : (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R)))))))) <=> ![R: $i, E: $i, M: $i] : (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R))))))))),
% 0.22/0.41 inference(quant_intro,[status(thm)],[53])).
% 0.22/0.41 tff(55,plain,
% 0.22/0.41 (^[R: $i, E: $i, M: $i] : rewrite((~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R)))))))) <=> (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R)))))))))),
% 0.22/0.41 inference(bind,[status(th)],[])).
% 0.22/0.41 tff(56,plain,
% 0.22/0.41 (![R: $i, E: $i, M: $i] : (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R)))))))) <=> ![R: $i, E: $i, M: $i] : (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R))))))))),
% 0.22/0.41 inference(quant_intro,[status(thm)],[55])).
% 0.22/0.41 tff(57,plain,
% 0.22/0.41 (![R: $i, E: $i, M: $i] : (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R)))))))) <=> ![R: $i, E: $i, M: $i] : (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R))))))))),
% 0.22/0.41 inference(transitivity,[status(thm)],[56, 54])).
% 0.22/0.41 tff(58,plain,
% 0.22/0.41 (^[R: $i, E: $i, M: $i] : trans(monotonicity(rewrite(((~least(M, R, E)) | (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) <=> ((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))), rewrite((least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R))))) <=> (least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R)))))), ((((~least(M, R, E)) | (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) & (least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R)))))) <=> (((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X)))))) & (least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R)))))))), rewrite((((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X)))))) & (least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R)))))) <=> (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R))))))))), ((((~least(M, R, E)) | (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) & (least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R)))))) <=> (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R))))))))))),
% 0.22/0.41 inference(bind,[status(th)],[])).
% 0.22/0.41 tff(59,plain,
% 0.22/0.41 (![R: $i, E: $i, M: $i] : (((~least(M, R, E)) | (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) & (least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R)))))) <=> ![R: $i, E: $i, M: $i] : (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R))))))))),
% 0.22/0.41 inference(quant_intro,[status(thm)],[58])).
% 0.22/0.41 tff(60,plain,
% 0.22/0.41 (^[R: $i, E: $i, M: $i] : rewrite((((~least(M, R, E)) | (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) & (least(M, R, E) | ((~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R))))))) <=> (((~least(M, R, E)) | (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) & (least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R)))))))),
% 0.22/0.41 inference(bind,[status(th)],[])).
% 0.22/0.41 tff(61,plain,
% 0.22/0.41 (![R: $i, E: $i, M: $i] : (((~least(M, R, E)) | (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) & (least(M, R, E) | ((~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R))))))) <=> ![R: $i, E: $i, M: $i] : (((~least(M, R, E)) | (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) & (least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R))))))),
% 0.22/0.41 inference(quant_intro,[status(thm)],[60])).
% 0.22/0.41 tff(62,plain,
% 0.22/0.41 (![R: $i, E: $i, M: $i] : (least(M, R, E) <=> (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) <=> ![R: $i, E: $i, M: $i] : (least(M, R, E) <=> (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X))))),
% 0.22/0.41 inference(rewrite,[status(thm)],[])).
% 0.22/0.41 tff(63,plain,
% 0.22/0.41 (^[R: $i, E: $i, M: $i] : rewrite((least(M, R, E) <=> (member(M, E) & ![X: $i] : (member(X, E) => apply(R, M, X)))) <=> (least(M, R, E) <=> (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))))),
% 0.22/0.41 inference(bind,[status(th)],[])).
% 0.22/0.41 tff(64,plain,
% 0.22/0.41 (![R: $i, E: $i, M: $i] : (least(M, R, E) <=> (member(M, E) & ![X: $i] : (member(X, E) => apply(R, M, X)))) <=> ![R: $i, E: $i, M: $i] : (least(M, R, E) <=> (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X))))),
% 0.22/0.41 inference(quant_intro,[status(thm)],[63])).
% 0.22/0.41 tff(65,axiom,(![R: $i, E: $i, M: $i] : (least(M, R, E) <=> (member(M, E) & ![X: $i] : (member(X, E) => apply(R, M, X))))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+3.ax','least')).
% 0.22/0.41 tff(66,plain,
% 0.22/0.41 (![R: $i, E: $i, M: $i] : (least(M, R, E) <=> (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X))))),
% 0.22/0.41 inference(modus_ponens,[status(thm)],[65, 64])).
% 0.22/0.41 tff(67,plain,
% 0.22/0.41 (![R: $i, E: $i, M: $i] : (least(M, R, E) <=> (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X))))),
% 0.22/0.41 inference(modus_ponens,[status(thm)],[66, 62])).
% 0.22/0.41 tff(68,plain,(
% 0.22/0.41 ![R: $i, E: $i, M: $i] : (((~least(M, R, E)) | (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) & (least(M, R, E) | ((~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R)))))))),
% 0.22/0.41 inference(skolemize,[status(sab)],[67])).
% 0.22/0.41 tff(69,plain,
% 0.22/0.41 (![R: $i, E: $i, M: $i] : (((~least(M, R, E)) | (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) & (least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R))))))),
% 0.22/0.41 inference(modus_ponens,[status(thm)],[68, 61])).
% 0.22/0.41 tff(70,plain,
% 0.22/0.41 (![R: $i, E: $i, M: $i] : (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R))))))))),
% 0.22/0.41 inference(modus_ponens,[status(thm)],[69, 59])).
% 0.22/0.41 tff(71,plain,
% 0.22/0.41 (![R: $i, E: $i, M: $i] : (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R))))))))),
% 0.22/0.41 inference(modus_ponens,[status(thm)],[70, 57])).
% 0.22/0.41 tff(72,plain,
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% 0.22/0.41 inference(modus_ponens,[status(thm)],[71, 52])).
% 0.22/0.41 tff(73,plain,
% 0.22/0.41 ((~![R: $i, E: $i, M: $i] : (~((~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R)))))) | (~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X)))))))))) | (~((~(least(M!18, R!15, E!14) | (~member(M!18, E!14)) | (~((~member(tptp_fun_X_9(M!18, E!14, R!15), E!14)) | apply(R!15, M!18, tptp_fun_X_9(M!18, E!14, R!15)))))) | (~((~least(M!18, R!15, E!14)) | (~((~member(M!18, E!14)) | (~![X: $i] : ((~member(X, E!14)) | apply(R!15, M!18, X)))))))))),
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% 0.22/0.42 ((~least(M!18, R!15, E!14)) | (~((~member(M!18, E!14)) | (~![X: $i] : ((~member(X, E!14)) | apply(R!15, M!18, X)))))),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[75, 74])).
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% 0.22/0.42 inference(tautology,[status(thm)],[])).
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% 0.22/0.42 inference(unit_resolution,[status(thm)],[81, 80])).
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% 0.22/0.42 inference(quant_inst,[status(thm)],[])).
% 0.22/0.42 tff(85,plain,
% 0.22/0.42 ((~![X: $i] : ((~member(X, E!14)) | apply(R!15, M!18, X))) | (~member(M1!17, E!14)) | apply(R!15, M!18, M1!17)),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[84, 83])).
% 0.22/0.42 tff(86,plain,
% 0.22/0.42 (apply(R!15, M!18, M1!17)),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[85, 82, 50])).
% 0.22/0.42 tff(87,plain,
% 0.22/0.42 (((~member(M1!17, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17))))) | ![X: $i] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17)))),
% 0.22/0.42 inference(tautology,[status(thm)],[])).
% 0.22/0.42 tff(88,plain,
% 0.22/0.42 (![X: $i] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17)))),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[87, 48])).
% 0.22/0.42 tff(89,plain,
% 0.22/0.42 (((~member(M!18, E!14)) | (~![X: $i] : ((~member(X, E!14)) | apply(R!15, M!18, X)))) | member(M!18, E!14)),
% 0.22/0.42 inference(tautology,[status(thm)],[])).
% 0.22/0.42 tff(90,plain,
% 0.22/0.42 (member(M!18, E!14)),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[89, 80])).
% 0.22/0.42 tff(91,plain,
% 0.22/0.42 (((~![X: $i] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17)))) | ((~member(M!18, E!14)) | (M1!17 = M!18) | (~apply(R!15, M!18, M1!17)))) <=> ((~![X: $i] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17)))) | (~member(M!18, E!14)) | (M1!17 = M!18) | (~apply(R!15, M!18, M1!17)))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(92,plain,
% 0.22/0.42 ((~![X: $i] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17)))) | ((~member(M!18, E!14)) | (M1!17 = M!18) | (~apply(R!15, M!18, M1!17)))),
% 0.22/0.42 inference(quant_inst,[status(thm)],[])).
% 0.22/0.42 tff(93,plain,
% 0.22/0.42 ((~![X: $i] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17)))) | (~member(M!18, E!14)) | (M1!17 = M!18) | (~apply(R!15, M!18, M1!17))),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[92, 91])).
% 0.22/0.42 tff(94,plain,
% 0.22/0.42 ((M1!17 = M!18) | (~apply(R!15, M!18, M1!17))),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[93, 90, 88])).
% 0.22/0.42 tff(95,plain,
% 0.22/0.42 (M1!17 = M!18),
% 0.22/0.42 inference(unit_resolution,[status(thm)],[94, 86])).
% 0.22/0.42 tff(96,plain,
% 0.22/0.42 (M!18 = M1!17),
% 0.22/0.42 inference(symmetry,[status(thm)],[95])).
% 0.22/0.42 tff(97,plain,
% 0.22/0.42 ((M2!16 = M!18) <=> (M2!16 = M1!17)),
% 0.22/0.42 inference(monotonicity,[status(thm)],[96])).
% 0.22/0.42 tff(98,plain,
% 0.22/0.42 ((M2!16 = M!18) <=> (M1!17 = M2!16)),
% 0.22/0.42 inference(transitivity,[status(thm)],[97, 1])).
% 0.22/0.42 tff(99,plain,
% 0.22/0.42 ((M1!17 = M2!16) <=> (M2!16 = M!18)),
% 0.22/0.42 inference(symmetry,[status(thm)],[98])).
% 0.22/0.42 tff(100,plain,
% 0.22/0.42 ((~(M1!17 = M2!16)) <=> (~(M2!16 = M!18))),
% 0.22/0.42 inference(monotonicity,[status(thm)],[99])).
% 0.22/0.42 tff(101,plain,
% 0.22/0.42 (~(M1!17 = M2!16)),
% 0.22/0.42 inference(and_elim,[status(thm)],[44])).
% 0.22/0.42 tff(102,plain,
% 0.22/0.42 (~(M2!16 = M!18)),
% 0.22/0.42 inference(modus_ponens,[status(thm)],[101, 100])).
% 0.22/0.42 tff(103,plain,
% 0.22/0.42 (((~![R: $i, E: $i, M: $i] : (~((~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))) | (~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M))))))))))) | (~((~(min(M2!16, R!15, E!14) | (~member(M2!16, E!14)) | (~((~apply(R!15, tptp_fun_X_11(M2!16, E!14, R!15), M2!16)) | (~member(tptp_fun_X_11(M2!16, E!14, R!15), E!14)) | (M2!16 = tptp_fun_X_11(M2!16, E!14, R!15)))))) | (~((~min(M2!16, R!15, E!14)) | (~((~member(M2!16, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16))))))))))) <=> ((~![R: $i, E: $i, M: $i] : (~((~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))) | (~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M))))))))))) | (~((~(min(M2!16, R!15, E!14) | (~member(M2!16, E!14)) | (~((~apply(R!15, tptp_fun_X_11(M2!16, E!14, R!15), M2!16)) | (~member(tptp_fun_X_11(M2!16, E!14, R!15), E!14)) | (M2!16 = tptp_fun_X_11(M2!16, E!14, R!15)))))) | (~((~min(M2!16, R!15, E!14)) | (~((~member(M2!16, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16)))))))))))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(104,plain,
% 0.22/0.42 ((~((~(min(M2!16, R!15, E!14) | (~member(M2!16, E!14)) | (~((M2!16 = tptp_fun_X_11(M2!16, E!14, R!15)) | (~member(tptp_fun_X_11(M2!16, E!14, R!15), E!14)) | (~apply(R!15, tptp_fun_X_11(M2!16, E!14, R!15), M2!16)))))) | (~((~min(M2!16, R!15, E!14)) | (~((~member(M2!16, E!14)) | (~![X: $i] : ((M2!16 = X) | (~member(X, E!14)) | (~apply(R!15, X, M2!16)))))))))) <=> (~((~(min(M2!16, R!15, E!14) | (~member(M2!16, E!14)) | (~((~apply(R!15, tptp_fun_X_11(M2!16, E!14, R!15), M2!16)) | (~member(tptp_fun_X_11(M2!16, E!14, R!15), E!14)) | (M2!16 = tptp_fun_X_11(M2!16, E!14, R!15)))))) | (~((~min(M2!16, R!15, E!14)) | (~((~member(M2!16, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16))))))))))),
% 0.22/0.42 inference(rewrite,[status(thm)],[])).
% 0.22/0.42 tff(105,plain,
% 0.22/0.42 (((~![R: $i, E: $i, M: $i] : (~((~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))) | (~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M))))))))))) | (~((~(min(M2!16, R!15, E!14) | (~member(M2!16, E!14)) | (~((M2!16 = tptp_fun_X_11(M2!16, E!14, R!15)) | (~member(tptp_fun_X_11(M2!16, E!14, R!15), E!14)) | (~apply(R!15, tptp_fun_X_11(M2!16, E!14, R!15), M2!16)))))) | (~((~min(M2!16, R!15, E!14)) | (~((~member(M2!16, E!14)) | (~![X: $i] : ((M2!16 = X) | (~member(X, E!14)) | (~apply(R!15, X, M2!16))))))))))) <=> ((~![R: $i, E: $i, M: $i] : (~((~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))) | (~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M))))))))))) | (~((~(min(M2!16, R!15, E!14) | (~member(M2!16, E!14)) | (~((~apply(R!15, tptp_fun_X_11(M2!16, E!14, R!15), M2!16)) | (~member(tptp_fun_X_11(M2!16, E!14, R!15), E!14)) | (M2!16 = tptp_fun_X_11(M2!16, E!14, R!15)))))) | (~((~min(M2!16, R!15, E!14)) | (~((~member(M2!16, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16)))))))))))),
% 0.22/0.43 inference(monotonicity,[status(thm)],[104])).
% 0.22/0.43 tff(106,plain,
% 0.22/0.43 (((~![R: $i, E: $i, M: $i] : (~((~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))) | (~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M))))))))))) | (~((~(min(M2!16, R!15, E!14) | (~member(M2!16, E!14)) | (~((M2!16 = tptp_fun_X_11(M2!16, E!14, R!15)) | (~member(tptp_fun_X_11(M2!16, E!14, R!15), E!14)) | (~apply(R!15, tptp_fun_X_11(M2!16, E!14, R!15), M2!16)))))) | (~((~min(M2!16, R!15, E!14)) | (~((~member(M2!16, E!14)) | (~![X: $i] : ((M2!16 = X) | (~member(X, E!14)) | (~apply(R!15, X, M2!16))))))))))) <=> ((~![R: $i, E: $i, M: $i] : (~((~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))) | (~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M))))))))))) | (~((~(min(M2!16, R!15, E!14) | (~member(M2!16, E!14)) | (~((~apply(R!15, tptp_fun_X_11(M2!16, E!14, R!15), M2!16)) | (~member(tptp_fun_X_11(M2!16, E!14, R!15), E!14)) | (M2!16 = tptp_fun_X_11(M2!16, E!14, R!15)))))) | (~((~min(M2!16, R!15, E!14)) | (~((~member(M2!16, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16)))))))))))),
% 0.22/0.43 inference(transitivity,[status(thm)],[105, 103])).
% 0.22/0.43 tff(107,plain,
% 0.22/0.43 ((~![R: $i, E: $i, M: $i] : (~((~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))) | (~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M))))))))))) | (~((~(min(M2!16, R!15, E!14) | (~member(M2!16, E!14)) | (~((M2!16 = tptp_fun_X_11(M2!16, E!14, R!15)) | (~member(tptp_fun_X_11(M2!16, E!14, R!15), E!14)) | (~apply(R!15, tptp_fun_X_11(M2!16, E!14, R!15), M2!16)))))) | (~((~min(M2!16, R!15, E!14)) | (~((~member(M2!16, E!14)) | (~![X: $i] : ((M2!16 = X) | (~member(X, E!14)) | (~apply(R!15, X, M2!16))))))))))),
% 0.22/0.43 inference(quant_inst,[status(thm)],[])).
% 0.22/0.43 tff(108,plain,
% 0.22/0.43 ((~![R: $i, E: $i, M: $i] : (~((~(min(M, R, E) | (~member(M, E)) | (~((M = tptp_fun_X_11(M, E, R)) | (~member(tptp_fun_X_11(M, E, R), E)) | (~apply(R, tptp_fun_X_11(M, E, R), M)))))) | (~((~min(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((M = X) | (~member(X, E)) | (~apply(R, X, M))))))))))) | (~((~(min(M2!16, R!15, E!14) | (~member(M2!16, E!14)) | (~((~apply(R!15, tptp_fun_X_11(M2!16, E!14, R!15), M2!16)) | (~member(tptp_fun_X_11(M2!16, E!14, R!15), E!14)) | (M2!16 = tptp_fun_X_11(M2!16, E!14, R!15)))))) | (~((~min(M2!16, R!15, E!14)) | (~((~member(M2!16, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16))))))))))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[107, 106])).
% 0.22/0.43 tff(109,plain,
% 0.22/0.43 (~((~(min(M2!16, R!15, E!14) | (~member(M2!16, E!14)) | (~((~apply(R!15, tptp_fun_X_11(M2!16, E!14, R!15), M2!16)) | (~member(tptp_fun_X_11(M2!16, E!14, R!15), E!14)) | (M2!16 = tptp_fun_X_11(M2!16, E!14, R!15)))))) | (~((~min(M2!16, R!15, E!14)) | (~((~member(M2!16, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16)))))))))),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[108, 23])).
% 0.22/0.43 tff(110,plain,
% 0.22/0.43 (((~(min(M2!16, R!15, E!14) | (~member(M2!16, E!14)) | (~((~apply(R!15, tptp_fun_X_11(M2!16, E!14, R!15), M2!16)) | (~member(tptp_fun_X_11(M2!16, E!14, R!15), E!14)) | (M2!16 = tptp_fun_X_11(M2!16, E!14, R!15)))))) | (~((~min(M2!16, R!15, E!14)) | (~((~member(M2!16, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16))))))))) | ((~min(M2!16, R!15, E!14)) | (~((~member(M2!16, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16)))))))),
% 0.22/0.43 inference(tautology,[status(thm)],[])).
% 0.22/0.43 tff(111,plain,
% 0.22/0.43 ((~min(M2!16, R!15, E!14)) | (~((~member(M2!16, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16))))))),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[110, 109])).
% 0.22/0.43 tff(112,plain,
% 0.22/0.43 (min(M2!16, R!15, E!14)),
% 0.22/0.43 inference(and_elim,[status(thm)],[44])).
% 0.22/0.43 tff(113,plain,
% 0.22/0.43 ((~((~min(M2!16, R!15, E!14)) | (~((~member(M2!16, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16)))))))) | (~min(M2!16, R!15, E!14)) | (~((~member(M2!16, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16))))))),
% 0.22/0.43 inference(tautology,[status(thm)],[])).
% 0.22/0.43 tff(114,plain,
% 0.22/0.43 ((~((~min(M2!16, R!15, E!14)) | (~((~member(M2!16, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16)))))))) | (~((~member(M2!16, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16))))))),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[113, 112])).
% 0.22/0.43 tff(115,plain,
% 0.22/0.43 (~((~member(M2!16, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16)))))),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[114, 111])).
% 0.22/0.43 tff(116,plain,
% 0.22/0.43 (((~member(M2!16, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16))))) | member(M2!16, E!14)),
% 0.22/0.43 inference(tautology,[status(thm)],[])).
% 0.22/0.43 tff(117,plain,
% 0.22/0.43 (member(M2!16, E!14)),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[116, 115])).
% 0.22/0.43 tff(118,plain,
% 0.22/0.43 (((~![X: $i] : ((~member(X, E!14)) | apply(R!15, M!18, X))) | ((~member(M2!16, E!14)) | apply(R!15, M!18, M2!16))) <=> ((~![X: $i] : ((~member(X, E!14)) | apply(R!15, M!18, X))) | (~member(M2!16, E!14)) | apply(R!15, M!18, M2!16))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(119,plain,
% 0.22/0.43 ((~![X: $i] : ((~member(X, E!14)) | apply(R!15, M!18, X))) | ((~member(M2!16, E!14)) | apply(R!15, M!18, M2!16))),
% 0.22/0.43 inference(quant_inst,[status(thm)],[])).
% 0.22/0.43 tff(120,plain,
% 0.22/0.43 ((~![X: $i] : ((~member(X, E!14)) | apply(R!15, M!18, X))) | (~member(M2!16, E!14)) | apply(R!15, M!18, M2!16)),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[119, 118])).
% 0.22/0.43 tff(121,plain,
% 0.22/0.43 (apply(R!15, M!18, M2!16)),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[120, 82, 117])).
% 0.22/0.43 tff(122,plain,
% 0.22/0.43 (((~member(M2!16, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16))))) | ![X: $i] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16)))),
% 0.22/0.43 inference(tautology,[status(thm)],[])).
% 0.22/0.43 tff(123,plain,
% 0.22/0.43 (![X: $i] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16)))),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[122, 115])).
% 0.22/0.43 tff(124,plain,
% 0.22/0.43 (((~![X: $i] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16)))) | ((~member(M!18, E!14)) | (M2!16 = M!18) | (~apply(R!15, M!18, M2!16)))) <=> ((~![X: $i] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16)))) | (~member(M!18, E!14)) | (M2!16 = M!18) | (~apply(R!15, M!18, M2!16)))),
% 0.22/0.43 inference(rewrite,[status(thm)],[])).
% 0.22/0.43 tff(125,plain,
% 0.22/0.43 ((~![X: $i] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16)))) | ((~member(M!18, E!14)) | (M2!16 = M!18) | (~apply(R!15, M!18, M2!16)))),
% 0.22/0.43 inference(quant_inst,[status(thm)],[])).
% 0.22/0.43 tff(126,plain,
% 0.22/0.43 ((~![X: $i] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16)))) | (~member(M!18, E!14)) | (M2!16 = M!18) | (~apply(R!15, M!18, M2!16))),
% 0.22/0.43 inference(modus_ponens,[status(thm)],[125, 124])).
% 0.22/0.43 tff(127,plain,
% 0.22/0.43 ((M2!16 = M!18) | (~apply(R!15, M!18, M2!16))),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[126, 90, 123])).
% 0.22/0.43 tff(128,plain,
% 0.22/0.43 (M2!16 = M!18),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[127, 121])).
% 0.22/0.43 tff(129,plain,
% 0.22/0.43 ($false),
% 0.22/0.43 inference(unit_resolution,[status(thm)],[128, 102])).
% 0.22/0.43 % SZS output end Proof
% 0.22/0.43 % E exiting
%------------------------------------------------------------------------------