TSTP Solution File: SET804+4 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SET804+4 : TPTP v8.1.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n025.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 20 05:08:05 EDT 2022
% Result : Theorem 0.20s 0.40s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : SET804+4 : TPTP v8.1.0. Released v3.2.0.
% 0.07/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.34 % Computer : n025.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Sat Sep 3 08:11:33 EDT 2022
% 0.13/0.35 % CPUTime :
% 0.13/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.13/0.35 Usage: tptp [options] [-file:]file
% 0.13/0.35 -h, -? prints this message.
% 0.13/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.13/0.35 -m, -model generate model.
% 0.13/0.35 -p, -proof generate proof.
% 0.13/0.35 -c, -core generate unsat core of named formulas.
% 0.13/0.35 -st, -statistics display statistics.
% 0.13/0.35 -t:timeout set timeout (in second).
% 0.13/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.13/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.13/0.35 -<param>:<value> configuration parameter and value.
% 0.13/0.35 -o:<output-file> file to place output in.
% 0.20/0.40 % SZS status Theorem
% 0.20/0.40 % SZS output start Proof
% 0.20/0.40 tff(tptp_fun_M_18_type, type, (
% 0.20/0.40 tptp_fun_M_18: $i)).
% 0.20/0.40 tff(tptp_fun_M2_16_type, type, (
% 0.20/0.40 tptp_fun_M2_16: $i)).
% 0.20/0.40 tff(tptp_fun_M1_17_type, type, (
% 0.20/0.40 tptp_fun_M1_17: $i)).
% 0.20/0.40 tff(apply_type, type, (
% 0.20/0.40 apply: ( $i * $i * $i ) > $o)).
% 0.20/0.40 tff(tptp_fun_R_15_type, type, (
% 0.20/0.40 tptp_fun_R_15: $i)).
% 0.20/0.40 tff(member_type, type, (
% 0.20/0.40 member: ( $i * $i ) > $o)).
% 0.20/0.40 tff(tptp_fun_E_14_type, type, (
% 0.20/0.40 tptp_fun_E_14: $i)).
% 0.20/0.40 tff(min_type, type, (
% 0.20/0.40 min: ( $i * $i * $i ) > $o)).
% 0.20/0.40 tff(tptp_fun_X_11_type, type, (
% 0.20/0.40 tptp_fun_X_11: ( $i * $i * $i ) > $i)).
% 0.20/0.40 tff(least_type, type, (
% 0.20/0.40 least: ( $i * $i * $i ) > $o)).
% 0.20/0.40 tff(order_type, type, (
% 0.20/0.40 order: ( $i * $i ) > $o)).
% 0.20/0.40 tff(tptp_fun_X_9_type, type, (
% 0.20/0.40 tptp_fun_X_9: ( $i * $i * $i ) > $i)).
% 0.20/0.40 tff(1,plain,
% 0.20/0.40 ((M2!16 = M1!17) <=> (M1!17 = M2!16)),
% 0.20/0.40 inference(commutativity,[status(thm)],[])).
% 0.20/0.40 tff(2,plain,
% 0.20/0.40 (^[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)))))))))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(3,plain,
% 0.20/0.41 (![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))))))))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[2])).
% 0.20/0.41 tff(4,plain,
% 0.20/0.41 (^[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)))))))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(5,plain,
% 0.20/0.41 (![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))))))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[4])).
% 0.20/0.41 tff(6,plain,
% 0.20/0.41 (^[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)))))))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(7,plain,
% 0.20/0.41 (![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))))))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[6])).
% 0.20/0.41 tff(8,plain,
% 0.20/0.41 (![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))))))))),
% 0.20/0.41 inference(transitivity,[status(thm)],[7, 5])).
% 0.20/0.41 tff(9,plain,
% 0.20/0.41 (^[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))))))))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(10,plain,
% 0.20/0.41 (![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))))))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[9])).
% 0.20/0.41 tff(11,plain,
% 0.20/0.41 (^[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)))))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(12,plain,
% 0.20/0.41 (![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))))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[11])).
% 0.20/0.41 tff(13,plain,
% 0.20/0.41 (![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))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(14,plain,
% 0.20/0.41 (^[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)))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(15,plain,
% 0.20/0.41 (![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))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[14])).
% 0.20/0.41 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/sandbox/benchmark/Axioms/SET006+3.ax','min')).
% 0.20/0.41 tff(17,plain,
% 0.20/0.41 (![R: $i, E: $i, M: $i] : (min(M, R, E) <=> (member(M, E) & ![X: $i] : ((~(member(X, E) & apply(R, X, M))) | (M = X))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[16, 15])).
% 0.20/0.41 tff(18,plain,
% 0.20/0.41 (![R: $i, E: $i, M: $i] : (min(M, R, E) <=> (member(M, E) & ![X: $i] : ((~(member(X, E) & apply(R, X, M))) | (M = X))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[17, 13])).
% 0.20/0.41 tff(19,plain,(
% 0.20/0.41 ![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)))))))),
% 0.20/0.41 inference(skolemize,[status(sab)],[18])).
% 0.20/0.41 tff(20,plain,
% 0.20/0.41 (![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))))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[19, 12])).
% 0.20/0.41 tff(21,plain,
% 0.20/0.41 (![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))))))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[20, 10])).
% 0.20/0.41 tff(22,plain,
% 0.20/0.41 (![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))))))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[21, 8])).
% 0.20/0.41 tff(23,plain,
% 0.20/0.41 (![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))))))))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[22, 3])).
% 0.20/0.41 tff(24,plain,
% 0.20/0.41 (((~![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] : ((~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)) | (~((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] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17)))))))))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(25,plain,
% 0.20/0.41 ((~((~(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)) | (~((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] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17))))))))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(26,plain,
% 0.20/0.41 (((~![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)) | (~((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] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17)))))))))))),
% 0.20/0.42 inference(monotonicity,[status(thm)],[25])).
% 0.20/0.42 tff(27,plain,
% 0.20/0.42 (((~![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(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)) | (~((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] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17)))))))))))),
% 0.20/0.42 inference(transitivity,[status(thm)],[26, 24])).
% 0.20/0.42 tff(28,plain,
% 0.20/0.42 ((~![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(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))))))))))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(29,plain,
% 0.20/0.42 ((~![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(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] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17))))))))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[28, 27])).
% 0.20/0.42 tff(30,plain,
% 0.20/0.42 (~((~(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] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17)))))))))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[29, 23])).
% 0.20/0.42 tff(31,plain,
% 0.20/0.42 (((~(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] : ((~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)))))))),
% 0.20/0.42 inference(tautology,[status(thm)],[])).
% 0.20/0.42 tff(32,plain,
% 0.20/0.42 ((~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))))))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[31, 30])).
% 0.20/0.42 tff(33,plain,
% 0.20/0.42 ((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)))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(34,plain,
% 0.20/0.42 ((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)))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(35,plain,
% 0.20/0.42 ((~(~(min(M1!17, R!15, E!14) & min(M2!16, R!15, E!14) & (~(M1!17 = M2!16))))) <=> (min(M1!17, R!15, E!14) & min(M2!16, R!15, E!14) & (~(M1!17 = M2!16)))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(36,plain,
% 0.20/0.42 ((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))))),
% 0.20/0.42 inference(monotonicity,[status(thm)],[35])).
% 0.20/0.42 tff(37,plain,
% 0.20/0.42 ((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)))),
% 0.20/0.42 inference(transitivity,[status(thm)],[36, 34])).
% 0.20/0.42 tff(38,plain,
% 0.20/0.42 ((~(~order(R!15, E!14))) <=> order(R!15, E!14)),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(39,plain,
% 0.20/0.42 (((~(~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))))),
% 0.20/0.42 inference(monotonicity,[status(thm)],[38, 37])).
% 0.20/0.42 tff(40,plain,
% 0.20/0.42 (((~(~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)))),
% 0.20/0.42 inference(transitivity,[status(thm)],[39, 33])).
% 0.20/0.42 tff(41,plain,
% 0.20/0.42 ((~![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)))))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(42,plain,
% 0.20/0.42 ((~![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)))))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(43,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/sandbox/benchmark/theBenchmark.p','thIV16')).
% 0.20/0.42 tff(44,plain,
% 0.20/0.42 (~![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))))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[43, 42])).
% 0.20/0.42 tff(45,plain,
% 0.20/0.42 (~![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))))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[44, 41])).
% 0.20/0.42 tff(46,plain,
% 0.20/0.42 (~![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))))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[45, 41])).
% 0.20/0.42 tff(47,plain,
% 0.20/0.42 (~![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))))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[46, 41])).
% 0.20/0.42 tff(48,plain,
% 0.20/0.42 (~![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))))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[47, 41])).
% 0.20/0.42 tff(49,plain,
% 0.20/0.42 (~![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))))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[48, 41])).
% 0.20/0.42 tff(50,plain,
% 0.20/0.42 (~![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))))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[49, 41])).
% 0.20/0.42 tff(51,plain,
% 0.20/0.42 (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))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[50, 40])).
% 0.20/0.42 tff(52,plain,
% 0.20/0.42 (min(M1!17, R!15, E!14)),
% 0.20/0.42 inference(and_elim,[status(thm)],[51])).
% 0.20/0.42 tff(53,plain,
% 0.20/0.42 ((~((~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))))))),
% 0.20/0.42 inference(tautology,[status(thm)],[])).
% 0.20/0.42 tff(54,plain,
% 0.20/0.42 ((~((~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))))))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[53, 52])).
% 0.20/0.42 tff(55,plain,
% 0.20/0.42 (~((~member(M1!17, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17)))))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[54, 32])).
% 0.20/0.42 tff(56,plain,
% 0.20/0.42 (((~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)),
% 0.20/0.42 inference(tautology,[status(thm)],[])).
% 0.20/0.42 tff(57,plain,
% 0.20/0.42 (member(M1!17, E!14)),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[56, 55])).
% 0.20/0.42 tff(58,plain,
% 0.20/0.42 (^[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))))))))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(59,plain,
% 0.20/0.42 (![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)))))))))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[58])).
% 0.20/0.42 tff(60,plain,
% 0.20/0.42 (^[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)))))))))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(61,plain,
% 0.20/0.43 (![R: $i, E: $i, M: $i] : (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_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))))))))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[60])).
% 0.20/0.43 tff(62,plain,
% 0.20/0.43 (^[R: $i, E: $i, M: $i] : rewrite((~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_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)))))))))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(63,plain,
% 0.20/0.43 (![R: $i, E: $i, M: $i] : (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_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))))))))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[62])).
% 0.20/0.43 tff(64,plain,
% 0.20/0.43 (![R: $i, E: $i, M: $i] : (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_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))))))))),
% 0.20/0.43 inference(transitivity,[status(thm)],[63, 61])).
% 0.20/0.43 tff(65,plain,
% 0.20/0.43 (^[R: $i, E: $i, M: $i] : trans(monotonicity(rewrite(((~least(M, R, E)) | (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) <=> ((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))), rewrite((least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_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))))))))))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(66,plain,
% 0.20/0.43 (![R: $i, E: $i, M: $i] : (((~least(M, R, E)) | (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) & (least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_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))))))))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[65])).
% 0.20/0.43 tff(67,plain,
% 0.20/0.43 (^[R: $i, E: $i, M: $i] : rewrite((((~least(M, R, E)) | (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) & (least(M, R, E) | ((~member(M, E)) | (~((~member(tptp_fun_X_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)))))))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(68,plain,
% 0.20/0.43 (![R: $i, E: $i, M: $i] : (((~least(M, R, E)) | (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) & (least(M, R, E) | ((~member(M, E)) | (~((~member(tptp_fun_X_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))))))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[67])).
% 0.20/0.43 tff(69,plain,
% 0.20/0.43 (![R: $i, E: $i, M: $i] : (least(M, R, E) <=> (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) <=> ![R: $i, E: $i, M: $i] : (least(M, R, E) <=> (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(70,plain,
% 0.20/0.43 (^[R: $i, E: $i, M: $i] : rewrite((least(M, R, E) <=> (member(M, E) & ![X: $i] : (member(X, E) => apply(R, M, X)))) <=> (least(M, R, E) <=> (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(71,plain,
% 0.20/0.43 (![R: $i, E: $i, M: $i] : (least(M, R, E) <=> (member(M, E) & ![X: $i] : (member(X, E) => apply(R, M, X)))) <=> ![R: $i, E: $i, M: $i] : (least(M, R, E) <=> (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X))))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[70])).
% 0.20/0.43 tff(72,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/sandbox/benchmark/Axioms/SET006+3.ax','least')).
% 0.20/0.43 tff(73,plain,
% 0.20/0.43 (![R: $i, E: $i, M: $i] : (least(M, R, E) <=> (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[72, 71])).
% 0.20/0.43 tff(74,plain,
% 0.20/0.43 (![R: $i, E: $i, M: $i] : (least(M, R, E) <=> (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[73, 69])).
% 0.20/0.43 tff(75,plain,(
% 0.20/0.43 ![R: $i, E: $i, M: $i] : (((~least(M, R, E)) | (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) & (least(M, R, E) | ((~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R)))))))),
% 0.20/0.43 inference(skolemize,[status(sab)],[74])).
% 0.20/0.43 tff(76,plain,
% 0.20/0.43 (![R: $i, E: $i, M: $i] : (((~least(M, R, E)) | (member(M, E) & ![X: $i] : ((~member(X, E)) | apply(R, M, X)))) & (least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R))))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[75, 68])).
% 0.20/0.43 tff(77,plain,
% 0.20/0.43 (![R: $i, E: $i, M: $i] : (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R))))))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[76, 66])).
% 0.20/0.43 tff(78,plain,
% 0.20/0.43 (![R: $i, E: $i, M: $i] : (~((~((~least(M, R, E)) | (~((~member(M, E)) | (~![X: $i] : ((~member(X, E)) | apply(R, M, X))))))) | (~(least(M, R, E) | (~member(M, E)) | (~((~member(tptp_fun_X_9(M, E, R), E)) | apply(R, M, tptp_fun_X_9(M, E, R))))))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[77, 64])).
% 0.20/0.43 tff(79,plain,
% 0.20/0.43 (![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)))))))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[78, 59])).
% 0.20/0.43 tff(80,plain,
% 0.20/0.43 ((~![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)))))))))),
% 0.20/0.43 inference(quant_inst,[status(thm)],[])).
% 0.20/0.43 tff(81,plain,
% 0.20/0.43 (~((~(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))))))))),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[80, 79])).
% 0.20/0.43 tff(82,plain,
% 0.20/0.43 (((~(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)))))))) | ((~least(M!18, R!15, E!14)) | (~((~member(M!18, E!14)) | (~![X: $i] : ((~member(X, E!14)) | apply(R!15, M!18, X))))))),
% 0.20/0.43 inference(tautology,[status(thm)],[])).
% 0.20/0.43 tff(83,plain,
% 0.20/0.43 ((~least(M!18, R!15, E!14)) | (~((~member(M!18, E!14)) | (~![X: $i] : ((~member(X, E!14)) | apply(R!15, M!18, X)))))),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[82, 81])).
% 0.20/0.43 tff(84,plain,
% 0.20/0.43 (least(M!18, R!15, E!14)),
% 0.20/0.43 inference(and_elim,[status(thm)],[51])).
% 0.20/0.43 tff(85,plain,
% 0.20/0.43 ((~((~least(M!18, R!15, E!14)) | (~((~member(M!18, E!14)) | (~![X: $i] : ((~member(X, E!14)) | apply(R!15, M!18, X))))))) | (~least(M!18, R!15, E!14)) | (~((~member(M!18, E!14)) | (~![X: $i] : ((~member(X, E!14)) | apply(R!15, M!18, X)))))),
% 0.20/0.43 inference(tautology,[status(thm)],[])).
% 0.20/0.43 tff(86,plain,
% 0.20/0.43 ((~((~least(M!18, R!15, E!14)) | (~((~member(M!18, E!14)) | (~![X: $i] : ((~member(X, E!14)) | apply(R!15, M!18, X))))))) | (~((~member(M!18, E!14)) | (~![X: $i] : ((~member(X, E!14)) | apply(R!15, M!18, X)))))),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[85, 84])).
% 0.20/0.43 tff(87,plain,
% 0.20/0.43 (~((~member(M!18, E!14)) | (~![X: $i] : ((~member(X, E!14)) | apply(R!15, M!18, X))))),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[86, 83])).
% 0.20/0.43 tff(88,plain,
% 0.20/0.43 (((~member(M!18, E!14)) | (~![X: $i] : ((~member(X, E!14)) | apply(R!15, M!18, X)))) | ![X: $i] : ((~member(X, E!14)) | apply(R!15, M!18, X))),
% 0.20/0.43 inference(tautology,[status(thm)],[])).
% 0.20/0.43 tff(89,plain,
% 0.20/0.43 (![X: $i] : ((~member(X, E!14)) | apply(R!15, M!18, X))),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[88, 87])).
% 0.20/0.43 tff(90,plain,
% 0.20/0.43 (((~![X: $i] : ((~member(X, E!14)) | apply(R!15, M!18, X))) | ((~member(M1!17, E!14)) | apply(R!15, M!18, M1!17))) <=> ((~![X: $i] : ((~member(X, E!14)) | apply(R!15, M!18, X))) | (~member(M1!17, E!14)) | apply(R!15, M!18, M1!17))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(91,plain,
% 0.20/0.43 ((~![X: $i] : ((~member(X, E!14)) | apply(R!15, M!18, X))) | ((~member(M1!17, E!14)) | apply(R!15, M!18, M1!17))),
% 0.20/0.43 inference(quant_inst,[status(thm)],[])).
% 0.20/0.43 tff(92,plain,
% 0.20/0.43 ((~![X: $i] : ((~member(X, E!14)) | apply(R!15, M!18, X))) | (~member(M1!17, E!14)) | apply(R!15, M!18, M1!17)),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[91, 90])).
% 0.20/0.43 tff(93,plain,
% 0.20/0.43 (apply(R!15, M!18, M1!17)),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[92, 89, 57])).
% 0.20/0.44 tff(94,plain,
% 0.20/0.44 (((~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)))),
% 0.20/0.44 inference(tautology,[status(thm)],[])).
% 0.20/0.44 tff(95,plain,
% 0.20/0.44 (![X: $i] : ((~member(X, E!14)) | (M1!17 = X) | (~apply(R!15, X, M1!17)))),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[94, 55])).
% 0.20/0.44 tff(96,plain,
% 0.20/0.44 (((~member(M!18, E!14)) | (~![X: $i] : ((~member(X, E!14)) | apply(R!15, M!18, X)))) | member(M!18, E!14)),
% 0.20/0.44 inference(tautology,[status(thm)],[])).
% 0.20/0.44 tff(97,plain,
% 0.20/0.44 (member(M!18, E!14)),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[96, 87])).
% 0.20/0.44 tff(98,plain,
% 0.20/0.44 (((~![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)))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(99,plain,
% 0.20/0.44 ((~![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)))),
% 0.20/0.44 inference(quant_inst,[status(thm)],[])).
% 0.20/0.44 tff(100,plain,
% 0.20/0.44 ((~![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))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[99, 98])).
% 0.20/0.44 tff(101,plain,
% 0.20/0.44 ((M1!17 = M!18) | (~apply(R!15, M!18, M1!17))),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[100, 97, 95])).
% 0.20/0.44 tff(102,plain,
% 0.20/0.44 (M1!17 = M!18),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[101, 93])).
% 0.20/0.44 tff(103,plain,
% 0.20/0.44 (M!18 = M1!17),
% 0.20/0.44 inference(symmetry,[status(thm)],[102])).
% 0.20/0.44 tff(104,plain,
% 0.20/0.44 ((M2!16 = M!18) <=> (M2!16 = M1!17)),
% 0.20/0.44 inference(monotonicity,[status(thm)],[103])).
% 0.20/0.44 tff(105,plain,
% 0.20/0.44 ((M2!16 = M!18) <=> (M1!17 = M2!16)),
% 0.20/0.44 inference(transitivity,[status(thm)],[104, 1])).
% 0.20/0.44 tff(106,plain,
% 0.20/0.44 ((M1!17 = M2!16) <=> (M2!16 = M!18)),
% 0.20/0.44 inference(symmetry,[status(thm)],[105])).
% 0.20/0.44 tff(107,plain,
% 0.20/0.44 ((~(M1!17 = M2!16)) <=> (~(M2!16 = M!18))),
% 0.20/0.44 inference(monotonicity,[status(thm)],[106])).
% 0.20/0.44 tff(108,plain,
% 0.20/0.44 (~(M1!17 = M2!16)),
% 0.20/0.44 inference(and_elim,[status(thm)],[51])).
% 0.20/0.44 tff(109,plain,
% 0.20/0.44 (~(M2!16 = M!18)),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[108, 107])).
% 0.20/0.44 tff(110,plain,
% 0.20/0.44 (((~![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] : ((~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)) | (~((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] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16)))))))))))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(111,plain,
% 0.20/0.44 ((~((~(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)) | (~((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] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16))))))))))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(112,plain,
% 0.20/0.44 (((~![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)) | (~((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] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16)))))))))))),
% 0.20/0.44 inference(monotonicity,[status(thm)],[111])).
% 0.20/0.44 tff(113,plain,
% 0.20/0.44 (((~![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)) | (~((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] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16)))))))))))),
% 0.20/0.44 inference(transitivity,[status(thm)],[112, 110])).
% 0.20/0.44 tff(114,plain,
% 0.20/0.44 ((~![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))))))))))),
% 0.20/0.44 inference(quant_inst,[status(thm)],[])).
% 0.20/0.44 tff(115,plain,
% 0.20/0.44 ((~![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] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16))))))))))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[114, 113])).
% 0.20/0.45 tff(116,plain,
% 0.20/0.45 (~((~(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] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16)))))))))),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[115, 23])).
% 0.20/0.45 tff(117,plain,
% 0.20/0.45 (((~(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] : ((~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)))))))),
% 0.20/0.45 inference(tautology,[status(thm)],[])).
% 0.20/0.45 tff(118,plain,
% 0.20/0.45 ((~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))))))),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[117, 116])).
% 0.20/0.45 tff(119,plain,
% 0.20/0.45 (min(M2!16, R!15, E!14)),
% 0.20/0.45 inference(and_elim,[status(thm)],[51])).
% 0.20/0.45 tff(120,plain,
% 0.20/0.45 ((~((~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))))))),
% 0.20/0.45 inference(tautology,[status(thm)],[])).
% 0.20/0.45 tff(121,plain,
% 0.20/0.45 ((~((~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))))))),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[120, 119])).
% 0.20/0.45 tff(122,plain,
% 0.20/0.45 (~((~member(M2!16, E!14)) | (~![X: $i] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16)))))),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[121, 118])).
% 0.20/0.45 tff(123,plain,
% 0.20/0.45 (((~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)),
% 0.20/0.45 inference(tautology,[status(thm)],[])).
% 0.20/0.45 tff(124,plain,
% 0.20/0.45 (member(M2!16, E!14)),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[123, 122])).
% 0.20/0.45 tff(125,plain,
% 0.20/0.45 (((~![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))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(126,plain,
% 0.20/0.45 ((~![X: $i] : ((~member(X, E!14)) | apply(R!15, M!18, X))) | ((~member(M2!16, E!14)) | apply(R!15, M!18, M2!16))),
% 0.20/0.45 inference(quant_inst,[status(thm)],[])).
% 0.20/0.45 tff(127,plain,
% 0.20/0.45 ((~![X: $i] : ((~member(X, E!14)) | apply(R!15, M!18, X))) | (~member(M2!16, E!14)) | apply(R!15, M!18, M2!16)),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[126, 125])).
% 0.20/0.45 tff(128,plain,
% 0.20/0.45 (apply(R!15, M!18, M2!16)),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[127, 89, 124])).
% 0.20/0.45 tff(129,plain,
% 0.20/0.45 (((~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)))),
% 0.20/0.45 inference(tautology,[status(thm)],[])).
% 0.20/0.45 tff(130,plain,
% 0.20/0.45 (![X: $i] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16)))),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[129, 122])).
% 0.20/0.46 tff(131,plain,
% 0.20/0.46 (((~![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)))),
% 0.20/0.46 inference(rewrite,[status(thm)],[])).
% 0.20/0.46 tff(132,plain,
% 0.20/0.46 ((~![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)))),
% 0.20/0.46 inference(quant_inst,[status(thm)],[])).
% 0.20/0.46 tff(133,plain,
% 0.20/0.46 ((~![X: $i] : ((~member(X, E!14)) | (M2!16 = X) | (~apply(R!15, X, M2!16)))) | (~member(M!18, E!14)) | (M2!16 = M!18) | (~apply(R!15, M!18, M2!16))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[132, 131])).
% 0.20/0.46 tff(134,plain,
% 0.20/0.46 ((M2!16 = M!18) | (~apply(R!15, M!18, M2!16))),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[133, 97, 130])).
% 0.20/0.46 tff(135,plain,
% 0.20/0.46 (M2!16 = M!18),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[134, 128])).
% 0.20/0.46 tff(136,plain,
% 0.20/0.46 ($false),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[135, 109])).
% 0.20/0.46 % SZS output end Proof
%------------------------------------------------------------------------------