TSTP Solution File: SET776+4 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : SET776+4 : TPTP v8.1.0. Released v2.2.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n013.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 05:07:58 EDT 2022
% Result : Theorem 0.20s 0.41s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SET776+4 : TPTP v8.1.0. Released v2.2.0.
% 0.13/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.34 % Computer : n013.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 07:48:47 EDT 2022
% 0.13/0.35 % CPUTime :
% 0.20/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.20/0.35 Usage: tptp [options] [-file:]file
% 0.20/0.35 -h, -? prints this message.
% 0.20/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.20/0.35 -m, -model generate model.
% 0.20/0.35 -p, -proof generate proof.
% 0.20/0.35 -c, -core generate unsat core of named formulas.
% 0.20/0.35 -st, -statistics display statistics.
% 0.20/0.35 -t:timeout set timeout (in second).
% 0.20/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.20/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.20/0.35 -<param>:<value> configuration parameter and value.
% 0.20/0.35 -o:<output-file> file to place output in.
% 0.20/0.41 % SZS status Theorem
% 0.20/0.41 % SZS output start Proof
% 0.20/0.41 tff(apply_type, type, (
% 0.20/0.41 apply: ( $i * $i * $i ) > $o)).
% 0.20/0.41 tff(tptp_fun_Y2_23_type, type, (
% 0.20/0.41 tptp_fun_Y2_23: $i)).
% 0.20/0.41 tff(tptp_fun_X1_26_type, type, (
% 0.20/0.41 tptp_fun_X1_26: $i)).
% 0.20/0.41 tff(tptp_fun_P_21_type, type, (
% 0.20/0.41 tptp_fun_P_21: $i)).
% 0.20/0.41 tff(tptp_fun_Y1_24_type, type, (
% 0.20/0.41 tptp_fun_Y1_24: $i)).
% 0.20/0.41 tff(tptp_fun_R_20_type, type, (
% 0.20/0.41 tptp_fun_R_20: $i)).
% 0.20/0.41 tff(member_type, type, (
% 0.20/0.41 member: ( $i * $i ) > $o)).
% 0.20/0.41 tff(tptp_fun_E_22_type, type, (
% 0.20/0.41 tptp_fun_E_22: $i)).
% 0.20/0.41 tff(tptp_fun_X2_25_type, type, (
% 0.20/0.41 tptp_fun_X2_25: $i)).
% 0.20/0.41 tff(pre_order_type, type, (
% 0.20/0.41 pre_order: ( $i * $i ) > $o)).
% 0.20/0.41 tff(tptp_fun_Z_17_type, type, (
% 0.20/0.41 tptp_fun_Z_17: ( $i * $i ) > $i)).
% 0.20/0.41 tff(tptp_fun_Y_18_type, type, (
% 0.20/0.41 tptp_fun_Y_18: ( $i * $i ) > $i)).
% 0.20/0.41 tff(tptp_fun_X_19_type, type, (
% 0.20/0.41 tptp_fun_X_19: ( $i * $i ) > $i)).
% 0.20/0.41 tff(tptp_fun_X_16_type, type, (
% 0.20/0.41 tptp_fun_X_16: ( $i * $i ) > $i)).
% 0.20/0.41 tff(1,plain,
% 0.20/0.41 (((pre_order(P!21, E!22) & ![A: $i, B: $i] : ((~(member(A, E!22) & member(B, E!22))) | (apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A))))) & (~(apply(P!21, Y1!24, Y2!23) | (~(apply(R!20, X1!26, Y1!24) & apply(R!20, X2!25, Y2!23) & apply(P!21, X1!26, X2!25))) | (~(member(X1!26, E!22) & member(X2!25, E!22) & member(Y1!24, E!22) & member(Y2!23, E!22)))))) <=> (pre_order(P!21, E!22) & ![A: $i, B: $i] : ((~(member(A, E!22) & member(B, E!22))) | (apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A)))) & (~(apply(P!21, Y1!24, Y2!23) | (~(apply(R!20, X1!26, Y1!24) & apply(R!20, X2!25, Y2!23) & apply(P!21, X1!26, X2!25))) | (~(member(X1!26, E!22) & member(X2!25, E!22) & member(Y1!24, E!22) & member(Y2!23, E!22))))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(2,plain,
% 0.20/0.41 ((~(apply(P!21, Y1!24, Y2!23) | (~(apply(R!20, X1!26, Y1!24) & apply(R!20, X2!25, Y2!23) & apply(P!21, X1!26, X2!25))) | (~(member(X1!26, E!22) & member(X2!25, E!22) & member(Y1!24, E!22) & member(Y2!23, E!22))))) <=> (~(apply(P!21, Y1!24, Y2!23) | (~(apply(R!20, X1!26, Y1!24) & apply(R!20, X2!25, Y2!23) & apply(P!21, X1!26, X2!25))) | (~(member(X1!26, E!22) & member(X2!25, E!22) & member(Y1!24, E!22) & member(Y2!23, E!22)))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(3,plain,
% 0.20/0.41 ((pre_order(P!21, E!22) & ![A: $i, B: $i] : ((~(member(A, E!22) & member(B, E!22))) | (apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A))))) <=> (pre_order(P!21, E!22) & ![A: $i, B: $i] : ((~(member(A, E!22) & member(B, E!22))) | (apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A)))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(4,plain,
% 0.20/0.41 (((pre_order(P!21, E!22) & ![A: $i, B: $i] : ((~(member(A, E!22) & member(B, E!22))) | (apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A))))) & (~(apply(P!21, Y1!24, Y2!23) | (~(apply(R!20, X1!26, Y1!24) & apply(R!20, X2!25, Y2!23) & apply(P!21, X1!26, X2!25))) | (~(member(X1!26, E!22) & member(X2!25, E!22) & member(Y1!24, E!22) & member(Y2!23, E!22)))))) <=> ((pre_order(P!21, E!22) & ![A: $i, B: $i] : ((~(member(A, E!22) & member(B, E!22))) | (apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A))))) & (~(apply(P!21, Y1!24, Y2!23) | (~(apply(R!20, X1!26, Y1!24) & apply(R!20, X2!25, Y2!23) & apply(P!21, X1!26, X2!25))) | (~(member(X1!26, E!22) & member(X2!25, E!22) & member(Y1!24, E!22) & member(Y2!23, E!22))))))),
% 0.20/0.41 inference(monotonicity,[status(thm)],[3, 2])).
% 0.20/0.41 tff(5,plain,
% 0.20/0.41 (((pre_order(P!21, E!22) & ![A: $i, B: $i] : ((~(member(A, E!22) & member(B, E!22))) | (apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A))))) & (~(apply(P!21, Y1!24, Y2!23) | (~(apply(R!20, X1!26, Y1!24) & apply(R!20, X2!25, Y2!23) & apply(P!21, X1!26, X2!25))) | (~(member(X1!26, E!22) & member(X2!25, E!22) & member(Y1!24, E!22) & member(Y2!23, E!22)))))) <=> (pre_order(P!21, E!22) & ![A: $i, B: $i] : ((~(member(A, E!22) & member(B, E!22))) | (apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A)))) & (~(apply(P!21, Y1!24, Y2!23) | (~(apply(R!20, X1!26, Y1!24) & apply(R!20, X2!25, Y2!23) & apply(P!21, X1!26, X2!25))) | (~(member(X1!26, E!22) & member(X2!25, E!22) & member(Y1!24, E!22) & member(Y2!23, E!22))))))),
% 0.20/0.41 inference(transitivity,[status(thm)],[4, 1])).
% 0.20/0.41 tff(6,plain,
% 0.20/0.41 ((~![E: $i, P: $i, R: $i] : ((~(pre_order(P, E) & ![A: $i, B: $i] : ((~(member(A, E) & member(B, E))) | (apply(R, A, B) <=> (apply(P, A, B) & apply(P, B, A)))))) | ![X1: $i, X2: $i, Y1: $i, Y2: $i] : (apply(P, Y1, Y2) | (~(apply(R, X1, Y1) & apply(R, X2, Y2) & apply(P, X1, X2))) | (~(member(X1, E) & member(X2, E) & member(Y1, E) & member(Y2, E)))))) <=> (~![E: $i, P: $i, R: $i] : ((~(pre_order(P, E) & ![A: $i, B: $i] : ((~(member(A, E) & member(B, E))) | (apply(R, A, B) <=> (apply(P, A, B) & apply(P, B, A)))))) | ![X1: $i, X2: $i, Y1: $i, Y2: $i] : (apply(P, Y1, Y2) | (~(apply(R, X1, Y1) & apply(R, X2, Y2) & apply(P, X1, X2))) | (~(member(X1, E) & member(X2, E) & member(Y1, E) & member(Y2, E))))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(7,plain,
% 0.20/0.41 ((~![E: $i, P: $i, R: $i] : ((pre_order(P, E) & ![A: $i, B: $i] : ((member(A, E) & member(B, E)) => (apply(R, A, B) <=> (apply(P, A, B) & apply(P, B, A))))) => ![X1: $i, X2: $i, Y1: $i, Y2: $i] : ((((member(X1, E) & member(X2, E)) & member(Y1, E)) & member(Y2, E)) => (((apply(R, X1, Y1) & apply(R, X2, Y2)) & apply(P, X1, X2)) => apply(P, Y1, Y2))))) <=> (~![E: $i, P: $i, R: $i] : ((~(pre_order(P, E) & ![A: $i, B: $i] : ((~(member(A, E) & member(B, E))) | (apply(R, A, B) <=> (apply(P, A, B) & apply(P, B, A)))))) | ![X1: $i, X2: $i, Y1: $i, Y2: $i] : (apply(P, Y1, Y2) | (~(apply(R, X1, Y1) & apply(R, X2, Y2) & apply(P, X1, X2))) | (~(member(X1, E) & member(X2, E) & member(Y1, E) & member(Y2, E))))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(8,axiom,(~![E: $i, P: $i, R: $i] : ((pre_order(P, E) & ![A: $i, B: $i] : ((member(A, E) & member(B, E)) => (apply(R, A, B) <=> (apply(P, A, B) & apply(P, B, A))))) => ![X1: $i, X2: $i, Y1: $i, Y2: $i] : ((((member(X1, E) & member(X2, E)) & member(Y1, E)) & member(Y2, E)) => (((apply(R, X1, Y1) & apply(R, X2, Y2)) & apply(P, X1, X2)) => apply(P, Y1, Y2))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','thIII12')).
% 0.20/0.41 tff(9,plain,
% 0.20/0.41 (~![E: $i, P: $i, R: $i] : ((~(pre_order(P, E) & ![A: $i, B: $i] : ((~(member(A, E) & member(B, E))) | (apply(R, A, B) <=> (apply(P, A, B) & apply(P, B, A)))))) | ![X1: $i, X2: $i, Y1: $i, Y2: $i] : (apply(P, Y1, Y2) | (~(apply(R, X1, Y1) & apply(R, X2, Y2) & apply(P, X1, X2))) | (~(member(X1, E) & member(X2, E) & member(Y1, E) & member(Y2, E)))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[8, 7])).
% 0.20/0.41 tff(10,plain,
% 0.20/0.41 (~![E: $i, P: $i, R: $i] : ((~(pre_order(P, E) & ![A: $i, B: $i] : ((~(member(A, E) & member(B, E))) | (apply(R, A, B) <=> (apply(P, A, B) & apply(P, B, A)))))) | ![X1: $i, X2: $i, Y1: $i, Y2: $i] : (apply(P, Y1, Y2) | (~(apply(R, X1, Y1) & apply(R, X2, Y2) & apply(P, X1, X2))) | (~(member(X1, E) & member(X2, E) & member(Y1, E) & member(Y2, E)))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[9, 6])).
% 0.20/0.41 tff(11,plain,
% 0.20/0.41 (~![E: $i, P: $i, R: $i] : ((~(pre_order(P, E) & ![A: $i, B: $i] : ((~(member(A, E) & member(B, E))) | (apply(R, A, B) <=> (apply(P, A, B) & apply(P, B, A)))))) | ![X1: $i, X2: $i, Y1: $i, Y2: $i] : (apply(P, Y1, Y2) | (~(apply(R, X1, Y1) & apply(R, X2, Y2) & apply(P, X1, X2))) | (~(member(X1, E) & member(X2, E) & member(Y1, E) & member(Y2, E)))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[10, 6])).
% 0.20/0.41 tff(12,plain,
% 0.20/0.41 (~![E: $i, P: $i, R: $i] : ((~(pre_order(P, E) & ![A: $i, B: $i] : ((~(member(A, E) & member(B, E))) | (apply(R, A, B) <=> (apply(P, A, B) & apply(P, B, A)))))) | ![X1: $i, X2: $i, Y1: $i, Y2: $i] : (apply(P, Y1, Y2) | (~(apply(R, X1, Y1) & apply(R, X2, Y2) & apply(P, X1, X2))) | (~(member(X1, E) & member(X2, E) & member(Y1, E) & member(Y2, E)))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[11, 6])).
% 0.20/0.41 tff(13,plain,
% 0.20/0.41 (~![E: $i, P: $i, R: $i] : ((~(pre_order(P, E) & ![A: $i, B: $i] : ((~(member(A, E) & member(B, E))) | (apply(R, A, B) <=> (apply(P, A, B) & apply(P, B, A)))))) | ![X1: $i, X2: $i, Y1: $i, Y2: $i] : (apply(P, Y1, Y2) | (~(apply(R, X1, Y1) & apply(R, X2, Y2) & apply(P, X1, X2))) | (~(member(X1, E) & member(X2, E) & member(Y1, E) & member(Y2, E)))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[12, 6])).
% 0.20/0.42 tff(14,plain,
% 0.20/0.42 (~![E: $i, P: $i, R: $i] : ((~(pre_order(P, E) & ![A: $i, B: $i] : ((~(member(A, E) & member(B, E))) | (apply(R, A, B) <=> (apply(P, A, B) & apply(P, B, A)))))) | ![X1: $i, X2: $i, Y1: $i, Y2: $i] : (apply(P, Y1, Y2) | (~(apply(R, X1, Y1) & apply(R, X2, Y2) & apply(P, X1, X2))) | (~(member(X1, E) & member(X2, E) & member(Y1, E) & member(Y2, E)))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[13, 6])).
% 0.20/0.42 tff(15,plain,
% 0.20/0.42 (~![E: $i, P: $i, R: $i] : ((~(pre_order(P, E) & ![A: $i, B: $i] : ((~(member(A, E) & member(B, E))) | (apply(R, A, B) <=> (apply(P, A, B) & apply(P, B, A)))))) | ![X1: $i, X2: $i, Y1: $i, Y2: $i] : (apply(P, Y1, Y2) | (~(apply(R, X1, Y1) & apply(R, X2, Y2) & apply(P, X1, X2))) | (~(member(X1, E) & member(X2, E) & member(Y1, E) & member(Y2, E)))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[14, 6])).
% 0.20/0.42 tff(16,plain,
% 0.20/0.42 (pre_order(P!21, E!22) & ![A: $i, B: $i] : ((~(member(A, E!22) & member(B, E!22))) | (apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A)))) & (~(apply(P!21, Y1!24, Y2!23) | (~(apply(R!20, X1!26, Y1!24) & apply(R!20, X2!25, Y2!23) & apply(P!21, X1!26, X2!25))) | (~(member(X1!26, E!22) & member(X2!25, E!22) & member(Y1!24, E!22) & member(Y2!23, E!22)))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[15, 5])).
% 0.20/0.42 tff(17,plain,
% 0.20/0.42 (~(apply(P!21, Y1!24, Y2!23) | (~(apply(R!20, X1!26, Y1!24) & apply(R!20, X2!25, Y2!23) & apply(P!21, X1!26, X2!25))) | (~(member(X1!26, E!22) & member(X2!25, E!22) & member(Y1!24, E!22) & member(Y2!23, E!22))))),
% 0.20/0.42 inference(and_elim,[status(thm)],[16])).
% 0.20/0.42 tff(18,plain,
% 0.20/0.42 (member(X1!26, E!22) & member(X2!25, E!22) & member(Y1!24, E!22) & member(Y2!23, E!22)),
% 0.20/0.42 inference(or_elim,[status(thm)],[17])).
% 0.20/0.42 tff(19,plain,
% 0.20/0.42 (member(Y1!24, E!22)),
% 0.20/0.42 inference(and_elim,[status(thm)],[18])).
% 0.20/0.42 tff(20,plain,
% 0.20/0.42 (member(X1!26, E!22)),
% 0.20/0.42 inference(and_elim,[status(thm)],[18])).
% 0.20/0.42 tff(21,plain,
% 0.20/0.42 (^[A: $i, B: $i] : refl(((~member(A, E!22)) | (~member(B, E!22)) | (apply(R!20, A, B) <=> (~((~apply(P!21, A, B)) | (~apply(P!21, B, A)))))) <=> ((~member(A, E!22)) | (~member(B, E!22)) | (apply(R!20, A, B) <=> (~((~apply(P!21, A, B)) | (~apply(P!21, B, A)))))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(22,plain,
% 0.20/0.42 (![A: $i, B: $i] : ((~member(A, E!22)) | (~member(B, E!22)) | (apply(R!20, A, B) <=> (~((~apply(P!21, A, B)) | (~apply(P!21, B, A)))))) <=> ![A: $i, B: $i] : ((~member(A, E!22)) | (~member(B, E!22)) | (apply(R!20, A, B) <=> (~((~apply(P!21, A, B)) | (~apply(P!21, B, A))))))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[21])).
% 0.20/0.42 tff(23,plain,
% 0.20/0.42 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(rewrite((member(A, E!22) & member(B, E!22)) <=> (~((~member(A, E!22)) | (~member(B, E!22))))), ((~(member(A, E!22) & member(B, E!22))) <=> (~(~((~member(A, E!22)) | (~member(B, E!22))))))), rewrite((~(~((~member(A, E!22)) | (~member(B, E!22))))) <=> ((~member(A, E!22)) | (~member(B, E!22)))), ((~(member(A, E!22) & member(B, E!22))) <=> ((~member(A, E!22)) | (~member(B, E!22))))), rewrite((apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A))) <=> (apply(R!20, A, B) <=> (~((~apply(P!21, A, B)) | (~apply(P!21, B, A)))))), (((~(member(A, E!22) & member(B, E!22))) | (apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A)))) <=> (((~member(A, E!22)) | (~member(B, E!22))) | (apply(R!20, A, B) <=> (~((~apply(P!21, A, B)) | (~apply(P!21, B, A)))))))), rewrite((((~member(A, E!22)) | (~member(B, E!22))) | (apply(R!20, A, B) <=> (~((~apply(P!21, A, B)) | (~apply(P!21, B, A)))))) <=> ((~member(A, E!22)) | (~member(B, E!22)) | (apply(R!20, A, B) <=> (~((~apply(P!21, A, B)) | (~apply(P!21, B, A))))))), (((~(member(A, E!22) & member(B, E!22))) | (apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A)))) <=> ((~member(A, E!22)) | (~member(B, E!22)) | (apply(R!20, A, B) <=> (~((~apply(P!21, A, B)) | (~apply(P!21, B, A))))))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(24,plain,
% 0.20/0.42 (![A: $i, B: $i] : ((~(member(A, E!22) & member(B, E!22))) | (apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A)))) <=> ![A: $i, B: $i] : ((~member(A, E!22)) | (~member(B, E!22)) | (apply(R!20, A, B) <=> (~((~apply(P!21, A, B)) | (~apply(P!21, B, A))))))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[23])).
% 0.20/0.42 tff(25,plain,
% 0.20/0.42 (![A: $i, B: $i] : ((~(member(A, E!22) & member(B, E!22))) | (apply(R!20, A, B) <=> (apply(P!21, A, B) & apply(P!21, B, A))))),
% 0.20/0.42 inference(and_elim,[status(thm)],[16])).
% 0.20/0.42 tff(26,plain,
% 0.20/0.42 (![A: $i, B: $i] : ((~member(A, E!22)) | (~member(B, E!22)) | (apply(R!20, A, B) <=> (~((~apply(P!21, A, B)) | (~apply(P!21, B, A))))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[25, 24])).
% 0.20/0.42 tff(27,plain,
% 0.20/0.42 (![A: $i, B: $i] : ((~member(A, E!22)) | (~member(B, E!22)) | (apply(R!20, A, B) <=> (~((~apply(P!21, A, B)) | (~apply(P!21, B, A))))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[26, 22])).
% 0.20/0.42 tff(28,plain,
% 0.20/0.42 (((~![A: $i, B: $i] : ((~member(A, E!22)) | (~member(B, E!22)) | (apply(R!20, A, B) <=> (~((~apply(P!21, A, B)) | (~apply(P!21, B, A))))))) | ((~member(X1!26, E!22)) | (~member(Y1!24, E!22)) | (apply(R!20, X1!26, Y1!24) <=> (~((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))))))) <=> ((~![A: $i, B: $i] : ((~member(A, E!22)) | (~member(B, E!22)) | (apply(R!20, A, B) <=> (~((~apply(P!21, A, B)) | (~apply(P!21, B, A))))))) | (~member(X1!26, E!22)) | (~member(Y1!24, E!22)) | (apply(R!20, X1!26, Y1!24) <=> (~((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(29,plain,
% 0.20/0.42 ((~![A: $i, B: $i] : ((~member(A, E!22)) | (~member(B, E!22)) | (apply(R!20, A, B) <=> (~((~apply(P!21, A, B)) | (~apply(P!21, B, A))))))) | ((~member(X1!26, E!22)) | (~member(Y1!24, E!22)) | (apply(R!20, X1!26, Y1!24) <=> (~((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))))))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(30,plain,
% 0.20/0.42 ((~![A: $i, B: $i] : ((~member(A, E!22)) | (~member(B, E!22)) | (apply(R!20, A, B) <=> (~((~apply(P!21, A, B)) | (~apply(P!21, B, A))))))) | (~member(X1!26, E!22)) | (~member(Y1!24, E!22)) | (apply(R!20, X1!26, Y1!24) <=> (~((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26)))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[29, 28])).
% 0.20/0.42 tff(31,plain,
% 0.20/0.42 (apply(R!20, X1!26, Y1!24) <=> (~((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[30, 27, 20, 19])).
% 0.20/0.42 tff(32,plain,
% 0.20/0.42 (apply(R!20, X1!26, Y1!24) & apply(R!20, X2!25, Y2!23) & apply(P!21, X1!26, X2!25)),
% 0.20/0.42 inference(or_elim,[status(thm)],[17])).
% 0.20/0.42 tff(33,plain,
% 0.20/0.42 (apply(R!20, X1!26, Y1!24)),
% 0.20/0.42 inference(and_elim,[status(thm)],[32])).
% 0.20/0.42 tff(34,plain,
% 0.20/0.42 ((~(apply(R!20, X1!26, Y1!24) <=> (~((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26)))))) | (~apply(R!20, X1!26, Y1!24)) | (~((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))))),
% 0.20/0.42 inference(tautology,[status(thm)],[])).
% 0.20/0.42 tff(35,plain,
% 0.20/0.42 ((~(apply(R!20, X1!26, Y1!24) <=> (~((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26)))))) | (~((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[34, 33])).
% 0.20/0.42 tff(36,plain,
% 0.20/0.42 (~((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26)))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[35, 31])).
% 0.20/0.42 tff(37,plain,
% 0.20/0.42 (((~apply(P!21, X1!26, Y1!24)) | (~apply(P!21, Y1!24, X1!26))) | apply(P!21, Y1!24, X1!26)),
% 0.20/0.42 inference(tautology,[status(thm)],[])).
% 0.20/0.42 tff(38,plain,
% 0.20/0.42 (apply(P!21, Y1!24, X1!26)),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[37, 36])).
% 0.20/0.42 tff(39,plain,
% 0.20/0.42 (^[R: $i, E: $i] : rewrite((~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))))) <=> (~((~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))) | (~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)))))))))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(40,plain,
% 0.20/0.42 (![R: $i, E: $i] : (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))))) <=> ![R: $i, E: $i] : (~((~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))) | (~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E))))))))))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[39])).
% 0.20/0.42 tff(41,plain,
% 0.20/0.42 (^[R: $i, E: $i] : refl((~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))))) <=> (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(42,plain,
% 0.20/0.42 (![R: $i, E: $i] : (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))))) <=> ![R: $i, E: $i] : (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E))))))))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[41])).
% 0.20/0.42 tff(43,plain,
% 0.20/0.42 (^[R: $i, E: $i] : rewrite((~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))))) <=> (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(44,plain,
% 0.20/0.42 (![R: $i, E: $i] : (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))))) <=> ![R: $i, E: $i] : (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E))))))))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[43])).
% 0.20/0.42 tff(45,plain,
% 0.20/0.42 (![R: $i, E: $i] : (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))))) <=> ![R: $i, E: $i] : (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E))))))))),
% 0.20/0.42 inference(transitivity,[status(thm)],[44, 42])).
% 0.20/0.42 tff(46,plain,
% 0.20/0.42 (^[R: $i, E: $i] : trans(monotonicity(rewrite(((~pre_order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) <=> ((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)))))))), trans(monotonicity(rewrite((~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R)) & apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R)))) | (~(member(tptp_fun_X_19(E, R), E) & member(tptp_fun_Y_18(E, R), E) & member(tptp_fun_Z_17(E, R), E))))) <=> (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E))))), ((pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R)) & apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R)))) | (~(member(tptp_fun_X_19(E, R), E) & member(tptp_fun_Y_18(E, R), E) & member(tptp_fun_Z_17(E, R), E)))))) <=> (pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E))))))), rewrite((pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E))))) <=> (pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))), ((pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R)) & apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R)))) | (~(member(tptp_fun_X_19(E, R), E) & member(tptp_fun_Y_18(E, R), E) & member(tptp_fun_Z_17(E, R), E)))))) <=> (pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E))))))), ((((~pre_order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R)) & apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R)))) | (~(member(tptp_fun_X_19(E, R), E) & member(tptp_fun_Y_18(E, R), E) & member(tptp_fun_Z_17(E, R), E))))))) <=> (((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E))))))) & (pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))))), rewrite((((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E))))))) & (pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))) <=> (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E))))))))), ((((~pre_order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R)) & apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R)))) | (~(member(tptp_fun_X_19(E, R), E) & member(tptp_fun_Y_18(E, R), E) & member(tptp_fun_Z_17(E, R), E))))))) <=> (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E))))))))))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(47,plain,
% 0.20/0.43 (![R: $i, E: $i] : (((~pre_order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R)) & apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R)))) | (~(member(tptp_fun_X_19(E, R), E) & member(tptp_fun_Y_18(E, R), E) & member(tptp_fun_Z_17(E, R), E))))))) <=> ![R: $i, E: $i] : (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E))))))))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[46])).
% 0.20/0.43 tff(48,plain,
% 0.20/0.43 (^[R: $i, E: $i] : rewrite((((~pre_order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (pre_order(R, E) | ((~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R)) & apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R)))) | (~(member(tptp_fun_X_19(E, R), E) & member(tptp_fun_Y_18(E, R), E) & member(tptp_fun_Z_17(E, R), E)))))))) <=> (((~pre_order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R)) & apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R)))) | (~(member(tptp_fun_X_19(E, R), E) & member(tptp_fun_Y_18(E, R), E) & member(tptp_fun_Z_17(E, R), E))))))))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(49,plain,
% 0.20/0.43 (![R: $i, E: $i] : (((~pre_order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (pre_order(R, E) | ((~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R)) & apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R)))) | (~(member(tptp_fun_X_19(E, R), E) & member(tptp_fun_Y_18(E, R), E) & member(tptp_fun_Z_17(E, R), E)))))))) <=> ![R: $i, E: $i] : (((~pre_order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R)) & apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R)))) | (~(member(tptp_fun_X_19(E, R), E) & member(tptp_fun_Y_18(E, R), E) & member(tptp_fun_Z_17(E, R), E)))))))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[48])).
% 0.20/0.43 tff(50,plain,
% 0.20/0.43 (![R: $i, E: $i] : (pre_order(R, E) <=> (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) <=> ![R: $i, E: $i] : (pre_order(R, E) <=> (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E))))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(51,plain,
% 0.20/0.43 (^[R: $i, E: $i] : rewrite((pre_order(R, E) <=> (![X: $i] : (member(X, E) => apply(R, X, X)) & ![X: $i, Y: $i, Z: $i] : (((member(X, E) & member(Y, E)) & member(Z, E)) => ((apply(R, X, Y) & apply(R, Y, Z)) => apply(R, X, Z))))) <=> (pre_order(R, E) <=> (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(52,plain,
% 0.20/0.43 (![R: $i, E: $i] : (pre_order(R, E) <=> (![X: $i] : (member(X, E) => apply(R, X, X)) & ![X: $i, Y: $i, Z: $i] : (((member(X, E) & member(Y, E)) & member(Z, E)) => ((apply(R, X, Y) & apply(R, Y, Z)) => apply(R, X, Z))))) <=> ![R: $i, E: $i] : (pre_order(R, E) <=> (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E))))))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[51])).
% 0.20/0.43 tff(53,axiom,(![R: $i, E: $i] : (pre_order(R, E) <=> (![X: $i] : (member(X, E) => apply(R, X, X)) & ![X: $i, Y: $i, Z: $i] : (((member(X, E) & member(Y, E)) & member(Z, E)) => ((apply(R, X, Y) & apply(R, Y, Z)) => apply(R, X, Z)))))), file('/export/starexec/sandbox/benchmark/Axioms/SET006+2.ax','pre_order')).
% 0.20/0.43 tff(54,plain,
% 0.20/0.43 (![R: $i, E: $i] : (pre_order(R, E) <=> (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E))))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[53, 52])).
% 0.20/0.43 tff(55,plain,
% 0.20/0.43 (![R: $i, E: $i] : (pre_order(R, E) <=> (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E))))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[54, 50])).
% 0.20/0.43 tff(56,plain,(
% 0.20/0.43 ![R: $i, E: $i] : (((~pre_order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (pre_order(R, E) | ((~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R)) & apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R)))) | (~(member(tptp_fun_X_19(E, R), E) & member(tptp_fun_Y_18(E, R), E) & member(tptp_fun_Z_17(E, R), E))))))))),
% 0.20/0.43 inference(skolemize,[status(sab)],[55])).
% 0.20/0.43 tff(57,plain,
% 0.20/0.43 (![R: $i, E: $i] : (((~pre_order(R, E)) | (![X: $i] : ((~member(X, E)) | apply(R, X, X)) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R)) & apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R)))) | (~(member(tptp_fun_X_19(E, R), E) & member(tptp_fun_Y_18(E, R), E) & member(tptp_fun_Z_17(E, R), E)))))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[56, 49])).
% 0.20/0.43 tff(58,plain,
% 0.20/0.43 (![R: $i, E: $i] : (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E))))))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[57, 47])).
% 0.20/0.43 tff(59,plain,
% 0.20/0.43 (![R: $i, E: $i] : (~((~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)))))))) | (~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E))))))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[58, 45])).
% 0.20/0.43 tff(60,plain,
% 0.20/0.43 (![R: $i, E: $i] : (~((~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))) | (~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E))))))))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[59, 40])).
% 0.20/0.43 tff(61,plain,
% 0.20/0.43 (((~![R: $i, E: $i] : (~((~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))) | (~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E))))))))))) | (~((~(pre_order(P!21, E!22) | (~((~member(tptp_fun_X_16(E!22, P!21), E!22)) | apply(P!21, tptp_fun_X_16(E!22, P!21), tptp_fun_X_16(E!22, P!21)))) | (~(apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Z_17(E!22, P!21)) | (~apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Y_18(E!22, P!21))) | (~apply(P!21, tptp_fun_Y_18(E!22, P!21), tptp_fun_Z_17(E!22, P!21))) | (~member(tptp_fun_X_19(E!22, P!21), E!22)) | (~member(tptp_fun_Y_18(E!22, P!21), E!22)) | (~member(tptp_fun_Z_17(E!22, P!21), E!22)))))) | (~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22))))))))))) <=> ((~![R: $i, E: $i] : (~((~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))) | (~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E))))))))))) | (~((~(pre_order(P!21, E!22) | (~((~member(tptp_fun_X_16(E!22, P!21), E!22)) | apply(P!21, tptp_fun_X_16(E!22, P!21), tptp_fun_X_16(E!22, P!21)))) | (~(apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Z_17(E!22, P!21)) | (~apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Y_18(E!22, P!21))) | (~apply(P!21, tptp_fun_Y_18(E!22, P!21), tptp_fun_Z_17(E!22, P!21))) | (~member(tptp_fun_X_19(E!22, P!21), E!22)) | (~member(tptp_fun_Y_18(E!22, P!21), E!22)) | (~member(tptp_fun_Z_17(E!22, P!21), E!22)))))) | (~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22)))))))))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(62,plain,
% 0.20/0.43 ((~((~(pre_order(P!21, E!22) | (~((~member(tptp_fun_X_16(E!22, P!21), E!22)) | apply(P!21, tptp_fun_X_16(E!22, P!21), tptp_fun_X_16(E!22, P!21)))) | (~(apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Z_17(E!22, P!21)) | (~apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Y_18(E!22, P!21))) | (~apply(P!21, tptp_fun_Y_18(E!22, P!21), tptp_fun_Z_17(E!22, P!21))) | (~member(tptp_fun_X_19(E!22, P!21), E!22)) | (~member(tptp_fun_Y_18(E!22, P!21), E!22)) | (~member(tptp_fun_Z_17(E!22, P!21), E!22)))))) | (~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(P!21, X, Z) | (~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, X, Y)) | (~apply(P!21, Y, Z)) | (~member(X, E!22)))))))))) <=> (~((~(pre_order(P!21, E!22) | (~((~member(tptp_fun_X_16(E!22, P!21), E!22)) | apply(P!21, tptp_fun_X_16(E!22, P!21), tptp_fun_X_16(E!22, P!21)))) | (~(apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Z_17(E!22, P!21)) | (~apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Y_18(E!22, P!21))) | (~apply(P!21, tptp_fun_Y_18(E!22, P!21), tptp_fun_Z_17(E!22, P!21))) | (~member(tptp_fun_X_19(E!22, P!21), E!22)) | (~member(tptp_fun_Y_18(E!22, P!21), E!22)) | (~member(tptp_fun_Z_17(E!22, P!21), E!22)))))) | (~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22))))))))))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(63,plain,
% 0.20/0.44 (((~![R: $i, E: $i] : (~((~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))) | (~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E))))))))))) | (~((~(pre_order(P!21, E!22) | (~((~member(tptp_fun_X_16(E!22, P!21), E!22)) | apply(P!21, tptp_fun_X_16(E!22, P!21), tptp_fun_X_16(E!22, P!21)))) | (~(apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Z_17(E!22, P!21)) | (~apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Y_18(E!22, P!21))) | (~apply(P!21, tptp_fun_Y_18(E!22, P!21), tptp_fun_Z_17(E!22, P!21))) | (~member(tptp_fun_X_19(E!22, P!21), E!22)) | (~member(tptp_fun_Y_18(E!22, P!21), E!22)) | (~member(tptp_fun_Z_17(E!22, P!21), E!22)))))) | (~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(P!21, X, Z) | (~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, X, Y)) | (~apply(P!21, Y, Z)) | (~member(X, E!22))))))))))) <=> ((~![R: $i, E: $i] : (~((~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))) | (~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E))))))))))) | (~((~(pre_order(P!21, E!22) | (~((~member(tptp_fun_X_16(E!22, P!21), E!22)) | apply(P!21, tptp_fun_X_16(E!22, P!21), tptp_fun_X_16(E!22, P!21)))) | (~(apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Z_17(E!22, P!21)) | (~apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Y_18(E!22, P!21))) | (~apply(P!21, tptp_fun_Y_18(E!22, P!21), tptp_fun_Z_17(E!22, P!21))) | (~member(tptp_fun_X_19(E!22, P!21), E!22)) | (~member(tptp_fun_Y_18(E!22, P!21), E!22)) | (~member(tptp_fun_Z_17(E!22, P!21), E!22)))))) | (~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22)))))))))))),
% 0.20/0.44 inference(monotonicity,[status(thm)],[62])).
% 0.20/0.44 tff(64,plain,
% 0.20/0.44 (((~![R: $i, E: $i] : (~((~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))) | (~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E))))))))))) | (~((~(pre_order(P!21, E!22) | (~((~member(tptp_fun_X_16(E!22, P!21), E!22)) | apply(P!21, tptp_fun_X_16(E!22, P!21), tptp_fun_X_16(E!22, P!21)))) | (~(apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Z_17(E!22, P!21)) | (~apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Y_18(E!22, P!21))) | (~apply(P!21, tptp_fun_Y_18(E!22, P!21), tptp_fun_Z_17(E!22, P!21))) | (~member(tptp_fun_X_19(E!22, P!21), E!22)) | (~member(tptp_fun_Y_18(E!22, P!21), E!22)) | (~member(tptp_fun_Z_17(E!22, P!21), E!22)))))) | (~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(P!21, X, Z) | (~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, X, Y)) | (~apply(P!21, Y, Z)) | (~member(X, E!22))))))))))) <=> ((~![R: $i, E: $i] : (~((~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))) | (~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E))))))))))) | (~((~(pre_order(P!21, E!22) | (~((~member(tptp_fun_X_16(E!22, P!21), E!22)) | apply(P!21, tptp_fun_X_16(E!22, P!21), tptp_fun_X_16(E!22, P!21)))) | (~(apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Z_17(E!22, P!21)) | (~apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Y_18(E!22, P!21))) | (~apply(P!21, tptp_fun_Y_18(E!22, P!21), tptp_fun_Z_17(E!22, P!21))) | (~member(tptp_fun_X_19(E!22, P!21), E!22)) | (~member(tptp_fun_Y_18(E!22, P!21), E!22)) | (~member(tptp_fun_Z_17(E!22, P!21), E!22)))))) | (~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22)))))))))))),
% 0.20/0.44 inference(transitivity,[status(thm)],[63, 61])).
% 0.20/0.44 tff(65,plain,
% 0.20/0.44 ((~![R: $i, E: $i] : (~((~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))) | (~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E))))))))))) | (~((~(pre_order(P!21, E!22) | (~((~member(tptp_fun_X_16(E!22, P!21), E!22)) | apply(P!21, tptp_fun_X_16(E!22, P!21), tptp_fun_X_16(E!22, P!21)))) | (~(apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Z_17(E!22, P!21)) | (~apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Y_18(E!22, P!21))) | (~apply(P!21, tptp_fun_Y_18(E!22, P!21), tptp_fun_Z_17(E!22, P!21))) | (~member(tptp_fun_X_19(E!22, P!21), E!22)) | (~member(tptp_fun_Y_18(E!22, P!21), E!22)) | (~member(tptp_fun_Z_17(E!22, P!21), E!22)))))) | (~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(P!21, X, Z) | (~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, X, Y)) | (~apply(P!21, Y, Z)) | (~member(X, E!22))))))))))),
% 0.20/0.44 inference(quant_inst,[status(thm)],[])).
% 0.20/0.44 tff(66,plain,
% 0.20/0.44 ((~![R: $i, E: $i] : (~((~(pre_order(R, E) | (~((~member(tptp_fun_X_16(E, R), E)) | apply(R, tptp_fun_X_16(E, R), tptp_fun_X_16(E, R)))) | (~(apply(R, tptp_fun_X_19(E, R), tptp_fun_Z_17(E, R)) | (~apply(R, tptp_fun_X_19(E, R), tptp_fun_Y_18(E, R))) | (~apply(R, tptp_fun_Y_18(E, R), tptp_fun_Z_17(E, R))) | (~member(tptp_fun_X_19(E, R), E)) | (~member(tptp_fun_Y_18(E, R), E)) | (~member(tptp_fun_Z_17(E, R), E)))))) | (~((~pre_order(R, E)) | (~((~![X: $i] : ((~member(X, E)) | apply(R, X, X))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~member(Y, E)) | (~member(Z, E)) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E))))))))))) | (~((~(pre_order(P!21, E!22) | (~((~member(tptp_fun_X_16(E!22, P!21), E!22)) | apply(P!21, tptp_fun_X_16(E!22, P!21), tptp_fun_X_16(E!22, P!21)))) | (~(apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Z_17(E!22, P!21)) | (~apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Y_18(E!22, P!21))) | (~apply(P!21, tptp_fun_Y_18(E!22, P!21), tptp_fun_Z_17(E!22, P!21))) | (~member(tptp_fun_X_19(E!22, P!21), E!22)) | (~member(tptp_fun_Y_18(E!22, P!21), E!22)) | (~member(tptp_fun_Z_17(E!22, P!21), E!22)))))) | (~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22))))))))))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[65, 64])).
% 0.20/0.44 tff(67,plain,
% 0.20/0.44 (~((~(pre_order(P!21, E!22) | (~((~member(tptp_fun_X_16(E!22, P!21), E!22)) | apply(P!21, tptp_fun_X_16(E!22, P!21), tptp_fun_X_16(E!22, P!21)))) | (~(apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Z_17(E!22, P!21)) | (~apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Y_18(E!22, P!21))) | (~apply(P!21, tptp_fun_Y_18(E!22, P!21), tptp_fun_Z_17(E!22, P!21))) | (~member(tptp_fun_X_19(E!22, P!21), E!22)) | (~member(tptp_fun_Y_18(E!22, P!21), E!22)) | (~member(tptp_fun_Z_17(E!22, P!21), E!22)))))) | (~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22)))))))))),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[66, 60])).
% 0.20/0.44 tff(68,plain,
% 0.20/0.44 (((~(pre_order(P!21, E!22) | (~((~member(tptp_fun_X_16(E!22, P!21), E!22)) | apply(P!21, tptp_fun_X_16(E!22, P!21), tptp_fun_X_16(E!22, P!21)))) | (~(apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Z_17(E!22, P!21)) | (~apply(P!21, tptp_fun_X_19(E!22, P!21), tptp_fun_Y_18(E!22, P!21))) | (~apply(P!21, tptp_fun_Y_18(E!22, P!21), tptp_fun_Z_17(E!22, P!21))) | (~member(tptp_fun_X_19(E!22, P!21), E!22)) | (~member(tptp_fun_Y_18(E!22, P!21), E!22)) | (~member(tptp_fun_Z_17(E!22, P!21), E!22)))))) | (~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22))))))))) | ((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22)))))))),
% 0.20/0.44 inference(tautology,[status(thm)],[])).
% 0.20/0.44 tff(69,plain,
% 0.20/0.44 ((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22))))))),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[68, 67])).
% 0.20/0.45 tff(70,plain,
% 0.20/0.45 (pre_order(P!21, E!22)),
% 0.20/0.45 inference(and_elim,[status(thm)],[16])).
% 0.20/0.45 tff(71,plain,
% 0.20/0.45 ((~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22)))))))) | (~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22))))))),
% 0.20/0.45 inference(tautology,[status(thm)],[])).
% 0.20/0.45 tff(72,plain,
% 0.20/0.45 ((~((~pre_order(P!21, E!22)) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22)))))))) | (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22))))))),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[71, 70])).
% 0.20/0.45 tff(73,plain,
% 0.20/0.45 (~((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22)))))),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[72, 69])).
% 0.20/0.45 tff(74,plain,
% 0.20/0.45 (((~![X: $i] : ((~member(X, E!22)) | apply(P!21, X, X))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22))))) | ![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22)))),
% 0.20/0.45 inference(tautology,[status(thm)],[])).
% 0.20/0.45 tff(75,plain,
% 0.20/0.45 (![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22)))),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[74, 73])).
% 0.20/0.45 tff(76,plain,
% 0.20/0.45 (member(Y2!23, E!22)),
% 0.20/0.45 inference(and_elim,[status(thm)],[18])).
% 0.20/0.45 tff(77,plain,
% 0.20/0.45 (~apply(P!21, Y1!24, Y2!23)),
% 0.20/0.45 inference(or_elim,[status(thm)],[17])).
% 0.20/0.45 tff(78,plain,
% 0.20/0.45 (((~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22)))) | (apply(P!21, Y1!24, Y2!23) | (~member(Y1!24, E!22)) | (~member(Y2!23, E!22)) | (~member(X1!26, E!22)) | (~apply(P!21, Y1!24, X1!26)) | (~apply(P!21, X1!26, Y2!23)))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22)))) | apply(P!21, Y1!24, Y2!23) | (~member(Y1!24, E!22)) | (~member(Y2!23, E!22)) | (~member(X1!26, E!22)) | (~apply(P!21, Y1!24, X1!26)) | (~apply(P!21, X1!26, Y2!23)))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(79,plain,
% 0.20/0.45 (((~member(X1!26, E!22)) | (~member(Y2!23, E!22)) | (~apply(P!21, X1!26, Y2!23)) | apply(P!21, Y1!24, Y2!23) | (~apply(P!21, Y1!24, X1!26)) | (~member(Y1!24, E!22))) <=> (apply(P!21, Y1!24, Y2!23) | (~member(Y1!24, E!22)) | (~member(Y2!23, E!22)) | (~member(X1!26, E!22)) | (~apply(P!21, Y1!24, X1!26)) | (~apply(P!21, X1!26, Y2!23)))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(80,plain,
% 0.20/0.45 (((~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22)))) | ((~member(X1!26, E!22)) | (~member(Y2!23, E!22)) | (~apply(P!21, X1!26, Y2!23)) | apply(P!21, Y1!24, Y2!23) | (~apply(P!21, Y1!24, X1!26)) | (~member(Y1!24, E!22)))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22)))) | (apply(P!21, Y1!24, Y2!23) | (~member(Y1!24, E!22)) | (~member(Y2!23, E!22)) | (~member(X1!26, E!22)) | (~apply(P!21, Y1!24, X1!26)) | (~apply(P!21, X1!26, Y2!23))))),
% 0.20/0.45 inference(monotonicity,[status(thm)],[79])).
% 0.20/0.45 tff(81,plain,
% 0.20/0.45 (((~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22)))) | ((~member(X1!26, E!22)) | (~member(Y2!23, E!22)) | (~apply(P!21, X1!26, Y2!23)) | apply(P!21, Y1!24, Y2!23) | (~apply(P!21, Y1!24, X1!26)) | (~member(Y1!24, E!22)))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22)))) | apply(P!21, Y1!24, Y2!23) | (~member(Y1!24, E!22)) | (~member(Y2!23, E!22)) | (~member(X1!26, E!22)) | (~apply(P!21, Y1!24, X1!26)) | (~apply(P!21, X1!26, Y2!23)))),
% 0.20/0.45 inference(transitivity,[status(thm)],[80, 78])).
% 0.20/0.45 tff(82,plain,
% 0.20/0.45 ((~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22)))) | ((~member(X1!26, E!22)) | (~member(Y2!23, E!22)) | (~apply(P!21, X1!26, Y2!23)) | apply(P!21, Y1!24, Y2!23) | (~apply(P!21, Y1!24, X1!26)) | (~member(Y1!24, E!22)))),
% 0.20/0.45 inference(quant_inst,[status(thm)],[])).
% 0.20/0.45 tff(83,plain,
% 0.20/0.45 ((~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22)))) | apply(P!21, Y1!24, Y2!23) | (~member(Y1!24, E!22)) | (~member(Y2!23, E!22)) | (~member(X1!26, E!22)) | (~apply(P!21, Y1!24, X1!26)) | (~apply(P!21, X1!26, Y2!23))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[82, 81])).
% 0.20/0.45 tff(84,plain,
% 0.20/0.45 (~apply(P!21, X1!26, Y2!23)),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[83, 77, 20, 19, 76, 75, 38])).
% 0.20/0.45 tff(85,plain,
% 0.20/0.45 (member(X2!25, E!22)),
% 0.20/0.45 inference(and_elim,[status(thm)],[18])).
% 0.20/0.45 tff(86,plain,
% 0.20/0.45 (((~![A: $i, B: $i] : ((~member(A, E!22)) | (~member(B, E!22)) | (apply(R!20, A, B) <=> (~((~apply(P!21, A, B)) | (~apply(P!21, B, A))))))) | ((~member(X2!25, E!22)) | (~member(Y2!23, E!22)) | (apply(R!20, X2!25, Y2!23) <=> (~((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))))))) <=> ((~![A: $i, B: $i] : ((~member(A, E!22)) | (~member(B, E!22)) | (apply(R!20, A, B) <=> (~((~apply(P!21, A, B)) | (~apply(P!21, B, A))))))) | (~member(X2!25, E!22)) | (~member(Y2!23, E!22)) | (apply(R!20, X2!25, Y2!23) <=> (~((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))))))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(87,plain,
% 0.20/0.45 ((~![A: $i, B: $i] : ((~member(A, E!22)) | (~member(B, E!22)) | (apply(R!20, A, B) <=> (~((~apply(P!21, A, B)) | (~apply(P!21, B, A))))))) | ((~member(X2!25, E!22)) | (~member(Y2!23, E!22)) | (apply(R!20, X2!25, Y2!23) <=> (~((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))))))),
% 0.20/0.45 inference(quant_inst,[status(thm)],[])).
% 0.20/0.45 tff(88,plain,
% 0.20/0.45 ((~![A: $i, B: $i] : ((~member(A, E!22)) | (~member(B, E!22)) | (apply(R!20, A, B) <=> (~((~apply(P!21, A, B)) | (~apply(P!21, B, A))))))) | (~member(X2!25, E!22)) | (~member(Y2!23, E!22)) | (apply(R!20, X2!25, Y2!23) <=> (~((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25)))))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[87, 86])).
% 0.20/0.45 tff(89,plain,
% 0.20/0.45 (apply(R!20, X2!25, Y2!23) <=> (~((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))))),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[88, 27, 85, 76])).
% 0.20/0.45 tff(90,plain,
% 0.20/0.45 (apply(R!20, X2!25, Y2!23)),
% 0.20/0.45 inference(and_elim,[status(thm)],[32])).
% 0.20/0.45 tff(91,plain,
% 0.20/0.45 ((~(apply(R!20, X2!25, Y2!23) <=> (~((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25)))))) | (~apply(R!20, X2!25, Y2!23)) | (~((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))))),
% 0.20/0.45 inference(tautology,[status(thm)],[])).
% 0.20/0.45 tff(92,plain,
% 0.20/0.45 ((~(apply(R!20, X2!25, Y2!23) <=> (~((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25)))))) | (~((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))))),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[91, 90])).
% 0.20/0.46 tff(93,plain,
% 0.20/0.46 (~((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25)))),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[92, 89])).
% 0.20/0.46 tff(94,plain,
% 0.20/0.46 (((~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, Y2!23, X2!25))) | apply(P!21, X2!25, Y2!23)),
% 0.20/0.46 inference(tautology,[status(thm)],[])).
% 0.20/0.46 tff(95,plain,
% 0.20/0.46 (apply(P!21, X2!25, Y2!23)),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[94, 93])).
% 0.20/0.46 tff(96,plain,
% 0.20/0.46 (apply(P!21, X1!26, X2!25)),
% 0.20/0.46 inference(and_elim,[status(thm)],[32])).
% 0.20/0.46 tff(97,plain,
% 0.20/0.46 (((~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22)))) | ((~member(Y2!23, E!22)) | (~member(X1!26, E!22)) | (~member(X2!25, E!22)) | (~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, X1!26, X2!25)) | apply(P!21, X1!26, Y2!23))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22)))) | (~member(Y2!23, E!22)) | (~member(X1!26, E!22)) | (~member(X2!25, E!22)) | (~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, X1!26, X2!25)) | apply(P!21, X1!26, Y2!23))),
% 0.20/0.46 inference(rewrite,[status(thm)],[])).
% 0.20/0.46 tff(98,plain,
% 0.20/0.46 (((~member(X2!25, E!22)) | (~member(Y2!23, E!22)) | (~apply(P!21, X2!25, Y2!23)) | apply(P!21, X1!26, Y2!23) | (~apply(P!21, X1!26, X2!25)) | (~member(X1!26, E!22))) <=> ((~member(Y2!23, E!22)) | (~member(X1!26, E!22)) | (~member(X2!25, E!22)) | (~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, X1!26, X2!25)) | apply(P!21, X1!26, Y2!23))),
% 0.20/0.46 inference(rewrite,[status(thm)],[])).
% 0.20/0.46 tff(99,plain,
% 0.20/0.46 (((~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22)))) | ((~member(X2!25, E!22)) | (~member(Y2!23, E!22)) | (~apply(P!21, X2!25, Y2!23)) | apply(P!21, X1!26, Y2!23) | (~apply(P!21, X1!26, X2!25)) | (~member(X1!26, E!22)))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22)))) | ((~member(Y2!23, E!22)) | (~member(X1!26, E!22)) | (~member(X2!25, E!22)) | (~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, X1!26, X2!25)) | apply(P!21, X1!26, Y2!23)))),
% 0.20/0.46 inference(monotonicity,[status(thm)],[98])).
% 0.20/0.46 tff(100,plain,
% 0.20/0.46 (((~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22)))) | ((~member(X2!25, E!22)) | (~member(Y2!23, E!22)) | (~apply(P!21, X2!25, Y2!23)) | apply(P!21, X1!26, Y2!23) | (~apply(P!21, X1!26, X2!25)) | (~member(X1!26, E!22)))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22)))) | (~member(Y2!23, E!22)) | (~member(X1!26, E!22)) | (~member(X2!25, E!22)) | (~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, X1!26, X2!25)) | apply(P!21, X1!26, Y2!23))),
% 0.20/0.46 inference(transitivity,[status(thm)],[99, 97])).
% 0.20/0.46 tff(101,plain,
% 0.20/0.46 ((~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22)))) | ((~member(X2!25, E!22)) | (~member(Y2!23, E!22)) | (~apply(P!21, X2!25, Y2!23)) | apply(P!21, X1!26, Y2!23) | (~apply(P!21, X1!26, X2!25)) | (~member(X1!26, E!22)))),
% 0.20/0.46 inference(quant_inst,[status(thm)],[])).
% 0.20/0.46 tff(102,plain,
% 0.20/0.46 ((~![X: $i, Y: $i, Z: $i] : ((~member(Y, E!22)) | (~member(Z, E!22)) | (~apply(P!21, Y, Z)) | apply(P!21, X, Z) | (~apply(P!21, X, Y)) | (~member(X, E!22)))) | (~member(Y2!23, E!22)) | (~member(X1!26, E!22)) | (~member(X2!25, E!22)) | (~apply(P!21, X2!25, Y2!23)) | (~apply(P!21, X1!26, X2!25)) | apply(P!21, X1!26, Y2!23)),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[101, 100])).
% 0.20/0.46 tff(103,plain,
% 0.20/0.46 ($false),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[102, 96, 20, 85, 76, 75, 95, 84])).
% 0.20/0.46 % SZS output end Proof
%------------------------------------------------------------------------------