%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET734+4 : TPTP v9.0.0. Bugfixed v2.2.1.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n002.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:26:14 AM UTC 2025
% Result : Theorem 0.21s 0.40s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.13 % Problem : SET734+4 : TPTP v9.0.0. Bugfixed v2.2.1.
% 0.08/0.13 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n002.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 : Fri Jun 20 05:40:46 EDT 2025
% 0.13/0.35 % CPUTime :
% 0.21/0.40 % SZS status Theorem
% 0.21/0.40 % SZS output start Proof
% 0.21/0.40 tff(apply_type, type, (
% 0.21/0.40 apply: ( $i * $i * $i ) > $o)).
% 0.21/0.40 tff(tptp_fun_Y_20_type, type, (
% 0.21/0.40 tptp_fun_Y_20: ( $i * $i * $i ) > $i)).
% 0.21/0.40 tff(tptp_fun_G_41_type, type, (
% 0.21/0.40 tptp_fun_G_41: $i)).
% 0.21/0.40 tff(tptp_fun_A_40_type, type, (
% 0.21/0.40 tptp_fun_A_40: $i)).
% 0.21/0.40 tff(tptp_fun_B_39_type, type, (
% 0.21/0.40 tptp_fun_B_39: $i)).
% 0.21/0.40 tff(tptp_fun_Y_8_type, type, (
% 0.21/0.40 tptp_fun_Y_8: ( $i * $i * $i * $i * $i ) > $i)).
% 0.21/0.40 tff(tptp_fun_F_42_type, type, (
% 0.21/0.40 tptp_fun_F_42: $i)).
% 0.21/0.40 tff(member_type, type, (
% 0.21/0.40 member: ( $i * $i ) > $o)).
% 0.21/0.40 tff(compose_function_type, type, (
% 0.21/0.40 compose_function: ( $i * $i * $i * $i * $i ) > $i)).
% 0.21/0.40 tff(surjective_type, type, (
% 0.21/0.40 surjective: ( $i * $i * $i ) > $o)).
% 0.21/0.40 tff(tptp_fun_E_19_type, type, (
% 0.21/0.40 tptp_fun_E_19: ( $i * $i * $i ) > $i)).
% 0.21/0.40 tff(identity_type, type, (
% 0.21/0.40 identity: ( $i * $i ) > $o)).
% 0.21/0.40 tff(maps_type, type, (
% 0.21/0.40 maps: ( $i * $i * $i ) > $o)).
% 0.21/0.40 tff(tptp_fun_X_15_type, type, (
% 0.21/0.40 tptp_fun_X_15: ( $i * $i ) > $i)).
% 0.21/0.40 tff(1,plain,
% 0.21/0.40 (^[F: $i, A: $i, B: $i] : refl((~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))))))))) <=> (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))))))))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(2,plain,
% 0.21/0.40 (![F: $i, A: $i, B: $i] : (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))))))))) <=> ![F: $i, A: $i, B: $i] : (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))))))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[1])).
% 0.21/0.40 tff(3,plain,
% 0.21/0.40 (^[F: $i, A: $i, B: $i] : rewrite((~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))))))))) <=> (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))))))))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(4,plain,
% 0.21/0.40 (![F: $i, A: $i, B: $i] : (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))))))))) <=> ![F: $i, A: $i, B: $i] : (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))))))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[3])).
% 0.21/0.40 tff(5,plain,
% 0.21/0.40 (![F: $i, A: $i, B: $i] : (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))))))))) <=> ![F: $i, A: $i, B: $i] : (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))))))),
% 0.21/0.40 inference(transitivity,[status(thm)],[4, 2])).
% 0.21/0.40 tff(6,plain,
% 0.21/0.40 (^[F: $i, A: $i, B: $i] : trans(monotonicity(rewrite(((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (member(tptp_fun_E_19(Y, A, F), A) & apply(F, tptp_fun_E_19(Y, A, F), Y)))) <=> ((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))), trans(monotonicity(trans(monotonicity(quant_intro(proof_bind(^[E: $i] : trans(monotonicity(rewrite((member(E, A) & apply(F, E, tptp_fun_Y_20(B, A, F))) <=> (~((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))), ((~(member(E, A) & apply(F, E, tptp_fun_Y_20(B, A, F)))) <=> (~(~((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))), rewrite((~(~((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))) <=> ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))), ((~(member(E, A) & apply(F, E, tptp_fun_Y_20(B, A, F)))) <=> ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))))), (![E: $i] : (~(member(E, A) & apply(F, E, tptp_fun_Y_20(B, A, F)))) <=> ![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))), ((member(tptp_fun_Y_20(B, A, F), B) & ![E: $i] : (~(member(E, A) & apply(F, E, tptp_fun_Y_20(B, A, F))))) <=> (member(tptp_fun_Y_20(B, A, F), B) & ![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))))), rewrite((member(tptp_fun_Y_20(B, A, F), B) & ![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))) <=> (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))), ((member(tptp_fun_Y_20(B, A, F), B) & ![E: $i] : (~(member(E, A) & apply(F, E, tptp_fun_Y_20(B, A, F))))) <=> (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))))))), ((surjective(F, A, B) | (member(tptp_fun_Y_20(B, A, F), B) & ![E: $i] : (~(member(E, A) & apply(F, E, tptp_fun_Y_20(B, A, F)))))) <=> (surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))))), rewrite((surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))) <=> (surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))))))), ((surjective(F, A, B) | (member(tptp_fun_Y_20(B, A, F), B) & ![E: $i] : (~(member(E, A) & apply(F, E, tptp_fun_Y_20(B, A, F)))))) <=> (surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))))), ((((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (member(tptp_fun_E_19(Y, A, F), A) & apply(F, tptp_fun_E_19(Y, A, F), Y)))) & (surjective(F, A, B) | (member(tptp_fun_Y_20(B, A, F), B) & ![E: $i] : (~(member(E, A) & apply(F, E, tptp_fun_Y_20(B, A, F))))))) <=> (((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y)))))) & (surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))))))))), rewrite((((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y)))))) & (surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F))))))))) <=> (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))))))), ((((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (member(tptp_fun_E_19(Y, A, F), A) & apply(F, tptp_fun_E_19(Y, A, F), Y)))) & (surjective(F, A, B) | (member(tptp_fun_Y_20(B, A, F), B) & ![E: $i] : (~(member(E, A) & apply(F, E, tptp_fun_Y_20(B, A, F))))))) <=> (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))))))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(7,plain,
% 0.21/0.41 (![F: $i, A: $i, B: $i] : (((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (member(tptp_fun_E_19(Y, A, F), A) & apply(F, tptp_fun_E_19(Y, A, F), Y)))) & (surjective(F, A, B) | (member(tptp_fun_Y_20(B, A, F), B) & ![E: $i] : (~(member(E, A) & apply(F, E, tptp_fun_Y_20(B, A, F))))))) <=> ![F: $i, A: $i, B: $i] : (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))))))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[6])).
% 0.21/0.41 tff(8,plain,
% 0.21/0.41 (![F: $i, A: $i, B: $i] : (surjective(F, A, B) <=> ![Y: $i] : ((~member(Y, B)) | ?[E: $i] : (member(E, A) & apply(F, E, Y)))) <=> ![F: $i, A: $i, B: $i] : (surjective(F, A, B) <=> ![Y: $i] : ((~member(Y, B)) | ?[E: $i] : (member(E, A) & apply(F, E, Y))))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(9,plain,
% 0.21/0.41 (^[F: $i, A: $i, B: $i] : rewrite((surjective(F, A, B) <=> ![Y: $i] : (member(Y, B) => ?[E: $i] : (member(E, A) & apply(F, E, Y)))) <=> (surjective(F, A, B) <=> ![Y: $i] : ((~member(Y, B)) | ?[E: $i] : (member(E, A) & apply(F, E, Y)))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(10,plain,
% 0.21/0.41 (![F: $i, A: $i, B: $i] : (surjective(F, A, B) <=> ![Y: $i] : (member(Y, B) => ?[E: $i] : (member(E, A) & apply(F, E, Y)))) <=> ![F: $i, A: $i, B: $i] : (surjective(F, A, B) <=> ![Y: $i] : ((~member(Y, B)) | ?[E: $i] : (member(E, A) & apply(F, E, Y))))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[9])).
% 0.21/0.41 tff(11,axiom,(![F: $i, A: $i, B: $i] : (surjective(F, A, B) <=> ![Y: $i] : (member(Y, B) => ?[E: $i] : (member(E, A) & apply(F, E, Y))))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+1.ax','surjective')).
% 0.21/0.41 tff(12,plain,
% 0.21/0.41 (![F: $i, A: $i, B: $i] : (surjective(F, A, B) <=> ![Y: $i] : ((~member(Y, B)) | ?[E: $i] : (member(E, A) & apply(F, E, Y))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[11, 10])).
% 0.21/0.41 tff(13,plain,
% 0.21/0.41 (![F: $i, A: $i, B: $i] : (surjective(F, A, B) <=> ![Y: $i] : ((~member(Y, B)) | ?[E: $i] : (member(E, A) & apply(F, E, Y))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[12, 8])).
% 0.21/0.41 tff(14,plain,(
% 0.21/0.41 ![F: $i, A: $i, B: $i] : (((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (member(tptp_fun_E_19(Y, A, F), A) & apply(F, tptp_fun_E_19(Y, A, F), Y)))) & (surjective(F, A, B) | (member(tptp_fun_Y_20(B, A, F), B) & ![E: $i] : (~(member(E, A) & apply(F, E, tptp_fun_Y_20(B, A, F)))))))),
% 0.21/0.41 inference(skolemize,[status(sab)],[13])).
% 0.21/0.41 tff(15,plain,
% 0.21/0.41 (![F: $i, A: $i, B: $i] : (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[14, 7])).
% 0.21/0.41 tff(16,plain,
% 0.21/0.41 (![F: $i, A: $i, B: $i] : (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[15, 5])).
% 0.21/0.41 tff(17,plain,
% 0.21/0.41 ((~![F: $i, A: $i, B: $i] : (~((~((~surjective(F, A, B)) | ![Y: $i] : ((~member(Y, B)) | (~((~member(tptp_fun_E_19(Y, A, F), A)) | (~apply(F, tptp_fun_E_19(Y, A, F), Y))))))) | (~(surjective(F, A, B) | (~((~member(tptp_fun_Y_20(B, A, F), B)) | (~![E: $i] : ((~member(E, A)) | (~apply(F, E, tptp_fun_Y_20(B, A, F)))))))))))) | (~((~((~surjective(G!41, A!40, B!39)) | ![Y: $i] : ((~member(Y, B!39)) | (~((~member(tptp_fun_E_19(Y, A!40, G!41), A!40)) | (~apply(G!41, tptp_fun_E_19(Y, A!40, G!41), Y))))))) | (~(surjective(G!41, A!40, B!39) | (~((~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(B!39, A!40, G!41)))))))))))),
% 0.21/0.41 inference(quant_inst,[status(thm)],[])).
% 0.21/0.41 tff(18,plain,
% 0.21/0.41 (~((~((~surjective(G!41, A!40, B!39)) | ![Y: $i] : ((~member(Y, B!39)) | (~((~member(tptp_fun_E_19(Y, A!40, G!41), A!40)) | (~apply(G!41, tptp_fun_E_19(Y, A!40, G!41), Y))))))) | (~(surjective(G!41, A!40, B!39) | (~((~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(B!39, A!40, G!41))))))))))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[17, 16])).
% 0.21/0.41 tff(19,plain,
% 0.21/0.41 (((~((~surjective(G!41, A!40, B!39)) | ![Y: $i] : ((~member(Y, B!39)) | (~((~member(tptp_fun_E_19(Y, A!40, G!41), A!40)) | (~apply(G!41, tptp_fun_E_19(Y, A!40, G!41), Y))))))) | (~(surjective(G!41, A!40, B!39) | (~((~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(B!39, A!40, G!41)))))))))) | (surjective(G!41, A!40, B!39) | (~((~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(B!39, A!40, G!41))))))))),
% 0.21/0.41 inference(tautology,[status(thm)],[])).
% 0.21/0.41 tff(20,plain,
% 0.21/0.41 (surjective(G!41, A!40, B!39) | (~((~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(B!39, A!40, G!41)))))))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[19, 18])).
% 0.21/0.41 tff(21,plain,
% 0.21/0.41 ((~![F: $i, G: $i, A: $i, B: $i] : ((~(maps(G, A, B) & maps(F, B, A) & identity(compose_function(G, F, B, A, B), B))) | surjective(G, A, B))) <=> (~![F: $i, G: $i, A: $i, B: $i] : ((~(maps(G, A, B) & maps(F, B, A) & identity(compose_function(G, F, B, A, B), B))) | surjective(G, A, B)))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(22,plain,
% 0.21/0.41 ((~![F: $i, G: $i, A: $i, B: $i] : (((maps(G, A, B) & maps(F, B, A)) & identity(compose_function(G, F, B, A, B), B)) => surjective(G, A, B))) <=> (~![F: $i, G: $i, A: $i, B: $i] : ((~(maps(G, A, B) & maps(F, B, A) & identity(compose_function(G, F, B, A, B), B))) | surjective(G, A, B)))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(23,axiom,(~![F: $i, G: $i, A: $i, B: $i] : (((maps(G, A, B) & maps(F, B, A)) & identity(compose_function(G, F, B, A, B), B)) => surjective(G, A, B))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','thII25')).
% 0.21/0.41 tff(24,plain,
% 0.21/0.41 (~![F: $i, G: $i, A: $i, B: $i] : ((~(maps(G, A, B) & maps(F, B, A) & identity(compose_function(G, F, B, A, B), B))) | surjective(G, A, B))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[23, 22])).
% 0.21/0.41 tff(25,plain,
% 0.21/0.41 (~![F: $i, G: $i, A: $i, B: $i] : ((~(maps(G, A, B) & maps(F, B, A) & identity(compose_function(G, F, B, A, B), B))) | surjective(G, A, B))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[24, 21])).
% 0.21/0.41 tff(26,plain,
% 0.21/0.41 (~![F: $i, G: $i, A: $i, B: $i] : ((~(maps(G, A, B) & maps(F, B, A) & identity(compose_function(G, F, B, A, B), B))) | surjective(G, A, B))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[25, 21])).
% 0.21/0.41 tff(27,plain,
% 0.21/0.41 (~![F: $i, G: $i, A: $i, B: $i] : ((~(maps(G, A, B) & maps(F, B, A) & identity(compose_function(G, F, B, A, B), B))) | surjective(G, A, B))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[26, 21])).
% 0.21/0.41 tff(28,plain,(
% 0.21/0.41 ~((~(maps(G!41, A!40, B!39) & maps(F!42, B!39, A!40) & identity(compose_function(G!41, F!42, B!39, A!40, B!39), B!39))) | surjective(G!41, A!40, B!39))),
% 0.21/0.41 inference(skolemize,[status(sab)],[27])).
% 0.21/0.41 tff(29,plain,
% 0.21/0.41 (~surjective(G!41, A!40, B!39)),
% 0.21/0.41 inference(or_elim,[status(thm)],[28])).
% 0.21/0.41 tff(30,plain,
% 0.21/0.41 ((~(surjective(G!41, A!40, B!39) | (~((~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(B!39, A!40, G!41))))))))) | surjective(G!41, A!40, B!39) | (~((~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(B!39, A!40, G!41)))))))),
% 0.21/0.41 inference(tautology,[status(thm)],[])).
% 0.21/0.41 tff(31,plain,
% 0.21/0.41 ((~(surjective(G!41, A!40, B!39) | (~((~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(B!39, A!40, G!41))))))))) | (~((~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(B!39, A!40, G!41)))))))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[30, 29])).
% 0.21/0.41 tff(32,plain,
% 0.21/0.41 (~((~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(B!39, A!40, G!41))))))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[31, 20])).
% 0.21/0.41 tff(33,plain,
% 0.21/0.41 (((~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(B!39, A!40, G!41)))))) | member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)),
% 0.21/0.41 inference(tautology,[status(thm)],[])).
% 0.21/0.41 tff(34,plain,
% 0.21/0.41 (member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[33, 32])).
% 0.21/0.41 tff(35,plain,
% 0.21/0.41 (^[G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : refl(((~member(Z, C)) | (~member(X, A)) | (~((~((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))) | (~(apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z)))))))) <=> ((~member(Z, C)) | (~member(X, A)) | (~((~((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))) | (~(apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z)))))))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(36,plain,
% 0.21/0.41 (![G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : ((~member(Z, C)) | (~member(X, A)) | (~((~((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))) | (~(apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z)))))))) <=> ![G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : ((~member(Z, C)) | (~member(X, A)) | (~((~((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))) | (~(apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z))))))))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[35])).
% 0.21/0.41 tff(37,plain,
% 0.21/0.41 (^[G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : rewrite(((~member(Z, C)) | (~member(X, A)) | (~((~((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))) | (~(apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z)))))))) <=> ((~member(Z, C)) | (~member(X, A)) | (~((~((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))) | (~(apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z)))))))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(38,plain,
% 0.21/0.41 (![G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : ((~member(Z, C)) | (~member(X, A)) | (~((~((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))) | (~(apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z)))))))) <=> ![G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : ((~member(Z, C)) | (~member(X, A)) | (~((~((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))) | (~(apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z))))))))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[37])).
% 0.21/0.41 tff(39,plain,
% 0.21/0.41 (![G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : ((~member(Z, C)) | (~member(X, A)) | (~((~((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))) | (~(apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z)))))))) <=> ![G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : ((~member(Z, C)) | (~member(X, A)) | (~((~((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))) | (~(apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z))))))))),
% 0.21/0.41 inference(transitivity,[status(thm)],[38, 36])).
% 0.21/0.41 tff(40,plain,
% 0.21/0.41 (^[G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : trans(monotonicity(trans(monotonicity(rewrite((member(X, A) & member(Z, C)) <=> (~((~member(Z, C)) | (~member(X, A))))), ((~(member(X, A) & member(Z, C))) <=> (~(~((~member(Z, C)) | (~member(X, A))))))), rewrite((~(~((~member(Z, C)) | (~member(X, A))))) <=> ((~member(Z, C)) | (~member(X, A)))), ((~(member(X, A) & member(Z, C))) <=> ((~member(Z, C)) | (~member(X, A))))), trans(monotonicity(rewrite(((~apply(compose_function(G, F, A, B, C), X, Z)) | (member(tptp_fun_Y_8(Z, X, B, F, G), B) & apply(F, X, tptp_fun_Y_8(Z, X, B, F, G)) & apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z))) <=> ((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))), rewrite((apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : (~(member(Y, B) & apply(F, X, Y) & apply(G, Y, Z)))) <=> (apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z))))), ((((~apply(compose_function(G, F, A, B, C), X, Z)) | (member(tptp_fun_Y_8(Z, X, B, F, G), B) & apply(F, X, tptp_fun_Y_8(Z, X, B, F, G)) & apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z))) & (apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : (~(member(Y, B) & apply(F, X, Y) & apply(G, Y, Z))))) <=> (((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z))))) & (apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z))))))), rewrite((((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z))))) & (apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z))))) <=> (~((~((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))) | (~(apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z)))))))), ((((~apply(compose_function(G, F, A, B, C), X, Z)) | (member(tptp_fun_Y_8(Z, X, B, F, G), B) & apply(F, X, tptp_fun_Y_8(Z, X, B, F, G)) & apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z))) & (apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : (~(member(Y, B) & apply(F, X, Y) & apply(G, Y, Z))))) <=> (~((~((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))) | (~(apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z))))))))), (((~(member(X, A) & member(Z, C))) | (((~apply(compose_function(G, F, A, B, C), X, Z)) | (member(tptp_fun_Y_8(Z, X, B, F, G), B) & apply(F, X, tptp_fun_Y_8(Z, X, B, F, G)) & apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z))) & (apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : (~(member(Y, B) & apply(F, X, Y) & apply(G, Y, Z)))))) <=> (((~member(Z, C)) | (~member(X, A))) | (~((~((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))) | (~(apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z)))))))))), rewrite((((~member(Z, C)) | (~member(X, A))) | (~((~((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))) | (~(apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z)))))))) <=> ((~member(Z, C)) | (~member(X, A)) | (~((~((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))) | (~(apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z))))))))), (((~(member(X, A) & member(Z, C))) | (((~apply(compose_function(G, F, A, B, C), X, Z)) | (member(tptp_fun_Y_8(Z, X, B, F, G), B) & apply(F, X, tptp_fun_Y_8(Z, X, B, F, G)) & apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z))) & (apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : (~(member(Y, B) & apply(F, X, Y) & apply(G, Y, Z)))))) <=> ((~member(Z, C)) | (~member(X, A)) | (~((~((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))) | (~(apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z))))))))))),
% 0.21/0.42 inference(bind,[status(th)],[])).
% 0.21/0.42 tff(41,plain,
% 0.21/0.42 (![G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : ((~(member(X, A) & member(Z, C))) | (((~apply(compose_function(G, F, A, B, C), X, Z)) | (member(tptp_fun_Y_8(Z, X, B, F, G), B) & apply(F, X, tptp_fun_Y_8(Z, X, B, F, G)) & apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z))) & (apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : (~(member(Y, B) & apply(F, X, Y) & apply(G, Y, Z)))))) <=> ![G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : ((~member(Z, C)) | (~member(X, A)) | (~((~((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))) | (~(apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z))))))))),
% 0.21/0.42 inference(quant_intro,[status(thm)],[40])).
% 0.21/0.42 tff(42,plain,
% 0.21/0.42 (![G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : ((~(member(X, A) & member(Z, C))) | (apply(compose_function(G, F, A, B, C), X, Z) <=> ?[Y: $i] : (member(Y, B) & apply(F, X, Y) & apply(G, Y, Z)))) <=> ![G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : ((~(member(X, A) & member(Z, C))) | (apply(compose_function(G, F, A, B, C), X, Z) <=> ?[Y: $i] : (member(Y, B) & apply(F, X, Y) & apply(G, Y, Z))))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(43,plain,
% 0.21/0.42 (^[G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : trans(monotonicity(rewrite((apply(compose_function(G, F, A, B, C), X, Z) <=> ?[Y: $i] : ((member(Y, B) & apply(F, X, Y)) & apply(G, Y, Z))) <=> (apply(compose_function(G, F, A, B, C), X, Z) <=> ?[Y: $i] : (member(Y, B) & apply(F, X, Y) & apply(G, Y, Z)))), (((member(X, A) & member(Z, C)) => (apply(compose_function(G, F, A, B, C), X, Z) <=> ?[Y: $i] : ((member(Y, B) & apply(F, X, Y)) & apply(G, Y, Z)))) <=> ((member(X, A) & member(Z, C)) => (apply(compose_function(G, F, A, B, C), X, Z) <=> ?[Y: $i] : (member(Y, B) & apply(F, X, Y) & apply(G, Y, Z)))))), rewrite(((member(X, A) & member(Z, C)) => (apply(compose_function(G, F, A, B, C), X, Z) <=> ?[Y: $i] : (member(Y, B) & apply(F, X, Y) & apply(G, Y, Z)))) <=> ((~(member(X, A) & member(Z, C))) | (apply(compose_function(G, F, A, B, C), X, Z) <=> ?[Y: $i] : (member(Y, B) & apply(F, X, Y) & apply(G, Y, Z))))), (((member(X, A) & member(Z, C)) => (apply(compose_function(G, F, A, B, C), X, Z) <=> ?[Y: $i] : ((member(Y, B) & apply(F, X, Y)) & apply(G, Y, Z)))) <=> ((~(member(X, A) & member(Z, C))) | (apply(compose_function(G, F, A, B, C), X, Z) <=> ?[Y: $i] : (member(Y, B) & apply(F, X, Y) & apply(G, Y, Z))))))),
% 0.21/0.42 inference(bind,[status(th)],[])).
% 0.21/0.42 tff(44,plain,
% 0.21/0.42 (![G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : ((member(X, A) & member(Z, C)) => (apply(compose_function(G, F, A, B, C), X, Z) <=> ?[Y: $i] : ((member(Y, B) & apply(F, X, Y)) & apply(G, Y, Z)))) <=> ![G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : ((~(member(X, A) & member(Z, C))) | (apply(compose_function(G, F, A, B, C), X, Z) <=> ?[Y: $i] : (member(Y, B) & apply(F, X, Y) & apply(G, Y, Z))))),
% 0.21/0.42 inference(quant_intro,[status(thm)],[43])).
% 0.21/0.42 tff(45,axiom,(![G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : ((member(X, A) & member(Z, C)) => (apply(compose_function(G, F, A, B, C), X, Z) <=> ?[Y: $i] : ((member(Y, B) & apply(F, X, Y)) & apply(G, Y, Z))))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+1.ax','compose_function')).
% 0.21/0.42 tff(46,plain,
% 0.21/0.42 (![G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : ((~(member(X, A) & member(Z, C))) | (apply(compose_function(G, F, A, B, C), X, Z) <=> ?[Y: $i] : (member(Y, B) & apply(F, X, Y) & apply(G, Y, Z))))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[45, 44])).
% 0.21/0.42 tff(47,plain,
% 0.21/0.42 (![G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : ((~(member(X, A) & member(Z, C))) | (apply(compose_function(G, F, A, B, C), X, Z) <=> ?[Y: $i] : (member(Y, B) & apply(F, X, Y) & apply(G, Y, Z))))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[46, 42])).
% 0.21/0.42 tff(48,plain,(
% 0.21/0.42 ![G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : ((~(member(X, A) & member(Z, C))) | (((~apply(compose_function(G, F, A, B, C), X, Z)) | (member(tptp_fun_Y_8(Z, X, B, F, G), B) & apply(F, X, tptp_fun_Y_8(Z, X, B, F, G)) & apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z))) & (apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : (~(member(Y, B) & apply(F, X, Y) & apply(G, Y, Z))))))),
% 0.21/0.42 inference(skolemize,[status(sab)],[47])).
% 0.21/0.42 tff(49,plain,
% 0.21/0.42 (![G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : ((~member(Z, C)) | (~member(X, A)) | (~((~((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))) | (~(apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z))))))))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[48, 41])).
% 0.21/0.42 tff(50,plain,
% 0.21/0.42 (![G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : ((~member(Z, C)) | (~member(X, A)) | (~((~((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))) | (~(apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z))))))))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[49, 39])).
% 0.21/0.42 tff(51,plain,
% 0.21/0.42 (((~![G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : ((~member(Z, C)) | (~member(X, A)) | (~((~((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))) | (~(apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z))))))))) | ((~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~((~(apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41)) | ![Y: $i] : ((~member(Y, A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), Y)) | (~apply(G!41, Y, tptp_fun_Y_20(B!39, A!40, G!41)))))) | (~((~apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41))) | (~((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41))) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41))))))))))) <=> ((~![G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : ((~member(Z, C)) | (~member(X, A)) | (~((~((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))) | (~(apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z))))))))) | (~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~((~(apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41)) | ![Y: $i] : ((~member(Y, A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), Y)) | (~apply(G!41, Y, tptp_fun_Y_20(B!39, A!40, G!41)))))) | (~((~apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41))) | (~((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41))) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41))))))))))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(52,plain,
% 0.21/0.42 (((~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~((~(apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41)) | ![Y: $i] : ((~member(Y, A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), Y)) | (~apply(G!41, Y, tptp_fun_Y_20(B!39, A!40, G!41)))))) | (~((~apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41))) | (~((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41))) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41)))))))))) <=> ((~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~((~(apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41)) | ![Y: $i] : ((~member(Y, A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), Y)) | (~apply(G!41, Y, tptp_fun_Y_20(B!39, A!40, G!41)))))) | (~((~apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41))) | (~((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41))) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41))))))))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(53,plain,
% 0.21/0.43 ((~((~((~apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41))) | (~((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41))) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41))))))) | (~(apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41)) | ![Y: $i] : ((~member(Y, A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), Y)) | (~apply(G!41, Y, tptp_fun_Y_20(B!39, A!40, G!41)))))))) <=> (~((~(apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41)) | ![Y: $i] : ((~member(Y, A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), Y)) | (~apply(G!41, Y, tptp_fun_Y_20(B!39, A!40, G!41)))))) | (~((~apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41))) | (~((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41))) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41)))))))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(54,plain,
% 0.21/0.43 (((~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~((~((~apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41))) | (~((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41))) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41))))))) | (~(apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41)) | ![Y: $i] : ((~member(Y, A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), Y)) | (~apply(G!41, Y, tptp_fun_Y_20(B!39, A!40, G!41))))))))) <=> ((~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~((~(apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41)) | ![Y: $i] : ((~member(Y, A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), Y)) | (~apply(G!41, Y, tptp_fun_Y_20(B!39, A!40, G!41)))))) | (~((~apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41))) | (~((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41))) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41))))))))))),
% 0.21/0.43 inference(monotonicity,[status(thm)],[53])).
% 0.21/0.43 tff(55,plain,
% 0.21/0.43 (((~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~((~((~apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41))) | (~((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41))) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41))))))) | (~(apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41)) | ![Y: $i] : ((~member(Y, A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), Y)) | (~apply(G!41, Y, tptp_fun_Y_20(B!39, A!40, G!41))))))))) <=> ((~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~((~(apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41)) | ![Y: $i] : ((~member(Y, A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), Y)) | (~apply(G!41, Y, tptp_fun_Y_20(B!39, A!40, G!41)))))) | (~((~apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41))) | (~((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41))) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41))))))))))),
% 0.21/0.43 inference(transitivity,[status(thm)],[54, 52])).
% 0.21/0.43 tff(56,plain,
% 0.21/0.43 (((~![G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : ((~member(Z, C)) | (~member(X, A)) | (~((~((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))) | (~(apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z))))))))) | ((~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~((~((~apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41))) | (~((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41))) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41))))))) | (~(apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41)) | ![Y: $i] : ((~member(Y, A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), Y)) | (~apply(G!41, Y, tptp_fun_Y_20(B!39, A!40, G!41)))))))))) <=> ((~![G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : ((~member(Z, C)) | (~member(X, A)) | (~((~((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))) | (~(apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z))))))))) | ((~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~((~(apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41)) | ![Y: $i] : ((~member(Y, A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), Y)) | (~apply(G!41, Y, tptp_fun_Y_20(B!39, A!40, G!41)))))) | (~((~apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41))) | (~((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41))) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41)))))))))))),
% 0.21/0.44 inference(monotonicity,[status(thm)],[55])).
% 0.21/0.44 tff(57,plain,
% 0.21/0.44 (((~![G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : ((~member(Z, C)) | (~member(X, A)) | (~((~((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))) | (~(apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z))))))))) | ((~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~((~((~apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41))) | (~((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41))) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41))))))) | (~(apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41)) | ![Y: $i] : ((~member(Y, A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), Y)) | (~apply(G!41, Y, tptp_fun_Y_20(B!39, A!40, G!41)))))))))) <=> ((~![G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : ((~member(Z, C)) | (~member(X, A)) | (~((~((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))) | (~(apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z))))))))) | (~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~((~(apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41)) | ![Y: $i] : ((~member(Y, A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), Y)) | (~apply(G!41, Y, tptp_fun_Y_20(B!39, A!40, G!41)))))) | (~((~apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41))) | (~((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41))) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41))))))))))),
% 0.21/0.44 inference(transitivity,[status(thm)],[56, 51])).
% 0.21/0.44 tff(58,plain,
% 0.21/0.44 ((~![G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : ((~member(Z, C)) | (~member(X, A)) | (~((~((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))) | (~(apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z))))))))) | ((~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~((~((~apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41))) | (~((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41))) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41))))))) | (~(apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41)) | ![Y: $i] : ((~member(Y, A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), Y)) | (~apply(G!41, Y, tptp_fun_Y_20(B!39, A!40, G!41)))))))))),
% 0.21/0.44 inference(quant_inst,[status(thm)],[])).
% 0.21/0.44 tff(59,plain,
% 0.21/0.44 ((~![G: $i, F: $i, A: $i, B: $i, C: $i, X: $i, Z: $i] : ((~member(Z, C)) | (~member(X, A)) | (~((~((~apply(compose_function(G, F, A, B, C), X, Z)) | (~((~member(tptp_fun_Y_8(Z, X, B, F, G), B)) | (~apply(F, X, tptp_fun_Y_8(Z, X, B, F, G))) | (~apply(G, tptp_fun_Y_8(Z, X, B, F, G), Z)))))) | (~(apply(compose_function(G, F, A, B, C), X, Z) | ![Y: $i] : ((~member(Y, B)) | (~apply(F, X, Y)) | (~apply(G, Y, Z))))))))) | (~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~((~(apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41)) | ![Y: $i] : ((~member(Y, A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), Y)) | (~apply(G!41, Y, tptp_fun_Y_20(B!39, A!40, G!41)))))) | (~((~apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41))) | (~((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41))) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41)))))))))),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[58, 57])).
% 0.21/0.44 tff(60,plain,
% 0.21/0.44 (~((~(apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41)) | ![Y: $i] : ((~member(Y, A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), Y)) | (~apply(G!41, Y, tptp_fun_Y_20(B!39, A!40, G!41)))))) | (~((~apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41))) | (~((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41))) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41))))))))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[59, 50, 34])).
% 0.21/0.44 tff(61,plain,
% 0.21/0.44 (((~(apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41)) | ![Y: $i] : ((~member(Y, A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), Y)) | (~apply(G!41, Y, tptp_fun_Y_20(B!39, A!40, G!41)))))) | (~((~apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41))) | (~((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41))) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41)))))))) | ((~apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41))) | (~((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41))) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41))))))),
% 0.21/0.44 inference(tautology,[status(thm)],[])).
% 0.21/0.44 tff(62,plain,
% 0.21/0.44 ((~apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41))) | (~((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41))) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41)))))),
% 0.21/0.44 inference(unit_resolution,[status(thm)],[61, 60])).
% 0.21/0.44 tff(63,plain,
% 0.21/0.44 (^[F: $i, A: $i] : refl((~((~((~identity(F, A)) | ![X: $i] : ((~member(X, A)) | apply(F, X, X)))) | (~(identity(F, A) | (~((~member(tptp_fun_X_15(A, F), A)) | apply(F, tptp_fun_X_15(A, F), tptp_fun_X_15(A, F)))))))) <=> (~((~((~identity(F, A)) | ![X: $i] : ((~member(X, A)) | apply(F, X, X)))) | (~(identity(F, A) | (~((~member(tptp_fun_X_15(A, F), A)) | apply(F, tptp_fun_X_15(A, F), tptp_fun_X_15(A, F)))))))))),
% 0.21/0.44 inference(bind,[status(th)],[])).
% 0.21/0.44 tff(64,plain,
% 0.21/0.44 (![F: $i, A: $i] : (~((~((~identity(F, A)) | ![X: $i] : ((~member(X, A)) | apply(F, X, X)))) | (~(identity(F, A) | (~((~member(tptp_fun_X_15(A, F), A)) | apply(F, tptp_fun_X_15(A, F), tptp_fun_X_15(A, F)))))))) <=> ![F: $i, A: $i] : (~((~((~identity(F, A)) | ![X: $i] : ((~member(X, A)) | apply(F, X, X)))) | (~(identity(F, A) | (~((~member(tptp_fun_X_15(A, F), A)) | apply(F, tptp_fun_X_15(A, F), tptp_fun_X_15(A, F))))))))),
% 0.21/0.44 inference(quant_intro,[status(thm)],[63])).
% 0.21/0.44 tff(65,plain,
% 0.21/0.44 (^[F: $i, A: $i] : rewrite((~((~((~identity(F, A)) | ![X: $i] : ((~member(X, A)) | apply(F, X, X)))) | (~(identity(F, A) | (~((~member(tptp_fun_X_15(A, F), A)) | apply(F, tptp_fun_X_15(A, F), tptp_fun_X_15(A, F)))))))) <=> (~((~((~identity(F, A)) | ![X: $i] : ((~member(X, A)) | apply(F, X, X)))) | (~(identity(F, A) | (~((~member(tptp_fun_X_15(A, F), A)) | apply(F, tptp_fun_X_15(A, F), tptp_fun_X_15(A, F)))))))))),
% 0.21/0.44 inference(bind,[status(th)],[])).
% 0.21/0.44 tff(66,plain,
% 0.21/0.44 (![F: $i, A: $i] : (~((~((~identity(F, A)) | ![X: $i] : ((~member(X, A)) | apply(F, X, X)))) | (~(identity(F, A) | (~((~member(tptp_fun_X_15(A, F), A)) | apply(F, tptp_fun_X_15(A, F), tptp_fun_X_15(A, F)))))))) <=> ![F: $i, A: $i] : (~((~((~identity(F, A)) | ![X: $i] : ((~member(X, A)) | apply(F, X, X)))) | (~(identity(F, A) | (~((~member(tptp_fun_X_15(A, F), A)) | apply(F, tptp_fun_X_15(A, F), tptp_fun_X_15(A, F))))))))),
% 0.21/0.44 inference(quant_intro,[status(thm)],[65])).
% 0.21/0.44 tff(67,plain,
% 0.21/0.44 (![F: $i, A: $i] : (~((~((~identity(F, A)) | ![X: $i] : ((~member(X, A)) | apply(F, X, X)))) | (~(identity(F, A) | (~((~member(tptp_fun_X_15(A, F), A)) | apply(F, tptp_fun_X_15(A, F), tptp_fun_X_15(A, F)))))))) <=> ![F: $i, A: $i] : (~((~((~identity(F, A)) | ![X: $i] : ((~member(X, A)) | apply(F, X, X)))) | (~(identity(F, A) | (~((~member(tptp_fun_X_15(A, F), A)) | apply(F, tptp_fun_X_15(A, F), tptp_fun_X_15(A, F))))))))),
% 0.21/0.44 inference(transitivity,[status(thm)],[66, 64])).
% 0.21/0.44 tff(68,plain,
% 0.21/0.44 (^[F: $i, A: $i] : trans(monotonicity(rewrite(((~identity(F, A)) | ![X: $i] : ((~member(X, A)) | apply(F, X, X))) <=> ((~identity(F, A)) | ![X: $i] : ((~member(X, A)) | apply(F, X, X)))), rewrite((identity(F, A) | (~((~member(tptp_fun_X_15(A, F), A)) | apply(F, tptp_fun_X_15(A, F), tptp_fun_X_15(A, F))))) <=> (identity(F, A) | (~((~member(tptp_fun_X_15(A, F), A)) | apply(F, tptp_fun_X_15(A, F), tptp_fun_X_15(A, F)))))), ((((~identity(F, A)) | ![X: $i] : ((~member(X, A)) | apply(F, X, X))) & (identity(F, A) | (~((~member(tptp_fun_X_15(A, F), A)) | apply(F, tptp_fun_X_15(A, F), tptp_fun_X_15(A, F)))))) <=> (((~identity(F, A)) | ![X: $i] : ((~member(X, A)) | apply(F, X, X))) & (identity(F, A) | (~((~member(tptp_fun_X_15(A, F), A)) | apply(F, tptp_fun_X_15(A, F), tptp_fun_X_15(A, F)))))))), rewrite((((~identity(F, A)) | ![X: $i] : ((~member(X, A)) | apply(F, X, X))) & (identity(F, A) | (~((~member(tptp_fun_X_15(A, F), A)) | apply(F, tptp_fun_X_15(A, F), tptp_fun_X_15(A, F)))))) <=> (~((~((~identity(F, A)) | ![X: $i] : ((~member(X, A)) | apply(F, X, X)))) | (~(identity(F, A) | (~((~member(tptp_fun_X_15(A, F), A)) | apply(F, tptp_fun_X_15(A, F), tptp_fun_X_15(A, F))))))))), ((((~identity(F, A)) | ![X: $i] : ((~member(X, A)) | apply(F, X, X))) & (identity(F, A) | (~((~member(tptp_fun_X_15(A, F), A)) | apply(F, tptp_fun_X_15(A, F), tptp_fun_X_15(A, F)))))) <=> (~((~((~identity(F, A)) | ![X: $i] : ((~member(X, A)) | apply(F, X, X)))) | (~(identity(F, A) | (~((~member(tptp_fun_X_15(A, F), A)) | apply(F, tptp_fun_X_15(A, F), tptp_fun_X_15(A, F))))))))))),
% 0.21/0.45 inference(bind,[status(th)],[])).
% 0.21/0.45 tff(69,plain,
% 0.21/0.45 (![F: $i, A: $i] : (((~identity(F, A)) | ![X: $i] : ((~member(X, A)) | apply(F, X, X))) & (identity(F, A) | (~((~member(tptp_fun_X_15(A, F), A)) | apply(F, tptp_fun_X_15(A, F), tptp_fun_X_15(A, F)))))) <=> ![F: $i, A: $i] : (~((~((~identity(F, A)) | ![X: $i] : ((~member(X, A)) | apply(F, X, X)))) | (~(identity(F, A) | (~((~member(tptp_fun_X_15(A, F), A)) | apply(F, tptp_fun_X_15(A, F), tptp_fun_X_15(A, F))))))))),
% 0.21/0.45 inference(quant_intro,[status(thm)],[68])).
% 0.21/0.45 tff(70,plain,
% 0.21/0.45 (![F: $i, A: $i] : (identity(F, A) <=> ![X: $i] : ((~member(X, A)) | apply(F, X, X))) <=> ![F: $i, A: $i] : (identity(F, A) <=> ![X: $i] : ((~member(X, A)) | apply(F, X, X)))),
% 0.21/0.45 inference(rewrite,[status(thm)],[])).
% 0.21/0.45 tff(71,plain,
% 0.21/0.45 (^[F: $i, A: $i] : rewrite((identity(F, A) <=> ![X: $i] : (member(X, A) => apply(F, X, X))) <=> (identity(F, A) <=> ![X: $i] : ((~member(X, A)) | apply(F, X, X))))),
% 0.21/0.45 inference(bind,[status(th)],[])).
% 0.21/0.45 tff(72,plain,
% 0.21/0.45 (![F: $i, A: $i] : (identity(F, A) <=> ![X: $i] : (member(X, A) => apply(F, X, X))) <=> ![F: $i, A: $i] : (identity(F, A) <=> ![X: $i] : ((~member(X, A)) | apply(F, X, X)))),
% 0.21/0.45 inference(quant_intro,[status(thm)],[71])).
% 0.21/0.45 tff(73,axiom,(![F: $i, A: $i] : (identity(F, A) <=> ![X: $i] : (member(X, A) => apply(F, X, X)))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+1.ax','identity')).
% 0.21/0.45 tff(74,plain,
% 0.21/0.45 (![F: $i, A: $i] : (identity(F, A) <=> ![X: $i] : ((~member(X, A)) | apply(F, X, X)))),
% 0.21/0.45 inference(modus_ponens,[status(thm)],[73, 72])).
% 0.21/0.45 tff(75,plain,
% 0.21/0.45 (![F: $i, A: $i] : (identity(F, A) <=> ![X: $i] : ((~member(X, A)) | apply(F, X, X)))),
% 0.21/0.45 inference(modus_ponens,[status(thm)],[74, 70])).
% 0.21/0.45 tff(76,plain,(
% 0.21/0.45 ![F: $i, A: $i] : (((~identity(F, A)) | ![X: $i] : ((~member(X, A)) | apply(F, X, X))) & (identity(F, A) | (~((~member(tptp_fun_X_15(A, F), A)) | apply(F, tptp_fun_X_15(A, F), tptp_fun_X_15(A, F))))))),
% 0.21/0.45 inference(skolemize,[status(sab)],[75])).
% 0.21/0.45 tff(77,plain,
% 0.21/0.45 (![F: $i, A: $i] : (~((~((~identity(F, A)) | ![X: $i] : ((~member(X, A)) | apply(F, X, X)))) | (~(identity(F, A) | (~((~member(tptp_fun_X_15(A, F), A)) | apply(F, tptp_fun_X_15(A, F), tptp_fun_X_15(A, F))))))))),
% 0.21/0.45 inference(modus_ponens,[status(thm)],[76, 69])).
% 0.21/0.45 tff(78,plain,
% 0.21/0.45 (![F: $i, A: $i] : (~((~((~identity(F, A)) | ![X: $i] : ((~member(X, A)) | apply(F, X, X)))) | (~(identity(F, A) | (~((~member(tptp_fun_X_15(A, F), A)) | apply(F, tptp_fun_X_15(A, F), tptp_fun_X_15(A, F))))))))),
% 0.21/0.45 inference(modus_ponens,[status(thm)],[77, 67])).
% 0.21/0.45 tff(79,plain,
% 0.21/0.45 ((~![F: $i, A: $i] : (~((~((~identity(F, A)) | ![X: $i] : ((~member(X, A)) | apply(F, X, X)))) | (~(identity(F, A) | (~((~member(tptp_fun_X_15(A, F), A)) | apply(F, tptp_fun_X_15(A, F), tptp_fun_X_15(A, F))))))))) | (~((~((~identity(compose_function(G!41, F!42, B!39, A!40, B!39), B!39)) | ![X: $i] : ((~member(X, B!39)) | apply(compose_function(G!41, F!42, B!39, A!40, B!39), X, X)))) | (~(identity(compose_function(G!41, F!42, B!39, A!40, B!39), B!39) | (~((~member(tptp_fun_X_15(B!39, compose_function(G!41, F!42, B!39, A!40, B!39)), B!39)) | apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_X_15(B!39, compose_function(G!41, F!42, B!39, A!40, B!39)), tptp_fun_X_15(B!39, compose_function(G!41, F!42, B!39, A!40, B!39)))))))))),
% 0.21/0.45 inference(quant_inst,[status(thm)],[])).
% 0.21/0.45 tff(80,plain,
% 0.21/0.45 (~((~((~identity(compose_function(G!41, F!42, B!39, A!40, B!39), B!39)) | ![X: $i] : ((~member(X, B!39)) | apply(compose_function(G!41, F!42, B!39, A!40, B!39), X, X)))) | (~(identity(compose_function(G!41, F!42, B!39, A!40, B!39), B!39) | (~((~member(tptp_fun_X_15(B!39, compose_function(G!41, F!42, B!39, A!40, B!39)), B!39)) | apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_X_15(B!39, compose_function(G!41, F!42, B!39, A!40, B!39)), tptp_fun_X_15(B!39, compose_function(G!41, F!42, B!39, A!40, B!39))))))))),
% 0.21/0.45 inference(unit_resolution,[status(thm)],[79, 78])).
% 0.21/0.45 tff(81,plain,
% 0.21/0.45 (((~((~identity(compose_function(G!41, F!42, B!39, A!40, B!39), B!39)) | ![X: $i] : ((~member(X, B!39)) | apply(compose_function(G!41, F!42, B!39, A!40, B!39), X, X)))) | (~(identity(compose_function(G!41, F!42, B!39, A!40, B!39), B!39) | (~((~member(tptp_fun_X_15(B!39, compose_function(G!41, F!42, B!39, A!40, B!39)), B!39)) | apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_X_15(B!39, compose_function(G!41, F!42, B!39, A!40, B!39)), tptp_fun_X_15(B!39, compose_function(G!41, F!42, B!39, A!40, B!39)))))))) | ((~identity(compose_function(G!41, F!42, B!39, A!40, B!39), B!39)) | ![X: $i] : ((~member(X, B!39)) | apply(compose_function(G!41, F!42, B!39, A!40, B!39), X, X)))),
% 0.21/0.45 inference(tautology,[status(thm)],[])).
% 0.21/0.45 tff(82,plain,
% 0.21/0.45 ((~identity(compose_function(G!41, F!42, B!39, A!40, B!39), B!39)) | ![X: $i] : ((~member(X, B!39)) | apply(compose_function(G!41, F!42, B!39, A!40, B!39), X, X))),
% 0.21/0.45 inference(unit_resolution,[status(thm)],[81, 80])).
% 0.21/0.45 tff(83,plain,
% 0.21/0.45 (maps(G!41, A!40, B!39) & maps(F!42, B!39, A!40) & identity(compose_function(G!41, F!42, B!39, A!40, B!39), B!39)),
% 0.21/0.45 inference(or_elim,[status(thm)],[28])).
% 0.21/0.45 tff(84,plain,
% 0.21/0.45 (identity(compose_function(G!41, F!42, B!39, A!40, B!39), B!39)),
% 0.21/0.45 inference(and_elim,[status(thm)],[83])).
% 0.21/0.45 tff(85,plain,
% 0.21/0.45 ((~((~identity(compose_function(G!41, F!42, B!39, A!40, B!39), B!39)) | ![X: $i] : ((~member(X, B!39)) | apply(compose_function(G!41, F!42, B!39, A!40, B!39), X, X)))) | (~identity(compose_function(G!41, F!42, B!39, A!40, B!39), B!39)) | ![X: $i] : ((~member(X, B!39)) | apply(compose_function(G!41, F!42, B!39, A!40, B!39), X, X))),
% 0.21/0.45 inference(tautology,[status(thm)],[])).
% 0.21/0.45 tff(86,plain,
% 0.21/0.45 ((~((~identity(compose_function(G!41, F!42, B!39, A!40, B!39), B!39)) | ![X: $i] : ((~member(X, B!39)) | apply(compose_function(G!41, F!42, B!39, A!40, B!39), X, X)))) | ![X: $i] : ((~member(X, B!39)) | apply(compose_function(G!41, F!42, B!39, A!40, B!39), X, X))),
% 0.21/0.45 inference(unit_resolution,[status(thm)],[85, 84])).
% 0.21/0.45 tff(87,plain,
% 0.21/0.45 (![X: $i] : ((~member(X, B!39)) | apply(compose_function(G!41, F!42, B!39, A!40, B!39), X, X))),
% 0.21/0.45 inference(unit_resolution,[status(thm)],[86, 82])).
% 0.21/0.45 tff(88,plain,
% 0.21/0.45 (((~![X: $i] : ((~member(X, B!39)) | apply(compose_function(G!41, F!42, B!39, A!40, B!39), X, X))) | ((~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41)))) <=> ((~![X: $i] : ((~member(X, B!39)) | apply(compose_function(G!41, F!42, B!39, A!40, B!39), X, X))) | (~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41)))),
% 0.21/0.45 inference(rewrite,[status(thm)],[])).
% 0.21/0.45 tff(89,plain,
% 0.21/0.45 ((~![X: $i] : ((~member(X, B!39)) | apply(compose_function(G!41, F!42, B!39, A!40, B!39), X, X))) | ((~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41)))),
% 0.21/0.45 inference(quant_inst,[status(thm)],[])).
% 0.21/0.45 tff(90,plain,
% 0.21/0.45 ((~![X: $i] : ((~member(X, B!39)) | apply(compose_function(G!41, F!42, B!39, A!40, B!39), X, X))) | (~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41))),
% 0.21/0.45 inference(modus_ponens,[status(thm)],[89, 88])).
% 0.21/0.45 tff(91,plain,
% 0.21/0.45 (apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41))),
% 0.21/0.45 inference(unit_resolution,[status(thm)],[90, 87, 34])).
% 0.21/0.45 tff(92,plain,
% 0.21/0.45 ((~((~apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41))) | (~((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41))) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41))))))) | (~apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41))) | (~((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41))) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41)))))),
% 0.21/0.46 inference(tautology,[status(thm)],[])).
% 0.21/0.46 tff(93,plain,
% 0.21/0.46 ((~((~apply(compose_function(G!41, F!42, B!39, A!40, B!39), tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41))) | (~((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41))) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41))))))) | (~((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41))) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41)))))),
% 0.21/0.46 inference(unit_resolution,[status(thm)],[92, 91])).
% 0.21/0.46 tff(94,plain,
% 0.21/0.46 (~((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41))) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41))))),
% 0.21/0.46 inference(unit_resolution,[status(thm)],[93, 62])).
% 0.21/0.46 tff(95,plain,
% 0.21/0.46 (((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41))) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41)))) | apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41))),
% 0.21/0.46 inference(tautology,[status(thm)],[])).
% 0.21/0.46 tff(96,plain,
% 0.21/0.46 (apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41))),
% 0.21/0.46 inference(unit_resolution,[status(thm)],[95, 94])).
% 0.21/0.46 tff(97,plain,
% 0.21/0.46 (((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(F!42, tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41))) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41)))) | member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)),
% 0.21/0.46 inference(tautology,[status(thm)],[])).
% 0.21/0.46 tff(98,plain,
% 0.21/0.46 (member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)),
% 0.21/0.46 inference(unit_resolution,[status(thm)],[97, 94])).
% 0.21/0.46 tff(99,plain,
% 0.21/0.46 (((~member(tptp_fun_Y_20(B!39, A!40, G!41), B!39)) | (~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(B!39, A!40, G!41)))))) | ![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(B!39, A!40, G!41))))),
% 0.21/0.46 inference(tautology,[status(thm)],[])).
% 0.21/0.46 tff(100,plain,
% 0.21/0.46 (![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(B!39, A!40, G!41))))),
% 0.21/0.46 inference(unit_resolution,[status(thm)],[99, 32])).
% 0.21/0.46 tff(101,plain,
% 0.21/0.46 (((~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(B!39, A!40, G!41))))) | ((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41))))) <=> ((~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(B!39, A!40, G!41))))) | (~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41))))),
% 0.21/0.46 inference(rewrite,[status(thm)],[])).
% 0.21/0.46 tff(102,plain,
% 0.21/0.46 ((~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(B!39, A!40, G!41))))) | ((~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41))))),
% 0.21/0.46 inference(quant_inst,[status(thm)],[])).
% 0.21/0.46 tff(103,plain,
% 0.21/0.46 ((~![E: $i] : ((~member(E, A!40)) | (~apply(G!41, E, tptp_fun_Y_20(B!39, A!40, G!41))))) | (~member(tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), A!40)) | (~apply(G!41, tptp_fun_Y_8(tptp_fun_Y_20(B!39, A!40, G!41), tptp_fun_Y_20(B!39, A!40, G!41), A!40, F!42, G!41), tptp_fun_Y_20(B!39, A!40, G!41)))),
% 0.21/0.46 inference(modus_ponens,[status(thm)],[102, 101])).
% 0.21/0.46 tff(104,plain,
% 0.21/0.46 ($false),
% 0.21/0.46 inference(unit_resolution,[status(thm)],[103, 100, 98, 96])).
% 0.21/0.46 % SZS output end Proof
% 0.21/0.46 % E exiting
%------------------------------------------------------------------------------