%------------------------------------------------------------------------------ % File : Satallax---3.5 % Problem : NUM641^4 : TPTP v9.2.0. Released v7.1.0. % Transfm : none % Format : tptp:raw % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % Computer : n028.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 : Thu Oct 9 02:11:35 PM UTC 2025 % Result : Theorem 92.27s 90.97s % Output : Proof 92.30s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.05/0.14 % Problem : NUM641^4 : TPTP v9.2.0. Released v7.1.0. % 0.05/0.15 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % 0.14/0.36 % Computer : n028.cluster.edu % 0.14/0.36 % Model : x86_64 x86_64 % 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.14/0.36 % Memory : 8042.1875MB % 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64 % 0.14/0.36 % CPULimit : 300 % 0.14/0.36 % WCLimit : 300 % 0.14/0.36 % DateTime : Wed Oct 8 22:38:08 EDT 2025 % 0.14/0.36 % CPUTime : % 92.27/90.97 % SZS status Theorem % 92.27/90.97 % Mode: mode94:USE_SINE=true:SINE_TOLERANCE=2.0:SINE_GENERALITY_THRESHOLD=4:SINE_RANK_LIMIT=2.:SINE_DEPTH=0 % 92.27/90.97 % Inferences: 6 % 92.27/90.97 % SZS output start Proof % 92.27/90.97 thf(ty_univof, type, univof : ($i>$i)). % 92.27/90.97 thf(ty_power, type, power : ($i>$i)). % 92.27/90.97 thf(ty_eps, type, eps : (($i>$o)>$i)). % 92.27/90.97 thf(ty_union, type, union : ($i>$i)). % 92.27/90.97 thf(ty_d_ReplSep, type, d_ReplSep : ($i>($i>$o)>($i>$i)>$i)). % 92.27/90.97 thf(ty_eigen__0, type, eigen__0 : $i). % 92.27/90.97 thf(ty_famunion, type, famunion : ($i>($i>$i)>$i)). % 92.27/90.97 thf(ty_emptyset, type, emptyset : $i). % 92.27/90.97 thf(ty_ind, type, ind : ($i>($i>$o)>$i)). % 92.27/90.97 thf(ty_pair, type, pair : ($i>$i>$i)). % 92.27/90.97 thf(ty_proj1, type, proj1 : ($i>$i)). % 92.27/90.97 thf(ty_d_24_prop2, type, d_24_prop2 : ($i>$i>$o)). % 92.27/90.97 thf(ty_repl, type, repl : ($i>($i>$i)>$i)). % 92.27/90.97 thf(ty_in, type, in : ($i>$i>$o)). % 92.27/90.97 thf(ty_d_UPair, type, d_UPair : ($i>$i>$i)). % 92.27/90.97 thf(ty_nat_p, type, nat_p : ($i>$o)). % 92.27/90.97 thf(sP1,plain,sP1 <=> (((in @ eigen__0) @ (eps @ (^[X1:$i]:(((~((![X2:$i]:(((in @ X2) @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))))) => (~((X3 = ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((nat_p @ X4) => (~((X5 = X4)))) => (~(((~((nat_p @ X4))) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ 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(~(((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))) => (~((X6 = emptyset)))))))))) => (X5 = emptyset))))) => (~((X4 = ((repl @ (eps @ (^[X5:$i]:(((~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))) => (~((X5 = ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(eps @ (^[X7:$i]:(((nat_p @ X6) => (~((X7 = X6)))) => (~(((~((nat_p @ X6))) => (~((X7 = (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8))))))))))))))))))))))) => (~(((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))) => (~((X5 = emptyset)))))))))) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((~((X5 = emptyset))) => (~((X6 = X5)))) => (~(((X5 = emptyset) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (eps @ (^[X8:$i]:(((~((![X9:$i]:(((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9))))))) => (~((X8 = ((repl @ (univof @ emptyset)) @ (^[X9:$i]:(eps @ (^[X10:$i]:(((nat_p @ X9) => (~((X10 = X9)))) => (~(((~((nat_p @ X9))) => (~((X10 = (eps @ (^[X11:$i]:(~((((in @ X11) @ (univof @ emptyset)) => (~((nat_p @ X11))))))))))))))))))))))) => (~(((![X9:$i]:(((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9))))) => (~((X8 = emptyset)))))))))) => (X7 = emptyset))))))))))))))))))))) => (~(((![X5:$i]:(((in @ X5) @ (eps @ (^[X6:$i]:(((~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))) => (~((X6 = ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(eps @ (^[X8:$i]:(((nat_p @ X7) => (~((X8 = X7)))) => (~(((~((nat_p @ X7))) => (~((X8 = (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9))))))))))))))))))))))) => (~(((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))) => (~((X6 = emptyset)))))))))) => (X5 = emptyset))) => (~((X4 = emptyset)))))))))))))))) => (~((X1 = emptyset)))))))))) @ (d_24_prop2 @ (union @ ((d_UPair @ emptyset) @ ((d_UPair @ emptyset) @ emptyset)))))) @ (^[X1:$i]:(~((![X2:$i]:(~((X1 = ((pair @ eigen__0) @ X2))))))))) @ proj1)),introduced(definition,[new_symbols(definition,[sP8])])). % 92.30/91.02 thf(def_is_of,definition,(is_of = (^[X1:$i]:(^[X2:$i>$o]:(X2 @ X1))))). % 92.30/91.02 thf(def_all_of,definition,(all_of = (^[X1:$i>$o]:(^[X2:$i>$o]:(![X3:$i]:(((is_of @ X3) @ X1) => (X2 @ X3))))))). % 92.30/91.02 thf(def_if,definition,(if = (^[X1:$o]:(^[X2:$i]:(^[X3:$i]:(eps @ (^[X4:$i]:((X1 => (~((X4 = X2)))) => (~(((~(X1)) => (~((X4 = X3)))))))))))))). % 92.30/91.02 thf(def_d_Sing,definition,(d_Sing = (^[X1:$i]:((d_UPair @ X1) @ X1)))). % 92.30/91.02 thf(def_binunion,definition,(binunion = (^[X1:$i]:(^[X2:$i]:(union @ ((d_UPair @ X1) @ X2)))))). % 92.30/91.02 thf(def_d_Sep,definition,(d_Sep = (^[X1:$i]:(^[X2:$i>$o]:(((if @ (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) @ ((repl @ X1) @ (^[X3:$i]:(((if @ (X2 @ X3)) @ X3) @ (eps @ (^[X4:$i]:(~((((in @ X4) @ X1) => (~((X2 @ X4)))))))))))) @ emptyset))))). % 92.30/91.02 thf(def_ordsucc,definition,(ordsucc = (^[X1:$i]:((binunion @ X1) @ (d_Sing @ X1))))). % 92.30/91.02 thf(def_omega,definition,(omega = ((d_Sep @ (univof @ emptyset)) @ nat_p))). % 92.30/91.02 thf(def_d_Sigma,definition,(d_Sigma = (^[X1:$i]:(^[X2:$i>$i]:((famunion @ X1) @ (^[X3:$i]:((repl @ (X2 @ X3)) @ (pair @ X3)))))))). % 92.30/91.02 thf(def_setprod,definition,(setprod = (^[X1:$i]:(^[X2:$i]:((d_Sigma @ X1) @ (^[X3:$i]:X2)))))). % 92.30/91.02 thf(def_ap,definition,(ap = (^[X1:$i]:(^[X2:$i]:(((d_ReplSep @ X1) @ (^[X3:$i]:(~((![X4:$i]:(~((X3 = ((pair @ X2) @ X4))))))))) @ proj1))))). % 92.30/91.02 thf(def_d_Pi,definition,(d_Pi = (^[X1:$i]:(^[X2:$i>$i]:((d_Sep @ (power @ ((d_Sigma @ X1) @ (^[X3:$i]:(union @ (X2 @ X3)))))) @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X1) => ((in @ ((ap @ X3) @ X4)) @ (X2 @ X4)))))))))). % 92.30/91.02 thf(def_d_and,definition,(d_and = (^[X1:$o]:(^[X2:$o]:(d_not @ ((l_ec @ X1) @ X2)))))). % 92.30/91.02 thf(def_e_is,definition,(e_is = (^[X1:$i]:(=)))). % 92.30/91.02 thf(def_one,definition,(one = (^[X1:$i]:(^[X2:$i>$o]:((d_and @ ((amone @ X1) @ X2)) @ ((l_some @ X1) @ X2)))))). % 92.30/91.02 thf(def_e_in,definition,(e_in = (^[X1:$i]:(^[X2:$i>$o]:(^[X3:$i]:X3))))). % 92.30/91.02 thf(def_d_pair,definition,(d_pair = (^[X1:$i]:(^[X2:$i]:pair)))). % 92.30/91.02 thf(def_first,definition,(first = (^[X1:$i]:(^[X2:$i]:proj0)))). % 92.30/91.02 thf(def_second,definition,(second = (^[X1:$i]:(^[X2:$i]:proj1)))). % 92.30/91.02 thf(def_esti,definition,(esti = (^[X1:$i]:in))). % 92.30/91.02 thf(def_nat,definition,(nat = ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))). % 92.30/91.02 thf(def_n_is,definition,(n_is = (e_is @ nat))). % 92.30/91.02 thf(def_nis,definition,(nis = (^[X1:$i]:(^[X2:$i]:(d_not @ ((n_is @ X1) @ X2)))))). % 92.30/91.02 thf(def_n_in,definition,(n_in = (esti @ nat))). % 92.30/91.02 thf(def_n_some,definition,(n_some = (l_some @ nat))). % 92.30/91.02 thf(def_n_all,definition,(n_all = (all @ nat))). % 92.30/91.02 thf(def_n_one,definition,(n_one = (one @ nat))). % 92.30/91.02 thf(def_n_1,definition,(n_1 = (ordsucc @ emptyset))). % 92.30/91.02 thf(def_cond1,definition,(cond1 = (n_in @ n_1))). % 92.30/91.02 thf(def_cond2,definition,(cond2 = (^[X1:$i]:(n_all @ (^[X2:$i]:((imp @ ((n_in @ X2) @ X1)) @ ((n_in @ (ordsucc @ X2)) @ X1))))))). % 92.30/91.02 thf(def_plus,definition,(plus = (^[X1:$i]:((ind @ ((d_Pi @ nat) @ (^[X2:$i]:nat))) @ (d_24_prop2 @ X1))))). % 92.30/91.02 thf(def_n_pl,definition,(n_pl = (^[X1:$i]:(ap @ (plus @ X1))))). % 92.30/91.02 thf(satz4g,conjecture,(![X1:$i]:(((in @ X1) @ (eps @ (^[X2:$i]:(((~((![X3:$i]:(((in @ X3) @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((nat_p @ X5) => (~((X6 = X5)))) => (~(((~((nat_p @ X5))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))))))))))))))))))) => (~(((~((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))))))) => (~((X4 = emptyset)))))))))) => (~((~((X3 = emptyset))))))))) => (~((X2 = ((repl @ (eps @ (^[X3:$i]:(((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))))) => (~((X3 = ((repl @ (univof @ emptyset)) @ (^[X4:$i]:(eps @ (^[X5:$i]:(((nat_p @ X4) => (~((X5 = X4)))) => (~(((~((nat_p @ X4))) => (~((X5 = (eps @ (^[X6:$i]:(~((((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))))))))))))))))))) => (~(((~((~((![X4:$i]:(((in @ X4) @ (univof @ emptyset)) => (~((nat_p @ X4))))))))) => (~((X3 = emptyset)))))))))) @ (^[X3:$i]:(eps @ (^[X4:$i]:(((~((X3 = emptyset))) => (~((X4 = X3)))) => (~(((~((~((X3 = emptyset))))) => (~((X4 = (eps @ (^[X5:$i]:(~((((in @ X5) @ (eps @ (^[X6:$i]:(((~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))) => (~((X6 = ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(eps @ (^[X8:$i]:(((nat_p @ X7) => (~((X8 = X7)))) => (~(((~((nat_p @ X7))) => (~((X8 = (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9))))))))))))))))))))))) => (~(((~((~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))))) => (~((X6 = emptyset)))))))))) => (~((~((X5 = emptyset))))))))))))))))))))))))) => (~(((~((~((![X3:$i]:(((in @ X3) @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))))) => (~((X4 = ((repl @ (univof @ emptyset)) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((nat_p @ X5) => (~((X6 = X5)))) => (~(((~((nat_p @ X5))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))))))))))))))))))) => (~(((~((~((![X5:$i]:(((in @ X5) @ (univof @ emptyset)) => (~((nat_p @ X5))))))))) => (~((X4 = emptyset)))))))))) => (~((~((X3 = emptyset))))))))))) => (~((X2 = emptyset)))))))))) => ((union @ ((d_UPair @ X1) @ ((d_UPair @ X1) @ X1))) = (((d_ReplSep @ ((ind @ (eps @ (^[X2:$i]:(((~((![X3:$i]:(((in @ X3) @ (power @ ((famunion @ (eps @ (^[X4:$i]:(((~((![X5:$i]:(((in @ X5) @ (eps @ (^[X6:$i]:(((~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))) => (~((X6 = ((repl @ (univof @ emptyset)) @ (^[X7:$i]:(eps @ (^[X8:$i]:(((nat_p @ X7) => (~((X8 = X7)))) => (~(((~((nat_p @ X7))) => (~((X8 = (eps @ (^[X9:$i]:(~((((in @ X9) @ (univof @ emptyset)) => (~((nat_p @ X9))))))))))))))))))))))) => (~(((~((~((![X7:$i]:(((in @ X7) @ (univof @ emptyset)) => (~((nat_p @ X7))))))))) => (~((X6 = emptyset)))))))))) => (~((~((X5 = emptyset))))))))) => (~((X4 = ((repl @ (eps @ (^[X5:$i]:(((~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))) => (~((X5 = ((repl @ (univof @ emptyset)) @ (^[X6:$i]:(eps @ (^[X7:$i]:(((nat_p @ X6) => (~((X7 = X6)))) => (~(((~((nat_p @ X6))) => (~((X7 = (eps @ (^[X8:$i]:(~((((in @ X8) @ (univof @ emptyset)) => (~((nat_p @ X8))))))))))))))))))))))) => (~(((~((~((![X6:$i]:(((in @ X6) @ (univof @ emptyset)) => (~((nat_p @ X6))))))))) => (~((X5 = emptyset)))))))))) @ (^[X5:$i]:(eps @ (^[X6:$i]:(((~((X5 = emptyset))) => (~((X6 = X5)))) => (~(((~((~((X5 = emptyset))))) => (~((X6 = (eps @ (^[X7:$i]:(~((((in @ X7) @ (eps @ (^[X8:$i]:(((~((![X9:$i]:(((in @ X9) @ (univof @ 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