%------------------------------------------------------------------------------ % File : Satallax---3.5 % Problem : NUM695^4 : 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 : n024.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 02:11:50 PM UTC 2025 % Result : Theorem 38.62s 38.80s % Output : Proof 38.62s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.07/0.13 % Problem : NUM695^4 : TPTP v9.2.0. Released v7.1.0. % 0.07/0.13 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % 0.12/0.35 % Computer : n024.cluster.edu % 0.12/0.35 % Model : x86_64 x86_64 % 0.12/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.12/0.35 % Memory : 8042.1875MB % 0.12/0.35 % OS : Linux 3.10.0-693.el7.x86_64 % 0.12/0.35 % CPULimit : 300 % 0.12/0.35 % WCLimit : 300 % 0.12/0.35 % DateTime : Wed Oct 8 22:20:23 EDT 2025 % 0.12/0.35 % CPUTime : % 38.62/38.80 % SZS status Theorem % 38.62/38.80 % Mode: mode478:USE_SINE=true:SINE_TOLERANCE=2.0:SINE_GENERALITY_THRESHOLD=16:SINE_RANK_LIMIT=1.:SINE_DEPTH=0 % 38.62/38.80 % Inferences: 1192 % 38.62/38.80 % SZS output start Proof % 38.62/38.80 thf(ty_is_of, type, is_of : ($i>($i>$o)>$o)). % 38.62/38.80 thf(ty_eigen__0, type, eigen__0 : $i). % 38.62/38.80 thf(ty_d_Sep, type, d_Sep : ($i>($i>$o)>$i)). % 38.62/38.80 thf(ty_emptyset, type, emptyset : $i). % 38.62/38.80 thf(ty_lessis, type, lessis : ($i>$i>$o)). % 38.62/38.80 thf(ty_n_some, type, n_some : (($i>$o)>$o)). % 38.62/38.80 thf(ty_diffprop, type, diffprop : ($i>$i>$i>$o)). % 38.62/38.80 thf(ty_omega, type, omega : $i). % 38.62/38.80 thf(ty_in, type, in : ($i>$i>$o)). % 38.62/38.80 thf(ty_d_Sing, type, d_Sing : ($i>$i)). % 38.62/38.80 thf(ty_iii, type, iii : ($i>$i>$o)). % 38.62/38.80 thf(ty_binunion, type, binunion : ($i>$i>$i)). % 38.62/38.80 thf(sP1,plain,sP1 <=> (n_some @ ((diffprop @ ((binunion @ eigen__0) @ (d_Sing @ eigen__0))) @ ((binunion @ emptyset) @ (d_Sing @ emptyset)))),introduced(definition,[new_symbols(definition,[sP1])])). % 38.62/38.80 thf(sP2,plain,sP2 <=> (![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (![X2:$i]:(((is_of @ X2) @ (^[X3:$i]:((in @ X3) @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))))) => (((iii @ X1) @ X2) => (n_some @ ((diffprop @ X2) @ X1))))))),introduced(definition,[new_symbols(definition,[sP2])])). % 38.62/38.80 thf(sP3,plain,sP3 <=> (![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => ((is_of @ ((binunion @ X1) @ (d_Sing @ X1))) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))))),introduced(definition,[new_symbols(definition,[sP3])])). % 38.62/38.80 thf(sP4,plain,sP4 <=> (((iii @ eigen__0) @ ((binunion @ eigen__0) @ (d_Sing @ eigen__0))) => ((iii @ ((binunion @ emptyset) @ (d_Sing @ emptyset))) @ ((binunion @ eigen__0) @ (d_Sing @ eigen__0)))),introduced(definition,[new_symbols(definition,[sP4])])). % 38.62/38.80 thf(sP5,plain,sP5 <=> (((is_of @ ((binunion @ eigen__0) @ (d_Sing @ eigen__0))) @ (^[X1:$i]:((in @ X1) @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) => (((iii @ ((binunion @ emptyset) @ (d_Sing @ emptyset))) @ ((binunion @ eigen__0) @ (d_Sing @ eigen__0))) => sP1)),introduced(definition,[new_symbols(definition,[sP5])])). % 38.62/38.80 thf(sP6,plain,sP6 <=> (![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (![X2:$i]:(((is_of @ X2) @ (^[X3:$i]:((in @ X3) @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))))) => (![X3:$i]:(((is_of @ X3) @ (^[X4:$i]:((in @ X4) @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))))) => (((lessis @ X1) @ X2) => (((iii @ X2) @ X3) => ((iii @ X1) @ X3))))))))),introduced(definition,[new_symbols(definition,[sP6])])). % 38.62/38.80 thf(sP7,plain,sP7 <=> (![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (((lessis @ ((binunion @ emptyset) @ (d_Sing @ emptyset))) @ eigen__0) => (((iii @ eigen__0) @ X1) => ((iii @ ((binunion @ emptyset) @ (d_Sing @ emptyset))) @ X1))))),introduced(definition,[new_symbols(definition,[sP7])])). % 38.62/38.80 thf(sP8,plain,sP8 <=> (((is_of @ eigen__0) @ (^[X1:$i]:((in @ X1) @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) => ((is_of @ ((binunion @ eigen__0) @ (d_Sing @ eigen__0))) @ (^[X1:$i]:((in @ X1) @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset))))))))),introduced(definition,[new_symbols(definition,[sP8])])). % 38.62/38.80 thf(sP9,plain,sP9 <=> (((lessis @ ((binunion @ emptyset) @ (d_Sing @ emptyset))) @ eigen__0) => sP4),introduced(definition,[new_symbols(definition,[sP9])])). % 38.62/38.80 thf(sP10,plain,sP10 <=> (((is_of @ ((binunion @ emptyset) @ (d_Sing @ emptyset))) @ (^[X1:$i]:((in @ X1) @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) => (![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (((iii @ ((binunion @ emptyset) @ (d_Sing @ emptyset))) @ X1) => (n_some @ ((diffprop @ X1) @ ((binunion @ emptyset) @ (d_Sing @ emptyset)))))))),introduced(definition,[new_symbols(definition,[sP10])])). % 38.62/38.80 thf(sP11,plain,sP11 <=> (((is_of @ ((binunion @ eigen__0) @ (d_Sing @ eigen__0))) @ (^[X1:$i]:((in @ X1) @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) => sP9),introduced(definition,[new_symbols(definition,[sP11])])). % 38.62/38.80 thf(sP12,plain,sP12 <=> ((iii @ eigen__0) @ ((binunion @ eigen__0) @ (d_Sing @ eigen__0))),introduced(definition,[new_symbols(definition,[sP12])])). % 38.62/38.80 thf(sP13,plain,sP13 <=> ((is_of @ eigen__0) @ (^[X1:$i]:((in @ X1) @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))),introduced(definition,[new_symbols(definition,[sP13])])). % 38.62/38.80 thf(sP14,plain,sP14 <=> (sP13 => sP7),introduced(definition,[new_symbols(definition,[sP14])])). % 38.62/38.80 thf(sP15,plain,sP15 <=> (![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (![X2:$i]:(((is_of @ X2) @ (^[X3:$i]:((in @ X3) @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))))) => (((lessis @ ((binunion @ emptyset) @ (d_Sing @ emptyset))) @ X1) => (((iii @ X1) @ X2) => ((iii @ ((binunion @ emptyset) @ (d_Sing @ emptyset))) @ X2))))))),introduced(definition,[new_symbols(definition,[sP15])])). % 38.62/38.80 thf(sP16,plain,sP16 <=> (![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => ((iii @ X1) @ ((binunion @ X1) @ (d_Sing @ X1))))),introduced(definition,[new_symbols(definition,[sP16])])). % 38.62/38.80 thf(sP17,plain,sP17 <=> (![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (((iii @ ((binunion @ emptyset) @ (d_Sing @ emptyset))) @ X1) => (n_some @ ((diffprop @ X1) @ ((binunion @ emptyset) @ (d_Sing @ emptyset))))))),introduced(definition,[new_symbols(definition,[sP17])])). % 38.62/38.80 thf(sP18,plain,sP18 <=> (sP13 => sP12),introduced(definition,[new_symbols(definition,[sP18])])). % 38.62/38.80 thf(sP19,plain,sP19 <=> ((is_of @ ((binunion @ emptyset) @ (d_Sing @ emptyset))) @ (^[X1:$i]:((in @ X1) @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))),introduced(definition,[new_symbols(definition,[sP19])])). % 38.62/38.80 thf(sP20,plain,sP20 <=> ((iii @ ((binunion @ emptyset) @ (d_Sing @ emptyset))) @ ((binunion @ eigen__0) @ (d_Sing @ eigen__0))),introduced(definition,[new_symbols(definition,[sP20])])). % 38.62/38.80 thf(sP21,plain,sP21 <=> (![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => ((lessis @ ((binunion @ emptyset) @ (d_Sing @ emptyset))) @ X1))),introduced(definition,[new_symbols(definition,[sP21])])). % 38.62/38.80 thf(sP22,plain,sP22 <=> (sP13 => ((lessis @ ((binunion @ emptyset) @ (d_Sing @ emptyset))) @ eigen__0)),introduced(definition,[new_symbols(definition,[sP22])])). % 38.62/38.80 thf(sP23,plain,sP23 <=> (sP20 => sP1),introduced(definition,[new_symbols(definition,[sP23])])). % 38.62/38.80 thf(sP24,plain,sP24 <=> (sP19 => sP15),introduced(definition,[new_symbols(definition,[sP24])])). % 38.62/38.80 thf(sP25,plain,sP25 <=> ((lessis @ ((binunion @ emptyset) @ (d_Sing @ emptyset))) @ eigen__0),introduced(definition,[new_symbols(definition,[sP25])])). % 38.62/38.80 thf(sP26,plain,sP26 <=> ((is_of @ ((binunion @ eigen__0) @ (d_Sing @ eigen__0))) @ (^[X1:$i]:((in @ X1) @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))),introduced(definition,[new_symbols(definition,[sP26])])). % 38.62/38.80 thf(def_all_of,definition,(all_of = (^[X1:$i>$o]:(^[X2:$i>$o]:(![X3:$i]:(((is_of @ X3) @ X1) => (X2 @ X3))))))). % 38.62/38.80 thf(def_ordsucc,definition,(ordsucc = (^[X1:$i]:((binunion @ X1) @ (d_Sing @ X1))))). % 38.62/38.80 thf(def_nat,definition,(nat = ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))). % 38.62/38.80 thf(def_n_1,definition,(n_1 = (ordsucc @ emptyset))). % 38.62/38.80 thf(def_d_29_ii,definition,(d_29_ii = (^[X1:$i]:(^[X2:$i]:(n_some @ ((diffprop @ X1) @ X2)))))). % 38.62/38.80 thf(satz24b,conjecture,(![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (n_some @ ((diffprop @ ((binunion @ X1) @ (d_Sing @ X1))) @ ((binunion @ emptyset) @ (d_Sing @ emptyset))))))). % 38.62/38.80 thf(h0,negated_conjecture,(~((![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (n_some @ ((diffprop @ ((binunion @ X1) @ (d_Sing @ X1))) @ ((binunion @ emptyset) @ (d_Sing @ emptyset)))))))),inference(assume_negation,[status(cth)],[satz24b])). % 38.62/38.80 thf(h1,assumption,(~((sP13 => sP1))),introduced(assumption,[])). % 38.62/38.80 thf(h2,assumption,sP13,introduced(assumption,[])). % 38.62/38.80 thf(h3,assumption,(~(sP1)),introduced(assumption,[])). % 38.62/38.80 thf(1,plain,(~(sP7) | sP11),inference(all_rule,[status(thm)],[])). % 38.62/38.80 thf(2,plain,((~(sP11) | ~(sP26)) | sP9),inference(prop_rule,[status(thm)],[])). % 38.62/38.80 thf(3,plain,((~(sP9) | ~(sP25)) | sP4),inference(prop_rule,[status(thm)],[])). % 38.62/38.80 thf(4,plain,((~(sP4) | ~(sP12)) | sP20),inference(prop_rule,[status(thm)],[])). % 38.62/38.80 thf(5,plain,(~(sP15) | sP14),inference(all_rule,[status(thm)],[])). % 38.62/38.80 thf(6,plain,((~(sP14) | ~(sP13)) | sP7),inference(prop_rule,[status(thm)],[])). % 38.62/38.80 thf(7,plain,((~(sP8) | ~(sP13)) | sP26),inference(prop_rule,[status(thm)],[])). % 38.62/38.80 thf(8,plain,((~(sP18) | ~(sP13)) | sP12),inference(prop_rule,[status(thm)],[])). % 38.62/38.80 thf(9,plain,((~(sP22) | ~(sP13)) | sP25),inference(prop_rule,[status(thm)],[])). % 38.62/38.80 thf(10,plain,(~(sP17) | sP5),inference(all_rule,[status(thm)],[])). % 38.62/38.80 thf(11,plain,((~(sP5) | ~(sP26)) | sP23),inference(prop_rule,[status(thm)],[])). % 38.62/38.80 thf(12,plain,((~(sP23) | ~(sP20)) | sP1),inference(prop_rule,[status(thm)],[])). % 38.62/38.80 thf(13,plain,(~(sP21) | sP22),inference(all_rule,[status(thm)],[])). % 38.62/38.80 thf(14,plain,(~(sP16) | sP18),inference(all_rule,[status(thm)],[])). % 38.62/38.80 thf(15,plain,(~(sP6) | sP24),inference(all_rule,[status(thm)],[])). % 38.62/38.80 thf(16,plain,((~(sP24) | ~(sP19)) | sP15),inference(prop_rule,[status(thm)],[])). % 38.62/38.80 thf(17,plain,(~(sP2) | sP10),inference(all_rule,[status(thm)],[])). % 38.62/38.80 thf(18,plain,((~(sP10) | ~(sP19)) | sP17),inference(prop_rule,[status(thm)],[])). % 38.62/38.80 thf(19,plain,(~(sP3) | sP8),inference(all_rule,[status(thm)],[])). % 38.62/38.80 thf(satz24a,axiom,((all_of @ (^[X1:$i]:((in @ X1) @ nat))) @ (lessis @ n_1))). % 38.62/38.80 thf(20,plain,sP21,inference(preprocess,[status(thm)],[satz24a]). % 38.62/38.80 thf(satz18c,axiom,((all_of @ (^[X1:$i]:((in @ X1) @ nat))) @ (^[X1:$i]:((iii @ X1) @ (ordsucc @ X1))))). % 38.62/38.80 thf(21,plain,sP16,inference(preprocess,[status(thm)],[satz18c]). % 38.62/38.80 thf(satz16a,axiom,((all_of @ (^[X1:$i]:((in @ X1) @ nat))) @ (^[X1:$i]:((all_of @ (^[X2:$i]:((in @ X2) @ nat))) @ (^[X2:$i]:((all_of @ (^[X3:$i]:((in @ X3) @ nat))) @ (^[X3:$i]:(((lessis @ X1) @ X2) => (((iii @ X2) @ X3) => ((iii @ X1) @ X3)))))))))). % 38.62/38.80 thf(22,plain,sP6,inference(preprocess,[status(thm)],[satz16a]). % 38.62/38.80 thf(satz12,axiom,((all_of @ (^[X1:$i]:((in @ X1) @ nat))) @ (^[X1:$i]:((all_of @ (^[X2:$i]:((in @ X2) @ nat))) @ (^[X2:$i]:(((iii @ X1) @ X2) => ((d_29_ii @ X2) @ X1))))))). % 38.62/38.80 thf(23,plain,sP2,inference(preprocess,[status(thm)],[satz12]). % 38.62/38.80 thf(suc_p,axiom,((all_of @ (^[X1:$i]:((in @ X1) @ nat))) @ (^[X1:$i]:((is_of @ (ordsucc @ X1)) @ (^[X2:$i]:((in @ X2) @ nat)))))). % 38.62/38.80 thf(24,plain,sP3,inference(preprocess,[status(thm)],[suc_p]). % 38.62/38.80 thf(n_1_p,axiom,((is_of @ n_1) @ (^[X1:$i]:((in @ X1) @ nat)))). % 38.62/38.80 thf(25,plain,sP19,inference(preprocess,[status(thm)],[n_1_p]). % 38.62/38.80 thf(26,plain,$false,inference(prop_unsat,[status(thm),assumptions([h2,h3,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,h2,h3,20,21,22,23,24,25])). % 38.62/38.80 thf(27,plain,$false,inference(tab_negimp,[status(thm),assumptions([h1,h0]),tab_negimp(discharge,[h2,h3])],[h1,26,h2,h3])). % 38.62/38.80 thf(28,plain,$false,inference(tab_negall,[status(thm),assumptions([h0]),tab_negall(discharge,[h1]),tab_negall(eigenvar,eigen__0)],[h0,27,h1])). % 38.62/38.80 thf(0,theorem,(![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (n_some @ ((diffprop @ ((binunion @ X1) @ (d_Sing @ X1))) @ ((binunion @ emptyset) @ (d_Sing @ emptyset)))))),inference(contra,[status(thm),contra(discharge,[h0])],[28,h0])). % 38.62/38.80 % SZS output end Proof %------------------------------------------------------------------------------