%------------------------------------------------------------------------------ % File : Satallax---3.5 % Problem : NUM703^4 : 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 : n023.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 02:11:53 PM UTC 2025 % Result : Theorem 11.39s 11.60s % Output : Proof 11.39s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.08/0.13 % Problem : NUM703^4 : TPTP v9.2.0. Released v7.1.0. % 0.08/0.13 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % 0.13/0.35 % Computer : n023.cluster.edu % 0.13/0.35 % Model : x86_64 x86_64 % 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.13/0.35 % Memory : 8042.1875MB % 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64 % 0.13/0.35 % CPULimit : 300 % 0.13/0.35 % WCLimit : 300 % 0.13/0.35 % DateTime : Wed Oct 8 22:28:53 EDT 2025 % 0.13/0.35 % CPUTime : % 11.39/11.60 % SZS status Theorem % 11.39/11.60 % Mode: mode507:USE_SINE=true:SINE_TOLERANCE=3.0:SINE_GENERALITY_THRESHOLD=0:SINE_RANK_LIMIT=1.:SINE_DEPTH=1 % 11.39/11.60 % Inferences: 274 % 11.39/11.60 % SZS output start Proof % 11.39/11.60 thf(ty_is_of, type, is_of : ($i>($i>$o)>$o)). % 11.39/11.60 thf(ty_eigen__1, type, eigen__1 : $i). % 11.39/11.60 thf(ty_eigen__0, type, eigen__0 : $i). % 11.39/11.60 thf(ty_d_Sep, type, d_Sep : ($i>($i>$o)>$i)). % 11.39/11.60 thf(ty_l_or, type, l_or : ($o>$o>$o)). % 11.39/11.60 thf(ty_emptyset, type, emptyset : $i). % 11.39/11.60 thf(ty_lessis, type, lessis : ($i>$i>$o)). % 11.39/11.60 thf(ty_n_some, type, n_some : (($i>$o)>$o)). % 11.39/11.60 thf(ty_diffprop, type, diffprop : ($i>$i>$i>$o)). % 11.39/11.60 thf(ty_omega, type, omega : $i). % 11.39/11.60 thf(ty_n_is, type, n_is : ($i>$i>$o)). % 11.39/11.60 thf(ty_in, type, in : ($i>$i>$o)). % 11.39/11.60 thf(ty_d_Sing, type, d_Sing : ($i>$i)). % 11.39/11.60 thf(ty_iii, type, iii : ($i>$i>$o)). % 11.39/11.60 thf(ty_binunion, type, binunion : ($i>$i>$i)). % 11.39/11.60 thf(sP1,plain,sP1 <=> ((n_some @ ((diffprop @ ((binunion @ eigen__1) @ (d_Sing @ eigen__1))) @ eigen__0)) => ((iii @ eigen__0) @ ((binunion @ eigen__1) @ (d_Sing @ eigen__1)))),introduced(definition,[new_symbols(definition,[sP1])])). % 11.39/11.60 thf(sP2,plain,sP2 <=> (((is_of @ eigen__0) @ (^[X1:$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)))))))) => (((lessis @ eigen__0) @ X1) => ((l_or @ (n_some @ ((diffprop @ X1) @ eigen__0))) @ ((n_is @ X1) @ eigen__0)))))),introduced(definition,[new_symbols(definition,[sP2])])). % 11.39/11.60 thf(sP3,plain,sP3 <=> (n_some @ ((diffprop @ ((binunion @ eigen__1) @ (d_Sing @ eigen__1))) @ eigen__0)),introduced(definition,[new_symbols(definition,[sP3])])). % 11.39/11.60 thf(sP4,plain,sP4 <=> ((is_of @ eigen__1) @ (^[X1:$i]:((in @ X1) @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))),introduced(definition,[new_symbols(definition,[sP4])])). % 11.39/11.60 thf(sP5,plain,sP5 <=> ((lessis @ eigen__0) @ eigen__1),introduced(definition,[new_symbols(definition,[sP5])])). % 11.39/11.60 thf(sP6,plain,sP6 <=> (sP4 => ((is_of @ ((binunion @ eigen__1) @ (d_Sing @ eigen__1))) @ (^[X1:$i]:((in @ X1) @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset))))))))),introduced(definition,[new_symbols(definition,[sP6])])). % 11.39/11.60 thf(sP7,plain,sP7 <=> (![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:$i]:(~((X3 = emptyset)))))))))),introduced(definition,[new_symbols(definition,[sP7])])). % 11.39/11.60 thf(sP8,plain,sP8 <=> (![X1:$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:$i]:(~((X4 = emptyset)))))))) => (((iii @ X2) @ ((binunion @ X1) @ (d_Sing @ X1))) => ((lessis @ X2) @ X1)))))),introduced(definition,[new_symbols(definition,[sP8])])). % 11.39/11.60 thf(sP9,plain,sP9 <=> ((is_of @ ((binunion @ eigen__1) @ (d_Sing @ eigen__1))) @ (^[X1:$i]:((in @ X1) @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))),introduced(definition,[new_symbols(definition,[sP9])])). % 11.39/11.60 thf(sP10,plain,sP10 <=> (![X1:$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:$i]:(~((X4 = emptyset)))))))) => (((lessis @ X1) @ X2) => ((l_or @ (n_some @ ((diffprop @ X2) @ X1))) @ ((n_is @ X2) @ X1))))))),introduced(definition,[new_symbols(definition,[sP10])])). % 11.39/11.60 thf(sP11,plain,sP11 <=> (sP5 => ((l_or @ (n_some @ ((diffprop @ eigen__1) @ eigen__0))) @ ((n_is @ eigen__1) @ eigen__0))),introduced(definition,[new_symbols(definition,[sP11])])). % 11.39/11.60 thf(sP12,plain,sP12 <=> (sP9 => (![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => ((n_some @ ((diffprop @ ((binunion @ eigen__1) @ (d_Sing @ eigen__1))) @ X1)) => ((iii @ X1) @ ((binunion @ eigen__1) @ (d_Sing @ eigen__1))))))),introduced(definition,[new_symbols(definition,[sP12])])). % 11.39/11.60 thf(sP13,plain,sP13 <=> (![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => ((n_some @ ((diffprop @ ((binunion @ eigen__1) @ (d_Sing @ eigen__1))) @ X1)) => ((iii @ X1) @ ((binunion @ eigen__1) @ (d_Sing @ eigen__1)))))),introduced(definition,[new_symbols(definition,[sP13])])). % 11.39/11.60 thf(sP14,plain,sP14 <=> (sP4 => sP11),introduced(definition,[new_symbols(definition,[sP14])])). % 11.39/11.60 thf(sP15,plain,sP15 <=> (![X1:$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:$i]:(~((X4 = emptyset)))))))) => ((n_some @ ((diffprop @ X1) @ X2)) => ((iii @ X2) @ X1)))))),introduced(definition,[new_symbols(definition,[sP15])])). % 11.39/11.60 thf(sP16,plain,sP16 <=> ((l_or @ (n_some @ ((diffprop @ eigen__1) @ eigen__0))) @ ((n_is @ eigen__1) @ eigen__0)),introduced(definition,[new_symbols(definition,[sP16])])). % 11.39/11.60 thf(sP17,plain,sP17 <=> (sP4 => (![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (((iii @ X1) @ ((binunion @ eigen__1) @ (d_Sing @ eigen__1))) => ((lessis @ X1) @ eigen__1))))),introduced(definition,[new_symbols(definition,[sP17])])). % 11.39/11.60 thf(sP18,plain,sP18 <=> (![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (((lessis @ eigen__0) @ X1) => ((l_or @ (n_some @ ((diffprop @ X1) @ eigen__0))) @ ((n_is @ X1) @ eigen__0))))),introduced(definition,[new_symbols(definition,[sP18])])). % 11.39/11.60 thf(sP19,plain,sP19 <=> (((is_of @ eigen__0) @ (^[X1:$i]:((in @ X1) @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) => (((iii @ eigen__0) @ ((binunion @ eigen__1) @ (d_Sing @ eigen__1))) => sP5)),introduced(definition,[new_symbols(definition,[sP19])])). % 11.39/11.60 thf(sP20,plain,sP20 <=> ((is_of @ eigen__0) @ (^[X1:$i]:((in @ X1) @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))),introduced(definition,[new_symbols(definition,[sP20])])). % 11.39/11.60 thf(sP21,plain,sP21 <=> (((iii @ eigen__0) @ ((binunion @ eigen__1) @ (d_Sing @ eigen__1))) => sP5),introduced(definition,[new_symbols(definition,[sP21])])). % 11.39/11.60 thf(sP22,plain,sP22 <=> (sP20 => sP1),introduced(definition,[new_symbols(definition,[sP22])])). % 11.39/11.60 thf(sP23,plain,sP23 <=> (![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (((iii @ X1) @ ((binunion @ eigen__1) @ (d_Sing @ eigen__1))) => ((lessis @ X1) @ eigen__1)))),introduced(definition,[new_symbols(definition,[sP23])])). % 11.39/11.60 thf(sP24,plain,sP24 <=> ((iii @ eigen__0) @ ((binunion @ eigen__1) @ (d_Sing @ eigen__1))),introduced(definition,[new_symbols(definition,[sP24])])). % 11.39/11.60 thf(def_all_of,definition,(all_of = (^[X1:$i>$o]:(^[X2:$i>$o]:(![X3:$i]:(((is_of @ X3) @ X1) => (X2 @ X3))))))). % 11.39/11.60 thf(def_ordsucc,definition,(ordsucc = (^[X1:$i]:((binunion @ X1) @ (d_Sing @ X1))))). % 11.39/11.60 thf(def_nat,definition,(nat = ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))). % 11.39/11.60 thf(def_d_29_ii,definition,(d_29_ii = (^[X1:$i]:(^[X2:$i]:(n_some @ ((diffprop @ X1) @ X2)))))). % 11.39/11.60 thf(def_moreis,definition,(moreis = (^[X1:$i]:(^[X2:$i]:((l_or @ ((d_29_ii @ X1) @ X2)) @ ((n_is @ X1) @ X2)))))). % 11.39/11.60 thf(satz26c,conjecture,(![X1:$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:$i]:(~((X4 = emptyset)))))))) => ((n_some @ ((diffprop @ ((binunion @ X2) @ (d_Sing @ X2))) @ X1)) => ((l_or @ (n_some @ ((diffprop @ X2) @ X1))) @ ((n_is @ X2) @ X1)))))))). % 11.39/11.60 thf(h0,negated_conjecture,(~((![X1:$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:$i]:(~((X4 = emptyset)))))))) => ((n_some @ ((diffprop @ ((binunion @ X2) @ (d_Sing @ X2))) @ X1)) => ((l_or @ (n_some @ ((diffprop @ X2) @ X1))) @ ((n_is @ X2) @ X1))))))))),inference(assume_negation,[status(cth)],[satz26c])). % 11.39/11.60 thf(h1,assumption,(~((sP20 => (![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => ((n_some @ ((diffprop @ ((binunion @ X1) @ (d_Sing @ X1))) @ eigen__0)) => ((l_or @ (n_some @ ((diffprop @ X1) @ eigen__0))) @ ((n_is @ X1) @ eigen__0)))))))),introduced(assumption,[])). % 11.39/11.60 thf(h2,assumption,sP20,introduced(assumption,[])). % 11.39/11.60 thf(h3,assumption,(~((![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => ((n_some @ ((diffprop @ ((binunion @ X1) @ (d_Sing @ X1))) @ eigen__0)) => ((l_or @ (n_some @ ((diffprop @ X1) @ eigen__0))) @ ((n_is @ X1) @ eigen__0))))))),introduced(assumption,[])). % 11.39/11.60 thf(h4,assumption,(~((sP4 => (sP3 => sP16)))),introduced(assumption,[])). % 11.39/11.60 thf(h5,assumption,sP4,introduced(assumption,[])). % 11.39/11.60 thf(h6,assumption,(~((sP3 => sP16))),introduced(assumption,[])). % 11.39/11.60 thf(h7,assumption,sP3,introduced(assumption,[])). % 11.39/11.60 thf(h8,assumption,(~(sP16)),introduced(assumption,[])). % 11.39/11.60 thf(1,plain,((~(sP17) | ~(sP4)) | sP23),inference(prop_rule,[status(thm)],[])). % 11.39/11.60 thf(2,plain,(~(sP23) | sP19),inference(all_rule,[status(thm)],[])). % 11.39/11.60 thf(3,plain,((~(sP19) | ~(sP20)) | sP21),inference(prop_rule,[status(thm)],[])). % 11.39/11.60 thf(4,plain,((~(sP21) | ~(sP24)) | sP5),inference(prop_rule,[status(thm)],[])). % 11.39/11.60 thf(5,plain,(~(sP7) | sP6),inference(all_rule,[status(thm)],[])). % 11.39/11.60 thf(6,plain,((~(sP6) | ~(sP4)) | sP9),inference(prop_rule,[status(thm)],[])). % 11.39/11.60 thf(7,plain,(~(sP15) | sP12),inference(all_rule,[status(thm)],[])). % 11.39/11.60 thf(8,plain,((~(sP12) | ~(sP9)) | sP13),inference(prop_rule,[status(thm)],[])). % 11.39/11.60 thf(9,plain,(~(sP13) | sP22),inference(all_rule,[status(thm)],[])). % 11.39/11.60 thf(10,plain,((~(sP22) | ~(sP20)) | sP1),inference(prop_rule,[status(thm)],[])). % 11.39/11.60 thf(11,plain,((~(sP1) | ~(sP3)) | sP24),inference(prop_rule,[status(thm)],[])). % 11.39/11.60 thf(12,plain,(~(sP10) | sP2),inference(all_rule,[status(thm)],[])). % 11.39/11.60 thf(13,plain,((~(sP2) | ~(sP20)) | sP18),inference(prop_rule,[status(thm)],[])). % 11.39/11.60 thf(14,plain,(~(sP18) | sP14),inference(all_rule,[status(thm)],[])). % 11.39/11.60 thf(15,plain,((~(sP14) | ~(sP4)) | sP11),inference(prop_rule,[status(thm)],[])). % 11.39/11.60 thf(16,plain,((~(sP11) | ~(sP5)) | sP16),inference(prop_rule,[status(thm)],[])). % 11.39/11.60 thf(17,plain,(~(sP8) | sP17),inference(all_rule,[status(thm)],[])). % 11.39/11.60 thf(satz26a,axiom,((all_of @ (^[X1:$i]:((in @ X1) @ nat))) @ (^[X1:$i]:((all_of @ (^[X2:$i]:((in @ X2) @ nat))) @ (^[X2:$i]:(((iii @ X2) @ (ordsucc @ X1)) => ((lessis @ X2) @ X1))))))). % 11.39/11.60 thf(18,plain,sP8,inference(preprocess,[status(thm)],[satz26a]). % 11.39/11.60 thf(satz14,axiom,((all_of @ (^[X1:$i]:((in @ X1) @ nat))) @ (^[X1:$i]:((all_of @ (^[X2:$i]:((in @ X2) @ nat))) @ (^[X2:$i]:(((lessis @ X1) @ X2) => ((moreis @ X2) @ X1))))))). % 11.39/11.60 thf(19,plain,sP10,inference(preprocess,[status(thm)],[satz14]). % 11.39/11.60 thf(satz11,axiom,((all_of @ (^[X1:$i]:((in @ X1) @ nat))) @ (^[X1:$i]:((all_of @ (^[X2:$i]:((in @ X2) @ nat))) @ (^[X2:$i]:(((d_29_ii @ X1) @ X2) => ((iii @ X2) @ X1))))))). % 11.39/11.60 thf(20,plain,sP15,inference(preprocess,[status(thm)],[satz11]). % 11.39/11.60 thf(suc_p,axiom,((all_of @ (^[X1:$i]:((in @ X1) @ nat))) @ (^[X1:$i]:((is_of @ (ordsucc @ X1)) @ (^[X2:$i]:((in @ X2) @ nat)))))). % 11.39/11.60 thf(21,plain,sP7,inference(preprocess,[status(thm)],[suc_p]). % 11.39/11.60 thf(22,plain,$false,inference(prop_unsat,[status(thm),assumptions([h7,h8,h5,h6,h4,h2,h3,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,h2,h5,h7,h8,18,19,20,21])). % 11.39/11.60 thf(23,plain,$false,inference(tab_negimp,[status(thm),assumptions([h5,h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h7,h8])],[h6,22,h7,h8])). % 11.39/11.60 thf(24,plain,$false,inference(tab_negimp,[status(thm),assumptions([h4,h2,h3,h1,h0]),tab_negimp(discharge,[h5,h6])],[h4,23,h5,h6])). % 11.39/11.60 thf(25,plain,$false,inference(tab_negall,[status(thm),assumptions([h2,h3,h1,h0]),tab_negall(discharge,[h4]),tab_negall(eigenvar,eigen__1)],[h3,24,h4])). % 11.39/11.60 thf(26,plain,$false,inference(tab_negimp,[status(thm),assumptions([h1,h0]),tab_negimp(discharge,[h2,h3])],[h1,25,h2,h3])). % 11.39/11.60 thf(27,plain,$false,inference(tab_negall,[status(thm),assumptions([h0]),tab_negall(discharge,[h1]),tab_negall(eigenvar,eigen__0)],[h0,26,h1])). % 11.39/11.60 thf(0,theorem,(![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)))))))) => ((n_some @ ((diffprop @ ((binunion @ X2) @ (d_Sing @ X2))) @ X1)) => ((l_or @ (n_some @ ((diffprop @ X2) @ X1))) @ ((n_is @ X2) @ X1))))))),inference(contra,[status(thm),contra(discharge,[h0])],[27,h0])). % 11.39/11.60 % SZS output end Proof %------------------------------------------------------------------------------