%------------------------------------------------------------------------------ % File : Satallax---3.5 % Problem : NUM684^4 : 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 : n013.cluster.edu % Model : x86_64 x86_64 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz % Memory : 8042.1875MB % OS : Linux 3.10.0-693.el7.x86_64 % CPULimit : 300s % WCLimit : 300s % DateTime : Thu Oct 9 02:11:47 PM UTC 2025 % Result : Theorem 0.68s 0.87s % Output : Proof 0.68s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.08/0.12 % Problem : NUM684^4 : 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 % 0.14/0.34 % Computer : n013.cluster.edu % 0.14/0.34 % Model : x86_64 x86_64 % 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.14/0.34 % Memory : 8042.1875MB % 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64 % 0.14/0.34 % CPULimit : 300 % 0.14/0.34 % WCLimit : 300 % 0.14/0.34 % DateTime : Wed Oct 8 22:51:23 EDT 2025 % 0.14/0.34 % CPUTime : % 0.68/0.87 % SZS status Theorem % 0.68/0.87 % Mode: mode507:USE_SINE=true:SINE_TOLERANCE=3.0:SINE_GENERALITY_THRESHOLD=0:SINE_RANK_LIMIT=1.:SINE_DEPTH=1 % 0.68/0.87 % Inferences: 54 % 0.68/0.87 % SZS output start Proof % 0.68/0.87 thf(ty_eigen__2, type, eigen__2 : $i). % 0.68/0.87 thf(ty_is_of, type, is_of : ($i>($i>$o)>$o)). % 0.68/0.87 thf(ty_eigen__1, type, eigen__1 : $i). % 0.68/0.87 thf(ty_eigen__0, type, eigen__0 : $i). % 0.68/0.87 thf(ty_d_Sep, type, d_Sep : ($i>($i>$o)>$i)). % 0.68/0.87 thf(ty_emptyset, type, emptyset : $i). % 0.68/0.87 thf(ty_plus, type, plus : ($i>$i)). % 0.68/0.87 thf(ty_ap, type, ap : ($i>$i>$i)). % 0.68/0.87 thf(ty_omega, type, omega : $i). % 0.68/0.87 thf(ty_in, type, in : ($i>$i>$o)). % 0.68/0.87 thf(ty_e_is, type, e_is : ($i>$i>$i>$o)). % 0.68/0.87 thf(sP1,plain,sP1 <=> (((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:$i]:(~((X3 = emptyset)))))))) => ((((e_is @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ ((ap @ (plus @ eigen__2)) @ eigen__0)) @ ((ap @ (plus @ eigen__2)) @ X1)) => (((e_is @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ eigen__0) @ X1))))),introduced(definition,[new_symbols(definition,[sP1])])). % 0.68/0.87 thf(sP2,plain,sP2 <=> ((is_of @ eigen__1) @ (^[X1:$i]:((in @ X1) @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))),introduced(definition,[new_symbols(definition,[sP2])])). % 0.68/0.87 thf(sP3,plain,sP3 <=> (((e_is @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ ((ap @ (plus @ eigen__2)) @ eigen__0)) @ ((ap @ (plus @ eigen__2)) @ eigen__1)),introduced(definition,[new_symbols(definition,[sP3])])). % 0.68/0.87 thf(sP4,plain,sP4 <=> (![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:$i]:(~((X5 = emptyset)))))))) => ((((e_is @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (plus @ X1)) @ X2)) @ ((ap @ (plus @ X1)) @ X3)) => (((e_is @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X2) @ X3)))))))),introduced(definition,[new_symbols(definition,[sP4])])). % 0.68/0.87 thf(sP5,plain,sP5 <=> (![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => ((((e_is @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ ((ap @ (plus @ eigen__2)) @ eigen__0)) @ ((ap @ (plus @ eigen__2)) @ X1)) => (((e_is @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ eigen__0) @ X1)))),introduced(definition,[new_symbols(definition,[sP5])])). % 0.68/0.87 thf(sP6,plain,sP6 <=> (((is_of @ eigen__2) @ (^[X1:$i]:((in @ X1) @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) => (![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (![X2:$i]:(((is_of @ X2) @ (^[X3:$i]:((in @ X3) @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))))) => ((((e_is @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (plus @ eigen__2)) @ X1)) @ ((ap @ (plus @ eigen__2)) @ X2)) => (((e_is @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1) @ X2))))))),introduced(definition,[new_symbols(definition,[sP6])])). % 0.68/0.87 thf(sP7,plain,sP7 <=> (sP2 => (sP3 => (((e_is @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__0) @ eigen__1))),introduced(definition,[new_symbols(definition,[sP7])])). % 0.68/0.87 thf(sP8,plain,sP8 <=> ((is_of @ eigen__2) @ (^[X1:$i]:((in @ X1) @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))),introduced(definition,[new_symbols(definition,[sP8])])). % 0.68/0.87 thf(sP9,plain,sP9 <=> ((is_of @ eigen__0) @ (^[X1:$i]:((in @ X1) @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))),introduced(definition,[new_symbols(definition,[sP9])])). % 0.68/0.87 thf(sP10,plain,sP10 <=> (((e_is @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__0) @ eigen__1),introduced(definition,[new_symbols(definition,[sP10])])). % 0.68/0.87 thf(sP11,plain,sP11 <=> (![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)))))))) => ((((e_is @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (plus @ eigen__2)) @ X1)) @ ((ap @ (plus @ eigen__2)) @ X2)) => (((e_is @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1) @ X2)))))),introduced(definition,[new_symbols(definition,[sP11])])). % 0.68/0.87 thf(sP12,plain,sP12 <=> (sP3 => sP10),introduced(definition,[new_symbols(definition,[sP12])])). % 0.68/0.87 thf(def_all_of,definition,(all_of = (^[X1:$i>$o]:(^[X2:$i>$o]:(![X3:$i]:(((is_of @ X3) @ X1) => (X2 @ X3))))))). % 0.68/0.87 thf(def_nat,definition,(nat = ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))). % 0.68/0.87 thf(def_n_is,definition,(n_is = (e_is @ nat))). % 0.68/0.87 thf(def_n_pl,definition,(n_pl = (^[X1:$i]:(ap @ (plus @ X1))))). % 0.68/0.87 thf(satz20e,conjecture,(![X1:$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:$i]:(~((X4 = emptyset)))))))) => (![X3:$i]:(((is_of @ X3) @ (^[X4:$i]:((in @ X4) @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))))) => ((((e_is @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (plus @ X3)) @ X1)) @ ((ap @ (plus @ X3)) @ X2)) => (((e_is @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1) @ X2))))))))). % 0.68/0.87 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)))))))) => (![X3:$i]:(((is_of @ X3) @ (^[X4:$i]:((in @ X4) @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))))) => ((((e_is @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (plus @ X3)) @ X1)) @ ((ap @ (plus @ X3)) @ X2)) => (((e_is @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1) @ X2)))))))))),inference(assume_negation,[status(cth)],[satz20e])). % 0.68/0.87 thf(h1,assumption,(~((sP9 => (![X1:$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:$i]:(~((X4 = emptyset)))))))) => ((((e_is @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (plus @ X2)) @ eigen__0)) @ ((ap @ (plus @ X2)) @ X1)) => (((e_is @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__0) @ X1))))))))),introduced(assumption,[])). % 0.68/0.87 thf(h2,assumption,sP9,introduced(assumption,[])). % 0.68/0.87 thf(h3,assumption,(~((![X1:$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:$i]:(~((X4 = emptyset)))))))) => ((((e_is @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (plus @ X2)) @ eigen__0)) @ ((ap @ (plus @ X2)) @ X1)) => (((e_is @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ eigen__0) @ X1)))))))),introduced(assumption,[])). % 0.68/0.87 thf(h4,assumption,(~((sP2 => (![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => ((((e_is @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ ((ap @ (plus @ X1)) @ eigen__0)) @ ((ap @ (plus @ X1)) @ eigen__1)) => sP10)))))),introduced(assumption,[])). % 0.68/0.87 thf(h5,assumption,sP2,introduced(assumption,[])). % 0.68/0.87 thf(h6,assumption,(~((![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => ((((e_is @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ ((ap @ (plus @ X1)) @ eigen__0)) @ ((ap @ (plus @ X1)) @ eigen__1)) => sP10))))),introduced(assumption,[])). % 0.68/0.87 thf(h7,assumption,(~((sP8 => sP12))),introduced(assumption,[])). % 0.68/0.87 thf(h8,assumption,sP8,introduced(assumption,[])). % 0.68/0.87 thf(h9,assumption,(~(sP12)),introduced(assumption,[])). % 0.68/0.87 thf(h10,assumption,sP3,introduced(assumption,[])). % 0.68/0.87 thf(h11,assumption,(~(sP10)),introduced(assumption,[])). % 0.68/0.87 thf(1,plain,(~(sP4) | sP6),inference(all_rule,[status(thm)],[])). % 0.68/0.87 thf(2,plain,((~(sP6) | ~(sP8)) | sP11),inference(prop_rule,[status(thm)],[])). % 0.68/0.87 thf(3,plain,(~(sP11) | sP1),inference(all_rule,[status(thm)],[])). % 0.68/0.87 thf(4,plain,((~(sP1) | ~(sP9)) | sP5),inference(prop_rule,[status(thm)],[])). % 0.68/0.87 thf(5,plain,(~(sP5) | sP7),inference(all_rule,[status(thm)],[])). % 0.68/0.87 thf(6,plain,((~(sP7) | ~(sP2)) | sP12),inference(prop_rule,[status(thm)],[])). % 0.68/0.87 thf(7,plain,((~(sP12) | ~(sP3)) | sP10),inference(prop_rule,[status(thm)],[])). % 0.68/0.87 thf(satz8a,axiom,((all_of @ (^[X1:$i]:((in @ X1) @ nat))) @ (^[X1:$i]:((all_of @ (^[X2:$i]:((in @ X2) @ nat))) @ (^[X2:$i]:((all_of @ (^[X3:$i]:((in @ X3) @ nat))) @ (^[X3:$i]:(((n_is @ ((n_pl @ X1) @ X2)) @ ((n_pl @ X1) @ X3)) => ((n_is @ X2) @ X3))))))))). % 0.68/0.87 thf(8,plain,sP4,inference(preprocess,[status(thm)],[satz8a]). % 0.68/0.87 thf(9,plain,$false,inference(prop_unsat,[status(thm),assumptions([h10,h11,h8,h9,h7,h5,h6,h4,h2,h3,h1,h0])],[1,2,3,4,5,6,7,h2,h5,h8,h10,h11,8])). % 0.68/0.87 thf(10,plain,$false,inference(tab_negimp,[status(thm),assumptions([h8,h9,h7,h5,h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h10,h11])],[h9,9,h10,h11])). % 0.68/0.87 thf(11,plain,$false,inference(tab_negimp,[status(thm),assumptions([h7,h5,h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h8,h9])],[h7,10,h8,h9])). % 0.68/0.87 thf(12,plain,$false,inference(tab_negall,[status(thm),assumptions([h5,h6,h4,h2,h3,h1,h0]),tab_negall(discharge,[h7]),tab_negall(eigenvar,eigen__2)],[h6,11,h7])). % 0.68/0.87 thf(13,plain,$false,inference(tab_negimp,[status(thm),assumptions([h4,h2,h3,h1,h0]),tab_negimp(discharge,[h5,h6])],[h4,12,h5,h6])). % 0.68/0.87 thf(14,plain,$false,inference(tab_negall,[status(thm),assumptions([h2,h3,h1,h0]),tab_negall(discharge,[h4]),tab_negall(eigenvar,eigen__1)],[h3,13,h4])). % 0.68/0.87 thf(15,plain,$false,inference(tab_negimp,[status(thm),assumptions([h1,h0]),tab_negimp(discharge,[h2,h3])],[h1,14,h2,h3])). % 0.68/0.87 thf(16,plain,$false,inference(tab_negall,[status(thm),assumptions([h0]),tab_negall(discharge,[h1]),tab_negall(eigenvar,eigen__0)],[h0,15,h1])). % 0.68/0.87 thf(0,theorem,(![X1:$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:$i]:(~((X4 = emptyset)))))))) => (![X3:$i]:(((is_of @ X3) @ (^[X4:$i]:((in @ X4) @ ((d_Sep @ omega) @ (^[X5:$i]:(~((X5 = emptyset)))))))) => ((((e_is @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ ((ap @ (plus @ X3)) @ X1)) @ ((ap @ (plus @ X3)) @ X2)) => (((e_is @ ((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))) @ X1) @ X2)))))))),inference(contra,[status(thm),contra(discharge,[h0])],[16,h0])). % 0.68/0.87 % SZS output end Proof 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