%------------------------------------------------------------------------------ % File : Satallax---3.5 % Problem : NUM749^4 : 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 : n001.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 02:12:02 PM UTC 2025 % Result : Theorem 86.49s 86.20s % Output : Proof 86.49s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.03/0.12 % Problem : NUM749^4 : TPTP v9.2.0. Released v7.1.0. % 0.03/0.12 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % 0.12/0.34 % Computer : n001.cluster.edu % 0.12/0.34 % Model : x86_64 x86_64 % 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.12/0.34 % Memory : 8042.1875MB % 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64 % 0.12/0.34 % CPULimit : 300 % 0.12/0.34 % WCLimit : 300 % 0.12/0.34 % DateTime : Wed Oct 8 22:32:38 EDT 2025 % 0.12/0.34 % CPUTime : % 86.49/86.20 % SZS status Theorem % 86.49/86.20 % Mode: mode94:USE_SINE=true:SINE_TOLERANCE=2.0:SINE_GENERALITY_THRESHOLD=4:SINE_RANK_LIMIT=2.:SINE_DEPTH=0 % 86.49/86.20 % Inferences: 4 % 86.49/86.20 % SZS output start Proof % 86.49/86.20 thf(ty_times, type, times : ($i>$i)). % 86.49/86.20 thf(ty_d_1to, type, d_1to : ($i>$i)). % 86.49/86.20 thf(ty_d_Pi, type, d_Pi : ($i>($i>$i)>$i)). % 86.49/86.20 thf(ty_eigen__1, type, eigen__1 : $i). % 86.49/86.20 thf(ty_eigen__0, type, eigen__0 : $i). % 86.49/86.20 thf(ty_d_Sep, type, d_Sep : ($i>($i>$o)>$i)). % 86.49/86.20 thf(ty_second1, type, second1 : ($i>$i>$i)). % 86.49/86.20 thf(ty_emptyset, type, emptyset : $i). % 86.49/86.20 thf(ty_first1, type, first1 : ($i>$i>$i)). % 86.49/86.20 thf(ty_plus, type, plus : ($i>$i)). % 86.49/86.20 thf(ty_ap, type, ap : ($i>$i>$i)). % 86.49/86.20 thf(ty_n_some, type, n_some : (($i>$o)>$o)). % 86.49/86.20 thf(ty_diffprop, type, diffprop : ($i>$i>$i>$o)). % 86.49/86.20 thf(ty_omega, type, omega : $i). % 86.49/86.20 thf(ty_pair1, type, pair1 : ($i>$i>$i>$i)). % 86.49/86.20 thf(ty_in, type, in : ($i>$i>$o)). % 86.49/86.20 thf(ty_n_2, type, n_2 : $i). % 86.49/86.20 thf(sP1,plain,sP1 <=> (![X1:$i]:(((in @ X1) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X2:$i]:((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ X1)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ eigen__0)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ X1)))))) @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ eigen__0))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ X1)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ eigen__0)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))) @ X1))))))))),introduced(definition,[new_symbols(definition,[sP1])])). % 86.49/86.20 thf(sP2,plain,sP2 <=> (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__1)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__0)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__1)))))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__0))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__1)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__0)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__0))) @ ((second1 @ ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset)))))) @ eigen__1))))))),introduced(definition,[new_symbols(definition,[sP2])])). % 86.49/86.20 thf(sP3,plain,sP3 <=> ((in @ eigen__1) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X1:$i]:((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))),introduced(definition,[new_symbols(definition,[sP3])])). % 86.49/86.20 thf(sP4,plain,sP4 <=> (![X1:$i]:(((in @ X1) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X2:$i]:((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (![X2:$i]:(((in @ X2) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X3:$i]:((d_Sep @ omega) @ (^[X4:$i]:(~((X4 = emptyset)))))))) => (n_some @ ((diffprop @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ (((pair1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ ((ap @ (plus @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2)))) @ ((ap @ (times @ ((first1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1)))) @ ((ap @ (times @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X1))) @ ((second1 @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))) @ X2))))))))))),introduced(definition,[new_symbols(definition,[sP4])])). % 86.49/86.20 thf(sP5,plain,sP5 <=> ((in @ eigen__0) @ ((d_Pi @ (d_1to @ n_2)) @ (^[X1:$i]:((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))),introduced(definition,[new_symbols(definition,[sP5])])). % 86.49/86.20 thf(sP6,plain,sP6 <=> (sP3 => sP2),introduced(definition,[new_symbols(definition,[sP6])])). % 86.49/86.20 thf(sP7,plain,sP7 <=> (sP5 => sP1),introduced(definition,[new_symbols(definition,[sP7])])). % 86.49/86.20 thf(def_is_of,definition,(is_of = (^[X1:$i]:(^[X2:$i>$o]:(X2 @ X1))))). % 86.49/86.20 thf(def_all_of,definition,(all_of = (^[X1:$i>$o]:(^[X2:$i>$o]:(![X3:$i]:(((is_of @ X3) @ X1) => (X2 @ X3))))))). % 86.49/86.20 thf(def_ordsucc,definition,(ordsucc = (^[X1:$i]:((binunion @ X1) @ (d_Sing @ X1))))). % 86.49/86.20 thf(def_d_not,definition,(d_not = (^[X1:$o]:((imp @ X1) @ $false)))). % 86.49/86.20 thf(def_d_and,definition,(d_and = (^[X1:$o]:(^[X2:$o]:(d_not @ ((l_ec @ X1) @ X2)))))). % 86.49/86.20 thf(def_l_some,definition,(l_some = (^[X1:$i]:(^[X2:$i>$o]:(d_not @ ((all_of @ (^[X3:$i]:((in @ X3) @ X1))) @ ((non @ X1) @ X2))))))). % 86.49/86.20 thf(def_or3,definition,(or3 = (^[X1:$o]:(^[X2:$o]:(^[X3:$o]:((l_or @ X1) @ ((l_or @ X2) @ X3))))))). % 86.49/86.20 thf(def_ec3,definition,(ec3 = (^[X1:$o]:(^[X2:$o]:(^[X3:$o]:(((and3 @ ((l_ec @ X1) @ X2)) @ ((l_ec @ X2) @ X3)) @ ((l_ec @ X3) @ X1))))))). % 86.49/86.20 thf(def_orec3,definition,(orec3 = (^[X1:$o]:(^[X2:$o]:(^[X3:$o]:((d_and @ (((or3 @ X1) @ X2) @ X3)) @ (((ec3 @ X1) @ X2) @ X3))))))). % 86.49/86.20 thf(def_nat,definition,(nat = ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))). % 86.49/86.20 thf(def_n_is,definition,(n_is = (e_is @ nat))). % 86.49/86.20 thf(def_n_1,definition,(n_1 = (ordsucc @ emptyset))). % 86.49/86.20 thf(def_n_pl,definition,(n_pl = (^[X1:$i]:(ap @ (plus @ X1))))). % 86.49/86.20 thf(def_d_29_ii,definition,(d_29_ii = (^[X1:$i]:(^[X2:$i]:(n_some @ ((diffprop @ X1) @ X2)))))). % 86.49/86.20 thf(def_iii,definition,(iii = (^[X1:$i]:(^[X2:$i]:(n_some @ ((diffprop @ X2) @ X1)))))). % 86.49/86.20 thf(def_moreis,definition,(moreis = (^[X1:$i]:(^[X2:$i]:((l_or @ ((d_29_ii @ X1) @ X2)) @ ((n_is @ X1) @ X2)))))). % 86.49/86.20 thf(def_lessis,definition,(lessis = (^[X1:$i]:(^[X2:$i]:((l_or @ ((iii @ X1) @ X2)) @ ((n_is @ X1) @ X2)))))). % 86.49/86.20 thf(def_n_ts,definition,(n_ts = (^[X1:$i]:(ap @ (times @ X1))))). % 86.49/86.20 thf(def_pair1type,definition,(pair1type = (^[X1:$i]:((d_Pi @ (d_1to @ n_2)) @ (^[X2:$i]:X1))))). % 86.49/86.20 thf(def_frac,definition,(frac = (pair1type @ nat))). % 86.49/86.20 thf(def_n_fr,definition,(n_fr = (pair1 @ nat))). % 86.49/86.20 thf(def_num,definition,(num = (first1 @ nat))). % 86.49/86.20 thf(def_den,definition,(den = (second1 @ nat))). % 86.49/86.20 thf(def_n_eq,definition,(n_eq = (^[X1:$i]:(^[X2:$i]:((n_is @ ((n_ts @ (num @ X1)) @ (den @ X2))) @ ((n_ts @ (num @ X2)) @ (den @ X1))))))). % 86.49/86.20 thf(def_moref,definition,(moref = (^[X1:$i]:(^[X2:$i]:((d_29_ii @ ((n_ts @ (num @ X1)) @ (den @ X2))) @ ((n_ts @ (num @ X2)) @ (den @ X1))))))). % 86.49/86.20 thf(def_lessf,definition,(lessf = (^[X1:$i]:(^[X2:$i]:((iii @ ((n_ts @ (num @ X1)) @ (den @ X2))) @ ((n_ts @ (num @ X2)) @ (den @ X1))))))). % 86.49/86.20 thf(def_moreq,definition,(moreq = (^[X1:$i]:(^[X2:$i]:((l_or @ ((moref @ X1) @ X2)) @ ((n_eq @ X1) @ X2)))))). % 86.49/86.20 thf(def_lesseq,definition,(lesseq = (^[X1:$i]:(^[X2:$i]:((l_or @ ((lessf @ X1) @ X2)) @ ((n_eq @ X1) @ X2)))))). % 86.49/86.20 thf(def_n_pf,definition,(n_pf = (^[X1:$i]:(^[X2:$i]:((n_fr @ ((n_pl @ ((n_ts @ (num @ X1)) @ (den @ X2))) @ ((n_ts @ (num @ X2)) @ (den @ X1)))) @ ((n_ts @ (den @ X1)) @ (den @ X2))))))). % 86.49/86.20 thf(satz60a,conjecture,sP4). % 86.49/86.20 thf(h0,negated_conjecture,(~(sP4)),inference(assume_negation,[status(cth)],[satz60a])). % 86.49/86.20 thf(h1,assumption,(~(sP7)),introduced(assumption,[])). % 86.49/86.20 thf(h2,assumption,sP5,introduced(assumption,[])). % 86.49/86.20 thf(h3,assumption,(~(sP1)),introduced(assumption,[])). % 86.49/86.20 thf(h4,assumption,(~(sP6)),introduced(assumption,[])). % 86.49/86.20 thf(h5,assumption,sP3,introduced(assumption,[])). % 86.49/86.20 thf(h6,assumption,(~(sP2)),introduced(assumption,[])). % 86.49/86.20 thf(1,plain,(~(sP1) | sP6),inference(all_rule,[status(thm)],[])). % 86.49/86.20 thf(2,plain,((~(sP6) | ~(sP3)) | sP2),inference(prop_rule,[status(thm)],[])). % 86.49/86.20 thf(3,plain,(~(sP4) | sP7),inference(all_rule,[status(thm)],[])). % 86.49/86.20 thf(4,plain,((~(sP7) | ~(sP5)) | sP1),inference(prop_rule,[status(thm)],[])). % 86.49/86.20 thf(satz60,axiom,((all_of @ (^[X1:$i]:((in @ X1) @ frac))) @ (^[X1:$i]:((all_of @ (^[X2:$i]:((in @ X2) @ frac))) @ (^[X2:$i]:((moref @ ((n_pf @ X1) @ X2)) @ X1)))))). % 86.49/86.20 thf(5,plain,sP4,inference(preprocess,[status(thm)],[satz60]). % 86.49/86.20 thf(6,plain,$false,inference(prop_unsat,[status(thm),assumptions([h5,h6,h4,h2,h3,h1,h0])],[1,2,3,4,h2,h5,h6,5])). % 86.49/86.20 thf(7,plain,$false,inference(tab_negimp,[status(thm),assumptions([h4,h2,h3,h1,h0]),tab_negimp(discharge,[h5,h6])],[h4,6,h5,h6])). % 86.49/86.20 thf(8,plain,$false,inference(tab_negall,[status(thm),assumptions([h2,h3,h1,h0]),tab_negall(discharge,[h4]),tab_negall(eigenvar,eigen__1)],[h3,7,h4])). % 86.49/86.20 thf(9,plain,$false,inference(tab_negimp,[status(thm),assumptions([h1,h0]),tab_negimp(discharge,[h2,h3])],[h1,8,h2,h3])). % 86.49/86.20 thf(10,plain,$false,inference(tab_negall,[status(thm),assumptions([h0]),tab_negall(discharge,[h1]),tab_negall(eigenvar,eigen__0)],[h0,9,h1])). % 86.49/86.20 thf(0,theorem,sP4,inference(contra,[status(thm),contra(discharge,[h0])],[10,h0])). % 86.49/86.20 % SZS output end Proof %------------------------------------------------------------------------------