%------------------------------------------------------------------------------ % File : Satallax---3.5 % Problem : NUM666^4 : 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 : n009.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 02:11:42 PM UTC 2025 % Result : Theorem 37.25s 37.57s % Output : Proof 37.25s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.07/0.12 % Problem : NUM666^4 : TPTP v9.2.0. Released v7.1.0. % 0.07/0.12 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % 0.13/0.33 % Computer : n009.cluster.edu % 0.13/0.33 % Model : x86_64 x86_64 % 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.13/0.33 % Memory : 8042.1875MB % 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64 % 0.13/0.33 % CPULimit : 300 % 0.13/0.33 % WCLimit : 300 % 0.13/0.33 % DateTime : Wed Oct 8 23:04:53 EDT 2025 % 0.13/0.33 % CPUTime : % 37.25/37.57 % SZS status Theorem % 37.25/37.57 % Mode: mode478:USE_SINE=true:SINE_TOLERANCE=2.0:SINE_GENERALITY_THRESHOLD=16:SINE_RANK_LIMIT=1.:SINE_DEPTH=0 % 37.25/37.57 % Inferences: 553 % 37.25/37.57 % SZS output start Proof % 37.25/37.57 thf(ty_eigen__2, type, eigen__2 : $i). % 37.25/37.57 thf(ty_is_of, type, is_of : ($i>($i>$o)>$o)). % 37.25/37.57 thf(ty_eigen__1, type, eigen__1 : $i). % 37.25/37.57 thf(ty_eigen__0, type, eigen__0 : $i). % 37.25/37.57 thf(ty_d_Sep, type, d_Sep : ($i>($i>$o)>$i)). % 37.25/37.57 thf(ty_l_or, type, l_or : ($o>$o>$o)). % 37.25/37.57 thf(ty_emptyset, type, emptyset : $i). % 37.25/37.57 thf(ty_lessis, type, lessis : ($i>$i>$o)). % 37.25/37.57 thf(ty_n_some, type, n_some : (($i>$o)>$o)). % 37.25/37.57 thf(ty_diffprop, type, diffprop : ($i>$i>$i>$o)). % 37.25/37.57 thf(ty_omega, type, omega : $i). % 37.25/37.57 thf(ty_n_is, type, n_is : ($i>$i>$o)). % 37.25/37.57 thf(ty_in, type, in : ($i>$i>$o)). % 37.25/37.57 thf(ty_iii, type, iii : ($i>$i>$o)). % 37.25/37.57 thf(sP1,plain,sP1 <=> (((is_of @ eigen__1) @ (^[X1:$i]:((in @ X1) @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) => (![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (((lessis @ eigen__2) @ eigen__1) => (((iii @ eigen__1) @ X1) => ((iii @ eigen__2) @ X1)))))),introduced(definition,[new_symbols(definition,[sP1])])). % 37.25/37.57 thf(sP2,plain,sP2 <=> (![X1:$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:$i]:(~((X4 = emptyset)))))))) => (((l_or @ (n_some @ ((diffprop @ X1) @ X2))) @ ((n_is @ X1) @ X2)) => ((lessis @ X2) @ X1)))))),introduced(definition,[new_symbols(definition,[sP2])])). % 37.25/37.57 thf(sP3,plain,sP3 <=> ((iii @ eigen__1) @ eigen__0),introduced(definition,[new_symbols(definition,[sP3])])). % 37.25/37.57 thf(sP4,plain,sP4 <=> (![X1:$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)))))))) => (((lessis @ X1) @ X2) => (((iii @ X2) @ X3) => ((iii @ X1) @ X3))))))))),introduced(definition,[new_symbols(definition,[sP4])])). % 37.25/37.57 thf(sP5,plain,sP5 <=> (((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:$i]:(~((X3 = emptyset)))))))) => (((iii @ eigen__2) @ X1) => (n_some @ ((diffprop @ X1) @ eigen__2)))))),introduced(definition,[new_symbols(definition,[sP5])])). % 37.25/37.57 thf(sP6,plain,sP6 <=> (((iii @ eigen__2) @ eigen__0) => (n_some @ ((diffprop @ eigen__0) @ eigen__2))),introduced(definition,[new_symbols(definition,[sP6])])). % 37.25/37.57 thf(sP7,plain,sP7 <=> (![X1:$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:$i]:(~((X4 = emptyset)))))))) => ((n_some @ ((diffprop @ X1) @ X2)) => ((iii @ X2) @ X1)))))),introduced(definition,[new_symbols(definition,[sP7])])). % 37.25/37.57 thf(sP8,plain,sP8 <=> ((is_of @ eigen__1) @ (^[X1:$i]:((in @ X1) @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))),introduced(definition,[new_symbols(definition,[sP8])])). % 37.25/37.57 thf(sP9,plain,sP9 <=> (((l_or @ (n_some @ ((diffprop @ eigen__1) @ eigen__2))) @ ((n_is @ eigen__1) @ eigen__2)) => ((lessis @ eigen__2) @ eigen__1)),introduced(definition,[new_symbols(definition,[sP9])])). % 37.25/37.57 thf(sP10,plain,sP10 <=> (![X1:$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:$i]:(~((X4 = emptyset)))))))) => (((iii @ X1) @ X2) => (n_some @ ((diffprop @ X2) @ X1))))))),introduced(definition,[new_symbols(definition,[sP10])])). % 37.25/37.57 thf(sP11,plain,sP11 <=> (((is_of @ eigen__2) @ (^[X1:$i]:((in @ X1) @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) => sP9),introduced(definition,[new_symbols(definition,[sP11])])). % 37.25/37.57 thf(sP12,plain,sP12 <=> (![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (((lessis @ eigen__2) @ eigen__1) => (((iii @ eigen__1) @ X1) => ((iii @ eigen__2) @ X1))))),introduced(definition,[new_symbols(definition,[sP12])])). % 37.25/37.57 thf(sP13,plain,sP13 <=> (sP8 => ((n_some @ ((diffprop @ eigen__0) @ eigen__1)) => sP3)),introduced(definition,[new_symbols(definition,[sP13])])). % 37.25/37.57 thf(sP14,plain,sP14 <=> (((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)))))))) => (((lessis @ eigen__2) @ X1) => (((iii @ X1) @ X2) => ((iii @ eigen__2) @ X2)))))))),introduced(definition,[new_symbols(definition,[sP14])])). % 37.25/37.57 thf(sP15,plain,sP15 <=> (n_some @ ((diffprop @ eigen__0) @ eigen__1)),introduced(definition,[new_symbols(definition,[sP15])])). % 37.25/37.57 thf(sP16,plain,sP16 <=> (sP3 => ((iii @ eigen__2) @ eigen__0)),introduced(definition,[new_symbols(definition,[sP16])])). % 37.25/37.57 thf(sP17,plain,sP17 <=> ((iii @ eigen__2) @ eigen__0),introduced(definition,[new_symbols(definition,[sP17])])). % 37.25/37.57 thf(sP18,plain,sP18 <=> (((is_of @ eigen__0) @ (^[X1:$i]:((in @ X1) @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))) => (((lessis @ eigen__2) @ eigen__1) => sP16)),introduced(definition,[new_symbols(definition,[sP18])])). % 37.25/37.57 thf(sP19,plain,sP19 <=> (![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 @ eigen__2) @ X1) => (((iii @ X1) @ X2) => ((iii @ eigen__2) @ X2))))))),introduced(definition,[new_symbols(definition,[sP19])])). % 37.25/37.57 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])])). % 37.25/37.57 thf(sP21,plain,sP21 <=> (![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => ((n_some @ ((diffprop @ eigen__0) @ X1)) => ((iii @ X1) @ eigen__0)))),introduced(definition,[new_symbols(definition,[sP21])])). % 37.25/37.57 thf(sP22,plain,sP22 <=> (sP15 => sP3),introduced(definition,[new_symbols(definition,[sP22])])). % 37.25/37.57 thf(sP23,plain,sP23 <=> (((lessis @ eigen__2) @ eigen__1) => sP16),introduced(definition,[new_symbols(definition,[sP23])])). % 37.25/37.57 thf(sP24,plain,sP24 <=> ((lessis @ eigen__2) @ eigen__1),introduced(definition,[new_symbols(definition,[sP24])])). % 37.25/37.57 thf(sP25,plain,sP25 <=> (sP20 => sP6),introduced(definition,[new_symbols(definition,[sP25])])). % 37.25/37.57 thf(sP26,plain,sP26 <=> (![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (((iii @ eigen__2) @ X1) => (n_some @ ((diffprop @ X1) @ eigen__2))))),introduced(definition,[new_symbols(definition,[sP26])])). % 37.25/37.57 thf(sP27,plain,sP27 <=> (n_some @ ((diffprop @ eigen__0) @ eigen__2)),introduced(definition,[new_symbols(definition,[sP27])])). % 37.25/37.57 thf(sP28,plain,sP28 <=> ((is_of @ eigen__2) @ (^[X1:$i]:((in @ X1) @ ((d_Sep @ omega) @ (^[X2:$i]:(~((X2 = emptyset)))))))),introduced(definition,[new_symbols(definition,[sP28])])). % 37.25/37.57 thf(sP29,plain,sP29 <=> (sP20 => sP21),introduced(definition,[new_symbols(definition,[sP29])])). % 37.25/37.57 thf(sP30,plain,sP30 <=> (![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (((l_or @ (n_some @ ((diffprop @ eigen__1) @ X1))) @ ((n_is @ eigen__1) @ X1)) => ((lessis @ X1) @ eigen__1)))),introduced(definition,[new_symbols(definition,[sP30])])). % 37.25/37.57 thf(sP31,plain,sP31 <=> ((l_or @ (n_some @ ((diffprop @ eigen__1) @ eigen__2))) @ ((n_is @ eigen__1) @ eigen__2)),introduced(definition,[new_symbols(definition,[sP31])])). % 37.25/37.57 thf(sP32,plain,sP32 <=> (sP8 => sP30),introduced(definition,[new_symbols(definition,[sP32])])). % 37.25/37.57 thf(def_all_of,definition,(all_of = (^[X1:$i>$o]:(^[X2:$i>$o]:(![X3:$i]:(((is_of @ X3) @ X1) => (X2 @ X3))))))). % 37.25/37.57 thf(def_nat,definition,(nat = ((d_Sep @ omega) @ (^[X1:$i]:(~((X1 = emptyset))))))). % 37.25/37.57 thf(def_d_29_ii,definition,(d_29_ii = (^[X1:$i]:(^[X2:$i]:(n_some @ ((diffprop @ X1) @ X2)))))). % 37.25/37.57 thf(def_moreis,definition,(moreis = (^[X1:$i]:(^[X2:$i]:((l_or @ ((d_29_ii @ X1) @ X2)) @ ((n_is @ X1) @ X2)))))). % 37.25/37.57 thf(satz16d,conjecture,(![X1:$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)))))))) => ((n_some @ ((diffprop @ X1) @ X2)) => (((l_or @ (n_some @ ((diffprop @ X2) @ X3))) @ ((n_is @ X2) @ X3)) => (n_some @ ((diffprop @ X1) @ X3))))))))))). % 37.25/37.57 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:$i]:(~((X5 = emptyset)))))))) => ((n_some @ ((diffprop @ X1) @ X2)) => (((l_or @ (n_some @ ((diffprop @ X2) @ X3))) @ ((n_is @ X2) @ X3)) => (n_some @ ((diffprop @ X1) @ X3)))))))))))),inference(assume_negation,[status(cth)],[satz16d])). % 37.25/37.57 thf(h1,assumption,(~((sP20 => (![X1:$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:$i]:(~((X4 = emptyset)))))))) => ((n_some @ ((diffprop @ eigen__0) @ X1)) => (((l_or @ (n_some @ ((diffprop @ X1) @ X2))) @ ((n_is @ X1) @ X2)) => (n_some @ ((diffprop @ eigen__0) @ X2))))))))))),introduced(assumption,[])). % 37.25/37.57 thf(h2,assumption,sP20,introduced(assumption,[])). % 37.25/37.57 thf(h3,assumption,(~((![X1:$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:$i]:(~((X4 = emptyset)))))))) => ((n_some @ ((diffprop @ eigen__0) @ X1)) => (((l_or @ (n_some @ ((diffprop @ X1) @ X2))) @ ((n_is @ X1) @ X2)) => (n_some @ ((diffprop @ eigen__0) @ X2)))))))))),introduced(assumption,[])). % 37.25/37.57 thf(h4,assumption,(~((sP8 => (![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (sP15 => (((l_or @ (n_some @ ((diffprop @ eigen__1) @ X1))) @ ((n_is @ eigen__1) @ X1)) => (n_some @ ((diffprop @ eigen__0) @ X1))))))))),introduced(assumption,[])). % 37.25/37.57 thf(h5,assumption,sP8,introduced(assumption,[])). % 37.25/37.57 thf(h6,assumption,(~((![X1:$i]:(((is_of @ X1) @ (^[X2:$i]:((in @ X2) @ ((d_Sep @ omega) @ (^[X3:$i]:(~((X3 = emptyset)))))))) => (sP15 => (((l_or @ (n_some @ ((diffprop @ eigen__1) @ X1))) @ ((n_is @ eigen__1) @ X1)) => (n_some @ ((diffprop @ eigen__0) @ X1)))))))),introduced(assumption,[])). % 37.25/37.57 thf(h7,assumption,(~((sP28 => (sP15 => (sP31 => sP27))))),introduced(assumption,[])). % 37.25/37.57 thf(h8,assumption,sP28,introduced(assumption,[])). % 37.25/37.57 thf(h9,assumption,(~((sP15 => (sP31 => sP27)))),introduced(assumption,[])). % 37.25/37.57 thf(h10,assumption,sP15,introduced(assumption,[])). % 37.25/37.57 thf(h11,assumption,(~((sP31 => sP27))),introduced(assumption,[])). % 37.25/37.57 thf(h12,assumption,sP31,introduced(assumption,[])). % 37.25/37.57 thf(h13,assumption,(~(sP27)),introduced(assumption,[])). % 37.25/37.57 thf(1,plain,(~(sP12) | sP18),inference(all_rule,[status(thm)],[])). % 37.25/37.57 thf(2,plain,((~(sP18) | ~(sP20)) | sP23),inference(prop_rule,[status(thm)],[])). % 37.25/37.57 thf(3,plain,((~(sP23) | ~(sP24)) | sP16),inference(prop_rule,[status(thm)],[])). % 37.25/37.57 thf(4,plain,((~(sP16) | ~(sP3)) | sP17),inference(prop_rule,[status(thm)],[])). % 37.25/37.57 thf(5,plain,(~(sP19) | sP1),inference(all_rule,[status(thm)],[])). % 37.25/37.57 thf(6,plain,((~(sP1) | ~(sP8)) | sP12),inference(prop_rule,[status(thm)],[])). % 37.25/37.57 thf(7,plain,((~(sP14) | ~(sP28)) | sP19),inference(prop_rule,[status(thm)],[])). % 37.25/37.57 thf(8,plain,((~(sP29) | ~(sP20)) | sP21),inference(prop_rule,[status(thm)],[])). % 37.25/37.57 thf(9,plain,(~(sP21) | sP13),inference(all_rule,[status(thm)],[])). % 37.25/37.57 thf(10,plain,((~(sP13) | ~(sP8)) | sP22),inference(prop_rule,[status(thm)],[])). % 37.25/37.57 thf(11,plain,((~(sP22) | ~(sP15)) | sP3),inference(prop_rule,[status(thm)],[])). % 37.25/37.57 thf(12,plain,((~(sP32) | ~(sP8)) | sP30),inference(prop_rule,[status(thm)],[])). % 37.25/37.57 thf(13,plain,(~(sP30) | sP11),inference(all_rule,[status(thm)],[])). % 37.25/37.57 thf(14,plain,((~(sP11) | ~(sP28)) | sP9),inference(prop_rule,[status(thm)],[])). % 37.25/37.57 thf(15,plain,((~(sP9) | ~(sP31)) | sP24),inference(prop_rule,[status(thm)],[])). % 37.25/37.57 thf(16,plain,((~(sP5) | ~(sP28)) | sP26),inference(prop_rule,[status(thm)],[])). % 37.25/37.57 thf(17,plain,(~(sP26) | sP25),inference(all_rule,[status(thm)],[])). % 37.25/37.57 thf(18,plain,((~(sP25) | ~(sP20)) | sP6),inference(prop_rule,[status(thm)],[])). % 37.25/37.57 thf(19,plain,((~(sP6) | ~(sP17)) | sP27),inference(prop_rule,[status(thm)],[])). % 37.25/37.57 thf(20,plain,(~(sP4) | sP14),inference(all_rule,[status(thm)],[])). % 37.25/37.57 thf(21,plain,(~(sP2) | sP32),inference(all_rule,[status(thm)],[])). % 37.25/37.57 thf(22,plain,(~(sP10) | sP5),inference(all_rule,[status(thm)],[])). % 37.25/37.57 thf(23,plain,(~(sP7) | sP29),inference(all_rule,[status(thm)],[])). % 37.25/37.57 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:$i]:(((lessis @ X1) @ X2) => (((iii @ X2) @ X3) => ((iii @ X1) @ X3)))))))))). % 37.25/37.57 thf(24,plain,sP4,inference(preprocess,[status(thm)],[satz16a]). % 37.25/37.57 thf(satz13,axiom,((all_of @ (^[X1:$i]:((in @ X1) @ nat))) @ (^[X1:$i]:((all_of @ (^[X2:$i]:((in @ X2) @ nat))) @ (^[X2:$i]:(((moreis @ X1) @ X2) => ((lessis @ X2) @ X1))))))). % 37.25/37.57 thf(25,plain,sP2,inference(preprocess,[status(thm)],[satz13]). % 37.25/37.57 thf(satz12,axiom,((all_of @ (^[X1:$i]:((in @ X1) @ nat))) @ (^[X1:$i]:((all_of @ (^[X2:$i]:((in @ X2) @ nat))) @ (^[X2:$i]:(((iii @ X1) @ X2) => ((d_29_ii @ X2) @ X1))))))). % 37.25/37.57 thf(26,plain,sP10,inference(preprocess,[status(thm)],[satz12]). % 37.25/37.57 thf(satz11,axiom,((all_of @ (^[X1:$i]:((in @ X1) @ nat))) @ (^[X1:$i]:((all_of @ (^[X2:$i]:((in @ X2) @ nat))) @ (^[X2:$i]:(((d_29_ii @ X1) @ X2) => ((iii @ X2) @ X1))))))). % 37.25/37.57 thf(27,plain,sP7,inference(preprocess,[status(thm)],[satz11]). % 37.25/37.57 thf(28,plain,$false,inference(prop_unsat,[status(thm),assumptions([h12,h13,h10,h11,h8,h9,h7,h5,h6,h4,h2,h3,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,h2,h5,h8,h10,h12,h13,24,25,26,27])). % 37.25/37.57 thf(29,plain,$false,inference(tab_negimp,[status(thm),assumptions([h10,h11,h8,h9,h7,h5,h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h12,h13])],[h11,28,h12,h13])). % 37.25/37.57 thf(30,plain,$false,inference(tab_negimp,[status(thm),assumptions([h8,h9,h7,h5,h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h10,h11])],[h9,29,h10,h11])). % 37.25/37.57 thf(31,plain,$false,inference(tab_negimp,[status(thm),assumptions([h7,h5,h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h8,h9])],[h7,30,h8,h9])). % 37.25/37.57 thf(32,plain,$false,inference(tab_negall,[status(thm),assumptions([h5,h6,h4,h2,h3,h1,h0]),tab_negall(discharge,[h7]),tab_negall(eigenvar,eigen__2)],[h6,31,h7])). % 37.25/37.57 thf(33,plain,$false,inference(tab_negimp,[status(thm),assumptions([h4,h2,h3,h1,h0]),tab_negimp(discharge,[h5,h6])],[h4,32,h5,h6])). % 37.25/37.57 thf(34,plain,$false,inference(tab_negall,[status(thm),assumptions([h2,h3,h1,h0]),tab_negall(discharge,[h4]),tab_negall(eigenvar,eigen__1)],[h3,33,h4])). % 37.25/37.57 thf(35,plain,$false,inference(tab_negimp,[status(thm),assumptions([h1,h0]),tab_negimp(discharge,[h2,h3])],[h1,34,h2,h3])). % 37.25/37.57 thf(36,plain,$false,inference(tab_negall,[status(thm),assumptions([h0]),tab_negall(discharge,[h1]),tab_negall(eigenvar,eigen__0)],[h0,35,h1])). % 37.25/37.57 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)))))))) => ((n_some @ ((diffprop @ X1) @ X2)) => (((l_or @ (n_some @ ((diffprop @ X2) @ X3))) @ ((n_is @ X2) @ X3)) => (n_some @ ((diffprop @ X1) @ X3)))))))))),inference(contra,[status(thm),contra(discharge,[h0])],[36,h0])). % 37.25/37.57 % SZS output end Proof %------------------------------------------------------------------------------