%------------------------------------------------------------------------------ % File : Satallax---3.5 % Problem : SET926^7 : TPTP v9.2.0. Released v5.5.0. % 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 02:26:35 PM UTC 2025 % Result : Theorem 93.43s 91.17s % Output : Proof 93.43s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.00/0.11 % Problem : SET926^7 : TPTP v9.2.0. Released v5.5.0. % 0.12/0.12 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % 0.12/0.33 % Computer : n001.cluster.edu % 0.12/0.33 % Model : x86_64 x86_64 % 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.12/0.33 % Memory : 8042.1875MB % 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64 % 0.12/0.33 % CPULimit : 300 % 0.12/0.33 % WCLimit : 300 % 0.12/0.33 % DateTime : Wed Oct 8 14:29:38 EDT 2025 % 0.12/0.33 % CPUTime : % 93.43/91.17 % SZS status Theorem % 93.43/91.17 % Mode: mode454:USE_SINE=true:SINE_TOLERANCE=2.0:SINE_GENERALITY_THRESHOLD=4:SINE_RANK_LIMIT=3.:SINE_DEPTH=0 % 93.43/91.17 % Inferences: 167 % 93.43/91.17 % SZS output start Proof % 93.43/91.17 thf(ty_mu, type, mu : $tType). % 93.43/91.17 thf(ty_singleton, type, singleton : (mu>mu)). % 93.43/91.17 thf(ty_eigen__2, type, eigen__2 : mu). % 93.43/91.17 thf(ty_eigen__1, type, eigen__1 : mu). % 93.43/91.17 thf(ty_eigen__0, type, eigen__0 : $i). % 93.43/91.17 thf(ty_exists_in_world, type, exists_in_world : (mu>$i>$o)). % 93.43/91.17 thf(ty_qmltpeq, type, qmltpeq : (mu>mu>$i>$o)). % 93.43/91.17 thf(ty_empty_set, type, empty_set : mu). % 93.43/91.17 thf(ty_set_difference, type, set_difference : (mu>mu>mu)). % 93.43/91.17 thf(ty_in, type, in : (mu>mu>$i>$o)). % 93.43/91.17 thf(h0, assumption, (![X1:mu>$o]:(![X2:mu]:((X1 @ X2) => (X1 @ (eps__0 @ X1))))),introduced(assumption,[])). % 93.43/91.17 thf(eigendef_eigen__1, definition, eigen__1 = (eps__0 @ (^[X1:mu]:(~((((exists_in_world @ X1) @ eigen__0) => (![X2:mu]:(((exists_in_world @ X2) @ eigen__0) => ((~((((qmltpeq @ ((set_difference @ (singleton @ X1)) @ X2)) @ empty_set) @ eigen__0))) => (((qmltpeq @ ((set_difference @ (singleton @ X1)) @ X2)) @ (singleton @ X1)) @ eigen__0))))))))), introduced(definition,[new_symbols(definition,[eigen__1])])). % 93.43/91.17 thf(h1, assumption, (![X1:$i>$o]:(![X2:$i]:((X1 @ X2) => (X1 @ (eps__1 @ X1))))),introduced(assumption,[])). % 93.43/91.17 thf(eigendef_eigen__0, definition, eigen__0 = (eps__1 @ (^[X1:$i]:(~((![X2:mu]:(((exists_in_world @ X2) @ X1) => (![X3:mu]:(((exists_in_world @ X3) @ X1) => ((~((((qmltpeq @ ((set_difference @ (singleton @ X2)) @ X3)) @ empty_set) @ X1))) => (((qmltpeq @ ((set_difference @ (singleton @ X2)) @ X3)) @ (singleton @ X2)) @ X1)))))))))), introduced(definition,[new_symbols(definition,[eigen__0])])). % 93.43/91.17 thf(eigendef_eigen__2, definition, eigen__2 = (eps__0 @ (^[X1:mu]:(~((((exists_in_world @ X1) @ eigen__0) => ((~((((qmltpeq @ ((set_difference @ (singleton @ eigen__1)) @ X1)) @ empty_set) @ eigen__0))) => (((qmltpeq @ ((set_difference @ (singleton @ eigen__1)) @ X1)) @ (singleton @ eigen__1)) @ eigen__0))))))), introduced(definition,[new_symbols(definition,[eigen__2])])). % 93.43/91.17 thf(sP1,plain,sP1 <=> ((~((((qmltpeq @ ((set_difference @ (singleton @ eigen__1)) @ eigen__2)) @ empty_set) @ eigen__0))) => (((qmltpeq @ ((set_difference @ (singleton @ eigen__1)) @ eigen__2)) @ (singleton @ eigen__1)) @ eigen__0)),introduced(definition,[new_symbols(definition,[sP1])])). % 93.43/91.17 thf(sP2,plain,sP2 <=> (((exists_in_world @ eigen__2) @ eigen__0) => (~((((((qmltpeq @ ((set_difference @ (singleton @ eigen__1)) @ eigen__2)) @ (singleton @ eigen__1)) @ eigen__0) => (~((((in @ eigen__1) @ eigen__2) @ eigen__0)))) => (~(((~((((in @ eigen__1) @ eigen__2) @ eigen__0))) => (((qmltpeq @ ((set_difference @ (singleton @ eigen__1)) @ eigen__2)) @ (singleton @ eigen__1)) @ eigen__0)))))))),introduced(definition,[new_symbols(definition,[sP2])])). % 93.43/91.17 thf(sP3,plain,sP3 <=> (((qmltpeq @ ((set_difference @ (singleton @ eigen__1)) @ eigen__2)) @ (singleton @ eigen__1)) @ eigen__0),introduced(definition,[new_symbols(definition,[sP3])])). % 93.43/91.17 thf(sP4,plain,sP4 <=> ((~((((in @ eigen__1) @ eigen__2) @ eigen__0))) => sP3),introduced(definition,[new_symbols(definition,[sP4])])). % 93.43/91.17 thf(sP5,plain,sP5 <=> (((exists_in_world @ eigen__2) @ eigen__0) => sP1),introduced(definition,[new_symbols(definition,[sP5])])). % 93.43/91.17 thf(sP6,plain,sP6 <=> (![X1:$i]:(![X2:mu]:(((exists_in_world @ X2) @ X1) => (![X3:mu]:(((exists_in_world @ X3) @ X1) => (~((((((qmltpeq @ ((set_difference @ (singleton @ X2)) @ X3)) @ (singleton @ X2)) @ X1) => (~((((in @ X2) @ X3) @ X1)))) => (~(((~((((in @ X2) @ X3) @ X1))) => (((qmltpeq @ ((set_difference @ (singleton @ X2)) @ X3)) @ (singleton @ X2)) @ X1)))))))))))),introduced(definition,[new_symbols(definition,[sP6])])). % 93.43/91.17 thf(sP7,plain,sP7 <=> (((qmltpeq @ ((set_difference @ (singleton @ eigen__1)) @ eigen__2)) @ empty_set) @ eigen__0),introduced(definition,[new_symbols(definition,[sP7])])). % 93.43/91.17 thf(sP8,plain,sP8 <=> ((exists_in_world @ eigen__2) @ eigen__0),introduced(definition,[new_symbols(definition,[sP8])])). % 93.43/91.17 thf(sP9,plain,sP9 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => (~((((((qmltpeq @ ((set_difference @ (singleton @ eigen__1)) @ X1)) @ (singleton @ eigen__1)) @ eigen__0) => (~((((in @ eigen__1) @ X1) @ eigen__0)))) => (~(((~((((in @ eigen__1) @ X1) @ eigen__0))) => (((qmltpeq @ ((set_difference @ (singleton @ eigen__1)) @ X1)) @ (singleton @ eigen__1)) @ eigen__0))))))))),introduced(definition,[new_symbols(definition,[sP9])])). % 93.43/91.17 thf(sP10,plain,sP10 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => (~((((((qmltpeq @ ((set_difference @ (singleton @ eigen__1)) @ X1)) @ empty_set) @ eigen__0) => (((in @ eigen__1) @ X1) @ eigen__0)) => (~(((((in @ eigen__1) @ X1) @ eigen__0) => (((qmltpeq @ ((set_difference @ (singleton @ eigen__1)) @ X1)) @ empty_set) @ eigen__0))))))))),introduced(definition,[new_symbols(definition,[sP10])])). % 93.43/91.17 thf(sP11,plain,sP11 <=> (sP8 => (~(((sP7 => (((in @ eigen__1) @ eigen__2) @ eigen__0)) => (~(((((in @ eigen__1) @ eigen__2) @ eigen__0) => sP7))))))),introduced(definition,[new_symbols(definition,[sP11])])). % 93.43/91.17 thf(sP12,plain,sP12 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => ((~((((qmltpeq @ ((set_difference @ (singleton @ eigen__1)) @ X1)) @ empty_set) @ eigen__0))) => (((qmltpeq @ ((set_difference @ (singleton @ eigen__1)) @ X1)) @ (singleton @ eigen__1)) @ eigen__0)))),introduced(definition,[new_symbols(definition,[sP12])])). % 93.43/91.17 thf(sP13,plain,sP13 <=> ((((in @ eigen__1) @ eigen__2) @ eigen__0) => sP7),introduced(definition,[new_symbols(definition,[sP13])])). % 93.43/91.17 thf(sP14,plain,sP14 <=> (((exists_in_world @ eigen__1) @ eigen__0) => sP9),introduced(definition,[new_symbols(definition,[sP14])])). % 93.43/91.17 thf(sP15,plain,sP15 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => (![X2:mu]:(((exists_in_world @ X2) @ eigen__0) => (~((((((qmltpeq @ ((set_difference @ (singleton @ X1)) @ X2)) @ (singleton @ X1)) @ eigen__0) => (~((((in @ X1) @ X2) @ eigen__0)))) => (~(((~((((in @ X1) @ X2) @ eigen__0))) => (((qmltpeq @ ((set_difference @ (singleton @ X1)) @ X2)) @ (singleton @ X1)) @ eigen__0))))))))))),introduced(definition,[new_symbols(definition,[sP15])])). % 93.43/91.17 thf(sP16,plain,sP16 <=> (((in @ eigen__1) @ eigen__2) @ eigen__0),introduced(definition,[new_symbols(definition,[sP16])])). % 93.43/91.17 thf(sP17,plain,sP17 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => (![X2:mu]:(((exists_in_world @ X2) @ eigen__0) => (~((((((qmltpeq @ ((set_difference @ (singleton @ X1)) @ X2)) @ empty_set) @ eigen__0) => (((in @ X1) @ X2) @ eigen__0)) => (~(((((in @ X1) @ X2) @ eigen__0) => (((qmltpeq @ ((set_difference @ (singleton @ X1)) @ X2)) @ empty_set) @ eigen__0))))))))))),introduced(definition,[new_symbols(definition,[sP17])])). % 93.43/91.17 thf(sP18,plain,sP18 <=> ((exists_in_world @ eigen__1) @ eigen__0),introduced(definition,[new_symbols(definition,[sP18])])). % 93.43/91.17 thf(sP19,plain,sP19 <=> ((sP7 => sP16) => (~(sP13))),introduced(definition,[new_symbols(definition,[sP19])])). % 93.43/91.17 thf(sP20,plain,sP20 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => (![X2:mu]:(((exists_in_world @ X2) @ eigen__0) => ((~((((qmltpeq @ ((set_difference @ (singleton @ X1)) @ X2)) @ empty_set) @ eigen__0))) => (((qmltpeq @ ((set_difference @ (singleton @ X1)) @ X2)) @ (singleton @ X1)) @ eigen__0)))))),introduced(definition,[new_symbols(definition,[sP20])])). % 93.43/91.17 thf(sP21,plain,sP21 <=> (![X1:$i]:(![X2:mu]:(((exists_in_world @ X2) @ X1) => (![X3:mu]:(((exists_in_world @ X3) @ X1) => ((~((((qmltpeq @ ((set_difference @ (singleton @ X2)) @ X3)) @ empty_set) @ X1))) => (((qmltpeq @ ((set_difference @ (singleton @ X2)) @ X3)) @ (singleton @ X2)) @ X1))))))),introduced(definition,[new_symbols(definition,[sP21])])). % 93.43/91.17 thf(sP22,plain,sP22 <=> ((sP3 => (~(sP16))) => (~(sP4))),introduced(definition,[new_symbols(definition,[sP22])])). % 93.43/91.17 thf(sP23,plain,sP23 <=> (sP18 => sP10),introduced(definition,[new_symbols(definition,[sP23])])). % 93.43/91.17 thf(sP24,plain,sP24 <=> (![X1:$i]:(![X2:mu]:(((exists_in_world @ X2) @ X1) => (![X3:mu]:(((exists_in_world @ X3) @ X1) => (~((((((qmltpeq @ ((set_difference @ (singleton @ X2)) @ X3)) @ empty_set) @ X1) => (((in @ X2) @ X3) @ X1)) => (~(((((in @ X2) @ X3) @ X1) => (((qmltpeq @ ((set_difference @ (singleton @ X2)) @ X3)) @ empty_set) @ X1)))))))))))),introduced(definition,[new_symbols(definition,[sP24])])). % 93.43/91.17 thf(sP25,plain,sP25 <=> (sP18 => sP12),introduced(definition,[new_symbols(definition,[sP25])])). % 93.43/91.17 thf(def_mnot,definition,(mnot = (^[X1:$i>$o]:(^[X2:$i]:(~((X1 @ X2))))))). % 93.43/91.17 thf(def_mor,definition,(mor = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((~((X1 @ X3))) => (X2 @ X3))))))). % 93.43/91.17 thf(def_mand,definition,(mand = (^[X1:$i>$o]:(^[X2:$i>$o]:(mnot @ ((mor @ (mnot @ X1)) @ (mnot @ X2))))))). % 93.43/91.17 thf(def_mimplies,definition,(mimplies = (^[X1:$i>$o]:(mor @ (mnot @ X1))))). % 93.43/91.17 thf(def_mequiv,definition,(mequiv = (^[X1:$i>$o]:(^[X2:$i>$o]:((mand @ ((mimplies @ X1) @ X2)) @ ((mimplies @ X2) @ X1)))))). % 93.43/91.17 thf(def_mforall_ind,definition,(mforall_ind = (^[X1:mu>$i>$o]:(^[X2:$i]:(![X3:mu]:(((exists_in_world @ X3) @ X2) => ((X1 @ X3) @ X2))))))). % 93.43/91.17 thf(def_mexists_ind,definition,(mexists_ind = (^[X1:mu>$i>$o]:(mnot @ (mforall_ind @ (^[X2:mu]:(mnot @ (X1 @ X2)))))))). % 93.43/91.17 thf(def_mvalid,definition,(mvalid = (!!))). % 93.43/91.17 thf(t69_zfmisc_1,conjecture,sP21). % 93.43/91.17 thf(h2,negated_conjecture,(~(sP21)),inference(assume_negation,[status(cth)],[t69_zfmisc_1])). % 93.43/91.17 thf(1,plain,((~(sP13) | ~(sP16)) | sP7),inference(prop_rule,[status(thm)],[])). % 93.43/91.17 thf(2,plain,(sP19 | sP13),inference(prop_rule,[status(thm)],[])). % 93.43/91.17 thf(3,plain,((~(sP4) | sP16) | sP3),inference(prop_rule,[status(thm)],[])). % 93.43/91.17 thf(4,plain,(sP22 | sP4),inference(prop_rule,[status(thm)],[])). % 93.43/91.17 thf(5,plain,(~(sP10) | sP11),inference(all_rule,[status(thm)],[])). % 93.43/91.17 thf(6,plain,((~(sP11) | ~(sP8)) | ~(sP19)),inference(prop_rule,[status(thm)],[])). % 93.43/91.17 thf(7,plain,(~(sP9) | sP2),inference(all_rule,[status(thm)],[])). % 93.43/91.17 thf(8,plain,((~(sP2) | ~(sP8)) | ~(sP22)),inference(prop_rule,[status(thm)],[])). % 93.43/91.17 thf(9,plain,(sP1 | ~(sP3)),inference(prop_rule,[status(thm)],[])). % 93.43/91.17 thf(10,plain,(sP1 | ~(sP7)),inference(prop_rule,[status(thm)],[])). % 93.43/91.17 thf(11,plain,(sP5 | ~(sP1)),inference(prop_rule,[status(thm)],[])). % 93.43/91.17 thf(12,plain,(sP5 | sP8),inference(prop_rule,[status(thm)],[])). % 93.43/91.17 thf(13,plain,(sP12 | ~(sP5)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__2])). % 93.43/91.17 thf(14,plain,(~(sP24) | sP17),inference(all_rule,[status(thm)],[])). % 93.43/91.17 thf(15,plain,(~(sP17) | sP23),inference(all_rule,[status(thm)],[])). % 93.43/91.17 thf(16,plain,((~(sP23) | ~(sP18)) | sP10),inference(prop_rule,[status(thm)],[])). % 93.43/91.17 thf(17,plain,(~(sP6) | sP15),inference(all_rule,[status(thm)],[])). % 93.43/91.17 thf(18,plain,(~(sP15) | sP14),inference(all_rule,[status(thm)],[])). % 93.43/91.17 thf(19,plain,((~(sP14) | ~(sP18)) | sP9),inference(prop_rule,[status(thm)],[])). % 93.43/91.17 thf(20,plain,(sP25 | ~(sP12)),inference(prop_rule,[status(thm)],[])). % 93.43/91.17 thf(21,plain,(sP25 | sP18),inference(prop_rule,[status(thm)],[])). % 93.43/91.17 thf(22,plain,(sP20 | ~(sP25)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__1])). % 93.43/91.17 thf(23,plain,(sP21 | ~(sP20)),inference(eigen_choice_rule,[status(thm),assumptions([h1])],[h1,eigendef_eigen__0])). % 93.43/91.17 thf(l34_zfmisc_1,axiom,(mvalid @ (mforall_ind @ (^[X1:mu]:(mforall_ind @ (^[X2:mu]:((mequiv @ ((qmltpeq @ ((set_difference @ (singleton @ X1)) @ X2)) @ (singleton @ X1))) @ (mnot @ ((in @ X1) @ X2))))))))). % 93.43/91.17 thf(24,plain,sP6,inference(preprocess,[status(thm)],[l34_zfmisc_1]). % 93.43/91.17 thf(l36_zfmisc_1,axiom,(mvalid @ (mforall_ind @ (^[X1:mu]:(mforall_ind @ (^[X2:mu]:((mequiv @ ((qmltpeq @ ((set_difference @ (singleton @ X1)) @ X2)) @ empty_set)) @ ((in @ X1) @ X2)))))))). % 93.43/91.17 thf(25,plain,sP24,inference(preprocess,[status(thm)],[l36_zfmisc_1]). % 93.43/91.17 thf(26,plain,$false,inference(prop_unsat,[status(thm),assumptions([h2,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,h2])). % 93.43/91.17 thf(27,plain,$false,inference(eigenvar_choice,[status(thm),assumptions([h2,h0]),eigenvar_choice(discharge,[h1])],[26,h1])). % 93.43/91.17 thf(28,plain,$false,inference(eigenvar_choice,[status(thm),assumptions([h2]),eigenvar_choice(discharge,[h0])],[27,h0])). % 93.43/91.17 thf(0,theorem,sP21,inference(contra,[status(thm),contra(discharge,[h2])],[26,h2])). % 93.43/91.17 % SZS output end Proof %------------------------------------------------------------------------------