%------------------------------------------------------------------------------ % File : Satallax---3.5 % Problem : PHI002^1 : TPTP v9.2.0. Released v6.1.0. % Transfm : none % Format : tptp:raw % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % Computer : n010.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 02:13:37 PM UTC 2025 % Result : Theorem 1.49s 1.71s % Output : Proof 1.49s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.04/0.13 % Problem : PHI002^1 : TPTP v9.2.0. Released v6.1.0. % 0.04/0.14 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % 0.14/0.36 % Computer : n010.cluster.edu % 0.14/0.36 % Model : x86_64 x86_64 % 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.14/0.36 % Memory : 8042.1875MB % 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64 % 0.14/0.36 % CPULimit : 300 % 0.14/0.36 % WCLimit : 300 % 0.14/0.36 % DateTime : Wed Oct 8 23:18:08 EDT 2025 % 0.14/0.36 % CPUTime : % 1.49/1.71 % SZS status Theorem % 1.49/1.71 % Mode: mode506 % 1.49/1.71 % Inferences: 49262 % 1.49/1.71 % SZS output start Proof % 1.49/1.71 thf(ty_mu, type, mu : $tType). % 1.49/1.71 thf(ty_eigen__1, type, eigen__1 : (mu>$i>$o)). % 1.49/1.71 thf(ty_eigen__0, type, eigen__0 : $i). % 1.49/1.71 thf(ty_positive, type, positive : ((mu>$i>$o)>$i>$o)). % 1.49/1.71 thf(ty_rel, type, rel : ($i>$i>$o)). % 1.49/1.71 thf(ty_eigen__50, type, eigen__50 : $i). % 1.49/1.71 thf(ty_eigen__57, type, eigen__57 : mu). % 1.49/1.71 thf(h0, assumption, (![X1:(mu>$i>$o)>$o]:(![X2:mu>$i>$o]:((X1 @ X2) => (X1 @ (eps__0 @ X1))))),introduced(assumption,[])). % 1.49/1.71 thf(eigendef_eigen__1, definition, eigen__1 = (eps__0 @ (^[X1:mu>$i>$o]:(~((((positive @ X1) @ eigen__0) => (~((![X2:$i]:(((rel @ eigen__0) @ X2) => (![X3:mu]:(~(((X1 @ X3) @ X2))))))))))))), introduced(definition,[new_symbols(definition,[eigen__1])])). % 1.49/1.71 thf(h1, assumption, (![X1:$i>$o]:(![X2:$i]:((X1 @ X2) => (X1 @ (eps__1 @ X1))))),introduced(assumption,[])). % 1.49/1.71 thf(eigendef_eigen__0, definition, eigen__0 = (eps__1 @ (^[X1:$i]:(~((![X2:mu>$i>$o]:(((positive @ X2) @ X1) => (~((![X3:$i]:(((rel @ X1) @ X3) => (![X4:mu]:(~(((X2 @ X4) @ X3)))))))))))))), introduced(definition,[new_symbols(definition,[eigen__0])])). % 1.49/1.71 thf(eigendef_eigen__50, definition, eigen__50 = (eps__1 @ (^[X1:$i]:(~((((rel @ eigen__0) @ X1) => (![X2:mu]:((![X3:mu>$i>$o]:(((positive @ X3) @ X1) => ((X3 @ X2) @ X1))) => (~($false))))))))), introduced(definition,[new_symbols(definition,[eigen__50])])). % 1.49/1.71 thf(h2, assumption, (![X1:mu>$o]:(![X2:mu]:((X1 @ X2) => (X1 @ (eps__2 @ X1))))),introduced(assumption,[])). % 1.49/1.71 thf(eigendef_eigen__57, definition, eigen__57 = (eps__2 @ (^[X1:mu]:(~(((![X2:mu>$i>$o]:(((positive @ X2) @ eigen__50) => ((X2 @ X1) @ eigen__50))) => (~($false))))))), introduced(definition,[new_symbols(definition,[eigen__57])])). % 1.49/1.71 thf(sP1,plain,sP1 <=> (![X1:$i]:(![X2:mu>$i>$o]:(((positive @ (^[X3:mu]:(^[X4:$i]:(~(((X2 @ X3) @ X4)))))) @ X1) = (~(((positive @ X2) @ X1)))))),introduced(definition,[new_symbols(definition,[sP1])])). % 1.49/1.71 thf(sP2,plain,sP2 <=> (![X1:mu>$i>$o]:(((positive @ (^[X2:mu]:(^[X3:$i]:(~(((X1 @ X2) @ X3)))))) @ eigen__0) = (~(((positive @ X1) @ eigen__0))))),introduced(definition,[new_symbols(definition,[sP2])])). % 1.49/1.71 thf(sP3,plain,sP3 <=> (![X1:$i]:(((rel @ eigen__0) @ X1) => (![X2:mu]:((![X3:mu>$i>$o]:(((positive @ X3) @ X1) => ((X3 @ X2) @ X1))) => (~($false)))))),introduced(definition,[new_symbols(definition,[sP3])])). % 1.49/1.71 thf(sP4,plain,sP4 <=> (![X1:mu>$i>$o]:(((positive @ X1) @ eigen__0) => (~((![X2:$i]:(((rel @ eigen__0) @ X2) => (![X3:mu]:(~(((X1 @ X3) @ X2)))))))))),introduced(definition,[new_symbols(definition,[sP4])])). % 1.49/1.71 thf(sP5,plain,sP5 <=> (![X1:mu>$i>$o]:((~((((positive @ (^[X2:mu]:(^[X3:$i]:(![X4:mu>$i>$o]:(((positive @ X4) @ X3) => ((X4 @ X2) @ X3)))))) @ eigen__0) => (~((![X2:$i]:(((rel @ eigen__0) @ X2) => (![X3:mu]:((![X4:mu>$i>$o]:(((positive @ X4) @ X2) => ((X4 @ X3) @ X2))) => ((X1 @ X3) @ X2)))))))))) => ((positive @ X1) @ eigen__0))),introduced(definition,[new_symbols(definition,[sP5])])). % 1.49/1.71 thf(sP6,plain,sP6 <=> ((~(((positive @ (^[X1:mu]:(^[X2:$i]:$false))) @ eigen__0))) = ((positive @ (^[X1:mu]:(^[X2:$i]:(~($false))))) @ eigen__0)),introduced(definition,[new_symbols(definition,[sP6])])). % 1.49/1.71 thf(sP7,plain,sP7 <=> (((positive @ (^[X1:mu]:(^[X2:$i]:(~($false))))) @ eigen__0) = (~(((positive @ (^[X1:mu]:(^[X2:$i]:$false))) @ eigen__0)))),introduced(definition,[new_symbols(definition,[sP7])])). % 1.49/1.71 thf(sP8,plain,sP8 <=> (![X1:$o]:(![X2:$o]:((X1 = X2) => (X2 = X1)))),introduced(definition,[new_symbols(definition,[sP8])])). % 1.49/1.71 thf(sP9,plain,sP9 <=> (sP7 => sP6),introduced(definition,[new_symbols(definition,[sP9])])). % 1.49/1.71 thf(sP10,plain,sP10 <=> $false,introduced(definition,[new_symbols(definition,[sP10])])). % 1.49/1.71 thf(sP11,plain,sP11 <=> (![X1:mu>$i>$o]:((~((((positive @ eigen__1) @ eigen__0) => (~((![X2:$i]:(((rel @ eigen__0) @ X2) => (![X3:mu]:(((eigen__1 @ X3) @ X2) => ((X1 @ X3) @ X2)))))))))) => ((positive @ X1) @ eigen__0))),introduced(definition,[new_symbols(definition,[sP11])])). % 1.49/1.71 thf(sP12,plain,sP12 <=> ((![X1:mu>$i>$o]:(((positive @ X1) @ eigen__50) => ((X1 @ eigen__57) @ eigen__50))) => (~(sP10))),introduced(definition,[new_symbols(definition,[sP12])])). % 1.49/1.71 thf(sP13,plain,sP13 <=> (![X1:mu]:((![X2:mu>$i>$o]:(((positive @ X2) @ eigen__50) => ((X2 @ X1) @ eigen__50))) => (~(sP10)))),introduced(definition,[new_symbols(definition,[sP13])])). % 1.49/1.71 thf(sP14,plain,sP14 <=> ((~((((positive @ eigen__1) @ eigen__0) => (~((![X1:$i]:(((rel @ eigen__0) @ X1) => (![X2:mu]:(~(((eigen__1 @ X2) @ X1))))))))))) => ((positive @ (^[X1:mu]:(^[X2:$i]:sP10))) @ eigen__0)),introduced(definition,[new_symbols(definition,[sP14])])). % 1.49/1.71 thf(sP15,plain,sP15 <=> (![X1:$i]:(![X2:mu>$i>$o]:(![X3:mu>$i>$o]:((~((((positive @ X2) @ X1) => (~((![X4:$i]:(((rel @ X1) @ X4) => (![X5:mu]:(((X2 @ X5) @ X4) => ((X3 @ X5) @ X4)))))))))) => ((positive @ X3) @ X1))))),introduced(definition,[new_symbols(definition,[sP15])])). % 1.49/1.71 thf(sP16,plain,sP16 <=> ((positive @ (^[X1:mu]:(^[X2:$i]:(![X3:mu>$i>$o]:(((positive @ X3) @ X2) => ((X3 @ X1) @ X2)))))) @ eigen__0),introduced(definition,[new_symbols(definition,[sP16])])). % 1.49/1.71 thf(sP17,plain,sP17 <=> (![X1:$i]:(![X2:mu>$i>$o]:(((positive @ X2) @ X1) => (~((![X3:$i]:(((rel @ X1) @ X3) => (![X4:mu]:(~(((X2 @ X4) @ X3))))))))))),introduced(definition,[new_symbols(definition,[sP17])])). % 1.49/1.71 thf(sP18,plain,sP18 <=> (![X1:$o]:((((positive @ (^[X2:mu]:(^[X3:$i]:(~(sP10))))) @ eigen__0) = X1) => (X1 = ((positive @ (^[X2:mu]:(^[X3:$i]:(~(sP10))))) @ eigen__0)))),introduced(definition,[new_symbols(definition,[sP18])])). % 1.49/1.71 thf(sP19,plain,sP19 <=> (((positive @ eigen__1) @ eigen__0) => (~((![X1:$i]:(((rel @ eigen__0) @ X1) => (![X2:mu]:(~(((eigen__1 @ X2) @ X1))))))))),introduced(definition,[new_symbols(definition,[sP19])])). % 1.49/1.71 thf(sP20,plain,sP20 <=> (sP16 => (~(sP3))),introduced(definition,[new_symbols(definition,[sP20])])). % 1.49/1.71 thf(sP21,plain,sP21 <=> ((positive @ (^[X1:mu]:(^[X2:$i]:sP10))) @ eigen__0),introduced(definition,[new_symbols(definition,[sP21])])). % 1.49/1.71 thf(sP22,plain,sP22 <=> ((positive @ (^[X1:mu]:(^[X2:$i]:(~(sP10))))) @ eigen__0),introduced(definition,[new_symbols(definition,[sP22])])). % 1.49/1.71 thf(sP23,plain,sP23 <=> ((!!) @ (positive @ (^[X1:mu]:(^[X2:$i]:(![X3:mu>$i>$o]:(((positive @ X3) @ X2) => ((X3 @ X1) @ X2))))))),introduced(definition,[new_symbols(definition,[sP23])])). % 1.49/1.71 thf(sP24,plain,sP24 <=> ((~(sP20)) => sP22),introduced(definition,[new_symbols(definition,[sP24])])). % 1.49/1.71 thf(sP25,plain,sP25 <=> (![X1:mu>$i>$o]:(![X2:mu>$i>$o]:((~((((positive @ X1) @ eigen__0) => (~((![X3:$i]:(((rel @ eigen__0) @ X3) => (![X4:mu]:(((X1 @ X4) @ X3) => ((X2 @ X4) @ X3)))))))))) => ((positive @ X2) @ eigen__0)))),introduced(definition,[new_symbols(definition,[sP25])])). % 1.49/1.71 thf(sP26,plain,sP26 <=> (((rel @ eigen__0) @ eigen__50) => sP13),introduced(definition,[new_symbols(definition,[sP26])])). % 1.49/1.71 thf(def_meq_ind,definition,(meq_ind = (^[X1:mu]:(^[X2:mu]:(^[X3:$i]:(X1 = X2)))))). % 1.49/1.71 thf(def_mtrue,definition,(mtrue = (^[X1:$i]:(~(sP10))))). % 1.49/1.71 thf(def_mfalse,definition,(mfalse = (^[X1:$i]:sP10))). % 1.49/1.71 thf(def_mnot,definition,(mnot = (^[X1:$i>$o]:(^[X2:$i]:(~((X1 @ X2))))))). % 1.49/1.71 thf(def_mor,definition,(mor = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((~((X1 @ X3))) => (X2 @ X3))))))). % 1.49/1.71 thf(def_mand,definition,(mand = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:(~(((X1 @ X3) => (~((X2 @ X3))))))))))). % 1.49/1.71 thf(def_mimplies,definition,(mimplies = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((X1 @ X3) => (X2 @ X3))))))). % 1.49/1.71 thf(def_mimplied,definition,(mimplied = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((X2 @ X3) => (X1 @ X3))))))). % 1.49/1.71 thf(def_mequiv,definition,(mequiv = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((X1 @ X3) = (X2 @ X3))))))). % 1.49/1.71 thf(def_mxor,definition,(mxor = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:(((X1 @ X3) => (X2 @ X3)) => (~(((~((X1 @ X3))) => (~((X2 @ X3)))))))))))). % 1.49/1.71 thf(def_mforall_ind,definition,(mforall_ind = (^[X1:mu>$i>$o]:(^[X2:$i]:(![X3:mu]:((X1 @ X3) @ X2)))))). % 1.49/1.71 thf(def_mforall_indset,definition,(mforall_indset = (^[X1:(mu>$i>$o)>$i>$o]:(^[X2:$i]:(![X3:mu>$i>$o]:((X1 @ X3) @ X2)))))). % 1.49/1.71 thf(def_mforall_prop,definition,(mforall_prop = (^[X1:($i>$o)>$i>$o]:(^[X2:$i]:(![X3:$i>$o]:((X1 @ X3) @ X2)))))). % 1.49/1.71 thf(def_mexists_ind,definition,(mexists_ind = (^[X1:mu>$i>$o]:(^[X2:$i]:(~((![X3:mu]:(~(((X1 @ X3) @ X2)))))))))). % 1.49/1.71 thf(def_mexists_indset,definition,(mexists_indset = (^[X1:(mu>$i>$o)>$i>$o]:(^[X2:$i]:(~((![X3:mu>$i>$o]:(~(((X1 @ X3) @ X2)))))))))). % 1.49/1.71 thf(def_mexists_prop,definition,(mexists_prop = (^[X1:($i>$o)>$i>$o]:(^[X2:$i]:(~((![X3:$i>$o]:(~(((X1 @ X3) @ X2)))))))))). % 1.49/1.71 thf(def_mbox_generic,definition,(mbox_generic = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:(![X4:$i]:(((X1 @ X3) @ X4) => (X2 @ X4)))))))). % 1.49/1.71 thf(def_mdia_generic,definition,(mdia_generic = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:(~((![X4:$i]:(((X1 @ X3) @ X4) => (~((X2 @ X4)))))))))))). % 1.49/1.71 thf(def_mbox,definition,(mbox = (mbox_generic @ rel))). % 1.49/1.71 thf(def_mdia,definition,(mdia = (mdia_generic @ rel))). % 1.49/1.71 thf(def_mvalid,definition,(mvalid = (!!))). % 1.49/1.71 thf(def_minvalid,definition,(minvalid = (^[X1:$i>$o]:(![X2:$i]:(~((X1 @ X2))))))). % 1.49/1.71 thf(def_god,definition,(god = (^[X1:mu]:(mforall_indset @ (^[X2:mu>$i>$o]:((mimplies @ (positive @ X2)) @ (X2 @ X1))))))). % 1.49/1.71 thf(def_essence,definition,(essence = (^[X1:mu>$i>$o]:(^[X2:mu]:((mand @ (X1 @ X2)) @ (mforall_indset @ (^[X3:mu>$i>$o]:((mimplies @ (X3 @ X2)) @ (mbox @ (mforall_ind @ (^[X4:mu]:((mimplies @ (X1 @ X4)) @ (X3 @ X4))))))))))))). % 1.49/1.71 thf(def_necessary_existence,definition,(necessary_existence = (^[X1:mu]:(mforall_indset @ (^[X2:mu>$i>$o]:((mimplies @ ((essence @ X2) @ X1)) @ (mbox @ (mexists_ind @ X2)))))))). % 1.49/1.71 thf(thmT1,conjecture,(![X1:$i]:(![X2:mu>$i>$o]:(((positive @ X2) @ X1) => (~((![X3:$i]:(((rel @ X1) @ X3) => (~((~((![X4:mu]:(~(((X2 @ X4) @ X3)))))))))))))))). % 1.49/1.71 thf(h3,negated_conjecture,(~(sP17)),inference(assume_negation,[status(cth)],[thmT1])). % 1.49/1.71 thf(1,plain,~(sP10),inference(prop_rule,[status(thm)],[])). % 1.49/1.71 thf(2,plain,(sP12 | sP10),inference(prop_rule,[status(thm)],[])). % 1.49/1.71 thf(3,plain,(sP13 | ~(sP12)),inference(eigen_choice_rule,[status(thm),assumptions([h2])],[h2,eigendef_eigen__57])). % 1.49/1.71 thf(4,plain,(sP26 | ~(sP13)),inference(prop_rule,[status(thm)],[])). % 1.49/1.71 thf(5,plain,((~(sP6) | ~(sP21)) | ~(sP22)),inference(prop_rule,[status(thm)],[])). % 1.49/1.71 thf(6,plain,(sP3 | ~(sP26)),inference(eigen_choice_rule,[status(thm),assumptions([h1])],[h1,eigendef_eigen__50])). % 1.49/1.71 thf(7,plain,((~(sP24) | sP20) | sP22),inference(prop_rule,[status(thm)],[])). % 1.49/1.71 thf(8,plain,((~(sP20) | ~(sP16)) | ~(sP3)),inference(prop_rule,[status(thm)],[])). % 1.49/1.71 thf(9,plain,((~(sP9) | ~(sP7)) | sP6),inference(prop_rule,[status(thm)],[])). % 1.49/1.71 thf(10,plain,(~(sP18) | sP9),inference(all_rule,[status(thm)],[])). % 1.49/1.71 thf(11,plain,(~(sP8) | sP18),inference(all_rule,[status(thm)],[])). % 1.49/1.71 thf(12,plain,(~(sP5) | sP24),inference(all_rule,[status(thm)],[])). % 1.49/1.71 thf(13,plain,sP8,inference(eq_sym,[status(thm)],[])). % 1.49/1.71 thf(14,plain,(~(sP2) | sP7),inference(all_rule,[status(thm)],[])). % 1.49/1.71 thf(15,plain,(~(sP25) | sP5),inference(all_rule,[status(thm)],[])). % 1.49/1.71 thf(16,plain,(~(sP1) | sP2),inference(all_rule,[status(thm)],[])). % 1.49/1.71 thf(17,plain,(~(sP23) | sP16),inference(all_rule,[status(thm)],[])). % 1.49/1.71 thf(18,plain,(~(sP15) | sP25),inference(all_rule,[status(thm)],[])). % 1.49/1.71 thf(19,plain,(~(sP25) | sP11),inference(all_rule,[status(thm)],[])). % 1.49/1.71 thf(20,plain,(~(sP11) | sP14),inference(all_rule,[status(thm)],[])). % 1.49/1.71 thf(21,plain,((~(sP14) | sP19) | sP21),inference(prop_rule,[status(thm)],[])). % 1.49/1.71 thf(22,plain,(sP4 | ~(sP19)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__1])). % 1.49/1.71 thf(23,plain,(sP17 | ~(sP4)),inference(eigen_choice_rule,[status(thm),assumptions([h1])],[h1,eigendef_eigen__0])). % 1.49/1.71 thf(axA3,axiom,(mvalid @ (positive @ god))). % 1.49/1.71 thf(24,plain,sP23,inference(preprocess,[status(thm)],[axA3]). % 1.49/1.71 thf(axA2,axiom,(mvalid @ (mforall_indset @ (^[X1:mu>$i>$o]:(mforall_indset @ (^[X2:mu>$i>$o]:((mimplies @ ((mand @ (positive @ X1)) @ (mbox @ (mforall_ind @ (^[X3:mu]:((mimplies @ (X1 @ X3)) @ (X2 @ X3))))))) @ (positive @ X2)))))))). % 1.49/1.71 thf(25,plain,sP15,inference(preprocess,[status(thm)],[axA2]). % 1.49/1.71 thf(axA1,axiom,(mvalid @ (mforall_indset @ (^[X1:mu>$i>$o]:((mequiv @ (positive @ (^[X2:mu]:(mnot @ (X1 @ X2))))) @ (mnot @ (positive @ X1))))))). % 1.49/1.71 thf(26,plain,sP1,inference(preprocess,[status(thm)],[axA1]). % 1.49/1.71 thf(27,plain,$false,inference(prop_unsat,[status(thm),assumptions([h3,h2,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,h3])). % 1.49/1.71 thf(28,plain,$false,inference(eigenvar_choice,[status(thm),assumptions([h3,h1,h0]),eigenvar_choice(discharge,[h2])],[27,h2])). % 1.49/1.71 thf(29,plain,$false,inference(eigenvar_choice,[status(thm),assumptions([h3,h0]),eigenvar_choice(discharge,[h1])],[28,h1])). % 1.49/1.71 thf(30,plain,$false,inference(eigenvar_choice,[status(thm),assumptions([h3]),eigenvar_choice(discharge,[h0])],[29,h0])). % 1.49/1.71 thf(0,theorem,(![X1:$i]:(![X2:mu>$i>$o]:(((positive @ X2) @ X1) => (~((![X3:$i]:(((rel @ X1) @ X3) => (~((~((![X4:mu]:(~(((X2 @ X4) @ X3))))))))))))))),inference(contra,[status(thm),contra(discharge,[h3])],[27,h3])). % 1.49/1.71 % SZS output end Proof %------------------------------------------------------------------------------