%------------------------------------------------------------------------------ % File : Satallax---3.5 % Problem : PHI003^1 : 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 : n021.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 13.14s 13.32s % Output : Proof 13.14s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.11/0.12 % Problem : PHI003^1 : TPTP v9.2.0. Released v6.1.0. % 0.11/0.13 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % 0.13/0.34 % Computer : n021.cluster.edu % 0.13/0.34 % Model : x86_64 x86_64 % 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.13/0.34 % Memory : 8042.1875MB % 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64 % 0.13/0.34 % CPULimit : 300 % 0.13/0.34 % WCLimit : 300 % 0.13/0.34 % DateTime : Wed Oct 8 23:17:23 EDT 2025 % 0.13/0.34 % CPUTime : % 13.14/13.32 % SZS status Theorem % 13.14/13.32 % Mode: mode454 % 13.14/13.32 % Inferences: 116 % 13.14/13.32 % SZS output start Proof % 13.14/13.32 thf(ty_mu, type, mu : $tType). % 13.14/13.32 thf(ty_eigen__0, type, eigen__0 : $i). % 13.14/13.32 thf(ty_eigen__4, type, eigen__4 : $i). % 13.14/13.32 thf(ty_positive, type, positive : ((mu>$i>$o)>$i>$o)). % 13.14/13.32 thf(ty_eigen__5, type, eigen__5 : mu). % 13.14/13.32 thf(ty_rel, type, rel : ($i>$i>$o)). % 13.14/13.32 thf(h0, assumption, (![X1:$i>$o]:(![X2:$i]:((X1 @ X2) => (X1 @ (eps__0 @ X1))))),introduced(assumption,[])). % 13.14/13.32 thf(eigendef_eigen__0, definition, eigen__0 = (eps__0 @ (^[X1:$i]:(~((~((![X2:$i]:(((rel @ X1) @ X2) => (![X3:mu]:(~((![X4:mu>$i>$o]:(((positive @ X4) @ X2) => ((X4 @ X3) @ X2)))))))))))))), introduced(definition,[new_symbols(definition,[eigen__0])])). % 13.14/13.32 thf(eigendef_eigen__4, definition, eigen__4 = (eps__0 @ (^[X1:$i]:(~((((rel @ eigen__0) @ X1) => (![X2:mu]:($false => (~((![X3:mu>$i>$o]:(((positive @ X3) @ X1) => ((X3 @ X2) @ X1)))))))))))), introduced(definition,[new_symbols(definition,[eigen__4])])). % 13.14/13.32 thf(h1, assumption, (![X1:mu>$o]:(![X2:mu]:((X1 @ X2) => (X1 @ (eps__1 @ X1))))),introduced(assumption,[])). % 13.14/13.32 thf(eigendef_eigen__5, definition, eigen__5 = (eps__1 @ (^[X1:mu]:(~(($false => (~((![X2:mu>$i>$o]:(((positive @ X2) @ eigen__4) => ((X2 @ X1) @ eigen__4)))))))))), introduced(definition,[new_symbols(definition,[eigen__5])])). % 13.14/13.32 thf(sP1,plain,sP1 <=> (![X1:$i]:(((rel @ eigen__0) @ X1) => (![X2:mu]:(~((![X3:mu>$i>$o]:(((positive @ X3) @ X1) => ((X3 @ X2) @ X1)))))))),introduced(definition,[new_symbols(definition,[sP1])])). % 13.14/13.32 thf(sP2,plain,sP2 <=> (((positive @ (^[X1:mu]:(^[X2:$i]:(![X3:mu>$i>$o]:(((positive @ X3) @ X2) => ((X3 @ X1) @ X2)))))) @ eigen__0) => (~(sP1))),introduced(definition,[new_symbols(definition,[sP2])])). % 13.14/13.32 thf(sP3,plain,sP3 <=> $false,introduced(definition,[new_symbols(definition,[sP3])])). % 13.14/13.32 thf(sP4,plain,sP4 <=> (((positive @ (^[X1:mu]:(^[X2:$i]:(~((![X3:mu>$i>$o]:(((positive @ X3) @ X2) => ((X3 @ X1) @ X2)))))))) @ eigen__0) = (~(((positive @ (^[X1:mu]:(^[X2:$i]:(![X3:mu>$i>$o]:(((positive @ X3) @ X2) => ((X3 @ X1) @ X2)))))) @ eigen__0)))),introduced(definition,[new_symbols(definition,[sP4])])). % 13.14/13.32 thf(sP5,plain,sP5 <=> ((positive @ (^[X1:mu]:(^[X2:$i]:(~((![X3:mu>$i>$o]:(((positive @ X3) @ X2) => ((X3 @ X1) @ X2)))))))) @ eigen__0),introduced(definition,[new_symbols(definition,[sP5])])). % 13.14/13.32 thf(sP6,plain,sP6 <=> (![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,[sP6])])). % 13.14/13.32 thf(sP7,plain,sP7 <=> (![X1:mu>$i>$o]:((~((((positive @ (^[X2:mu]:(^[X3:$i]:sP3))) @ eigen__0) => (~((![X2:$i]:(((rel @ eigen__0) @ X2) => (![X3:mu]:(sP3 => ((X1 @ X3) @ X2)))))))))) => ((positive @ X1) @ eigen__0))),introduced(definition,[new_symbols(definition,[sP7])])). % 13.14/13.32 thf(sP8,plain,sP8 <=> (![X1:mu>$i>$o]:(((positive @ (^[X2:mu]:(^[X3:$i]:(~(((X1 @ X2) @ X3)))))) @ eigen__0) = (~(((positive @ X1) @ eigen__0))))),introduced(definition,[new_symbols(definition,[sP8])])). % 13.14/13.32 thf(sP9,plain,sP9 <=> ((~((((positive @ (^[X1:mu]:(^[X2:$i]:sP3))) @ eigen__0) => (~((![X1:$i]:(((rel @ eigen__0) @ X1) => (![X2:mu]:(sP3 => (~((![X3:mu>$i>$o]:(((positive @ X3) @ X1) => ((X3 @ X2) @ X1)))))))))))))) => sP5),introduced(definition,[new_symbols(definition,[sP9])])). % 13.14/13.32 thf(sP10,plain,sP10 <=> (![X1:$i]:(![X2:mu>$i>$o]:(((positive @ (^[X3:mu]:(^[X4:$i]:(~(((X2 @ X3) @ X4)))))) @ X1) = (~(((positive @ X2) @ X1)))))),introduced(definition,[new_symbols(definition,[sP10])])). % 13.14/13.32 thf(sP11,plain,sP11 <=> (![X1:$i]:(((rel @ eigen__0) @ X1) => (![X2:mu]:(sP3 => (~((![X3:mu>$i>$o]:(((positive @ X3) @ X1) => ((X3 @ X2) @ X1))))))))),introduced(definition,[new_symbols(definition,[sP11])])). % 13.14/13.32 thf(sP12,plain,sP12 <=> (sP3 => (~((![X1:mu>$i>$o]:(((positive @ X1) @ eigen__4) => ((X1 @ eigen__5) @ eigen__4)))))),introduced(definition,[new_symbols(definition,[sP12])])). % 13.14/13.32 thf(sP13,plain,sP13 <=> (![X1:mu>$i>$o]:(![X2:mu>$i>$o]:((~((((positive @ X1) @ eigen__0) => (~((![X3:$i]:(((rel @ eigen__0) @ X3) => (![X4:mu]:(((X1 @ X4) @ X3) => ((X2 @ X4) @ X3)))))))))) => ((positive @ X2) @ eigen__0)))),introduced(definition,[new_symbols(definition,[sP13])])). % 13.14/13.32 thf(sP14,plain,sP14 <=> ((positive @ (^[X1:mu]:(^[X2:$i]:(![X3:mu>$i>$o]:(((positive @ X3) @ X2) => ((X3 @ X1) @ X2)))))) @ eigen__0),introduced(definition,[new_symbols(definition,[sP14])])). % 13.14/13.32 thf(sP15,plain,sP15 <=> ((~(sP2)) => ((positive @ (^[X1:mu]:(^[X2:$i]:sP3))) @ eigen__0)),introduced(definition,[new_symbols(definition,[sP15])])). % 13.14/13.32 thf(sP16,plain,sP16 <=> (((rel @ eigen__0) @ eigen__4) => (![X1:mu]:(sP3 => (~((![X2:mu>$i>$o]:(((positive @ X2) @ eigen__4) => ((X2 @ X1) @ eigen__4)))))))),introduced(definition,[new_symbols(definition,[sP16])])). % 13.14/13.32 thf(sP17,plain,sP17 <=> (![X1:$i]:(~((![X2:$i]:(((rel @ X1) @ X2) => (![X3:mu]:(~((![X4:mu>$i>$o]:(((positive @ X4) @ X2) => ((X4 @ X3) @ X2))))))))))),introduced(definition,[new_symbols(definition,[sP17])])). % 13.14/13.32 thf(sP18,plain,sP18 <=> ((positive @ (^[X1:mu]:(^[X2:$i]:sP3))) @ eigen__0),introduced(definition,[new_symbols(definition,[sP18])])). % 13.14/13.32 thf(sP19,plain,sP19 <=> (![X1:mu>$i>$o]:((~((sP14 => (~((![X2:$i]:(((rel @ eigen__0) @ X2) => (![X3:mu]:((![X4:mu>$i>$o]:(((positive @ X4) @ X2) => ((X4 @ X3) @ X2))) => ((X1 @ X3) @ X2)))))))))) => ((positive @ X1) @ eigen__0))),introduced(definition,[new_symbols(definition,[sP19])])). % 13.14/13.32 thf(sP20,plain,sP20 <=> (![X1:mu]:(sP3 => (~((![X2:mu>$i>$o]:(((positive @ X2) @ eigen__4) => ((X2 @ X1) @ eigen__4))))))),introduced(definition,[new_symbols(definition,[sP20])])). % 13.14/13.32 thf(sP21,plain,sP21 <=> (sP18 => (~(sP11))),introduced(definition,[new_symbols(definition,[sP21])])). % 13.14/13.32 thf(sP22,plain,sP22 <=> ((!!) @ (positive @ (^[X1:mu]:(^[X2:$i]:(![X3:mu>$i>$o]:(((positive @ X3) @ X2) => ((X3 @ X1) @ X2))))))),introduced(definition,[new_symbols(definition,[sP22])])). % 13.14/13.32 thf(def_meq_ind,definition,(meq_ind = (^[X1:mu]:(^[X2:mu]:(^[X3:$i]:(X1 = X2)))))). % 13.14/13.32 thf(def_mtrue,definition,(mtrue = (^[X1:$i]:(~(sP3))))). % 13.14/13.32 thf(def_mfalse,definition,(mfalse = (^[X1:$i]:sP3))). % 13.14/13.32 thf(def_mnot,definition,(mnot = (^[X1:$i>$o]:(^[X2:$i]:(~((X1 @ X2))))))). % 13.14/13.32 thf(def_mor,definition,(mor = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((~((X1 @ X3))) => (X2 @ X3))))))). % 13.14/13.32 thf(def_mand,definition,(mand = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:(~(((X1 @ X3) => (~((X2 @ X3))))))))))). % 13.14/13.32 thf(def_mimplies,definition,(mimplies = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((X1 @ X3) => (X2 @ X3))))))). % 13.14/13.32 thf(def_mimplied,definition,(mimplied = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((X2 @ X3) => (X1 @ X3))))))). % 13.14/13.32 thf(def_mequiv,definition,(mequiv = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((X1 @ X3) = (X2 @ X3))))))). % 13.14/13.32 thf(def_mxor,definition,(mxor = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:(((X1 @ X3) => (X2 @ X3)) => (~(((~((X1 @ X3))) => (~((X2 @ X3)))))))))))). % 13.14/13.32 thf(def_mforall_ind,definition,(mforall_ind = (^[X1:mu>$i>$o]:(^[X2:$i]:(![X3:mu]:((X1 @ X3) @ X2)))))). % 13.14/13.32 thf(def_mforall_indset,definition,(mforall_indset = (^[X1:(mu>$i>$o)>$i>$o]:(^[X2:$i]:(![X3:mu>$i>$o]:((X1 @ X3) @ X2)))))). % 13.14/13.32 thf(def_mforall_prop,definition,(mforall_prop = (^[X1:($i>$o)>$i>$o]:(^[X2:$i]:(![X3:$i>$o]:((X1 @ X3) @ X2)))))). % 13.14/13.32 thf(def_mexists_ind,definition,(mexists_ind = (^[X1:mu>$i>$o]:(^[X2:$i]:(~((![X3:mu]:(~(((X1 @ X3) @ X2)))))))))). % 13.14/13.32 thf(def_mexists_indset,definition,(mexists_indset = (^[X1:(mu>$i>$o)>$i>$o]:(^[X2:$i]:(~((![X3:mu>$i>$o]:(~(((X1 @ X3) @ X2)))))))))). % 13.14/13.32 thf(def_mexists_prop,definition,(mexists_prop = (^[X1:($i>$o)>$i>$o]:(^[X2:$i]:(~((![X3:$i>$o]:(~(((X1 @ X3) @ X2)))))))))). % 13.14/13.32 thf(def_mbox_generic,definition,(mbox_generic = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:(![X4:$i]:(((X1 @ X3) @ X4) => (X2 @ X4)))))))). % 13.14/13.32 thf(def_mdia_generic,definition,(mdia_generic = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:(~((![X4:$i]:(((X1 @ X3) @ X4) => (~((X2 @ X4)))))))))))). % 13.14/13.32 thf(def_mbox,definition,(mbox = (mbox_generic @ rel))). % 13.14/13.32 thf(def_mdia,definition,(mdia = (mdia_generic @ rel))). % 13.14/13.32 thf(def_mvalid,definition,(mvalid = (!!))). % 13.14/13.32 thf(def_minvalid,definition,(minvalid = (^[X1:$i>$o]:(![X2:$i]:(~((X1 @ X2))))))). % 13.14/13.32 thf(def_god,definition,(god = (^[X1:mu]:(mforall_indset @ (^[X2:mu>$i>$o]:((mimplies @ (positive @ X2)) @ (X2 @ X1))))))). % 13.14/13.32 thf(def_essence,definition,(essence = (^[X1:mu>$i>$o]:(^[X2:mu]:((mand @ (X1 @ X2)) @ (mforall_indset @ (^[X3:mu>$i>$o]:((mimplies @ (X3 @ X2)) @ (mbox @ (mforall_ind @ (^[X4:mu]:((mimplies @ (X1 @ X4)) @ (X3 @ X4))))))))))))). % 13.14/13.32 thf(def_necessary_existence,definition,(necessary_existence = (^[X1:mu]:(mforall_indset @ (^[X2:mu>$i>$o]:((mimplies @ ((essence @ X2) @ X1)) @ (mbox @ (mexists_ind @ X2)))))))). % 13.14/13.32 thf(corC,conjecture,(![X1:$i]:(~((![X2:$i]:(((rel @ X1) @ X2) => (~((~((![X3:mu]:(~((![X4:mu>$i>$o]:(((positive @ X4) @ X2) => ((X4 @ X3) @ X2)))))))))))))))). % 13.14/13.32 thf(h2,negated_conjecture,(~(sP17)),inference(assume_negation,[status(cth)],[corC])). % 13.14/13.32 thf(1,plain,(sP12 | sP3),inference(prop_rule,[status(thm)],[])). % 13.14/13.32 thf(2,plain,(sP20 | ~(sP12)),inference(eigen_choice_rule,[status(thm),assumptions([h1])],[h1,eigendef_eigen__5])). % 13.14/13.32 thf(3,plain,(sP16 | ~(sP20)),inference(prop_rule,[status(thm)],[])). % 13.14/13.32 thf(4,plain,(sP11 | ~(sP16)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__4])). % 13.14/13.32 thf(5,plain,(~(sP7) | sP9),inference(all_rule,[status(thm)],[])). % 13.14/13.32 thf(6,plain,((~(sP9) | sP21) | sP5),inference(prop_rule,[status(thm)],[])). % 13.14/13.32 thf(7,plain,((~(sP21) | ~(sP18)) | ~(sP11)),inference(prop_rule,[status(thm)],[])). % 13.14/13.32 thf(8,plain,((~(sP4) | ~(sP5)) | ~(sP14)),inference(prop_rule,[status(thm)],[])). % 13.14/13.32 thf(9,plain,(~(sP13) | sP7),inference(all_rule,[status(thm)],[])). % 13.14/13.32 thf(10,plain,(~(sP8) | sP4),inference(all_rule,[status(thm)],[])). % 13.14/13.32 thf(11,plain,(~(sP22) | sP14),inference(all_rule,[status(thm)],[])). % 13.14/13.32 thf(12,plain,(~(sP10) | sP8),inference(all_rule,[status(thm)],[])). % 13.14/13.32 thf(13,plain,~(sP3),inference(prop_rule,[status(thm)],[])). % 13.14/13.32 thf(14,plain,(~(sP6) | sP13),inference(all_rule,[status(thm)],[])). % 13.14/13.32 thf(15,plain,(~(sP13) | sP19),inference(all_rule,[status(thm)],[])). % 13.14/13.32 thf(16,plain,(~(sP19) | sP15),inference(all_rule,[status(thm)],[])). % 13.14/13.32 thf(17,plain,((~(sP15) | sP2) | sP18),inference(prop_rule,[status(thm)],[])). % 13.14/13.32 thf(18,plain,((~(sP2) | ~(sP14)) | ~(sP1)),inference(prop_rule,[status(thm)],[])). % 13.14/13.32 thf(19,plain,(sP17 | sP1),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__0])). % 13.14/13.32 thf(axA1,axiom,(mvalid @ (mforall_indset @ (^[X1:mu>$i>$o]:((mequiv @ (positive @ (^[X2:mu]:(mnot @ (X1 @ X2))))) @ (mnot @ (positive @ X1))))))). % 13.14/13.32 thf(20,plain,sP10,inference(preprocess,[status(thm)],[axA1]). % 13.14/13.32 thf(axA2,axiom,(mvalid @ (mforall_indset @ (^[X1:mu>$i>$o]:(mforall_indset @ (^[X2:mu>$i>$o]:((mimplies @ ((mand @ (positive @ X1)) @ (mbox @ (mforall_ind @ (^[X3:mu]:((mimplies @ (X1 @ X3)) @ (X2 @ X3))))))) @ (positive @ X2)))))))). % 13.14/13.32 thf(21,plain,sP6,inference(preprocess,[status(thm)],[axA2]). % 13.14/13.32 thf(axA3,axiom,(mvalid @ (positive @ god))). % 13.14/13.32 thf(22,plain,sP22,inference(preprocess,[status(thm)],[axA3]). % 13.14/13.32 thf(23,plain,$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,h2])). % 13.14/13.32 thf(24,plain,$false,inference(eigenvar_choice,[status(thm),assumptions([h2,h0]),eigenvar_choice(discharge,[h1])],[23,h1])). % 13.14/13.32 thf(25,plain,$false,inference(eigenvar_choice,[status(thm),assumptions([h2]),eigenvar_choice(discharge,[h0])],[24,h0])). % 13.14/13.32 thf(0,theorem,(![X1:$i]:(~((![X2:$i]:(((rel @ X1) @ X2) => (~((~((![X3:mu]:(~((![X4:mu>$i>$o]:(((positive @ X4) @ X2) => ((X4 @ X3) @ X2))))))))))))))),inference(contra,[status(thm),contra(discharge,[h2])],[23,h2])). % 13.14/13.32 % SZS output end Proof %------------------------------------------------------------------------------