%------------------------------------------------------------------------------ % File : Satallax---3.5 % Problem : SYO450^5 : TPTP v9.2.0. Released v4.0.0. % Transfm : none % Format : tptp:raw % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % Computer : n004.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 03:25:37 PM UTC 2025 % Result : Theorem 1.07s 1.31s % Output : Proof 1.07s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.00/0.12 % Problem : SYO450^5 : TPTP v9.2.0. Released v4.0.0. % 0.12/0.12 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % 0.12/0.33 % Computer : n004.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 % 0.12/0.34 % DateTime : Thu Oct 9 00:41:23 EDT 2025 % 0.18/0.34 % CPUTime : % 1.07/1.31 % SZS status Theorem % 1.07/1.31 % Mode: mode506 % 1.07/1.31 % Inferences: 39941 % 1.07/1.31 % SZS output start Proof % 1.07/1.31 thf(ty_p, type, p : ($i>$o)). % 1.07/1.31 thf(ty_rel_s4, type, rel_s4 : ($i>$i>$o)). % 1.07/1.31 thf(ty_eigen__2, type, eigen__2 : $i). % 1.07/1.31 thf(ty_q, type, q : ($i>$o)). % 1.07/1.31 thf(ty_eigen__0, type, eigen__0 : $i). % 1.07/1.31 thf(ty_r, type, r : ($i>$o)). % 1.07/1.31 thf(h0, assumption, (![X1:$i>$o]:(![X2:$i]:((X1 @ X2) => (X1 @ (eps__0 @ X1))))),introduced(assumption,[])). % 1.07/1.31 thf(eigendef_eigen__0, definition, eigen__0 = (eps__0 @ (^[X1:$i]:(~(((![X2:$i]:(((rel_s4 @ X1) @ X2) => ((p @ X2) => (~((![X3:$i]:(((rel_s4 @ X2) @ X3) => (~(((q @ X3) => (r @ X3))))))))))) => (~((![X2:$i]:(((rel_s4 @ X1) @ X2) => (~(((q @ X2) => ((![X3:$i]:(((rel_s4 @ X2) @ X3) => (p @ X3))) => (~((![X3:$i]:(((rel_s4 @ X2) @ X3) => (~((r @ X3)))))))))))))))))))), introduced(definition,[new_symbols(definition,[eigen__0])])). % 1.07/1.31 thf(eigendef_eigen__2, definition, eigen__2 = (eps__0 @ (^[X1:$i]:(~((((rel_s4 @ eigen__0) @ X1) => (~(((q @ X1) => (r @ X1))))))))), introduced(definition,[new_symbols(definition,[eigen__2])])). % 1.07/1.31 thf(sP1,plain,sP1 <=> (![X1:$i]:((rel_s4 @ X1) @ X1)),introduced(definition,[new_symbols(definition,[sP1])])). % 1.07/1.31 thf(sP2,plain,sP2 <=> ((q @ eigen__2) => ((![X1:$i]:(((rel_s4 @ eigen__2) @ X1) => (p @ X1))) => (~((![X1:$i]:(((rel_s4 @ eigen__2) @ X1) => (~((r @ X1))))))))),introduced(definition,[new_symbols(definition,[sP2])])). % 1.07/1.31 thf(sP3,plain,sP3 <=> (![X1:$i]:(((rel_s4 @ eigen__0) @ X1) => ((p @ X1) => (~((![X2:$i]:(((rel_s4 @ X1) @ X2) => (~(((q @ X2) => (r @ X2))))))))))),introduced(definition,[new_symbols(definition,[sP3])])). % 1.07/1.31 thf(sP4,plain,sP4 <=> (((rel_s4 @ eigen__0) @ eigen__2) => (~((r @ eigen__2)))),introduced(definition,[new_symbols(definition,[sP4])])). % 1.07/1.31 thf(sP5,plain,sP5 <=> (((rel_s4 @ eigen__0) @ eigen__0) => ((p @ eigen__0) => (~((![X1:$i]:(((rel_s4 @ eigen__0) @ X1) => (~(((q @ X1) => (r @ X1)))))))))),introduced(definition,[new_symbols(definition,[sP5])])). % 1.07/1.31 thf(sP6,plain,sP6 <=> (r @ eigen__2),introduced(definition,[new_symbols(definition,[sP6])])). % 1.07/1.31 thf(sP7,plain,sP7 <=> (![X1:$i]:(((rel_s4 @ eigen__0) @ X1) => (~(((q @ X1) => ((![X2:$i]:(((rel_s4 @ X1) @ X2) => (p @ X2))) => (~((![X2:$i]:(((rel_s4 @ X1) @ X2) => (~((r @ X2))))))))))))),introduced(definition,[new_symbols(definition,[sP7])])). % 1.07/1.31 thf(sP8,plain,sP8 <=> ((rel_s4 @ eigen__0) @ eigen__2),introduced(definition,[new_symbols(definition,[sP8])])). % 1.07/1.31 thf(sP9,plain,sP9 <=> (![X1:$i]:(((rel_s4 @ eigen__0) @ X1) => (~((r @ X1))))),introduced(definition,[new_symbols(definition,[sP9])])). % 1.07/1.31 thf(sP10,plain,sP10 <=> (![X1:$i]:(((rel_s4 @ eigen__0) @ X1) => (~(((q @ X1) => (r @ X1)))))),introduced(definition,[new_symbols(definition,[sP10])])). % 1.07/1.31 thf(sP11,plain,sP11 <=> ((q @ eigen__0) => ((![X1:$i]:(((rel_s4 @ eigen__0) @ X1) => (p @ X1))) => (~(sP9)))),introduced(definition,[new_symbols(definition,[sP11])])). % 1.07/1.31 thf(sP12,plain,sP12 <=> (((rel_s4 @ eigen__0) @ eigen__0) => (~(sP11))),introduced(definition,[new_symbols(definition,[sP12])])). % 1.07/1.31 thf(sP13,plain,sP13 <=> (sP8 => (~(sP2))),introduced(definition,[new_symbols(definition,[sP13])])). % 1.07/1.31 thf(sP14,plain,sP14 <=> (q @ eigen__2),introduced(definition,[new_symbols(definition,[sP14])])). % 1.07/1.31 thf(sP15,plain,sP15 <=> (((rel_s4 @ eigen__0) @ eigen__0) => (p @ eigen__0)),introduced(definition,[new_symbols(definition,[sP15])])). % 1.07/1.31 thf(sP16,plain,sP16 <=> (![X1:$i]:(((rel_s4 @ eigen__0) @ X1) => (p @ X1))),introduced(definition,[new_symbols(definition,[sP16])])). % 1.07/1.31 thf(sP17,plain,sP17 <=> (p @ eigen__0),introduced(definition,[new_symbols(definition,[sP17])])). % 1.07/1.31 thf(sP18,plain,sP18 <=> (![X1:$i]:((![X2:$i]:(((rel_s4 @ X1) @ X2) => ((p @ X2) => (~((![X3:$i]:(((rel_s4 @ X2) @ X3) => (~(((q @ X3) => (r @ X3))))))))))) => (~((![X2:$i]:(((rel_s4 @ X1) @ X2) => (~(((q @ X2) => ((![X3:$i]:(((rel_s4 @ X2) @ X3) => (p @ X3))) => (~((![X3:$i]:(((rel_s4 @ X2) @ X3) => (~((r @ X3))))))))))))))))),introduced(definition,[new_symbols(definition,[sP18])])). % 1.07/1.31 thf(sP19,plain,sP19 <=> (sP8 => (~((sP14 => sP6)))),introduced(definition,[new_symbols(definition,[sP19])])). % 1.07/1.31 thf(sP20,plain,sP20 <=> ((rel_s4 @ eigen__0) @ eigen__0),introduced(definition,[new_symbols(definition,[sP20])])). % 1.07/1.31 thf(sP21,plain,sP21 <=> (sP14 => sP6),introduced(definition,[new_symbols(definition,[sP21])])). % 1.07/1.31 thf(sP22,plain,sP22 <=> (sP16 => (~(sP9))),introduced(definition,[new_symbols(definition,[sP22])])). % 1.07/1.31 thf(sP23,plain,sP23 <=> (sP3 => (~(sP7))),introduced(definition,[new_symbols(definition,[sP23])])). % 1.07/1.31 thf(sP24,plain,sP24 <=> (sP17 => (~(sP10))),introduced(definition,[new_symbols(definition,[sP24])])). % 1.07/1.31 thf(def_meq_ind,definition,(meq_ind = (^[X1:mu]:(^[X2:mu]:(^[X3:$i]:(X1 = X2)))))). % 1.07/1.31 thf(def_meq_prop,definition,(meq_prop = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((X1 @ X3) = (X2 @ X3))))))). % 1.07/1.31 thf(def_mnot,definition,(mnot = (^[X1:$i>$o]:(^[X2:$i]:(~((X1 @ X2))))))). % 1.07/1.31 thf(def_mor,definition,(mor = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((~((X1 @ X3))) => (X2 @ X3))))))). % 1.07/1.31 thf(def_mand,definition,(mand = (^[X1:$i>$o]:(^[X2:$i>$o]:(mnot @ ((mor @ (mnot @ X1)) @ (mnot @ X2))))))). % 1.07/1.31 thf(def_mimplies,definition,(mimplies = (^[X1:$i>$o]:(mor @ (mnot @ X1))))). % 1.07/1.31 thf(def_mimplied,definition,(mimplied = (^[X1:$i>$o]:(^[X2:$i>$o]:((mor @ (mnot @ X2)) @ X1))))). % 1.07/1.31 thf(def_mequiv,definition,(mequiv = (^[X1:$i>$o]:(^[X2:$i>$o]:((mand @ ((mimplies @ X1) @ X2)) @ ((mimplies @ X2) @ X1)))))). % 1.07/1.31 thf(def_mxor,definition,(mxor = (^[X1:$i>$o]:(^[X2:$i>$o]:(mnot @ ((mequiv @ X1) @ X2)))))). % 1.07/1.31 thf(def_mforall_ind,definition,(mforall_ind = (^[X1:mu>$i>$o]:(^[X2:$i]:(![X3:mu]:((X1 @ X3) @ X2)))))). % 1.07/1.31 thf(def_mforall_prop,definition,(mforall_prop = (^[X1:($i>$o)>$i>$o]:(^[X2:$i]:(![X3:$i>$o]:((X1 @ X3) @ X2)))))). % 1.07/1.31 thf(def_mexists_ind,definition,(mexists_ind = (^[X1:mu>$i>$o]:(mnot @ (mforall_ind @ (^[X2:mu]:(mnot @ (X1 @ X2)))))))). % 1.07/1.31 thf(def_mexists_prop,definition,(mexists_prop = (^[X1:($i>$o)>$i>$o]:(mnot @ (mforall_prop @ (^[X2:$i>$o]:(mnot @ (X1 @ X2)))))))). % 1.07/1.31 thf(def_mtrue,definition,(mtrue = (^[X1:$i]:(~($false))))). % 1.07/1.31 thf(def_mfalse,definition,(mfalse = (mnot @ mtrue))). % 1.07/1.31 thf(def_mbox,definition,(mbox = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:(![X4:$i]:(((X1 @ X3) @ X4) => (X2 @ X4)))))))). % 1.07/1.31 thf(def_mdia,definition,(mdia = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(mnot @ ((mbox @ X1) @ (mnot @ X2))))))). % 1.07/1.31 thf(def_mreflexive,definition,(mreflexive = (^[X1:$i>$i>$o]:(![X2:$i]:((X1 @ X2) @ X2))))). % 1.07/1.31 thf(def_msymmetric,definition,(msymmetric = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(((X1 @ X2) @ X3) => ((X1 @ X3) @ X2))))))). % 1.07/1.31 thf(def_mserial,definition,(mserial = (^[X1:$i>$i>$o]:(![X2:$i]:(~((![X3:$i]:(~(((X1 @ X2) @ X3)))))))))). % 1.07/1.31 thf(def_mtransitive,definition,(mtransitive = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X3) @ X4)))))) => ((X1 @ X2) @ X4)))))))). % 1.07/1.31 thf(def_meuclidean,definition,(meuclidean = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => ((X1 @ X3) @ X4)))))))). % 1.07/1.31 thf(def_mpartially_functional,definition,(mpartially_functional = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => (X3 = X4)))))))). % 1.07/1.31 thf(def_mfunctional,definition,(mfunctional = (^[X1:$i>$i>$o]:(![X2:$i]:(~((![X3:$i]:(((X1 @ X2) @ X3) => (~((![X4:$i]:(((X1 @ X2) @ X4) => (X3 = X4))))))))))))). % 1.07/1.31 thf(def_mweakly_dense,definition,(mweakly_dense = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((X1 @ X2) @ X3) => (~((![X5:$i]:(((X1 @ X2) @ X5) => (~(((X1 @ X5) @ X3)))))))))))))). % 1.07/1.31 thf(def_mweakly_connected,definition,(mweakly_connected = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => ((~(((~(((X1 @ X3) @ X4))) => (X3 = X4)))) => ((X1 @ X4) @ X3))))))))). % 1.07/1.31 thf(def_mweakly_directed,definition,(mweakly_directed = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => (~((![X5:$i]:(((X1 @ X3) @ X5) => (~(((X1 @ X4) @ X5)))))))))))))). % 1.07/1.31 thf(def_mvalid,definition,(mvalid = (!!))). % 1.07/1.31 thf(def_minvalid,definition,(minvalid = (^[X1:$i>$o]:(![X2:$i]:(~((X1 @ X2))))))). % 1.07/1.31 thf(def_msatisfiable,definition,(msatisfiable = (^[X1:$i>$o]:(~((![X2:$i]:(~((X1 @ X2))))))))). % 1.07/1.31 thf(def_mcountersatisfiable,definition,(mcountersatisfiable = (^[X1:$i>$o]:(~(((!!) @ X1)))))). % 1.07/1.31 thf(def_mbox_s4,definition,(mbox_s4 = (^[X1:$i>$o]:(^[X2:$i]:(![X3:$i]:(((rel_s4 @ X2) @ X3) => (X1 @ X3))))))). % 1.07/1.31 thf(def_mdia_s4,definition,(mdia_s4 = (^[X1:$i>$o]:(mnot @ (mbox_s4 @ (mnot @ X1)))))). % 1.07/1.31 thf(prove,conjecture,(![X1:$i]:((~((~((![X2:$i]:(((rel_s4 @ X1) @ X2) => ((~((~((p @ X2))))) => (~((![X3:$i]:(((rel_s4 @ X2) @ X3) => (~(((~((~((q @ X3))))) => (r @ X3))))))))))))))) => (~((![X2:$i]:(((rel_s4 @ X1) @ X2) => (~(((~((~((q @ X2))))) => ((~((~((![X3:$i]:(((rel_s4 @ X2) @ X3) => (p @ X3))))))) => (~((![X3:$i]:(((rel_s4 @ X2) @ X3) => (~((r @ X3)))))))))))))))))). % 1.07/1.31 thf(h1,negated_conjecture,(~(sP18)),inference(assume_negation,[status(cth)],[prove])). % 1.07/1.31 thf(1,plain,(sP2 | sP14),inference(prop_rule,[status(thm)],[])). % 1.07/1.31 thf(2,plain,((~(sP13) | ~(sP8)) | ~(sP2)),inference(prop_rule,[status(thm)],[])). % 1.07/1.31 thf(3,plain,(~(sP7) | sP13),inference(all_rule,[status(thm)],[])). % 1.07/1.31 thf(4,plain,(~(sP9) | sP4),inference(all_rule,[status(thm)],[])). % 1.07/1.31 thf(5,plain,((~(sP4) | ~(sP8)) | ~(sP6)),inference(prop_rule,[status(thm)],[])). % 1.07/1.31 thf(6,plain,((~(sP21) | ~(sP14)) | sP6),inference(prop_rule,[status(thm)],[])). % 1.07/1.31 thf(7,plain,(sP19 | sP21),inference(prop_rule,[status(thm)],[])). % 1.07/1.31 thf(8,plain,(sP19 | sP8),inference(prop_rule,[status(thm)],[])). % 1.07/1.31 thf(9,plain,(sP10 | ~(sP19)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__2])). % 1.07/1.31 thf(10,plain,(~(sP3) | sP5),inference(all_rule,[status(thm)],[])). % 1.07/1.31 thf(11,plain,((~(sP5) | ~(sP20)) | sP24),inference(prop_rule,[status(thm)],[])). % 1.07/1.31 thf(12,plain,((~(sP24) | ~(sP17)) | ~(sP10)),inference(prop_rule,[status(thm)],[])). % 1.07/1.31 thf(13,plain,(~(sP16) | sP15),inference(all_rule,[status(thm)],[])). % 1.07/1.31 thf(14,plain,((~(sP15) | ~(sP20)) | sP17),inference(prop_rule,[status(thm)],[])). % 1.07/1.31 thf(15,plain,(sP22 | sP9),inference(prop_rule,[status(thm)],[])). % 1.07/1.31 thf(16,plain,(sP22 | sP16),inference(prop_rule,[status(thm)],[])). % 1.07/1.31 thf(17,plain,(sP11 | ~(sP22)),inference(prop_rule,[status(thm)],[])). % 1.07/1.31 thf(18,plain,((~(sP12) | ~(sP20)) | ~(sP11)),inference(prop_rule,[status(thm)],[])). % 1.07/1.31 thf(19,plain,(~(sP1) | sP20),inference(all_rule,[status(thm)],[])). % 1.07/1.31 thf(20,plain,(~(sP7) | sP12),inference(all_rule,[status(thm)],[])). % 1.07/1.31 thf(21,plain,(sP23 | sP7),inference(prop_rule,[status(thm)],[])). % 1.07/1.31 thf(22,plain,(sP23 | sP3),inference(prop_rule,[status(thm)],[])). % 1.07/1.31 thf(23,plain,(sP18 | ~(sP23)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__0])). % 1.07/1.31 thf(a1,axiom,(mreflexive @ rel_s4)). % 1.07/1.31 thf(24,plain,sP1,inference(preprocess,[status(thm)],[a1]). % 1.07/1.31 thf(25,plain,$false,inference(prop_unsat,[status(thm),assumptions([h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,h1])). % 1.07/1.31 thf(26,plain,$false,inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[25,h0])). % 1.07/1.31 thf(0,theorem,(![X1:$i]:((~((~((![X2:$i]:(((rel_s4 @ X1) @ X2) => ((~((~((p @ X2))))) => (~((![X3:$i]:(((rel_s4 @ X2) @ X3) => (~(((~((~((q @ X3))))) => (r @ X3))))))))))))))) => (~((![X2:$i]:(((rel_s4 @ X1) @ X2) => (~(((~((~((q @ X2))))) => ((~((~((![X3:$i]:(((rel_s4 @ X2) @ X3) => (p @ X3))))))) => (~((![X3:$i]:(((rel_s4 @ X2) @ X3) => (~((r @ X3))))))))))))))))),inference(contra,[status(thm),contra(discharge,[h1])],[25,h1])). % 1.07/1.31 % SZS output end Proof %------------------------------------------------------------------------------