%------------------------------------------------------------------------------ % File : Satallax---3.5 % Problem : SYO469^5 : 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 : n031.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 03:25:47 PM UTC 2025 % Result : Theorem 0.14s 0.38s % Output : Proof 0.14s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.08/0.13 % Problem : SYO469^5 : TPTP v9.2.0. Released v4.0.0. % 0.08/0.13 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % 0.14/0.34 % Computer : n031.cluster.edu % 0.14/0.34 % Model : x86_64 x86_64 % 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.14/0.34 % Memory : 8042.1875MB % 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64 % 0.14/0.35 % CPULimit : 300 % 0.14/0.35 % WCLimit : 300 % 0.14/0.35 % DateTime : Thu Oct 9 00:55:53 EDT 2025 % 0.14/0.35 % CPUTime : % 0.14/0.38 % SZS status Theorem % 0.14/0.38 % Mode: mode213 % 0.14/0.38 % Inferences: 42 % 0.14/0.38 % SZS output start Proof % 0.14/0.38 thf(ty_p, type, p : ($i>$o)). % 0.14/0.38 thf(ty_rel_s4, type, rel_s4 : ($i>$i>$o)). % 0.14/0.38 thf(ty_eigen__2, type, eigen__2 : $i). % 0.14/0.38 thf(ty_q, type, q : ($i>$o)). % 0.14/0.38 thf(ty_eigen__1, type, eigen__1 : $i). % 0.14/0.38 thf(ty_eigen__0, type, eigen__0 : $i). % 0.14/0.38 thf(ty_eigen__4, type, eigen__4 : $i). % 0.14/0.38 thf(h0, assumption, (![X1:$i>$o]:(![X2:$i]:((X1 @ X2) => (X1 @ (eps__0 @ X1))))),introduced(assumption,[])). % 0.14/0.38 thf(eigendef_eigen__4, definition, eigen__4 = (eps__0 @ (^[X1:$i]:(~((((rel_s4 @ eigen__2) @ X1) => (p @ X1)))))), introduced(definition,[new_symbols(definition,[eigen__4])])). % 0.14/0.38 thf(sP1,plain,sP1 <=> ((rel_s4 @ eigen__1) @ eigen__2),introduced(definition,[new_symbols(definition,[sP1])])). % 0.14/0.38 thf(sP2,plain,sP2 <=> (((rel_s4 @ eigen__1) @ eigen__4) => (p @ eigen__4)),introduced(definition,[new_symbols(definition,[sP2])])). % 0.14/0.38 thf(sP3,plain,sP3 <=> (![X1:$i]:((~((sP1 => (~(((rel_s4 @ eigen__2) @ X1)))))) => ((rel_s4 @ eigen__1) @ X1))),introduced(definition,[new_symbols(definition,[sP3])])). % 0.14/0.38 thf(sP4,plain,sP4 <=> (![X1:$i]:(((rel_s4 @ eigen__0) @ X1) => ((![X2:$i]:(((rel_s4 @ X1) @ X2) => (p @ X2))) => (q @ X1)))),introduced(definition,[new_symbols(definition,[sP4])])). % 0.14/0.38 thf(sP5,plain,sP5 <=> (sP1 => (~(((rel_s4 @ eigen__2) @ eigen__4)))),introduced(definition,[new_symbols(definition,[sP5])])). % 0.14/0.38 thf(sP6,plain,sP6 <=> (![X1:$i]:(![X2:$i]:((~((((rel_s4 @ eigen__1) @ X1) => (~(((rel_s4 @ X1) @ X2)))))) => ((rel_s4 @ eigen__1) @ X2)))),introduced(definition,[new_symbols(definition,[sP6])])). % 0.14/0.38 thf(sP7,plain,sP7 <=> ((rel_s4 @ eigen__1) @ eigen__4),introduced(definition,[new_symbols(definition,[sP7])])). % 0.14/0.38 thf(sP8,plain,sP8 <=> (![X1:$i]:((~((((rel_s4 @ eigen__0) @ eigen__1) => (~(((rel_s4 @ eigen__1) @ X1)))))) => ((rel_s4 @ eigen__0) @ X1))),introduced(definition,[new_symbols(definition,[sP8])])). % 0.14/0.38 thf(sP9,plain,sP9 <=> ((~((((rel_s4 @ eigen__0) @ eigen__1) => (~(sP1))))) => ((rel_s4 @ eigen__0) @ eigen__2)),introduced(definition,[new_symbols(definition,[sP9])])). % 0.14/0.38 thf(sP10,plain,sP10 <=> (![X1:$i]:(![X2:$i]:(![X3:$i]:((~((((rel_s4 @ X1) @ X2) => (~(((rel_s4 @ X2) @ X3)))))) => ((rel_s4 @ X1) @ X3))))),introduced(definition,[new_symbols(definition,[sP10])])). % 0.14/0.38 thf(sP11,plain,sP11 <=> (![X1:$i]:(![X2:$i]:((~((((rel_s4 @ eigen__0) @ X1) => (~(((rel_s4 @ X1) @ X2)))))) => ((rel_s4 @ eigen__0) @ X2)))),introduced(definition,[new_symbols(definition,[sP11])])). % 0.14/0.38 thf(sP12,plain,sP12 <=> (((rel_s4 @ eigen__0) @ eigen__1) => (~(sP1))),introduced(definition,[new_symbols(definition,[sP12])])). % 0.14/0.38 thf(sP13,plain,sP13 <=> (![X1:$i]:(((rel_s4 @ eigen__2) @ X1) => (p @ X1))),introduced(definition,[new_symbols(definition,[sP13])])). % 0.14/0.38 thf(sP14,plain,sP14 <=> (((rel_s4 @ eigen__0) @ eigen__2) => (sP13 => (q @ eigen__2))),introduced(definition,[new_symbols(definition,[sP14])])). % 0.14/0.38 thf(sP15,plain,sP15 <=> ((rel_s4 @ eigen__2) @ eigen__4),introduced(definition,[new_symbols(definition,[sP15])])). % 0.14/0.38 thf(sP16,plain,sP16 <=> (![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => (p @ X1))),introduced(definition,[new_symbols(definition,[sP16])])). % 0.14/0.38 thf(sP17,plain,sP17 <=> (q @ eigen__2),introduced(definition,[new_symbols(definition,[sP17])])). % 0.14/0.38 thf(sP18,plain,sP18 <=> ((~(sP5)) => sP7),introduced(definition,[new_symbols(definition,[sP18])])). % 0.14/0.38 thf(sP19,plain,sP19 <=> (sP13 => sP17),introduced(definition,[new_symbols(definition,[sP19])])). % 0.14/0.38 thf(sP20,plain,sP20 <=> ((rel_s4 @ eigen__0) @ eigen__2),introduced(definition,[new_symbols(definition,[sP20])])). % 0.14/0.38 thf(sP21,plain,sP21 <=> ((rel_s4 @ eigen__0) @ eigen__1),introduced(definition,[new_symbols(definition,[sP21])])). % 0.14/0.38 thf(sP22,plain,sP22 <=> (p @ eigen__4),introduced(definition,[new_symbols(definition,[sP22])])). % 0.14/0.38 thf(sP23,plain,sP23 <=> (sP15 => sP22),introduced(definition,[new_symbols(definition,[sP23])])). % 0.14/0.38 thf(def_meq_ind,definition,(meq_ind = (^[X1:mu]:(^[X2:mu]:(^[X3:$i]:(X1 = X2)))))). % 0.14/0.38 thf(def_meq_prop,definition,(meq_prop = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((X1 @ X3) = (X2 @ X3))))))). % 0.14/0.38 thf(def_mnot,definition,(mnot = (^[X1:$i>$o]:(^[X2:$i]:(~((X1 @ X2))))))). % 0.14/0.38 thf(def_mor,definition,(mor = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((~((X1 @ X3))) => (X2 @ X3))))))). % 0.14/0.38 thf(def_mand,definition,(mand = (^[X1:$i>$o]:(^[X2:$i>$o]:(mnot @ ((mor @ (mnot @ X1)) @ (mnot @ X2))))))). % 0.14/0.38 thf(def_mimplies,definition,(mimplies = (^[X1:$i>$o]:(mor @ (mnot @ X1))))). % 0.14/0.38 thf(def_mimplied,definition,(mimplied = (^[X1:$i>$o]:(^[X2:$i>$o]:((mor @ (mnot @ X2)) @ X1))))). % 0.14/0.38 thf(def_mequiv,definition,(mequiv = (^[X1:$i>$o]:(^[X2:$i>$o]:((mand @ ((mimplies @ X1) @ X2)) @ ((mimplies @ X2) @ X1)))))). % 0.14/0.38 thf(def_mxor,definition,(mxor = (^[X1:$i>$o]:(^[X2:$i>$o]:(mnot @ ((mequiv @ X1) @ X2)))))). % 0.14/0.38 thf(def_mforall_ind,definition,(mforall_ind = (^[X1:mu>$i>$o]:(^[X2:$i]:(![X3:mu]:((X1 @ X3) @ X2)))))). % 0.14/0.38 thf(def_mforall_prop,definition,(mforall_prop = (^[X1:($i>$o)>$i>$o]:(^[X2:$i]:(![X3:$i>$o]:((X1 @ X3) @ X2)))))). % 0.14/0.38 thf(def_mexists_ind,definition,(mexists_ind = (^[X1:mu>$i>$o]:(mnot @ (mforall_ind @ (^[X2:mu]:(mnot @ (X1 @ X2)))))))). % 0.14/0.38 thf(def_mexists_prop,definition,(mexists_prop = (^[X1:($i>$o)>$i>$o]:(mnot @ (mforall_prop @ (^[X2:$i>$o]:(mnot @ (X1 @ X2)))))))). % 0.14/0.38 thf(def_mtrue,definition,(mtrue = (^[X1:$i]:(~($false))))). % 0.14/0.38 thf(def_mfalse,definition,(mfalse = (mnot @ mtrue))). % 0.14/0.38 thf(def_mbox,definition,(mbox = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:(![X4:$i]:(((X1 @ X3) @ X4) => (X2 @ X4)))))))). % 0.14/0.38 thf(def_mdia,definition,(mdia = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(mnot @ ((mbox @ X1) @ (mnot @ X2))))))). % 0.14/0.38 thf(def_mreflexive,definition,(mreflexive = (^[X1:$i>$i>$o]:(![X2:$i]:((X1 @ X2) @ X2))))). % 0.14/0.38 thf(def_msymmetric,definition,(msymmetric = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(((X1 @ X2) @ X3) => ((X1 @ X3) @ X2))))))). % 0.14/0.38 thf(def_mserial,definition,(mserial = (^[X1:$i>$i>$o]:(![X2:$i]:(~((![X3:$i]:(~(((X1 @ X2) @ X3)))))))))). % 0.14/0.38 thf(def_mtransitive,definition,(mtransitive = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X3) @ X4)))))) => ((X1 @ X2) @ X4)))))))). % 0.14/0.38 thf(def_meuclidean,definition,(meuclidean = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => ((X1 @ X3) @ X4)))))))). % 0.14/0.38 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)))))))). % 0.14/0.38 thf(def_mfunctional,definition,(mfunctional = (^[X1:$i>$i>$o]:(![X2:$i]:(~((![X3:$i]:(((X1 @ X2) @ X3) => (~((![X4:$i]:(((X1 @ X2) @ X4) => (X3 = X4))))))))))))). % 0.14/0.38 thf(def_mweakly_dense,definition,(mweakly_dense = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((X1 @ X2) @ X3) => (~((![X5:$i]:(((X1 @ X2) @ X5) => (~(((X1 @ X5) @ X3)))))))))))))). % 0.14/0.38 thf(def_mweakly_connected,definition,(mweakly_connected = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => ((~(((~(((X1 @ X3) @ X4))) => (X3 = X4)))) => ((X1 @ X4) @ X3))))))))). % 0.14/0.38 thf(def_mweakly_directed,definition,(mweakly_directed = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => (~((![X5:$i]:(((X1 @ X3) @ X5) => (~(((X1 @ X4) @ X5)))))))))))))). % 0.14/0.38 thf(def_mvalid,definition,(mvalid = (!!))). % 0.14/0.38 thf(def_minvalid,definition,(minvalid = (^[X1:$i>$o]:(![X2:$i]:(~((X1 @ X2))))))). % 0.14/0.38 thf(def_msatisfiable,definition,(msatisfiable = (^[X1:$i>$o]:(~((![X2:$i]:(~((X1 @ X2))))))))). % 0.14/0.38 thf(def_mcountersatisfiable,definition,(mcountersatisfiable = (^[X1:$i>$o]:(~(((!!) @ X1)))))). % 0.14/0.38 thf(def_mbox_s4,definition,(mbox_s4 = (^[X1:$i>$o]:(^[X2:$i]:(![X3:$i]:(((rel_s4 @ X2) @ X3) => (X1 @ X3))))))). % 0.14/0.38 thf(def_mdia_s4,definition,(mdia_s4 = (^[X1:$i>$o]:(mnot @ (mbox_s4 @ (mnot @ X1)))))). % 0.14/0.38 thf(prove,conjecture,(![X1:$i]:((~((~((![X2:$i]:(((rel_s4 @ X1) @ X2) => ((~((~((![X3:$i]:(((rel_s4 @ X2) @ X3) => (p @ X3))))))) => (q @ X2)))))))) => (![X2:$i]:(((rel_s4 @ X1) @ X2) => ((~((~((![X3:$i]:(((rel_s4 @ X2) @ X3) => (p @ X3))))))) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => (q @ X3))))))))). % 0.14/0.38 thf(h1,negated_conjecture,(~((![X1:$i]:((![X2:$i]:(((rel_s4 @ X1) @ X2) => ((![X3:$i]:(((rel_s4 @ X2) @ X3) => (p @ X3))) => (q @ X2)))) => (![X2:$i]:(((rel_s4 @ X1) @ X2) => ((![X3:$i]:(((rel_s4 @ X2) @ X3) => (p @ X3))) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => (q @ X3)))))))))),inference(assume_negation,[status(cth)],[prove])). % 0.14/0.38 thf(h2,assumption,(~((sP4 => (![X1:$i]:(((rel_s4 @ eigen__0) @ X1) => ((![X2:$i]:(((rel_s4 @ X1) @ X2) => (p @ X2))) => (![X2:$i]:(((rel_s4 @ X1) @ X2) => (q @ X2))))))))),introduced(assumption,[])). % 0.14/0.38 thf(h3,assumption,sP4,introduced(assumption,[])). % 0.14/0.38 thf(h4,assumption,(~((![X1:$i]:(((rel_s4 @ eigen__0) @ X1) => ((![X2:$i]:(((rel_s4 @ X1) @ X2) => (p @ X2))) => (![X2:$i]:(((rel_s4 @ X1) @ X2) => (q @ X2)))))))),introduced(assumption,[])). % 0.14/0.38 thf(h5,assumption,(~((sP21 => (sP16 => (![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => (q @ X1))))))),introduced(assumption,[])). % 0.14/0.38 thf(h6,assumption,sP21,introduced(assumption,[])). % 0.14/0.38 thf(h7,assumption,(~((sP16 => (![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => (q @ X1)))))),introduced(assumption,[])). % 0.14/0.38 thf(h8,assumption,sP16,introduced(assumption,[])). % 0.14/0.38 thf(h9,assumption,(~((![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => (q @ X1))))),introduced(assumption,[])). % 0.14/0.38 thf(h10,assumption,(~((sP1 => sP17))),introduced(assumption,[])). % 0.14/0.38 thf(h11,assumption,sP1,introduced(assumption,[])). % 0.14/0.38 thf(h12,assumption,(~(sP17)),introduced(assumption,[])). % 0.14/0.38 thf(1,plain,(~(sP8) | sP9),inference(all_rule,[status(thm)],[])). % 0.14/0.38 thf(2,plain,((~(sP9) | sP12) | sP20),inference(prop_rule,[status(thm)],[])). % 0.14/0.38 thf(3,plain,((~(sP12) | ~(sP21)) | ~(sP1)),inference(prop_rule,[status(thm)],[])). % 0.14/0.38 thf(4,plain,(~(sP10) | sP11),inference(all_rule,[status(thm)],[])). % 0.14/0.38 thf(5,plain,(~(sP11) | sP8),inference(all_rule,[status(thm)],[])). % 0.14/0.38 thf(6,plain,(~(sP10) | sP6),inference(all_rule,[status(thm)],[])). % 0.14/0.38 thf(7,plain,(~(sP6) | sP3),inference(all_rule,[status(thm)],[])). % 0.14/0.38 thf(8,plain,(~(sP3) | sP18),inference(all_rule,[status(thm)],[])). % 0.14/0.38 thf(9,plain,((~(sP18) | sP5) | sP7),inference(prop_rule,[status(thm)],[])). % 0.14/0.38 thf(10,plain,((~(sP5) | ~(sP1)) | ~(sP15)),inference(prop_rule,[status(thm)],[])). % 0.14/0.38 thf(11,plain,(~(sP16) | sP2),inference(all_rule,[status(thm)],[])). % 0.14/0.38 thf(12,plain,((~(sP2) | ~(sP7)) | sP22),inference(prop_rule,[status(thm)],[])). % 0.14/0.38 thf(13,plain,(sP23 | ~(sP22)),inference(prop_rule,[status(thm)],[])). % 0.14/0.38 thf(14,plain,(sP23 | sP15),inference(prop_rule,[status(thm)],[])). % 0.14/0.38 thf(15,plain,(sP13 | ~(sP23)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__4])). % 0.14/0.38 thf(16,plain,(~(sP4) | sP14),inference(all_rule,[status(thm)],[])). % 0.14/0.38 thf(17,plain,((~(sP14) | ~(sP20)) | sP19),inference(prop_rule,[status(thm)],[])). % 0.14/0.38 thf(18,plain,((~(sP19) | ~(sP13)) | sP17),inference(prop_rule,[status(thm)],[])). % 0.14/0.38 thf(a2,axiom,(mtransitive @ rel_s4)). % 0.14/0.38 thf(19,plain,sP10,inference(preprocess,[status(thm)],[a2]). % 0.14/0.38 thf(20,plain,$false,inference(prop_unsat,[status(thm),assumptions([h11,h12,h10,h8,h9,h6,h7,h5,h3,h4,h2,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,h3,h6,h8,h11,h12])). % 0.14/0.38 thf(21,plain,$false,inference(tab_negimp,[status(thm),assumptions([h10,h8,h9,h6,h7,h5,h3,h4,h2,h1,h0]),tab_negimp(discharge,[h11,h12])],[h10,20,h11,h12])). % 0.14/0.38 thf(22,plain,$false,inference(tab_negall,[status(thm),assumptions([h8,h9,h6,h7,h5,h3,h4,h2,h1,h0]),tab_negall(discharge,[h10]),tab_negall(eigenvar,eigen__2)],[h9,21,h10])). % 0.14/0.38 thf(23,plain,$false,inference(tab_negimp,[status(thm),assumptions([h6,h7,h5,h3,h4,h2,h1,h0]),tab_negimp(discharge,[h8,h9])],[h7,22,h8,h9])). % 0.14/0.38 thf(24,plain,$false,inference(tab_negimp,[status(thm),assumptions([h5,h3,h4,h2,h1,h0]),tab_negimp(discharge,[h6,h7])],[h5,23,h6,h7])). % 0.14/0.38 thf(25,plain,$false,inference(tab_negall,[status(thm),assumptions([h3,h4,h2,h1,h0]),tab_negall(discharge,[h5]),tab_negall(eigenvar,eigen__1)],[h4,24,h5])). % 0.14/0.38 thf(26,plain,$false,inference(tab_negimp,[status(thm),assumptions([h2,h1,h0]),tab_negimp(discharge,[h3,h4])],[h2,25,h3,h4])). % 0.14/0.38 thf(27,plain,$false,inference(tab_negall,[status(thm),assumptions([h1,h0]),tab_negall(discharge,[h2]),tab_negall(eigenvar,eigen__0)],[h1,26,h2])). % 0.14/0.38 thf(28,plain,$false,inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[27,h0])). % 0.14/0.38 thf(0,theorem,(![X1:$i]:((~((~((![X2:$i]:(((rel_s4 @ X1) @ X2) => ((~((~((![X3:$i]:(((rel_s4 @ X2) @ X3) => (p @ X3))))))) => (q @ X2)))))))) => (![X2:$i]:(((rel_s4 @ X1) @ X2) => ((~((~((![X3:$i]:(((rel_s4 @ X2) @ X3) => (p @ X3))))))) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => (q @ X3)))))))),inference(contra,[status(thm),contra(discharge,[h1])],[27,h1])). % 0.14/0.38 % SZS output end Proof %------------------------------------------------------------------------------