%------------------------------------------------------------------------------ % File : Satallax---3.5 % Problem : SET002^7 : 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 : n026.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 02:25:00 PM UTC 2025 % Result : Theorem 1.09s 1.29s % Output : Proof 1.09s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.07/0.13 % Problem : SET002^7 : TPTP v9.2.0. Released v5.5.0. % 0.07/0.13 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % 0.14/0.34 % Computer : n026.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 % 0.14/0.34 % CPULimit : 300 % 0.14/0.34 % WCLimit : 300 % 0.14/0.35 % DateTime : Wed Oct 8 14:34:38 EDT 2025 % 0.14/0.35 % CPUTime : % 1.09/1.29 % SZS status Theorem % 1.09/1.29 % Mode: mode506 % 1.09/1.29 % Inferences: 8177 % 1.09/1.29 % SZS output start Proof % 1.09/1.29 thf(ty_mu, type, mu : $tType). % 1.09/1.29 thf(ty_subset, type, subset : (mu>mu>$i>$o)). % 1.09/1.29 thf(ty_union, type, union : (mu>mu>mu)). % 1.09/1.29 thf(ty_eigen__1, type, eigen__1 : mu). % 1.09/1.29 thf(ty_eigen__0, type, eigen__0 : $i). % 1.09/1.29 thf(ty_exists_in_world, type, exists_in_world : (mu>$i>$o)). % 1.09/1.29 thf(ty_qmltpeq, type, qmltpeq : (mu>mu>$i>$o)). % 1.09/1.29 thf(h0, assumption, (![X1:mu>$o]:(![X2:mu]:((X1 @ X2) => (X1 @ (eps__0 @ X1))))),introduced(assumption,[])). % 1.09/1.29 thf(eigendef_eigen__1, definition, eigen__1 = (eps__0 @ (^[X1:mu]:(~((((exists_in_world @ X1) @ eigen__0) => (((qmltpeq @ ((union @ X1) @ X1)) @ X1) @ eigen__0)))))), introduced(definition,[new_symbols(definition,[eigen__1])])). % 1.09/1.29 thf(h1, assumption, (![X1:$i>$o]:(![X2:$i]:((X1 @ X2) => (X1 @ (eps__1 @ X1))))),introduced(assumption,[])). % 1.09/1.29 thf(eigendef_eigen__0, definition, eigen__0 = (eps__1 @ (^[X1:$i]:(~((![X2:mu]:(((exists_in_world @ X2) @ X1) => (((qmltpeq @ ((union @ X2) @ X2)) @ X2) @ X1))))))), introduced(definition,[new_symbols(definition,[eigen__0])])). % 1.09/1.29 thf(sP1,plain,sP1 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => ((((subset @ eigen__1) @ X1) @ eigen__0) => (((qmltpeq @ ((union @ eigen__1) @ X1)) @ X1) @ eigen__0)))),introduced(definition,[new_symbols(definition,[sP1])])). % 1.09/1.29 thf(sP2,plain,sP2 <=> (((exists_in_world @ eigen__1) @ eigen__0) => (((subset @ eigen__1) @ eigen__1) @ eigen__0)),introduced(definition,[new_symbols(definition,[sP2])])). % 1.09/1.29 thf(sP3,plain,sP3 <=> (((exists_in_world @ eigen__1) @ eigen__0) => sP1),introduced(definition,[new_symbols(definition,[sP3])])). % 1.09/1.29 thf(sP4,plain,sP4 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => (![X2:mu]:(((exists_in_world @ X2) @ eigen__0) => ((((subset @ X1) @ X2) @ eigen__0) => (((qmltpeq @ ((union @ X1) @ X2)) @ X2) @ eigen__0)))))),introduced(definition,[new_symbols(definition,[sP4])])). % 1.09/1.29 thf(sP5,plain,sP5 <=> (((subset @ eigen__1) @ eigen__1) @ eigen__0),introduced(definition,[new_symbols(definition,[sP5])])). % 1.09/1.29 thf(sP6,plain,sP6 <=> (((qmltpeq @ ((union @ eigen__1) @ eigen__1)) @ eigen__1) @ eigen__0),introduced(definition,[new_symbols(definition,[sP6])])). % 1.09/1.29 thf(sP7,plain,sP7 <=> (((exists_in_world @ eigen__1) @ eigen__0) => (sP5 => sP6)),introduced(definition,[new_symbols(definition,[sP7])])). % 1.09/1.29 thf(sP8,plain,sP8 <=> (![X1:$i]:(![X2:mu]:(((exists_in_world @ X2) @ X1) => (((subset @ X2) @ X2) @ X1)))),introduced(definition,[new_symbols(definition,[sP8])])). % 1.09/1.29 thf(sP9,plain,sP9 <=> (((exists_in_world @ eigen__1) @ eigen__0) => sP6),introduced(definition,[new_symbols(definition,[sP9])])). % 1.09/1.29 thf(sP10,plain,sP10 <=> (![X1:$i]:(![X2:mu]:(((exists_in_world @ X2) @ X1) => (![X3:mu]:(((exists_in_world @ X3) @ X1) => ((((subset @ X2) @ X3) @ X1) => (((qmltpeq @ ((union @ X2) @ X3)) @ X3) @ X1))))))),introduced(definition,[new_symbols(definition,[sP10])])). % 1.09/1.29 thf(sP11,plain,sP11 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => (((qmltpeq @ ((union @ X1) @ X1)) @ X1) @ eigen__0))),introduced(definition,[new_symbols(definition,[sP11])])). % 1.09/1.29 thf(sP12,plain,sP12 <=> (![X1:$i]:(![X2:mu]:(((exists_in_world @ X2) @ X1) => (((qmltpeq @ ((union @ X2) @ X2)) @ X2) @ X1)))),introduced(definition,[new_symbols(definition,[sP12])])). % 1.09/1.29 thf(sP13,plain,sP13 <=> (sP5 => sP6),introduced(definition,[new_symbols(definition,[sP13])])). % 1.09/1.29 thf(sP14,plain,sP14 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => (((subset @ X1) @ X1) @ eigen__0))),introduced(definition,[new_symbols(definition,[sP14])])). % 1.09/1.29 thf(sP15,plain,sP15 <=> ((exists_in_world @ eigen__1) @ eigen__0),introduced(definition,[new_symbols(definition,[sP15])])). % 1.09/1.29 thf(def_meq_prop,definition,(meq_prop = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((X1 @ X3) = (X2 @ X3))))))). % 1.09/1.29 thf(def_mnot,definition,(mnot = (^[X1:$i>$o]:(^[X2:$i]:(~((X1 @ X2))))))). % 1.09/1.29 thf(def_mor,definition,(mor = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((~((X1 @ X3))) => (X2 @ X3))))))). % 1.09/1.29 thf(def_mbox,definition,(mbox = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:(![X4:$i]:(((X1 @ X3) @ X4) => (X2 @ X4)))))))). % 1.09/1.29 thf(def_mforall_prop,definition,(mforall_prop = (^[X1:($i>$o)>$i>$o]:(^[X2:$i]:(![X3:$i>$o]:((X1 @ X3) @ X2)))))). % 1.09/1.29 thf(def_mtrue,definition,(mtrue = (^[X1:$i]:(~($false))))). % 1.09/1.29 thf(def_mfalse,definition,(mfalse = (mnot @ mtrue))). % 1.09/1.29 thf(def_mand,definition,(mand = (^[X1:$i>$o]:(^[X2:$i>$o]:(mnot @ ((mor @ (mnot @ X1)) @ (mnot @ X2))))))). % 1.09/1.29 thf(def_mimplies,definition,(mimplies = (^[X1:$i>$o]:(mor @ (mnot @ X1))))). % 1.09/1.29 thf(def_mimplied,definition,(mimplied = (^[X1:$i>$o]:(^[X2:$i>$o]:((mor @ (mnot @ X2)) @ X1))))). % 1.09/1.29 thf(def_mequiv,definition,(mequiv = (^[X1:$i>$o]:(^[X2:$i>$o]:((mand @ ((mimplies @ X1) @ X2)) @ ((mimplies @ X2) @ X1)))))). % 1.09/1.29 thf(def_mxor,definition,(mxor = (^[X1:$i>$o]:(^[X2:$i>$o]:(mnot @ ((mequiv @ X1) @ X2)))))). % 1.09/1.29 thf(def_mdia,definition,(mdia = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(mnot @ ((mbox @ X1) @ (mnot @ X2))))))). % 1.09/1.29 thf(def_mforall_ind,definition,(mforall_ind = (^[X1:mu>$i>$o]:(^[X2:$i]:(![X3:mu]:(((exists_in_world @ X3) @ X2) => ((X1 @ X3) @ X2))))))). % 1.09/1.29 thf(def_mexists_ind,definition,(mexists_ind = (^[X1:mu>$i>$o]:(mnot @ (mforall_ind @ (^[X2:mu]:(mnot @ (X1 @ X2)))))))). % 1.09/1.29 thf(def_mexists_prop,definition,(mexists_prop = (^[X1:($i>$o)>$i>$o]:(mnot @ (mforall_prop @ (^[X2:$i>$o]:(mnot @ (X1 @ X2)))))))). % 1.09/1.29 thf(def_mreflexive,definition,(mreflexive = (^[X1:$i>$i>$o]:(![X2:$i]:((X1 @ X2) @ X2))))). % 1.09/1.29 thf(def_msymmetric,definition,(msymmetric = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(((X1 @ X2) @ X3) => ((X1 @ X3) @ X2))))))). % 1.09/1.29 thf(def_mserial,definition,(mserial = (^[X1:$i>$i>$o]:(![X2:$i]:(~((![X3:$i]:(~(((X1 @ X2) @ X3)))))))))). % 1.09/1.29 thf(def_mtransitive,definition,(mtransitive = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X3) @ X4)))))) => ((X1 @ X2) @ X4)))))))). % 1.09/1.29 thf(def_meuclidean,definition,(meuclidean = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => ((X1 @ X3) @ X4)))))))). % 1.09/1.29 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)))))))). % 1.09/1.29 thf(def_mfunctional,definition,(mfunctional = (^[X1:$i>$i>$o]:(![X2:$i]:(~((![X3:$i]:(((X1 @ X2) @ X3) => (~((![X4:$i]:(((X1 @ X2) @ X4) => (X3 = X4))))))))))))). % 1.09/1.29 thf(def_mweakly_dense,definition,(mweakly_dense = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((X1 @ X2) @ X3) => (~((![X5:$i]:(((X1 @ X2) @ X5) => (~(((X1 @ X5) @ X3)))))))))))))). % 1.09/1.29 thf(def_mweakly_connected,definition,(mweakly_connected = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => ((~(((~(((X1 @ X3) @ X4))) => (X3 = X4)))) => ((X1 @ X4) @ X3))))))))). % 1.09/1.29 thf(def_mweakly_directed,definition,(mweakly_directed = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => (~((![X5:$i]:(((X1 @ X3) @ X5) => (~(((X1 @ X4) @ X5)))))))))))))). % 1.09/1.29 thf(def_mvalid,definition,(mvalid = (!!))). % 1.09/1.29 thf(def_msatisfiable,definition,(msatisfiable = (^[X1:$i>$o]:(~((![X2:$i]:(~((X1 @ X2))))))))). % 1.09/1.29 thf(def_mcountersatisfiable,definition,(mcountersatisfiable = (^[X1:$i>$o]:(~(((!!) @ X1)))))). % 1.09/1.29 thf(def_minvalid,definition,(minvalid = (^[X1:$i>$o]:(![X2:$i]:(~((X1 @ X2))))))). % 1.09/1.29 thf(def_mbox_s4,definition,(mbox_s4 = (^[X1:$i>$o]:(^[X2:$i]:(![X3:$i]:(((rel_s4 @ X2) @ X3) => (X1 @ X3))))))). % 1.09/1.29 thf(def_mdia_s4,definition,(mdia_s4 = (^[X1:$i>$o]:(mnot @ (mbox_s4 @ (mnot @ X1)))))). % 1.09/1.29 thf(prove_idempotency_of_union,conjecture,sP12). % 1.09/1.29 thf(h2,negated_conjecture,(~(sP12)),inference(assume_negation,[status(cth)],[prove_idempotency_of_union])). % 1.09/1.29 thf(1,plain,(~(sP14) | sP2),inference(all_rule,[status(thm)],[])). % 1.09/1.29 thf(2,plain,((~(sP2) | ~(sP15)) | sP5),inference(prop_rule,[status(thm)],[])). % 1.09/1.29 thf(3,plain,(~(sP10) | sP4),inference(all_rule,[status(thm)],[])). % 1.09/1.29 thf(4,plain,(~(sP4) | sP3),inference(all_rule,[status(thm)],[])). % 1.09/1.29 thf(5,plain,((~(sP3) | ~(sP15)) | sP1),inference(prop_rule,[status(thm)],[])). % 1.09/1.29 thf(6,plain,(~(sP1) | sP7),inference(all_rule,[status(thm)],[])). % 1.09/1.29 thf(7,plain,((~(sP7) | ~(sP15)) | sP13),inference(prop_rule,[status(thm)],[])). % 1.09/1.29 thf(8,plain,((~(sP13) | ~(sP5)) | sP6),inference(prop_rule,[status(thm)],[])). % 1.09/1.29 thf(9,plain,(~(sP8) | sP14),inference(all_rule,[status(thm)],[])). % 1.09/1.29 thf(10,plain,(sP9 | ~(sP6)),inference(prop_rule,[status(thm)],[])). % 1.09/1.29 thf(11,plain,(sP9 | sP15),inference(prop_rule,[status(thm)],[])). % 1.09/1.29 thf(12,plain,(sP11 | ~(sP9)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__1])). % 1.09/1.29 thf(13,plain,(sP12 | ~(sP11)),inference(eigen_choice_rule,[status(thm),assumptions([h1])],[h1,eigendef_eigen__0])). % 1.09/1.29 thf(reflexivity_of_subset,axiom,(mvalid @ (mforall_ind @ (^[X1:mu]:((subset @ X1) @ X1))))). % 1.09/1.29 thf(14,plain,sP8,inference(preprocess,[status(thm)],[reflexivity_of_subset]). % 1.09/1.29 thf(subset_union,axiom,(mvalid @ (mforall_ind @ (^[X1:mu]:(mforall_ind @ (^[X2:mu]:((mimplies @ ((subset @ X1) @ X2)) @ ((qmltpeq @ ((union @ X1) @ X2)) @ X2)))))))). % 1.09/1.29 thf(15,plain,sP10,inference(preprocess,[status(thm)],[subset_union]). % 1.09/1.29 thf(16,plain,$false,inference(prop_unsat,[status(thm),assumptions([h2,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,h2])). % 1.09/1.29 thf(17,plain,$false,inference(eigenvar_choice,[status(thm),assumptions([h2,h0]),eigenvar_choice(discharge,[h1])],[16,h1])). % 1.09/1.29 thf(18,plain,$false,inference(eigenvar_choice,[status(thm),assumptions([h2]),eigenvar_choice(discharge,[h0])],[17,h0])). % 1.09/1.29 thf(0,theorem,sP12,inference(contra,[status(thm),contra(discharge,[h2])],[16,h2])). % 1.09/1.29 % SZS output end Proof %------------------------------------------------------------------------------