%------------------------------------------------------------------------------ % File : Satallax---3.5 % Problem : SET027^7 : 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 : n006.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 02:25:07 PM UTC 2025 % Result : Theorem 0.19s 0.43s % Output : Proof 0.19s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.07/0.12 % Problem : SET027^7 : 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 % 0.12/0.34 % Computer : n006.cluster.edu % 0.12/0.34 % Model : x86_64 x86_64 % 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.12/0.34 % Memory : 8042.1875MB % 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64 % 0.12/0.34 % CPULimit : 300 % 0.12/0.34 % WCLimit : 300 % 0.12/0.34 % DateTime : Wed Oct 8 14:50:23 EDT 2025 % 0.12/0.34 % CPUTime : % 0.19/0.43 % SZS status Theorem % 0.19/0.43 % Mode: mode213 % 0.19/0.43 % Inferences: 384 % 0.19/0.43 % SZS output start Proof % 0.19/0.43 thf(ty_mu, type, mu : $tType). % 0.19/0.43 thf(ty_subset, type, subset : (mu>mu>$i>$o)). % 0.19/0.43 thf(ty_eigen__2, type, eigen__2 : mu). % 0.19/0.43 thf(ty_eigen__1, type, eigen__1 : mu). % 0.19/0.43 thf(ty_eigen__0, type, eigen__0 : $i). % 0.19/0.43 thf(ty_member, type, member : (mu>mu>$i>$o)). % 0.19/0.43 thf(ty_eigen__3, type, eigen__3 : mu). % 0.19/0.43 thf(ty_eigen__8, type, eigen__8 : mu). % 0.19/0.43 thf(ty_exists_in_world, type, exists_in_world : (mu>$i>$o)). % 0.19/0.43 thf(h0, assumption, (![X1:mu>$o]:(![X2:mu]:((X1 @ X2) => (X1 @ (eps__0 @ X1))))),introduced(assumption,[])). % 0.19/0.43 thf(eigendef_eigen__8, definition, eigen__8 = (eps__0 @ (^[X1:mu]:(~((((exists_in_world @ X1) @ eigen__0) => ((((member @ X1) @ eigen__1) @ eigen__0) => (((member @ X1) @ eigen__3) @ eigen__0))))))), introduced(definition,[new_symbols(definition,[eigen__8])])). % 0.19/0.43 thf(sP1,plain,sP1 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => (![X2:mu]:(((exists_in_world @ X2) @ eigen__0) => (~((((((subset @ X1) @ X2) @ eigen__0) => (![X3:mu]:(((exists_in_world @ X3) @ eigen__0) => ((((member @ X3) @ X1) @ eigen__0) => (((member @ X3) @ X2) @ eigen__0))))) => (~(((![X3:mu]:(((exists_in_world @ X3) @ eigen__0) => ((((member @ X3) @ X1) @ eigen__0) => (((member @ X3) @ X2) @ eigen__0)))) => (((subset @ X1) @ X2) @ eigen__0))))))))))),introduced(definition,[new_symbols(definition,[sP1])])). % 0.19/0.43 thf(sP2,plain,sP2 <=> (((subset @ eigen__2) @ eigen__3) @ eigen__0),introduced(definition,[new_symbols(definition,[sP2])])). % 0.19/0.43 thf(sP3,plain,sP3 <=> ((((member @ eigen__8) @ eigen__2) @ eigen__0) => (((member @ eigen__8) @ eigen__3) @ eigen__0)),introduced(definition,[new_symbols(definition,[sP3])])). % 0.19/0.43 thf(sP4,plain,sP4 <=> ((sP2 => (![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => ((((member @ X1) @ eigen__2) @ eigen__0) => (((member @ X1) @ eigen__3) @ eigen__0))))) => (~(((![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => ((((member @ X1) @ eigen__2) @ eigen__0) => (((member @ X1) @ eigen__3) @ eigen__0)))) => sP2)))),introduced(definition,[new_symbols(definition,[sP4])])). % 0.19/0.43 thf(sP5,plain,sP5 <=> (((((subset @ eigen__1) @ eigen__3) @ eigen__0) => (![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => ((((member @ X1) @ eigen__1) @ eigen__0) => (((member @ X1) @ eigen__3) @ eigen__0))))) => (~(((![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => ((((member @ X1) @ eigen__1) @ eigen__0) => (((member @ X1) @ eigen__3) @ eigen__0)))) => (((subset @ eigen__1) @ eigen__3) @ eigen__0))))),introduced(definition,[new_symbols(definition,[sP5])])). % 0.19/0.43 thf(sP6,plain,sP6 <=> ((exists_in_world @ eigen__2) @ eigen__0),introduced(definition,[new_symbols(definition,[sP6])])). % 0.19/0.43 thf(sP7,plain,sP7 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => ((((member @ X1) @ eigen__2) @ eigen__0) => (((member @ X1) @ eigen__3) @ eigen__0)))),introduced(definition,[new_symbols(definition,[sP7])])). % 0.19/0.43 thf(sP8,plain,sP8 <=> (sP6 => (![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => (~((((((subset @ eigen__2) @ X1) @ eigen__0) => (![X2:mu]:(((exists_in_world @ X2) @ eigen__0) => ((((member @ X2) @ eigen__2) @ eigen__0) => (((member @ X2) @ X1) @ eigen__0))))) => (~(((![X2:mu]:(((exists_in_world @ X2) @ eigen__0) => ((((member @ X2) @ eigen__2) @ eigen__0) => (((member @ X2) @ X1) @ eigen__0)))) => (((subset @ eigen__2) @ X1) @ eigen__0)))))))))),introduced(definition,[new_symbols(definition,[sP8])])). % 0.19/0.43 thf(sP9,plain,sP9 <=> (((exists_in_world @ eigen__3) @ eigen__0) => (~(sP5))),introduced(definition,[new_symbols(definition,[sP9])])). % 0.19/0.43 thf(sP10,plain,sP10 <=> (((exists_in_world @ eigen__8) @ eigen__0) => ((((member @ eigen__8) @ eigen__1) @ eigen__0) => (((member @ eigen__8) @ eigen__3) @ eigen__0))),introduced(definition,[new_symbols(definition,[sP10])])). % 0.19/0.43 thf(sP11,plain,sP11 <=> (((member @ eigen__8) @ eigen__1) @ eigen__0),introduced(definition,[new_symbols(definition,[sP11])])). % 0.19/0.43 thf(sP12,plain,sP12 <=> (sP11 => (((member @ eigen__8) @ eigen__3) @ eigen__0)),introduced(definition,[new_symbols(definition,[sP12])])). % 0.19/0.43 thf(sP13,plain,sP13 <=> ((exists_in_world @ eigen__1) @ eigen__0),introduced(definition,[new_symbols(definition,[sP13])])). % 0.19/0.43 thf(sP14,plain,sP14 <=> (((member @ eigen__8) @ eigen__2) @ eigen__0),introduced(definition,[new_symbols(definition,[sP14])])). % 0.19/0.43 thf(sP15,plain,sP15 <=> (((exists_in_world @ eigen__8) @ eigen__0) => (sP11 => sP14)),introduced(definition,[new_symbols(definition,[sP15])])). % 0.19/0.43 thf(sP16,plain,sP16 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => ((((member @ X1) @ eigen__1) @ eigen__0) => (((member @ X1) @ eigen__2) @ eigen__0)))),introduced(definition,[new_symbols(definition,[sP16])])). % 0.19/0.43 thf(sP17,plain,sP17 <=> (((exists_in_world @ eigen__3) @ eigen__0) => (~(sP4))),introduced(definition,[new_symbols(definition,[sP17])])). % 0.19/0.43 thf(sP18,plain,sP18 <=> (((member @ eigen__8) @ eigen__3) @ eigen__0),introduced(definition,[new_symbols(definition,[sP18])])). % 0.19/0.43 thf(sP19,plain,sP19 <=> (sP13 => (![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => (~((((((subset @ eigen__1) @ X1) @ eigen__0) => (![X2:mu]:(((exists_in_world @ X2) @ eigen__0) => ((((member @ X2) @ eigen__1) @ eigen__0) => (((member @ X2) @ X1) @ eigen__0))))) => (~(((![X2:mu]:(((exists_in_world @ X2) @ eigen__0) => ((((member @ X2) @ eigen__1) @ eigen__0) => (((member @ X2) @ X1) @ eigen__0)))) => (((subset @ eigen__1) @ X1) @ eigen__0)))))))))),introduced(definition,[new_symbols(definition,[sP19])])). % 0.19/0.43 thf(sP20,plain,sP20 <=> (sP6 => (~((((((subset @ eigen__1) @ eigen__2) @ eigen__0) => sP16) => (~((sP16 => (((subset @ eigen__1) @ eigen__2) @ eigen__0)))))))),introduced(definition,[new_symbols(definition,[sP20])])). % 0.19/0.43 thf(sP21,plain,sP21 <=> ((exists_in_world @ eigen__8) @ eigen__0),introduced(definition,[new_symbols(definition,[sP21])])). % 0.19/0.43 thf(sP22,plain,sP22 <=> (sP21 => sP3),introduced(definition,[new_symbols(definition,[sP22])])). % 0.19/0.43 thf(sP23,plain,sP23 <=> (sP11 => sP14),introduced(definition,[new_symbols(definition,[sP23])])). % 0.19/0.43 thf(sP24,plain,sP24 <=> (sP2 => sP7),introduced(definition,[new_symbols(definition,[sP24])])). % 0.19/0.43 thf(sP25,plain,sP25 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => ((((member @ X1) @ eigen__1) @ eigen__0) => (((member @ X1) @ eigen__3) @ eigen__0)))),introduced(definition,[new_symbols(definition,[sP25])])). % 0.19/0.43 thf(sP26,plain,sP26 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => (~((((((subset @ eigen__2) @ X1) @ eigen__0) => (![X2:mu]:(((exists_in_world @ X2) @ eigen__0) => ((((member @ X2) @ eigen__2) @ eigen__0) => (((member @ X2) @ X1) @ eigen__0))))) => (~(((![X2:mu]:(((exists_in_world @ X2) @ eigen__0) => ((((member @ X2) @ eigen__2) @ eigen__0) => (((member @ X2) @ X1) @ eigen__0)))) => (((subset @ eigen__2) @ X1) @ eigen__0))))))))),introduced(definition,[new_symbols(definition,[sP26])])). % 0.19/0.43 thf(sP27,plain,sP27 <=> (((subset @ eigen__1) @ eigen__2) @ eigen__0),introduced(definition,[new_symbols(definition,[sP27])])). % 0.19/0.43 thf(sP28,plain,sP28 <=> (((subset @ eigen__1) @ eigen__3) @ eigen__0),introduced(definition,[new_symbols(definition,[sP28])])). % 0.19/0.43 thf(sP29,plain,sP29 <=> (![X1:$i]:(![X2:mu]:(((exists_in_world @ X2) @ X1) => (![X3:mu]:(((exists_in_world @ X3) @ X1) => (~((((((subset @ X2) @ X3) @ X1) => (![X4:mu]:(((exists_in_world @ X4) @ X1) => ((((member @ X4) @ X2) @ X1) => (((member @ X4) @ X3) @ X1))))) => (~(((![X4:mu]:(((exists_in_world @ X4) @ X1) => ((((member @ X4) @ X2) @ X1) => (((member @ X4) @ X3) @ X1)))) => (((subset @ X2) @ X3) @ X1)))))))))))),introduced(definition,[new_symbols(definition,[sP29])])). % 0.19/0.43 thf(sP30,plain,sP30 <=> ((sP27 => sP16) => (~((sP16 => sP27)))),introduced(definition,[new_symbols(definition,[sP30])])). % 0.19/0.43 thf(sP31,plain,sP31 <=> ((exists_in_world @ eigen__3) @ eigen__0),introduced(definition,[new_symbols(definition,[sP31])])). % 0.19/0.43 thf(sP32,plain,sP32 <=> (sP27 => sP16),introduced(definition,[new_symbols(definition,[sP32])])). % 0.19/0.43 thf(sP33,plain,sP33 <=> (sP25 => sP28),introduced(definition,[new_symbols(definition,[sP33])])). % 0.19/0.43 thf(sP34,plain,sP34 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => (~((((((subset @ eigen__1) @ X1) @ eigen__0) => (![X2:mu]:(((exists_in_world @ X2) @ eigen__0) => ((((member @ X2) @ eigen__1) @ eigen__0) => (((member @ X2) @ X1) @ eigen__0))))) => (~(((![X2:mu]:(((exists_in_world @ X2) @ eigen__0) => ((((member @ X2) @ eigen__1) @ eigen__0) => (((member @ X2) @ X1) @ eigen__0)))) => (((subset @ eigen__1) @ X1) @ eigen__0))))))))),introduced(definition,[new_symbols(definition,[sP34])])). % 0.19/0.43 thf(def_meq_prop,definition,(meq_prop = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((X1 @ X3) = (X2 @ X3))))))). % 0.19/0.43 thf(def_mnot,definition,(mnot = (^[X1:$i>$o]:(^[X2:$i]:(~((X1 @ X2))))))). % 0.19/0.43 thf(def_mor,definition,(mor = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((~((X1 @ X3))) => (X2 @ X3))))))). % 0.19/0.43 thf(def_mbox,definition,(mbox = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:(![X4:$i]:(((X1 @ X3) @ X4) => (X2 @ X4)))))))). % 0.19/0.43 thf(def_mforall_prop,definition,(mforall_prop = (^[X1:($i>$o)>$i>$o]:(^[X2:$i]:(![X3:$i>$o]:((X1 @ X3) @ X2)))))). % 0.19/0.43 thf(def_mtrue,definition,(mtrue = (^[X1:$i]:(~($false))))). % 0.19/0.43 thf(def_mfalse,definition,(mfalse = (mnot @ mtrue))). % 0.19/0.43 thf(def_mand,definition,(mand = (^[X1:$i>$o]:(^[X2:$i>$o]:(mnot @ ((mor @ (mnot @ X1)) @ (mnot @ X2))))))). % 0.19/0.43 thf(def_mimplies,definition,(mimplies = (^[X1:$i>$o]:(mor @ (mnot @ X1))))). % 0.19/0.43 thf(def_mimplied,definition,(mimplied = (^[X1:$i>$o]:(^[X2:$i>$o]:((mor @ (mnot @ X2)) @ X1))))). % 0.19/0.43 thf(def_mequiv,definition,(mequiv = (^[X1:$i>$o]:(^[X2:$i>$o]:((mand @ ((mimplies @ X1) @ X2)) @ ((mimplies @ X2) @ X1)))))). % 0.19/0.43 thf(def_mxor,definition,(mxor = (^[X1:$i>$o]:(^[X2:$i>$o]:(mnot @ ((mequiv @ X1) @ X2)))))). % 0.19/0.43 thf(def_mdia,definition,(mdia = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(mnot @ ((mbox @ X1) @ (mnot @ X2))))))). % 0.19/0.43 thf(def_mforall_ind,definition,(mforall_ind = (^[X1:mu>$i>$o]:(^[X2:$i]:(![X3:mu]:(((exists_in_world @ X3) @ X2) => ((X1 @ X3) @ X2))))))). % 0.19/0.43 thf(def_mexists_ind,definition,(mexists_ind = (^[X1:mu>$i>$o]:(mnot @ (mforall_ind @ (^[X2:mu]:(mnot @ (X1 @ X2)))))))). % 0.19/0.43 thf(def_mexists_prop,definition,(mexists_prop = (^[X1:($i>$o)>$i>$o]:(mnot @ (mforall_prop @ (^[X2:$i>$o]:(mnot @ (X1 @ X2)))))))). % 0.19/0.43 thf(def_mreflexive,definition,(mreflexive = (^[X1:$i>$i>$o]:(![X2:$i]:((X1 @ X2) @ X2))))). % 0.19/0.43 thf(def_msymmetric,definition,(msymmetric = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(((X1 @ X2) @ X3) => ((X1 @ X3) @ X2))))))). % 0.19/0.43 thf(def_mserial,definition,(mserial = (^[X1:$i>$i>$o]:(![X2:$i]:(~((![X3:$i]:(~(((X1 @ X2) @ X3)))))))))). % 0.19/0.43 thf(def_mtransitive,definition,(mtransitive = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X3) @ X4)))))) => ((X1 @ X2) @ X4)))))))). % 0.19/0.43 thf(def_meuclidean,definition,(meuclidean = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => ((X1 @ X3) @ X4)))))))). % 0.19/0.43 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)))))))). % 0.19/0.43 thf(def_mfunctional,definition,(mfunctional = (^[X1:$i>$i>$o]:(![X2:$i]:(~((![X3:$i]:(((X1 @ X2) @ X3) => (~((![X4:$i]:(((X1 @ X2) @ X4) => (X3 = X4))))))))))))). % 0.19/0.43 thf(def_mweakly_dense,definition,(mweakly_dense = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((X1 @ X2) @ X3) => (~((![X5:$i]:(((X1 @ X2) @ X5) => (~(((X1 @ X5) @ X3)))))))))))))). % 0.19/0.43 thf(def_mweakly_connected,definition,(mweakly_connected = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => ((~(((~(((X1 @ X3) @ X4))) => (X3 = X4)))) => ((X1 @ X4) @ X3))))))))). % 0.19/0.43 thf(def_mweakly_directed,definition,(mweakly_directed = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => (~((![X5:$i]:(((X1 @ X3) @ X5) => (~(((X1 @ X4) @ X5)))))))))))))). % 0.19/0.43 thf(def_mvalid,definition,(mvalid = (!!))). % 0.19/0.43 thf(def_msatisfiable,definition,(msatisfiable = (^[X1:$i>$o]:(~((![X2:$i]:(~((X1 @ X2))))))))). % 0.19/0.43 thf(def_mcountersatisfiable,definition,(mcountersatisfiable = (^[X1:$i>$o]:(~(((!!) @ X1)))))). % 0.19/0.43 thf(def_minvalid,definition,(minvalid = (^[X1:$i>$o]:(![X2:$i]:(~((X1 @ X2))))))). % 0.19/0.43 thf(def_mbox_s4,definition,(mbox_s4 = (^[X1:$i>$o]:(^[X2:$i]:(![X3:$i]:(((rel_s4 @ X2) @ X3) => (X1 @ X3))))))). % 0.19/0.43 thf(def_mdia_s4,definition,(mdia_s4 = (^[X1:$i>$o]:(mnot @ (mbox_s4 @ (mnot @ X1)))))). % 0.19/0.43 thf(prove_transitivity_of_subset,conjecture,(![X1:$i]:(![X2:mu]:(((exists_in_world @ X2) @ X1) => (![X3:mu]:(((exists_in_world @ X3) @ X1) => (![X4:mu]:(((exists_in_world @ X4) @ X1) => ((~((~((~(((~((~((((subset @ X2) @ X3) @ X1))))) => (~((((subset @ X3) @ X4) @ X1)))))))))) => (((subset @ X2) @ X4) @ X1)))))))))). % 0.19/0.43 thf(h1,negated_conjecture,(~((![X1:$i]:(![X2:mu]:(((exists_in_world @ X2) @ X1) => (![X3:mu]:(((exists_in_world @ X3) @ X1) => (![X4:mu]:(((exists_in_world @ X4) @ X1) => ((~(((((subset @ X2) @ X3) @ X1) => (~((((subset @ X3) @ X4) @ X1)))))) => (((subset @ X2) @ X4) @ X1))))))))))),inference(assume_negation,[status(cth)],[prove_transitivity_of_subset])). % 0.19/0.43 thf(h2,assumption,(~((![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => (![X2:mu]:(((exists_in_world @ X2) @ eigen__0) => (![X3:mu]:(((exists_in_world @ X3) @ eigen__0) => ((~(((((subset @ X1) @ X2) @ eigen__0) => (~((((subset @ X2) @ X3) @ eigen__0)))))) => (((subset @ X1) @ X3) @ eigen__0)))))))))),introduced(assumption,[])). % 0.19/0.43 thf(h3,assumption,(~((sP13 => (![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => (![X2:mu]:(((exists_in_world @ X2) @ eigen__0) => ((~(((((subset @ eigen__1) @ X1) @ eigen__0) => (~((((subset @ X1) @ X2) @ eigen__0)))))) => (((subset @ eigen__1) @ X2) @ eigen__0))))))))),introduced(assumption,[])). % 0.19/0.43 thf(h4,assumption,sP13,introduced(assumption,[])). % 0.19/0.43 thf(h5,assumption,(~((![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => (![X2:mu]:(((exists_in_world @ X2) @ eigen__0) => ((~(((((subset @ eigen__1) @ X1) @ eigen__0) => (~((((subset @ X1) @ X2) @ eigen__0)))))) => (((subset @ eigen__1) @ X2) @ eigen__0)))))))),introduced(assumption,[])). % 0.19/0.43 thf(h6,assumption,(~((sP6 => (![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => ((~((sP27 => (~((((subset @ eigen__2) @ X1) @ eigen__0)))))) => (((subset @ eigen__1) @ X1) @ eigen__0))))))),introduced(assumption,[])). % 0.19/0.43 thf(h7,assumption,sP6,introduced(assumption,[])). % 0.19/0.43 thf(h8,assumption,(~((![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => ((~((sP27 => (~((((subset @ eigen__2) @ X1) @ eigen__0)))))) => (((subset @ eigen__1) @ X1) @ eigen__0)))))),introduced(assumption,[])). % 0.19/0.43 thf(h9,assumption,(~((sP31 => ((~((sP27 => (~(sP2))))) => sP28)))),introduced(assumption,[])). % 0.19/0.43 thf(h10,assumption,sP31,introduced(assumption,[])). % 0.19/0.43 thf(h11,assumption,(~(((~((sP27 => (~(sP2))))) => sP28))),introduced(assumption,[])). % 0.19/0.43 thf(h12,assumption,(~((sP27 => (~(sP2))))),introduced(assumption,[])). % 0.19/0.43 thf(h13,assumption,(~(sP28)),introduced(assumption,[])). % 0.19/0.43 thf(h14,assumption,sP27,introduced(assumption,[])). % 0.19/0.43 thf(h15,assumption,sP2,introduced(assumption,[])). % 0.19/0.43 thf(1,plain,(sP12 | ~(sP18)),inference(prop_rule,[status(thm)],[])). % 0.19/0.43 thf(2,plain,(sP12 | sP11),inference(prop_rule,[status(thm)],[])). % 0.19/0.43 thf(3,plain,(~(sP16) | sP15),inference(all_rule,[status(thm)],[])). % 0.19/0.43 thf(4,plain,((~(sP15) | ~(sP21)) | sP23),inference(prop_rule,[status(thm)],[])). % 0.19/0.43 thf(5,plain,((~(sP23) | ~(sP11)) | sP14),inference(prop_rule,[status(thm)],[])). % 0.19/0.43 thf(6,plain,(~(sP7) | sP22),inference(all_rule,[status(thm)],[])). % 0.19/0.43 thf(7,plain,((~(sP22) | ~(sP21)) | sP3),inference(prop_rule,[status(thm)],[])). % 0.19/0.43 thf(8,plain,((~(sP3) | ~(sP14)) | sP18),inference(prop_rule,[status(thm)],[])). % 0.19/0.43 thf(9,plain,(sP10 | ~(sP12)),inference(prop_rule,[status(thm)],[])). % 0.19/0.43 thf(10,plain,(sP10 | sP21),inference(prop_rule,[status(thm)],[])). % 0.19/0.43 thf(11,plain,(sP25 | ~(sP10)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__8])). % 0.19/0.43 thf(12,plain,((~(sP24) | ~(sP2)) | sP7),inference(prop_rule,[status(thm)],[])). % 0.19/0.43 thf(13,plain,((~(sP33) | ~(sP25)) | sP28),inference(prop_rule,[status(thm)],[])). % 0.19/0.43 thf(14,plain,((~(sP32) | ~(sP27)) | sP16),inference(prop_rule,[status(thm)],[])). % 0.19/0.43 thf(15,plain,(sP4 | sP24),inference(prop_rule,[status(thm)],[])). % 0.19/0.43 thf(16,plain,(sP5 | sP33),inference(prop_rule,[status(thm)],[])). % 0.19/0.43 thf(17,plain,(sP30 | sP32),inference(prop_rule,[status(thm)],[])). % 0.19/0.43 thf(18,plain,(~(sP26) | sP17),inference(all_rule,[status(thm)],[])). % 0.19/0.43 thf(19,plain,((~(sP17) | ~(sP31)) | ~(sP4)),inference(prop_rule,[status(thm)],[])). % 0.19/0.43 thf(20,plain,(~(sP34) | sP9),inference(all_rule,[status(thm)],[])). % 0.19/0.43 thf(21,plain,((~(sP9) | ~(sP31)) | ~(sP5)),inference(prop_rule,[status(thm)],[])). % 0.19/0.43 thf(22,plain,(~(sP34) | sP20),inference(all_rule,[status(thm)],[])). % 0.19/0.43 thf(23,plain,((~(sP20) | ~(sP6)) | ~(sP30)),inference(prop_rule,[status(thm)],[])). % 0.19/0.43 thf(24,plain,(~(sP1) | sP8),inference(all_rule,[status(thm)],[])). % 0.19/0.43 thf(25,plain,((~(sP8) | ~(sP6)) | sP26),inference(prop_rule,[status(thm)],[])). % 0.19/0.43 thf(26,plain,(~(sP29) | sP1),inference(all_rule,[status(thm)],[])). % 0.19/0.43 thf(27,plain,(~(sP1) | sP19),inference(all_rule,[status(thm)],[])). % 0.19/0.43 thf(28,plain,((~(sP19) | ~(sP13)) | sP34),inference(prop_rule,[status(thm)],[])). % 0.19/0.43 thf(subset_defn,axiom,(mvalid @ (mforall_ind @ (^[X1:mu]:(mforall_ind @ (^[X2:mu]:((mequiv @ ((subset @ X1) @ X2)) @ (mforall_ind @ (^[X3:mu]:((mimplies @ ((member @ X3) @ X1)) @ ((member @ X3) @ X2))))))))))). % 0.19/0.43 thf(29,plain,sP29,inference(preprocess,[status(thm)],[subset_defn]). % 0.19/0.43 thf(30,plain,$false,inference(prop_unsat,[status(thm),assumptions([h14,h15,h12,h13,h10,h11,h9,h7,h8,h6,h4,h5,h3,h2,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,h4,h7,h10,h14,h15,h13])). % 0.19/0.43 thf(31,plain,$false,inference(tab_negimp,[status(thm),assumptions([h12,h13,h10,h11,h9,h7,h8,h6,h4,h5,h3,h2,h1,h0]),tab_negimp(discharge,[h14,h15])],[h12,30,h14,h15])). % 0.19/0.43 thf(32,plain,$false,inference(tab_negimp,[status(thm),assumptions([h10,h11,h9,h7,h8,h6,h4,h5,h3,h2,h1,h0]),tab_negimp(discharge,[h12,h13])],[h11,31,h12,h13])). % 0.19/0.43 thf(33,plain,$false,inference(tab_negimp,[status(thm),assumptions([h9,h7,h8,h6,h4,h5,h3,h2,h1,h0]),tab_negimp(discharge,[h10,h11])],[h9,32,h10,h11])). % 0.19/0.43 thf(34,plain,$false,inference(tab_negall,[status(thm),assumptions([h7,h8,h6,h4,h5,h3,h2,h1,h0]),tab_negall(discharge,[h9]),tab_negall(eigenvar,eigen__3)],[h8,33,h9])). % 0.19/0.43 thf(35,plain,$false,inference(tab_negimp,[status(thm),assumptions([h6,h4,h5,h3,h2,h1,h0]),tab_negimp(discharge,[h7,h8])],[h6,34,h7,h8])). % 0.19/0.43 thf(36,plain,$false,inference(tab_negall,[status(thm),assumptions([h4,h5,h3,h2,h1,h0]),tab_negall(discharge,[h6]),tab_negall(eigenvar,eigen__2)],[h5,35,h6])). % 0.19/0.43 thf(37,plain,$false,inference(tab_negimp,[status(thm),assumptions([h3,h2,h1,h0]),tab_negimp(discharge,[h4,h5])],[h3,36,h4,h5])). % 0.19/0.43 thf(38,plain,$false,inference(tab_negall,[status(thm),assumptions([h2,h1,h0]),tab_negall(discharge,[h3]),tab_negall(eigenvar,eigen__1)],[h2,37,h3])). % 0.19/0.43 thf(39,plain,$false,inference(tab_negall,[status(thm),assumptions([h1,h0]),tab_negall(discharge,[h2]),tab_negall(eigenvar,eigen__0)],[h1,38,h2])). % 0.19/0.43 thf(40,plain,$false,inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[39,h0])). % 0.19/0.43 thf(0,theorem,(![X1:$i]:(![X2:mu]:(((exists_in_world @ X2) @ X1) => (![X3:mu]:(((exists_in_world @ X3) @ X1) => (![X4:mu]:(((exists_in_world @ X4) @ X1) => ((~((~((~(((~((~((((subset @ X2) @ X3) @ X1))))) => (~((((subset @ X3) @ X4) @ X1)))))))))) => (((subset @ X2) @ X4) @ X1))))))))),inference(contra,[status(thm),contra(discharge,[h1])],[39,h1])). % 0.19/0.43 % SZS output end Proof %------------------------------------------------------------------------------