%------------------------------------------------------------------------------ % File : Satallax---3.5 % Problem : SYN397^7 : 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 : n029.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 03:18:12 PM UTC 2025 % Result : Theorem 4.67s 5.04s % Output : Proof 4.67s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.11/0.11 % Problem : SYN397^7 : TPTP v9.2.0. Released v5.5.0. % 0.11/0.12 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % 0.12/0.33 % Computer : n029.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 % 0.12/0.33 % DateTime : Wed Oct 8 15:04:53 EDT 2025 % 0.12/0.33 % CPUTime : % 4.67/5.04 % SZS status Theorem % 4.67/5.04 % Mode: mode506 % 4.67/5.04 % Inferences: 64548 % 4.67/5.04 % SZS output start Proof % 4.67/5.04 thf(ty_mu, type, mu : $tType). % 4.67/5.04 thf(ty_eigen__6, type, eigen__6 : $i). % 4.67/5.04 thf(ty_rel_s4, type, rel_s4 : ($i>$i>$o)). % 4.67/5.04 thf(ty_eigen__16, type, eigen__16 : $i). % 4.67/5.04 thf(ty_eigen__7, type, eigen__7 : mu). % 4.67/5.04 thf(ty_eigen__15, type, eigen__15 : mu). % 4.67/5.04 thf(ty_eigen__0, type, eigen__0 : $i). % 4.67/5.04 thf(ty_eigen__4, type, eigen__4 : $i). % 4.67/5.04 thf(ty_eigen__5, type, eigen__5 : $i). % 4.67/5.04 thf(ty_eigen__10, type, eigen__10 : $i). % 4.67/5.04 thf(ty_exists_in_world, type, exists_in_world : (mu>$i>$o)). % 4.67/5.04 thf(ty_f, type, f : (mu>$i>$o)). % 4.67/5.04 thf(h0, assumption, (![X1:mu>$o]:(![X2:mu]:((X1 @ X2) => (X1 @ (eps__0 @ X1))))),introduced(assumption,[])). % 4.67/5.04 thf(eigendef_eigen__15, definition, eigen__15 = (eps__0 @ (^[X1:mu]:(~((((exists_in_world @ X1) @ eigen__10) => (![X2:$i]:(((rel_s4 @ eigen__10) @ X2) => (~((![X3:$i]:(((rel_s4 @ X2) @ X3) => ((f @ X1) @ X3)))))))))))), introduced(definition,[new_symbols(definition,[eigen__15])])). % 4.67/5.04 thf(h1, assumption, (![X1:$i>$o]:(![X2:$i]:((X1 @ X2) => (X1 @ (eps__1 @ X1))))),introduced(assumption,[])). % 4.67/5.04 thf(eigendef_eigen__6, definition, eigen__6 = (eps__1 @ (^[X1:$i]:(~((((rel_s4 @ eigen__4) @ X1) => (![X2:mu]:(((exists_in_world @ X2) @ X1) => (~((![X3:$i]:(((rel_s4 @ X1) @ X3) => ((f @ X2) @ X3)))))))))))), introduced(definition,[new_symbols(definition,[eigen__6])])). % 4.67/5.04 thf(eigendef_eigen__0, definition, eigen__0 = (eps__1 @ (^[X1:$i]:(~((~(((![X2:$i]:(((rel_s4 @ X1) @ X2) => ((![X3:$i]:(((rel_s4 @ X2) @ X3) => (![X4:mu]:(((exists_in_world @ X4) @ X3) => (~((![X5:$i]:(((rel_s4 @ X3) @ X5) => ((f @ X4) @ X5))))))))) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => (![X4:mu]:(((exists_in_world @ X4) @ X3) => (![X5:$i]:(((rel_s4 @ X3) @ X5) => (~((![X6:$i]:(((rel_s4 @ X5) @ X6) => ((f @ X4) @ X6)))))))))))))) => (~((![X2:$i]:(((rel_s4 @ X1) @ X2) => ((![X3:$i]:(((rel_s4 @ X2) @ X3) => (![X4:mu]:(((exists_in_world @ X4) @ X3) => (![X5:$i]:(((rel_s4 @ X3) @ X5) => (~((![X6:$i]:(((rel_s4 @ X5) @ X6) => ((f @ X4) @ X6))))))))))) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => (![X4:mu]:(((exists_in_world @ X4) @ X3) => (~((![X5:$i]:(((rel_s4 @ X3) @ X5) => ((f @ X4) @ X5))))))))))))))))))))), introduced(definition,[new_symbols(definition,[eigen__0])])). % 4.67/5.04 thf(eigendef_eigen__10, definition, eigen__10 = (eps__1 @ (^[X1:$i]:(~((((rel_s4 @ eigen__5) @ X1) => (![X2:mu]:(((exists_in_world @ X2) @ X1) => (![X3:$i]:(((rel_s4 @ X1) @ X3) => (~((![X4:$i]:(((rel_s4 @ X3) @ X4) => ((f @ X2) @ X4)))))))))))))), introduced(definition,[new_symbols(definition,[eigen__10])])). % 4.67/5.04 thf(eigendef_eigen__4, definition, eigen__4 = (eps__1 @ (^[X1:$i]:(~((((rel_s4 @ eigen__0) @ X1) => ((![X2:$i]:(((rel_s4 @ X1) @ X2) => (![X3:mu]:(((exists_in_world @ X3) @ X2) => (![X4:$i]:(((rel_s4 @ X2) @ X4) => (~((![X5:$i]:(((rel_s4 @ X4) @ X5) => ((f @ X3) @ X5))))))))))) => (![X2:$i]:(((rel_s4 @ X1) @ X2) => (![X3:mu]:(((exists_in_world @ X3) @ X2) => (~((![X4:$i]:(((rel_s4 @ X2) @ X4) => ((f @ X3) @ X4))))))))))))))), introduced(definition,[new_symbols(definition,[eigen__4])])). % 4.67/5.04 thf(eigendef_eigen__16, definition, eigen__16 = (eps__1 @ (^[X1:$i]:(~((((rel_s4 @ eigen__10) @ X1) => (~((![X2:$i]:(((rel_s4 @ X1) @ X2) => ((f @ eigen__15) @ X2)))))))))), introduced(definition,[new_symbols(definition,[eigen__16])])). % 4.67/5.04 thf(eigendef_eigen__7, definition, eigen__7 = (eps__0 @ (^[X1:mu]:(~((((exists_in_world @ X1) @ eigen__6) => (~((![X2:$i]:(((rel_s4 @ eigen__6) @ X2) => ((f @ X1) @ X2)))))))))), introduced(definition,[new_symbols(definition,[eigen__7])])). % 4.67/5.04 thf(eigendef_eigen__5, definition, eigen__5 = (eps__1 @ (^[X1:$i]:(~((((rel_s4 @ eigen__0) @ X1) => ((![X2:$i]:(((rel_s4 @ X1) @ X2) => (![X3:mu]:(((exists_in_world @ X3) @ X2) => (~((![X4:$i]:(((rel_s4 @ X2) @ X4) => ((f @ X3) @ X4))))))))) => (![X2:$i]:(((rel_s4 @ X1) @ X2) => (![X3:mu]:(((exists_in_world @ X3) @ X2) => (![X4:$i]:(((rel_s4 @ X2) @ X4) => (~((![X5:$i]:(((rel_s4 @ X4) @ X5) => ((f @ X3) @ X5))))))))))))))))), introduced(definition,[new_symbols(definition,[eigen__5])])). % 4.67/5.04 thf(sP1,plain,sP1 <=> ((rel_s4 @ eigen__6) @ eigen__6),introduced(definition,[new_symbols(definition,[sP1])])). % 4.67/5.04 thf(sP2,plain,sP2 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__6) => (~((![X2:$i]:(((rel_s4 @ eigen__6) @ X2) => ((f @ X1) @ X2))))))),introduced(definition,[new_symbols(definition,[sP2])])). % 4.67/5.04 thf(sP3,plain,sP3 <=> (((exists_in_world @ eigen__15) @ eigen__10) => (~(((rel_s4 @ eigen__10) @ eigen__16)))),introduced(definition,[new_symbols(definition,[sP3])])). % 4.67/5.04 thf(sP4,plain,sP4 <=> ((exists_in_world @ eigen__7) @ eigen__6),introduced(definition,[new_symbols(definition,[sP4])])). % 4.67/5.04 thf(sP5,plain,sP5 <=> (![X1:$i]:((~((((rel_s4 @ eigen__5) @ eigen__10) => (~(((rel_s4 @ eigen__10) @ X1)))))) => ((rel_s4 @ eigen__5) @ X1))),introduced(definition,[new_symbols(definition,[sP5])])). % 4.67/5.04 thf(sP6,plain,sP6 <=> (![X1:$i]:(((rel_s4 @ eigen__0) @ X1) => ((![X2:$i]:(((rel_s4 @ X1) @ X2) => (![X3:mu]:(((exists_in_world @ X3) @ X2) => (~((![X4:$i]:(((rel_s4 @ X2) @ X4) => ((f @ X3) @ X4))))))))) => (![X2:$i]:(((rel_s4 @ X1) @ X2) => (![X3:mu]:(((exists_in_world @ X3) @ X2) => (![X4:$i]:(((rel_s4 @ X2) @ X4) => (~((![X5:$i]:(((rel_s4 @ X4) @ X5) => ((f @ X3) @ X5)))))))))))))),introduced(definition,[new_symbols(definition,[sP6])])). % 4.67/5.04 thf(sP7,plain,sP7 <=> (((exists_in_world @ eigen__15) @ eigen__16) => (~((![X1:$i]:(((rel_s4 @ eigen__16) @ X1) => ((f @ eigen__15) @ X1)))))),introduced(definition,[new_symbols(definition,[sP7])])). % 4.67/5.04 thf(sP8,plain,sP8 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__10) => (![X2:$i]:(((rel_s4 @ eigen__10) @ X2) => (~((![X3:$i]:(((rel_s4 @ X2) @ X3) => ((f @ X1) @ X3))))))))),introduced(definition,[new_symbols(definition,[sP8])])). % 4.67/5.04 thf(sP9,plain,sP9 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__6) => (![X2:$i]:(((rel_s4 @ eigen__6) @ X2) => (~((![X3:$i]:(((rel_s4 @ X2) @ X3) => ((f @ X1) @ X3))))))))),introduced(definition,[new_symbols(definition,[sP9])])). % 4.67/5.04 thf(sP10,plain,sP10 <=> ((rel_s4 @ eigen__10) @ eigen__16),introduced(definition,[new_symbols(definition,[sP10])])). % 4.67/5.04 thf(sP11,plain,sP11 <=> ((rel_s4 @ eigen__5) @ eigen__10),introduced(definition,[new_symbols(definition,[sP11])])). % 4.67/5.04 thf(sP12,plain,sP12 <=> (![X1:$i]:(((rel_s4 @ eigen__5) @ X1) => (![X2:mu]:(((exists_in_world @ X2) @ X1) => (~((![X3:$i]:(((rel_s4 @ X1) @ X3) => ((f @ X2) @ X3))))))))),introduced(definition,[new_symbols(definition,[sP12])])). % 4.67/5.04 thf(sP13,plain,sP13 <=> (![X1:mu]:(![X2:$i]:(![X3:$i]:((~((((exists_in_world @ X1) @ X2) => (~(((rel_s4 @ X2) @ X3)))))) => ((exists_in_world @ X1) @ X3))))),introduced(definition,[new_symbols(definition,[sP13])])). % 4.67/5.04 thf(sP14,plain,sP14 <=> ((rel_s4 @ eigen__4) @ eigen__6),introduced(definition,[new_symbols(definition,[sP14])])). % 4.67/5.04 thf(sP15,plain,sP15 <=> ((rel_s4 @ eigen__5) @ eigen__16),introduced(definition,[new_symbols(definition,[sP15])])). % 4.67/5.04 thf(sP16,plain,sP16 <=> (((rel_s4 @ eigen__0) @ eigen__5) => (sP12 => (![X1:$i]:(((rel_s4 @ eigen__5) @ X1) => (![X2:mu]:(((exists_in_world @ X2) @ X1) => (![X3:$i]:(((rel_s4 @ X1) @ X3) => (~((![X4:$i]:(((rel_s4 @ X3) @ X4) => ((f @ X2) @ X4))))))))))))),introduced(definition,[new_symbols(definition,[sP16])])). % 4.67/5.04 thf(sP17,plain,sP17 <=> (![X1:$i]:(((rel_s4 @ eigen__5) @ X1) => (![X2:mu]:(((exists_in_world @ X2) @ X1) => (![X3:$i]:(((rel_s4 @ X1) @ X3) => (~((![X4:$i]:(((rel_s4 @ X3) @ X4) => ((f @ X2) @ X4))))))))))),introduced(definition,[new_symbols(definition,[sP17])])). % 4.67/5.04 thf(sP18,plain,sP18 <=> (((exists_in_world @ eigen__15) @ eigen__10) => (![X1:$i]:(((rel_s4 @ eigen__10) @ X1) => (~((![X2:$i]:(((rel_s4 @ X1) @ X2) => ((f @ eigen__15) @ X2)))))))),introduced(definition,[new_symbols(definition,[sP18])])). % 4.67/5.04 thf(sP19,plain,sP19 <=> (![X1:$i]:(![X2:$i]:((~((((rel_s4 @ eigen__5) @ X1) => (~(((rel_s4 @ X1) @ X2)))))) => ((rel_s4 @ eigen__5) @ X2)))),introduced(definition,[new_symbols(definition,[sP19])])). % 4.67/5.04 thf(sP20,plain,sP20 <=> (sP15 => (![X1:mu]:(((exists_in_world @ X1) @ eigen__16) => (~((![X2:$i]:(((rel_s4 @ eigen__16) @ X2) => ((f @ X1) @ X2)))))))),introduced(definition,[new_symbols(definition,[sP20])])). % 4.67/5.04 thf(sP21,plain,sP21 <=> (sP14 => sP2),introduced(definition,[new_symbols(definition,[sP21])])). % 4.67/5.04 thf(sP22,plain,sP22 <=> (sP1 => (~((![X1:$i]:(((rel_s4 @ eigen__6) @ X1) => ((f @ eigen__7) @ X1)))))),introduced(definition,[new_symbols(definition,[sP22])])). % 4.67/5.04 thf(sP23,plain,sP23 <=> (sP6 => (~((![X1:$i]:(((rel_s4 @ eigen__0) @ X1) => ((![X2:$i]:(((rel_s4 @ X1) @ X2) => (![X3:mu]:(((exists_in_world @ X3) @ X2) => (![X4:$i]:(((rel_s4 @ X2) @ X4) => (~((![X5:$i]:(((rel_s4 @ X4) @ X5) => ((f @ X3) @ X5))))))))))) => (![X2:$i]:(((rel_s4 @ X1) @ X2) => (![X3:mu]:(((exists_in_world @ X3) @ X2) => (~((![X4:$i]:(((rel_s4 @ X2) @ X4) => ((f @ X3) @ X4))))))))))))))),introduced(definition,[new_symbols(definition,[sP23])])). % 4.67/5.04 thf(sP24,plain,sP24 <=> (![X1:$i]:(((rel_s4 @ eigen__6) @ X1) => (~((![X2:$i]:(((rel_s4 @ X1) @ X2) => ((f @ eigen__7) @ X2))))))),introduced(definition,[new_symbols(definition,[sP24])])). % 4.67/5.04 thf(sP25,plain,sP25 <=> (![X1:$i]:(![X2:$i]:((~((((exists_in_world @ eigen__15) @ X1) => (~(((rel_s4 @ X1) @ X2)))))) => ((exists_in_world @ eigen__15) @ X2)))),introduced(definition,[new_symbols(definition,[sP25])])). % 4.67/5.04 thf(sP26,plain,sP26 <=> (![X1:$i]:((~((((exists_in_world @ eigen__15) @ eigen__10) => (~(((rel_s4 @ eigen__10) @ X1)))))) => ((exists_in_world @ eigen__15) @ X1))),introduced(definition,[new_symbols(definition,[sP26])])). % 4.67/5.04 thf(sP27,plain,sP27 <=> (sP4 => (~((![X1:$i]:(((rel_s4 @ eigen__6) @ X1) => ((f @ eigen__7) @ X1)))))),introduced(definition,[new_symbols(definition,[sP27])])). % 4.67/5.04 thf(sP28,plain,sP28 <=> (sP10 => (~((![X1:$i]:(((rel_s4 @ eigen__16) @ X1) => ((f @ eigen__15) @ X1)))))),introduced(definition,[new_symbols(definition,[sP28])])). % 4.67/5.04 thf(sP29,plain,sP29 <=> ((~(sP3)) => ((exists_in_world @ eigen__15) @ eigen__16)),introduced(definition,[new_symbols(definition,[sP29])])). % 4.67/5.04 thf(sP30,plain,sP30 <=> (![X1:$i]:(((rel_s4 @ eigen__4) @ X1) => (![X2:mu]:(((exists_in_world @ X2) @ X1) => (~((![X3:$i]:(((rel_s4 @ X1) @ X3) => ((f @ X2) @ X3))))))))),introduced(definition,[new_symbols(definition,[sP30])])). % 4.67/5.04 thf(sP31,plain,sP31 <=> ((~((sP11 => (~(sP10))))) => sP15),introduced(definition,[new_symbols(definition,[sP31])])). % 4.67/5.04 thf(sP32,plain,sP32 <=> ((exists_in_world @ eigen__15) @ eigen__10),introduced(definition,[new_symbols(definition,[sP32])])). % 4.67/5.04 thf(sP33,plain,sP33 <=> (sP14 => sP9),introduced(definition,[new_symbols(definition,[sP33])])). % 4.67/5.04 thf(sP34,plain,sP34 <=> (sP4 => sP24),introduced(definition,[new_symbols(definition,[sP34])])). % 4.67/5.04 thf(sP35,plain,sP35 <=> (![X1:$i]:(((rel_s4 @ eigen__6) @ X1) => ((f @ eigen__7) @ X1))),introduced(definition,[new_symbols(definition,[sP35])])). % 4.67/5.04 thf(sP36,plain,sP36 <=> (sP11 => (~(sP10))),introduced(definition,[new_symbols(definition,[sP36])])). % 4.67/5.04 thf(sP37,plain,sP37 <=> (((rel_s4 @ eigen__0) @ eigen__4) => ((![X1:$i]:(((rel_s4 @ eigen__4) @ X1) => (![X2:mu]:(((exists_in_world @ X2) @ X1) => (![X3:$i]:(((rel_s4 @ X1) @ X3) => (~((![X4:$i]:(((rel_s4 @ X3) @ X4) => ((f @ X2) @ X4))))))))))) => sP30)),introduced(definition,[new_symbols(definition,[sP37])])). % 4.67/5.04 thf(sP38,plain,sP38 <=> ((![X1:$i]:(((rel_s4 @ eigen__4) @ X1) => (![X2:mu]:(((exists_in_world @ X2) @ X1) => (![X3:$i]:(((rel_s4 @ X1) @ X3) => (~((![X4:$i]:(((rel_s4 @ X3) @ X4) => ((f @ X2) @ X4))))))))))) => sP30),introduced(definition,[new_symbols(definition,[sP38])])). % 4.67/5.04 thf(sP39,plain,sP39 <=> (![X1:$i]:((rel_s4 @ X1) @ X1)),introduced(definition,[new_symbols(definition,[sP39])])). % 4.67/5.04 thf(sP40,plain,sP40 <=> (sP11 => sP8),introduced(definition,[new_symbols(definition,[sP40])])). % 4.67/5.04 thf(sP41,plain,sP41 <=> ((exists_in_world @ eigen__15) @ eigen__16),introduced(definition,[new_symbols(definition,[sP41])])). % 4.67/5.04 thf(sP42,plain,sP42 <=> (![X1:$i]:(~(((![X2:$i]:(((rel_s4 @ X1) @ X2) => ((![X3:$i]:(((rel_s4 @ X2) @ X3) => (![X4:mu]:(((exists_in_world @ X4) @ X3) => (~((![X5:$i]:(((rel_s4 @ X3) @ X5) => ((f @ X4) @ X5))))))))) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => (![X4:mu]:(((exists_in_world @ X4) @ X3) => (![X5:$i]:(((rel_s4 @ X3) @ X5) => (~((![X6:$i]:(((rel_s4 @ X5) @ X6) => ((f @ X4) @ X6)))))))))))))) => (~((![X2:$i]:(((rel_s4 @ X1) @ X2) => ((![X3:$i]:(((rel_s4 @ X2) @ X3) => (![X4:mu]:(((exists_in_world @ X4) @ X3) => (![X5:$i]:(((rel_s4 @ X3) @ X5) => (~((![X6:$i]:(((rel_s4 @ X5) @ X6) => ((f @ X4) @ X6))))))))))) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => (![X4:mu]:(((exists_in_world @ X4) @ X3) => (~((![X5:$i]:(((rel_s4 @ X3) @ X5) => ((f @ X4) @ X5)))))))))))))))))),introduced(definition,[new_symbols(definition,[sP42])])). % 4.67/5.04 thf(sP43,plain,sP43 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__16) => (~((![X2:$i]:(((rel_s4 @ eigen__16) @ X2) => ((f @ X1) @ X2))))))),introduced(definition,[new_symbols(definition,[sP43])])). % 4.67/5.04 thf(sP44,plain,sP44 <=> (![X1:$i]:(((rel_s4 @ eigen__0) @ X1) => ((![X2:$i]:(((rel_s4 @ X1) @ X2) => (![X3:mu]:(((exists_in_world @ X3) @ X2) => (![X4:$i]:(((rel_s4 @ X2) @ X4) => (~((![X5:$i]:(((rel_s4 @ X4) @ X5) => ((f @ X3) @ X5))))))))))) => (![X2:$i]:(((rel_s4 @ X1) @ X2) => (![X3:mu]:(((exists_in_world @ X3) @ X2) => (~((![X4:$i]:(((rel_s4 @ X2) @ X4) => ((f @ X3) @ X4)))))))))))),introduced(definition,[new_symbols(definition,[sP44])])). % 4.67/5.04 thf(sP45,plain,sP45 <=> (sP12 => sP17),introduced(definition,[new_symbols(definition,[sP45])])). % 4.67/5.04 thf(sP46,plain,sP46 <=> (![X1:$i]:(((rel_s4 @ eigen__16) @ X1) => ((f @ eigen__15) @ X1))),introduced(definition,[new_symbols(definition,[sP46])])). % 4.67/5.04 thf(sP47,plain,sP47 <=> (![X1:$i]:(((rel_s4 @ eigen__10) @ X1) => (~((![X2:$i]:(((rel_s4 @ X1) @ X2) => ((f @ eigen__15) @ X2))))))),introduced(definition,[new_symbols(definition,[sP47])])). % 4.67/5.04 thf(sP48,plain,sP48 <=> (![X1:$i]:(![X2:$i]:(![X3:$i]:((~((((rel_s4 @ X1) @ X2) => (~(((rel_s4 @ X2) @ X3)))))) => ((rel_s4 @ X1) @ X3))))),introduced(definition,[new_symbols(definition,[sP48])])). % 4.67/5.04 thf(sP49,plain,sP49 <=> (![X1:$i]:(((rel_s4 @ eigen__4) @ X1) => (![X2:mu]:(((exists_in_world @ X2) @ X1) => (![X3:$i]:(((rel_s4 @ X1) @ X3) => (~((![X4:$i]:(((rel_s4 @ X3) @ X4) => ((f @ X2) @ X4))))))))))),introduced(definition,[new_symbols(definition,[sP49])])). % 4.67/5.04 thf(def_meq_prop,definition,(meq_prop = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((X1 @ X3) = (X2 @ X3))))))). % 4.67/5.04 thf(def_mnot,definition,(mnot = (^[X1:$i>$o]:(^[X2:$i]:(~((X1 @ X2))))))). % 4.67/5.04 thf(def_mor,definition,(mor = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((~((X1 @ X3))) => (X2 @ X3))))))). % 4.67/5.04 thf(def_mbox,definition,(mbox = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:(![X4:$i]:(((X1 @ X3) @ X4) => (X2 @ X4)))))))). % 4.67/5.04 thf(def_mforall_prop,definition,(mforall_prop = (^[X1:($i>$o)>$i>$o]:(^[X2:$i]:(![X3:$i>$o]:((X1 @ X3) @ X2)))))). % 4.67/5.04 thf(def_mtrue,definition,(mtrue = (^[X1:$i]:(~($false))))). % 4.67/5.04 thf(def_mfalse,definition,(mfalse = (mnot @ mtrue))). % 4.67/5.04 thf(def_mand,definition,(mand = (^[X1:$i>$o]:(^[X2:$i>$o]:(mnot @ ((mor @ (mnot @ X1)) @ (mnot @ X2))))))). % 4.67/5.04 thf(def_mimplies,definition,(mimplies = (^[X1:$i>$o]:(mor @ (mnot @ X1))))). % 4.67/5.04 thf(def_mimplied,definition,(mimplied = (^[X1:$i>$o]:(^[X2:$i>$o]:((mor @ (mnot @ X2)) @ X1))))). % 4.67/5.04 thf(def_mequiv,definition,(mequiv = (^[X1:$i>$o]:(^[X2:$i>$o]:((mand @ ((mimplies @ X1) @ X2)) @ ((mimplies @ X2) @ X1)))))). % 4.67/5.04 thf(def_mxor,definition,(mxor = (^[X1:$i>$o]:(^[X2:$i>$o]:(mnot @ ((mequiv @ X1) @ X2)))))). % 4.67/5.04 thf(def_mdia,definition,(mdia = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(mnot @ ((mbox @ X1) @ (mnot @ X2))))))). % 4.67/5.04 thf(def_mforall_ind,definition,(mforall_ind = (^[X1:mu>$i>$o]:(^[X2:$i]:(![X3:mu]:(((exists_in_world @ X3) @ X2) => ((X1 @ X3) @ X2))))))). % 4.67/5.04 thf(def_mexists_ind,definition,(mexists_ind = (^[X1:mu>$i>$o]:(mnot @ (mforall_ind @ (^[X2:mu]:(mnot @ (X1 @ X2)))))))). % 4.67/5.04 thf(def_mexists_prop,definition,(mexists_prop = (^[X1:($i>$o)>$i>$o]:(mnot @ (mforall_prop @ (^[X2:$i>$o]:(mnot @ (X1 @ X2)))))))). % 4.67/5.04 thf(def_mreflexive,definition,(mreflexive = (^[X1:$i>$i>$o]:(![X2:$i]:((X1 @ X2) @ X2))))). % 4.67/5.04 thf(def_msymmetric,definition,(msymmetric = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(((X1 @ X2) @ X3) => ((X1 @ X3) @ X2))))))). % 4.67/5.04 thf(def_mserial,definition,(mserial = (^[X1:$i>$i>$o]:(![X2:$i]:(~((![X3:$i]:(~(((X1 @ X2) @ X3)))))))))). % 4.67/5.04 thf(def_mtransitive,definition,(mtransitive = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X3) @ X4)))))) => ((X1 @ X2) @ X4)))))))). % 4.67/5.04 thf(def_meuclidean,definition,(meuclidean = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => ((X1 @ X3) @ X4)))))))). % 4.67/5.04 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)))))))). % 4.67/5.04 thf(def_mfunctional,definition,(mfunctional = (^[X1:$i>$i>$o]:(![X2:$i]:(~((![X3:$i]:(((X1 @ X2) @ X3) => (~((![X4:$i]:(((X1 @ X2) @ X4) => (X3 = X4))))))))))))). % 4.67/5.04 thf(def_mweakly_dense,definition,(mweakly_dense = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((X1 @ X2) @ X3) => (~((![X5:$i]:(((X1 @ X2) @ X5) => (~(((X1 @ X5) @ X3)))))))))))))). % 4.67/5.04 thf(def_mweakly_connected,definition,(mweakly_connected = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => ((~(((~(((X1 @ X3) @ X4))) => (X3 = X4)))) => ((X1 @ X4) @ X3))))))))). % 4.67/5.04 thf(def_mweakly_directed,definition,(mweakly_directed = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => (~((![X5:$i]:(((X1 @ X3) @ X5) => (~(((X1 @ X4) @ X5)))))))))))))). % 4.67/5.04 thf(def_mvalid,definition,(mvalid = (!!))). % 4.67/5.04 thf(def_msatisfiable,definition,(msatisfiable = (^[X1:$i>$o]:(~((![X2:$i]:(~((X1 @ X2))))))))). % 4.67/5.04 thf(def_mcountersatisfiable,definition,(mcountersatisfiable = (^[X1:$i>$o]:(~(((!!) @ X1)))))). % 4.67/5.04 thf(def_minvalid,definition,(minvalid = (^[X1:$i>$o]:(![X2:$i]:(~((X1 @ X2))))))). % 4.67/5.04 thf(def_mbox_s4,definition,(mbox_s4 = (^[X1:$i>$o]:(^[X2:$i]:(![X3:$i]:(((rel_s4 @ X2) @ X3) => (X1 @ X3))))))). % 4.67/5.04 thf(def_mdia_s4,definition,(mdia_s4 = (^[X1:$i>$o]:(mnot @ (mbox_s4 @ (mnot @ X1)))))). % 4.67/5.04 thf(kalish204,conjecture,(![X1:$i]:(~(((~((~((![X2:$i]:(((rel_s4 @ X1) @ X2) => ((~((~((![X3:$i]:(((rel_s4 @ X2) @ X3) => (~((~((![X4:mu]:(((exists_in_world @ X4) @ X3) => (~((![X5:$i]:(((rel_s4 @ X3) @ X5) => ((f @ X4) @ X5))))))))))))))))) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => (![X4:mu]:(((exists_in_world @ X4) @ X3) => (![X5:$i]:(((rel_s4 @ X3) @ X5) => (~((![X6:$i]:(((rel_s4 @ X5) @ X6) => ((f @ X4) @ X6)))))))))))))))))) => (~((![X2:$i]:(((rel_s4 @ X1) @ X2) => ((~((~((![X3:$i]:(((rel_s4 @ X2) @ X3) => (![X4:mu]:(((exists_in_world @ X4) @ X3) => (![X5:$i]:(((rel_s4 @ X3) @ X5) => (~((![X6:$i]:(((rel_s4 @ X5) @ X6) => ((f @ X4) @ X6))))))))))))))) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => (~((~((![X4:mu]:(((exists_in_world @ X4) @ X3) => (~((![X5:$i]:(((rel_s4 @ X3) @ X5) => ((f @ X4) @ X5))))))))))))))))))))))). % 4.67/5.04 thf(h2,negated_conjecture,(~(sP42)),inference(assume_negation,[status(cth)],[kalish204])). % 4.67/5.04 thf(1,plain,(~(sP5) | sP31),inference(all_rule,[status(thm)],[])). % 4.67/5.04 thf(2,plain,((~(sP31) | sP36) | sP15),inference(prop_rule,[status(thm)],[])). % 4.67/5.04 thf(3,plain,((~(sP36) | ~(sP11)) | ~(sP10)),inference(prop_rule,[status(thm)],[])). % 4.67/5.04 thf(4,plain,(~(sP26) | sP29),inference(all_rule,[status(thm)],[])). % 4.67/5.04 thf(5,plain,((~(sP29) | sP3) | sP41),inference(prop_rule,[status(thm)],[])). % 4.67/5.04 thf(6,plain,((~(sP3) | ~(sP32)) | ~(sP10)),inference(prop_rule,[status(thm)],[])). % 4.67/5.04 thf(7,plain,(~(sP25) | sP26),inference(all_rule,[status(thm)],[])). % 4.67/5.04 thf(8,plain,(~(sP19) | sP5),inference(all_rule,[status(thm)],[])). % 4.67/5.04 thf(9,plain,(~(sP13) | sP25),inference(all_rule,[status(thm)],[])). % 4.67/5.04 thf(10,plain,(~(sP12) | sP20),inference(all_rule,[status(thm)],[])). % 4.67/5.04 thf(11,plain,((~(sP20) | ~(sP15)) | sP43),inference(prop_rule,[status(thm)],[])). % 4.67/5.04 thf(12,plain,(~(sP43) | sP7),inference(all_rule,[status(thm)],[])). % 4.67/5.04 thf(13,plain,((~(sP7) | ~(sP41)) | ~(sP46)),inference(prop_rule,[status(thm)],[])). % 4.67/5.04 thf(14,plain,(sP28 | sP46),inference(prop_rule,[status(thm)],[])). % 4.67/5.04 thf(15,plain,(sP28 | sP10),inference(prop_rule,[status(thm)],[])). % 4.67/5.04 thf(16,plain,(sP47 | ~(sP28)),inference(eigen_choice_rule,[status(thm),assumptions([h1])],[h1,eigendef_eigen__16])). % 4.67/5.04 thf(17,plain,(sP18 | ~(sP47)),inference(prop_rule,[status(thm)],[])). % 4.67/5.04 thf(18,plain,(sP18 | sP32),inference(prop_rule,[status(thm)],[])). % 4.67/5.04 thf(19,plain,(sP8 | ~(sP18)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__15])). % 4.67/5.04 thf(20,plain,(sP40 | ~(sP8)),inference(prop_rule,[status(thm)],[])). % 4.67/5.04 thf(21,plain,(sP40 | sP11),inference(prop_rule,[status(thm)],[])). % 4.67/5.04 thf(22,plain,(~(sP9) | sP34),inference(all_rule,[status(thm)],[])). % 4.67/5.04 thf(23,plain,((~(sP34) | ~(sP4)) | sP24),inference(prop_rule,[status(thm)],[])). % 4.67/5.04 thf(24,plain,(~(sP24) | sP22),inference(all_rule,[status(thm)],[])). % 4.67/5.04 thf(25,plain,((~(sP22) | ~(sP1)) | ~(sP35)),inference(prop_rule,[status(thm)],[])). % 4.67/5.04 thf(26,plain,((~(sP33) | ~(sP14)) | sP9),inference(prop_rule,[status(thm)],[])). % 4.67/5.04 thf(27,plain,(~(sP48) | sP19),inference(all_rule,[status(thm)],[])). % 4.67/5.04 thf(28,plain,(sP17 | ~(sP40)),inference(eigen_choice_rule,[status(thm),assumptions([h1])],[h1,eigendef_eigen__10])). % 4.67/5.04 thf(29,plain,(~(sP49) | sP33),inference(all_rule,[status(thm)],[])). % 4.67/5.04 thf(30,plain,(~(sP39) | sP1),inference(all_rule,[status(thm)],[])). % 4.67/5.04 thf(31,plain,(sP27 | sP35),inference(prop_rule,[status(thm)],[])). % 4.67/5.04 thf(32,plain,(sP27 | sP4),inference(prop_rule,[status(thm)],[])). % 4.67/5.04 thf(33,plain,(sP45 | ~(sP17)),inference(prop_rule,[status(thm)],[])). % 4.67/5.04 thf(34,plain,(sP45 | sP12),inference(prop_rule,[status(thm)],[])). % 4.67/5.04 thf(35,plain,(sP2 | ~(sP27)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__7])). % 4.67/5.04 thf(36,plain,(sP16 | ~(sP45)),inference(prop_rule,[status(thm)],[])). % 4.67/5.04 thf(37,plain,(sP21 | ~(sP2)),inference(prop_rule,[status(thm)],[])). % 4.67/5.04 thf(38,plain,(sP21 | sP14),inference(prop_rule,[status(thm)],[])). % 4.67/5.04 thf(39,plain,(sP30 | ~(sP21)),inference(eigen_choice_rule,[status(thm),assumptions([h1])],[h1,eigendef_eigen__6])). % 4.67/5.04 thf(40,plain,(sP38 | ~(sP30)),inference(prop_rule,[status(thm)],[])). % 4.67/5.04 thf(41,plain,(sP38 | sP49),inference(prop_rule,[status(thm)],[])). % 4.67/5.04 thf(42,plain,(sP37 | ~(sP38)),inference(prop_rule,[status(thm)],[])). % 4.67/5.04 thf(43,plain,(sP6 | ~(sP16)),inference(eigen_choice_rule,[status(thm),assumptions([h1])],[h1,eigendef_eigen__5])). % 4.67/5.04 thf(44,plain,(sP44 | ~(sP37)),inference(eigen_choice_rule,[status(thm),assumptions([h1])],[h1,eigendef_eigen__4])). % 4.67/5.04 thf(45,plain,((~(sP23) | ~(sP6)) | ~(sP44)),inference(prop_rule,[status(thm)],[])). % 4.67/5.04 thf(46,plain,(sP42 | sP23),inference(eigen_choice_rule,[status(thm),assumptions([h1])],[h1,eigendef_eigen__0])). % 4.67/5.04 thf(cumulative_ax,axiom,sP13). % 4.67/5.04 thf(a2,axiom,(mtransitive @ rel_s4)). % 4.67/5.04 thf(47,plain,sP48,inference(preprocess,[status(thm)],[a2]). % 4.67/5.04 thf(a1,axiom,(mreflexive @ rel_s4)). % 4.67/5.04 thf(48,plain,sP39,inference(preprocess,[status(thm)],[a1]). % 4.67/5.04 thf(49,plain,$false,inference(prop_unsat,[status(thm),assumptions([h2,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,cumulative_ax,47,48,h2])). % 4.67/5.04 thf(50,plain,$false,inference(eigenvar_choice,[status(thm),assumptions([h2,h0]),eigenvar_choice(discharge,[h1])],[49,h1])). % 4.67/5.04 thf(51,plain,$false,inference(eigenvar_choice,[status(thm),assumptions([h2]),eigenvar_choice(discharge,[h0])],[50,h0])). % 4.67/5.04 thf(0,theorem,(![X1:$i]:(~(((~((~((![X2:$i]:(((rel_s4 @ X1) @ X2) => ((~((~((![X3:$i]:(((rel_s4 @ X2) @ X3) => (~((~((![X4:mu]:(((exists_in_world @ X4) @ X3) => (~((![X5:$i]:(((rel_s4 @ X3) @ X5) => ((f @ X4) @ X5))))))))))))))))) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => (![X4:mu]:(((exists_in_world @ X4) @ X3) => (![X5:$i]:(((rel_s4 @ X3) @ X5) => (~((![X6:$i]:(((rel_s4 @ X5) @ X6) => ((f @ X4) @ X6)))))))))))))))))) => (~((![X2:$i]:(((rel_s4 @ X1) @ X2) => ((~((~((![X3:$i]:(((rel_s4 @ X2) @ X3) => (![X4:mu]:(((exists_in_world @ X4) @ X3) => (![X5:$i]:(((rel_s4 @ X3) @ X5) => (~((![X6:$i]:(((rel_s4 @ X5) @ X6) => ((f @ X4) @ X6))))))))))))))) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => (~((~((![X4:mu]:(((exists_in_world @ X4) @ X3) => (~((![X5:$i]:(((rel_s4 @ X3) @ X5) => ((f @ X4) @ X5)))))))))))))))))))))),inference(contra,[status(thm),contra(discharge,[h2])],[49,h2])). % 4.67/5.04 % SZS output end Proof %------------------------------------------------------------------------------