%------------------------------------------------------------------------------ % File : Satallax---3.5 % Problem : SYO441^1 : 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 : n017.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 03:25:36 PM UTC 2025 % Result : Theorem 4.36s 4.60s % Output : Proof 4.36s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.04/0.13 % Problem : SYO441^1 : TPTP v9.2.0. Released v4.0.0. % 0.04/0.14 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % 0.13/0.35 % Computer : n017.cluster.edu % 0.13/0.35 % Model : x86_64 x86_64 % 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.13/0.35 % Memory : 8042.1875MB % 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64 % 0.13/0.35 % CPULimit : 300 % 0.13/0.35 % WCLimit : 300 % 0.13/0.35 % DateTime : Thu Oct 9 00:58:53 EDT 2025 % 0.13/0.35 % CPUTime : % 4.36/4.60 % SZS status Theorem % 4.36/4.60 % Mode: mode506 % 4.36/4.60 % Inferences: 83084 % 4.36/4.60 % SZS output start Proof % 4.36/4.60 thf(ty_p, type, p : ($i>$o)). % 4.36/4.60 thf(ty_eigen__7, type, eigen__7 : $i). % 4.36/4.60 thf(ty_eigen__0, type, eigen__0 : $i). % 4.36/4.60 thf(ty_eigen__5, type, eigen__5 : $i). % 4.36/4.60 thf(ty_eigen__3, type, eigen__3 : $i). % 4.36/4.60 thf(ty_rel_s5, type, rel_s5 : ($i>$i>$o)). % 4.36/4.60 thf(ty_eigen__9, type, eigen__9 : $i). % 4.36/4.60 thf(h0, assumption, (![X1:$i>$o]:(![X2:$i]:((X1 @ X2) => (X1 @ (eps__0 @ X1))))),introduced(assumption,[])). % 4.36/4.60 thf(eigendef_eigen__3, definition, eigen__3 = (eps__0 @ (^[X1:$i]:(~((((rel_s5 @ eigen__0) @ X1) => (p @ X1)))))), introduced(definition,[new_symbols(definition,[eigen__3])])). % 4.36/4.60 thf(eigendef_eigen__0, definition, eigen__0 = (eps__0 @ (^[X1:$i]:(~((~((((~((![X2:$i]:(((rel_s5 @ X1) @ X2) => (![X3:$i]:(((rel_s5 @ X2) @ X3) => (![X4:$i]:(((rel_s5 @ X3) @ X4) => (~((![X5:$i]:(((rel_s5 @ X4) @ X5) => (p @ X5))))))))))))) => (![X2:$i]:(((rel_s5 @ X1) @ X2) => (p @ X2)))) => (~(((![X2:$i]:(((rel_s5 @ X1) @ X2) => (p @ X2))) => (~((![X2:$i]:(((rel_s5 @ X1) @ X2) => (![X3:$i]:(((rel_s5 @ X2) @ X3) => (![X4:$i]:(((rel_s5 @ X3) @ X4) => (~((![X5:$i]:(((rel_s5 @ X4) @ X5) => (p @ X5))))))))))))))))))))))), introduced(definition,[new_symbols(definition,[eigen__0])])). % 4.36/4.60 thf(eigendef_eigen__9, definition, eigen__9 = (eps__0 @ (^[X1:$i]:(~((((rel_s5 @ eigen__7) @ X1) => (~((![X2:$i]:(((rel_s5 @ X1) @ X2) => (p @ X2)))))))))), introduced(definition,[new_symbols(definition,[eigen__9])])). % 4.36/4.60 thf(eigendef_eigen__7, definition, eigen__7 = (eps__0 @ (^[X1:$i]:(~((((rel_s5 @ eigen__5) @ X1) => (![X2:$i]:(((rel_s5 @ X1) @ X2) => (~((![X3:$i]:(((rel_s5 @ X2) @ X3) => (p @ X3)))))))))))), introduced(definition,[new_symbols(definition,[eigen__7])])). % 4.36/4.60 thf(eigendef_eigen__5, definition, eigen__5 = (eps__0 @ (^[X1:$i]:(~((((rel_s5 @ eigen__0) @ X1) => (![X2:$i]:(((rel_s5 @ X1) @ X2) => (![X3:$i]:(((rel_s5 @ X2) @ X3) => (~((![X4:$i]:(((rel_s5 @ X3) @ X4) => (p @ X4)))))))))))))), introduced(definition,[new_symbols(definition,[eigen__5])])). % 4.36/4.60 thf(sP1,plain,sP1 <=> (![X1:$i]:((rel_s5 @ X1) @ X1)),introduced(definition,[new_symbols(definition,[sP1])])). % 4.36/4.60 thf(sP2,plain,sP2 <=> (((rel_s5 @ eigen__0) @ eigen__0) => (![X1:$i]:(((rel_s5 @ eigen__0) @ X1) => (![X2:$i]:(((rel_s5 @ X1) @ X2) => (~((![X3:$i]:(((rel_s5 @ X2) @ X3) => (p @ X3)))))))))),introduced(definition,[new_symbols(definition,[sP2])])). % 4.36/4.60 thf(sP3,plain,sP3 <=> (![X1:$i]:(((rel_s5 @ eigen__0) @ X1) => (![X2:$i]:(((rel_s5 @ X1) @ X2) => (~((![X3:$i]:(((rel_s5 @ X2) @ X3) => (p @ X3))))))))),introduced(definition,[new_symbols(definition,[sP3])])). % 4.36/4.60 thf(sP4,plain,sP4 <=> ((![X1:$i]:(((rel_s5 @ eigen__0) @ X1) => (p @ X1))) => (~((![X1:$i]:(((rel_s5 @ eigen__0) @ X1) => (![X2:$i]:(((rel_s5 @ X1) @ X2) => (![X3:$i]:(((rel_s5 @ X2) @ X3) => (~((![X4:$i]:(((rel_s5 @ X3) @ X4) => (p @ X4)))))))))))))),introduced(definition,[new_symbols(definition,[sP4])])). % 4.36/4.60 thf(sP5,plain,sP5 <=> (((rel_s5 @ eigen__0) @ eigen__9) => ((rel_s5 @ eigen__9) @ eigen__0)),introduced(definition,[new_symbols(definition,[sP5])])). % 4.36/4.60 thf(sP6,plain,sP6 <=> ((~((((rel_s5 @ eigen__9) @ eigen__7) => (~(((rel_s5 @ eigen__7) @ eigen__3)))))) => ((rel_s5 @ eigen__9) @ eigen__3)),introduced(definition,[new_symbols(definition,[sP6])])). % 4.36/4.60 thf(sP7,plain,sP7 <=> (![X1:$i]:((~((((rel_s5 @ eigen__7) @ eigen__0) => (~(((rel_s5 @ eigen__0) @ X1)))))) => ((rel_s5 @ eigen__7) @ X1))),introduced(definition,[new_symbols(definition,[sP7])])). % 4.36/4.60 thf(sP8,plain,sP8 <=> (((rel_s5 @ eigen__0) @ eigen__7) => (~(((rel_s5 @ eigen__7) @ eigen__9)))),introduced(definition,[new_symbols(definition,[sP8])])). % 4.36/4.60 thf(sP9,plain,sP9 <=> (![X1:$i]:((~((((rel_s5 @ eigen__0) @ eigen__7) => (~(((rel_s5 @ eigen__7) @ X1)))))) => ((rel_s5 @ eigen__0) @ X1))),introduced(definition,[new_symbols(definition,[sP9])])). % 4.36/4.60 thf(sP10,plain,sP10 <=> ((rel_s5 @ eigen__9) @ eigen__0),introduced(definition,[new_symbols(definition,[sP10])])). % 4.36/4.60 thf(sP11,plain,sP11 <=> (![X1:$i]:(((rel_s5 @ eigen__9) @ X1) => (p @ X1))),introduced(definition,[new_symbols(definition,[sP11])])). % 4.36/4.60 thf(sP12,plain,sP12 <=> (((~((![X1:$i]:(((rel_s5 @ eigen__0) @ X1) => (![X2:$i]:(((rel_s5 @ X1) @ X2) => (![X3:$i]:(((rel_s5 @ X2) @ X3) => (~((![X4:$i]:(((rel_s5 @ X3) @ X4) => (p @ X4))))))))))))) => (![X1:$i]:(((rel_s5 @ eigen__0) @ X1) => (p @ X1)))) => (~(sP4))),introduced(definition,[new_symbols(definition,[sP12])])). % 4.36/4.60 thf(sP13,plain,sP13 <=> (((rel_s5 @ eigen__0) @ eigen__0) => (![X1:$i]:(((rel_s5 @ eigen__0) @ X1) => (~((![X2:$i]:(((rel_s5 @ X1) @ X2) => (p @ X2)))))))),introduced(definition,[new_symbols(definition,[sP13])])). % 4.36/4.60 thf(sP14,plain,sP14 <=> ((rel_s5 @ eigen__0) @ eigen__9),introduced(definition,[new_symbols(definition,[sP14])])). % 4.36/4.60 thf(sP15,plain,sP15 <=> (![X1:$i]:(((rel_s5 @ eigen__7) @ X1) => (~((![X2:$i]:(((rel_s5 @ X1) @ X2) => (p @ X2))))))),introduced(definition,[new_symbols(definition,[sP15])])). % 4.36/4.60 thf(sP16,plain,sP16 <=> ((~(sP8)) => sP14),introduced(definition,[new_symbols(definition,[sP16])])). % 4.36/4.60 thf(sP17,plain,sP17 <=> ((rel_s5 @ eigen__7) @ eigen__9),introduced(definition,[new_symbols(definition,[sP17])])). % 4.36/4.60 thf(sP18,plain,sP18 <=> (![X1:$i]:((~((((rel_s5 @ eigen__0) @ eigen__5) => (~(((rel_s5 @ eigen__5) @ X1)))))) => ((rel_s5 @ eigen__0) @ X1))),introduced(definition,[new_symbols(definition,[sP18])])). % 4.36/4.60 thf(sP19,plain,sP19 <=> ((rel_s5 @ eigen__0) @ eigen__5),introduced(definition,[new_symbols(definition,[sP19])])). % 4.36/4.60 thf(sP20,plain,sP20 <=> (((rel_s5 @ eigen__9) @ eigen__7) => (~(((rel_s5 @ eigen__7) @ eigen__3)))),introduced(definition,[new_symbols(definition,[sP20])])). % 4.36/4.60 thf(sP21,plain,sP21 <=> (![X1:$i]:(![X2:$i]:((~((((rel_s5 @ eigen__9) @ X1) => (~(((rel_s5 @ X1) @ X2)))))) => ((rel_s5 @ eigen__9) @ X2)))),introduced(definition,[new_symbols(definition,[sP21])])). % 4.36/4.60 thf(sP22,plain,sP22 <=> (![X1:$i]:((~((sP10 => (~(((rel_s5 @ eigen__0) @ X1)))))) => ((rel_s5 @ eigen__9) @ X1))),introduced(definition,[new_symbols(definition,[sP22])])). % 4.36/4.60 thf(sP23,plain,sP23 <=> ((~((![X1:$i]:(((rel_s5 @ eigen__0) @ X1) => (![X2:$i]:(((rel_s5 @ X1) @ X2) => (![X3:$i]:(((rel_s5 @ X2) @ X3) => (~((![X4:$i]:(((rel_s5 @ X3) @ X4) => (p @ X4))))))))))))) => (![X1:$i]:(((rel_s5 @ eigen__0) @ X1) => (p @ X1)))),introduced(definition,[new_symbols(definition,[sP23])])). % 4.36/4.60 thf(sP24,plain,sP24 <=> (((rel_s5 @ eigen__9) @ eigen__3) => (p @ eigen__3)),introduced(definition,[new_symbols(definition,[sP24])])). % 4.36/4.60 thf(sP25,plain,sP25 <=> (((rel_s5 @ eigen__5) @ eigen__7) => sP15),introduced(definition,[new_symbols(definition,[sP25])])). % 4.36/4.60 thf(sP26,plain,sP26 <=> ((rel_s5 @ eigen__0) @ eigen__3),introduced(definition,[new_symbols(definition,[sP26])])). % 4.36/4.60 thf(sP27,plain,sP27 <=> (p @ eigen__3),introduced(definition,[new_symbols(definition,[sP27])])). % 4.36/4.60 thf(sP28,plain,sP28 <=> (sP26 => sP27),introduced(definition,[new_symbols(definition,[sP28])])). % 4.36/4.60 thf(sP29,plain,sP29 <=> (sP19 => (~(((rel_s5 @ eigen__5) @ eigen__7)))),introduced(definition,[new_symbols(definition,[sP29])])). % 4.36/4.60 thf(sP30,plain,sP30 <=> (((rel_s5 @ eigen__0) @ eigen__7) => ((rel_s5 @ eigen__7) @ eigen__0)),introduced(definition,[new_symbols(definition,[sP30])])). % 4.36/4.60 thf(sP31,plain,sP31 <=> ((rel_s5 @ eigen__5) @ eigen__7),introduced(definition,[new_symbols(definition,[sP31])])). % 4.36/4.60 thf(sP32,plain,sP32 <=> (![X1:$i]:(((rel_s5 @ eigen__0) @ X1) => (p @ X1))),introduced(definition,[new_symbols(definition,[sP32])])). % 4.36/4.60 thf(sP33,plain,sP33 <=> (sP17 => (~(sP11))),introduced(definition,[new_symbols(definition,[sP33])])). % 4.36/4.60 thf(sP34,plain,sP34 <=> ((rel_s5 @ eigen__9) @ eigen__7),introduced(definition,[new_symbols(definition,[sP34])])). % 4.36/4.60 thf(sP35,plain,sP35 <=> ((rel_s5 @ eigen__9) @ eigen__3),introduced(definition,[new_symbols(definition,[sP35])])). % 4.36/4.60 thf(sP36,plain,sP36 <=> (![X1:$i]:((~((sP34 => (~(((rel_s5 @ eigen__7) @ X1)))))) => ((rel_s5 @ eigen__9) @ X1))),introduced(definition,[new_symbols(definition,[sP36])])). % 4.36/4.60 thf(sP37,plain,sP37 <=> (![X1:$i]:(((rel_s5 @ eigen__0) @ X1) => ((rel_s5 @ X1) @ eigen__0))),introduced(definition,[new_symbols(definition,[sP37])])). % 4.36/4.60 thf(sP38,plain,sP38 <=> (![X1:$i]:(![X2:$i]:(![X3:$i]:((~((((rel_s5 @ X1) @ X2) => (~(((rel_s5 @ X2) @ X3)))))) => ((rel_s5 @ X1) @ X3))))),introduced(definition,[new_symbols(definition,[sP38])])). % 4.36/4.60 thf(sP39,plain,sP39 <=> ((~(sP29)) => ((rel_s5 @ eigen__0) @ eigen__7)),introduced(definition,[new_symbols(definition,[sP39])])). % 4.36/4.60 thf(sP40,plain,sP40 <=> ((rel_s5 @ eigen__0) @ eigen__0),introduced(definition,[new_symbols(definition,[sP40])])). % 4.36/4.60 thf(sP41,plain,sP41 <=> (![X1:$i]:(~((((~((![X2:$i]:(((rel_s5 @ X1) @ X2) => (![X3:$i]:(((rel_s5 @ X2) @ X3) => (![X4:$i]:(((rel_s5 @ X3) @ X4) => (~((![X5:$i]:(((rel_s5 @ X4) @ X5) => (p @ X5))))))))))))) => (![X2:$i]:(((rel_s5 @ X1) @ X2) => (p @ X2)))) => (~(((![X2:$i]:(((rel_s5 @ X1) @ X2) => (p @ X2))) => (~((![X2:$i]:(((rel_s5 @ X1) @ X2) => (![X3:$i]:(((rel_s5 @ X2) @ X3) => (![X4:$i]:(((rel_s5 @ X3) @ X4) => (~((![X5:$i]:(((rel_s5 @ X4) @ X5) => (p @ X5)))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP41])])). % 4.36/4.60 thf(sP42,plain,sP42 <=> (sP19 => (![X1:$i]:(((rel_s5 @ eigen__5) @ X1) => (![X2:$i]:(((rel_s5 @ X1) @ X2) => (~((![X3:$i]:(((rel_s5 @ X2) @ X3) => (p @ X3)))))))))),introduced(definition,[new_symbols(definition,[sP42])])). % 4.36/4.60 thf(sP43,plain,sP43 <=> ((rel_s5 @ eigen__0) @ eigen__7),introduced(definition,[new_symbols(definition,[sP43])])). % 4.36/4.60 thf(sP44,plain,sP44 <=> (![X1:$i]:(((rel_s5 @ eigen__5) @ X1) => (![X2:$i]:(((rel_s5 @ X1) @ X2) => (~((![X3:$i]:(((rel_s5 @ X2) @ X3) => (p @ X3))))))))),introduced(definition,[new_symbols(definition,[sP44])])). % 4.36/4.60 thf(sP45,plain,sP45 <=> ((rel_s5 @ eigen__7) @ eigen__0),introduced(definition,[new_symbols(definition,[sP45])])). % 4.36/4.60 thf(sP46,plain,sP46 <=> ((~((sP45 => (~(sP26))))) => ((rel_s5 @ eigen__7) @ eigen__3)),introduced(definition,[new_symbols(definition,[sP46])])). % 4.36/4.60 thf(sP47,plain,sP47 <=> (![X1:$i]:(((rel_s5 @ eigen__0) @ X1) => (![X2:$i]:(((rel_s5 @ X1) @ X2) => (![X3:$i]:(((rel_s5 @ X2) @ X3) => (~((![X4:$i]:(((rel_s5 @ X3) @ X4) => (p @ X4))))))))))),introduced(definition,[new_symbols(definition,[sP47])])). % 4.36/4.60 thf(sP48,plain,sP48 <=> ((~((sP10 => (~(sP43))))) => sP34),introduced(definition,[new_symbols(definition,[sP48])])). % 4.36/4.60 thf(sP49,plain,sP49 <=> (![X1:$i]:(![X2:$i]:(((rel_s5 @ X1) @ X2) => ((rel_s5 @ X2) @ X1)))),introduced(definition,[new_symbols(definition,[sP49])])). % 4.36/4.60 thf(sP50,plain,sP50 <=> (![X1:$i]:(![X2:$i]:((~((((rel_s5 @ eigen__7) @ X1) => (~(((rel_s5 @ X1) @ X2)))))) => ((rel_s5 @ eigen__7) @ X2)))),introduced(definition,[new_symbols(definition,[sP50])])). % 4.36/4.60 thf(sP51,plain,sP51 <=> (sP40 => (~(sP32))),introduced(definition,[new_symbols(definition,[sP51])])). % 4.36/4.60 thf(sP52,plain,sP52 <=> (![X1:$i]:(((rel_s5 @ eigen__0) @ X1) => (~((![X2:$i]:(((rel_s5 @ X1) @ X2) => (p @ X2))))))),introduced(definition,[new_symbols(definition,[sP52])])). % 4.36/4.60 thf(sP53,plain,sP53 <=> (sP10 => (~(sP43))),introduced(definition,[new_symbols(definition,[sP53])])). % 4.36/4.60 thf(sP54,plain,sP54 <=> (![X1:$i]:(![X2:$i]:((~((((rel_s5 @ eigen__0) @ X1) => (~(((rel_s5 @ X1) @ X2)))))) => ((rel_s5 @ eigen__0) @ X2)))),introduced(definition,[new_symbols(definition,[sP54])])). % 4.36/4.60 thf(sP55,plain,sP55 <=> ((rel_s5 @ eigen__7) @ eigen__3),introduced(definition,[new_symbols(definition,[sP55])])). % 4.36/4.60 thf(sP56,plain,sP56 <=> (sP45 => (~(sP26))),introduced(definition,[new_symbols(definition,[sP56])])). % 4.36/4.60 thf(def_meq_ind,definition,(meq_ind = (^[X1:mu]:(^[X2:mu]:(^[X3:$i]:(X1 = X2)))))). % 4.36/4.60 thf(def_meq_prop,definition,(meq_prop = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((X1 @ X3) = (X2 @ X3))))))). % 4.36/4.60 thf(def_mnot,definition,(mnot = (^[X1:$i>$o]:(^[X2:$i]:(~((X1 @ X2))))))). % 4.36/4.60 thf(def_mor,definition,(mor = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((~((X1 @ X3))) => (X2 @ X3))))))). % 4.36/4.60 thf(def_mand,definition,(mand = (^[X1:$i>$o]:(^[X2:$i>$o]:(mnot @ ((mor @ (mnot @ X1)) @ (mnot @ X2))))))). % 4.36/4.60 thf(def_mimplies,definition,(mimplies = (^[X1:$i>$o]:(mor @ (mnot @ X1))))). % 4.36/4.60 thf(def_mimplied,definition,(mimplied = (^[X1:$i>$o]:(^[X2:$i>$o]:((mor @ (mnot @ X2)) @ X1))))). % 4.36/4.60 thf(def_mequiv,definition,(mequiv = (^[X1:$i>$o]:(^[X2:$i>$o]:((mand @ ((mimplies @ X1) @ X2)) @ ((mimplies @ X2) @ X1)))))). % 4.36/4.60 thf(def_mxor,definition,(mxor = (^[X1:$i>$o]:(^[X2:$i>$o]:(mnot @ ((mequiv @ X1) @ X2)))))). % 4.36/4.60 thf(def_mforall_ind,definition,(mforall_ind = (^[X1:mu>$i>$o]:(^[X2:$i]:(![X3:mu]:((X1 @ X3) @ X2)))))). % 4.36/4.60 thf(def_mforall_prop,definition,(mforall_prop = (^[X1:($i>$o)>$i>$o]:(^[X2:$i]:(![X3:$i>$o]:((X1 @ X3) @ X2)))))). % 4.36/4.60 thf(def_mexists_ind,definition,(mexists_ind = (^[X1:mu>$i>$o]:(mnot @ (mforall_ind @ (^[X2:mu]:(mnot @ (X1 @ X2)))))))). % 4.36/4.60 thf(def_mexists_prop,definition,(mexists_prop = (^[X1:($i>$o)>$i>$o]:(mnot @ (mforall_prop @ (^[X2:$i>$o]:(mnot @ (X1 @ X2)))))))). % 4.36/4.60 thf(def_mtrue,definition,(mtrue = (^[X1:$i]:(~($false))))). % 4.36/4.60 thf(def_mfalse,definition,(mfalse = (mnot @ mtrue))). % 4.36/4.60 thf(def_mbox,definition,(mbox = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:(![X4:$i]:(((X1 @ X3) @ X4) => (X2 @ X4)))))))). % 4.36/4.60 thf(def_mdia,definition,(mdia = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(mnot @ ((mbox @ X1) @ (mnot @ X2))))))). % 4.36/4.60 thf(def_mreflexive,definition,(mreflexive = (^[X1:$i>$i>$o]:(![X2:$i]:((X1 @ X2) @ X2))))). % 4.36/4.60 thf(def_msymmetric,definition,(msymmetric = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(((X1 @ X2) @ X3) => ((X1 @ X3) @ X2))))))). % 4.36/4.60 thf(def_mserial,definition,(mserial = (^[X1:$i>$i>$o]:(![X2:$i]:(~((![X3:$i]:(~(((X1 @ X2) @ X3)))))))))). % 4.36/4.60 thf(def_mtransitive,definition,(mtransitive = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X3) @ X4)))))) => ((X1 @ X2) @ X4)))))))). % 4.36/4.60 thf(def_meuclidean,definition,(meuclidean = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => ((X1 @ X3) @ X4)))))))). % 4.36/4.60 thf(def_mpartially_functional,definition,(mpartially_functional = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => (X3 = X4)))))))). % 4.36/4.60 thf(def_mfunctional,definition,(mfunctional = (^[X1:$i>$i>$o]:(![X2:$i]:(~((![X3:$i]:(((X1 @ X2) @ X3) => (~((![X4:$i]:(((X1 @ X2) @ X4) => (X3 = X4))))))))))))). % 4.36/4.60 thf(def_mweakly_dense,definition,(mweakly_dense = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((X1 @ X2) @ X3) => (~((![X5:$i]:(((X1 @ X2) @ X5) => (~(((X1 @ X5) @ X3)))))))))))))). % 4.36/4.60 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))))))))). % 4.36/4.60 thf(def_mweakly_directed,definition,(mweakly_directed = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => (~((![X5:$i]:(((X1 @ X3) @ X5) => (~(((X1 @ X4) @ X5)))))))))))))). % 4.36/4.60 thf(def_mvalid,definition,(mvalid = (!!))). % 4.36/4.60 thf(def_minvalid,definition,(minvalid = (^[X1:$i>$o]:(![X2:$i]:(~((X1 @ X2))))))). % 4.36/4.60 thf(def_msatisfiable,definition,(msatisfiable = (^[X1:$i>$o]:(~((![X2:$i]:(~((X1 @ X2))))))))). % 4.36/4.60 thf(def_mcountersatisfiable,definition,(mcountersatisfiable = (^[X1:$i>$o]:(~(((!!) @ X1)))))). % 4.36/4.60 thf(def_mbox_s5,definition,(mbox_s5 = (^[X1:$i>$o]:(^[X2:$i]:(![X3:$i]:(((rel_s5 @ X2) @ X3) => (X1 @ X3))))))). % 4.36/4.60 thf(def_mdia_s5,definition,(mdia_s5 = (^[X1:$i>$o]:(mnot @ (mbox_s5 @ (mnot @ X1)))))). % 4.36/4.60 thf(prove,conjecture,(![X1:$i]:(~(((~((~(((~((~((~((![X2:$i]:(((rel_s5 @ X1) @ X2) => (~((~((![X3:$i]:(((rel_s5 @ X2) @ X3) => (~((~((![X4:$i]:(((rel_s5 @ X3) @ X4) => (~((![X5:$i]:(((rel_s5 @ X4) @ X5) => (p @ X5))))))))))))))))))))))))) => (![X2:$i]:(((rel_s5 @ X1) @ X2) => (p @ X2)))))))) => (~(((~((~((![X2:$i]:(((rel_s5 @ X1) @ X2) => (p @ X2))))))) => (~((![X2:$i]:(((rel_s5 @ X1) @ X2) => (~((~((![X3:$i]:(((rel_s5 @ X2) @ X3) => (~((~((![X4:$i]:(((rel_s5 @ X3) @ X4) => (~((![X5:$i]:(((rel_s5 @ X4) @ X5) => (p @ X5))))))))))))))))))))))))))))). % 4.36/4.60 thf(h1,negated_conjecture,(~(sP41)),inference(assume_negation,[status(cth)],[prove])). % 4.36/4.60 thf(1,plain,(~(sP21) | sP22),inference(all_rule,[status(thm)],[])). % 4.36/4.60 thf(2,plain,(~(sP22) | sP48),inference(all_rule,[status(thm)],[])). % 4.36/4.60 thf(3,plain,((~(sP48) | sP53) | sP34),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(4,plain,((~(sP53) | ~(sP10)) | ~(sP43)),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(5,plain,(~(sP36) | sP6),inference(all_rule,[status(thm)],[])). % 4.36/4.60 thf(6,plain,((~(sP6) | sP20) | sP35),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(7,plain,((~(sP20) | ~(sP34)) | ~(sP55)),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(8,plain,(~(sP21) | sP36),inference(all_rule,[status(thm)],[])). % 4.36/4.60 thf(9,plain,(~(sP18) | sP39),inference(all_rule,[status(thm)],[])). % 4.36/4.60 thf(10,plain,((~(sP39) | sP29) | sP43),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(11,plain,((~(sP29) | ~(sP19)) | ~(sP31)),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(12,plain,(~(sP9) | sP16),inference(all_rule,[status(thm)],[])). % 4.36/4.60 thf(13,plain,((~(sP16) | sP8) | sP14),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(14,plain,((~(sP8) | ~(sP43)) | ~(sP17)),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(15,plain,(~(sP7) | sP46),inference(all_rule,[status(thm)],[])). % 4.36/4.60 thf(16,plain,((~(sP46) | sP56) | sP55),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(17,plain,((~(sP56) | ~(sP45)) | ~(sP26)),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(18,plain,(~(sP50) | sP7),inference(all_rule,[status(thm)],[])). % 4.36/4.60 thf(19,plain,(~(sP54) | sP18),inference(all_rule,[status(thm)],[])). % 4.36/4.60 thf(20,plain,((~(sP5) | ~(sP14)) | sP10),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(21,plain,(~(sP38) | sP21),inference(all_rule,[status(thm)],[])). % 4.36/4.60 thf(22,plain,(~(sP37) | sP5),inference(all_rule,[status(thm)],[])). % 4.36/4.60 thf(23,plain,(~(sP11) | sP24),inference(all_rule,[status(thm)],[])). % 4.36/4.60 thf(24,plain,((~(sP24) | ~(sP35)) | sP27),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(25,plain,(sP33 | sP11),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(26,plain,(sP33 | sP17),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(27,plain,((~(sP30) | ~(sP43)) | sP45),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(28,plain,(~(sP38) | sP50),inference(all_rule,[status(thm)],[])). % 4.36/4.60 thf(29,plain,(~(sP54) | sP9),inference(all_rule,[status(thm)],[])). % 4.36/4.60 thf(30,plain,(~(sP37) | sP30),inference(all_rule,[status(thm)],[])). % 4.36/4.60 thf(31,plain,(sP15 | ~(sP33)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__9])). % 4.36/4.60 thf(32,plain,(sP25 | ~(sP15)),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(33,plain,(sP25 | sP31),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(34,plain,(sP44 | ~(sP25)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__7])). % 4.36/4.60 thf(35,plain,(sP42 | ~(sP44)),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(36,plain,(sP42 | sP19),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(37,plain,(sP47 | ~(sP42)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__5])). % 4.36/4.60 thf(38,plain,(sP23 | ~(sP32)),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(39,plain,(sP23 | ~(sP47)),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(40,plain,(sP28 | ~(sP27)),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(41,plain,(sP28 | sP26),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(42,plain,(sP32 | ~(sP28)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__3])). % 4.36/4.60 thf(43,plain,(~(sP3) | sP13),inference(all_rule,[status(thm)],[])). % 4.36/4.60 thf(44,plain,((~(sP13) | ~(sP40)) | sP52),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(45,plain,(~(sP52) | sP51),inference(all_rule,[status(thm)],[])). % 4.36/4.60 thf(46,plain,((~(sP51) | ~(sP40)) | ~(sP32)),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(47,plain,((~(sP2) | ~(sP40)) | sP3),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(48,plain,(~(sP38) | sP54),inference(all_rule,[status(thm)],[])). % 4.36/4.60 thf(49,plain,(~(sP49) | sP37),inference(all_rule,[status(thm)],[])). % 4.36/4.60 thf(50,plain,(~(sP1) | sP40),inference(all_rule,[status(thm)],[])). % 4.36/4.60 thf(51,plain,(~(sP47) | sP2),inference(all_rule,[status(thm)],[])). % 4.36/4.60 thf(52,plain,(sP4 | sP47),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(53,plain,(sP4 | sP32),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(54,plain,((~(sP12) | ~(sP23)) | ~(sP4)),inference(prop_rule,[status(thm)],[])). % 4.36/4.60 thf(55,plain,(sP41 | sP12),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__0])). % 4.36/4.60 thf(a3,axiom,(msymmetric @ rel_s5)). % 4.36/4.60 thf(56,plain,sP49,inference(preprocess,[status(thm)],[a3]). % 4.36/4.60 thf(a2,axiom,(mtransitive @ rel_s5)). % 4.36/4.60 thf(57,plain,sP38,inference(preprocess,[status(thm)],[a2]). % 4.36/4.60 thf(a1,axiom,(mreflexive @ rel_s5)). % 4.36/4.60 thf(58,plain,sP1,inference(preprocess,[status(thm)],[a1]). % 4.36/4.60 thf(59,plain,$false,inference(prop_unsat,[status(thm),assumptions([h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,h1])). % 4.36/4.60 thf(60,plain,$false,inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[59,h0])). % 4.36/4.60 thf(0,theorem,(![X1:$i]:(~(((~((~(((~((~((~((![X2:$i]:(((rel_s5 @ X1) @ X2) => (~((~((![X3:$i]:(((rel_s5 @ X2) @ X3) => (~((~((![X4:$i]:(((rel_s5 @ X3) @ X4) => (~((![X5:$i]:(((rel_s5 @ X4) @ X5) => (p @ X5))))))))))))))))))))))))) => (![X2:$i]:(((rel_s5 @ X1) @ X2) => (p @ X2)))))))) => (~(((~((~((![X2:$i]:(((rel_s5 @ X1) @ X2) => (p @ X2))))))) => (~((![X2:$i]:(((rel_s5 @ X1) @ X2) => (~((~((![X3:$i]:(((rel_s5 @ X2) @ X3) => (~((~((![X4:$i]:(((rel_s5 @ X3) @ X4) => (~((![X5:$i]:(((rel_s5 @ X4) @ X5) => (p @ X5)))))))))))))))))))))))))))),inference(contra,[status(thm),contra(discharge,[h1])],[59,h1])). % 4.36/4.60 % SZS output end Proof %------------------------------------------------------------------------------