%------------------------------------------------------------------------------ % File : Satallax---3.5 % Problem : PLA033^7 : 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 : 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 02:14:08 PM UTC 2025 % Result : Theorem 47.04s 47.35s % Output : Proof 47.04s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.04/0.12 % Problem : PLA033^7 : TPTP v9.2.0. Released v5.5.0. % 0.04/0.13 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % 0.12/0.34 % Computer : n017.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 12:21:08 EDT 2025 % 0.12/0.34 % CPUTime : % 47.04/47.35 % SZS status Theorem % 47.04/47.35 % Mode: mode484 % 47.04/47.35 % Inferences: 11504 % 47.04/47.35 % SZS output start Proof % 47.04/47.35 thf(ty_mu, type, mu : $tType). % 47.04/47.35 thf(ty_h, type, h : (mu>$i>$o)). % 47.04/47.35 thf(ty_rel_s4, type, rel_s4 : ($i>$i>$o)). % 47.04/47.35 thf(ty_d, type, d : mu). % 47.04/47.35 thf(ty_n, type, n : mu). % 47.04/47.35 thf(ty_eigen__1, type, eigen__1 : $i). % 47.04/47.35 thf(ty_eigen__0, type, eigen__0 : $i). % 47.04/47.35 thf(ty_eigen__10, type, eigen__10 : mu). % 47.04/47.35 thf(ty_open, type, open : (mu>$i>$o)). % 47.04/47.35 thf(ty_closed, type, closed : (mu>$i>$o)). % 47.04/47.35 thf(ty_exists_in_world, type, exists_in_world : (mu>$i>$o)). % 47.04/47.35 thf(ty_combo, type, combo : (mu>mu>$i>$o)). % 47.04/47.35 thf(ty_o, type, o : mu). % 47.04/47.35 thf(h0, assumption, (![X1:mu>$o]:(![X2:mu]:((X1 @ X2) => (X1 @ (eps__0 @ X1))))),introduced(assumption,[])). % 47.04/47.35 thf(eigendef_eigen__10, definition, eigen__10 = (eps__0 @ (^[X1:mu]:(~((((exists_in_world @ X1) @ eigen__0) => ((![X2:$i]:(((rel_s4 @ eigen__0) @ X2) => ((~((((closed @ d) @ X2) => ((((combo @ d) @ n) @ X2) => (~(((h @ X1) @ X2))))))) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => ((open @ d) @ X3)))))) => (~((![X2:$i]:(((rel_s4 @ eigen__0) @ X2) => ((~((((closed @ d) @ X2) => ((~((((combo @ d) @ n) @ X2))) => (~(((h @ o) @ X2))))))) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => ((closed @ d) @ X3)))))))))))))), introduced(definition,[new_symbols(definition,[eigen__10])])). % 47.04/47.35 thf(sP1,plain,sP1 <=> ((rel_s4 @ eigen__0) @ eigen__1),introduced(definition,[new_symbols(definition,[sP1])])). % 47.04/47.35 thf(sP2,plain,sP2 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__1) => (~(((![X2:$i]:(((rel_s4 @ eigen__1) @ X2) => (~(((((combo @ d) @ n) @ X2) => (~(((h @ X1) @ X2)))))))) => (![X2:$i]:(((rel_s4 @ eigen__1) @ X2) => ((open @ d) @ X2)))))))),introduced(definition,[new_symbols(definition,[sP2])])). % 47.04/47.35 thf(sP3,plain,sP3 <=> ((exists_in_world @ d) @ eigen__0),introduced(definition,[new_symbols(definition,[sP3])])). % 47.04/47.35 thf(sP4,plain,sP4 <=> (![X1:$i]:(![X2:$i]:(((rel_s4 @ X1) @ X2) => (![X3:mu]:(((exists_in_world @ X3) @ X2) => (![X4:mu]:(((exists_in_world @ X4) @ X2) => (~((![X5:mu]:(((exists_in_world @ X5) @ X2) => ((![X6:$i]:(((rel_s4 @ X2) @ X6) => ((~((((closed @ X3) @ X6) => ((((combo @ X3) @ X4) @ X6) => (~(((h @ X5) @ X6))))))) => (![X7:$i]:(((rel_s4 @ X6) @ X7) => ((open @ X3) @ X7)))))) => (~((![X6:$i]:(((rel_s4 @ X2) @ X6) => ((~((((closed @ X3) @ X6) => ((~((((combo @ X3) @ X4) @ X6))) => (~(((h @ o) @ X6))))))) => (![X7:$i]:(((rel_s4 @ X6) @ X7) => ((closed @ X3) @ X7)))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP4])])). % 47.04/47.35 thf(sP5,plain,sP5 <=> ((closed @ d) @ eigen__1),introduced(definition,[new_symbols(definition,[sP5])])). % 47.04/47.35 thf(sP6,plain,sP6 <=> ((rel_s4 @ eigen__0) @ eigen__0),introduced(definition,[new_symbols(definition,[sP6])])). % 47.04/47.35 thf(sP7,plain,sP7 <=> ((exists_in_world @ n) @ eigen__0),introduced(definition,[new_symbols(definition,[sP7])])). % 47.04/47.35 thf(sP8,plain,sP8 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => (![X2:mu]:(((exists_in_world @ X2) @ eigen__0) => (~((![X3:mu]:(((exists_in_world @ X3) @ eigen__0) => ((![X4:$i]:(((rel_s4 @ eigen__0) @ X4) => ((~((((closed @ X1) @ X4) => ((((combo @ X1) @ X2) @ X4) => (~(((h @ X3) @ X4))))))) => (![X5:$i]:(((rel_s4 @ X4) @ X5) => ((open @ X1) @ X5)))))) => (~((![X4:$i]:(((rel_s4 @ eigen__0) @ X4) => ((~((((closed @ X1) @ X4) => ((~((((combo @ X1) @ X2) @ X4))) => (~(((h @ o) @ X4))))))) => (![X5:$i]:(((rel_s4 @ X4) @ X5) => ((closed @ X1) @ X5))))))))))))))))),introduced(definition,[new_symbols(definition,[sP8])])). % 47.04/47.35 thf(sP9,plain,sP9 <=> ((exists_in_world @ eigen__10) @ eigen__0),introduced(definition,[new_symbols(definition,[sP9])])). % 47.04/47.35 thf(sP10,plain,sP10 <=> (sP1 => ((~((sP5 => ((((combo @ d) @ n) @ eigen__1) => (~(((h @ eigen__10) @ eigen__1))))))) => (![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => ((open @ d) @ X1))))),introduced(definition,[new_symbols(definition,[sP10])])). % 47.04/47.35 thf(sP11,plain,sP11 <=> ((rel_s4 @ eigen__1) @ eigen__1),introduced(definition,[new_symbols(definition,[sP11])])). % 47.04/47.35 thf(sP12,plain,sP12 <=> ((exists_in_world @ n) @ eigen__1),introduced(definition,[new_symbols(definition,[sP12])])). % 47.04/47.35 thf(sP13,plain,sP13 <=> (![X1:$i]:((~((sP9 => (~(((rel_s4 @ eigen__0) @ X1)))))) => ((exists_in_world @ eigen__10) @ X1))),introduced(definition,[new_symbols(definition,[sP13])])). % 47.04/47.35 thf(sP14,plain,sP14 <=> ((![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => (~(((((combo @ d) @ n) @ X1) => (~(((h @ n) @ X1)))))))) => (![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => ((open @ d) @ X1)))),introduced(definition,[new_symbols(definition,[sP14])])). % 47.04/47.35 thf(sP15,plain,sP15 <=> (![X1:$i]:(![X2:$i]:(((rel_s4 @ X1) @ X2) => ((closed @ d) @ X2)))),introduced(definition,[new_symbols(definition,[sP15])])). % 47.04/47.35 thf(sP16,plain,sP16 <=> (sP6 => sP8),introduced(definition,[new_symbols(definition,[sP16])])). % 47.04/47.35 thf(sP17,plain,sP17 <=> (![X1:$i]:(((rel_s4 @ eigen__0) @ X1) => ((~((((closed @ d) @ X1) => ((((combo @ d) @ n) @ X1) => (~(((h @ eigen__10) @ X1))))))) => (![X2:$i]:(((rel_s4 @ X1) @ X2) => ((open @ d) @ X2)))))),introduced(definition,[new_symbols(definition,[sP17])])). % 47.04/47.35 thf(sP18,plain,sP18 <=> (![X1:$i]:(![X2:$i]:((~((((exists_in_world @ eigen__10) @ X1) => (~(((rel_s4 @ X1) @ X2)))))) => ((exists_in_world @ eigen__10) @ X2)))),introduced(definition,[new_symbols(definition,[sP18])])). % 47.04/47.35 thf(sP19,plain,sP19 <=> (sP11 => (~(((((combo @ d) @ n) @ eigen__1) => (~(((h @ eigen__10) @ eigen__1))))))),introduced(definition,[new_symbols(definition,[sP19])])). % 47.04/47.35 thf(sP20,plain,sP20 <=> (sP9 => (~(sP1))),introduced(definition,[new_symbols(definition,[sP20])])). % 47.04/47.35 thf(sP21,plain,sP21 <=> (![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => ((open @ d) @ X1))),introduced(definition,[new_symbols(definition,[sP21])])). % 47.04/47.35 thf(sP22,plain,sP22 <=> (![X1:$i]:((rel_s4 @ X1) @ X1)),introduced(definition,[new_symbols(definition,[sP22])])). % 47.04/47.35 thf(sP23,plain,sP23 <=> ((~(sP20)) => ((exists_in_world @ eigen__10) @ eigen__1)),introduced(definition,[new_symbols(definition,[sP23])])). % 47.04/47.35 thf(sP24,plain,sP24 <=> ((!!) @ (exists_in_world @ n)),introduced(definition,[new_symbols(definition,[sP24])])). % 47.04/47.35 thf(sP25,plain,sP25 <=> (![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,[sP25])])). % 47.04/47.35 thf(sP26,plain,sP26 <=> (![X1:$i]:(((rel_s4 @ eigen__0) @ X1) => (![X2:mu]:(((exists_in_world @ X2) @ X1) => (![X3:mu]:(((exists_in_world @ X3) @ X1) => (~((![X4:mu]:(((exists_in_world @ X4) @ X1) => ((![X5:$i]:(((rel_s4 @ X1) @ X5) => ((~((((closed @ X2) @ X5) => ((((combo @ X2) @ X3) @ X5) => (~(((h @ X4) @ X5))))))) => (![X6:$i]:(((rel_s4 @ X5) @ X6) => ((open @ X2) @ X6)))))) => (~((![X5:$i]:(((rel_s4 @ X1) @ X5) => ((~((((closed @ X2) @ X5) => ((~((((combo @ X2) @ X3) @ X5))) => (~(((h @ o) @ X5))))))) => (![X6:$i]:(((rel_s4 @ X5) @ X6) => ((closed @ X2) @ X6))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP26])])). % 47.04/47.35 thf(sP27,plain,sP27 <=> (((exists_in_world @ eigen__10) @ eigen__1) => (~(((![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => (~(((((combo @ d) @ n) @ X1) => (~(((h @ eigen__10) @ X1)))))))) => sP21)))),introduced(definition,[new_symbols(definition,[sP27])])). % 47.04/47.35 thf(sP28,plain,sP28 <=> (sP7 => (~((![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => ((![X2:$i]:(((rel_s4 @ eigen__0) @ X2) => ((~((((closed @ d) @ X2) => ((((combo @ d) @ n) @ X2) => (~(((h @ X1) @ X2))))))) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => ((open @ d) @ X3)))))) => (~((![X2:$i]:(((rel_s4 @ eigen__0) @ X2) => ((~((((closed @ d) @ X2) => ((~((((combo @ d) @ n) @ X2))) => (~(((h @ o) @ X2))))))) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => ((closed @ d) @ X3)))))))))))))),introduced(definition,[new_symbols(definition,[sP28])])). % 47.04/47.35 thf(sP29,plain,sP29 <=> (sP1 => sP5),introduced(definition,[new_symbols(definition,[sP29])])). % 47.04/47.35 thf(sP30,plain,sP30 <=> (sP17 => (~((![X1:$i]:(((rel_s4 @ eigen__0) @ X1) => ((~((((closed @ d) @ X1) => ((~((((combo @ d) @ n) @ X1))) => (~(((h @ o) @ X1))))))) => (![X2:$i]:(((rel_s4 @ X1) @ X2) => ((closed @ d) @ X2))))))))),introduced(definition,[new_symbols(definition,[sP30])])). % 47.04/47.35 thf(sP31,plain,sP31 <=> ((exists_in_world @ eigen__10) @ eigen__1),introduced(definition,[new_symbols(definition,[sP31])])). % 47.04/47.35 thf(sP32,plain,sP32 <=> (![X1:$i]:(((rel_s4 @ eigen__0) @ X1) => ((closed @ d) @ X1))),introduced(definition,[new_symbols(definition,[sP32])])). % 47.04/47.35 thf(sP33,plain,sP33 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => ((![X2:$i]:(((rel_s4 @ eigen__0) @ X2) => ((~((((closed @ d) @ X2) => ((((combo @ d) @ n) @ X2) => (~(((h @ X1) @ X2))))))) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => ((open @ d) @ X3)))))) => (~((![X2:$i]:(((rel_s4 @ eigen__0) @ X2) => ((~((((closed @ d) @ X2) => ((~((((combo @ d) @ n) @ X2))) => (~(((h @ o) @ X2))))))) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => ((closed @ d) @ X3))))))))))),introduced(definition,[new_symbols(definition,[sP33])])). % 47.04/47.35 thf(sP34,plain,sP34 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__1) => (![X2:mu]:(((exists_in_world @ X2) @ eigen__1) => (~(((![X3:$i]:(((rel_s4 @ eigen__1) @ X3) => (~(((((combo @ d) @ X1) @ X3) => (~(((h @ X2) @ X3)))))))) => sP21))))))),introduced(definition,[new_symbols(definition,[sP34])])). % 47.04/47.35 thf(sP35,plain,sP35 <=> ((~((sP5 => ((((combo @ d) @ n) @ eigen__1) => (~(((h @ eigen__10) @ eigen__1))))))) => sP21),introduced(definition,[new_symbols(definition,[sP35])])). % 47.04/47.35 thf(sP36,plain,sP36 <=> (sP9 => sP30),introduced(definition,[new_symbols(definition,[sP36])])). % 47.04/47.35 thf(sP37,plain,sP37 <=> (sP5 => ((((combo @ d) @ n) @ eigen__1) => (~(((h @ eigen__10) @ eigen__1))))),introduced(definition,[new_symbols(definition,[sP37])])). % 47.04/47.35 thf(sP38,plain,sP38 <=> (![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => (~(((((combo @ d) @ n) @ X1) => (~(((h @ eigen__10) @ X1)))))))),introduced(definition,[new_symbols(definition,[sP38])])). % 47.04/47.35 thf(sP39,plain,sP39 <=> ((!!) @ (exists_in_world @ d)),introduced(definition,[new_symbols(definition,[sP39])])). % 47.04/47.35 thf(sP40,plain,sP40 <=> (sP3 => (![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => (~((![X2:mu]:(((exists_in_world @ X2) @ eigen__0) => ((![X3:$i]:(((rel_s4 @ eigen__0) @ X3) => ((~((((closed @ d) @ X3) => ((((combo @ d) @ X1) @ X3) => (~(((h @ X2) @ X3))))))) => (![X4:$i]:(((rel_s4 @ X3) @ X4) => ((open @ d) @ X4)))))) => (~((![X3:$i]:(((rel_s4 @ eigen__0) @ X3) => ((~((((closed @ d) @ X3) => ((~((((combo @ d) @ X1) @ X3))) => (~(((h @ o) @ X3))))))) => (![X4:$i]:(((rel_s4 @ X3) @ X4) => ((closed @ d) @ X4)))))))))))))))),introduced(definition,[new_symbols(definition,[sP40])])). % 47.04/47.35 thf(sP41,plain,sP41 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__0) => (~((![X2:mu]:(((exists_in_world @ X2) @ eigen__0) => ((![X3:$i]:(((rel_s4 @ eigen__0) @ X3) => ((~((((closed @ d) @ X3) => ((((combo @ d) @ X1) @ X3) => (~(((h @ X2) @ X3))))))) => (![X4:$i]:(((rel_s4 @ X3) @ X4) => ((open @ d) @ X4)))))) => (~((![X3:$i]:(((rel_s4 @ eigen__0) @ X3) => ((~((((closed @ d) @ X3) => ((~((((combo @ d) @ X1) @ X3))) => (~(((h @ o) @ X3))))))) => (![X4:$i]:(((rel_s4 @ X3) @ X4) => ((closed @ d) @ X4))))))))))))))),introduced(definition,[new_symbols(definition,[sP41])])). % 47.04/47.35 thf(sP42,plain,sP42 <=> (sP12 => sP2),introduced(definition,[new_symbols(definition,[sP42])])). % 47.04/47.35 thf(sP43,plain,sP43 <=> (sP38 => sP21),introduced(definition,[new_symbols(definition,[sP43])])). % 47.04/47.35 thf(sP44,plain,sP44 <=> (sP12 => (~(sP14))),introduced(definition,[new_symbols(definition,[sP44])])). % 47.04/47.35 thf(sP45,plain,sP45 <=> ((((combo @ d) @ n) @ eigen__1) => (~(((h @ eigen__10) @ eigen__1)))),introduced(definition,[new_symbols(definition,[sP45])])). % 47.04/47.35 thf(def_meq_prop,definition,(meq_prop = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((X1 @ X3) = (X2 @ X3))))))). % 47.04/47.35 thf(def_mnot,definition,(mnot = (^[X1:$i>$o]:(^[X2:$i]:(~((X1 @ X2))))))). % 47.04/47.35 thf(def_mor,definition,(mor = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((~((X1 @ X3))) => (X2 @ X3))))))). % 47.04/47.35 thf(def_mbox,definition,(mbox = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:(![X4:$i]:(((X1 @ X3) @ X4) => (X2 @ X4)))))))). % 47.04/47.35 thf(def_mforall_prop,definition,(mforall_prop = (^[X1:($i>$o)>$i>$o]:(^[X2:$i]:(![X3:$i>$o]:((X1 @ X3) @ X2)))))). % 47.04/47.35 thf(def_mtrue,definition,(mtrue = (^[X1:$i]:(~($false))))). % 47.04/47.35 thf(def_mfalse,definition,(mfalse = (mnot @ mtrue))). % 47.04/47.35 thf(def_mand,definition,(mand = (^[X1:$i>$o]:(^[X2:$i>$o]:(mnot @ ((mor @ (mnot @ X1)) @ (mnot @ X2))))))). % 47.04/47.35 thf(def_mimplies,definition,(mimplies = (^[X1:$i>$o]:(mor @ (mnot @ X1))))). % 47.04/47.35 thf(def_mimplied,definition,(mimplied = (^[X1:$i>$o]:(^[X2:$i>$o]:((mor @ (mnot @ X2)) @ X1))))). % 47.04/47.35 thf(def_mequiv,definition,(mequiv = (^[X1:$i>$o]:(^[X2:$i>$o]:((mand @ ((mimplies @ X1) @ X2)) @ ((mimplies @ X2) @ X1)))))). % 47.04/47.35 thf(def_mxor,definition,(mxor = (^[X1:$i>$o]:(^[X2:$i>$o]:(mnot @ ((mequiv @ X1) @ X2)))))). % 47.04/47.35 thf(def_mdia,definition,(mdia = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(mnot @ ((mbox @ X1) @ (mnot @ X2))))))). % 47.04/47.35 thf(def_mforall_ind,definition,(mforall_ind = (^[X1:mu>$i>$o]:(^[X2:$i]:(![X3:mu]:(((exists_in_world @ X3) @ X2) => ((X1 @ X3) @ X2))))))). % 47.04/47.35 thf(def_mexists_ind,definition,(mexists_ind = (^[X1:mu>$i>$o]:(mnot @ (mforall_ind @ (^[X2:mu]:(mnot @ (X1 @ X2)))))))). % 47.04/47.35 thf(def_mexists_prop,definition,(mexists_prop = (^[X1:($i>$o)>$i>$o]:(mnot @ (mforall_prop @ (^[X2:$i>$o]:(mnot @ (X1 @ X2)))))))). % 47.04/47.35 thf(def_mreflexive,definition,(mreflexive = (^[X1:$i>$i>$o]:(![X2:$i]:((X1 @ X2) @ X2))))). % 47.04/47.35 thf(def_msymmetric,definition,(msymmetric = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(((X1 @ X2) @ X3) => ((X1 @ X3) @ X2))))))). % 47.04/47.35 thf(def_mserial,definition,(mserial = (^[X1:$i>$i>$o]:(![X2:$i]:(~((![X3:$i]:(~(((X1 @ X2) @ X3)))))))))). % 47.04/47.35 thf(def_mtransitive,definition,(mtransitive = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X3) @ X4)))))) => ((X1 @ X2) @ X4)))))))). % 47.04/47.35 thf(def_meuclidean,definition,(meuclidean = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => ((X1 @ X3) @ X4)))))))). % 47.04/47.35 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)))))))). % 47.04/47.35 thf(def_mfunctional,definition,(mfunctional = (^[X1:$i>$i>$o]:(![X2:$i]:(~((![X3:$i]:(((X1 @ X2) @ X3) => (~((![X4:$i]:(((X1 @ X2) @ X4) => (X3 = X4))))))))))))). % 47.04/47.35 thf(def_mweakly_dense,definition,(mweakly_dense = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((X1 @ X2) @ X3) => (~((![X5:$i]:(((X1 @ X2) @ X5) => (~(((X1 @ X5) @ X3)))))))))))))). % 47.04/47.35 thf(def_mweakly_connected,definition,(mweakly_connected = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => ((~(((~(((X1 @ X3) @ X4))) => (X3 = X4)))) => ((X1 @ X4) @ X3))))))))). % 47.04/47.35 thf(def_mweakly_directed,definition,(mweakly_directed = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => (~((![X5:$i]:(((X1 @ X3) @ X5) => (~(((X1 @ X4) @ X5)))))))))))))). % 47.04/47.35 thf(def_mvalid,definition,(mvalid = (!!))). % 47.04/47.35 thf(def_msatisfiable,definition,(msatisfiable = (^[X1:$i>$o]:(~((![X2:$i]:(~((X1 @ X2))))))))). % 47.04/47.35 thf(def_mcountersatisfiable,definition,(mcountersatisfiable = (^[X1:$i>$o]:(~(((!!) @ X1)))))). % 47.04/47.35 thf(def_minvalid,definition,(minvalid = (^[X1:$i>$o]:(![X2:$i]:(~((X1 @ X2))))))). % 47.04/47.35 thf(def_mbox_s4,definition,(mbox_s4 = (^[X1:$i>$o]:(^[X2:$i]:(![X3:$i]:(((rel_s4 @ X2) @ X3) => (X1 @ X3))))))). % 47.04/47.35 thf(def_mdia_s4,definition,(mdia_s4 = (^[X1:$i>$o]:(mnot @ (mbox_s4 @ (mnot @ X1)))))). % 47.04/47.35 thf(con,conjecture,(![X1:$i]:(![X2:$i]:(((rel_s4 @ X1) @ X2) => (~((![X3:mu]:(((exists_in_world @ X3) @ X2) => (~((~((![X4:mu]:(((exists_in_world @ X4) @ X2) => (~(((~((~((![X5:$i]:(((rel_s4 @ X2) @ X5) => (~(((~((~((((combo @ d) @ X3) @ X5))))) => (~(((h @ X4) @ X5)))))))))))) => (![X5:$i]:(((rel_s4 @ X2) @ X5) => ((open @ d) @ X5)))))))))))))))))))). % 47.04/47.35 thf(h1,negated_conjecture,(~((![X1:$i]:(![X2:$i]:(((rel_s4 @ X1) @ X2) => (~((![X3:mu]:(((exists_in_world @ X3) @ X2) => (![X4:mu]:(((exists_in_world @ X4) @ X2) => (~(((![X5:$i]:(((rel_s4 @ X2) @ X5) => (~(((((combo @ d) @ X3) @ X5) => (~(((h @ X4) @ X5)))))))) => (![X5:$i]:(((rel_s4 @ X2) @ X5) => ((open @ d) @ X5))))))))))))))))),inference(assume_negation,[status(cth)],[con])). % 47.04/47.35 thf(h2,assumption,(~((![X1:$i]:(((rel_s4 @ eigen__0) @ X1) => (~((![X2:mu]:(((exists_in_world @ X2) @ X1) => (![X3:mu]:(((exists_in_world @ X3) @ X1) => (~(((![X4:$i]:(((rel_s4 @ X1) @ X4) => (~(((((combo @ d) @ X2) @ X4) => (~(((h @ X3) @ X4)))))))) => (![X4:$i]:(((rel_s4 @ X1) @ X4) => ((open @ d) @ X4)))))))))))))))),introduced(assumption,[])). % 47.04/47.35 thf(h3,assumption,(~((sP1 => (~(sP34))))),introduced(assumption,[])). % 47.04/47.35 thf(h4,assumption,sP1,introduced(assumption,[])). % 47.04/47.35 thf(h5,assumption,sP34,introduced(assumption,[])). % 47.04/47.35 thf(1,plain,((~(sP20) | ~(sP9)) | ~(sP1)),inference(prop_rule,[status(thm)],[])). % 47.04/47.35 thf(2,plain,((~(sP23) | sP20) | sP31),inference(prop_rule,[status(thm)],[])). % 47.04/47.35 thf(3,plain,((~(sP19) | ~(sP11)) | ~(sP45)),inference(prop_rule,[status(thm)],[])). % 47.04/47.35 thf(4,plain,(~(sP13) | sP23),inference(all_rule,[status(thm)],[])). % 47.04/47.35 thf(5,plain,(~(sP38) | sP19),inference(all_rule,[status(thm)],[])). % 47.04/47.35 thf(6,plain,(sP43 | sP38),inference(prop_rule,[status(thm)],[])). % 47.04/47.35 thf(7,plain,((~(sP37) | ~(sP5)) | sP45),inference(prop_rule,[status(thm)],[])). % 47.04/47.35 thf(8,plain,((~(sP35) | sP37) | sP21),inference(prop_rule,[status(thm)],[])). % 47.04/47.35 thf(9,plain,(~(sP18) | sP13),inference(all_rule,[status(thm)],[])). % 47.04/47.35 thf(10,plain,((~(sP27) | ~(sP31)) | ~(sP43)),inference(prop_rule,[status(thm)],[])). % 47.04/47.35 thf(11,plain,((~(sP10) | ~(sP1)) | sP35),inference(prop_rule,[status(thm)],[])). % 47.04/47.35 thf(12,plain,(~(sP25) | sP18),inference(all_rule,[status(thm)],[])). % 47.04/47.35 thf(13,plain,(~(sP2) | sP27),inference(all_rule,[status(thm)],[])). % 47.04/47.35 thf(14,plain,(~(sP17) | sP10),inference(all_rule,[status(thm)],[])). % 47.04/47.35 thf(15,plain,(sP30 | sP17),inference(prop_rule,[status(thm)],[])). % 47.04/47.35 thf(16,plain,(sP36 | ~(sP30)),inference(prop_rule,[status(thm)],[])). % 47.04/47.35 thf(17,plain,(sP36 | sP9),inference(prop_rule,[status(thm)],[])). % 47.04/47.35 thf(18,plain,(sP33 | ~(sP36)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__10])). % 47.04/47.35 thf(19,plain,((~(sP28) | ~(sP7)) | ~(sP33)),inference(prop_rule,[status(thm)],[])). % 47.04/47.35 thf(20,plain,(~(sP41) | sP28),inference(all_rule,[status(thm)],[])). % 47.04/47.35 thf(21,plain,((~(sP40) | ~(sP3)) | sP41),inference(prop_rule,[status(thm)],[])). % 47.04/47.35 thf(22,plain,(sP14 | ~(sP21)),inference(prop_rule,[status(thm)],[])). % 47.04/47.35 thf(23,plain,(~(sP8) | sP40),inference(all_rule,[status(thm)],[])). % 47.04/47.35 thf(24,plain,((~(sP44) | ~(sP12)) | ~(sP14)),inference(prop_rule,[status(thm)],[])). % 47.04/47.35 thf(25,plain,((~(sP16) | ~(sP6)) | sP8),inference(prop_rule,[status(thm)],[])). % 47.04/47.35 thf(26,plain,((~(sP29) | ~(sP1)) | sP5),inference(prop_rule,[status(thm)],[])). % 47.04/47.35 thf(27,plain,(~(sP2) | sP44),inference(all_rule,[status(thm)],[])). % 47.04/47.35 thf(28,plain,(~(sP26) | sP16),inference(all_rule,[status(thm)],[])). % 47.04/47.35 thf(29,plain,(~(sP32) | sP29),inference(all_rule,[status(thm)],[])). % 47.04/47.35 thf(30,plain,((~(sP42) | ~(sP12)) | sP2),inference(prop_rule,[status(thm)],[])). % 47.04/47.35 thf(31,plain,(~(sP24) | sP12),inference(all_rule,[status(thm)],[])). % 47.04/47.35 thf(32,plain,(~(sP24) | sP7),inference(all_rule,[status(thm)],[])). % 47.04/47.35 thf(33,plain,(~(sP39) | sP3),inference(all_rule,[status(thm)],[])). % 47.04/47.35 thf(34,plain,(~(sP22) | sP11),inference(all_rule,[status(thm)],[])). % 47.04/47.35 thf(35,plain,(~(sP22) | sP6),inference(all_rule,[status(thm)],[])). % 47.04/47.35 thf(36,plain,(~(sP4) | sP26),inference(all_rule,[status(thm)],[])). % 47.04/47.35 thf(37,plain,(~(sP15) | sP32),inference(all_rule,[status(thm)],[])). % 47.04/47.35 thf(38,plain,(~(sP34) | sP42),inference(all_rule,[status(thm)],[])). % 47.04/47.35 thf(a1,axiom,(mreflexive @ rel_s4)). % 47.04/47.35 thf(39,plain,sP22,inference(preprocess,[status(thm)],[a1]). % 47.04/47.35 thf(cumulative_ax,axiom,sP25). % 47.04/47.35 thf(existence_of_n_ax,axiom,sP24). % 47.04/47.35 thf(existence_of_d_ax,axiom,sP39). % 47.04/47.35 thf(ax1,axiom,(mvalid @ (mbox_s4 @ (mforall_ind @ (^[X1:mu]:(mforall_ind @ (^[X2:mu]:(mexists_ind @ (^[X3:mu]:((mand @ (mbox_s4 @ ((mimplies @ ((mand @ (closed @ X1)) @ ((mand @ ((combo @ X1) @ X2)) @ (h @ X3)))) @ (mbox_s4 @ (open @ X1))))) @ (mbox_s4 @ ((mimplies @ ((mand @ (closed @ X1)) @ ((mand @ (mnot @ ((combo @ X1) @ X2))) @ (h @ o)))) @ (mbox_s4 @ (closed @ X1)))))))))))))). % 47.04/47.35 thf(40,plain,sP4,inference(preprocess,[status(thm)],[ax1]). % 47.04/47.35 thf(ax2,axiom,(mvalid @ (mbox_s4 @ (closed @ d)))). % 47.04/47.35 thf(41,plain,sP15,inference(preprocess,[status(thm)],[ax2]). % 47.04/47.35 thf(42,plain,$false,inference(prop_unsat,[status(thm),assumptions([h4,h5,h3,h2,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,cumulative_ax,existence_of_n_ax,existence_of_d_ax,40,41,h4,h5])). % 47.04/47.35 thf(43,plain,$false,inference(tab_negimp,[status(thm),assumptions([h3,h2,h1,h0]),tab_negimp(discharge,[h4,h5])],[h3,42,h4,h5])). % 47.04/47.35 thf(44,plain,$false,inference(tab_negall,[status(thm),assumptions([h2,h1,h0]),tab_negall(discharge,[h3]),tab_negall(eigenvar,eigen__1)],[h2,43,h3])). % 47.04/47.35 thf(45,plain,$false,inference(tab_negall,[status(thm),assumptions([h1,h0]),tab_negall(discharge,[h2]),tab_negall(eigenvar,eigen__0)],[h1,44,h2])). % 47.04/47.35 thf(46,plain,$false,inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[45,h0])). % 47.04/47.35 thf(0,theorem,(![X1:$i]:(![X2:$i]:(((rel_s4 @ X1) @ X2) => (~((![X3:mu]:(((exists_in_world @ X3) @ X2) => (~((~((![X4:mu]:(((exists_in_world @ X4) @ X2) => (~(((~((~((![X5:$i]:(((rel_s4 @ X2) @ X5) => (~(((~((~((((combo @ d) @ X3) @ X5))))) => (~(((h @ X4) @ X5)))))))))))) => (![X5:$i]:(((rel_s4 @ X2) @ X5) => ((open @ d) @ X5))))))))))))))))))),inference(contra,[status(thm),contra(discharge,[h1])],[45,h1])). % 47.04/47.35 % SZS output end Proof %------------------------------------------------------------------------------