%------------------------------------------------------------------------------ % File : Satallax---3.5 % Problem : LCL460^7 : TPTP v9.3.1. Released v5.5.0. % Transfm : none % Format : tptp:raw % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % Computer : n007.cluster.edu % Model : x86_64 x86_64 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz % Memory : 8046.5625MB % OS : Linux 6.8.0-71-generic % CPULimit : 300s % WCLimit : 300s % DateTime : Mon Sep 7 01:01:56 PM UTC 2026 % Result : Theorem 7.08s 8.00s % Output : Proof 7.08s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.00/0.02 % Problem : LCL460^7 : TPTP v9.3.1. Released v5.5.0. % 0.00/0.03 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % 0.09/0.35 % Computer : n007.cluster.edu % 0.09/0.35 % Model : x86_64 x86_64 % 0.09/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.09/0.35 % Memory : 8046.5625MB % 0.09/0.35 % OS : Linux 6.8.0-71-generic % 0.09/0.35 % CPULimit : 300 % 0.09/0.35 % WCLimit : 300 % 0.09/0.35 % DateTime : Fri Sep 4 17:07:24 UTC 2026 % 0.09/0.35 % CPUTime : % 7.08/8.00 % SZS status Theorem % 7.08/8.00 % Mode: mode8a:USE_SINE=true:SINE_TOLERANCE=1.2:SINE_GENERALITY_THRESHOLD=16:SINE_RANK_LIMIT=3.:SINE_DEPTH=0 % 7.08/8.00 % Inferences: 18007 % 7.08/8.00 % SZS output start Proof % 7.08/8.00 thf(ty_mu, type, mu : $tType). % 7.08/8.00 thf(ty_rel_s4, type, rel_s4 : ($i>$i>$o)). % 7.08/8.00 thf(ty_and_1, type, and_1 : ($i>$o)). % 7.08/8.00 thf(ty_eigen__1, type, eigen__1 : $i). % 7.08/8.00 thf(ty_eigen__0, type, eigen__0 : $i). % 7.08/8.00 thf(ty_is_a_theorem, type, is_a_theorem : (mu>$i>$o)). % 7.08/8.00 thf(ty_implies, type, implies : (mu>mu>mu)). % 7.08/8.00 thf(ty_and, type, and : (mu>mu>mu)). % 7.08/8.00 thf(ty_exists_in_world, type, exists_in_world : (mu>$i>$o)). % 7.08/8.00 thf(ty_kn2, type, kn2 : ($i>$o)). % 7.08/8.00 thf(sP1,plain,sP1 <=> (![X1:$i]:((rel_s4 @ X1) @ X1)),introduced(definition,[new_symbols(definition,[sP1])])). % 7.08/8.00 thf(sP2,plain,sP2 <=> (![X1:$i]:(~(((![X2:$i]:(((rel_s4 @ X1) @ X2) => ((![X3:$i]:(((rel_s4 @ X2) @ X3) => (kn2 @ X3))) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => (![X4:mu]:(((exists_in_world @ X4) @ X3) => (![X5:$i]:(((rel_s4 @ X3) @ X5) => (![X6:mu]:(((exists_in_world @ X6) @ X5) => (![X7:$i]:(((rel_s4 @ X5) @ X7) => ((is_a_theorem @ ((implies @ ((and @ X4) @ X6)) @ X4)) @ X7)))))))))))))) => (~((![X2:$i]:(((rel_s4 @ X1) @ X2) => ((![X3:$i]:(((rel_s4 @ X2) @ X3) => (![X4:mu]:(((exists_in_world @ X4) @ X3) => (![X5:$i]:(((rel_s4 @ X3) @ X5) => (![X6:mu]:(((exists_in_world @ X6) @ X5) => (![X7:$i]:(((rel_s4 @ X5) @ X7) => ((is_a_theorem @ ((implies @ ((and @ X4) @ X6)) @ X4)) @ X7))))))))))) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => (kn2 @ X3)))))))))))),introduced(definition,[new_symbols(definition,[sP2])])). % 7.08/8.00 thf(sP3,plain,sP3 <=> ((![X1:$i]:(((rel_s4 @ eigen__0) @ X1) => ((![X2:$i]:(((rel_s4 @ X1) @ X2) => (and_1 @ X2))) => (![X2:$i]:(((rel_s4 @ X1) @ X2) => (![X3:mu]:(((exists_in_world @ X3) @ X2) => (![X4:$i]:(((rel_s4 @ X2) @ X4) => (![X5:mu]:(((exists_in_world @ X5) @ X4) => (![X6:$i]:(((rel_s4 @ X4) @ X6) => ((is_a_theorem @ ((implies @ ((and @ X3) @ X5)) @ X3)) @ X6)))))))))))))) => (~((![X1:$i]:(((rel_s4 @ eigen__0) @ X1) => ((![X2:$i]:(((rel_s4 @ X1) @ X2) => (![X3:mu]:(((exists_in_world @ X3) @ X2) => (![X4:$i]:(((rel_s4 @ X2) @ X4) => (![X5:mu]:(((exists_in_world @ X5) @ X4) => (![X6:$i]:(((rel_s4 @ X4) @ X6) => ((is_a_theorem @ ((implies @ ((and @ X3) @ X5)) @ X3)) @ X6))))))))))) => (![X2:$i]:(((rel_s4 @ X1) @ X2) => (and_1 @ X2))))))))),introduced(definition,[new_symbols(definition,[sP3])])). % 7.08/8.00 thf(sP4,plain,sP4 <=> (![X1:$i]:(((rel_s4 @ eigen__0) @ X1) => (kn2 @ X1))),introduced(definition,[new_symbols(definition,[sP4])])). % 7.08/8.00 thf(sP5,plain,sP5 <=> (![X1:$i]:(((rel_s4 @ eigen__0) @ X1) => ((![X2:$i]:(((rel_s4 @ X1) @ X2) => (![X3:mu]:(((exists_in_world @ X3) @ X2) => (![X4:$i]:(((rel_s4 @ X2) @ X4) => (![X5:mu]:(((exists_in_world @ X5) @ X4) => (![X6:$i]:(((rel_s4 @ X4) @ X6) => ((is_a_theorem @ ((implies @ ((and @ X3) @ X5)) @ X3)) @ X6))))))))))) => (![X2:$i]:(((rel_s4 @ X1) @ X2) => (kn2 @ X2)))))),introduced(definition,[new_symbols(definition,[sP5])])). % 7.08/8.00 thf(sP6,plain,sP6 <=> (![X1:$i]:(((rel_s4 @ eigen__0) @ X1) => (![X2:mu]:(((exists_in_world @ X2) @ X1) => (![X3:$i]:(((rel_s4 @ X1) @ X3) => (![X4:mu]:(((exists_in_world @ X4) @ X3) => (![X5:$i]:(((rel_s4 @ X3) @ X5) => ((is_a_theorem @ ((implies @ ((and @ X2) @ X4)) @ X2)) @ X5))))))))))),introduced(definition,[new_symbols(definition,[sP6])])). % 7.08/8.00 thf(sP7,plain,sP7 <=> (((rel_s4 @ eigen__0) @ eigen__1) => (kn2 @ eigen__1)),introduced(definition,[new_symbols(definition,[sP7])])). % 7.08/8.00 thf(sP8,plain,sP8 <=> (kn2 @ eigen__1),introduced(definition,[new_symbols(definition,[sP8])])). % 7.08/8.00 thf(sP9,plain,sP9 <=> (sP6 => sP4),introduced(definition,[new_symbols(definition,[sP9])])). % 7.08/8.00 thf(sP10,plain,sP10 <=> (![X1:$i]:(((rel_s4 @ eigen__0) @ X1) => (and_1 @ X1))),introduced(definition,[new_symbols(definition,[sP10])])). % 7.08/8.00 thf(sP11,plain,sP11 <=> (![X1:$i]:(((rel_s4 @ eigen__0) @ X1) => ((![X2:$i]:(((rel_s4 @ X1) @ X2) => (and_1 @ X2))) => (![X2:$i]:(((rel_s4 @ X1) @ X2) => (![X3:mu]:(((exists_in_world @ X3) @ X2) => (![X4:$i]:(((rel_s4 @ X2) @ X4) => (![X5:mu]:(((exists_in_world @ X5) @ X4) => (![X6:$i]:(((rel_s4 @ X4) @ X6) => ((is_a_theorem @ ((implies @ ((and @ X3) @ X5)) @ X3)) @ X6)))))))))))))),introduced(definition,[new_symbols(definition,[sP11])])). % 7.08/8.00 thf(sP12,plain,sP12 <=> (((rel_s4 @ eigen__0) @ eigen__0) => (sP10 => sP6)),introduced(definition,[new_symbols(definition,[sP12])])). % 7.08/8.00 thf(sP13,plain,sP13 <=> ((![X1:$i]:(((rel_s4 @ eigen__0) @ X1) => ((![X2:$i]:(((rel_s4 @ X1) @ X2) => (kn2 @ X2))) => (![X2:$i]:(((rel_s4 @ X1) @ X2) => (![X3:mu]:(((exists_in_world @ X3) @ X2) => (![X4:$i]:(((rel_s4 @ X2) @ X4) => (![X5:mu]:(((exists_in_world @ X5) @ X4) => (![X6:$i]:(((rel_s4 @ X4) @ X6) => ((is_a_theorem @ ((implies @ ((and @ X3) @ X5)) @ X3)) @ X6)))))))))))))) => (~(sP5))),introduced(definition,[new_symbols(definition,[sP13])])). % 7.08/8.00 thf(sP14,plain,sP14 <=> ((rel_s4 @ eigen__0) @ eigen__1),introduced(definition,[new_symbols(definition,[sP14])])). % 7.08/8.00 thf(sP15,plain,sP15 <=> ((rel_s4 @ eigen__0) @ eigen__0),introduced(definition,[new_symbols(definition,[sP15])])). % 7.08/8.00 thf(sP16,plain,sP16 <=> (![X1:$i]:(![X2:$i]:(((rel_s4 @ X1) @ X2) => (and_1 @ X2)))),introduced(definition,[new_symbols(definition,[sP16])])). % 7.08/8.00 thf(sP17,plain,sP17 <=> (![X1:$i]:(~(((![X2:$i]:(((rel_s4 @ X1) @ X2) => ((![X3:$i]:(((rel_s4 @ X2) @ X3) => (and_1 @ X3))) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => (![X4:mu]:(((exists_in_world @ X4) @ X3) => (![X5:$i]:(((rel_s4 @ X3) @ X5) => (![X6:mu]:(((exists_in_world @ X6) @ X5) => (![X7:$i]:(((rel_s4 @ X5) @ X7) => ((is_a_theorem @ ((implies @ ((and @ X4) @ X6)) @ X4)) @ X7)))))))))))))) => (~((![X2:$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:mu]:(((exists_in_world @ X6) @ X5) => (![X7:$i]:(((rel_s4 @ X5) @ X7) => ((is_a_theorem @ ((implies @ ((and @ X4) @ X6)) @ X4)) @ X7))))))))))) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => (and_1 @ X3)))))))))))),introduced(definition,[new_symbols(definition,[sP17])])). % 7.08/8.00 thf(sP18,plain,sP18 <=> (sP10 => sP6),introduced(definition,[new_symbols(definition,[sP18])])). % 7.08/8.00 thf(sP19,plain,sP19 <=> (sP15 => sP9),introduced(definition,[new_symbols(definition,[sP19])])). % 7.08/8.00 thf(def_mnot,definition,(mnot = (^[X1:$i>$o]:(^[X2:$i]:(~((X1 @ X2))))))). % 7.08/8.00 thf(def_mor,definition,(mor = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((~((X1 @ X3))) => (X2 @ X3))))))). % 7.08/8.00 thf(def_mand,definition,(mand = (^[X1:$i>$o]:(^[X2:$i>$o]:(mnot @ ((mor @ (mnot @ X1)) @ (mnot @ X2))))))). % 7.08/8.00 thf(def_mimplies,definition,(mimplies = (^[X1:$i>$o]:(mor @ (mnot @ X1))))). % 7.08/8.00 thf(def_mforall_ind,definition,(mforall_ind = (^[X1:mu>$i>$o]:(^[X2:$i]:(![X3:mu]:(((exists_in_world @ X3) @ X2) => ((X1 @ X3) @ X2))))))). % 7.08/8.00 thf(def_mreflexive,definition,(mreflexive = (^[X1:$i>$i>$o]:(![X2:$i]:((X1 @ X2) @ X2))))). % 7.08/8.00 thf(def_mtransitive,definition,(mtransitive = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X3) @ X4)))))) => ((X1 @ X2) @ X4)))))))). % 7.08/8.00 thf(def_mvalid,definition,(mvalid = (!!))). % 7.08/8.00 thf(def_mbox_s4,definition,(mbox_s4 = (^[X1:$i>$o]:(^[X2:$i]:(![X3:$i]:(((rel_s4 @ X2) @ X3) => (X1 @ X3))))))). % 7.08/8.00 thf(rosser_kn2,conjecture,(![X1:$i]:(![X2:$i]:(((rel_s4 @ X1) @ X2) => (kn2 @ X2))))). % 7.08/8.00 thf(h0,negated_conjecture,(~((![X1:$i]:(![X2:$i]:(((rel_s4 @ X1) @ X2) => (kn2 @ X2)))))),inference(assume_negation,[status(cth)],[rosser_kn2])). % 7.08/8.00 thf(h1,assumption,(~(sP4)),introduced(assumption,[])). % 7.08/8.00 thf(h2,assumption,(~(sP7)),introduced(assumption,[])). % 7.08/8.00 thf(h3,assumption,sP14,introduced(assumption,[])). % 7.08/8.00 thf(h4,assumption,(~(sP8)),introduced(assumption,[])). % 7.08/8.00 thf(1,plain,((~(sP7) | ~(sP14)) | sP8),inference(prop_rule,[status(thm)],[])). % 7.08/8.00 thf(2,plain,(~(sP4) | sP7),inference(all_rule,[status(thm)],[])). % 7.08/8.00 thf(3,plain,((~(sP18) | ~(sP10)) | sP6),inference(prop_rule,[status(thm)],[])). % 7.08/8.00 thf(4,plain,((~(sP9) | ~(sP6)) | sP4),inference(prop_rule,[status(thm)],[])). % 7.08/8.00 thf(5,plain,((~(sP12) | ~(sP15)) | sP18),inference(prop_rule,[status(thm)],[])). % 7.08/8.00 thf(6,plain,((~(sP19) | ~(sP15)) | sP9),inference(prop_rule,[status(thm)],[])). % 7.08/8.00 thf(7,plain,(~(sP5) | sP19),inference(all_rule,[status(thm)],[])). % 7.08/8.00 thf(8,plain,(~(sP11) | sP12),inference(all_rule,[status(thm)],[])). % 7.08/8.00 thf(9,plain,(sP3 | sP11),inference(prop_rule,[status(thm)],[])). % 7.08/8.00 thf(10,plain,(sP13 | sP5),inference(prop_rule,[status(thm)],[])). % 7.08/8.00 thf(11,plain,(~(sP16) | sP10),inference(all_rule,[status(thm)],[])). % 7.08/8.00 thf(12,plain,(~(sP2) | ~(sP13)),inference(all_rule,[status(thm)],[])). % 7.08/8.00 thf(13,plain,(~(sP17) | ~(sP3)),inference(all_rule,[status(thm)],[])). % 7.08/8.00 thf(14,plain,(~(sP1) | sP15),inference(all_rule,[status(thm)],[])). % 7.08/8.00 thf(hilbert_and_1,axiom,(mvalid @ (mbox_s4 @ and_1))). % 7.08/8.00 thf(15,plain,sP16,inference(preprocess,[status(thm)],[hilbert_and_1]). % 7.08/8.00 thf(kn2,axiom,(mvalid @ ((mand @ (mbox_s4 @ ((mimplies @ (mbox_s4 @ kn2)) @ (mbox_s4 @ (mforall_ind @ (^[X1:mu]:(mbox_s4 @ (mforall_ind @ (^[X2:mu]:(mbox_s4 @ (is_a_theorem @ ((implies @ ((and @ X1) @ X2)) @ X1)))))))))))) @ (mbox_s4 @ ((mimplies @ (mbox_s4 @ (mforall_ind @ (^[X1:mu]:(mbox_s4 @ (mforall_ind @ (^[X2:mu]:(mbox_s4 @ (is_a_theorem @ ((implies @ ((and @ X1) @ X2)) @ X1)))))))))) @ (mbox_s4 @ kn2)))))). % 7.08/8.00 thf(16,plain,sP2,inference(preprocess,[status(thm)],[kn2]). % 7.08/8.00 thf(and_1,axiom,(mvalid @ ((mand @ (mbox_s4 @ ((mimplies @ (mbox_s4 @ and_1)) @ (mbox_s4 @ (mforall_ind @ (^[X1:mu]:(mbox_s4 @ (mforall_ind @ (^[X2:mu]:(mbox_s4 @ (is_a_theorem @ ((implies @ ((and @ X1) @ X2)) @ X1)))))))))))) @ (mbox_s4 @ ((mimplies @ (mbox_s4 @ (mforall_ind @ (^[X1:mu]:(mbox_s4 @ (mforall_ind @ (^[X2:mu]:(mbox_s4 @ (is_a_theorem @ ((implies @ ((and @ X1) @ X2)) @ X1)))))))))) @ (mbox_s4 @ and_1)))))). % 7.08/8.00 thf(17,plain,sP17,inference(preprocess,[status(thm)],[and_1]). % 7.08/8.00 thf(a1,axiom,(mreflexive @ rel_s4)). % 7.08/8.00 thf(18,plain,sP1,inference(preprocess,[status(thm)],[a1]). % 7.08/8.00 thf(19,plain,$false,inference(prop_unsat,[status(thm),assumptions([h3,h4,h2,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,h3,h4,15,16,17,18])). % 7.08/8.00 thf(20,plain,$false,inference(tab_negimp,[status(thm),assumptions([h2,h1,h0]),tab_negimp(discharge,[h3,h4])],[h2,19,h3,h4])). % 7.08/8.00 thf(21,plain,$false,inference(tab_negall,[status(thm),assumptions([h1,h0]),tab_negall(discharge,[h2]),tab_negall(eigenvar,eigen__1)],[h1,20,h2])). % 7.08/8.00 thf(22,plain,$false,inference(tab_negall,[status(thm),assumptions([h0]),tab_negall(discharge,[h1]),tab_negall(eigenvar,eigen__0)],[h0,21,h1])). % 7.08/8.00 thf(0,theorem,(![X1:$i]:(![X2:$i]:(((rel_s4 @ X1) @ X2) => (kn2 @ X2)))),inference(contra,[status(thm),contra(discharge,[h0])],[22,h0])). % 7.08/8.00 % SZS output end Proof %------------------------------------------------------------------------------