%------------------------------------------------------------------------------ % File : Satallax---3.5 % Problem : LCL724^1 : TPTP v9.3.1. Bugfixed v5.0.0. % 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 : 8046.5625MB % OS : Linux 6.8.0-71-generic % CPULimit : 300s % WCLimit : 300s % DateTime : Mon Sep 7 01:02:30 PM UTC 2026 % Result : Theorem 0.08s 0.39s % Output : Proof 0.08s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.00/0.03 % Problem : LCL724^1 : TPTP v9.3.1. Bugfixed v5.0.0. % 0.00/0.04 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % 0.08/0.36 % Computer : n006.cluster.edu % 0.08/0.36 % Model : x86_64 x86_64 % 0.08/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.08/0.36 % Memory : 8046.5625MB % 0.08/0.36 % OS : Linux 6.8.0-71-generic % 0.08/0.36 % CPULimit : 300 % 0.08/0.36 % WCLimit : 300 % 0.08/0.36 % DateTime : Sat Sep 5 16:25:31 UTC 2026 % 0.08/0.36 % CPUTime : % 0.08/0.39 % SZS status Theorem % 0.08/0.39 % Mode: mode213 % 0.08/0.39 % Inferences: 11 % 0.08/0.39 % SZS output start Proof % 0.08/0.39 thf(ty_rel_s4, type, rel_s4 : ($i>$i>$o)). % 0.08/0.39 thf(ty_eigen__2, type, eigen__2 : $i). % 0.08/0.39 thf(ty_eigen__1, type, eigen__1 : $i). % 0.08/0.39 thf(ty_phi, type, phi : ($i>$o)). % 0.08/0.39 thf(ty_eigen__0, type, eigen__0 : $i). % 0.08/0.39 thf(ty_eigen__3, type, eigen__3 : $i). % 0.08/0.39 thf(sP1,plain,sP1 <=> (((rel_s4 @ eigen__1) @ eigen__3) => (phi @ eigen__3)),introduced(definition,[new_symbols(definition,[sP1])])). % 0.08/0.39 thf(sP2,plain,sP2 <=> (![X1:$i]:(((rel_s4 @ eigen__0) @ X1) => (phi @ X1))),introduced(definition,[new_symbols(definition,[sP2])])). % 0.08/0.39 thf(sP3,plain,sP3 <=> (phi @ eigen__3),introduced(definition,[new_symbols(definition,[sP3])])). % 0.08/0.39 thf(sP4,plain,sP4 <=> (![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => (phi @ X1))),introduced(definition,[new_symbols(definition,[sP4])])). % 0.08/0.39 thf(sP5,plain,sP5 <=> ((rel_s4 @ eigen__0) @ eigen__0),introduced(definition,[new_symbols(definition,[sP5])])). % 0.08/0.39 thf(sP6,plain,sP6 <=> (sP5 => (phi @ eigen__0)),introduced(definition,[new_symbols(definition,[sP6])])). % 0.08/0.39 thf(sP7,plain,sP7 <=> (![X1:$i]:(![X2:$i]:((~((((rel_s4 @ eigen__1) @ X1) => (~(((rel_s4 @ X1) @ X2)))))) => ((rel_s4 @ eigen__1) @ X2)))),introduced(definition,[new_symbols(definition,[sP7])])). % 0.08/0.39 thf(sP8,plain,sP8 <=> ((rel_s4 @ eigen__1) @ eigen__3),introduced(definition,[new_symbols(definition,[sP8])])). % 0.08/0.39 thf(sP9,plain,sP9 <=> ((rel_s4 @ eigen__1) @ eigen__2),introduced(definition,[new_symbols(definition,[sP9])])). % 0.08/0.39 thf(sP10,plain,sP10 <=> (![X1:$i]:((~((sP9 => (~(((rel_s4 @ eigen__2) @ X1)))))) => ((rel_s4 @ eigen__1) @ X1))),introduced(definition,[new_symbols(definition,[sP10])])). % 0.08/0.39 thf(sP11,plain,sP11 <=> (![X1:$i]:(![X2:$i]:(![X3:$i]:((~((((rel_s4 @ X1) @ X2) => (~(((rel_s4 @ X2) @ X3)))))) => ((rel_s4 @ X1) @ X3))))),introduced(definition,[new_symbols(definition,[sP11])])). % 0.08/0.39 thf(sP12,plain,sP12 <=> (phi @ eigen__0),introduced(definition,[new_symbols(definition,[sP12])])). % 0.08/0.39 thf(sP13,plain,sP13 <=> ((rel_s4 @ eigen__2) @ eigen__3),introduced(definition,[new_symbols(definition,[sP13])])). % 0.08/0.39 thf(sP14,plain,sP14 <=> ((~((sP9 => (~(sP13))))) => sP8),introduced(definition,[new_symbols(definition,[sP14])])). % 0.08/0.39 thf(sP15,plain,sP15 <=> (sP9 => (~(sP13))),introduced(definition,[new_symbols(definition,[sP15])])). % 0.08/0.39 thf(sP16,plain,sP16 <=> (![X1:$i]:((rel_s4 @ X1) @ X1)),introduced(definition,[new_symbols(definition,[sP16])])). % 0.08/0.39 thf(def_meq_ind,definition,(meq_ind = (^[X1:mu]:(^[X2:mu]:(^[X3:$i]:(X1 = X2)))))). % 0.08/0.39 thf(def_meq_prop,definition,(meq_prop = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((X1 @ X3) = (X2 @ X3))))))). % 0.08/0.39 thf(def_mnot,definition,(mnot = (^[X1:$i>$o]:(^[X2:$i]:(~((X1 @ X2))))))). % 0.08/0.39 thf(def_mor,definition,(mor = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((~((X1 @ X3))) => (X2 @ X3))))))). % 0.08/0.39 thf(def_mand,definition,(mand = (^[X1:$i>$o]:(^[X2:$i>$o]:(mnot @ ((mor @ (mnot @ X1)) @ (mnot @ X2))))))). % 0.08/0.39 thf(def_mimplies,definition,(mimplies = (^[X1:$i>$o]:(mor @ (mnot @ X1))))). % 0.08/0.39 thf(def_mimplied,definition,(mimplied = (^[X1:$i>$o]:(^[X2:$i>$o]:((mor @ (mnot @ X2)) @ X1))))). % 0.08/0.39 thf(def_mequiv,definition,(mequiv = (^[X1:$i>$o]:(^[X2:$i>$o]:((mand @ ((mimplies @ X1) @ X2)) @ ((mimplies @ X2) @ X1)))))). % 0.08/0.39 thf(def_mxor,definition,(mxor = (^[X1:$i>$o]:(^[X2:$i>$o]:(mnot @ ((mequiv @ X1) @ X2)))))). % 0.08/0.39 thf(def_mforall_ind,definition,(mforall_ind = (^[X1:mu>$i>$o]:(^[X2:$i]:(![X3:mu]:((X1 @ X3) @ X2)))))). % 0.08/0.39 thf(def_mforall_prop,definition,(mforall_prop = (^[X1:($i>$o)>$i>$o]:(^[X2:$i]:(![X3:$i>$o]:((X1 @ X3) @ X2)))))). % 0.08/0.39 thf(def_mexists_ind,definition,(mexists_ind = (^[X1:mu>$i>$o]:(mnot @ (mforall_ind @ (^[X2:mu]:(mnot @ (X1 @ X2)))))))). % 0.08/0.39 thf(def_mexists_prop,definition,(mexists_prop = (^[X1:($i>$o)>$i>$o]:(mnot @ (mforall_prop @ (^[X2:$i>$o]:(mnot @ (X1 @ X2)))))))). % 0.08/0.39 thf(def_mtrue,definition,(mtrue = (^[X1:$i]:(~($false))))). % 0.08/0.39 thf(def_mfalse,definition,(mfalse = (mnot @ mtrue))). % 0.08/0.39 thf(def_mbox,definition,(mbox = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:(![X4:$i]:(((X1 @ X3) @ X4) => (X2 @ X4)))))))). % 0.08/0.39 thf(def_mdia,definition,(mdia = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(mnot @ ((mbox @ X1) @ (mnot @ X2))))))). % 0.08/0.39 thf(def_mreflexive,definition,(mreflexive = (^[X1:$i>$i>$o]:(![X2:$i]:((X1 @ X2) @ X2))))). % 0.08/0.39 thf(def_msymmetric,definition,(msymmetric = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(((X1 @ X2) @ X3) => ((X1 @ X3) @ X2))))))). % 0.08/0.39 thf(def_mserial,definition,(mserial = (^[X1:$i>$i>$o]:(![X2:$i]:(~((![X3:$i]:(~(((X1 @ X2) @ X3)))))))))). % 0.08/0.39 thf(def_mtransitive,definition,(mtransitive = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X3) @ X4)))))) => ((X1 @ X2) @ X4)))))))). % 0.08/0.39 thf(def_meuclidean,definition,(meuclidean = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => ((X1 @ X3) @ X4)))))))). % 0.08/0.39 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)))))))). % 0.08/0.39 thf(def_mfunctional,definition,(mfunctional = (^[X1:$i>$i>$o]:(![X2:$i]:(~((![X3:$i]:(((X1 @ X2) @ X3) => (~((![X4:$i]:(((X1 @ X2) @ X4) => (X3 = X4))))))))))))). % 0.08/0.39 thf(def_mweakly_dense,definition,(mweakly_dense = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((X1 @ X2) @ X3) => (~((![X5:$i]:(((X1 @ X2) @ X5) => (~(((X1 @ X5) @ X3)))))))))))))). % 0.08/0.39 thf(def_mweakly_connected,definition,(mweakly_connected = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => ((~(((~(((X1 @ X3) @ X4))) => (X3 = X4)))) => ((X1 @ X4) @ X3))))))))). % 0.08/0.39 thf(def_mweakly_directed,definition,(mweakly_directed = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => (~((![X5:$i]:(((X1 @ X3) @ X5) => (~(((X1 @ X4) @ X5)))))))))))))). % 0.08/0.39 thf(def_mvalid,definition,(mvalid = (!!))). % 0.08/0.39 thf(def_minvalid,definition,(minvalid = (^[X1:$i>$o]:(![X2:$i]:(~((X1 @ X2))))))). % 0.08/0.39 thf(def_msatisfiable,definition,(msatisfiable = (^[X1:$i>$o]:(~((![X2:$i]:(~((X1 @ X2))))))))). % 0.08/0.39 thf(def_mcountersatisfiable,definition,(mcountersatisfiable = (^[X1:$i>$o]:(~(((!!) @ X1)))))). % 0.08/0.39 thf(def_mbox_s4,definition,(mbox_s4 = (^[X1:$i>$o]:(^[X2:$i]:(![X3:$i]:(((rel_s4 @ X2) @ X3) => (X1 @ X3))))))). % 0.08/0.39 thf(def_mdia_s4,definition,(mdia_s4 = (^[X1:$i>$o]:(mnot @ (mbox_s4 @ (mnot @ X1)))))). % 0.08/0.39 thf(conj,conjecture,(~(((![X1:$i]:((~((~((![X2:$i]:(((rel_s4 @ X1) @ X2) => (phi @ X2))))))) => (phi @ X1))) => (~((![X1:$i]:((~((~((![X2:$i]:(((rel_s4 @ X1) @ X2) => (phi @ X2))))))) => (![X2:$i]:(((rel_s4 @ X1) @ X2) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => (phi @ X3))))))))))))). % 0.08/0.39 thf(h0,negated_conjecture,((![X1:$i]:((![X2:$i]:(((rel_s4 @ X1) @ X2) => (phi @ X2))) => (phi @ X1))) => (~((![X1:$i]:((![X2:$i]:(((rel_s4 @ X1) @ X2) => (phi @ X2))) => (![X2:$i]:(((rel_s4 @ X1) @ X2) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => (phi @ X3)))))))))),inference(assume_negation,[status(cth)],[conj])). % 0.08/0.39 thf(h1,assumption,(~((![X1:$i]:((![X2:$i]:(((rel_s4 @ X1) @ X2) => (phi @ X2))) => (phi @ X1))))),introduced(assumption,[])). % 0.08/0.39 thf(h2,assumption,(~((![X1:$i]:((![X2:$i]:(((rel_s4 @ X1) @ X2) => (phi @ X2))) => (![X2:$i]:(((rel_s4 @ X1) @ X2) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => (phi @ X3))))))))),introduced(assumption,[])). % 0.08/0.39 thf(h3,assumption,(~((sP2 => sP12))),introduced(assumption,[])). % 0.08/0.39 thf(h4,assumption,sP2,introduced(assumption,[])). % 0.08/0.39 thf(h5,assumption,(~(sP12)),introduced(assumption,[])). % 0.08/0.39 thf(1,plain,(~(sP16) | sP5),inference(all_rule,[status(thm)],[])). % 0.08/0.39 thf(2,plain,(~(sP2) | sP6),inference(all_rule,[status(thm)],[])). % 0.08/0.39 thf(3,plain,((~(sP6) | ~(sP5)) | sP12),inference(prop_rule,[status(thm)],[])). % 0.08/0.39 thf(a1,axiom,(mreflexive @ rel_s4)). % 0.08/0.39 thf(4,plain,sP16,inference(preprocess,[status(thm)],[a1]). % 0.08/0.39 thf(5,plain,$false,inference(prop_unsat,[status(thm),assumptions([h4,h5,h3,h1,h0])],[1,2,3,4,h4,h5])). % 0.08/0.39 thf(6,plain,$false,inference(tab_negimp,[status(thm),assumptions([h3,h1,h0]),tab_negimp(discharge,[h4,h5])],[h3,5,h4,h5])). % 0.08/0.39 thf(7,plain,$false,inference(tab_negall,[status(thm),assumptions([h1,h0]),tab_negall(discharge,[h3]),tab_negall(eigenvar,eigen__0)],[h1,6,h3])). % 0.08/0.39 thf(h6,assumption,(~((sP4 => (![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => (![X2:$i]:(((rel_s4 @ X1) @ X2) => (phi @ X2)))))))),introduced(assumption,[])). % 0.08/0.39 thf(h7,assumption,sP4,introduced(assumption,[])). % 0.08/0.39 thf(h8,assumption,(~((![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => (![X2:$i]:(((rel_s4 @ X1) @ X2) => (phi @ X2))))))),introduced(assumption,[])). % 0.08/0.39 thf(h9,assumption,(~((sP9 => (![X1:$i]:(((rel_s4 @ eigen__2) @ X1) => (phi @ X1)))))),introduced(assumption,[])). % 0.08/0.39 thf(h10,assumption,sP9,introduced(assumption,[])). % 0.08/0.39 thf(h11,assumption,(~((![X1:$i]:(((rel_s4 @ eigen__2) @ X1) => (phi @ X1))))),introduced(assumption,[])). % 0.08/0.39 thf(h12,assumption,(~((sP13 => sP3))),introduced(assumption,[])). % 0.08/0.39 thf(h13,assumption,sP13,introduced(assumption,[])). % 0.08/0.39 thf(h14,assumption,(~(sP3)),introduced(assumption,[])). % 0.08/0.39 thf(8,plain,(~(sP4) | sP1),inference(all_rule,[status(thm)],[])). % 0.08/0.39 thf(9,plain,((~(sP1) | ~(sP8)) | sP3),inference(prop_rule,[status(thm)],[])). % 0.08/0.39 thf(10,plain,(~(sP11) | sP7),inference(all_rule,[status(thm)],[])). % 0.08/0.39 thf(11,plain,(~(sP7) | sP10),inference(all_rule,[status(thm)],[])). % 0.08/0.39 thf(12,plain,(~(sP10) | sP14),inference(all_rule,[status(thm)],[])). % 0.08/0.39 thf(13,plain,((~(sP14) | sP15) | sP8),inference(prop_rule,[status(thm)],[])). % 0.08/0.39 thf(14,plain,((~(sP15) | ~(sP9)) | ~(sP13)),inference(prop_rule,[status(thm)],[])). % 0.08/0.39 thf(a2,axiom,(mtransitive @ rel_s4)). % 0.08/0.39 thf(15,plain,sP11,inference(preprocess,[status(thm)],[a2]). % 0.08/0.39 thf(16,plain,$false,inference(prop_unsat,[status(thm),assumptions([h13,h14,h12,h10,h11,h9,h7,h8,h6,h2,h0])],[8,9,10,11,12,13,14,15,h7,h10,h13,h14])). % 0.08/0.39 thf(17,plain,$false,inference(tab_negimp,[status(thm),assumptions([h12,h10,h11,h9,h7,h8,h6,h2,h0]),tab_negimp(discharge,[h13,h14])],[h12,16,h13,h14])). % 0.08/0.39 thf(18,plain,$false,inference(tab_negall,[status(thm),assumptions([h10,h11,h9,h7,h8,h6,h2,h0]),tab_negall(discharge,[h12]),tab_negall(eigenvar,eigen__3)],[h11,17,h12])). % 0.08/0.39 thf(19,plain,$false,inference(tab_negimp,[status(thm),assumptions([h9,h7,h8,h6,h2,h0]),tab_negimp(discharge,[h10,h11])],[h9,18,h10,h11])). % 0.08/0.39 thf(20,plain,$false,inference(tab_negall,[status(thm),assumptions([h7,h8,h6,h2,h0]),tab_negall(discharge,[h9]),tab_negall(eigenvar,eigen__2)],[h8,19,h9])). % 0.08/0.39 thf(21,plain,$false,inference(tab_negimp,[status(thm),assumptions([h6,h2,h0]),tab_negimp(discharge,[h7,h8])],[h6,20,h7,h8])). % 0.08/0.39 thf(22,plain,$false,inference(tab_negall,[status(thm),assumptions([h2,h0]),tab_negall(discharge,[h6]),tab_negall(eigenvar,eigen__1)],[h2,21,h6])). % 0.08/0.39 thf(23,plain,$false,inference(tab_imp,[status(thm),assumptions([h0]),tab_imp(discharge,[h1]),tab_imp(discharge,[h2])],[h0,7,22,h1,h2])). % 0.08/0.39 thf(0,theorem,(~(((![X1:$i]:((~((~((![X2:$i]:(((rel_s4 @ X1) @ X2) => (phi @ X2))))))) => (phi @ X1))) => (~((![X1:$i]:((~((~((![X2:$i]:(((rel_s4 @ X1) @ X2) => (phi @ X2))))))) => (![X2:$i]:(((rel_s4 @ X1) @ X2) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => (phi @ X3)))))))))))),inference(contra,[status(thm),contra(discharge,[h0])],[23,h0])). % 0.08/0.39 % SZS output end Proof %------------------------------------------------------------------------------