%------------------------------------------------------------------------------ % File : Satallax---3.5 % Problem : LCL626^1 : TPTP v9.3.1. Bugfixed v7.3.0. % Transfm : none % Format : tptp:raw % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % Computer : n005.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 01:02:09 PM UTC 2026 % Result : Theorem 0.13s 0.38s % Output : Proof 0.13s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.00/0.03 % Problem : LCL626^1 : TPTP v9.3.1. Bugfixed v7.3.0. % 0.00/0.04 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % 0.10/0.35 % Computer : n005.cluster.edu % 0.10/0.35 % Model : x86_64 x86_64 % 0.10/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.10/0.35 % Memory : 8046.5625MB % 0.10/0.35 % OS : Linux 6.8.0-71-generic % 0.10/0.35 % CPULimit : 300 % 0.10/0.35 % WCLimit : 300 % 0.10/0.35 % DateTime : Fri Sep 4 23:03:36 UTC 2026 % 0.10/0.35 % CPUTime : % 0.13/0.38 % SZS status Theorem % 0.13/0.38 % Mode: mode213 % 0.13/0.38 % Inferences: 13 % 0.13/0.38 % SZS output start Proof % 0.13/0.38 thf(ty_eigen__2, type, eigen__2 : $i). % 0.13/0.38 thf(ty_eigen__1, type, eigen__1 : $i). % 0.13/0.38 thf(ty_eigen__0, type, eigen__0 : ($i>$o)). % 0.13/0.38 thf(ty_eigen__4, type, eigen__4 : $i). % 0.13/0.38 thf(ty_r, type, r : ($i>$i>$o)). % 0.13/0.38 thf(ty_eigen__3, type, eigen__3 : $i). % 0.13/0.38 thf(sP1,plain,sP1 <=> ((r @ eigen__3) @ eigen__4),introduced(definition,[new_symbols(definition,[sP1])])). % 0.13/0.38 thf(sP2,plain,sP2 <=> (![X1:$i]:(![X2:$i]:(![X3:$i]:((~((((r @ X1) @ X2) => (~(((r @ X2) @ X3)))))) => ((r @ X1) @ X3))))),introduced(definition,[new_symbols(definition,[sP2])])). % 0.13/0.38 thf(sP3,plain,sP3 <=> ((r @ eigen__1) @ eigen__4),introduced(definition,[new_symbols(definition,[sP3])])). % 0.13/0.38 thf(sP4,plain,sP4 <=> (![X1:$i]:(((r @ eigen__1) @ X1) => (eigen__0 @ X1))),introduced(definition,[new_symbols(definition,[sP4])])). % 0.13/0.38 thf(sP5,plain,sP5 <=> ((~((((r @ eigen__1) @ eigen__2) => (~(((r @ eigen__2) @ eigen__3)))))) => ((r @ eigen__1) @ eigen__3)),introduced(definition,[new_symbols(definition,[sP5])])). % 0.13/0.38 thf(sP6,plain,sP6 <=> (![X1:$i]:(![X2:$i]:((~((((r @ eigen__1) @ X1) => (~(((r @ X1) @ X2)))))) => ((r @ eigen__1) @ X2)))),introduced(definition,[new_symbols(definition,[sP6])])). % 0.13/0.38 thf(sP7,plain,sP7 <=> (((r @ eigen__1) @ eigen__2) => (~(((r @ eigen__2) @ eigen__3)))),introduced(definition,[new_symbols(definition,[sP7])])). % 0.13/0.38 thf(sP8,plain,sP8 <=> (sP3 => (eigen__0 @ eigen__4)),introduced(definition,[new_symbols(definition,[sP8])])). % 0.13/0.38 thf(sP9,plain,sP9 <=> (![X1:$i]:((~((((r @ eigen__1) @ eigen__3) => (~(((r @ eigen__3) @ X1)))))) => ((r @ eigen__1) @ X1))),introduced(definition,[new_symbols(definition,[sP9])])). % 0.13/0.38 thf(sP10,plain,sP10 <=> (![X1:$i]:((~((((r @ eigen__1) @ eigen__2) => (~(((r @ eigen__2) @ X1)))))) => ((r @ eigen__1) @ X1))),introduced(definition,[new_symbols(definition,[sP10])])). % 0.13/0.38 thf(sP11,plain,sP11 <=> (((r @ eigen__1) @ eigen__3) => (~(sP1))),introduced(definition,[new_symbols(definition,[sP11])])). % 0.13/0.38 thf(sP12,plain,sP12 <=> ((r @ eigen__2) @ eigen__3),introduced(definition,[new_symbols(definition,[sP12])])). % 0.13/0.38 thf(sP13,plain,sP13 <=> ((r @ eigen__1) @ eigen__3),introduced(definition,[new_symbols(definition,[sP13])])). % 0.13/0.38 thf(sP14,plain,sP14 <=> (eigen__0 @ eigen__4),introduced(definition,[new_symbols(definition,[sP14])])). % 0.13/0.38 thf(sP15,plain,sP15 <=> ((r @ eigen__1) @ eigen__2),introduced(definition,[new_symbols(definition,[sP15])])). % 0.13/0.38 thf(sP16,plain,sP16 <=> ((~(sP11)) => sP3),introduced(definition,[new_symbols(definition,[sP16])])). % 0.13/0.38 thf(def_mfalse,definition,(mfalse = (^[X1:$i]:$false))). % 0.13/0.38 thf(def_mtrue,definition,(mtrue = (^[X1:$i]:(~($false))))). % 0.13/0.38 thf(def_mnot,definition,(mnot = (^[X1:$i>$o]:(^[X2:$i]:(~((X1 @ X2))))))). % 0.13/0.38 thf(def_mor,definition,(mor = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((~((X1 @ X3))) => (X2 @ X3))))))). % 0.13/0.38 thf(def_mand,definition,(mand = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:(~(((X1 @ X3) => (~((X2 @ X3))))))))))). % 0.13/0.38 thf(def_mimpl,definition,(mimpl = (^[X1:$i>$o]:(mor @ (mnot @ X1))))). % 0.13/0.38 thf(def_miff,definition,(miff = (^[X1:$i>$o]:(^[X2:$i>$o]:((mand @ ((mimpl @ X1) @ X2)) @ ((mimpl @ X2) @ X1)))))). % 0.13/0.38 thf(def_mbox,definition,(mbox = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:(![X4:$i]:(((X1 @ X3) @ X4) => (X2 @ X4)))))))). % 0.13/0.38 thf(def_mdia,definition,(mdia = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:(~((![X4:$i]:(((X1 @ X3) @ X4) => (~((X2 @ X4)))))))))))). % 0.13/0.38 thf(def_mall,definition,(mall = (^[X1:individuals>$i>$o]:(^[X2:$i]:(![X3:individuals]:((X1 @ X3) @ X2)))))). % 0.13/0.38 thf(def_mexists,definition,(mexists = (^[X1:individuals>$i>$o]:(^[X2:$i]:(~((![X3:individuals]:(~(((X1 @ X3) @ X2)))))))))). % 0.13/0.38 thf(def_mvalid,definition,(mvalid = (!!))). % 0.13/0.38 thf(def_msatisfiable,definition,(msatisfiable = (^[X1:$i>$o]:(~((![X2:$i]:(~((X1 @ X2))))))))). % 0.13/0.38 thf(def_mcountersatisfiable,definition,(mcountersatisfiable = (^[X1:$i>$o]:(~(((!!) @ X1)))))). % 0.13/0.38 thf(def_minvalid,definition,(minvalid = (^[X1:$i>$o]:(![X2:$i]:(~((X1 @ X2))))))). % 0.13/0.38 thf(def_cartesian_product,definition,(cartesian_product = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:(^[X4:$i]:(~(((X1 @ X3) => (~((X2 @ X4)))))))))))). % 0.13/0.38 thf(def_pair_rel,definition,(pair_rel = (^[X1:$i]:(^[X2:$i]:(^[X3:$i]:(^[X4:$i]:((~((X3 = X1))) => (X4 = X2)))))))). % 0.13/0.38 thf(def_id_rel,definition,(id_rel = (^[X1:$i>$o]:(^[X2:$i]:(^[X3:$i]:(~(((X1 @ X2) => (~((X2 = X3))))))))))). % 0.13/0.38 thf(def_sub_rel,definition,(sub_rel = (^[X1:$i>$i>$o]:(^[X2:$i>$i>$o]:(![X3:$i]:(![X4:$i]:(((X1 @ X3) @ X4) => ((X2 @ X3) @ X4)))))))). % 0.13/0.38 thf(def_is_rel_on,definition,(is_rel_on = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(^[X3:$i>$o]:(![X4:$i]:(![X5:$i]:(((X1 @ X4) @ X5) => (~(((X2 @ X4) => (~((X3 @ X5)))))))))))))). % 0.13/0.38 thf(def_restrict_rel_domain,definition,(restrict_rel_domain = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:(^[X4:$i]:(~(((X2 @ X3) => (~(((X1 @ X3) @ X4)))))))))))). % 0.13/0.38 thf(def_rel_diagonal,definition,(rel_diagonal = (=))). % 0.13/0.38 thf(def_rel_composition,definition,(rel_composition = (^[X1:$i>$i>$o]:(^[X2:$i>$i>$o]:(^[X3:$i]:(^[X4:$i]:(~((![X5:$i]:(((X1 @ X3) @ X5) => (~(((X2 @ X5) @ X4))))))))))))). % 0.13/0.38 thf(def_reflexive,definition,(reflexive = (^[X1:$i>$i>$o]:(![X2:$i]:((X1 @ X2) @ X2))))). % 0.13/0.38 thf(def_irreflexive,definition,(irreflexive = (^[X1:$i>$i>$o]:(![X2:$i]:(~(((X1 @ X2) @ X2))))))). % 0.13/0.38 thf(def_symmetric,definition,(symmetric = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(((X1 @ X2) @ X3) => ((X1 @ X3) @ X2))))))). % 0.13/0.38 thf(def_transitive,definition,(transitive = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X3) @ X4)))))) => ((X1 @ X2) @ X4)))))))). % 0.13/0.38 thf(def_equiv_rel,definition,(equiv_rel = (^[X1:$i>$i>$o]:(~(((~(((reflexive @ X1) => (~((symmetric @ X1)))))) => (~((transitive @ X1))))))))). % 0.13/0.38 thf(def_rel_codomain,definition,(rel_codomain = (^[X1:$i>$i>$o]:(^[X2:$i]:(~((![X3:$i]:(~(((X1 @ X3) @ X2)))))))))). % 0.13/0.38 thf(def_rel_domain,definition,(rel_domain = (^[X1:$i>$i>$o]:(^[X2:$i]:(~((![X3:$i]:(~(((X1 @ X2) @ X3)))))))))). % 0.13/0.38 thf(def_rel_inverse,definition,(rel_inverse = (^[X1:$i>$i>$o]:(^[X2:$i]:(^[X3:$i]:((X1 @ X3) @ X2)))))). % 0.13/0.38 thf(def_equiv_classes,definition,(equiv_classes = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(~((![X3:$i]:((X2 @ X3) => (~((![X4:$i]:((X2 @ X4) = ((X1 @ X3) @ X4))))))))))))). % 0.13/0.38 thf(def_restrict_rel_codomain,definition,(restrict_rel_codomain = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:(^[X4:$i]:(~(((X2 @ X4) => (~(((X1 @ X3) @ X4)))))))))))). % 0.13/0.38 thf(def_rel_field,definition,(rel_field = (^[X1:$i>$i>$o]:(^[X2:$i]:((~(((rel_domain @ X1) @ X2))) => ((rel_codomain @ X1) @ X2)))))). % 0.13/0.38 thf(def_well_founded,definition,(well_founded = (^[X1:$i>$i>$o]:(![X2:$i>$o]:(![X3:$i]:((X2 @ X3) => (~((![X4:$i]:((X2 @ X4) => (~((![X5:$i]:(((X1 @ X4) @ X5) => (~((X2 @ X5))))))))))))))))). % 0.13/0.38 thf(def_upwards_well_founded,definition,(upwards_well_founded = (^[X1:$i>$i>$o]:(![X2:$i>$o]:(![X3:$i]:((X2 @ X3) => (~((![X4:$i]:((X2 @ X4) => (~((![X5:$i]:(((X1 @ X4) @ X4) => (~((X2 @ X5))))))))))))))))). % 0.13/0.38 thf(k4,conjecture,(![X1:$i>$o]:(![X2:$i]:((~((~((![X3:$i]:(((r @ X2) @ X3) => (X1 @ X3))))))) => (![X3:$i]:(((r @ X2) @ X3) => (![X4:$i]:(((r @ X3) @ X4) => (![X5:$i]:(((r @ X4) @ X5) => (X1 @ X5))))))))))). % 0.13/0.38 thf(h0,negated_conjecture,(~((![X1:$i>$o]:(![X2:$i]:((![X3:$i]:(((r @ X2) @ X3) => (X1 @ X3))) => (![X3:$i]:(((r @ X2) @ X3) => (![X4:$i]:(((r @ X3) @ X4) => (![X5:$i]:(((r @ X4) @ X5) => (X1 @ X5)))))))))))),inference(assume_negation,[status(cth)],[k4])). % 0.13/0.38 thf(h1,assumption,(~((![X1:$i]:((![X2:$i]:(((r @ X1) @ X2) => (eigen__0 @ X2))) => (![X2:$i]:(((r @ X1) @ X2) => (![X3:$i]:(((r @ X2) @ X3) => (![X4:$i]:(((r @ X3) @ X4) => (eigen__0 @ X4))))))))))),introduced(assumption,[])). % 0.13/0.38 thf(h2,assumption,(~((sP4 => (![X1:$i]:(((r @ eigen__1) @ X1) => (![X2:$i]:(((r @ X1) @ X2) => (![X3:$i]:(((r @ X2) @ X3) => (eigen__0 @ X3)))))))))),introduced(assumption,[])). % 0.13/0.38 thf(h3,assumption,sP4,introduced(assumption,[])). % 0.13/0.38 thf(h4,assumption,(~((![X1:$i]:(((r @ eigen__1) @ X1) => (![X2:$i]:(((r @ X1) @ X2) => (![X3:$i]:(((r @ X2) @ X3) => (eigen__0 @ X3))))))))),introduced(assumption,[])). % 0.13/0.38 thf(h5,assumption,(~((sP15 => (![X1:$i]:(((r @ eigen__2) @ X1) => (![X2:$i]:(((r @ X1) @ X2) => (eigen__0 @ X2)))))))),introduced(assumption,[])). % 0.13/0.38 thf(h6,assumption,sP15,introduced(assumption,[])). % 0.13/0.38 thf(h7,assumption,(~((![X1:$i]:(((r @ eigen__2) @ X1) => (![X2:$i]:(((r @ X1) @ X2) => (eigen__0 @ X2))))))),introduced(assumption,[])). % 0.13/0.38 thf(h8,assumption,(~((sP12 => (![X1:$i]:(((r @ eigen__3) @ X1) => (eigen__0 @ X1)))))),introduced(assumption,[])). % 0.13/0.38 thf(h9,assumption,sP12,introduced(assumption,[])). % 0.13/0.38 thf(h10,assumption,(~((![X1:$i]:(((r @ eigen__3) @ X1) => (eigen__0 @ X1))))),introduced(assumption,[])). % 0.13/0.38 thf(h11,assumption,(~((sP1 => sP14))),introduced(assumption,[])). % 0.13/0.38 thf(h12,assumption,sP1,introduced(assumption,[])). % 0.13/0.38 thf(h13,assumption,(~(sP14)),introduced(assumption,[])). % 0.13/0.38 thf(h14,assumption,sP2,introduced(assumption,[])). % 0.13/0.38 thf(h15,assumption,(![X1:$i>$o]:(![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (~((![X4:$i]:(((r @ X3) @ X3) => (~((X1 @ X4)))))))))))))),introduced(assumption,[])). % 0.13/0.38 thf(1,plain,(~(sP10) | sP5),inference(all_rule,[status(thm)],[])). % 0.13/0.38 thf(2,plain,((~(sP5) | sP7) | sP13),inference(prop_rule,[status(thm)],[])). % 0.13/0.38 thf(3,plain,((~(sP7) | ~(sP15)) | ~(sP12)),inference(prop_rule,[status(thm)],[])). % 0.13/0.38 thf(4,plain,(~(sP6) | sP9),inference(all_rule,[status(thm)],[])). % 0.13/0.38 thf(5,plain,(~(sP9) | sP16),inference(all_rule,[status(thm)],[])). % 0.13/0.38 thf(6,plain,((~(sP16) | sP11) | sP3),inference(prop_rule,[status(thm)],[])). % 0.13/0.38 thf(7,plain,((~(sP11) | ~(sP13)) | ~(sP1)),inference(prop_rule,[status(thm)],[])). % 0.13/0.38 thf(8,plain,(~(sP2) | sP6),inference(all_rule,[status(thm)],[])). % 0.13/0.38 thf(9,plain,(~(sP6) | sP10),inference(all_rule,[status(thm)],[])). % 0.13/0.38 thf(10,plain,(~(sP4) | sP8),inference(all_rule,[status(thm)],[])). % 0.13/0.38 thf(11,plain,((~(sP8) | ~(sP3)) | sP14),inference(prop_rule,[status(thm)],[])). % 0.13/0.38 thf(12,plain,$false,inference(prop_unsat,[status(thm),assumptions([h14,h15,h12,h13,h11,h9,h10,h8,h6,h7,h5,h3,h4,h2,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,h14,h3,h6,h9,h12,h13])). % 0.13/0.38 thf(upwf_trans,axiom,(~(((transitive @ r) => (~((upwards_well_founded @ r))))))). % 0.13/0.38 thf(13,plain,(~((sP2 => (~((![X1:$i>$o]:(![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (~((![X4:$i]:(((r @ X3) @ X3) => (~((X1 @ X4))))))))))))))))))),inference(preprocess,[status(thm)],[upwf_trans]). % 0.13/0.38 thf(14,plain,$false,inference(tab_negimp,[status(thm),assumptions([h12,h13,h11,h9,h10,h8,h6,h7,h5,h3,h4,h2,h1,h0]),tab_negimp(discharge,[h14,h15])],[13,12,h14,h15])). % 0.13/0.38 thf(15,plain,$false,inference(tab_negimp,[status(thm),assumptions([h11,h9,h10,h8,h6,h7,h5,h3,h4,h2,h1,h0]),tab_negimp(discharge,[h12,h13])],[h11,14,h12,h13])). % 0.13/0.38 thf(16,plain,$false,inference(tab_negall,[status(thm),assumptions([h9,h10,h8,h6,h7,h5,h3,h4,h2,h1,h0]),tab_negall(discharge,[h11]),tab_negall(eigenvar,eigen__4)],[h10,15,h11])). % 0.13/0.38 thf(17,plain,$false,inference(tab_negimp,[status(thm),assumptions([h8,h6,h7,h5,h3,h4,h2,h1,h0]),tab_negimp(discharge,[h9,h10])],[h8,16,h9,h10])). % 0.13/0.38 thf(18,plain,$false,inference(tab_negall,[status(thm),assumptions([h6,h7,h5,h3,h4,h2,h1,h0]),tab_negall(discharge,[h8]),tab_negall(eigenvar,eigen__3)],[h7,17,h8])). % 0.13/0.38 thf(19,plain,$false,inference(tab_negimp,[status(thm),assumptions([h5,h3,h4,h2,h1,h0]),tab_negimp(discharge,[h6,h7])],[h5,18,h6,h7])). % 0.13/0.38 thf(20,plain,$false,inference(tab_negall,[status(thm),assumptions([h3,h4,h2,h1,h0]),tab_negall(discharge,[h5]),tab_negall(eigenvar,eigen__2)],[h4,19,h5])). % 0.13/0.38 thf(21,plain,$false,inference(tab_negimp,[status(thm),assumptions([h2,h1,h0]),tab_negimp(discharge,[h3,h4])],[h2,20,h3,h4])). % 0.13/0.38 thf(22,plain,$false,inference(tab_negall,[status(thm),assumptions([h1,h0]),tab_negall(discharge,[h2]),tab_negall(eigenvar,eigen__1)],[h1,21,h2])). % 0.13/0.38 thf(23,plain,$false,inference(tab_negall,[status(thm),assumptions([h0]),tab_negall(discharge,[h1]),tab_negall(eigenvar,eigen__0)],[h0,22,h1])). % 0.13/0.38 thf(0,theorem,(![X1:$i>$o]:(![X2:$i]:((~((~((![X3:$i]:(((r @ X2) @ X3) => (X1 @ X3))))))) => (![X3:$i]:(((r @ X2) @ X3) => (![X4:$i]:(((r @ X3) @ X4) => (![X5:$i]:(((r @ X4) @ X5) => (X1 @ X5)))))))))),inference(contra,[status(thm),contra(discharge,[h0])],[23,h0])). % 0.13/0.38 % SZS output end Proof %------------------------------------------------------------------------------