%------------------------------------------------------------------------------ % File : Satallax---3.5 % Problem : SET575^7 : 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 : n027.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 02:25:56 PM UTC 2025 % Result : Theorem 4.52s 4.72s % Output : Proof 4.52s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.08/0.12 % Problem : SET575^7 : TPTP v9.2.0. Released v5.5.0. % 0.08/0.13 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s % 0.12/0.34 % Computer : n027.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 14:32:23 EDT 2025 % 0.12/0.34 % CPUTime : % 4.52/4.72 % SZS status Theorem % 4.52/4.72 % Mode: mode506 % 4.52/4.72 % Inferences: 21962 % 4.52/4.72 % SZS output start Proof % 4.52/4.72 thf(ty_mu, type, mu : $tType). % 4.52/4.72 thf(ty_rel_s4, type, rel_s4 : ($i>$i>$o)). % 4.52/4.72 thf(ty_eigen__2, type, eigen__2 : mu). % 4.52/4.72 thf(ty_eigen__1, type, eigen__1 : $i). % 4.52/4.72 thf(ty_eigen__0, type, eigen__0 : $i). % 4.52/4.72 thf(ty_member, type, member : (mu>mu>$i>$o)). % 4.52/4.72 thf(ty_eigen__4, type, eigen__4 : mu). % 4.52/4.72 thf(ty_eigen__3, type, eigen__3 : $i). % 4.52/4.72 thf(ty_exists_in_world, type, exists_in_world : (mu>$i>$o)). % 4.52/4.72 thf(ty_intersect, type, intersect : (mu>mu>$i>$o)). % 4.52/4.72 thf(h0, assumption, (![X1:$i>$o]:(![X2:$i]:((X1 @ X2) => (X1 @ (eps__0 @ X1))))),introduced(assumption,[])). % 4.52/4.72 thf(eigendef_eigen__3, definition, eigen__3 = (eps__0 @ (^[X1:$i]:(~((((rel_s4 @ eigen__1) @ X1) => (![X2:mu]:(((exists_in_world @ X2) @ X1) => (![X3:$i]:(((rel_s4 @ X1) @ X3) => ((![X4:$i]:(((rel_s4 @ X3) @ X4) => (((intersect @ eigen__2) @ X2) @ X4))) => (~((![X4:mu]:(((exists_in_world @ X4) @ X3) => ((![X5:$i]:(((rel_s4 @ X3) @ X5) => (((member @ X4) @ eigen__2) @ X5))) => (~((![X5:$i]:(((rel_s4 @ X3) @ X5) => (((member @ X4) @ X2) @ X5)))))))))))))))))))), introduced(definition,[new_symbols(definition,[eigen__3])])). % 4.52/4.72 thf(eigendef_eigen__1, definition, eigen__1 = (eps__0 @ (^[X1:$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) => ((![X6:$i]:(((rel_s4 @ X5) @ X6) => (((intersect @ X2) @ X4) @ X6))) => (~((![X6:mu]:(((exists_in_world @ X6) @ X5) => ((![X7:$i]:(((rel_s4 @ X5) @ X7) => (((member @ X6) @ X2) @ X7))) => (~((![X7:$i]:(((rel_s4 @ X5) @ X7) => (((member @ X6) @ X4) @ X7)))))))))))))))))))))))), introduced(definition,[new_symbols(definition,[eigen__1])])). % 4.52/4.72 thf(eigendef_eigen__0, definition, eigen__0 = (eps__0 @ (^[X1:$i]:(~((![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:$i]:(((rel_s4 @ X4) @ X6) => ((![X7:$i]:(((rel_s4 @ X6) @ X7) => (((intersect @ X3) @ X5) @ X7))) => (~((![X7:mu]:(((exists_in_world @ X7) @ X6) => ((![X8:$i]:(((rel_s4 @ X6) @ X8) => (((member @ X7) @ X3) @ X8))) => (~((![X8:$i]:(((rel_s4 @ X6) @ X8) => (((member @ X7) @ X5) @ X8))))))))))))))))))))))))), introduced(definition,[new_symbols(definition,[eigen__0])])). % 4.52/4.72 thf(h1, assumption, (![X1:mu>$o]:(![X2:mu]:((X1 @ X2) => (X1 @ (eps__1 @ X1))))),introduced(assumption,[])). % 4.52/4.72 thf(eigendef_eigen__2, definition, eigen__2 = (eps__1 @ (^[X1:mu]:(~((((exists_in_world @ X1) @ eigen__1) => (![X2:$i]:(((rel_s4 @ eigen__1) @ X2) => (![X3:mu]:(((exists_in_world @ X3) @ X2) => (![X4:$i]:(((rel_s4 @ X2) @ X4) => ((![X5:$i]:(((rel_s4 @ X4) @ X5) => (((intersect @ X1) @ X3) @ X5))) => (~((![X5:mu]:(((exists_in_world @ X5) @ X4) => ((![X6:$i]:(((rel_s4 @ X4) @ X6) => (((member @ X5) @ X1) @ X6))) => (~((![X6:$i]:(((rel_s4 @ X4) @ X6) => (((member @ X5) @ X3) @ X6)))))))))))))))))))))), introduced(definition,[new_symbols(definition,[eigen__2])])). % 4.52/4.72 thf(eigendef_eigen__4, definition, eigen__4 = (eps__1 @ (^[X1:mu]:(~((((exists_in_world @ X1) @ eigen__3) => (![X2:$i]:(((rel_s4 @ eigen__3) @ X2) => ((![X3:$i]:(((rel_s4 @ X2) @ X3) => (((intersect @ eigen__2) @ X1) @ X3))) => (~((![X3:mu]:(((exists_in_world @ X3) @ X2) => ((![X4:$i]:(((rel_s4 @ X2) @ X4) => (((member @ X3) @ eigen__2) @ X4))) => (~((![X4:$i]:(((rel_s4 @ X2) @ X4) => (((member @ X3) @ X1) @ X4)))))))))))))))))), introduced(definition,[new_symbols(definition,[eigen__4])])). % 4.52/4.72 thf(sP1,plain,sP1 <=> (![X1:$i]:(((rel_s4 @ eigen__3) @ X1) => ((![X2:$i]:(((rel_s4 @ X1) @ X2) => (((intersect @ eigen__2) @ eigen__4) @ X2))) => (~((![X2:mu]:(((exists_in_world @ X2) @ X1) => ((![X3:$i]:(((rel_s4 @ X1) @ X3) => (((member @ X2) @ eigen__2) @ X3))) => (~((![X3:$i]:(((rel_s4 @ X1) @ X3) => (((member @ X2) @ eigen__4) @ X3))))))))))))),introduced(definition,[new_symbols(definition,[sP1])])). % 4.52/4.72 thf(sP2,plain,sP2 <=> (((rel_s4 @ eigen__0) @ eigen__1) => (![X1:mu]:(((exists_in_world @ X1) @ eigen__1) => (![X2:$i]:(((rel_s4 @ eigen__1) @ X2) => (![X3:mu]:(((exists_in_world @ X3) @ X2) => (~(((![X4:$i]:(((rel_s4 @ X2) @ X4) => ((![X5:$i]:(((rel_s4 @ X4) @ X5) => (((intersect @ X1) @ X3) @ X5))) => (~((![X5:mu]:(((exists_in_world @ X5) @ X4) => ((![X6:$i]:(((rel_s4 @ X4) @ X6) => (((member @ X5) @ X1) @ X6))) => (~((![X6:$i]:(((rel_s4 @ X4) @ X6) => (((member @ X5) @ X3) @ X6))))))))))))) => (~((![X4:$i]:(((rel_s4 @ X2) @ X4) => ((~((![X5:mu]:(((exists_in_world @ X5) @ X4) => ((![X6:$i]:(((rel_s4 @ X4) @ X6) => (((member @ X5) @ X1) @ X6))) => (~((![X6:$i]:(((rel_s4 @ X4) @ X6) => (((member @ X5) @ X3) @ X6)))))))))) => (![X5:$i]:(((rel_s4 @ X4) @ X5) => (((intersect @ X1) @ X3) @ X5)))))))))))))))))),introduced(definition,[new_symbols(definition,[sP2])])). % 4.52/4.72 thf(sP3,plain,sP3 <=> (((exists_in_world @ eigen__4) @ eigen__3) => (~((sP1 => (~((![X1:$i]:(((rel_s4 @ eigen__3) @ X1) => ((~((![X2:mu]:(((exists_in_world @ X2) @ X1) => ((![X3:$i]:(((rel_s4 @ X1) @ X3) => (((member @ X2) @ eigen__2) @ X3))) => (~((![X3:$i]:(((rel_s4 @ X1) @ X3) => (((member @ X2) @ eigen__4) @ X3)))))))))) => (![X2:$i]:(((rel_s4 @ X1) @ X2) => (((intersect @ eigen__2) @ eigen__4) @ X2)))))))))))),introduced(definition,[new_symbols(definition,[sP3])])). % 4.52/4.72 thf(sP4,plain,sP4 <=> ((rel_s4 @ eigen__0) @ eigen__1),introduced(definition,[new_symbols(definition,[sP4])])). % 4.52/4.72 thf(sP5,plain,sP5 <=> (((exists_in_world @ eigen__2) @ eigen__1) => (![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => (![X2:mu]:(((exists_in_world @ X2) @ X1) => (~(((![X3:$i]:(((rel_s4 @ X1) @ X3) => ((![X4:$i]:(((rel_s4 @ X3) @ X4) => (((intersect @ eigen__2) @ X2) @ X4))) => (~((![X4:mu]:(((exists_in_world @ X4) @ X3) => ((![X5:$i]:(((rel_s4 @ X3) @ X5) => (((member @ X4) @ eigen__2) @ X5))) => (~((![X5:$i]:(((rel_s4 @ X3) @ X5) => (((member @ X4) @ X2) @ X5))))))))))))) => (~((![X3:$i]:(((rel_s4 @ X1) @ X3) => ((~((![X4:mu]:(((exists_in_world @ X4) @ X3) => ((![X5:$i]:(((rel_s4 @ X3) @ X5) => (((member @ X4) @ eigen__2) @ X5))) => (~((![X5:$i]:(((rel_s4 @ X3) @ X5) => (((member @ X4) @ X2) @ X5)))))))))) => (![X4:$i]:(((rel_s4 @ X3) @ X4) => (((intersect @ eigen__2) @ X2) @ X4)))))))))))))))),introduced(definition,[new_symbols(definition,[sP5])])). % 4.52/4.72 thf(sP6,plain,sP6 <=> (![X1:$i]:(![X2:$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) => ((![X7:$i]:(((rel_s4 @ X6) @ X7) => (((intersect @ X3) @ X5) @ X7))) => (~((![X7:mu]:(((exists_in_world @ X7) @ X6) => ((![X8:$i]:(((rel_s4 @ X6) @ X8) => (((member @ X7) @ X3) @ X8))) => (~((![X8:$i]:(((rel_s4 @ X6) @ X8) => (((member @ X7) @ X5) @ X8))))))))))))) => (~((![X6:$i]:(((rel_s4 @ X4) @ X6) => ((~((![X7:mu]:(((exists_in_world @ X7) @ X6) => ((![X8:$i]:(((rel_s4 @ X6) @ X8) => (((member @ X7) @ X3) @ X8))) => (~((![X8:$i]:(((rel_s4 @ X6) @ X8) => (((member @ X7) @ X5) @ X8)))))))))) => (![X7:$i]:(((rel_s4 @ X6) @ X7) => (((intersect @ X3) @ X5) @ X7)))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP6])])). % 4.52/4.72 thf(sP7,plain,sP7 <=> (((rel_s4 @ eigen__1) @ eigen__3) => (![X1:mu]:(((exists_in_world @ X1) @ eigen__3) => (~(((![X2:$i]:(((rel_s4 @ eigen__3) @ X2) => ((![X3:$i]:(((rel_s4 @ X2) @ X3) => (((intersect @ eigen__2) @ X1) @ X3))) => (~((![X3:mu]:(((exists_in_world @ X3) @ X2) => ((![X4:$i]:(((rel_s4 @ X2) @ X4) => (((member @ X3) @ eigen__2) @ X4))) => (~((![X4:$i]:(((rel_s4 @ X2) @ X4) => (((member @ X3) @ X1) @ X4))))))))))))) => (~((![X2:$i]:(((rel_s4 @ eigen__3) @ X2) => ((~((![X3:mu]:(((exists_in_world @ X3) @ X2) => ((![X4:$i]:(((rel_s4 @ X2) @ X4) => (((member @ X3) @ eigen__2) @ X4))) => (~((![X4:$i]:(((rel_s4 @ X2) @ X4) => (((member @ X3) @ X1) @ X4)))))))))) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => (((intersect @ eigen__2) @ X1) @ X3)))))))))))))),introduced(definition,[new_symbols(definition,[sP7])])). % 4.52/4.72 thf(sP8,plain,sP8 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__3) => (![X2:$i]:(((rel_s4 @ eigen__3) @ X2) => ((![X3:$i]:(((rel_s4 @ X2) @ X3) => (((intersect @ eigen__2) @ X1) @ X3))) => (~((![X3:mu]:(((exists_in_world @ X3) @ X2) => ((![X4:$i]:(((rel_s4 @ X2) @ X4) => (((member @ X3) @ eigen__2) @ X4))) => (~((![X4:$i]:(((rel_s4 @ X2) @ X4) => (((member @ X3) @ X1) @ X4))))))))))))))),introduced(definition,[new_symbols(definition,[sP8])])). % 4.52/4.72 thf(sP9,plain,sP9 <=> ((exists_in_world @ eigen__4) @ eigen__3),introduced(definition,[new_symbols(definition,[sP9])])). % 4.52/4.72 thf(sP10,plain,sP10 <=> (![X1:$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) => ((![X6:$i]:(((rel_s4 @ X5) @ X6) => (((intersect @ X2) @ X4) @ X6))) => (~((![X6:mu]:(((exists_in_world @ X6) @ X5) => ((![X7:$i]:(((rel_s4 @ X5) @ X7) => (((member @ X6) @ X2) @ X7))) => (~((![X7:$i]:(((rel_s4 @ X5) @ X7) => (((member @ X6) @ X4) @ X7))))))))))))) => (~((![X5:$i]:(((rel_s4 @ X3) @ X5) => ((~((![X6:mu]:(((exists_in_world @ X6) @ X5) => ((![X7:$i]:(((rel_s4 @ X5) @ X7) => (((member @ X6) @ X2) @ X7))) => (~((![X7:$i]:(((rel_s4 @ X5) @ X7) => (((member @ X6) @ X4) @ X7)))))))))) => (![X6:$i]:(((rel_s4 @ X5) @ X6) => (((intersect @ X2) @ X4) @ X6))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP10])])). % 4.52/4.72 thf(sP11,plain,sP11 <=> ((exists_in_world @ eigen__2) @ eigen__1),introduced(definition,[new_symbols(definition,[sP11])])). % 4.52/4.72 thf(sP12,plain,sP12 <=> (sP4 => (![X1:mu]:(((exists_in_world @ X1) @ eigen__1) => (![X2:$i]:(((rel_s4 @ eigen__1) @ X2) => (![X3:mu]:(((exists_in_world @ X3) @ X2) => (![X4:$i]:(((rel_s4 @ X2) @ X4) => ((![X5:$i]:(((rel_s4 @ X4) @ X5) => (((intersect @ X1) @ X3) @ X5))) => (~((![X5:mu]:(((exists_in_world @ X5) @ X4) => ((![X6:$i]:(((rel_s4 @ X4) @ X6) => (((member @ X5) @ X1) @ X6))) => (~((![X6:$i]:(((rel_s4 @ X4) @ X6) => (((member @ X5) @ X3) @ X6)))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP12])])). % 4.52/4.72 thf(sP13,plain,sP13 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__1) => (![X2:$i]:(((rel_s4 @ eigen__1) @ X2) => (![X3:mu]:(((exists_in_world @ X3) @ X2) => (~(((![X4:$i]:(((rel_s4 @ X2) @ X4) => ((![X5:$i]:(((rel_s4 @ X4) @ X5) => (((intersect @ X1) @ X3) @ X5))) => (~((![X5:mu]:(((exists_in_world @ X5) @ X4) => ((![X6:$i]:(((rel_s4 @ X4) @ X6) => (((member @ X5) @ X1) @ X6))) => (~((![X6:$i]:(((rel_s4 @ X4) @ X6) => (((member @ X5) @ X3) @ X6))))))))))))) => (~((![X4:$i]:(((rel_s4 @ X2) @ X4) => ((~((![X5:mu]:(((exists_in_world @ X5) @ X4) => ((![X6:$i]:(((rel_s4 @ X4) @ X6) => (((member @ X5) @ X1) @ X6))) => (~((![X6:$i]:(((rel_s4 @ X4) @ X6) => (((member @ X5) @ X3) @ X6)))))))))) => (![X5:$i]:(((rel_s4 @ X4) @ X5) => (((intersect @ X1) @ X3) @ X5))))))))))))))))),introduced(definition,[new_symbols(definition,[sP13])])). % 4.52/4.72 thf(sP14,plain,sP14 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__1) => (![X2:$i]:(((rel_s4 @ eigen__1) @ X2) => (![X3:mu]:(((exists_in_world @ X3) @ X2) => (![X4:$i]:(((rel_s4 @ X2) @ X4) => ((![X5:$i]:(((rel_s4 @ X4) @ X5) => (((intersect @ X1) @ X3) @ X5))) => (~((![X5:mu]:(((exists_in_world @ X5) @ X4) => ((![X6:$i]:(((rel_s4 @ X4) @ X6) => (((member @ X5) @ X1) @ X6))) => (~((![X6:$i]:(((rel_s4 @ X4) @ X6) => (((member @ X5) @ X3) @ X6))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP14])])). % 4.52/4.72 thf(sP15,plain,sP15 <=> (sP1 => (~((![X1:$i]:(((rel_s4 @ eigen__3) @ X1) => ((~((![X2:mu]:(((exists_in_world @ X2) @ X1) => ((![X3:$i]:(((rel_s4 @ X1) @ X3) => (((member @ X2) @ eigen__2) @ X3))) => (~((![X3:$i]:(((rel_s4 @ X1) @ X3) => (((member @ X2) @ eigen__4) @ X3)))))))))) => (![X2:$i]:(((rel_s4 @ X1) @ X2) => (((intersect @ eigen__2) @ eigen__4) @ X2))))))))),introduced(definition,[new_symbols(definition,[sP15])])). % 4.52/4.72 thf(sP16,plain,sP16 <=> ((rel_s4 @ eigen__1) @ eigen__3),introduced(definition,[new_symbols(definition,[sP16])])). % 4.52/4.72 thf(sP17,plain,sP17 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__3) => (~(((![X2:$i]:(((rel_s4 @ eigen__3) @ X2) => ((![X3:$i]:(((rel_s4 @ X2) @ X3) => (((intersect @ eigen__2) @ X1) @ X3))) => (~((![X3:mu]:(((exists_in_world @ X3) @ X2) => ((![X4:$i]:(((rel_s4 @ X2) @ X4) => (((member @ X3) @ eigen__2) @ X4))) => (~((![X4:$i]:(((rel_s4 @ X2) @ X4) => (((member @ X3) @ X1) @ X4))))))))))))) => (~((![X2:$i]:(((rel_s4 @ eigen__3) @ X2) => ((~((![X3:mu]:(((exists_in_world @ X3) @ X2) => ((![X4:$i]:(((rel_s4 @ X2) @ X4) => (((member @ X3) @ eigen__2) @ X4))) => (~((![X4:$i]:(((rel_s4 @ X2) @ X4) => (((member @ X3) @ X1) @ X4)))))))))) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => (((intersect @ eigen__2) @ X1) @ X3))))))))))))),introduced(definition,[new_symbols(definition,[sP17])])). % 4.52/4.72 thf(sP18,plain,sP18 <=> (![X1:$i]:(![X2:$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) => ((![X7:$i]:(((rel_s4 @ X6) @ X7) => (((intersect @ X3) @ X5) @ X7))) => (~((![X7:mu]:(((exists_in_world @ X7) @ X6) => ((![X8:$i]:(((rel_s4 @ X6) @ X8) => (((member @ X7) @ X3) @ X8))) => (~((![X8:$i]:(((rel_s4 @ X6) @ X8) => (((member @ X7) @ X5) @ X8)))))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP18])])). % 4.52/4.72 thf(sP19,plain,sP19 <=> (![X1:$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) => ((![X6:$i]:(((rel_s4 @ X5) @ X6) => (((intersect @ X2) @ X4) @ X6))) => (~((![X6:mu]:(((exists_in_world @ X6) @ X5) => ((![X7:$i]:(((rel_s4 @ X5) @ X7) => (((member @ X6) @ X2) @ X7))) => (~((![X7:$i]:(((rel_s4 @ X5) @ X7) => (((member @ X6) @ X4) @ X7))))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP19])])). % 4.52/4.72 thf(sP20,plain,sP20 <=> (sP11 => (![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => (![X2:mu]:(((exists_in_world @ X2) @ X1) => (![X3:$i]:(((rel_s4 @ X1) @ X3) => ((![X4:$i]:(((rel_s4 @ X3) @ X4) => (((intersect @ eigen__2) @ X2) @ X4))) => (~((![X4:mu]:(((exists_in_world @ X4) @ X3) => ((![X5:$i]:(((rel_s4 @ X3) @ X5) => (((member @ X4) @ eigen__2) @ X5))) => (~((![X5:$i]:(((rel_s4 @ X3) @ X5) => (((member @ X4) @ X2) @ X5)))))))))))))))))),introduced(definition,[new_symbols(definition,[sP20])])). % 4.52/4.72 thf(sP21,plain,sP21 <=> (sP16 => sP8),introduced(definition,[new_symbols(definition,[sP21])])). % 4.52/4.72 thf(sP22,plain,sP22 <=> (![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => (![X2:mu]:(((exists_in_world @ X2) @ X1) => (![X3:$i]:(((rel_s4 @ X1) @ X3) => ((![X4:$i]:(((rel_s4 @ X3) @ X4) => (((intersect @ eigen__2) @ X2) @ X4))) => (~((![X4:mu]:(((exists_in_world @ X4) @ X3) => ((![X5:$i]:(((rel_s4 @ X3) @ X5) => (((member @ X4) @ eigen__2) @ X5))) => (~((![X5:$i]:(((rel_s4 @ X3) @ X5) => (((member @ X4) @ X2) @ X5))))))))))))))))),introduced(definition,[new_symbols(definition,[sP22])])). % 4.52/4.72 thf(sP23,plain,sP23 <=> (![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => (![X2:mu]:(((exists_in_world @ X2) @ X1) => (~(((![X3:$i]:(((rel_s4 @ X1) @ X3) => ((![X4:$i]:(((rel_s4 @ X3) @ X4) => (((intersect @ eigen__2) @ X2) @ X4))) => (~((![X4:mu]:(((exists_in_world @ X4) @ X3) => ((![X5:$i]:(((rel_s4 @ X3) @ X5) => (((member @ X4) @ eigen__2) @ X5))) => (~((![X5:$i]:(((rel_s4 @ X3) @ X5) => (((member @ X4) @ X2) @ X5))))))))))))) => (~((![X3:$i]:(((rel_s4 @ X1) @ X3) => ((~((![X4:mu]:(((exists_in_world @ X4) @ X3) => ((![X5:$i]:(((rel_s4 @ X3) @ X5) => (((member @ X4) @ eigen__2) @ X5))) => (~((![X5:$i]:(((rel_s4 @ X3) @ X5) => (((member @ X4) @ X2) @ X5)))))))))) => (![X4:$i]:(((rel_s4 @ X3) @ X4) => (((intersect @ eigen__2) @ X2) @ X4))))))))))))))),introduced(definition,[new_symbols(definition,[sP23])])). % 4.52/4.72 thf(sP24,plain,sP24 <=> (sP9 => sP1),introduced(definition,[new_symbols(definition,[sP24])])). % 4.52/4.72 thf(def_meq_prop,definition,(meq_prop = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((X1 @ X3) = (X2 @ X3))))))). % 4.52/4.72 thf(def_mnot,definition,(mnot = (^[X1:$i>$o]:(^[X2:$i]:(~((X1 @ X2))))))). % 4.52/4.72 thf(def_mor,definition,(mor = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((~((X1 @ X3))) => (X2 @ X3))))))). % 4.52/4.72 thf(def_mbox,definition,(mbox = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:(![X4:$i]:(((X1 @ X3) @ X4) => (X2 @ X4)))))))). % 4.52/4.72 thf(def_mforall_prop,definition,(mforall_prop = (^[X1:($i>$o)>$i>$o]:(^[X2:$i]:(![X3:$i>$o]:((X1 @ X3) @ X2)))))). % 4.52/4.72 thf(def_mtrue,definition,(mtrue = (^[X1:$i]:(~($false))))). % 4.52/4.72 thf(def_mfalse,definition,(mfalse = (mnot @ mtrue))). % 4.52/4.72 thf(def_mand,definition,(mand = (^[X1:$i>$o]:(^[X2:$i>$o]:(mnot @ ((mor @ (mnot @ X1)) @ (mnot @ X2))))))). % 4.52/4.72 thf(def_mimplies,definition,(mimplies = (^[X1:$i>$o]:(mor @ (mnot @ X1))))). % 4.52/4.72 thf(def_mimplied,definition,(mimplied = (^[X1:$i>$o]:(^[X2:$i>$o]:((mor @ (mnot @ X2)) @ X1))))). % 4.52/4.72 thf(def_mequiv,definition,(mequiv = (^[X1:$i>$o]:(^[X2:$i>$o]:((mand @ ((mimplies @ X1) @ X2)) @ ((mimplies @ X2) @ X1)))))). % 4.52/4.72 thf(def_mxor,definition,(mxor = (^[X1:$i>$o]:(^[X2:$i>$o]:(mnot @ ((mequiv @ X1) @ X2)))))). % 4.52/4.72 thf(def_mdia,definition,(mdia = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(mnot @ ((mbox @ X1) @ (mnot @ X2))))))). % 4.52/4.72 thf(def_mforall_ind,definition,(mforall_ind = (^[X1:mu>$i>$o]:(^[X2:$i]:(![X3:mu]:(((exists_in_world @ X3) @ X2) => ((X1 @ X3) @ X2))))))). % 4.52/4.72 thf(def_mexists_ind,definition,(mexists_ind = (^[X1:mu>$i>$o]:(mnot @ (mforall_ind @ (^[X2:mu]:(mnot @ (X1 @ X2)))))))). % 4.52/4.72 thf(def_mexists_prop,definition,(mexists_prop = (^[X1:($i>$o)>$i>$o]:(mnot @ (mforall_prop @ (^[X2:$i>$o]:(mnot @ (X1 @ X2)))))))). % 4.52/4.72 thf(def_mreflexive,definition,(mreflexive = (^[X1:$i>$i>$o]:(![X2:$i]:((X1 @ X2) @ X2))))). % 4.52/4.72 thf(def_msymmetric,definition,(msymmetric = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(((X1 @ X2) @ X3) => ((X1 @ X3) @ X2))))))). % 4.52/4.72 thf(def_mserial,definition,(mserial = (^[X1:$i>$i>$o]:(![X2:$i]:(~((![X3:$i]:(~(((X1 @ X2) @ X3)))))))))). % 4.52/4.72 thf(def_mtransitive,definition,(mtransitive = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X3) @ X4)))))) => ((X1 @ X2) @ X4)))))))). % 4.52/4.72 thf(def_meuclidean,definition,(meuclidean = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => ((X1 @ X3) @ X4)))))))). % 4.52/4.72 thf(def_mpartially_functional,definition,(mpartially_functional = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => (X3 = X4)))))))). % 4.52/4.72 thf(def_mfunctional,definition,(mfunctional = (^[X1:$i>$i>$o]:(![X2:$i]:(~((![X3:$i]:(((X1 @ X2) @ X3) => (~((![X4:$i]:(((X1 @ X2) @ X4) => (X3 = X4))))))))))))). % 4.52/4.72 thf(def_mweakly_dense,definition,(mweakly_dense = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((X1 @ X2) @ X3) => (~((![X5:$i]:(((X1 @ X2) @ X5) => (~(((X1 @ X5) @ X3)))))))))))))). % 4.52/4.72 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))))))))). % 4.52/4.72 thf(def_mweakly_directed,definition,(mweakly_directed = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => (~((![X5:$i]:(((X1 @ X3) @ X5) => (~(((X1 @ X4) @ X5)))))))))))))). % 4.52/4.72 thf(def_mvalid,definition,(mvalid = (!!))). % 4.52/4.72 thf(def_msatisfiable,definition,(msatisfiable = (^[X1:$i>$o]:(~((![X2:$i]:(~((X1 @ X2))))))))). % 4.52/4.72 thf(def_mcountersatisfiable,definition,(mcountersatisfiable = (^[X1:$i>$o]:(~(((!!) @ X1)))))). % 4.52/4.72 thf(def_minvalid,definition,(minvalid = (^[X1:$i>$o]:(![X2:$i]:(~((X1 @ X2))))))). % 4.52/4.72 thf(def_mbox_s4,definition,(mbox_s4 = (^[X1:$i>$o]:(^[X2:$i]:(![X3:$i]:(((rel_s4 @ X2) @ X3) => (X1 @ X3))))))). % 4.52/4.72 thf(def_mdia_s4,definition,(mdia_s4 = (^[X1:$i>$o]:(mnot @ (mbox_s4 @ (mnot @ X1)))))). % 4.52/4.72 thf(prove_th15,conjecture,(![X1:$i]:(![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:$i]:(((rel_s4 @ X4) @ X6) => ((~((~((![X7:$i]:(((rel_s4 @ X6) @ X7) => (((intersect @ X3) @ X5) @ X7))))))) => (~((![X7:mu]:(((exists_in_world @ X7) @ X6) => (~((~(((~((~((![X8:$i]:(((rel_s4 @ X6) @ X8) => (((member @ X7) @ X3) @ X8))))))) => (~((![X8:$i]:(((rel_s4 @ X6) @ X8) => (((member @ X7) @ X5) @ X8))))))))))))))))))))))))))). % 4.52/4.72 thf(h2,negated_conjecture,(~(sP18)),inference(assume_negation,[status(cth)],[prove_th15])). % 4.52/4.72 thf(1,plain,(sP15 | sP1),inference(prop_rule,[status(thm)],[])). % 4.52/4.72 thf(2,plain,(~(sP17) | sP3),inference(all_rule,[status(thm)],[])). % 4.52/4.72 thf(3,plain,((~(sP3) | ~(sP9)) | ~(sP15)),inference(prop_rule,[status(thm)],[])). % 4.52/4.72 thf(4,plain,(~(sP23) | sP7),inference(all_rule,[status(thm)],[])). % 4.52/4.72 thf(5,plain,((~(sP7) | ~(sP16)) | sP17),inference(prop_rule,[status(thm)],[])). % 4.52/4.72 thf(6,plain,(~(sP6) | sP10),inference(all_rule,[status(thm)],[])). % 4.52/4.72 thf(7,plain,(~(sP10) | sP2),inference(all_rule,[status(thm)],[])). % 4.52/4.72 thf(8,plain,((~(sP2) | ~(sP4)) | sP13),inference(prop_rule,[status(thm)],[])). % 4.52/4.72 thf(9,plain,(~(sP13) | sP5),inference(all_rule,[status(thm)],[])). % 4.52/4.72 thf(10,plain,((~(sP5) | ~(sP11)) | sP23),inference(prop_rule,[status(thm)],[])). % 4.52/4.72 thf(11,plain,(sP24 | ~(sP1)),inference(prop_rule,[status(thm)],[])). % 4.52/4.72 thf(12,plain,(sP24 | sP9),inference(prop_rule,[status(thm)],[])). % 4.52/4.72 thf(13,plain,(sP8 | ~(sP24)),inference(eigen_choice_rule,[status(thm),assumptions([h1])],[h1,eigendef_eigen__4])). % 4.52/4.72 thf(14,plain,(sP21 | ~(sP8)),inference(prop_rule,[status(thm)],[])). % 4.52/4.72 thf(15,plain,(sP21 | sP16),inference(prop_rule,[status(thm)],[])). % 4.52/4.72 thf(16,plain,(sP22 | ~(sP21)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__3])). % 4.52/4.72 thf(17,plain,(sP20 | ~(sP22)),inference(prop_rule,[status(thm)],[])). % 4.52/4.72 thf(18,plain,(sP20 | sP11),inference(prop_rule,[status(thm)],[])). % 4.52/4.72 thf(19,plain,(sP14 | ~(sP20)),inference(eigen_choice_rule,[status(thm),assumptions([h1])],[h1,eigendef_eigen__2])). % 4.52/4.72 thf(20,plain,(sP12 | ~(sP14)),inference(prop_rule,[status(thm)],[])). % 4.52/4.72 thf(21,plain,(sP12 | sP4),inference(prop_rule,[status(thm)],[])). % 4.52/4.72 thf(22,plain,(sP19 | ~(sP12)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__1])). % 4.52/4.72 thf(23,plain,(sP18 | ~(sP19)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__0])). % 4.52/4.72 thf(intersect_defn,axiom,(mvalid @ (mbox_s4 @ (mforall_ind @ (^[X1:mu]:(mbox_s4 @ (mforall_ind @ (^[X2:mu]:((mand @ (mbox_s4 @ ((mimplies @ (mbox_s4 @ ((intersect @ X1) @ X2))) @ (mexists_ind @ (^[X3:mu]:((mand @ (mbox_s4 @ ((member @ X3) @ X1))) @ (mbox_s4 @ ((member @ X3) @ X2)))))))) @ (mbox_s4 @ ((mimplies @ (mexists_ind @ (^[X3:mu]:((mand @ (mbox_s4 @ ((member @ X3) @ X1))) @ (mbox_s4 @ ((member @ X3) @ X2)))))) @ (mbox_s4 @ ((intersect @ X1) @ X2))))))))))))). % 4.52/4.72 thf(24,plain,sP6,inference(preprocess,[status(thm)],[intersect_defn]). % 4.52/4.72 thf(25,plain,$false,inference(prop_unsat,[status(thm),assumptions([h2,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,h2])). % 4.52/4.72 thf(26,plain,$false,inference(eigenvar_choice,[status(thm),assumptions([h2,h0]),eigenvar_choice(discharge,[h1])],[25,h1])). % 4.52/4.72 thf(27,plain,$false,inference(eigenvar_choice,[status(thm),assumptions([h2]),eigenvar_choice(discharge,[h0])],[26,h0])). % 4.52/4.72 thf(0,theorem,(![X1:$i]:(![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:$i]:(((rel_s4 @ X4) @ X6) => ((~((~((![X7:$i]:(((rel_s4 @ X6) @ X7) => (((intersect @ X3) @ X5) @ X7))))))) => (~((![X7:mu]:(((exists_in_world @ X7) @ X6) => (~((~(((~((~((![X8:$i]:(((rel_s4 @ X6) @ X8) => (((member @ X7) @ X3) @ X8))))))) => (~((![X8:$i]:(((rel_s4 @ X6) @ X8) => (((member @ X7) @ X5) @ X8)))))))))))))))))))))))))),inference(contra,[status(thm),contra(discharge,[h2])],[25,h2])). % 4.52/4.72 % SZS output end Proof %------------------------------------------------------------------------------