TSTP Solution File: SET045^7 by Satallax---3.5
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- Process Solution
%------------------------------------------------------------------------------
% File : Satallax---3.5
% Problem : SET045^7 : TPTP v8.1.0. Released v5.5.0.
% Transfm : none
% Format : tptp:raw
% Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% Computer : n026.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 : 600s
% DateTime : Tue Jul 19 04:50:18 EDT 2022
% Result : Theorem 11.73s 12.12s
% Output : Proof 11.73s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.12 % Problem : SET045^7 : TPTP v8.1.0. Released v5.5.0.
% 0.10/0.13 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% 0.12/0.34 % Computer : n026.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 : 600
% 0.12/0.34 % DateTime : Mon Jul 11 06:41:12 EDT 2022
% 0.12/0.34 % CPUTime :
% 11.73/12.12 % SZS status Theorem
% 11.73/12.12 % Mode: mode515
% 11.73/12.12 % Inferences: 3848
% 11.73/12.12 % SZS output start Proof
% 11.73/12.12 thf(ty_mu, type, mu : $tType).
% 11.73/12.12 thf(ty_rel_s4, type, rel_s4 : ($i>$i>$o)).
% 11.73/12.12 thf(ty_element, type, element : (mu>mu>$i>$o)).
% 11.73/12.12 thf(ty_eigen__2, type, eigen__2 : mu).
% 11.73/12.12 thf(ty_eigen__1, type, eigen__1 : $i).
% 11.73/12.12 thf(ty_eigen__0, type, eigen__0 : $i).
% 11.73/12.12 thf(ty_eigen__5, type, eigen__5 : $i).
% 11.73/12.12 thf(ty_eigen__3, type, eigen__3 : mu).
% 11.73/12.12 thf(ty_exists_in_world, type, exists_in_world : (mu>$i>$o)).
% 11.73/12.12 thf(h0, assumption, (![X1:mu>$o]:(![X2:mu]:((X1 @ X2) => (X1 @ (eps__0 @ X1))))),introduced(assumption,[])).
% 11.73/12.12 thf(eigendef_eigen__3, definition, eigen__3 = (eps__0 @ (^[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:$i]:(((rel_s4 @ X4) @ X5) => (((element @ X3) @ X1) @ X5))) => (~(((![X5:$i]:(((rel_s4 @ X4) @ X5) => (((element @ X3) @ eigen__2) @ X5))) => (~((![X5:$i]:(((rel_s4 @ X4) @ X5) => (~((![X6:$i]:(((rel_s4 @ X5) @ X6) => (((element @ X3) @ X3) @ X6))))))))))))))) => (~((![X4:$i]:(((rel_s4 @ X2) @ X4) => ((~(((![X5:$i]:(((rel_s4 @ X4) @ X5) => (((element @ X3) @ eigen__2) @ X5))) => (~((![X5:$i]:(((rel_s4 @ X4) @ X5) => (~((![X6:$i]:(((rel_s4 @ X5) @ X6) => (((element @ X3) @ X3) @ X6)))))))))))) => (![X5:$i]:(((rel_s4 @ X4) @ X5) => (((element @ X3) @ X1) @ X5)))))))))))))))))))))), introduced(definition,[new_symbols(definition,[eigen__3])])).
% 11.73/12.12 thf(h1, assumption, (![X1:$i>$o]:(![X2:$i]:((X1 @ X2) => (X1 @ (eps__1 @ X1))))),introduced(assumption,[])).
% 11.73/12.12 thf(eigendef_eigen__5, definition, eigen__5 = (eps__1 @ (^[X1:$i]:(~((((rel_s4 @ eigen__1) @ X1) => (~((![X2:$i]:(((rel_s4 @ X1) @ X2) => (((element @ eigen__3) @ eigen__3) @ X2)))))))))), introduced(definition,[new_symbols(definition,[eigen__5])])).
% 11.73/12.12 thf(sP1,plain,sP1 <=> ((exists_in_world @ eigen__2) @ eigen__1),introduced(definition,[new_symbols(definition,[sP1])])).
% 11.73/12.12 thf(sP2,plain,sP2 <=> (((exists_in_world @ eigen__3) @ eigen__1) => (~((![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => (![X2:mu]:(((exists_in_world @ X2) @ X1) => (~(((![X3:$i]:(((rel_s4 @ X1) @ X3) => ((![X4:$i]:(((rel_s4 @ X3) @ X4) => (((element @ X2) @ eigen__3) @ X4))) => (~(((![X4:$i]:(((rel_s4 @ X3) @ X4) => (((element @ X2) @ eigen__2) @ X4))) => (~((![X4:$i]:(((rel_s4 @ X3) @ X4) => (~((![X5:$i]:(((rel_s4 @ X4) @ X5) => (((element @ X2) @ X2) @ X5))))))))))))))) => (~((![X3:$i]:(((rel_s4 @ X1) @ X3) => ((~(((![X4:$i]:(((rel_s4 @ X3) @ X4) => (((element @ X2) @ eigen__2) @ X4))) => (~((![X4:$i]:(((rel_s4 @ X3) @ X4) => (~((![X5:$i]:(((rel_s4 @ X4) @ X5) => (((element @ X2) @ X2) @ X5)))))))))))) => (![X4:$i]:(((rel_s4 @ X3) @ X4) => (((element @ X2) @ eigen__3) @ X4)))))))))))))))))),introduced(definition,[new_symbols(definition,[sP2])])).
% 11.73/12.12 thf(sP3,plain,sP3 <=> ((![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => (((element @ eigen__3) @ eigen__2) @ X1))) => (~((![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => (~((![X2:$i]:(((rel_s4 @ X1) @ X2) => (((element @ eigen__3) @ eigen__3) @ X2)))))))))),introduced(definition,[new_symbols(definition,[sP3])])).
% 11.73/12.12 thf(sP4,plain,sP4 <=> (((exists_in_world @ eigen__3) @ eigen__1) => (![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => (((element @ eigen__3) @ eigen__2) @ X1)))),introduced(definition,[new_symbols(definition,[sP4])])).
% 11.73/12.12 thf(sP5,plain,sP5 <=> (![X1:$i]:(((rel_s4 @ eigen__5) @ X1) => (((element @ eigen__3) @ eigen__3) @ X1))),introduced(definition,[new_symbols(definition,[sP5])])).
% 11.73/12.12 thf(sP6,plain,sP6 <=> (((rel_s4 @ eigen__1) @ eigen__5) => (sP5 => (~(((![X1:$i]:(((rel_s4 @ eigen__5) @ X1) => (((element @ eigen__3) @ eigen__2) @ X1))) => (~((![X1:$i]:(((rel_s4 @ eigen__5) @ X1) => (~((![X2:$i]:(((rel_s4 @ X1) @ X2) => (((element @ eigen__3) @ eigen__3) @ X2)))))))))))))),introduced(definition,[new_symbols(definition,[sP6])])).
% 11.73/12.12 thf(sP7,plain,sP7 <=> (![X1:$i]:((rel_s4 @ X1) @ X1)),introduced(definition,[new_symbols(definition,[sP7])])).
% 11.73/12.12 thf(sP8,plain,sP8 <=> (((rel_s4 @ eigen__1) @ eigen__1) => (~((![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => (((element @ eigen__3) @ eigen__3) @ X1)))))),introduced(definition,[new_symbols(definition,[sP8])])).
% 11.73/12.12 thf(sP9,plain,sP9 <=> (![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => (~((![X2:$i]:(((rel_s4 @ X1) @ X2) => (((element @ eigen__3) @ eigen__3) @ X2))))))),introduced(definition,[new_symbols(definition,[sP9])])).
% 11.73/12.12 thf(sP10,plain,sP10 <=> (![X1:$i]:(((rel_s4 @ eigen__5) @ X1) => (~((![X2:$i]:(((rel_s4 @ X1) @ X2) => (((element @ eigen__3) @ eigen__3) @ X2))))))),introduced(definition,[new_symbols(definition,[sP10])])).
% 11.73/12.12 thf(sP11,plain,sP11 <=> (![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => (((element @ eigen__3) @ eigen__3) @ X1))),introduced(definition,[new_symbols(definition,[sP11])])).
% 11.73/12.12 thf(sP12,plain,sP12 <=> (((rel_s4 @ eigen__1) @ eigen__1) => (![X1:mu]:(((exists_in_world @ X1) @ eigen__1) => (~(((![X2:$i]:(((rel_s4 @ eigen__1) @ X2) => ((![X3:$i]:(((rel_s4 @ X2) @ X3) => (((element @ X1) @ eigen__3) @ X3))) => (~(((![X3:$i]:(((rel_s4 @ X2) @ X3) => (((element @ X1) @ eigen__2) @ X3))) => (~((![X3:$i]:(((rel_s4 @ X2) @ X3) => (~((![X4:$i]:(((rel_s4 @ X3) @ X4) => (((element @ X1) @ X1) @ X4))))))))))))))) => (~((![X2:$i]:(((rel_s4 @ eigen__1) @ X2) => ((~(((![X3:$i]:(((rel_s4 @ X2) @ X3) => (((element @ X1) @ eigen__2) @ X3))) => (~((![X3:$i]:(((rel_s4 @ X2) @ X3) => (~((![X4:$i]:(((rel_s4 @ X3) @ X4) => (((element @ X1) @ X1) @ X4)))))))))))) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => (((element @ X1) @ eigen__3) @ X3)))))))))))))),introduced(definition,[new_symbols(definition,[sP12])])).
% 11.73/12.12 thf(sP13,plain,sP13 <=> (![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => ((![X2:$i]:(((rel_s4 @ X1) @ X2) => (((element @ eigen__3) @ eigen__3) @ X2))) => (~(((![X2:$i]:(((rel_s4 @ X1) @ X2) => (((element @ eigen__3) @ eigen__2) @ X2))) => (~((![X2:$i]:(((rel_s4 @ X1) @ X2) => (~((![X3:$i]:(((rel_s4 @ X2) @ X3) => (((element @ eigen__3) @ eigen__3) @ X3))))))))))))))),introduced(definition,[new_symbols(definition,[sP13])])).
% 11.73/12.12 thf(sP14,plain,sP14 <=> ((~(sP3)) => sP11),introduced(definition,[new_symbols(definition,[sP14])])).
% 11.73/12.12 thf(sP15,plain,sP15 <=> (![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => (((element @ eigen__3) @ eigen__2) @ X1))),introduced(definition,[new_symbols(definition,[sP15])])).
% 11.73/12.12 thf(sP16,plain,sP16 <=> ((![X1:$i]:(((rel_s4 @ eigen__5) @ X1) => (((element @ eigen__3) @ eigen__2) @ X1))) => (~(sP10))),introduced(definition,[new_symbols(definition,[sP16])])).
% 11.73/12.12 thf(sP17,plain,sP17 <=> ((rel_s4 @ eigen__1) @ eigen__5),introduced(definition,[new_symbols(definition,[sP17])])).
% 11.73/12.12 thf(sP18,plain,sP18 <=> (![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => ((~(((![X2:$i]:(((rel_s4 @ X1) @ X2) => (((element @ eigen__3) @ eigen__2) @ X2))) => (~((![X2:$i]:(((rel_s4 @ X1) @ X2) => (~((![X3:$i]:(((rel_s4 @ X2) @ X3) => (((element @ eigen__3) @ eigen__3) @ X3)))))))))))) => (![X2:$i]:(((rel_s4 @ X1) @ X2) => (((element @ eigen__3) @ eigen__3) @ X2)))))),introduced(definition,[new_symbols(definition,[sP18])])).
% 11.73/12.12 thf(sP19,plain,sP19 <=> (![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) => (![X5:mu]:(((exists_in_world @ X5) @ X4) => (~(((![X6:$i]:(((rel_s4 @ X4) @ X6) => ((![X7:$i]:(((rel_s4 @ X6) @ X7) => (((element @ X5) @ X3) @ X7))) => (~(((![X7:$i]:(((rel_s4 @ X6) @ X7) => (((element @ X5) @ X2) @ X7))) => (~((![X7:$i]:(((rel_s4 @ X6) @ X7) => (~((![X8:$i]:(((rel_s4 @ X7) @ X8) => (((element @ X5) @ X5) @ X8))))))))))))))) => (~((![X6:$i]:(((rel_s4 @ X4) @ X6) => ((~(((![X7:$i]:(((rel_s4 @ X6) @ X7) => (((element @ X5) @ X2) @ X7))) => (~((![X7:$i]:(((rel_s4 @ X6) @ X7) => (~((![X8:$i]:(((rel_s4 @ X7) @ X8) => (((element @ X5) @ X5) @ X8)))))))))))) => (![X7:$i]:(((rel_s4 @ X6) @ X7) => (((element @ X5) @ X3) @ X7))))))))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP19])])).
% 11.73/12.12 thf(sP20,plain,sP20 <=> (sP1 => (~((![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:$i]:(((rel_s4 @ X4) @ X5) => (((element @ X3) @ X1) @ X5))) => (~(((![X5:$i]:(((rel_s4 @ X4) @ X5) => (((element @ X3) @ eigen__2) @ X5))) => (~((![X5:$i]:(((rel_s4 @ X4) @ X5) => (~((![X6:$i]:(((rel_s4 @ X5) @ X6) => (((element @ X3) @ X3) @ X6))))))))))))))) => (~((![X4:$i]:(((rel_s4 @ X2) @ X4) => ((~(((![X5:$i]:(((rel_s4 @ X4) @ X5) => (((element @ X3) @ eigen__2) @ X5))) => (~((![X5:$i]:(((rel_s4 @ X4) @ X5) => (~((![X6:$i]:(((rel_s4 @ X5) @ X6) => (((element @ X3) @ X3) @ X6)))))))))))) => (![X5:$i]:(((rel_s4 @ X4) @ X5) => (((element @ X3) @ X1) @ X5)))))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP20])])).
% 11.73/12.12 thf(sP21,plain,sP21 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__1) => (~(((![X2:$i]:(((rel_s4 @ eigen__1) @ X2) => ((![X3:$i]:(((rel_s4 @ X2) @ X3) => (((element @ X1) @ eigen__3) @ X3))) => (~(((![X3:$i]:(((rel_s4 @ X2) @ X3) => (((element @ X1) @ eigen__2) @ X3))) => (~((![X3:$i]:(((rel_s4 @ X2) @ X3) => (~((![X4:$i]:(((rel_s4 @ X3) @ X4) => (((element @ X1) @ X1) @ X4))))))))))))))) => (~((![X2:$i]:(((rel_s4 @ eigen__1) @ X2) => ((~(((![X3:$i]:(((rel_s4 @ X2) @ X3) => (((element @ X1) @ eigen__2) @ X3))) => (~((![X3:$i]:(((rel_s4 @ X2) @ X3) => (~((![X4:$i]:(((rel_s4 @ X3) @ X4) => (((element @ X1) @ X1) @ X4)))))))))))) => (![X3:$i]:(((rel_s4 @ X2) @ X3) => (((element @ X1) @ eigen__3) @ X3))))))))))))),introduced(definition,[new_symbols(definition,[sP21])])).
% 11.73/12.12 thf(sP22,plain,sP22 <=> (((rel_s4 @ eigen__5) @ eigen__5) => (~(sP5))),introduced(definition,[new_symbols(definition,[sP22])])).
% 11.73/12.12 thf(sP23,plain,sP23 <=> (((rel_s4 @ eigen__1) @ eigen__1) => (![X1:mu]:(((exists_in_world @ X1) @ eigen__1) => (![X2:$i]:(((rel_s4 @ eigen__1) @ X2) => (((element @ X1) @ eigen__2) @ X2)))))),introduced(definition,[new_symbols(definition,[sP23])])).
% 11.73/12.12 thf(sP24,plain,sP24 <=> ((rel_s4 @ eigen__1) @ eigen__1),introduced(definition,[new_symbols(definition,[sP24])])).
% 11.73/12.12 thf(sP25,plain,sP25 <=> (sP5 => (~(sP16))),introduced(definition,[new_symbols(definition,[sP25])])).
% 11.73/12.12 thf(sP26,plain,sP26 <=> (((rel_s4 @ eigen__0) @ eigen__1) => (![X1:mu]:(((exists_in_world @ X1) @ eigen__1) => (~((![X2:mu]:(((exists_in_world @ X2) @ eigen__1) => (~((![X3:$i]:(((rel_s4 @ eigen__1) @ X3) => (![X4:mu]:(((exists_in_world @ X4) @ X3) => (~(((![X5:$i]:(((rel_s4 @ X3) @ X5) => ((![X6:$i]:(((rel_s4 @ X5) @ X6) => (((element @ X4) @ X2) @ X6))) => (~(((![X6:$i]:(((rel_s4 @ X5) @ X6) => (((element @ X4) @ X1) @ X6))) => (~((![X6:$i]:(((rel_s4 @ X5) @ X6) => (~((![X7:$i]:(((rel_s4 @ X6) @ X7) => (((element @ X4) @ X4) @ X7))))))))))))))) => (~((![X5:$i]:(((rel_s4 @ X3) @ X5) => ((~(((![X6:$i]:(((rel_s4 @ X5) @ X6) => (((element @ X4) @ X1) @ X6))) => (~((![X6:$i]:(((rel_s4 @ X5) @ X6) => (~((![X7:$i]:(((rel_s4 @ X6) @ X7) => (((element @ X4) @ X4) @ X7)))))))))))) => (![X6:$i]:(((rel_s4 @ X5) @ X6) => (((element @ X4) @ X2) @ X6)))))))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP26])])).
% 11.73/12.12 thf(sP27,plain,sP27 <=> ((exists_in_world @ eigen__3) @ eigen__1),introduced(definition,[new_symbols(definition,[sP27])])).
% 11.73/12.12 thf(sP28,plain,sP28 <=> ((rel_s4 @ eigen__0) @ eigen__1),introduced(definition,[new_symbols(definition,[sP28])])).
% 11.73/12.12 thf(sP29,plain,sP29 <=> (![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => (![X2:mu]:(((exists_in_world @ X2) @ X1) => (![X3:$i]:(((rel_s4 @ X1) @ X3) => (((element @ X2) @ eigen__2) @ X3))))))),introduced(definition,[new_symbols(definition,[sP29])])).
% 11.73/12.12 thf(sP30,plain,sP30 <=> (sP24 => sP14),introduced(definition,[new_symbols(definition,[sP30])])).
% 11.73/12.12 thf(sP31,plain,sP31 <=> (sP17 => (~(sP5))),introduced(definition,[new_symbols(definition,[sP31])])).
% 11.73/12.12 thf(sP32,plain,sP32 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__1) => (![X2:$i]:(((rel_s4 @ eigen__1) @ X2) => (((element @ X1) @ eigen__2) @ X2))))),introduced(definition,[new_symbols(definition,[sP32])])).
% 11.73/12.12 thf(sP33,plain,sP33 <=> (sP27 => (~((sP13 => (~(sP18)))))),introduced(definition,[new_symbols(definition,[sP33])])).
% 11.73/12.12 thf(sP34,plain,sP34 <=> ((rel_s4 @ eigen__5) @ eigen__5),introduced(definition,[new_symbols(definition,[sP34])])).
% 11.73/12.12 thf(sP35,plain,sP35 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__1) => (~((![X2:mu]:(((exists_in_world @ X2) @ eigen__1) => (~((![X3:$i]:(((rel_s4 @ eigen__1) @ X3) => (![X4:mu]:(((exists_in_world @ X4) @ X3) => (~(((![X5:$i]:(((rel_s4 @ X3) @ X5) => ((![X6:$i]:(((rel_s4 @ X5) @ X6) => (((element @ X4) @ X2) @ X6))) => (~(((![X6:$i]:(((rel_s4 @ X5) @ X6) => (((element @ X4) @ X1) @ X6))) => (~((![X6:$i]:(((rel_s4 @ X5) @ X6) => (~((![X7:$i]:(((rel_s4 @ X6) @ X7) => (((element @ X4) @ X4) @ X7))))))))))))))) => (~((![X5:$i]:(((rel_s4 @ X3) @ X5) => ((~(((![X6:$i]:(((rel_s4 @ X5) @ X6) => (((element @ X4) @ X1) @ X6))) => (~((![X6:$i]:(((rel_s4 @ X5) @ X6) => (~((![X7:$i]:(((rel_s4 @ X6) @ X7) => (((element @ X4) @ X4) @ X7)))))))))))) => (![X6:$i]:(((rel_s4 @ X5) @ X6) => (((element @ X4) @ X2) @ X6))))))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP35])])).
% 11.73/12.12 thf(sP36,plain,sP36 <=> (sP13 => (~(sP18))),introduced(definition,[new_symbols(definition,[sP36])])).
% 11.73/12.12 thf(sP37,plain,sP37 <=> (![X1:$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) => (![X6:mu]:(((exists_in_world @ X6) @ X5) => (~(((![X7:$i]:(((rel_s4 @ X5) @ X7) => ((![X8:$i]:(((rel_s4 @ X7) @ X8) => (((element @ X6) @ X4) @ X8))) => (~(((![X8:$i]:(((rel_s4 @ X7) @ X8) => (((element @ X6) @ X3) @ X8))) => (~((![X8:$i]:(((rel_s4 @ X7) @ X8) => (~((![X9:$i]:(((rel_s4 @ X8) @ X9) => (((element @ X6) @ X6) @ X9))))))))))))))) => (~((![X7:$i]:(((rel_s4 @ X5) @ X7) => ((~(((![X8:$i]:(((rel_s4 @ X7) @ X8) => (((element @ X6) @ X3) @ X8))) => (~((![X8:$i]:(((rel_s4 @ X7) @ X8) => (~((![X9:$i]:(((rel_s4 @ X8) @ X9) => (((element @ X6) @ X6) @ X9)))))))))))) => (![X8:$i]:(((rel_s4 @ X7) @ X8) => (((element @ X6) @ X4) @ X8)))))))))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP37])])).
% 11.73/12.12 thf(sP38,plain,sP38 <=> (![X1:$i]:(((rel_s4 @ eigen__1) @ X1) => (![X2:mu]:(((exists_in_world @ X2) @ X1) => (~(((![X3:$i]:(((rel_s4 @ X1) @ X3) => ((![X4:$i]:(((rel_s4 @ X3) @ X4) => (((element @ X2) @ eigen__3) @ X4))) => (~(((![X4:$i]:(((rel_s4 @ X3) @ X4) => (((element @ X2) @ eigen__2) @ X4))) => (~((![X4:$i]:(((rel_s4 @ X3) @ X4) => (~((![X5:$i]:(((rel_s4 @ X4) @ X5) => (((element @ X2) @ X2) @ X5))))))))))))))) => (~((![X3:$i]:(((rel_s4 @ X1) @ X3) => ((~(((![X4:$i]:(((rel_s4 @ X3) @ X4) => (((element @ X2) @ eigen__2) @ X4))) => (~((![X4:$i]:(((rel_s4 @ X3) @ X4) => (~((![X5:$i]:(((rel_s4 @ X4) @ X5) => (((element @ X2) @ X2) @ X5)))))))))))) => (![X4:$i]:(((rel_s4 @ X3) @ X4) => (((element @ X2) @ eigen__3) @ X4))))))))))))))),introduced(definition,[new_symbols(definition,[sP38])])).
% 11.73/12.12 thf(sP39,plain,sP39 <=> (![X1:mu]:(((exists_in_world @ X1) @ eigen__1) => (~((![X2:$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) => (((element @ X3) @ X1) @ X5))) => (~(((![X5:$i]:(((rel_s4 @ X4) @ X5) => (((element @ X3) @ eigen__2) @ X5))) => (~((![X5:$i]:(((rel_s4 @ X4) @ X5) => (~((![X6:$i]:(((rel_s4 @ X5) @ X6) => (((element @ X3) @ X3) @ X6))))))))))))))) => (~((![X4:$i]:(((rel_s4 @ X2) @ X4) => ((~(((![X5:$i]:(((rel_s4 @ X4) @ X5) => (((element @ X3) @ eigen__2) @ X5))) => (~((![X5:$i]:(((rel_s4 @ X4) @ X5) => (~((![X6:$i]:(((rel_s4 @ X5) @ X6) => (((element @ X3) @ X3) @ X6)))))))))))) => (![X5:$i]:(((rel_s4 @ X4) @ X5) => (((element @ X3) @ X1) @ X5))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP39])])).
% 11.73/12.12 thf(def_meq_prop,definition,(meq_prop = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((X1 @ X3) = (X2 @ X3))))))).
% 11.73/12.12 thf(def_mnot,definition,(mnot = (^[X1:$i>$o]:(^[X2:$i]:(~((X1 @ X2))))))).
% 11.73/12.12 thf(def_mor,definition,(mor = (^[X1:$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:((~((X1 @ X3))) => (X2 @ X3))))))).
% 11.73/12.12 thf(def_mbox,definition,(mbox = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(^[X3:$i]:(![X4:$i]:(((X1 @ X3) @ X4) => (X2 @ X4)))))))).
% 11.73/12.12 thf(def_mforall_prop,definition,(mforall_prop = (^[X1:($i>$o)>$i>$o]:(^[X2:$i]:(![X3:$i>$o]:((X1 @ X3) @ X2)))))).
% 11.73/12.12 thf(def_mtrue,definition,(mtrue = (^[X1:$i]:(~($false))))).
% 11.73/12.12 thf(def_mfalse,definition,(mfalse = (mnot @ mtrue))).
% 11.73/12.12 thf(def_mand,definition,(mand = (^[X1:$i>$o]:(^[X2:$i>$o]:(mnot @ ((mor @ (mnot @ X1)) @ (mnot @ X2))))))).
% 11.73/12.12 thf(def_mimplies,definition,(mimplies = (^[X1:$i>$o]:(mor @ (mnot @ X1))))).
% 11.73/12.12 thf(def_mimplied,definition,(mimplied = (^[X1:$i>$o]:(^[X2:$i>$o]:((mor @ (mnot @ X2)) @ X1))))).
% 11.73/12.12 thf(def_mequiv,definition,(mequiv = (^[X1:$i>$o]:(^[X2:$i>$o]:((mand @ ((mimplies @ X1) @ X2)) @ ((mimplies @ X2) @ X1)))))).
% 11.73/12.12 thf(def_mxor,definition,(mxor = (^[X1:$i>$o]:(^[X2:$i>$o]:(mnot @ ((mequiv @ X1) @ X2)))))).
% 11.73/12.12 thf(def_mdia,definition,(mdia = (^[X1:$i>$i>$o]:(^[X2:$i>$o]:(mnot @ ((mbox @ X1) @ (mnot @ X2))))))).
% 11.73/12.12 thf(def_mforall_ind,definition,(mforall_ind = (^[X1:mu>$i>$o]:(^[X2:$i]:(![X3:mu]:(((exists_in_world @ X3) @ X2) => ((X1 @ X3) @ X2))))))).
% 11.73/12.12 thf(def_mexists_ind,definition,(mexists_ind = (^[X1:mu>$i>$o]:(mnot @ (mforall_ind @ (^[X2:mu]:(mnot @ (X1 @ X2)))))))).
% 11.73/12.12 thf(def_mexists_prop,definition,(mexists_prop = (^[X1:($i>$o)>$i>$o]:(mnot @ (mforall_prop @ (^[X2:$i>$o]:(mnot @ (X1 @ X2)))))))).
% 11.73/12.12 thf(def_mreflexive,definition,(mreflexive = (^[X1:$i>$i>$o]:(![X2:$i]:((X1 @ X2) @ X2))))).
% 11.73/12.12 thf(def_msymmetric,definition,(msymmetric = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(((X1 @ X2) @ X3) => ((X1 @ X3) @ X2))))))).
% 11.73/12.12 thf(def_mserial,definition,(mserial = (^[X1:$i>$i>$o]:(![X2:$i]:(~((![X3:$i]:(~(((X1 @ X2) @ X3)))))))))).
% 11.73/12.12 thf(def_mtransitive,definition,(mtransitive = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X3) @ X4)))))) => ((X1 @ X2) @ X4)))))))).
% 11.73/12.12 thf(def_meuclidean,definition,(meuclidean = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => ((X1 @ X3) @ X4)))))))).
% 11.73/12.12 thf(def_mpartially_functional,definition,(mpartially_functional = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => (X3 = X4)))))))).
% 11.73/12.12 thf(def_mfunctional,definition,(mfunctional = (^[X1:$i>$i>$o]:(![X2:$i]:(~((![X3:$i]:(((X1 @ X2) @ X3) => (~((![X4:$i]:(((X1 @ X2) @ X4) => (X3 = X4))))))))))))).
% 11.73/12.12 thf(def_mweakly_dense,definition,(mweakly_dense = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((X1 @ X2) @ X3) => (~((![X5:$i]:(((X1 @ X2) @ X5) => (~(((X1 @ X5) @ X3)))))))))))))).
% 11.73/12.12 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))))))))).
% 11.73/12.12 thf(def_mweakly_directed,definition,(mweakly_directed = (^[X1:$i>$i>$o]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((~((((X1 @ X2) @ X3) => (~(((X1 @ X2) @ X4)))))) => (~((![X5:$i]:(((X1 @ X3) @ X5) => (~(((X1 @ X4) @ X5)))))))))))))).
% 11.73/12.12 thf(def_mvalid,definition,(mvalid = (!!))).
% 11.73/12.12 thf(def_msatisfiable,definition,(msatisfiable = (^[X1:$i>$o]:(~((![X2:$i]:(~((X1 @ X2))))))))).
% 11.73/12.12 thf(def_mcountersatisfiable,definition,(mcountersatisfiable = (^[X1:$i>$o]:(~(((!!) @ X1)))))).
% 11.73/12.12 thf(def_minvalid,definition,(minvalid = (^[X1:$i>$o]:(![X2:$i]:(~((X1 @ X2))))))).
% 11.73/12.12 thf(def_mbox_s4,definition,(mbox_s4 = (^[X1:$i>$o]:(^[X2:$i]:(![X3:$i]:(((rel_s4 @ X2) @ X3) => (X1 @ X3))))))).
% 11.73/12.12 thf(def_mdia_s4,definition,(mdia_s4 = (^[X1:$i>$o]:(mnot @ (mbox_s4 @ (mnot @ X1)))))).
% 11.73/12.12 thf(pel41,conjecture,(![X1:$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:$i]:(((rel_s4 @ X4) @ X6) => (((element @ X5) @ X3) @ X6))))))))))))))))))).
% 11.73/12.12 thf(h2,negated_conjecture,(~((![X1:$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:$i]:(((rel_s4 @ X4) @ X6) => (((element @ X5) @ X3) @ X6)))))))))))))))),inference(assume_negation,[status(cth)],[pel41])).
% 11.73/12.12 thf(h3,assumption,(~((![X1:$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) => (((element @ X4) @ X2) @ X5))))))))))))))),introduced(assumption,[])).
% 11.73/12.12 thf(h4,assumption,(~((sP28 => (![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) => (((element @ X3) @ X1) @ X4)))))))))))))),introduced(assumption,[])).
% 11.73/12.12 thf(h5,assumption,sP28,introduced(assumption,[])).
% 11.73/12.12 thf(h6,assumption,(~((![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) => (((element @ X3) @ X1) @ X4))))))))))))),introduced(assumption,[])).
% 11.73/12.12 thf(h7,assumption,(~((sP1 => (~(sP29))))),introduced(assumption,[])).
% 11.73/12.12 thf(h8,assumption,sP1,introduced(assumption,[])).
% 11.73/12.12 thf(h9,assumption,sP29,introduced(assumption,[])).
% 11.73/12.12 thf(1,plain,(~(sP7) | sP24),inference(all_rule,[status(thm)],[])).
% 11.73/12.12 thf(2,plain,(~(sP7) | sP34),inference(all_rule,[status(thm)],[])).
% 11.73/12.12 thf(3,plain,(~(sP10) | sP22),inference(all_rule,[status(thm)],[])).
% 11.73/12.12 thf(4,plain,((~(sP22) | ~(sP34)) | ~(sP5)),inference(prop_rule,[status(thm)],[])).
% 11.73/12.12 thf(5,plain,(sP16 | sP10),inference(prop_rule,[status(thm)],[])).
% 11.73/12.12 thf(6,plain,(~(sP13) | sP6),inference(all_rule,[status(thm)],[])).
% 11.73/12.12 thf(7,plain,((~(sP6) | ~(sP17)) | sP25),inference(prop_rule,[status(thm)],[])).
% 11.73/12.12 thf(8,plain,((~(sP25) | ~(sP5)) | ~(sP16)),inference(prop_rule,[status(thm)],[])).
% 11.73/12.12 thf(9,plain,(sP31 | sP5),inference(prop_rule,[status(thm)],[])).
% 11.73/12.12 thf(10,plain,(sP31 | sP17),inference(prop_rule,[status(thm)],[])).
% 11.73/12.12 thf(11,plain,(~(sP9) | sP8),inference(all_rule,[status(thm)],[])).
% 11.73/12.12 thf(12,plain,((~(sP8) | ~(sP24)) | ~(sP11)),inference(prop_rule,[status(thm)],[])).
% 11.73/12.12 thf(13,plain,(sP9 | ~(sP31)),inference(eigen_choice_rule,[status(thm),assumptions([h1])],[h1,eigendef_eigen__5])).
% 11.73/12.12 thf(14,plain,(~(sP18) | sP30),inference(all_rule,[status(thm)],[])).
% 11.73/12.12 thf(15,plain,((~(sP30) | ~(sP24)) | sP14),inference(prop_rule,[status(thm)],[])).
% 11.73/12.12 thf(16,plain,((~(sP14) | sP3) | sP11),inference(prop_rule,[status(thm)],[])).
% 11.73/12.12 thf(17,plain,((~(sP3) | ~(sP15)) | ~(sP9)),inference(prop_rule,[status(thm)],[])).
% 11.73/12.12 thf(18,plain,(sP36 | sP18),inference(prop_rule,[status(thm)],[])).
% 11.73/12.12 thf(19,plain,(sP36 | sP13),inference(prop_rule,[status(thm)],[])).
% 11.73/12.12 thf(20,plain,(~(sP38) | sP12),inference(all_rule,[status(thm)],[])).
% 11.73/12.12 thf(21,plain,((~(sP12) | ~(sP24)) | sP21),inference(prop_rule,[status(thm)],[])).
% 11.73/12.12 thf(22,plain,(~(sP21) | sP33),inference(all_rule,[status(thm)],[])).
% 11.73/12.12 thf(23,plain,((~(sP33) | ~(sP27)) | ~(sP36)),inference(prop_rule,[status(thm)],[])).
% 11.73/12.12 thf(24,plain,(~(sP32) | sP4),inference(all_rule,[status(thm)],[])).
% 11.73/12.12 thf(25,plain,((~(sP4) | ~(sP27)) | sP15),inference(prop_rule,[status(thm)],[])).
% 11.73/12.12 thf(26,plain,(sP2 | sP38),inference(prop_rule,[status(thm)],[])).
% 11.73/12.12 thf(27,plain,(sP2 | sP27),inference(prop_rule,[status(thm)],[])).
% 11.73/12.12 thf(28,plain,(sP39 | ~(sP2)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__3])).
% 11.73/12.12 thf(29,plain,(~(sP35) | sP20),inference(all_rule,[status(thm)],[])).
% 11.73/12.12 thf(30,plain,((~(sP20) | ~(sP1)) | ~(sP39)),inference(prop_rule,[status(thm)],[])).
% 11.73/12.12 thf(31,plain,(~(sP37) | sP19),inference(all_rule,[status(thm)],[])).
% 11.73/12.12 thf(32,plain,(~(sP19) | sP26),inference(all_rule,[status(thm)],[])).
% 11.73/12.12 thf(33,plain,((~(sP26) | ~(sP28)) | sP35),inference(prop_rule,[status(thm)],[])).
% 11.73/12.12 thf(34,plain,(~(sP29) | sP23),inference(all_rule,[status(thm)],[])).
% 11.73/12.12 thf(35,plain,((~(sP23) | ~(sP24)) | sP32),inference(prop_rule,[status(thm)],[])).
% 11.73/12.12 thf(a1,axiom,(mreflexive @ rel_s4)).
% 11.73/12.12 thf(36,plain,sP7,inference(preprocess,[status(thm)],[a1]).
% 11.73/12.12 thf(pel41_1,axiom,(mvalid @ (mbox_s4 @ (mforall_ind @ (^[X1:mu]:(mexists_ind @ (^[X2:mu]:(mbox_s4 @ (mforall_ind @ (^[X3:mu]:((mand @ (mbox_s4 @ ((mimplies @ (mbox_s4 @ ((element @ X3) @ X2))) @ ((mand @ (mbox_s4 @ ((element @ X3) @ X1))) @ (mbox_s4 @ (mnot @ (mbox_s4 @ ((element @ X3) @ X3)))))))) @ (mbox_s4 @ ((mimplies @ ((mand @ (mbox_s4 @ ((element @ X3) @ X1))) @ (mbox_s4 @ (mnot @ (mbox_s4 @ ((element @ X3) @ X3)))))) @ (mbox_s4 @ ((element @ X3) @ X2))))))))))))))).
% 11.73/12.12 thf(37,plain,sP37,inference(preprocess,[status(thm)],[pel41_1]).
% 11.73/12.12 thf(38,plain,$false,inference(prop_unsat,[status(thm),assumptions([h8,h9,h7,h5,h6,h4,h3,h2,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,h5,h8,h9])).
% 11.73/12.12 thf(39,plain,$false,inference(tab_negimp,[status(thm),assumptions([h7,h5,h6,h4,h3,h2,h1,h0]),tab_negimp(discharge,[h8,h9])],[h7,38,h8,h9])).
% 11.73/12.12 thf(40,plain,$false,inference(tab_negall,[status(thm),assumptions([h5,h6,h4,h3,h2,h1,h0]),tab_negall(discharge,[h7]),tab_negall(eigenvar,eigen__2)],[h6,39,h7])).
% 11.73/12.12 thf(41,plain,$false,inference(tab_negimp,[status(thm),assumptions([h4,h3,h2,h1,h0]),tab_negimp(discharge,[h5,h6])],[h4,40,h5,h6])).
% 11.73/12.12 thf(42,plain,$false,inference(tab_negall,[status(thm),assumptions([h3,h2,h1,h0]),tab_negall(discharge,[h4]),tab_negall(eigenvar,eigen__1)],[h3,41,h4])).
% 11.73/12.12 thf(43,plain,$false,inference(tab_negall,[status(thm),assumptions([h2,h1,h0]),tab_negall(discharge,[h3]),tab_negall(eigenvar,eigen__0)],[h2,42,h3])).
% 11.73/12.12 thf(44,plain,$false,inference(eigenvar_choice,[status(thm),assumptions([h2,h0]),eigenvar_choice(discharge,[h1])],[43,h1])).
% 11.73/12.12 thf(45,plain,$false,inference(eigenvar_choice,[status(thm),assumptions([h2]),eigenvar_choice(discharge,[h0])],[44,h0])).
% 11.73/12.12 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) => (((element @ X5) @ X3) @ X6)))))))))))))))))),inference(contra,[status(thm),contra(discharge,[h2])],[43,h2])).
% 11.73/12.12 % SZS output end Proof
%------------------------------------------------------------------------------