0.02/0.09 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.02/0.10 % Command : run_portfolio.sh /export/starexec/sandbox2/benchmark/theBenchmark.p /export/starexec/sandbox2/tmp/tmp.h1Ibdw7UAh 0.10/0.30 % Computer : n025.cluster.edu 0.10/0.30 % Model : x86_64 x86_64 0.10/0.30 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.10/0.30 % Memory : 8042.1875MB 0.10/0.30 % OS : Linux 3.10.0-693.el7.x86_64 0.10/0.30 % CPULimit : 1200 0.10/0.30 % DateTime : Tue Jul 13 14:38:57 EDT 2021 0.10/0.30 % CPUTime : 0.10/0.30 % Running portfolio for 120 s 0.10/0.30 % File : /export/starexec/sandbox2/benchmark/theBenchmark.p 0.10/0.30 % Number of cores: 8 0.10/0.30 % Python version: Python 3.6.8 0.10/0.31 % Running in HO mode 0.16/0.57 % Estimated wc time : 1286 0.16/0.57 % Estimated cpu time (7 cpus) : 183.71428571428572 1.01/0.67 % /export/starexec/sandbox2/solver/bin/lams/new_b3.2.norw.sh running for 15s 1.01/0.67 % /export/starexec/sandbox2/solver/bin/lams/s6.sh running for 21s 1.01/0.67 % /export/starexec/sandbox2/solver/bin/lams/b.rpo.sh running for 21s 1.01/0.67 % /export/starexec/sandbox2/solver/bin/lams/s6.ds.sh running for 10s 1.01/0.67 % /export/starexec/sandbox2/solver/bin/lams/c.s.2.sh running for 68s 1.01/0.67 % /export/starexec/sandbox2/solver/bin/lams/c.easy.sh running for 26s 1.01/0.68 % /export/starexec/sandbox2/solver/bin/lams/old_b3.avatar.sh running for 15s 1.25/1.04 % Solved by lams/s6.ds.sh. 1.25/1.04 % done 494 iterations in 0.331s 1.25/1.04 % SZS status Theorem for '/export/starexec/sandbox2/benchmark/theBenchmark.p' 1.25/1.04 % SZS output start Refutation 1.25/1.04 tff(mbox_s4, axiom, mbox_s4() = 1.25/1.04 ^[Phi:(($i > $o)),W]: (![V]: (Phi(V) | (~rel_s4(W,V))))). 1.25/1.04 tff('0', plain, 1.25/1.04 mbox_s4 = (^[Phi:(($i > $o)),W]: (![V]: (Phi(V) | (~rel_s4(W,V))))), 1.25/1.04 inference('simplify_rw_rule', [status(thm)], [mbox_s4])). 1.25/1.04 tff('1', plain, 1.25/1.04 mbox_s4 = 1.25/1.04 (^[V_1:(($i > $o)),V_2]: (![X4]: (V_1(X4) | (~rel_s4(V_2,X4))))), 1.25/1.04 define([status(thm)])). 1.25/1.04 tff(mvalid, axiom, mvalid() = ^[Phi:(($i > $o))]: (![W]: Phi(W))). 1.25/1.04 tff('2', plain, mvalid = (^[Phi:(($i > $o))]: (![W]: Phi(W))), 1.25/1.04 inference('simplify_rw_rule', [status(thm)], [mvalid])). 1.25/1.04 tff('3', plain, mvalid = (^[V_1:(($i > $o))]: (![X4]: V_1(X4))), 1.25/1.04 define([status(thm)])). 1.25/1.04 tff(mexists_ind, axiom, mexists_ind() = 1.25/1.04 ^[Phi:((mu * $i) > $o)]: mnot(mforall_ind(^[X:mu]: mnot(Phi(X))))). 1.25/1.04 tff(mforall_ind, axiom, mforall_ind() = 1.25/1.04 ^[Phi:((mu * $i) > $o),W]: (![X:mu]: (exists_in_world(X,W) => Phi(X,W)))). 1.25/1.04 tff('4', plain, 1.25/1.04 mforall_ind = 1.25/1.04 (^[Phi:((mu * $i) > $o),W]: (![X:mu]: (exists_in_world(X,W) => Phi(X,W)))), 1.25/1.04 inference('simplify_rw_rule', [status(thm)], [mforall_ind])). 1.25/1.04 tff('5', plain, 1.25/1.04 mforall_ind = 1.25/1.04 (^[V_1:((mu * $i) > $o),V_2]: 1.25/1.04 (![X4:mu]: (exists_in_world(X4,V_2) => V_1(X4,V_2)))), 1.25/1.04 define([status(thm)])). 1.25/1.04 tff(mnot, axiom, mnot() = ^[Phi:(($i > $o)),W]: (~Phi(W))). 1.25/1.04 tff('6', plain, mnot = (^[Phi:(($i > $o)),W]: (~Phi(W))), 1.25/1.04 inference('simplify_rw_rule', [status(thm)], [mnot])). 1.25/1.04 tff('7', plain, mnot = (^[V_1:(($i > $o)),V_2]: (~V_1(V_2))), 1.25/1.04 define([status(thm)])). 1.25/1.04 tff('8', plain, 1.25/1.04 mexists_ind = 1.25/1.04 (^[Phi:((mu * $i) > $o)]: mnot(mforall_ind(^[X:mu]: mnot(Phi(X))))), 1.25/1.04 inference('simplify_rw_rule', [status(thm)], [mexists_ind, '5', '7'])). 1.25/1.04 tff('9', plain, 1.25/1.04 mexists_ind = 1.25/1.04 (^[V_1:((mu * $i) > $o)]: mnot(mforall_ind(^[V_2:mu]: mnot(V_1(V_2))))), 1.25/1.04 define([status(thm)])). 1.25/1.04 tff(mimplies, axiom, mimplies() = 1.25/1.04 ^[Phi:(($i > $o)),Psi:(($i > $o))]: mor(mnot(Phi),Psi)). 1.25/1.04 tff(mor, axiom, mor() = 1.25/1.04 ^[Phi:(($i > $o)),Psi:(($i > $o)),W]: (Phi(W) | Psi(W))). 1.25/1.04 tff('10', plain, 1.25/1.04 mor = (^[Phi:(($i > $o)),Psi:(($i > $o)),W]: (Phi(W) | Psi(W))), 1.25/1.04 inference('simplify_rw_rule', [status(thm)], [mor])). 1.25/1.04 tff('11', plain, 1.25/1.04 mor = (^[V_1:(($i > $o)),V_2:(($i > $o)),V_3]: (V_1(V_3) | V_2(V_3))), 1.25/1.04 define([status(thm)])). 1.25/1.04 tff('12', plain, 1.25/1.04 mimplies = (^[Phi:(($i > $o)),Psi:(($i > $o))]: mor(mnot(Phi),Psi)), 1.25/1.04 inference('simplify_rw_rule', [status(thm)], [mimplies, '11', '7'])). 1.25/1.04 tff('13', plain, 1.25/1.04 mimplies = (^[V_1:(($i > $o)),V_2:(($i > $o))]: mor(mnot(V_1),V_2)), 1.25/1.04 define([status(thm)])). 1.25/1.04 tff(mand, axiom, mand() = 1.25/1.04 ^[Phi:(($i > $o)),Psi:(($i > $o))]: mnot(mor(mnot(Phi),mnot(Psi)))). 1.25/1.04 tff('14', plain, 1.25/1.04 mand = 1.25/1.04 (^[Phi:(($i > $o)),Psi:(($i > $o))]: mnot(mor(mnot(Phi),mnot(Psi)))), 1.25/1.04 inference('simplify_rw_rule', [status(thm)], [mand, '11', '7'])). 1.25/1.04 tff('15', plain, 1.25/1.04 mand = 1.25/1.04 (^[V_1:(($i > $o)),V_2:(($i > $o))]: mnot(mor(mnot(V_1),mnot(V_2)))), 1.25/1.04 define([status(thm)])). 1.25/1.04 tff(kalish204, conjecture, 1.25/1.04 (mvalid(mand(mbox_s4(mimplies(mbox_s4(mnot(mexists_ind(^[X:mu]: 1.25/1.04 mbox_s4(f(X))))), 1.25/1.04 mbox_s4(mforall_ind(^[Y:mu]: 1.25/1.04 mbox_s4(mnot(mbox_s4( 1.25/1.04 f(Y)))))))), 1.25/1.04 mbox_s4(mimplies(mbox_s4(mforall_ind(^[Y:mu]: 1.25/1.04 mbox_s4(mnot(mbox_s4( 1.25/1.04 f(Y)))))), 1.25/1.04 mbox_s4(mnot(mexists_ind(^[X:mu]: 1.25/1.04 mbox_s4(f(X))))))))))). 1.25/1.04 tff(zf_stmt_0, conjecture, 1.25/1.04 (![X4]: 1.25/1.04 (~ 1.25/1.04 ((~ 1.25/1.04 (![X22]: 1.25/1.04 ((~rel_s4(X4,X22)) | 1.25/1.04 (~ 1.25/1.04 (![X24]: 1.25/1.04 ((~rel_s4(X22,X24)) | 1.25/1.04 (![X26:mu]: 1.25/1.04 (exists_in_world(X26,X24) => 1.25/1.04 (![X28]: 1.25/1.04 ((~rel_s4(X24,X28)) | 1.25/1.04 (~(![X30]: ((~rel_s4(X28,X30)) | f(X26,X30))))))))))) | 1.25/1.04 (![X32]: 1.25/1.04 ((~rel_s4(X22,X32)) | 1.25/1.04 (![X34:mu]: 1.25/1.04 (exists_in_world(X34,X32) => 1.25/1.04 (~(![X36]: ((~rel_s4(X32,X36)) | f(X34,X36))))))))))) | 1.25/1.04 (~ 1.25/1.04 (![X6]: 1.25/1.04 ((~rel_s4(X4,X6)) | 1.25/1.04 (~ 1.25/1.04 (![X8]: 1.25/1.04 ((~rel_s4(X6,X8)) | 1.25/1.04 (![X10:mu]: 1.25/1.04 (exists_in_world(X10,X8) => 1.25/1.04 (~(![X12]: ((~rel_s4(X8,X12)) | f(X10,X12))))))))) | 1.25/1.04 (![X14]: 1.25/1.04 ((~rel_s4(X6,X14)) | 1.25/1.04 (![X16:mu]: 1.25/1.04 (exists_in_world(X16,X14) => 1.25/1.04 (![X18]: 1.25/1.04 ((~rel_s4(X14,X18)) | 1.25/1.04 (~(![X20]: ((~rel_s4(X18,X20)) | f(X16,X20))))))))))))))))). 1.25/1.04 tff(zf_stmt_1, negated_conjecture, 1.25/1.04 (~ 1.25/1.04 (![X4]: 1.25/1.04 (~ 1.25/1.04 ((~ 1.25/1.04 (![X22]: 1.25/1.04 ((~rel_s4(X4,X22)) | 1.25/1.04 (~ 1.25/1.04 (![X24]: 1.25/1.04 ((~rel_s4(X22,X24)) | 1.25/1.04 (![X26:mu]: 1.25/1.04 (exists_in_world(X26,X24) => 1.25/1.04 (![X28]: 1.25/1.04 ((~rel_s4(X24,X28)) | 1.25/1.04 (~(![X30]: ((~rel_s4(X28,X30)) | f(X26,X30))))))))))) | 1.25/1.04 (![X32]: 1.25/1.04 ((~rel_s4(X22,X32)) | 1.25/1.04 (![X34:mu]: 1.25/1.04 (exists_in_world(X34,X32) => 1.25/1.04 (~(![X36]: ((~rel_s4(X32,X36)) | f(X34,X36))))))))))) | 1.25/1.04 (~ 1.25/1.04 (![X6]: 1.25/1.04 ((~rel_s4(X4,X6)) | 1.25/1.04 (~ 1.25/1.04 (![X8]: 1.25/1.04 ((~rel_s4(X6,X8)) | 1.25/1.04 (![X10:mu]: 1.25/1.04 (exists_in_world(X10,X8) => 1.25/1.04 (~(![X12]: ((~rel_s4(X8,X12)) | f(X10,X12))))))))) | 1.25/1.04 (![X14]: 1.25/1.04 ((~rel_s4(X6,X14)) | 1.25/1.04 (![X16:mu]: 1.25/1.04 (exists_in_world(X16,X14) => 1.25/1.04 (![X18]: 1.25/1.04 ((~rel_s4(X14,X18)) | 1.25/1.04 (~(![X20]: ((~rel_s4(X18,X20)) | f(X16,X20)))))))))))))))))). 1.25/1.04 tff('16', plain, 1.25/1.04 ~ (!!((^[Y0 : $i]: 1.25/1.04 (~(((~((!!((^[Y1 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y1)))| 1.25/1.04 (~((!!((^[Y2 : $i]: 1.25/1.04 ((~(rel_s4(Y1, Y2))) | 1.25/1.04 (!!((^[Y3 : mu]: 1.25/1.04 (exists_in_world(Y3, Y2) => 1.25/1.04 (!!((^[Y4 : $i]: 1.25/1.04 ((~(rel_s4(Y2, Y4))) | 1.25/1.04 (~((!!((^[Y5 : $i]: 1.25/1.04 ((~( 1.25/1.04 rel_s4( 1.25/1.04 Y4, Y5))) | 1.25/1.04 f(Y3, Y5)))))))))))))))))))))| 1.25/1.04 (!!((^[Y2 : $i]: 1.25/1.04 ((~(rel_s4(Y1, Y2))) | 1.25/1.04 (!!((^[Y3 : mu]: 1.25/1.04 (exists_in_world(Y3, Y2) => 1.25/1.04 (~((!!((^[Y4 : $i]: 1.25/1.04 ((~(rel_s4(Y2, Y4))) | 1.25/1.04 f(Y3, Y4))))))))))))))))))))) | 1.25/1.04 (~((!!((^[Y1 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y1)))| 1.25/1.04 (~((!!((^[Y2 : $i]: 1.25/1.04 ((~(rel_s4(Y1, Y2))) | 1.25/1.04 (!!((^[Y3 : mu]: 1.25/1.04 (exists_in_world(Y3, Y2) => 1.25/1.04 (~((!!((^[Y4 : $i]: 1.25/1.04 ((~(rel_s4(Y2, Y4))) | 1.25/1.04 f(Y3, Y4)))))))))))))))))| 1.25/1.04 (!!((^[Y2 : $i]: 1.25/1.04 ((~(rel_s4(Y1, Y2))) | 1.25/1.04 (!!((^[Y3 : mu]: 1.25/1.04 (exists_in_world(Y3, Y2) => 1.25/1.04 (!!((^[Y4 : $i]: 1.25/1.04 ((~(rel_s4(Y2, Y4))) | 1.25/1.04 (~((!!((^[Y5 : $i]: 1.25/1.04 ((~(rel_s4(Y4, 1.25/1.04 Y5))) | 1.25/1.04 f(Y3, Y5))))))))))))))))))))))))))))))), 1.25/1.04 inference('cnf', [status(esa)], [zf_stmt_1])). 1.25/1.04 tff('17', plain, 1.25/1.04 ((~((!!((^[Y0 : $i]: 1.25/1.04 ((~(rel_s4('#sk2', Y0)))| 1.25/1.04 (~((!!((^[Y1 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y1))) | 1.25/1.04 (!!((^[Y2 : mu]: 1.25/1.04 (exists_in_world(Y2, Y1) => 1.25/1.04 (!!((^[Y3 : $i]: 1.25/1.04 ((~(rel_s4(Y1, Y3))) | 1.25/1.04 (~((!!((^[Y4 : $i]: 1.25/1.04 ((~(rel_s4(Y3, Y4))) | 1.25/1.04 f(Y2, Y4)))))))))))))))))))))| 1.25/1.04 (!!((^[Y1 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y1))) | 1.25/1.04 (!!((^[Y2 : mu]: 1.25/1.04 (exists_in_world(Y2, Y1) => 1.25/1.04 (~((!!((^[Y3 : $i]: 1.25/1.04 ((~(rel_s4(Y1, Y3))) | f(Y2, Y3))))))))))))))))))))) | 1.25/1.04 (~((!!((^[Y0 : $i]: 1.25/1.04 ((~(rel_s4('#sk2', Y0)))| 1.25/1.04 (~((!!((^[Y1 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y1))) | 1.25/1.04 (!!((^[Y2 : mu]: 1.25/1.04 (exists_in_world(Y2, Y1) => 1.25/1.04 (~((!!((^[Y3 : $i]: 1.25/1.04 ((~(rel_s4(Y1, Y3))) | 1.25/1.04 f(Y2, Y3)))))))))))))))))| 1.25/1.04 (!!((^[Y1 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y1))) | 1.25/1.04 (!!((^[Y2 : mu]: 1.25/1.04 (exists_in_world(Y2, Y1) => 1.25/1.04 (!!((^[Y3 : $i]: 1.25/1.04 ((~(rel_s4(Y1, Y3))) | 1.25/1.04 (~((!!((^[Y4 : $i]: 1.25/1.04 ((~(rel_s4(Y3, Y4))) | 1.25/1.04 f(Y2, Y4)))))))))))))))))))))))))), 1.25/1.04 inference('lazy_cnf_exists', [status(thm)], ['16'])). 1.25/1.04 tff('18', plain, 1.25/1.04 (~ (!!((^[Y0 : $i]: 1.25/1.04 ((~(rel_s4('#sk2', Y0)))| 1.25/1.04 (~((!!((^[Y1 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y1))) | 1.25/1.04 (!!((^[Y2 : mu]: 1.25/1.04 (exists_in_world(Y2, Y1) => 1.25/1.04 (!!((^[Y3 : $i]: 1.25/1.04 ((~(rel_s4(Y1, Y3))) | 1.25/1.04 (~((!!((^[Y4 : $i]: 1.25/1.04 ((~(rel_s4(Y3, Y4))) | 1.25/1.04 f(Y2, Y4)))))))))))))))))))))| 1.25/1.04 (!!((^[Y1 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y1))) | 1.25/1.04 (!!((^[Y2 : mu]: 1.25/1.04 (exists_in_world(Y2, Y1) => 1.25/1.04 (~((!!((^[Y3 : $i]: 1.25/1.04 ((~(rel_s4(Y1, Y3))) | f(Y2, Y3))))))))))))))))))) 1.25/1.04 | ~ (!!((^[Y0 : $i]: 1.25/1.04 ((~(rel_s4('#sk2', Y0)))| 1.25/1.04 (~((!!((^[Y1 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y1))) | 1.25/1.04 (!!((^[Y2 : mu]: 1.25/1.04 (exists_in_world(Y2, Y1) => 1.25/1.04 (~((!!((^[Y3 : $i]: 1.25/1.04 ((~(rel_s4(Y1, Y3))) | 1.25/1.04 f(Y2, Y3)))))))))))))))))| 1.25/1.04 (!!((^[Y1 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y1))) | 1.25/1.04 (!!((^[Y2 : mu]: 1.25/1.04 (exists_in_world(Y2, Y1) => 1.25/1.04 (!!((^[Y3 : $i]: 1.25/1.04 ((~(rel_s4(Y1, Y3))) | 1.25/1.04 (~((!!((^[Y4 : $i]: 1.25/1.04 ((~(rel_s4(Y3, Y4))) | 1.25/1.04 f(Y2, Y4)))))))))))))))))))))))), 1.25/1.04 inference('lazy_cnf_or', [status(thm)], ['17'])). 1.25/1.04 tff('19', plain, 1.25/1.04 (~ ((~(rel_s4('#sk2', '#sk3')))| 1.25/1.04 (~((!!((^[Y0 : $i]: 1.25/1.04 ((~(rel_s4('#sk3', Y0))) | 1.25/1.04 (!!((^[Y1 : mu]: 1.25/1.04 (exists_in_world(Y1, Y0) => 1.25/1.04 (!!((^[Y2 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y2))) | 1.25/1.04 (~((!!((^[Y3 : $i]: 1.25/1.04 ((~(rel_s4(Y2, Y3))) | f(Y1, Y3)))))))))))))))))))))| 1.25/1.04 (!!((^[Y0 : $i]: 1.25/1.04 ((~(rel_s4('#sk3', Y0))) | 1.25/1.04 (!!((^[Y1 : mu]: 1.25/1.04 (exists_in_world(Y1, Y0) => 1.25/1.04 (~((!!((^[Y2 : $i]: ((~(rel_s4(Y0, Y2))) | f(Y1, Y2)))))))))))))))) 1.25/1.04 | ~ (!!((^[Y0 : $i]: 1.25/1.04 ((~(rel_s4('#sk2', Y0)))| 1.25/1.04 (~((!!((^[Y1 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y1))) | 1.25/1.04 (!!((^[Y2 : mu]: 1.25/1.04 (exists_in_world(Y2, Y1) => 1.25/1.04 (~((!!((^[Y3 : $i]: 1.25/1.04 ((~(rel_s4(Y1, Y3))) | 1.25/1.04 f(Y2, Y3)))))))))))))))))| 1.25/1.04 (!!((^[Y1 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y1))) | 1.25/1.04 (!!((^[Y2 : mu]: 1.25/1.04 (exists_in_world(Y2, Y1) => 1.25/1.04 (!!((^[Y3 : $i]: 1.25/1.04 ((~(rel_s4(Y1, Y3))) | 1.25/1.04 (~((!!((^[Y4 : $i]: 1.25/1.04 ((~(rel_s4(Y3, Y4))) | 1.25/1.04 f(Y2, Y4)))))))))))))))))))))))), 1.25/1.04 inference('lazy_cnf_exists', [status(thm)], ['18'])). 1.25/1.04 tff('20', plain, 1.25/1.04 (~ (!!((^[Y0 : $i]: 1.25/1.04 ((~(rel_s4('#sk3', Y0))) | 1.25/1.04 (!!((^[Y1 : mu]: 1.25/1.04 (exists_in_world(Y1, Y0) => 1.25/1.04 (~((!!((^[Y2 : $i]: ((~(rel_s4(Y0, Y2))) | f(Y1, Y2))))))))))))))) 1.25/1.04 | ~ (!!((^[Y0 : $i]: 1.25/1.04 ((~(rel_s4('#sk2', Y0)))| 1.25/1.04 (~((!!((^[Y1 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y1))) | 1.25/1.04 (!!((^[Y2 : mu]: 1.25/1.04 (exists_in_world(Y2, Y1) => 1.25/1.04 (~((!!((^[Y3 : $i]: 1.25/1.04 ((~(rel_s4(Y1, Y3))) | 1.25/1.04 f(Y2, Y3)))))))))))))))))| 1.25/1.04 (!!((^[Y1 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y1))) | 1.25/1.04 (!!((^[Y2 : mu]: 1.25/1.04 (exists_in_world(Y2, Y1) => 1.25/1.04 (!!((^[Y3 : $i]: 1.25/1.04 ((~(rel_s4(Y1, Y3))) | 1.25/1.04 (~((!!((^[Y4 : $i]: 1.25/1.04 ((~(rel_s4(Y3, Y4))) | 1.25/1.04 f(Y2, Y4)))))))))))))))))))))))), 1.25/1.04 inference('lazy_cnf_or', [status(thm)], ['19'])). 1.25/1.04 tff('21', plain, 1.25/1.04 (~ ((~(rel_s4('#sk3', '#sk5'))) | 1.25/1.04 (!!((^[Y0 : mu]: 1.25/1.04 (exists_in_world(Y0, '#sk5') => 1.25/1.04 (~((!!((^[Y1 : $i]: ((~(rel_s4('#sk5', Y1))) | f(Y0, Y1)))))))))))) 1.25/1.04 | ~ (!!((^[Y0 : $i]: 1.25/1.04 ((~(rel_s4('#sk2', Y0)))| 1.25/1.04 (~((!!((^[Y1 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y1))) | 1.25/1.04 (!!((^[Y2 : mu]: 1.25/1.04 (exists_in_world(Y2, Y1) => 1.25/1.04 (~((!!((^[Y3 : $i]: 1.25/1.04 ((~(rel_s4(Y1, Y3))) | 1.25/1.04 f(Y2, Y3)))))))))))))))))| 1.25/1.04 (!!((^[Y1 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y1))) | 1.25/1.04 (!!((^[Y2 : mu]: 1.25/1.04 (exists_in_world(Y2, Y1) => 1.25/1.04 (!!((^[Y3 : $i]: 1.25/1.04 ((~(rel_s4(Y1, Y3))) | 1.25/1.04 (~((!!((^[Y4 : $i]: 1.25/1.04 ((~(rel_s4(Y3, Y4))) | 1.25/1.04 f(Y2, Y4)))))))))))))))))))))))), 1.25/1.04 inference('lazy_cnf_exists', [status(thm)], ['20'])). 1.25/1.04 tff('22', plain, 1.25/1.04 (~ ((~(rel_s4('#sk3', '#sk5'))) | 1.25/1.04 (!!((^[Y0 : mu]: 1.25/1.04 (exists_in_world(Y0, '#sk5') => 1.25/1.04 (~((!!((^[Y1 : $i]: ((~(rel_s4('#sk5', Y1))) | f(Y0, Y1)))))))))))) 1.25/1.04 | ~ (!!((^[Y0 : $i]: 1.25/1.04 ((~(rel_s4('#sk2', Y0)))| 1.25/1.04 (~((!!((^[Y1 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y1))) | 1.25/1.04 (!!((^[Y2 : mu]: 1.25/1.04 (exists_in_world(Y2, Y1) => 1.25/1.04 (~((!!((^[Y3 : $i]: 1.25/1.04 ((~(rel_s4(Y1, Y3))) | 1.25/1.04 f(Y2, Y3)))))))))))))))))| 1.25/1.04 (!!((^[Y1 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y1))) | 1.25/1.04 (!!((^[Y2 : mu]: 1.25/1.04 (exists_in_world(Y2, Y1) => 1.25/1.04 (!!((^[Y3 : $i]: 1.25/1.04 ((~(rel_s4(Y1, Y3))) | 1.25/1.04 (~((!!((^[Y4 : $i]: 1.25/1.04 ((~(rel_s4(Y3, Y4))) | 1.25/1.04 f(Y2, Y4)))))))))))))))))))))))), 1.25/1.04 inference('lazy_cnf_exists', [status(thm)], ['20'])). 1.25/1.04 tff('23', plain, 1.25/1.04 ( '#form6' 1.25/1.04 | ~ (!!((^[Y0 : $i]: 1.25/1.04 ((~(rel_s4('#sk2', Y0)))| 1.25/1.04 (~((!!((^[Y1 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y1))) | 1.25/1.04 (!!((^[Y2 : mu]: 1.25/1.04 (exists_in_world(Y2, Y1) => 1.25/1.04 (~((!!((^[Y3 : $i]: 1.25/1.04 ((~(rel_s4(Y1, Y3))) | 1.25/1.04 f(Y2, Y3)))))))))))))))))| 1.25/1.04 (!!((^[Y1 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y1))) | 1.25/1.04 (!!((^[Y2 : mu]: 1.25/1.04 (exists_in_world(Y2, Y1) => 1.25/1.04 (!!((^[Y3 : $i]: 1.25/1.04 ((~(rel_s4(Y1, Y3))) | 1.25/1.04 (~((!!((^[Y4 : $i]: 1.25/1.04 ((~(rel_s4(Y3, Y4))) | 1.25/1.04 f(Y2, Y4)))))))))))))))))))))))), 1.25/1.04 define([status(thm)])). 1.25/1.04 tff('24', plain, 1.25/1.04 (~ '#form6' 1.25/1.04 | ~ ((~(rel_s4('#sk3', '#sk5'))) | 1.25/1.04 (!!((^[Y0 : mu]: 1.25/1.04 (exists_in_world(Y0, '#sk5') => 1.25/1.04 (~((!!((^[Y1 : $i]: ((~(rel_s4('#sk5', Y1))) | f(Y0, Y1))))))))))))), 1.25/1.04 inference('renaming', [status(thm)], ['21', '23'])). 1.25/1.04 tff('25', plain, ( rel_s4('#sk3', '#sk5') | ~ '#form6'), 1.25/1.04 inference('lazy_cnf_or', [status(thm)], ['24'])). 1.25/1.04 tff('26', plain, 1.25/1.04 ( '#form6' 1.25/1.04 | ~ (!!((^[Y0 : $i]: 1.25/1.04 ((~(rel_s4('#sk2', Y0)))| 1.25/1.04 (~((!!((^[Y1 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y1))) | 1.25/1.04 (!!((^[Y2 : mu]: 1.25/1.04 (exists_in_world(Y2, Y1) => 1.25/1.04 (~((!!((^[Y3 : $i]: 1.25/1.04 ((~(rel_s4(Y1, Y3))) | 1.25/1.04 f(Y2, Y3)))))))))))))))))| 1.25/1.04 (!!((^[Y1 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y1))) | 1.25/1.04 (!!((^[Y2 : mu]: 1.25/1.04 (exists_in_world(Y2, Y1) => 1.25/1.04 (!!((^[Y3 : $i]: 1.25/1.04 ((~(rel_s4(Y1, Y3))) | 1.25/1.04 (~((!!((^[Y4 : $i]: 1.25/1.04 ((~(rel_s4(Y3, Y4))) | 1.25/1.04 f(Y2, Y4)))))))))))))))))))))))), 1.25/1.04 define([status(thm)])). 1.25/1.04 tff('27', plain, 1.25/1.04 (~ ((~(rel_s4('#sk2', '#sk4')))| 1.25/1.04 (~((!!((^[Y0 : $i]: 1.25/1.04 ((~(rel_s4('#sk4', Y0))) | 1.25/1.04 (!!((^[Y1 : mu]: 1.25/1.04 (exists_in_world(Y1, Y0) => 1.25/1.04 (~((!!((^[Y2 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y2))) | f(Y1, Y2)))))))))))))))))| 1.25/1.04 (!!((^[Y0 : $i]: 1.25/1.04 ((~(rel_s4('#sk4', Y0))) | 1.25/1.04 (!!((^[Y1 : mu]: 1.25/1.04 (exists_in_world(Y1, Y0) => 1.25/1.04 (!!((^[Y2 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y2))) | 1.25/1.04 (~((!!((^[Y3 : $i]: 1.25/1.04 ((~(rel_s4(Y2, Y3))) | f(Y1, Y3)))))))))))))))))))) 1.25/1.04 | '#form6'), 1.25/1.04 inference('lazy_cnf_exists', [status(thm)], ['26'])). 1.25/1.04 tff('28', plain, 1.25/1.04 (~ (!!((^[Y0 : $i]: 1.25/1.04 ((~(rel_s4('#sk4', Y0))) | 1.25/1.04 (!!((^[Y1 : mu]: 1.25/1.04 (exists_in_world(Y1, Y0) => 1.25/1.04 (!!((^[Y2 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y2))) | 1.25/1.04 (~((!!((^[Y3 : $i]: 1.25/1.04 ((~(rel_s4(Y2, Y3))) | f(Y1, Y3))))))))))))))))))) 1.25/1.04 | '#form6'), 1.25/1.04 inference('lazy_cnf_or', [status(thm)], ['27'])). 1.25/1.04 tff('29', plain, 1.25/1.04 (~ ((~(rel_s4('#sk4', '#sk7'))) | 1.25/1.04 (!!((^[Y0 : mu]: 1.25/1.04 (exists_in_world(Y0, '#sk7') => 1.25/1.04 (!!((^[Y1 : $i]: 1.25/1.04 ((~(rel_s4('#sk7', Y1))) | 1.25/1.04 (~((!!((^[Y2 : $i]: ((~(rel_s4(Y1, Y2))) | f(Y0, Y2)))))))))))))))) 1.25/1.04 | '#form6'), 1.25/1.04 inference('lazy_cnf_exists', [status(thm)], ['28'])). 1.25/1.04 tff('30', plain, 1.25/1.04 (~ (!!((^[Y0 : mu]: 1.25/1.04 (exists_in_world(Y0, '#sk7') => 1.25/1.04 (!!((^[Y1 : $i]: 1.25/1.04 ((~(rel_s4('#sk7', Y1))) | 1.25/1.04 (~((!!((^[Y2 : $i]: ((~(rel_s4(Y1, Y2))) | f(Y0, Y2))))))))))))))) 1.25/1.04 | '#form6'), 1.25/1.04 inference('lazy_cnf_or', [status(thm)], ['29'])). 1.25/1.04 tff('31', plain, 1.25/1.04 (~ (exists_in_world('#sk9', '#sk7') => 1.25/1.04 (!!((^[Y0 : $i]: 1.25/1.04 ((~(rel_s4('#sk7', Y0))) | 1.25/1.04 (~((!!((^[Y1 : $i]: ((~(rel_s4(Y0, Y1))) | f('#sk9', Y1)))))))))))) 1.25/1.04 | '#form6'), 1.25/1.04 inference('lazy_cnf_exists', [status(thm)], ['30'])). 1.25/1.04 tff('32', plain, ( exists_in_world('#sk9', '#sk7') | '#form6'), 1.25/1.04 inference('lazy_cnf_imply', [status(thm)], ['31'])). 1.25/1.04 tff('33', plain, 1.25/1.04 (~ (!!((^[Y0 : $i]: 1.25/1.04 ((~(rel_s4('#sk7', Y0))) | 1.25/1.04 (~((!!((^[Y1 : $i]: ((~(rel_s4(Y0, Y1))) | f('#sk9', Y1))))))))))) 1.25/1.04 | '#form6'), 1.25/1.04 inference('lazy_cnf_imply', [status(thm)], ['31'])). 1.25/1.04 tff('34', plain, 1.25/1.04 (~ ((~(rel_s4('#sk7', '#sk11'))) | 1.25/1.04 (~((!!((^[Y0 : $i]: ((~(rel_s4('#sk11', Y0))) | f('#sk9', Y0)))))))) 1.25/1.04 | '#form6'), 1.25/1.04 inference('lazy_cnf_exists', [status(thm)], ['33'])). 1.25/1.04 tff('35', plain, ( rel_s4('#sk7', '#sk11') | '#form6'), 1.25/1.04 inference('lazy_cnf_or', [status(thm)], ['34'])). 1.25/1.04 tff(cumulative_ax, axiom, 1.25/1.04 (![X:mu,V,W]: 1.25/1.04 ((exists_in_world(X,V) & rel_s4(V,W)) => exists_in_world(X,W)))). 1.25/1.04 tff('36', plain, 1.25/1.04 (!!((^[Y0 : mu]: 1.25/1.04 (!!((^[Y1 : $i]: 1.25/1.04 (!!((^[Y2 : $i]: 1.25/1.04 ((exists_in_world(Y0, Y1) & rel_s4(Y1, Y2)) => 1.25/1.04 exists_in_world(Y0, Y2))))))))))), 1.25/1.04 inference('cnf', [status(esa)], [cumulative_ax])). 1.25/1.04 tff('37', plain, 1.25/1.04 ![X1 : mu]: 1.25/1.04 (!!((^[Y0 : $i]: 1.25/1.04 (!!((^[Y1 : $i]: 1.25/1.04 ((exists_in_world(X1, Y0) & rel_s4(Y0, Y1)) => 1.25/1.04 exists_in_world(X1, Y1)))))))), 1.25/1.04 inference('lazy_cnf_forall', [status(thm)], ['36'])). 1.25/1.04 tff('38', plain, 1.25/1.04 ![X1 : mu, X2]: 1.25/1.04 (!!((^[Y0 : $i]: 1.25/1.04 ((exists_in_world(X1, X2) & rel_s4(X2, Y0)) => 1.25/1.04 exists_in_world(X1, Y0))))), 1.25/1.04 inference('lazy_cnf_forall', [status(thm)], ['37'])). 1.25/1.04 tff('39', plain, 1.25/1.04 ![X1 : mu, X2, X3]: 1.25/1.04 ((exists_in_world(X1, X2) & rel_s4(X2, X3)) => exists_in_world(X1, X3)), 1.25/1.04 inference('lazy_cnf_forall', [status(thm)], ['38'])). 1.25/1.04 tff('40', plain, 1.25/1.04 ![X1 : mu, X2, X3]: 1.25/1.04 (~ (exists_in_world(X1, X2) & rel_s4(X2, X3)) 1.25/1.04 | exists_in_world(X1, X3)), 1.25/1.04 inference('lazy_cnf_imply', [status(thm)], ['39'])). 1.25/1.04 tff('41', plain, 1.25/1.04 ![X1 : mu, X2, X3]: 1.25/1.04 (~ exists_in_world(X1, X2) 1.25/1.04 | ~ rel_s4(X2, X3) 1.25/1.04 | exists_in_world(X1, X3)), 1.25/1.04 inference('lazy_cnf_and', [status(thm)], ['40'])). 1.25/1.04 tff('42', plain, 1.25/1.04 ![X0 : mu]: 1.25/1.04 ( '#form6' 1.25/1.04 | exists_in_world(X0, '#sk11') 1.25/1.04 | ~ exists_in_world(X0, '#sk7')), 1.25/1.04 inference('sup-', [status(thm)], ['35', '41'])). 1.25/1.04 tff('43', plain, ( '#form6' | exists_in_world('#sk9', '#sk11') | '#form6'), 1.25/1.04 inference('sup-', [status(thm)], ['32', '42'])). 1.25/1.04 tff('44', plain, ( exists_in_world('#sk9', '#sk11') | '#form6'), 1.25/1.04 inference('simplify', [status(thm)], ['43'])). 1.25/1.04 tff('45', plain, 1.25/1.04 ( (!!((^[Y0 : $i]: 1.25/1.04 ((~(rel_s4('#sk4', Y0))) | 1.25/1.04 (!!((^[Y1 : mu]: 1.25/1.04 (exists_in_world(Y1, Y0) => 1.25/1.04 (~((!!((^[Y2 : $i]: ((~(rel_s4(Y0, Y2))) | f(Y1, Y2))))))))))))))) 1.25/1.04 | '#form6'), 1.25/1.04 inference('lazy_cnf_or', [status(thm)], ['27'])). 1.25/1.04 tff('46', plain, 1.25/1.04 ![X1]: 1.25/1.04 ( ((~(rel_s4('#sk4', X1))) | 1.25/1.04 (!!((^[Y0 : mu]: 1.25/1.04 (exists_in_world(Y0, X1) => 1.25/1.04 (~((!!((^[Y1 : $i]: ((~(rel_s4(X1, Y1))) | f(Y0, Y1)))))))))))) 1.25/1.04 | '#form6'), 1.25/1.04 inference('lazy_cnf_forall', [status(thm)], ['45'])). 1.25/1.04 tff('47', plain, 1.25/1.04 ![X1]: 1.25/1.04 (~ rel_s4('#sk4', X1) 1.25/1.04 | (!!((^[Y0 : mu]: 1.25/1.04 (exists_in_world(Y0, X1) => 1.25/1.04 (~((!!((^[Y1 : $i]: ((~(rel_s4(X1, Y1))) | f(Y0, Y1))))))))))) 1.25/1.04 | '#form6'), 1.25/1.04 inference('lazy_cnf_or', [status(thm)], ['46'])). 1.25/1.04 tff('48', plain, 1.25/1.04 ![X1, X2 : mu]: 1.25/1.04 ( (exists_in_world(X2, X1) => 1.25/1.04 (~((!!((^[Y0 : $i]: ((~(rel_s4(X1, Y0))) | f(X2, Y0)))))))) 1.25/1.04 | '#form6' 1.25/1.04 | ~ rel_s4('#sk4', X1)), 1.25/1.04 inference('lazy_cnf_forall', [status(thm)], ['47'])). 1.25/1.04 tff('49', plain, 1.25/1.04 ![X1, X2 : mu]: 1.25/1.04 (~ exists_in_world(X2, X1) 1.25/1.04 | ~ (!!((^[Y0 : $i]: ((~(rel_s4(X1, Y0))) | f(X2, Y0))))) 1.25/1.04 | ~ rel_s4('#sk4', X1) 1.25/1.04 | '#form6'), 1.25/1.04 inference('lazy_cnf_imply', [status(thm)], ['48'])). 1.25/1.04 tff('50', plain, 1.25/1.04 ![X1, X2 : mu]: 1.25/1.04 (~ ((~(rel_s4(X1, '#sk10'(X1, X2)))) | f(X2, '#sk10'(X1, X2))) 1.25/1.04 | '#form6' 1.25/1.04 | ~ rel_s4('#sk4', X1) 1.25/1.04 | ~ exists_in_world(X2, X1)), 1.25/1.04 inference('lazy_cnf_exists', [status(thm)], ['49'])). 1.25/1.04 tff('51', plain, 1.25/1.04 ![X1, X2 : mu]: 1.25/1.04 ( rel_s4(X1, '#sk10'(X1, X2)) 1.25/1.04 | ~ exists_in_world(X2, X1) 1.25/1.04 | ~ rel_s4('#sk4', X1) 1.25/1.04 | '#form6'), 1.25/1.04 inference('lazy_cnf_or', [status(thm)], ['50'])). 1.25/1.04 tff('52', plain, 1.25/1.04 ( '#form6' 1.25/1.04 | '#form6' 1.25/1.04 | ~ rel_s4('#sk4', '#sk11') 1.25/1.04 | rel_s4('#sk11', '#sk10'('#sk11', '#sk9'))), 1.25/1.04 inference('sup-', [status(thm)], ['44', '51'])). 1.25/1.04 tff('53', plain, 1.25/1.04 ( rel_s4('#sk11', '#sk10'('#sk11', '#sk9')) 1.25/1.04 | ~ rel_s4('#sk4', '#sk11') 1.25/1.04 | '#form6'), 1.25/1.04 inference('simplify', [status(thm)], ['52'])). 1.25/1.04 tff('54', plain, ( rel_s4('#sk4', '#sk7') | '#form6'), 1.25/1.04 inference('lazy_cnf_or', [status(thm)], ['29'])). 1.25/1.04 tff('55', plain, ( rel_s4('#sk7', '#sk11') | '#form6'), 1.25/1.04 inference('lazy_cnf_or', [status(thm)], ['34'])). 1.25/1.04 tff(mtransitive, axiom, mtransitive() = 1.25/1.04 ^[R:(($i * $i) > $o)]: (![S,T,U]: ((R(S,T) & R(T,U)) => R(S,U)))). 1.25/1.04 tff('56', plain, 1.25/1.04 mtransitive = 1.25/1.04 (^[R:(($i * $i) > $o)]: (![S,T,U]: ((R(S,T) & R(T,U)) => R(S,U)))), 1.25/1.04 inference('simplify_rw_rule', [status(thm)], [mtransitive])). 1.25/1.04 tff('57', plain, 1.25/1.04 mtransitive = 1.25/1.04 (^[V_1:(($i * $i) > $o)]: 1.25/1.04 (![X4,X6,X8]: ((V_1(X4,X6) & V_1(X6,X8)) => V_1(X4,X8)))), 1.25/1.04 define([status(thm)])). 1.25/1.04 tff(a2, axiom, (mtransitive(rel_s4))). 1.25/1.04 tff(zf_stmt_2, axiom, 1.25/1.04 (![X4,X6,X8]: ((rel_s4(X6,X8) & rel_s4(X4,X6)) => rel_s4(X4,X8)))). 1.25/1.04 tff('58', plain, 1.25/1.04 (!!((^[Y0 : $i]: 1.25/1.04 (!!((^[Y1 : $i]: 1.25/1.04 (!!((^[Y2 : $i]: 1.25/1.04 ((rel_s4(Y1, Y2) & rel_s4(Y0, Y1)) => 1.25/1.04 rel_s4(Y0, Y2))))))))))), 1.25/1.04 inference('cnf', [status(esa)], [zf_stmt_2])). 1.25/1.04 tff('59', plain, 1.25/1.04 ![X1]: 1.25/1.04 (!!((^[Y0 : $i]: 1.25/1.04 (!!((^[Y1 : $i]: 1.25/1.04 ((rel_s4(Y0, Y1) & rel_s4(X1, Y0)) => rel_s4(X1, Y1)))))))), 1.25/1.04 inference('lazy_cnf_forall', [status(thm)], ['58'])). 1.25/1.04 tff('60', plain, 1.25/1.04 ![X1, X2]: 1.25/1.04 (!!((^[Y0 : $i]: ((rel_s4(X2, Y0) & rel_s4(X1, X2)) => rel_s4(X1, Y0))))), 1.25/1.04 inference('lazy_cnf_forall', [status(thm)], ['59'])). 1.25/1.04 tff('61', plain, 1.25/1.04 ![X1, X2, X3]: ((rel_s4(X2, X3) & rel_s4(X1, X2)) => rel_s4(X1, X3)), 1.25/1.04 inference('lazy_cnf_forall', [status(thm)], ['60'])). 1.25/1.04 tff('62', plain, 1.25/1.04 ![X1, X2, X3]: (~ (rel_s4(X2, X3) & rel_s4(X1, X2)) | rel_s4(X1, X3)), 1.25/1.04 inference('lazy_cnf_imply', [status(thm)], ['61'])). 1.25/1.04 tff('63', plain, 1.25/1.04 ![X1, X2, X3]: (~ rel_s4(X2, X3) | ~ rel_s4(X1, X2) | rel_s4(X1, X3)), 1.25/1.04 inference('lazy_cnf_and', [status(thm)], ['62'])). 1.25/1.04 tff('64', plain, 1.25/1.04 ![X0]: ( '#form6' | rel_s4(X0, '#sk11') | ~ rel_s4(X0, '#sk7')), 1.25/1.04 inference('sup-', [status(thm)], ['55', '63'])). 1.25/1.04 tff('65', plain, ( '#form6' | rel_s4('#sk4', '#sk11') | '#form6'), 1.25/1.04 inference('sup-', [status(thm)], ['54', '64'])). 1.25/1.04 tff('66', plain, ( rel_s4('#sk4', '#sk11') | '#form6'), 1.25/1.04 inference('simplify', [status(thm)], ['65'])). 1.25/1.04 tff('67', plain, ( '#form6' | rel_s4('#sk11', '#sk10'('#sk11', '#sk9'))), 1.25/1.04 inference('clc', [status(thm)], ['53', '66'])). 1.25/1.04 tff('68', plain, 1.25/1.04 ( (!!((^[Y0 : $i]: ((~(rel_s4('#sk11', Y0))) | f('#sk9', Y0))))) 1.25/1.04 | '#form6'), 1.25/1.04 inference('lazy_cnf_or', [status(thm)], ['34'])). 1.25/1.04 tff('69', plain, 1.25/1.04 ![X1]: ( ((~(rel_s4('#sk11', X1))) | f('#sk9', X1)) | '#form6'), 1.25/1.04 inference('lazy_cnf_forall', [status(thm)], ['68'])). 1.25/1.04 tff('70', plain, ![X1]: (~ rel_s4('#sk11', X1) | f('#sk9', X1) | '#form6'), 1.25/1.04 inference('lazy_cnf_or', [status(thm)], ['69'])). 1.25/1.04 tff('71', plain, 1.25/1.04 ( '#form6' | '#form6' | f('#sk9', '#sk10'('#sk11', '#sk9'))), 1.25/1.04 inference('sup-', [status(thm)], ['67', '70'])). 1.25/1.04 tff('72', plain, ( f('#sk9', '#sk10'('#sk11', '#sk9')) | '#form6'), 1.25/1.04 inference('simplify', [status(thm)], ['71'])). 1.25/1.04 tff('73', plain, 1.25/1.04 ![X1, X2 : mu]: 1.25/1.04 (~ f(X2, '#sk10'(X1, X2)) 1.25/1.04 | ~ exists_in_world(X2, X1) 1.25/1.04 | ~ rel_s4('#sk4', X1) 1.25/1.04 | '#form6'), 1.25/1.04 inference('lazy_cnf_or', [status(thm)], ['50'])). 1.25/1.04 tff('74', plain, 1.25/1.04 ( '#form6' 1.25/1.04 | '#form6' 1.25/1.04 | ~ rel_s4('#sk4', '#sk11') 1.25/1.04 | ~ exists_in_world('#sk9', '#sk11')), 1.25/1.04 inference('sup-', [status(thm)], ['72', '73'])). 1.25/1.04 tff('75', plain, 1.25/1.04 (~ exists_in_world('#sk9', '#sk11') 1.25/1.04 | ~ rel_s4('#sk4', '#sk11') 1.25/1.04 | '#form6'), 1.25/1.04 inference('simplify', [status(thm)], ['74'])). 1.25/1.04 tff('76', plain, ( exists_in_world('#sk9', '#sk11') | '#form6'), 1.25/1.04 inference('simplify', [status(thm)], ['43'])). 1.25/1.04 tff('77', plain, ( '#form6' | ~ rel_s4('#sk4', '#sk11')), 1.25/1.04 inference('clc', [status(thm)], ['75', '76'])). 1.25/1.04 tff('78', plain, ( rel_s4('#sk4', '#sk11') | '#form6'), 1.25/1.04 inference('simplify', [status(thm)], ['65'])). 1.25/1.04 tff('79', plain, '#form6', inference('clc', [status(thm)], ['77', '78'])). 1.25/1.04 tff('80', plain, rel_s4('#sk3', '#sk5'), 1.25/1.04 inference('demod', [status(thm)], ['25', '79'])). 1.25/1.04 tff(mreflexive, axiom, mreflexive() = ^[R:(($i * $i) > $o)]: (![S]: R(S,S))). 1.25/1.04 tff('81', plain, mreflexive = (^[R:(($i * $i) > $o)]: (![S]: R(S,S))), 1.25/1.04 inference('simplify_rw_rule', [status(thm)], [mreflexive])). 1.25/1.04 tff('82', plain, mreflexive = (^[V_1:(($i * $i) > $o)]: (![X4]: V_1(X4,X4))), 1.25/1.04 define([status(thm)])). 1.25/1.04 tff(a1, axiom, (mreflexive(rel_s4))). 1.25/1.04 tff(zf_stmt_3, axiom, (![X4]: rel_s4(X4,X4))). 1.25/1.04 tff('83', plain, (!!((^[Y0 : $i]: rel_s4(Y0, Y0)))), 1.25/1.04 inference('cnf', [status(esa)], [zf_stmt_3])). 1.25/1.04 tff('84', plain, ![X1]: rel_s4(X1, X1), 1.25/1.04 inference('lazy_cnf_forall', [status(thm)], ['83'])). 1.25/1.04 tff('85', plain, 1.25/1.04 (~ (!!((^[Y0 : mu]: 1.25/1.04 (exists_in_world(Y0, '#sk5') => 1.25/1.04 (~((!!((^[Y1 : $i]: ((~(rel_s4('#sk5', Y1))) | f(Y0, Y1))))))))))) 1.25/1.04 | ~ '#form6'), 1.25/1.04 inference('lazy_cnf_or', [status(thm)], ['24'])). 1.25/1.04 tff('86', plain, 1.25/1.04 (~ (exists_in_world('#sk8', '#sk5') => 1.25/1.04 (~((!!((^[Y0 : $i]: ((~(rel_s4('#sk5', Y0))) | f('#sk8', Y0)))))))) 1.25/1.04 | ~ '#form6'), 1.25/1.04 inference('lazy_cnf_exists', [status(thm)], ['85'])). 1.25/1.04 tff('87', plain, ( exists_in_world('#sk8', '#sk5') | ~ '#form6'), 1.25/1.04 inference('lazy_cnf_imply', [status(thm)], ['86'])). 1.25/1.04 tff('88', plain, '#form6', inference('clc', [status(thm)], ['77', '78'])). 1.25/1.04 tff('89', plain, exists_in_world('#sk8', '#sk5'), 1.25/1.04 inference('demod', [status(thm)], ['87', '88'])). 1.25/1.04 tff('90', plain, 1.25/1.04 ( (!!((^[Y0 : $i]: 1.25/1.04 ((~(rel_s4('#sk3', Y0))) | 1.25/1.04 (!!((^[Y1 : mu]: 1.25/1.04 (exists_in_world(Y1, Y0) => 1.25/1.04 (!!((^[Y2 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y2))) | 1.25/1.04 (~((!!((^[Y3 : $i]: 1.25/1.04 ((~(rel_s4(Y2, Y3))) | f(Y1, Y3))))))))))))))))))) 1.25/1.04 | ~ (!!((^[Y0 : $i]: 1.25/1.04 ((~(rel_s4('#sk2', Y0)))| 1.25/1.04 (~((!!((^[Y1 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y1))) | 1.25/1.04 (!!((^[Y2 : mu]: 1.25/1.04 (exists_in_world(Y2, Y1) => 1.25/1.04 (~((!!((^[Y3 : $i]: 1.25/1.04 ((~(rel_s4(Y1, Y3))) | 1.25/1.04 f(Y2, Y3)))))))))))))))))| 1.25/1.04 (!!((^[Y1 : $i]: 1.25/1.04 ((~(rel_s4(Y0, Y1))) | 1.25/1.04 (!!((^[Y2 : mu]: 1.25/1.04 (exists_in_world(Y2, Y1) => 1.25/1.04 (!!((^[Y3 : $i]: 1.25/1.04 ((~(rel_s4(Y1, Y3))) | 1.60/1.04 (~((!!((^[Y4 : $i]: 1.60/1.04 ((~(rel_s4(Y3, Y4))) | 1.60/1.04 f(Y2, Y4)))))))))))))))))))))))), 1.60/1.04 inference('lazy_cnf_or', [status(thm)], ['19'])). 1.60/1.04 tff('91', plain, 1.60/1.04 ![X1]: 1.60/1.04 ( ((~(rel_s4('#sk3', X1))) | 1.60/1.04 (!!((^[Y0 : mu]: 1.60/1.04 (exists_in_world(Y0, X1) => 1.60/1.04 (!!((^[Y1 : $i]: 1.60/1.04 ((~(rel_s4(X1, Y1))) | 1.60/1.04 (~((!!((^[Y2 : $i]: 1.60/1.04 ((~(rel_s4(Y1, Y2))) | f(Y0, Y2)))))))))))))))) 1.60/1.04 | ~ (!!((^[Y0 : $i]: 1.60/1.04 ((~(rel_s4('#sk2', Y0)))| 1.60/1.04 (~((!!((^[Y1 : $i]: 1.60/1.04 ((~(rel_s4(Y0, Y1))) | 1.60/1.04 (!!((^[Y2 : mu]: 1.60/1.04 (exists_in_world(Y2, Y1) => 1.60/1.04 (~((!!((^[Y3 : $i]: 1.60/1.04 ((~(rel_s4(Y1, Y3))) | 1.60/1.04 f(Y2, Y3)))))))))))))))))| 1.60/1.04 (!!((^[Y1 : $i]: 1.60/1.04 ((~(rel_s4(Y0, Y1))) | 1.60/1.04 (!!((^[Y2 : mu]: 1.60/1.04 (exists_in_world(Y2, Y1) => 1.60/1.04 (!!((^[Y3 : $i]: 1.60/1.04 ((~(rel_s4(Y1, Y3))) | 1.60/1.04 (~((!!((^[Y4 : $i]: 1.60/1.04 ((~(rel_s4(Y3, Y4))) | 1.60/1.04 f(Y2, Y4)))))))))))))))))))))))), 1.60/1.04 inference('lazy_cnf_forall', [status(thm)], ['90'])). 1.60/1.04 tff('92', plain, 1.60/1.04 ![X1]: 1.60/1.04 (~ rel_s4('#sk3', X1) 1.60/1.04 | (!!((^[Y0 : mu]: 1.60/1.04 (exists_in_world(Y0, X1) => 1.60/1.04 (!!((^[Y1 : $i]: 1.60/1.04 ((~(rel_s4(X1, Y1))) | 1.60/1.04 (~((!!((^[Y2 : $i]: 1.60/1.04 ((~(rel_s4(Y1, Y2))) | f(Y0, Y2))))))))))))))) 1.60/1.04 | ~ (!!((^[Y0 : $i]: 1.60/1.04 ((~(rel_s4('#sk2', Y0)))| 1.60/1.04 (~((!!((^[Y1 : $i]: 1.60/1.04 ((~(rel_s4(Y0, Y1))) | 1.60/1.04 (!!((^[Y2 : mu]: 1.60/1.04 (exists_in_world(Y2, Y1) => 1.60/1.04 (~((!!((^[Y3 : $i]: 1.60/1.04 ((~(rel_s4(Y1, Y3))) | 1.60/1.04 f(Y2, Y3)))))))))))))))))| 1.60/1.04 (!!((^[Y1 : $i]: 1.60/1.04 ((~(rel_s4(Y0, Y1))) | 1.60/1.04 (!!((^[Y2 : mu]: 1.60/1.04 (exists_in_world(Y2, Y1) => 1.60/1.04 (!!((^[Y3 : $i]: 1.60/1.04 ((~(rel_s4(Y1, Y3))) | 1.60/1.04 (~((!!((^[Y4 : $i]: 1.60/1.04 ((~(rel_s4(Y3, Y4))) | 1.60/1.04 f(Y2, Y4)))))))))))))))))))))))), 1.60/1.04 inference('lazy_cnf_or', [status(thm)], ['91'])). 1.60/1.04 tff('93', plain, 1.60/1.04 ( '#form6' 1.60/1.04 | ~ (!!((^[Y0 : $i]: 1.60/1.04 ((~(rel_s4('#sk2', Y0)))| 1.60/1.04 (~((!!((^[Y1 : $i]: 1.60/1.04 ((~(rel_s4(Y0, Y1))) | 1.60/1.04 (!!((^[Y2 : mu]: 1.60/1.04 (exists_in_world(Y2, Y1) => 1.60/1.04 (~((!!((^[Y3 : $i]: 1.60/1.04 ((~(rel_s4(Y1, Y3))) | 1.60/1.04 f(Y2, Y3)))))))))))))))))| 1.60/1.04 (!!((^[Y1 : $i]: 1.60/1.04 ((~(rel_s4(Y0, Y1))) | 1.60/1.04 (!!((^[Y2 : mu]: 1.60/1.04 (exists_in_world(Y2, Y1) => 1.60/1.04 (!!((^[Y3 : $i]: 1.60/1.04 ((~(rel_s4(Y1, Y3))) | 1.60/1.04 (~((!!((^[Y4 : $i]: 1.60/1.04 ((~(rel_s4(Y3, Y4))) | 1.60/1.04 f(Y2, Y4)))))))))))))))))))))))), 1.60/1.04 define([status(thm)])). 1.60/1.04 tff('94', plain, 1.60/1.04 ![X1]: 1.60/1.04 (~ '#form6' 1.60/1.04 | (!!((^[Y0 : mu]: 1.60/1.04 (exists_in_world(Y0, X1) => 1.60/1.04 (!!((^[Y1 : $i]: 1.60/1.04 ((~(rel_s4(X1, Y1))) | 1.60/1.04 (~((!!((^[Y2 : $i]: 1.60/1.04 ((~(rel_s4(Y1, Y2))) | f(Y0, Y2))))))))))))))) 1.60/1.04 | ~ rel_s4('#sk3', X1)), 1.60/1.04 inference('renaming', [status(thm)], ['92', '93'])). 1.60/1.04 tff('95', plain, 1.60/1.04 ![X1, X2 : mu]: 1.60/1.04 ( (exists_in_world(X2, X1) => 1.60/1.04 (!!((^[Y0 : $i]: 1.60/1.04 ((~(rel_s4(X1, Y0))) | 1.60/1.04 (~((!!((^[Y1 : $i]: ((~(rel_s4(Y0, Y1))) | f(X2, Y1)))))))))))) 1.60/1.04 | ~ rel_s4('#sk3', X1) 1.60/1.04 | ~ '#form6'), 1.60/1.04 inference('lazy_cnf_forall', [status(thm)], ['94'])). 1.60/1.04 tff('96', plain, 1.60/1.04 ![X1, X2 : mu]: 1.60/1.04 (~ exists_in_world(X2, X1) 1.60/1.04 | (!!((^[Y0 : $i]: 1.60/1.04 ((~(rel_s4(X1, Y0))) | 1.60/1.04 (~((!!((^[Y1 : $i]: ((~(rel_s4(Y0, Y1))) | f(X2, Y1))))))))))) 1.60/1.04 | ~ '#form6' 1.60/1.04 | ~ rel_s4('#sk3', X1)), 1.60/1.04 inference('lazy_cnf_imply', [status(thm)], ['95'])). 1.60/1.04 tff('97', plain, 1.60/1.04 ![X1, X2 : mu, X3]: 1.60/1.04 ( ((~(rel_s4(X1, X3))) | 1.60/1.04 (~((!!((^[Y0 : $i]: ((~(rel_s4(X3, Y0))) | f(X2, Y0)))))))) 1.60/1.04 | ~ rel_s4('#sk3', X1) 1.60/1.04 | ~ '#form6' 1.60/1.04 | ~ exists_in_world(X2, X1)), 1.60/1.04 inference('lazy_cnf_forall', [status(thm)], ['96'])). 1.60/1.04 tff('98', plain, 1.60/1.04 ![X1, X2 : mu, X3]: 1.60/1.04 (~ rel_s4(X1, X3) 1.60/1.04 | ~ (!!((^[Y0 : $i]: ((~(rel_s4(X3, Y0))) | f(X2, Y0))))) 1.60/1.04 | ~ exists_in_world(X2, X1) 1.60/1.04 | ~ '#form6' 1.60/1.04 | ~ rel_s4('#sk3', X1)), 1.60/1.04 inference('lazy_cnf_or', [status(thm)], ['97'])). 1.60/1.04 tff('99', plain, 1.60/1.04 ![X1, X2 : mu, X3]: 1.60/1.04 (~ ((~(rel_s4(X3, '#sk10'(X3, X2)))) | f(X2, '#sk10'(X3, X2))) 1.60/1.04 | ~ rel_s4('#sk3', X1) 1.60/1.04 | ~ '#form6' 1.60/1.04 | ~ exists_in_world(X2, X1) 1.60/1.04 | ~ rel_s4(X1, X3)), 1.60/1.04 inference('lazy_cnf_exists', [status(thm)], ['98'])). 1.60/1.04 tff('100', plain, 1.60/1.04 ![X1, X2 : mu, X3]: 1.60/1.04 ( rel_s4(X3, '#sk10'(X3, X2)) 1.60/1.04 | ~ rel_s4(X1, X3) 1.60/1.04 | ~ exists_in_world(X2, X1) 1.60/1.04 | ~ '#form6' 1.60/1.04 | ~ rel_s4('#sk3', X1)), 1.60/1.04 inference('lazy_cnf_or', [status(thm)], ['99'])). 1.60/1.04 tff('101', plain, '#form6', inference('clc', [status(thm)], ['77', '78'])). 1.60/1.04 tff('102', plain, 1.60/1.04 ![X1, X2 : mu, X3]: 1.60/1.04 ( rel_s4(X3, '#sk10'(X3, X2)) 1.60/1.04 | ~ rel_s4(X1, X3) 1.60/1.04 | ~ exists_in_world(X2, X1) 1.60/1.04 | ~ rel_s4('#sk3', X1)), 1.60/1.04 inference('demod', [status(thm)], ['100', '101'])). 1.60/1.04 tff('103', plain, 1.60/1.04 ![X0]: 1.60/1.04 (~ rel_s4('#sk3', '#sk5') 1.60/1.04 | ~ rel_s4('#sk5', X0) 1.60/1.04 | rel_s4(X0, '#sk10'(X0, '#sk8'))), 1.60/1.04 inference('sup-', [status(thm)], ['89', '102'])). 1.60/1.04 tff('104', plain, rel_s4('#sk3', '#sk5'), 1.60/1.04 inference('demod', [status(thm)], ['25', '79'])). 1.60/1.04 tff('105', plain, 1.60/1.04 ![X0]: (~ rel_s4('#sk5', X0) | rel_s4(X0, '#sk10'(X0, '#sk8'))), 1.60/1.04 inference('demod', [status(thm)], ['103', '104'])). 1.60/1.04 tff('106', plain, rel_s4('#sk5', '#sk10'('#sk5', '#sk8')), 1.60/1.04 inference('sup-', [status(thm)], ['84', '105'])). 1.60/1.04 tff('107', plain, 1.60/1.04 ( (!!((^[Y0 : $i]: ((~(rel_s4('#sk5', Y0))) | f('#sk8', Y0))))) 1.60/1.04 | ~ '#form6'), 1.60/1.04 inference('lazy_cnf_imply', [status(thm)], ['86'])). 1.60/1.04 tff('108', plain, 1.60/1.04 ![X1]: ( ((~(rel_s4('#sk5', X1))) | f('#sk8', X1)) | ~ '#form6'), 1.60/1.04 inference('lazy_cnf_forall', [status(thm)], ['107'])). 1.60/1.04 tff('109', plain, 1.60/1.04 ![X1]: (~ rel_s4('#sk5', X1) | f('#sk8', X1) | ~ '#form6'), 1.60/1.04 inference('lazy_cnf_or', [status(thm)], ['108'])). 1.60/1.04 tff('110', plain, '#form6', inference('clc', [status(thm)], ['77', '78'])). 1.60/1.04 tff('111', plain, ![X1]: (~ rel_s4('#sk5', X1) | f('#sk8', X1)), 1.60/1.04 inference('demod', [status(thm)], ['109', '110'])). 1.60/1.04 tff('112', plain, f('#sk8', '#sk10'('#sk5', '#sk8')), 1.60/1.04 inference('sup-', [status(thm)], ['106', '111'])). 1.60/1.04 tff('113', plain, 1.60/1.04 ![X1, X2 : mu, X3]: 1.60/1.04 (~ f(X2, '#sk10'(X3, X2)) 1.60/1.04 | ~ rel_s4(X1, X3) 1.60/1.04 | ~ exists_in_world(X2, X1) 1.60/1.04 | ~ '#form6' 1.60/1.04 | ~ rel_s4('#sk3', X1)), 1.60/1.04 inference('lazy_cnf_or', [status(thm)], ['99'])). 1.60/1.04 tff('114', plain, '#form6', inference('clc', [status(thm)], ['77', '78'])). 1.60/1.04 tff('115', plain, 1.60/1.04 ![X1, X2 : mu, X3]: 1.60/1.04 (~ f(X2, '#sk10'(X3, X2)) 1.60/1.04 | ~ rel_s4(X1, X3) 1.60/1.04 | ~ exists_in_world(X2, X1) 1.60/1.04 | ~ rel_s4('#sk3', X1)), 1.60/1.04 inference('demod', [status(thm)], ['113', '114'])). 1.60/1.04 tff('116', plain, 1.60/1.04 ![X0]: 1.60/1.04 (~ rel_s4('#sk3', X0) 1.60/1.04 | ~ exists_in_world('#sk8', X0) 1.60/1.04 | ~ rel_s4(X0, '#sk5')), 1.60/1.04 inference('sup-', [status(thm)], ['112', '115'])). 1.60/1.04 tff('117', plain, 1.60/1.04 (~ rel_s4('#sk5', '#sk5') | ~ exists_in_world('#sk8', '#sk5')), 1.60/1.04 inference('sup-', [status(thm)], ['80', '116'])). 1.60/1.04 tff('118', plain, ![X1]: rel_s4(X1, X1), 1.60/1.04 inference('lazy_cnf_forall', [status(thm)], ['83'])). 1.60/1.04 tff('119', plain, exists_in_world('#sk8', '#sk5'), 1.60/1.04 inference('demod', [status(thm)], ['87', '88'])). 1.60/1.04 tff('120', plain, $false, 1.60/1.04 inference('demod', [status(thm)], ['117', '118', '119'])). 1.60/1.04 1.60/1.04 % SZS output end Refutation 1.60/1.04 1.60/1.04 1.60/1.04 % Terminating... 1.60/1.09 % Runner terminated. 1.60/1.09 % Zipperpin 1.5 exiting 1.60/1.11 EOF