0.06/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.06/0.13 % Command : run_portfolio.sh /export/starexec/sandbox/benchmark/theBenchmark.p /export/starexec/sandbox/tmp/tmp.ZqNbW2K6hw 0.14/0.34 % Computer : n014.cluster.edu 0.14/0.34 % Model : x86_64 x86_64 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.14/0.34 % Memory : 8042.1875MB 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64 0.14/0.34 % CPULimit : 960 0.14/0.34 % DateTime : Thu Jul 2 08:53:55 EDT 2020 0.14/0.34 % CPUTime : 0.14/0.34 % Running portfolio for 120 s 0.14/0.34 % File : /export/starexec/sandbox/benchmark/theBenchmark.p 0.14/0.34 % Number of cores: 8 0.14/0.35 % Python version: Python 3.6.8 0.14/0.35 % Running in HO mode 0.53/0.62 % Estimated wc time : 1286 0.53/0.62 % Estimated cpu time (7 cpus) : 183.71428571428572 0.55/0.70 % /export/starexec/sandbox/solver/bin/lams/s6.sh running for 21s 0.55/0.73 % /export/starexec/sandbox/solver/bin/lams/new_b3.2.norw.sh running for 15s 0.55/0.73 % /export/starexec/sandbox/solver/bin/lams/b.rpo.sh running for 21s 0.55/0.75 % /export/starexec/sandbox/solver/bin/lams/s6.ds.sh running for 10s 0.55/0.75 % /export/starexec/sandbox/solver/bin/lams/c.s.2.sh running for 68s 0.55/0.75 % /export/starexec/sandbox/solver/bin/lams/c.easy.sh running for 26s 0.55/0.75 % /export/starexec/sandbox/solver/bin/lams/old_b3.avatar.sh running for 15s 69.15/10.57 % /export/starexec/sandbox/solver/bin/lams/old_sp2.sh running for 17s 104.57/15.60 % /export/starexec/sandbox/solver/bin/lams/b.sh running for 31s 106.07/15.81 % /export/starexec/sandbox/solver/bin/lams/sh1.sh running for 26s 139.61/20.78 % Solved by lams/b.sh. 139.61/20.78 % Running : timeout 14 /export/starexec/sandbox/solver/bin/lams/eprover-ho --pos-ext=all --neg-ext=all /export/starexec/sandbox/tmp/tmp.ZqNbW2K6hw/e_inputd2f9ba --cpu-limit=12 --auto -s -p. 139.61/20.78 % done 2623 iterations in 5.145s 139.61/20.78 % SZS status Theorem for '/export/starexec/sandbox/benchmark/theBenchmark.p' 139.61/20.78 % SZS output start Refutation 139.61/20.78 tff(nonempty_ax, axiom, (![V]: (?[X:mu]: exists_in_world(X,V)))). 139.61/20.78 tff('0', plain, 139.61/20.78 (!!((^[Y0 : $i]: (??((^[Y1 : mu]: exists_in_world(Y1, Y0))))))), 139.61/20.78 inference('cnf', [status(esa)], [nonempty_ax])). 139.61/20.78 tff('1', plain, ![X1]: (??((^[Y0 : mu]: exists_in_world(Y0, X1)))), 139.61/20.78 inference('lazy_cnf_forall', [status(thm)], ['0'])). 139.61/20.78 tff('2', plain, ![X1]: exists_in_world('#sk1'(X1), X1), 139.61/20.78 inference('lazy_cnf_exists', [status(thm)], ['1'])). 139.61/20.78 tff(x2128, conjecture, 139.61/20.78 (mvalid(mand(mbox_s4(mimplies(mbox_s4(mforall_ind(^[X:mu]: 139.61/20.78 mand(mbox_s4(mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 big_p(X)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 ^[Y:mu]: 139.61/20.78 mbox_s4( 139.61/20.78 big_p(Y)))))), 139.61/20.78 mbox_s4(mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 ^[Y:mu]: 139.61/20.78 mbox_s4( 139.61/20.78 big_p(Y)))), 139.61/20.78 mbox_s4( 139.61/20.78 big_p(X))))))), 139.61/20.78 mand(mbox_s4(mimplies(mexists_ind(^[X:mu]: 139.61/20.78 mbox_s4( 139.61/20.78 big_p(X))), 139.61/20.78 mbox_s4(mforall_ind( 139.61/20.78 ^[Y:mu]: 139.61/20.78 mbox_s4( 139.61/20.78 big_p(Y)))))), 139.61/20.78 mbox_s4(mimplies(mbox_s4(mforall_ind( 139.61/20.78 ^[Y:mu]: 139.61/20.78 mbox_s4( 139.61/20.78 big_p(Y)))), 139.61/20.78 mexists_ind(^[X:mu]: 139.61/20.78 mbox_s4( 139.61/20.78 big_p(X)))))))), 139.61/20.78 mbox_s4(mimplies(mand(mbox_s4(mimplies(mexists_ind(^[X:mu]: 139.61/20.78 mbox_s4( 139.61/20.78 big_p(X))), 139.61/20.78 mbox_s4(mforall_ind( 139.61/20.78 ^[Y:mu]: 139.61/20.78 mbox_s4( 139.61/20.78 big_p(Y)))))), 139.61/20.78 mbox_s4(mimplies(mbox_s4(mforall_ind( 139.61/20.78 ^[Y:mu]: 139.61/20.78 mbox_s4( 139.61/20.78 big_p(Y)))), 139.61/20.78 mexists_ind(^[X:mu]: 139.61/20.78 mbox_s4( 139.61/20.78 big_p(X)))))), 139.61/20.78 mbox_s4(mforall_ind(^[X:mu]: 139.61/20.78 mand(mbox_s4(mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 big_p(X)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 ^[Y:mu]: 139.61/20.78 mbox_s4( 139.61/20.78 big_p(Y)))))), 139.61/20.78 mbox_s4(mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 ^[Y:mu]: 139.61/20.78 mbox_s4( 139.61/20.78 big_p(Y)))), 139.61/20.78 mbox_s4( 139.61/20.78 big_p(X))))))))))))). 139.61/20.78 tff(zf_stmt_0, negated_conjecture, 139.61/20.78 (~ 139.61/20.78 mvalid(mand(mbox_s4(mimplies(mbox_s4(mforall_ind(^[X:mu]: 139.61/20.78 mand(mbox_s4(mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 big_p(X)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 ^[Y:mu]: 139.61/20.78 mbox_s4( 139.61/20.78 big_p(Y)))))), 139.61/20.78 mbox_s4(mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 ^[Y:mu]: 139.61/20.78 mbox_s4( 139.61/20.78 big_p(Y)))), 139.61/20.78 mbox_s4( 139.61/20.78 big_p(X))))))), 139.61/20.78 mand(mbox_s4(mimplies(mexists_ind(^[X:mu]: 139.61/20.78 mbox_s4( 139.61/20.78 big_p(X))), 139.61/20.78 mbox_s4(mforall_ind( 139.61/20.78 ^[Y:mu]: 139.61/20.78 mbox_s4( 139.61/20.78 big_p(Y)))))), 139.61/20.78 mbox_s4(mimplies(mbox_s4(mforall_ind( 139.61/20.78 ^[Y:mu]: 139.61/20.78 mbox_s4( 139.61/20.78 big_p(Y)))), 139.61/20.78 mexists_ind(^[X:mu]: 139.61/20.78 mbox_s4( 139.61/20.78 big_p(X)))))))), 139.61/20.78 mbox_s4(mimplies(mand(mbox_s4(mimplies(mexists_ind(^[X:mu]: 139.61/20.78 mbox_s4( 139.61/20.78 big_p(X))), 139.61/20.78 mbox_s4(mforall_ind( 139.61/20.78 ^[Y:mu]: 139.61/20.78 mbox_s4( 139.61/20.78 big_p(Y)))))), 139.61/20.78 mbox_s4(mimplies(mbox_s4(mforall_ind( 139.61/20.78 ^[Y:mu]: 139.61/20.78 mbox_s4( 139.61/20.78 big_p(Y)))), 139.61/20.78 mexists_ind(^[X:mu]: 139.61/20.78 mbox_s4( 139.61/20.78 big_p(X)))))), 139.61/20.78 mbox_s4(mforall_ind(^[X:mu]: 139.61/20.78 mand(mbox_s4(mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 big_p(X)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 ^[Y:mu]: 139.61/20.78 mbox_s4( 139.61/20.78 big_p(Y)))))), 139.61/20.78 mbox_s4(mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 ^[Y:mu]: 139.61/20.78 mbox_s4( 139.61/20.78 big_p(Y)))), 139.61/20.78 mbox_s4( 139.61/20.78 big_p(X))))))))))))). 139.61/20.78 tff('3', plain, 139.61/20.78 ~ mvalid( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y0 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0)))))))), 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y0 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0)))))))))))), 139.61/20.78 inference('cnf', [status(esa)], [zf_stmt_0])). 139.61/20.78 tff(mvalid, axiom, ~ mvalid(X23) => ((!!((^[Y0 : $i]: X23(Y0)))) != $true)). 139.61/20.78 tff('4', plain, 139.61/20.78 ![X23:(($i > $o))]: ((~mvalid(X23)) <=> (~(![X28]: X23(X28)))), 139.61/20.78 inference('rw.lit', [status(esa)], [mvalid])). 139.61/20.78 tff('5', plain, 139.61/20.78 ~ (!!((^[Y0 : $i]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))))), 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))))))), Y0)))), 139.61/20.78 inference('rw_clause', [status(thm)], ['3', '4'])). 139.61/20.78 tff('6', plain, 139.61/20.78 ~ (!!(mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y0 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0)))))))), 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y0 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0))))))))))))), 139.61/20.78 inference('ho_norm', [status(thm)], ['5'])). 139.61/20.78 tff('7', plain, 139.61/20.78 ~ mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y0 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0)))))))), 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y0 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0)))))))))), '#sk2'), 139.61/20.78 inference('lazy_cnf_exists', [status(thm)], ['6'])). 139.61/20.78 tff(mand, axiom, mand(X5, X6, X7) = mnot(mor(mnot(X5), mnot(X6)), X7)). 139.61/20.78 tff('8', plain, 139.61/20.78 ![X7,X6:(($i > $o)),X5:(($i > $o))]: 139.61/20.78 (mand(X5,X6,X7) = mnot(mor(mnot(X5),mnot(X6)),X7)), 139.61/20.78 define([status(thm)])). 139.61/20.78 tff('9', plain, 139.61/20.78 ~ mnot( 139.61/20.78 mor( 139.61/20.78 mnot( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y0 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0)))))))), 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))))))), 139.61/20.78 mnot( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y0 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0)))))))))))), '#sk2'), 139.61/20.78 inference('rw', [status(thm)], ['7', '8'])). 139.61/20.78 tff(mnot, axiom, ~ mnot(X0, X1) => (X0(X1) = $true)). 139.61/20.78 tff('10', plain, ![X0:(($i > $o)),X1]: ((~mnot(X0,X1)) <=> X0(X1)), 139.61/20.78 inference('rw.lit', [status(esa)], [mnot])). 139.61/20.78 tff('11', plain, 139.61/20.78 mor( 139.61/20.78 mnot( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y0 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0)))))))), 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))))))), 139.61/20.78 mnot( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y0 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0))))))))))), '#sk2'), 139.61/20.78 inference('rw_clause', [status(thm)], ['9', '10'])). 139.61/20.78 tff(mor, axiom, mor(X2, X3, X4) => ((X3(X4) | X2(X4)) = $true)). 139.61/20.78 tff('12', plain, 139.61/20.78 ![X2:(($i > $o)),X3:(($i > $o)),X4]: 139.61/20.78 (mor(X2,X3,X4) <=> (X2(X4) | X3(X4))), 139.61/20.78 inference('rw.lit', [status(esa)], [mor])). 139.61/20.78 tff('13', plain, 139.61/20.78 (mnot( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y0 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0)))))))))), '#sk2') | 139.61/20.78 mnot( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y0 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0)))))))), 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))))), '#sk2')), 139.61/20.78 inference('rw_clause', [status(thm)], ['11', '12'])). 139.61/20.78 tff('14', plain, 139.61/20.78 ( mnot( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y0 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0)))))))))), '#sk2') 139.61/20.78 | mnot( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y0 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0)))))))), 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))))), 139.61/20.78 '#sk2')), 139.61/20.78 inference('lazy_cnf_or', [status(thm)], ['13'])). 139.61/20.78 tff(mnot, axiom, mnot(X0, X1) => (X0(X1) != $true)). 139.61/20.78 tff('15', plain, ![X0:(($i > $o)),X1]: (mnot(X0,X1) <=> (~X0(X1))), 139.61/20.78 inference('rw.lit', [status(esa)], [mnot])). 139.61/20.78 tff('16', plain, 139.61/20.78 (~ mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y0 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0))))))))), '#sk2') 139.61/20.78 | mnot( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y0 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0)))))))), 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))))), 139.61/20.78 '#sk2')), 139.61/20.78 inference('rw_clause', [status(thm)], ['14', '15'])). 139.61/20.78 tff(mbox_s4, axiom, ~ mbox_s4(X24, X25) => 139.61/20.78 ((!!((^[Y0 : $i]: ((~(rel_s4(X25, Y0))) | X24(Y0))))) != $true)). 139.61/20.78 tff('17', plain, 139.61/20.78 ![X24:(($i > $o)),X25]: 139.61/20.78 ((~mbox_s4(X24,X25)) <=> (~(![X30]: ((~rel_s4(X25,X30)) | X24(X30))))), 139.61/20.78 inference('rw.lit', [status(esa)], [mbox_s4])). 139.61/20.78 tff('18', plain, 139.61/20.78 (~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk2', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))))), Y0))))) 139.61/20.78 | mnot( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y0 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0)))))))), 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))))), 139.61/20.78 '#sk2')), 139.61/20.78 inference('rw_clause', [status(thm)], ['16', '17'])). 139.61/20.78 tff('19', plain, 139.61/20.78 (~ mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y0 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0)))))))), 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))))), '#sk2') 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk2', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))))), Y0)))))), 139.61/20.78 inference('rw_clause', [status(thm)], ['18', '15'])). 139.61/20.78 tff('20', plain, 139.61/20.78 (~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk2', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))))), 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), Y0))))) 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk2', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))))), Y0)))))), 139.61/20.78 inference('rw_clause', [status(thm)], ['19', '17'])). 139.61/20.78 tff('21', plain, 139.61/20.78 (~ ((~(rel_s4('#sk2', '#sk3'))) | 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y0 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0)))))))), 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), '#sk3')) 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk2', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))))), Y0)))))), 139.61/20.78 inference('lazy_cnf_exists', [status(thm)], ['20'])). 139.61/20.78 tff(mimplies, axiom, mimplies(X8, X9, X10) = mor(mnot(X8), X9, X10)). 139.61/20.78 tff('22', plain, 139.61/20.78 ![X10,X9:(($i > $o)),X8:(($i > $o))]: 139.61/20.78 (mimplies(X8,X9,X10) = mor(mnot(X8),X9,X10)), 139.61/20.78 define([status(thm)])). 139.61/20.78 tff('23', plain, 139.61/20.78 (~ ((~(rel_s4('#sk2', '#sk3'))) | 139.61/20.78 mor( 139.61/20.78 mnot( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y0 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0))))))))), 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), '#sk3')) 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk2', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))))), Y0)))))), 139.61/20.78 inference('rw', [status(thm)], ['21', '22'])). 139.61/20.78 tff('24', plain, 139.61/20.78 (~ mor( 139.61/20.78 mnot( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y0 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0))))))))), 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), '#sk3') 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk2', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))))), Y0)))))), 139.61/20.78 inference('lazy_cnf_or', [status(thm)], ['23'])). 139.61/20.78 tff(mor, axiom, ~ mor(X2, X3, X4) => ((X3(X4) | X2(X4)) != $true)). 139.61/20.78 tff('25', plain, 139.61/20.78 ![X2:(($i > $o)),X3:(($i > $o)),X4]: 139.61/20.78 ((~mor(X2,X3,X4)) <=> (~(X3(X4) | X2(X4)))), 139.61/20.78 inference('rw.lit', [status(esa)], [mor])). 139.61/20.78 tff('26', plain, 139.61/20.78 (~ (mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))), '#sk3') | 139.61/20.78 mnot( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y0 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0)))))))), '#sk3')) 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk2', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))))), Y0)))))), 139.61/20.78 inference('rw_clause', [status(thm)], ['24', '25'])). 139.61/20.78 tff('27', plain, 139.61/20.78 (~ (mnot( 139.61/20.78 mor( 139.61/20.78 mnot( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))))), 139.61/20.78 mnot( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))))), '#sk3') | 139.61/20.78 mnot( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y0 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0)))))))), '#sk3')) 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk2', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))))), Y0)))))), 139.61/20.78 inference('rw', [status(thm)], ['26', '8'])). 139.61/20.78 tff('28', plain, 139.61/20.78 (~ mnot( 139.61/20.78 mor( 139.61/20.78 mnot( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))))), 139.61/20.78 mnot( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))))), '#sk3') 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk2', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))))), Y0)))))), 139.61/20.78 inference('lazy_cnf_or', [status(thm)], ['27'])). 139.61/20.78 tff('29', plain, 139.61/20.78 ( mor( 139.61/20.78 mnot( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))))), 139.61/20.78 mnot( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), '#sk3') 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk2', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))))), Y0)))))), 139.61/20.78 inference('rw_clause', [status(thm)], ['28', '10'])). 139.61/20.78 tff('30', plain, 139.61/20.78 ( (mnot( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))), '#sk3') | 139.61/20.78 mnot( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 '#sk3')) 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk2', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))))), Y0)))))), 139.61/20.78 inference('rw_clause', [status(thm)], ['29', '12'])). 139.61/20.78 tff('31', plain, 139.61/20.78 ( mnot( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))), '#sk3') 139.61/20.78 | mnot( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 '#sk3') 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk2', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))))), Y0)))))), 139.61/20.78 inference('lazy_cnf_or', [status(thm)], ['30'])). 139.61/20.78 tff('32', plain, 139.61/20.78 (~ mbox_s4( 139.61/20.78 mimplies(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), '#sk3') 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk2', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))))), Y0))))) 139.61/20.78 | mnot( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 '#sk3')), 139.61/20.78 inference('rw_clause', [status(thm)], ['31', '15'])). 139.61/20.78 tff('33', plain, 139.61/20.78 (~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk3', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), Y0))))) 139.61/20.78 | mnot( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.78 '#sk3') 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk2', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))))), Y0)))))), 139.61/20.78 inference('rw_clause', [status(thm)], ['32', '17'])). 139.61/20.78 tff('34', plain, 139.61/20.78 (~ mbox_s4( 139.61/20.78 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))), '#sk3') 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk2', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))))), Y0))))) 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk3', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), Y0)))))), 139.61/20.78 inference('rw_clause', [status(thm)], ['33', '15'])). 139.61/20.78 tff('35', plain, 139.61/20.78 (~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk3', Y0))) | 139.61/20.78 mimplies(mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), Y0))))) 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk3', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), Y0))))) 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk2', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))))), Y0)))))), 139.61/20.78 inference('rw_clause', [status(thm)], ['34', '17'])). 139.61/20.78 tff('36', plain, 139.61/20.78 (~ mnot( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y0 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0)))))))), '#sk3') 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk2', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))))), Y0)))))), 139.61/20.78 inference('lazy_cnf_or', [status(thm)], ['27'])). 139.61/20.78 tff('37', plain, 139.61/20.78 ( mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y0 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0))))))), '#sk3') 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk2', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))))), Y0)))))), 139.61/20.78 inference('rw_clause', [status(thm)], ['36', '10'])). 139.61/20.78 tff(mbox_s4, axiom, mbox_s4(X24, X25) => 139.61/20.78 ((!!((^[Y0 : $i]: ((~(rel_s4(X25, Y0))) | X24(Y0))))) = $true)). 139.61/20.78 tff('38', plain, 139.61/20.78 ![X24:(($i > $o)),X25]: 139.61/20.78 (mbox_s4(X24,X25) <=> (![X30]: ((~rel_s4(X25,X30)) | X24(X30)))), 139.61/20.78 inference('rw.lit', [status(esa)], [mbox_s4])). 139.61/20.78 tff('39', plain, 139.61/20.78 ( (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk3', Y0))) | 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))), Y0))))) 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk2', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))))), Y0)))))), 139.61/20.78 inference('rw_clause', [status(thm)], ['37', '38'])). 139.61/20.78 tff('40', plain, 139.61/20.78 ![X1]: 139.61/20.78 ( ((~(rel_s4('#sk3', X1))) | 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y0 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0)))))), X1)) 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk2', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))))), Y0)))))), 139.61/20.78 inference('lazy_cnf_forall', [status(thm)], ['39'])). 139.61/20.78 tff('41', plain, 139.61/20.78 ![X1]: 139.61/20.78 (~ rel_s4('#sk3', X1) 139.61/20.78 | mforall_ind( 139.61/20.78 (^[Y0 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0)))))), X1) 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk2', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))))), Y0)))))), 139.61/20.78 inference('lazy_cnf_or', [status(thm)], ['40'])). 139.61/20.78 tff(mforall_ind, axiom, mforall_ind(X17, X18) => 139.61/20.78 ((!!((^[Y0 : mu]: (exists_in_world(Y0, X18) => X17(Y0, X18))))) = $true)). 139.61/20.78 tff('42', plain, 139.61/20.78 ![X17:((mu * $i) > $o),X18]: 139.61/20.78 (mforall_ind(X17,X18) <=> 139.61/20.78 (![X23:mu]: (exists_in_world(X23,X18) => X17(X23,X18)))), 139.61/20.78 inference('rw.lit', [status(esa)], [mforall_ind])). 139.61/20.78 tff('43', plain, 139.61/20.78 ![X0]: 139.61/20.78 ( (!!((^[Y0 : mu]: 139.61/20.78 (exists_in_world(Y0, X0) => 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1))))))(Y0, X0))))) 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk2', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))))), Y0))))) 139.61/20.78 | ~ rel_s4('#sk3', X0)), 139.61/20.78 inference('rw_clause', [status(thm)], ['41', '42'])). 139.61/20.78 tff('44', plain, 139.61/20.78 ![X0]: 139.61/20.78 ( (!!((^[Y0 : mu]: 139.61/20.78 (exists_in_world(Y0, X0) => 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y0)), 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mbox_s4(big_p(Y0)))), X0))))) 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk2', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))))), Y0))))) 139.61/20.78 | ~ rel_s4('#sk3', X0)), 139.61/20.78 inference('ho_norm', [status(thm)], ['43'])). 139.61/20.78 tff('45', plain, 139.61/20.78 ( '#form6' 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk2', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y1 : mu]: 139.61/20.78 mand( 139.61/20.78 mbox_s4( 139.61/20.78 mimplies(mbox_s4(big_p(Y1)), 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.78 mbox_s4( 139.61/20.78 mimplies( 139.61/20.78 mbox_s4( 139.61/20.78 mforall_ind( 139.61/20.78 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.78 mbox_s4(big_p(Y1)))))))), Y0)))))), 139.61/20.78 define([status(thm)])). 139.61/20.78 tff('46', plain, 139.61/20.78 (~ '#form6' 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk3', Y0))) | 139.61/20.78 mimplies( 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), Y0))))) 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk3', Y0))) | 139.61/20.78 mimplies(mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 Y0)))))), 139.61/20.78 inference('renaming', [status(thm)], ['35', '45'])). 139.61/20.78 tff('47', plain, 139.61/20.78 (~ ((~(rel_s4('#sk3', '#sk8'))) | 139.61/20.78 mimplies(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), '#sk8')) 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk3', Y0))) | 139.61/20.78 mimplies(mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 Y0))))) 139.61/20.78 | ~ '#form6'), 139.61/20.78 inference('lazy_cnf_exists', [status(thm)], ['46'])). 139.61/20.78 tff('48', plain, 139.61/20.78 (~ ((~(rel_s4('#sk3', '#sk8'))) | 139.61/20.78 mor(mnot(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), '#sk8')) 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk3', Y0))) | 139.61/20.78 mimplies(mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 Y0))))) 139.61/20.78 | ~ '#form6'), 139.61/20.78 inference('rw', [status(thm)], ['47', '22'])). 139.61/20.78 tff('49', plain, 139.61/20.78 (~ mor(mnot(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))), 139.61/20.78 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), '#sk8') 139.61/20.78 | ~ '#form6' 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk3', Y0))) | 139.61/20.78 mimplies(mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 Y0)))))), 139.61/20.78 inference('lazy_cnf_or', [status(thm)], ['48'])). 139.61/20.78 tff('50', plain, 139.61/20.78 (~ (mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))), '#sk8') | 139.61/20.78 mnot(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), '#sk8')) 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk3', Y0))) | 139.61/20.78 mimplies(mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 Y0))))) 139.61/20.78 | ~ '#form6'), 139.61/20.78 inference('rw_clause', [status(thm)], ['49', '25'])). 139.61/20.78 tff(mexists_ind, axiom, mexists_ind(X19, X20) = 139.61/20.78 mnot(mforall_ind((^[Y0 : mu]: mnot(X19(Y0)))), X20)). 139.61/20.78 tff('51', plain, 139.61/20.78 ![X20,X19:((mu * $i) > $o)]: 139.61/20.78 (mexists_ind(X19,X20) = 139.61/20.78 mnot(mforall_ind(^[V_1:mu]: mnot(X19(V_1))),X20)), 139.61/20.78 define([status(thm)])). 139.61/20.78 tff('52', plain, 139.61/20.78 (~ (mnot(mforall_ind((^[Y0 : mu]: mnot(mbox_s4(big_p(Y0))))), '#sk8') | 139.61/20.78 mnot(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), '#sk8')) 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk3', Y0))) | 139.61/20.78 mimplies(mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 Y0))))) 139.61/20.78 | ~ '#form6'), 139.61/20.78 inference('rw', [status(thm)], ['50', '51'])). 139.61/20.78 tff('53', plain, 139.61/20.78 (~ mnot(mforall_ind((^[Y0 : mu]: mnot(mbox_s4(big_p(Y0))))), '#sk8') 139.61/20.78 | ~ '#form6' 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk3', Y0))) | 139.61/20.78 mimplies(mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 Y0)))))), 139.61/20.78 inference('lazy_cnf_or', [status(thm)], ['52'])). 139.61/20.78 tff('54', plain, 139.61/20.78 ( mforall_ind((^[Y0 : mu]: mnot(mbox_s4(big_p(Y0)))), '#sk8') 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk3', Y0))) | 139.61/20.78 mimplies(mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 Y0))))) 139.61/20.78 | ~ '#form6'), 139.61/20.78 inference('rw_clause', [status(thm)], ['53', '10'])). 139.61/20.78 tff('55', plain, 139.61/20.78 ( (!!((^[Y0 : mu]: 139.61/20.78 (exists_in_world(Y0, '#sk8') => 139.61/20.78 (^[Y1 : mu]: mnot(mbox_s4(big_p(Y1))))(Y0, '#sk8'))))) 139.61/20.78 | ~ '#form6' 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk3', Y0))) | 139.61/20.78 mimplies(mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 Y0)))))), 139.61/20.78 inference('rw_clause', [status(thm)], ['54', '42'])). 139.61/20.78 tff('56', plain, 139.61/20.78 ( (!!((^[Y0 : mu]: 139.61/20.78 (exists_in_world(Y0, '#sk8') => mnot(mbox_s4(big_p(Y0)), '#sk8'))))) 139.61/20.78 | ~ '#form6' 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk3', Y0))) | 139.61/20.78 mimplies(mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 Y0)))))), 139.61/20.78 inference('ho_norm', [status(thm)], ['55'])). 139.61/20.78 tff('57', plain, 139.61/20.78 ![X1 : mu]: 139.61/20.78 ( (exists_in_world(X1, '#sk8') => mnot(mbox_s4(big_p(X1)), '#sk8')) 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk3', Y0))) | 139.61/20.78 mimplies(mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 Y0))))) 139.61/20.78 | ~ '#form6'), 139.61/20.78 inference('lazy_cnf_forall', [status(thm)], ['56'])). 139.61/20.78 tff('58', plain, 139.61/20.78 ![X1 : mu]: 139.61/20.78 (~ exists_in_world(X1, '#sk8') 139.61/20.78 | mnot(mbox_s4(big_p(X1)), '#sk8') 139.61/20.78 | ~ '#form6' 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk3', Y0))) | 139.61/20.78 mimplies(mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 Y0)))))), 139.61/20.78 inference('lazy_cnf_imply', [status(thm)], ['57'])). 139.61/20.78 tff('59', plain, 139.61/20.78 ![X0 : mu]: 139.61/20.78 (~ mbox_s4(big_p(X0), '#sk8') 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk3', Y0))) | 139.61/20.78 mimplies(mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.78 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.78 Y0))))) 139.61/20.78 | ~ '#form6' 139.61/20.78 | ~ exists_in_world(X0, '#sk8')), 139.61/20.78 inference('rw_clause', [status(thm)], ['58', '15'])). 139.61/20.78 tff('60', plain, 139.61/20.78 ![X0 : mu]: 139.61/20.78 (~ (!!((^[Y0 : $i]: ((~(rel_s4('#sk8', Y0))) | big_p(X0, Y0))))) 139.61/20.78 | ~ exists_in_world(X0, '#sk8') 139.61/20.78 | ~ '#form6' 139.61/20.78 | ~ (!!((^[Y0 : $i]: 139.61/20.78 ((~(rel_s4('#sk3', Y0))) | 139.61/20.78 mimplies(mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 Y0)))))), 139.61/20.79 inference('rw_clause', [status(thm)], ['59', '17'])). 139.61/20.79 tff('61', plain, 139.61/20.79 (~ mnot(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), '#sk8') 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk3', Y0))) | 139.61/20.79 mimplies(mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 Y0)))))), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['52'])). 139.61/20.79 tff('62', plain, 139.61/20.79 ( mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), '#sk8') 139.61/20.79 | ~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk3', Y0))) | 139.61/20.79 mimplies(mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 Y0))))) 139.61/20.79 | ~ '#form6'), 139.61/20.79 inference('rw_clause', [status(thm)], ['61', '10'])). 139.61/20.79 tff('63', plain, 139.61/20.79 ( (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk8', Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0))))) 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk3', Y0))) | 139.61/20.79 mimplies(mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 Y0)))))), 139.61/20.79 inference('rw_clause', [status(thm)], ['62', '38'])). 139.61/20.79 tff('64', plain, 139.61/20.79 ![X1]: 139.61/20.79 ( ((~(rel_s4('#sk8', X1))) | 139.61/20.79 mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))), X1)) 139.61/20.79 | ~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk3', Y0))) | 139.61/20.79 mimplies(mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 Y0))))) 139.61/20.79 | ~ '#form6'), 139.61/20.79 inference('lazy_cnf_forall', [status(thm)], ['63'])). 139.61/20.79 tff('65', plain, 139.61/20.79 ( '#form12' 139.61/20.79 | ~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk3', Y0))) | 139.61/20.79 mimplies(mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 Y0)))))), 139.61/20.79 define([status(thm)])). 139.61/20.79 tff('66', plain, 139.61/20.79 ![X0 : mu]: 139.61/20.79 (~ '#form12' 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ exists_in_world(X0, '#sk8') 139.61/20.79 | ~ (!!((^[Y0 : $i]: ((~(rel_s4('#sk8', Y0))) | big_p(X0, Y0)))))), 139.61/20.79 inference('renaming', [status(thm)], ['60', '65'])). 139.61/20.79 tff('67', plain, 139.61/20.79 ![X0 : mu]: 139.61/20.79 (~ ((~(rel_s4('#sk8', '#sk19'(X0)))) | big_p(X0, '#sk19'(X0))) 139.61/20.79 | ~ exists_in_world(X0, '#sk8') 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ '#form12'), 139.61/20.79 inference('lazy_cnf_exists', [status(thm)], ['66'])). 139.61/20.79 tff('68', plain, 139.61/20.79 ![X0 : mu]: 139.61/20.79 ( rel_s4('#sk8', '#sk19'(X0)) 139.61/20.79 | ~ '#form12' 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ exists_in_world(X0, '#sk8')), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['67'])). 139.61/20.79 tff('69', plain, 139.61/20.79 ![X0 : mu]: 139.61/20.79 (~ big_p(X0, '#sk19'(X0)) 139.61/20.79 | ~ '#form12' 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ exists_in_world(X0, '#sk8')), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['67'])). 139.61/20.79 tff('70', plain, 139.61/20.79 ![X1]: 139.61/20.79 ( ((~(rel_s4('#sk8', X1))) | 139.61/20.79 mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))), X1)) 139.61/20.79 | ~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk3', Y0))) | 139.61/20.79 mimplies(mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 Y0))))) 139.61/20.79 | ~ '#form6'), 139.61/20.79 inference('lazy_cnf_forall', [status(thm)], ['63'])). 139.61/20.79 tff('71', plain, 139.61/20.79 ( '#form12' 139.61/20.79 | ~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk3', Y0))) | 139.61/20.79 mimplies(mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 Y0)))))), 139.61/20.79 define([status(thm)])). 139.61/20.79 tff('72', plain, 139.61/20.79 ![X1]: 139.61/20.79 (~ '#form12' 139.61/20.79 | ~ '#form6' 139.61/20.79 | ((~(rel_s4('#sk8', X1))) | 139.61/20.79 mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))), X1))), 139.61/20.79 inference('renaming', [status(thm)], ['70', '71'])). 139.61/20.79 tff('73', plain, 139.61/20.79 ![X1]: 139.61/20.79 (~ rel_s4('#sk8', X1) 139.61/20.79 | mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))), X1) 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ '#form12'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['72'])). 139.61/20.79 tff('74', plain, 139.61/20.79 ![X0]: 139.61/20.79 ( (!!((^[Y0 : mu]: 139.61/20.79 (exists_in_world(Y0, X0) => 139.61/20.79 (^[Y1 : mu]: mbox_s4(big_p(Y1)))(Y0, X0))))) 139.61/20.79 | ~ '#form12' 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ rel_s4('#sk8', X0)), 139.61/20.79 inference('rw_clause', [status(thm)], ['73', '42'])). 139.61/20.79 tff('75', plain, 139.61/20.79 ![X0]: 139.61/20.79 ( (!!((^[Y0 : mu]: (exists_in_world(Y0, X0) => mbox_s4(big_p(Y0), X0))))) 139.61/20.79 | ~ '#form12' 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ rel_s4('#sk8', X0)), 139.61/20.79 inference('ho_norm', [status(thm)], ['74'])). 139.61/20.79 tff('76', plain, 139.61/20.79 ![X0, X1 : mu]: 139.61/20.79 ( (exists_in_world(X1, X0) => mbox_s4(big_p(X1), X0)) 139.61/20.79 | ~ rel_s4('#sk8', X0) 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ '#form12'), 139.61/20.79 inference('lazy_cnf_forall', [status(thm)], ['75'])). 139.61/20.79 tff('77', plain, 139.61/20.79 ![X0, X1 : mu]: 139.61/20.79 (~ exists_in_world(X1, X0) 139.61/20.79 | mbox_s4(big_p(X1), X0) 139.61/20.79 | ~ '#form12' 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ rel_s4('#sk8', X0)), 139.61/20.79 inference('lazy_cnf_imply', [status(thm)], ['76'])). 139.61/20.79 tff('78', plain, 139.61/20.79 ![X0, X1 : mu]: 139.61/20.79 ( (!!((^[Y0 : $i]: ((~(rel_s4(X0, Y0))) | big_p(X1, Y0))))) 139.61/20.79 | ~ rel_s4('#sk8', X0) 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ '#form12' 139.61/20.79 | ~ exists_in_world(X1, X0)), 139.61/20.79 inference('rw_clause', [status(thm)], ['77', '38'])). 139.61/20.79 tff('79', plain, 139.61/20.79 ![X0, X1 : mu, X2]: 139.61/20.79 ( ((~(rel_s4(X0, X2))) | big_p(X1, X2)) 139.61/20.79 | ~ exists_in_world(X1, X0) 139.61/20.79 | ~ '#form12' 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ rel_s4('#sk8', X0)), 139.61/20.79 inference('lazy_cnf_forall', [status(thm)], ['78'])). 139.61/20.79 tff('80', plain, 139.61/20.79 ![X0, X1 : mu, X2]: 139.61/20.79 (~ rel_s4(X0, X2) 139.61/20.79 | big_p(X1, X2) 139.61/20.79 | ~ rel_s4('#sk8', X0) 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ '#form12' 139.61/20.79 | ~ exists_in_world(X1, X0)), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['79'])). 139.61/20.79 tff('81', plain, 139.61/20.79 ( '#form12' 139.61/20.79 | ~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk3', Y0))) | 139.61/20.79 mimplies(mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 Y0)))))), 139.61/20.79 define([status(thm)])). 139.61/20.79 tff('82', plain, 139.61/20.79 (~ ((~(rel_s4('#sk3', '#sk9'))) | 139.61/20.79 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), '#sk9')) 139.61/20.79 | '#form12'), 139.61/20.79 inference('lazy_cnf_exists', [status(thm)], ['81'])). 139.61/20.79 tff('83', plain, 139.61/20.79 (~ ((~(rel_s4('#sk3', '#sk9'))) | 139.61/20.79 mor(mnot(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), '#sk9')) 139.61/20.79 | '#form12'), 139.61/20.79 inference('rw', [status(thm)], ['82', '22'])). 139.61/20.79 tff('84', plain, 139.61/20.79 (~ mor(mnot(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), '#sk9') 139.61/20.79 | '#form12'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['83'])). 139.61/20.79 tff('85', plain, 139.61/20.79 (~ (mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), '#sk9') | 139.61/20.79 mnot(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), '#sk9')) 139.61/20.79 | '#form12'), 139.61/20.79 inference('rw_clause', [status(thm)], ['84', '25'])). 139.61/20.79 tff('86', plain, 139.61/20.79 (~ mnot(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), '#sk9') 139.61/20.79 | '#form12'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['85'])). 139.61/20.79 tff('87', plain, 139.61/20.79 ( mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))), '#sk9') | '#form12'), 139.61/20.79 inference('rw_clause', [status(thm)], ['86', '10'])). 139.61/20.79 tff('88', plain, 139.61/20.79 ( mnot(mforall_ind((^[Y0 : mu]: mnot(mbox_s4(big_p(Y0))))), '#sk9') 139.61/20.79 | '#form12'), 139.61/20.79 inference('rw', [status(thm)], ['87', '51'])). 139.61/20.79 tff('89', plain, 139.61/20.79 (~ mforall_ind((^[Y0 : mu]: mnot(mbox_s4(big_p(Y0)))), '#sk9') 139.61/20.79 | '#form12'), 139.61/20.79 inference('rw_clause', [status(thm)], ['88', '15'])). 139.61/20.79 tff(mforall_ind, axiom, ~ mforall_ind(X17, X18) => 139.61/20.79 ((!!((^[Y0 : mu]: (exists_in_world(Y0, X18) => X17(Y0, X18))))) != $true)). 139.61/20.79 tff('90', plain, 139.61/20.79 ![X17:((mu * $i) > $o),X18]: 139.61/20.79 ((~mforall_ind(X17,X18)) <=> 139.61/20.79 (~(![X23:mu]: (exists_in_world(X23,X18) => X17(X23,X18))))), 139.61/20.79 inference('rw.lit', [status(esa)], [mforall_ind])). 139.61/20.79 tff('91', plain, 139.61/20.79 (~ (!!((^[Y0 : mu]: 139.61/20.79 (exists_in_world(Y0, '#sk9') => 139.61/20.79 (^[Y1 : mu]: mnot(mbox_s4(big_p(Y1))))(Y0, '#sk9'))))) 139.61/20.79 | '#form12'), 139.61/20.79 inference('rw_clause', [status(thm)], ['89', '90'])). 139.61/20.79 tff('92', plain, 139.61/20.79 (~ (!!((^[Y0 : mu]: 139.61/20.79 (exists_in_world(Y0, '#sk9') => 139.61/20.79 mnot(mbox_s4(big_p(Y0)), '#sk9'))))) 139.61/20.79 | '#form12'), 139.61/20.79 inference('ho_norm', [status(thm)], ['91'])). 139.61/20.79 tff('93', plain, 139.61/20.79 (~ (exists_in_world('#sk15', '#sk9') => 139.61/20.79 mnot(mbox_s4(big_p('#sk15')), '#sk9')) 139.61/20.79 | '#form12'), 139.61/20.79 inference('lazy_cnf_exists', [status(thm)], ['92'])). 139.61/20.79 tff('94', plain, (~ mnot(mbox_s4(big_p('#sk15')), '#sk9') | '#form12'), 139.61/20.79 inference('lazy_cnf_imply', [status(thm)], ['93'])). 139.61/20.79 tff('95', plain, ( mbox_s4(big_p('#sk15'), '#sk9') | '#form12'), 139.61/20.79 inference('rw_clause', [status(thm)], ['94', '10'])). 139.61/20.79 tff('96', plain, 139.61/20.79 ( (!!((^[Y0 : $i]: ((~(rel_s4('#sk9', Y0))) | big_p('#sk15', Y0))))) 139.61/20.79 | '#form12'), 139.61/20.79 inference('rw_clause', [status(thm)], ['95', '38'])). 139.61/20.79 tff('97', plain, 139.61/20.79 ![X1]: ( ((~(rel_s4('#sk9', X1))) | big_p('#sk15', X1)) | '#form12'), 139.61/20.79 inference('lazy_cnf_forall', [status(thm)], ['96'])). 139.61/20.79 tff('98', plain, 139.61/20.79 ![X1]: (~ rel_s4('#sk9', X1) | big_p('#sk15', X1) | '#form12'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['97'])). 139.61/20.79 tff('99', plain, 139.61/20.79 ![X0]: 139.61/20.79 ( (!!((^[Y0 : mu]: 139.61/20.79 (exists_in_world(Y0, X0) => 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(Y0)), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p(Y0)))), X0))))) 139.61/20.79 | ~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk2', Y0))) | 139.61/20.79 mimplies( 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind( 139.61/20.79 (^[Y1 : mu]: 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(Y1)), 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind( 139.61/20.79 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind( 139.61/20.79 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.79 mbox_s4(big_p(Y1)))))))), Y0))))) 139.61/20.79 | ~ rel_s4('#sk3', X0)), 139.61/20.79 inference('ho_norm', [status(thm)], ['43'])). 139.61/20.79 tff('100', plain, 139.61/20.79 ( '#form6' 139.61/20.79 | ~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk2', Y0))) | 139.61/20.79 mimplies( 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind( 139.61/20.79 (^[Y1 : mu]: 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(Y1)), 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind( 139.61/20.79 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind( 139.61/20.79 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.79 mbox_s4(big_p(Y1)))))))), Y0)))))), 139.61/20.79 define([status(thm)])). 139.61/20.79 tff('101', plain, 139.61/20.79 ![X0]: 139.61/20.79 (~ '#form6' 139.61/20.79 | ~ rel_s4('#sk3', X0) 139.61/20.79 | (!!((^[Y0 : mu]: 139.61/20.79 (exists_in_world(Y0, X0) => 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(Y0)), 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p(Y0)))), X0)))))), 139.61/20.79 inference('renaming', [status(thm)], ['99', '100'])). 139.61/20.79 tff('102', plain, 139.61/20.79 ![X0, X1 : mu]: 139.61/20.79 ( (exists_in_world(X1, X0) => 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(X1)), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mbox_s4(big_p(X1)))), X0)) 139.61/20.79 | ~ rel_s4('#sk3', X0) 139.61/20.79 | ~ '#form6'), 139.61/20.79 inference('lazy_cnf_forall', [status(thm)], ['101'])). 139.61/20.79 tff('103', plain, 139.61/20.79 ![X0, X1 : mu]: 139.61/20.79 ( (exists_in_world(X1, X0) => 139.61/20.79 mnot( 139.61/20.79 mor( 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(X1)), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))))), 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mbox_s4(big_p(X1)))))), X0)) 139.61/20.79 | ~ rel_s4('#sk3', X0) 139.61/20.79 | ~ '#form6'), 139.61/20.79 inference('rw', [status(thm)], ['102', '8'])). 139.61/20.79 tff('104', plain, 139.61/20.79 ![X0, X1 : mu]: 139.61/20.79 (~ exists_in_world(X1, X0) 139.61/20.79 | mnot( 139.61/20.79 mor( 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(X1)), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))))), 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mbox_s4(big_p(X1)))))), X0) 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ rel_s4('#sk3', X0)), 139.61/20.79 inference('lazy_cnf_imply', [status(thm)], ['103'])). 139.61/20.79 tff('105', plain, 139.61/20.79 ![X0, X1 : mu]: 139.61/20.79 (~ mor( 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(X1)), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))))), 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mbox_s4(big_p(X1))))), X0) 139.61/20.79 | ~ rel_s4('#sk3', X0) 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ exists_in_world(X1, X0)), 139.61/20.79 inference('rw_clause', [status(thm)], ['104', '15'])). 139.61/20.79 tff('106', plain, 139.61/20.79 ![X0, X1 : mu]: 139.61/20.79 (~ (mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mbox_s4(big_p(X1)))), X0) | 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(X1)), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), X0)) 139.61/20.79 | ~ exists_in_world(X1, X0) 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ rel_s4('#sk3', X0)), 139.61/20.79 inference('rw_clause', [status(thm)], ['105', '25'])). 139.61/20.79 tff('107', plain, 139.61/20.79 ![X0, X1 : mu]: 139.61/20.79 (~ mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(X1)), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), X0) 139.61/20.79 | ~ rel_s4('#sk3', X0) 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ exists_in_world(X1, X0)), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['106'])). 139.61/20.79 tff('108', plain, 139.61/20.79 ![X0, X1 : mu]: 139.61/20.79 ( mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(X1)), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))), X0) 139.61/20.79 | ~ exists_in_world(X1, X0) 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ rel_s4('#sk3', X0)), 139.61/20.79 inference('rw_clause', [status(thm)], ['107', '10'])). 139.61/20.79 tff('109', plain, 139.61/20.79 ![X0, X1 : mu]: 139.61/20.79 ( (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mimplies(mbox_s4(big_p(X1)), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), Y0))))) 139.61/20.79 | ~ rel_s4('#sk3', X0) 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ exists_in_world(X1, X0)), 139.61/20.79 inference('rw_clause', [status(thm)], ['108', '38'])). 139.61/20.79 tff('110', plain, 139.61/20.79 ![X0, X1 : mu, X2]: 139.61/20.79 ( ((~(rel_s4(X0, X2))) | 139.61/20.79 mimplies(mbox_s4(big_p(X1)), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), X2)) 139.61/20.79 | ~ exists_in_world(X1, X0) 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ rel_s4('#sk3', X0)), 139.61/20.79 inference('lazy_cnf_forall', [status(thm)], ['109'])). 139.61/20.79 tff('111', plain, 139.61/20.79 ![X0, X1 : mu, X2]: 139.61/20.79 ( ((~(rel_s4(X0, X2))) | 139.61/20.79 mor(mnot(mbox_s4(big_p(X1))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), X2)) 139.61/20.79 | ~ exists_in_world(X1, X0) 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ rel_s4('#sk3', X0)), 139.61/20.79 inference('rw', [status(thm)], ['110', '22'])). 139.61/20.79 tff('112', plain, 139.61/20.79 ![X0, X1 : mu, X2]: 139.61/20.79 (~ rel_s4(X0, X2) 139.61/20.79 | mor(mnot(mbox_s4(big_p(X1))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), X2) 139.61/20.79 | ~ rel_s4('#sk3', X0) 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ exists_in_world(X1, X0)), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['111'])). 139.61/20.79 tff('113', plain, 139.61/20.79 ![X0, X1 : mu, X2]: 139.61/20.79 ( (mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), X2) | 139.61/20.79 mnot(mbox_s4(big_p(X1)), X2)) 139.61/20.79 | ~ exists_in_world(X1, X0) 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ rel_s4('#sk3', X0) 139.61/20.79 | ~ rel_s4(X0, X2)), 139.61/20.79 inference('rw_clause', [status(thm)], ['112', '12'])). 139.61/20.79 tff('114', plain, 139.61/20.79 ![X0, X1 : mu, X2]: 139.61/20.79 ( mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), X2) 139.61/20.79 | mnot(mbox_s4(big_p(X1)), X2) 139.61/20.79 | ~ rel_s4(X0, X2) 139.61/20.79 | ~ rel_s4('#sk3', X0) 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ exists_in_world(X1, X0)), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['113'])). 139.61/20.79 tff('115', plain, 139.61/20.79 ![X0, X1 : mu, X2]: 139.61/20.79 ( (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X2, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0))))) 139.61/20.79 | ~ exists_in_world(X1, X0) 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ rel_s4('#sk3', X0) 139.61/20.79 | ~ rel_s4(X0, X2) 139.61/20.79 | mnot(mbox_s4(big_p(X1)), X2)), 139.61/20.79 inference('rw_clause', [status(thm)], ['114', '38'])). 139.61/20.79 tff('116', plain, 139.61/20.79 ![X0, X1, X2 : mu]: 139.61/20.79 (~ mbox_s4(big_p(X2), X0) 139.61/20.79 | ~ rel_s4(X1, X0) 139.61/20.79 | ~ rel_s4('#sk3', X1) 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ exists_in_world(X2, X1) 139.61/20.79 | (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0)))))), 139.61/20.79 inference('rw_clause', [status(thm)], ['115', '15'])). 139.61/20.79 tff('117', plain, 139.61/20.79 ![X0, X1, X2 : mu]: 139.61/20.79 (~ (!!((^[Y0 : $i]: ((~(rel_s4(X0, Y0))) | big_p(X2, Y0))))) 139.61/20.79 | (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0))))) 139.61/20.79 | ~ exists_in_world(X2, X1) 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ rel_s4('#sk3', X1) 139.61/20.79 | ~ rel_s4(X1, X0)), 139.61/20.79 inference('rw_clause', [status(thm)], ['116', '17'])). 139.61/20.79 tff('118', plain, 139.61/20.79 ![X0, X1, X2 : mu]: 139.61/20.79 (~ ((~(rel_s4(X0, '#sk20'(X0, X2)))) | big_p(X2, '#sk20'(X0, X2))) 139.61/20.79 | ~ rel_s4(X1, X0) 139.61/20.79 | ~ rel_s4('#sk3', X1) 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ exists_in_world(X2, X1) 139.61/20.79 | (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0)))))), 139.61/20.79 inference('lazy_cnf_exists', [status(thm)], ['117'])). 139.61/20.79 tff('119', plain, 139.61/20.79 ![X0, X1, X2 : mu]: 139.61/20.79 (~ big_p(X2, '#sk20'(X0, X2)) 139.61/20.79 | (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0))))) 139.61/20.79 | ~ exists_in_world(X2, X1) 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ rel_s4('#sk3', X1) 139.61/20.79 | ~ rel_s4(X1, X0)), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['118'])). 139.61/20.79 tff('120', plain, 139.61/20.79 ( rel_s4('#sk2', '#sk3') 139.61/20.79 | ~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk2', Y0))) | 139.61/20.79 mimplies( 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind( 139.61/20.79 (^[Y1 : mu]: 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(Y1)), 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind( 139.61/20.79 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind( 139.61/20.79 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.79 mbox_s4(big_p(Y1)))))))), Y0)))))), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['23'])). 139.61/20.79 tff('121', plain, 139.61/20.79 (~ ((~(rel_s4('#sk2', '#sk4'))) | 139.61/20.79 mimplies( 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind( 139.61/20.79 (^[Y0 : mu]: 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(Y0)), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p(Y0)))))))), '#sk4')) 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('lazy_cnf_exists', [status(thm)], ['120'])). 139.61/20.79 tff('122', plain, 139.61/20.79 (~ ((~(rel_s4('#sk2', '#sk4'))) | 139.61/20.79 mor( 139.61/20.79 mnot( 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind( 139.61/20.79 (^[Y0 : mu]: 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(Y0)), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p(Y0)))))))), '#sk4')) 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('rw', [status(thm)], ['121', '22'])). 139.61/20.79 tff('123', plain, 139.61/20.79 (~ mor( 139.61/20.79 mnot( 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind( 139.61/20.79 (^[Y0 : mu]: 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(Y0)), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p(Y0)))))))), '#sk4') 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['122'])). 139.61/20.79 tff('124', plain, 139.61/20.79 (~ (mbox_s4( 139.61/20.79 mforall_ind( 139.61/20.79 (^[Y0 : mu]: 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(Y0)), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p(Y0))))))), '#sk4') | 139.61/20.79 mnot( 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), '#sk4')) 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('rw_clause', [status(thm)], ['123', '25'])). 139.61/20.79 tff('125', plain, 139.61/20.79 (~ mnot( 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), '#sk4') 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['124'])). 139.61/20.79 tff('126', plain, 139.61/20.79 ( mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))), '#sk4') 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('rw_clause', [status(thm)], ['125', '10'])). 139.61/20.79 tff('127', plain, 139.61/20.79 ( mnot( 139.61/20.79 mor( 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))))), 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))))), '#sk4') 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('rw', [status(thm)], ['126', '8'])). 139.61/20.79 tff('128', plain, 139.61/20.79 (~ mor( 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))))), 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), '#sk4') 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('rw_clause', [status(thm)], ['127', '15'])). 139.61/20.79 tff('129', plain, 139.61/20.79 (~ (mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))), '#sk4') | 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 '#sk4')) 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('rw_clause', [status(thm)], ['128', '25'])). 139.61/20.79 tff('130', plain, 139.61/20.79 (~ mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 '#sk4') 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['129'])). 139.61/20.79 tff('131', plain, 139.61/20.79 ( mbox_s4( 139.61/20.79 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))), '#sk4') 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('rw_clause', [status(thm)], ['130', '10'])). 139.61/20.79 tff('132', plain, 139.61/20.79 ( (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk4', Y0))) | 139.61/20.79 mimplies(mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), Y0))))) 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('rw_clause', [status(thm)], ['131', '38'])). 139.61/20.79 tff('133', plain, 139.61/20.79 ![X1]: 139.61/20.79 ( ((~(rel_s4('#sk4', X1))) | 139.61/20.79 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), X1)) 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('lazy_cnf_forall', [status(thm)], ['132'])). 139.61/20.79 tff('134', plain, 139.61/20.79 ![X1]: 139.61/20.79 ( ((~(rel_s4('#sk4', X1))) | 139.61/20.79 mor(mnot(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), X1)) 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('rw', [status(thm)], ['133', '22'])). 139.61/20.79 tff('135', plain, 139.61/20.79 ![X1]: 139.61/20.79 (~ rel_s4('#sk4', X1) 139.61/20.79 | mor(mnot(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), X1) 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['134'])). 139.61/20.79 tff('136', plain, 139.61/20.79 ![X0]: 139.61/20.79 ( (mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), X0) | 139.61/20.79 mnot(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), X0)) 139.61/20.79 | rel_s4('#sk2', '#sk3') 139.61/20.79 | ~ rel_s4('#sk4', X0)), 139.61/20.79 inference('rw_clause', [status(thm)], ['135', '12'])). 139.61/20.79 tff('137', plain, 139.61/20.79 ![X0]: 139.61/20.79 ( mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), X0) 139.61/20.79 | mnot(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), X0) 139.61/20.79 | ~ rel_s4('#sk4', X0) 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['136'])). 139.61/20.79 tff('138', plain, 139.61/20.79 ![X0]: 139.61/20.79 ( (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0))))) 139.61/20.79 | rel_s4('#sk2', '#sk3') 139.61/20.79 | ~ rel_s4('#sk4', X0) 139.61/20.79 | mnot(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), X0)), 139.61/20.79 inference('rw_clause', [status(thm)], ['137', '38'])). 139.61/20.79 tff('139', plain, 139.61/20.79 ![X0]: 139.61/20.79 (~ mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))), X0) 139.61/20.79 | ~ rel_s4('#sk4', X0) 139.61/20.79 | rel_s4('#sk2', '#sk3') 139.61/20.79 | (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0)))))), 139.61/20.79 inference('rw_clause', [status(thm)], ['138', '15'])). 139.61/20.79 tff('140', plain, 139.61/20.79 ![X0]: 139.61/20.79 (~ mnot(mforall_ind((^[Y0 : mu]: mnot(mbox_s4(big_p(Y0))))), X0) 139.61/20.79 | ~ rel_s4('#sk4', X0) 139.61/20.79 | rel_s4('#sk2', '#sk3') 139.61/20.79 | (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0)))))), 139.61/20.79 inference('rw', [status(thm)], ['139', '51'])). 139.61/20.79 tff('141', plain, 139.61/20.79 ![X0]: 139.61/20.79 ( mforall_ind((^[Y0 : mu]: mnot(mbox_s4(big_p(Y0)))), X0) 139.61/20.79 | (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0))))) 139.61/20.79 | rel_s4('#sk2', '#sk3') 139.61/20.79 | ~ rel_s4('#sk4', X0)), 139.61/20.79 inference('rw_clause', [status(thm)], ['140', '10'])). 139.61/20.79 tff('142', plain, 139.61/20.79 ![X0]: 139.61/20.79 ( (!!((^[Y0 : mu]: 139.61/20.79 (exists_in_world(Y0, X0) => 139.61/20.79 (^[Y1 : mu]: mnot(mbox_s4(big_p(Y1))))(Y0, X0))))) 139.61/20.79 | ~ rel_s4('#sk4', X0) 139.61/20.79 | rel_s4('#sk2', '#sk3') 139.61/20.79 | (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0)))))), 139.61/20.79 inference('rw_clause', [status(thm)], ['141', '42'])). 139.61/20.79 tff('143', plain, 139.61/20.79 ![X0]: 139.61/20.79 ( (!!((^[Y0 : mu]: 139.61/20.79 (exists_in_world(Y0, X0) => mnot(mbox_s4(big_p(Y0)), X0))))) 139.61/20.79 | ~ rel_s4('#sk4', X0) 139.61/20.79 | rel_s4('#sk2', '#sk3') 139.61/20.79 | (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0)))))), 139.61/20.79 inference('ho_norm', [status(thm)], ['142'])). 139.61/20.79 tff('144', plain, 139.61/20.79 ![X0, X1 : mu]: 139.61/20.79 ( (exists_in_world(X1, X0) => mnot(mbox_s4(big_p(X1)), X0)) 139.61/20.79 | (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0))))) 139.61/20.79 | rel_s4('#sk2', '#sk3') 139.61/20.79 | ~ rel_s4('#sk4', X0)), 139.61/20.79 inference('lazy_cnf_forall', [status(thm)], ['143'])). 139.61/20.79 tff('145', plain, 139.61/20.79 ![X0, X1 : mu]: 139.61/20.79 (~ exists_in_world(X1, X0) 139.61/20.79 | mnot(mbox_s4(big_p(X1)), X0) 139.61/20.79 | ~ rel_s4('#sk4', X0) 139.61/20.79 | rel_s4('#sk2', '#sk3') 139.61/20.79 | (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0)))))), 139.61/20.79 inference('lazy_cnf_imply', [status(thm)], ['144'])). 139.61/20.79 tff('146', plain, 139.61/20.79 ![X0, X1 : mu]: 139.61/20.79 (~ mbox_s4(big_p(X1), X0) 139.61/20.79 | (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0))))) 139.61/20.79 | rel_s4('#sk2', '#sk3') 139.61/20.79 | ~ rel_s4('#sk4', X0) 139.61/20.79 | ~ exists_in_world(X1, X0)), 139.61/20.79 inference('rw_clause', [status(thm)], ['145', '15'])). 139.61/20.79 tff('147', plain, 139.61/20.79 ![X0, X1 : mu]: 139.61/20.79 (~ (!!((^[Y0 : $i]: ((~(rel_s4(X0, Y0))) | big_p(X1, Y0))))) 139.61/20.79 | ~ exists_in_world(X1, X0) 139.61/20.79 | ~ rel_s4('#sk4', X0) 139.61/20.79 | rel_s4('#sk2', '#sk3') 139.61/20.79 | (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0)))))), 139.61/20.79 inference('rw_clause', [status(thm)], ['146', '17'])). 139.61/20.79 tff('148', plain, 139.61/20.79 ![X0, X1 : mu]: 139.61/20.79 (~ ((~(rel_s4(X0, '#sk20'(X0, X1)))) | big_p(X1, '#sk20'(X0, X1))) 139.61/20.79 | (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0))))) 139.61/20.79 | rel_s4('#sk2', '#sk3') 139.61/20.79 | ~ rel_s4('#sk4', X0) 139.61/20.79 | ~ exists_in_world(X1, X0)), 139.61/20.79 inference('lazy_cnf_exists', [status(thm)], ['147'])). 139.61/20.79 tff('149', plain, 139.61/20.79 ![X0, X1 : mu]: 139.61/20.79 ( rel_s4(X0, '#sk20'(X0, X1)) 139.61/20.79 | ~ exists_in_world(X1, X0) 139.61/20.79 | ~ rel_s4('#sk4', X0) 139.61/20.79 | rel_s4('#sk2', '#sk3') 139.61/20.79 | (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0)))))), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['148'])). 139.61/20.79 tff('150', plain, 139.61/20.79 ![X0]: 139.61/20.79 (~ '#form22'(X0) 139.61/20.79 | (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0)))))), 139.61/20.79 define([status(thm)])). 139.61/20.79 tff('151', plain, 139.61/20.79 ![X0, X1, X2 : mu]: 139.61/20.79 ( '#form22'(X0) 139.61/20.79 | ~ rel_s4(X1, X0) 139.61/20.79 | ~ rel_s4('#sk3', X1) 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ exists_in_world(X2, X1) 139.61/20.79 | ~ big_p(X2, '#sk20'(X0, X2))), 139.61/20.79 inference('renaming', [status(thm)], ['119', '150'])). 139.61/20.79 tff('152', plain, 139.61/20.79 (~ mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), '#sk9') 139.61/20.79 | '#form12'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['85'])). 139.61/20.79 tff('153', plain, 139.61/20.79 (~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk9', Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0))))) 139.61/20.79 | '#form12'), 139.61/20.79 inference('rw_clause', [status(thm)], ['152', '17'])). 139.61/20.79 tff('154', plain, 139.61/20.79 (~ ((~(rel_s4('#sk9', '#sk11'))) | 139.61/20.79 mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))), '#sk11')) 139.61/20.79 | '#form12'), 139.61/20.79 inference('lazy_cnf_exists', [status(thm)], ['153'])). 139.61/20.79 tff('155', plain, 139.61/20.79 (~ mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))), '#sk11') | '#form12'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['154'])). 139.61/20.79 tff('156', plain, 139.61/20.79 (~ (!!((^[Y0 : mu]: 139.61/20.79 (exists_in_world(Y0, '#sk11') => 139.61/20.79 (^[Y1 : mu]: mbox_s4(big_p(Y1)))(Y0, '#sk11'))))) 139.61/20.79 | '#form12'), 139.61/20.79 inference('rw_clause', [status(thm)], ['155', '90'])). 139.61/20.79 tff('157', plain, 139.61/20.79 (~ (!!((^[Y0 : mu]: 139.61/20.79 (exists_in_world(Y0, '#sk11') => mbox_s4(big_p(Y0), '#sk11'))))) 139.61/20.79 | '#form12'), 139.61/20.79 inference('ho_norm', [status(thm)], ['156'])). 139.61/20.79 tff('158', plain, 139.61/20.79 (~ (exists_in_world('#sk17', '#sk11') => mbox_s4(big_p('#sk17'), '#sk11')) 139.61/20.79 | '#form12'), 139.61/20.79 inference('lazy_cnf_exists', [status(thm)], ['157'])). 139.61/20.79 tff('159', plain, ( exists_in_world('#sk17', '#sk11') | '#form12'), 139.61/20.79 inference('lazy_cnf_imply', [status(thm)], ['158'])). 139.61/20.79 tff('160', plain, (~ mbox_s4(big_p('#sk17'), '#sk11') | '#form12'), 139.61/20.79 inference('lazy_cnf_imply', [status(thm)], ['158'])). 139.61/20.79 tff('161', plain, 139.61/20.79 (~ (!!((^[Y0 : $i]: ((~(rel_s4('#sk11', Y0))) | big_p('#sk17', Y0))))) 139.61/20.79 | '#form12'), 139.61/20.79 inference('rw_clause', [status(thm)], ['160', '17'])). 139.61/20.79 tff('162', plain, 139.61/20.79 (~ ((~(rel_s4('#sk11', '#sk20'('#sk11', '#sk17')))) | 139.61/20.79 big_p('#sk17', '#sk20'('#sk11', '#sk17'))) 139.61/20.79 | '#form12'), 139.61/20.79 inference('lazy_cnf_exists', [status(thm)], ['161'])). 139.61/20.79 tff('163', plain, (~ big_p('#sk17', '#sk20'('#sk11', '#sk17')) | '#form12'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['162'])). 139.61/20.79 tff('164', plain, ( rel_s4('#sk11', '#sk20'('#sk11', '#sk17')) | '#form12'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['162'])). 139.61/20.79 tff('165', plain, ( exists_in_world('#sk15', '#sk9') | '#form12'), 139.61/20.79 inference('lazy_cnf_imply', [status(thm)], ['93'])). 139.61/20.79 tff('166', plain, 139.61/20.79 ![X0]: 139.61/20.79 (~ '#form22'(X0) 139.61/20.79 | (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0)))))), 139.61/20.79 define([status(thm)])). 139.61/20.79 tff('167', plain, 139.61/20.79 ![X0, X1]: 139.61/20.79 ( ((~(rel_s4(X0, X1))) | 139.61/20.79 mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))), X1)) 139.61/20.79 | ~ '#form22'(X0)), 139.61/20.79 inference('lazy_cnf_forall', [status(thm)], ['166'])). 139.61/20.79 tff('168', plain, 139.61/20.79 ![X0, X1]: 139.61/20.79 (~ rel_s4(X0, X1) 139.61/20.79 | mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))), X1) 139.61/20.79 | ~ '#form22'(X0)), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['167'])). 139.61/20.79 tff('169', plain, 139.61/20.79 ![X0, X1]: 139.61/20.79 ( (!!((^[Y0 : mu]: 139.61/20.79 (exists_in_world(Y0, X1) => 139.61/20.79 (^[Y1 : mu]: mbox_s4(big_p(Y1)))(Y0, X1))))) 139.61/20.79 | ~ '#form22'(X0) 139.61/20.79 | ~ rel_s4(X0, X1)), 139.61/20.79 inference('rw_clause', [status(thm)], ['168', '42'])). 139.61/20.79 tff('170', plain, 139.61/20.79 ![X0, X1]: 139.61/20.79 ( (!!((^[Y0 : mu]: (exists_in_world(Y0, X1) => mbox_s4(big_p(Y0), X1))))) 139.61/20.79 | ~ '#form22'(X0) 139.61/20.79 | ~ rel_s4(X0, X1)), 139.61/20.79 inference('ho_norm', [status(thm)], ['169'])). 139.61/20.79 tff('171', plain, 139.61/20.79 ![X0, X1, X2 : mu]: 139.61/20.79 ( (exists_in_world(X2, X1) => mbox_s4(big_p(X2), X1)) 139.61/20.79 | ~ rel_s4(X0, X1) 139.61/20.79 | ~ '#form22'(X0)), 139.61/20.79 inference('lazy_cnf_forall', [status(thm)], ['170'])). 139.61/20.79 tff('172', plain, 139.61/20.79 ![X0, X1, X2 : mu]: 139.61/20.79 (~ exists_in_world(X2, X1) 139.61/20.79 | mbox_s4(big_p(X2), X1) 139.61/20.79 | ~ '#form22'(X0) 139.61/20.79 | ~ rel_s4(X0, X1)), 139.61/20.79 inference('lazy_cnf_imply', [status(thm)], ['171'])). 139.61/20.79 tff('173', plain, 139.61/20.79 ![X0, X1, X2 : mu]: 139.61/20.79 ( (!!((^[Y0 : $i]: ((~(rel_s4(X0, Y0))) | big_p(X2, Y0))))) 139.61/20.79 | ~ rel_s4(X1, X0) 139.61/20.79 | ~ '#form22'(X1) 139.61/20.79 | ~ exists_in_world(X2, X0)), 139.61/20.79 inference('rw_clause', [status(thm)], ['172', '38'])). 139.61/20.79 tff('174', plain, 139.61/20.79 ![X0, X1, X2 : mu, X3]: 139.61/20.79 ( ((~(rel_s4(X0, X3))) | big_p(X2, X3)) 139.61/20.79 | ~ exists_in_world(X2, X0) 139.61/20.79 | ~ '#form22'(X1) 139.61/20.79 | ~ rel_s4(X1, X0)), 139.61/20.79 inference('lazy_cnf_forall', [status(thm)], ['173'])). 139.61/20.79 tff('175', plain, 139.61/20.79 ![X0, X1, X2 : mu, X3]: 139.61/20.79 (~ rel_s4(X0, X3) 139.61/20.79 | big_p(X2, X3) 139.61/20.79 | ~ rel_s4(X1, X0) 139.61/20.79 | ~ '#form22'(X1) 139.61/20.79 | ~ exists_in_world(X2, X0)), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['174'])). 139.61/20.79 tff('176', plain, ( rel_s4('#sk9', '#sk11') | '#form12'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['154'])). 139.61/20.79 tff('177', plain, ( rel_s4('#sk3', '#sk9') | '#form12'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['83'])). 139.61/20.79 tff('178', plain, 139.61/20.79 ![X0, X1, X2 : mu]: 139.61/20.79 ( rel_s4(X0, '#sk20'(X0, X2)) 139.61/20.79 | (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0))))) 139.61/20.79 | ~ exists_in_world(X2, X1) 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ rel_s4('#sk3', X1) 139.61/20.79 | ~ rel_s4(X1, X0)), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['118'])). 139.61/20.79 tff('179', plain, 139.61/20.79 ![X0]: 139.61/20.79 (~ '#form22'(X0) 139.61/20.79 | (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0)))))), 139.61/20.79 define([status(thm)])). 139.61/20.79 tff('180', plain, 139.61/20.79 ![X0, X1, X2 : mu]: 139.61/20.79 ( '#form22'(X0) 139.61/20.79 | ~ rel_s4(X1, X0) 139.61/20.79 | ~ rel_s4('#sk3', X1) 139.61/20.79 | ~ '#form6' 139.61/20.79 | ~ exists_in_world(X2, X1) 139.61/20.79 | rel_s4(X0, '#sk20'(X0, X2))), 139.61/20.79 inference('renaming', [status(thm)], ['178', '179'])). 139.61/20.79 tff('181', plain, 139.61/20.79 ( '#form6' 139.61/20.79 | ~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk2', Y0))) | 139.61/20.79 mimplies( 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind( 139.61/20.79 (^[Y1 : mu]: 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(Y1)), 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind( 139.61/20.79 (^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind( 139.61/20.79 (^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.79 mbox_s4(big_p(Y1)))))))), Y0)))))), 139.61/20.79 define([status(thm)])). 139.61/20.79 tff('182', plain, 139.61/20.79 (~ ((~(rel_s4('#sk2', '#sk4'))) | 139.61/20.79 mimplies( 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind( 139.61/20.79 (^[Y0 : mu]: 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(Y0)), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p(Y0)))))))), '#sk4')) 139.61/20.79 | '#form6'), 139.61/20.79 inference('lazy_cnf_exists', [status(thm)], ['181'])). 139.61/20.79 tff('183', plain, 139.61/20.79 (~ ((~(rel_s4('#sk2', '#sk4'))) | 139.61/20.79 mor( 139.61/20.79 mnot( 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind( 139.61/20.79 (^[Y0 : mu]: 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(Y0)), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p(Y0)))))))), '#sk4')) 139.61/20.79 | '#form6'), 139.61/20.79 inference('rw', [status(thm)], ['182', '22'])). 139.61/20.79 tff('184', plain, 139.61/20.79 (~ mor( 139.61/20.79 mnot( 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind( 139.61/20.79 (^[Y0 : mu]: 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(Y0)), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p(Y0)))))))), '#sk4') 139.61/20.79 | '#form6'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['183'])). 139.61/20.79 tff('185', plain, 139.61/20.79 (~ (mbox_s4( 139.61/20.79 mforall_ind( 139.61/20.79 (^[Y0 : mu]: 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(Y0)), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p(Y0))))))), '#sk4') | 139.61/20.79 mnot( 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), '#sk4')) 139.61/20.79 | '#form6'), 139.61/20.79 inference('rw_clause', [status(thm)], ['184', '25'])). 139.61/20.79 tff('186', plain, 139.61/20.79 (~ mnot( 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), '#sk4') 139.61/20.79 | '#form6'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['185'])). 139.61/20.79 tff('187', plain, 139.61/20.79 ( mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))), '#sk4') 139.61/20.79 | '#form6'), 139.61/20.79 inference('rw_clause', [status(thm)], ['186', '10'])). 139.61/20.79 tff('188', plain, 139.61/20.79 ( mnot( 139.61/20.79 mor( 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))))), 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))))), '#sk4') 139.61/20.79 | '#form6'), 139.61/20.79 inference('rw', [status(thm)], ['187', '8'])). 139.61/20.79 tff('189', plain, 139.61/20.79 (~ mor( 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))))), 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), '#sk4') 139.61/20.79 | '#form6'), 139.61/20.79 inference('rw_clause', [status(thm)], ['188', '15'])). 139.61/20.79 tff('190', plain, 139.61/20.79 (~ (mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))), '#sk4') | 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 '#sk4')) 139.61/20.79 | '#form6'), 139.61/20.79 inference('rw_clause', [status(thm)], ['189', '25'])). 139.61/20.79 tff('191', plain, 139.61/20.79 (~ mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 '#sk4') 139.61/20.79 | '#form6'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['190'])). 139.61/20.79 tff('192', plain, 139.61/20.79 ( mbox_s4( 139.61/20.79 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))), '#sk4') 139.61/20.79 | '#form6'), 139.61/20.79 inference('rw_clause', [status(thm)], ['191', '10'])). 139.61/20.79 tff('193', plain, 139.61/20.79 ( (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk4', Y0))) | 139.61/20.79 mimplies(mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), Y0))))) 139.61/20.79 | '#form6'), 139.61/20.79 inference('rw_clause', [status(thm)], ['192', '38'])). 139.61/20.79 tff('194', plain, 139.61/20.79 ![X1]: 139.61/20.79 ( ((~(rel_s4('#sk4', X1))) | 139.61/20.79 mimplies(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), X1)) 139.61/20.79 | '#form6'), 139.61/20.79 inference('lazy_cnf_forall', [status(thm)], ['193'])). 139.61/20.79 tff('195', plain, 139.61/20.79 ![X1]: 139.61/20.79 ( ((~(rel_s4('#sk4', X1))) | 139.61/20.79 mor(mnot(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), X1)) 139.61/20.79 | '#form6'), 139.61/20.79 inference('rw', [status(thm)], ['194', '22'])). 139.61/20.79 tff('196', plain, 139.61/20.79 ![X1]: 139.61/20.79 (~ rel_s4('#sk4', X1) 139.61/20.79 | mor(mnot(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), X1) 139.61/20.79 | '#form6'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['195'])). 139.61/20.79 tff('197', plain, 139.61/20.79 ![X0]: 139.61/20.79 ( (mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), X0) | 139.61/20.79 mnot(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), X0)) 139.61/20.79 | '#form6' 139.61/20.79 | ~ rel_s4('#sk4', X0)), 139.61/20.79 inference('rw_clause', [status(thm)], ['196', '12'])). 139.61/20.79 tff('198', plain, 139.61/20.79 ![X0]: 139.61/20.79 ( mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), X0) 139.61/20.79 | mnot(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), X0) 139.61/20.79 | ~ rel_s4('#sk4', X0) 139.61/20.79 | '#form6'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['197'])). 139.61/20.79 tff('199', plain, 139.61/20.79 ![X0]: 139.61/20.79 ( (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0))))) 139.61/20.79 | '#form6' 139.61/20.79 | ~ rel_s4('#sk4', X0) 139.61/20.79 | mnot(mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), X0)), 139.61/20.79 inference('rw_clause', [status(thm)], ['198', '38'])). 139.61/20.79 tff('200', plain, 139.61/20.79 ![X0]: 139.61/20.79 (~ mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))), X0) 139.61/20.79 | ~ rel_s4('#sk4', X0) 139.61/20.79 | '#form6' 139.61/20.79 | (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0)))))), 139.61/20.79 inference('rw_clause', [status(thm)], ['199', '15'])). 139.61/20.79 tff('201', plain, 139.61/20.79 ![X0]: 139.61/20.79 (~ mnot(mforall_ind((^[Y0 : mu]: mnot(mbox_s4(big_p(Y0))))), X0) 139.61/20.79 | ~ rel_s4('#sk4', X0) 139.61/20.79 | '#form6' 139.61/20.79 | (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0)))))), 139.61/20.79 inference('rw', [status(thm)], ['200', '51'])). 139.61/20.79 tff('202', plain, 139.61/20.79 ![X0]: 139.61/20.79 ( mforall_ind((^[Y0 : mu]: mnot(mbox_s4(big_p(Y0)))), X0) 139.61/20.79 | (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0))))) 139.61/20.79 | '#form6' 139.61/20.79 | ~ rel_s4('#sk4', X0)), 139.61/20.79 inference('rw_clause', [status(thm)], ['201', '10'])). 139.61/20.79 tff('203', plain, 139.61/20.79 ![X0]: 139.61/20.79 ( (!!((^[Y0 : mu]: 139.61/20.79 (exists_in_world(Y0, X0) => 139.61/20.79 (^[Y1 : mu]: mnot(mbox_s4(big_p(Y1))))(Y0, X0))))) 139.61/20.79 | ~ rel_s4('#sk4', X0) 139.61/20.79 | '#form6' 139.61/20.79 | (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0)))))), 139.61/20.79 inference('rw_clause', [status(thm)], ['202', '42'])). 139.61/20.79 tff('204', plain, 139.61/20.79 ![X0]: 139.61/20.79 ( (!!((^[Y0 : mu]: 139.61/20.79 (exists_in_world(Y0, X0) => mnot(mbox_s4(big_p(Y0)), X0))))) 139.61/20.79 | ~ rel_s4('#sk4', X0) 139.61/20.79 | '#form6' 139.61/20.79 | (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0)))))), 139.61/20.79 inference('ho_norm', [status(thm)], ['203'])). 139.61/20.79 tff('205', plain, 139.61/20.79 ![X0]: 139.61/20.79 (~ '#form23'(X0) 139.61/20.79 | (!!((^[Y0 : mu]: 139.61/20.79 (exists_in_world(Y0, X0) => mnot(mbox_s4(big_p(Y0)), X0)))))), 139.61/20.79 define([status(thm)])). 139.61/20.79 tff('206', plain, 139.61/20.79 ![X0, X1 : mu]: 139.61/20.79 ( (exists_in_world(X1, X0) => mnot(mbox_s4(big_p(X1)), X0)) 139.61/20.79 | ~ '#form23'(X0)), 139.61/20.79 inference('lazy_cnf_forall', [status(thm)], ['205'])). 139.61/20.79 tff('207', plain, 139.61/20.79 ![X0, X1 : mu]: 139.61/20.79 (~ exists_in_world(X1, X0) 139.61/20.79 | mnot(mbox_s4(big_p(X1)), X0) 139.61/20.79 | ~ '#form23'(X0)), 139.61/20.79 inference('lazy_cnf_imply', [status(thm)], ['206'])). 139.61/20.79 tff('208', plain, 139.61/20.79 ![X0, X1 : mu]: 139.61/20.79 (~ mbox_s4(big_p(X1), X0) | ~ '#form23'(X0) | ~ exists_in_world(X1, X0)), 139.61/20.79 inference('rw_clause', [status(thm)], ['207', '15'])). 139.61/20.79 tff('209', plain, 139.61/20.79 ![X0, X1 : mu]: 139.61/20.79 (~ (!!((^[Y0 : $i]: ((~(rel_s4(X0, Y0))) | big_p(X1, Y0))))) 139.61/20.79 | ~ exists_in_world(X1, X0) 139.61/20.79 | ~ '#form23'(X0)), 139.61/20.79 inference('rw_clause', [status(thm)], ['208', '17'])). 139.61/20.79 tff('210', plain, 139.61/20.79 ![X0, X1 : mu]: 139.61/20.79 (~ ((~(rel_s4(X0, '#sk20'(X0, X1)))) | big_p(X1, '#sk20'(X0, X1))) 139.61/20.79 | ~ '#form23'(X0) 139.61/20.79 | ~ exists_in_world(X1, X0)), 139.61/20.79 inference('lazy_cnf_exists', [status(thm)], ['209'])). 139.61/20.79 tff('211', plain, 139.61/20.79 ![X0, X1 : mu]: 139.61/20.79 (~ big_p(X1, '#sk20'(X0, X1)) 139.61/20.79 | ~ exists_in_world(X1, X0) 139.61/20.79 | ~ '#form23'(X0)), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['210'])). 139.61/20.79 tff('212', plain, 139.61/20.79 (~ mbox_s4( 139.61/20.79 mforall_ind( 139.61/20.79 (^[Y0 : mu]: 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(Y0)), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p(Y0))))))), '#sk4') 139.61/20.79 | '#form6'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['185'])). 139.61/20.79 tff('213', plain, 139.61/20.79 (~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk4', Y0))) | 139.61/20.79 mforall_ind( 139.61/20.79 (^[Y1 : mu]: 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(Y1)), 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind((^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind((^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.79 mbox_s4(big_p(Y1)))))), Y0))))) 139.61/20.79 | '#form6'), 139.61/20.79 inference('rw_clause', [status(thm)], ['212', '17'])). 139.61/20.79 tff('214', plain, 139.61/20.79 (~ ((~(rel_s4('#sk4', '#sk5'))) | 139.61/20.79 mforall_ind( 139.61/20.79 (^[Y0 : mu]: 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(Y0)), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p(Y0)))))), '#sk5')) 139.61/20.79 | '#form6'), 139.61/20.79 inference('lazy_cnf_exists', [status(thm)], ['213'])). 139.61/20.79 tff('215', plain, 139.61/20.79 (~ mforall_ind( 139.61/20.79 (^[Y0 : mu]: 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(Y0)), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p(Y0)))))), '#sk5') 139.61/20.79 | '#form6'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['214'])). 139.61/20.79 tff('216', plain, 139.61/20.79 (~ (!!((^[Y0 : mu]: 139.61/20.79 (exists_in_world(Y0, '#sk5') => 139.61/20.79 (^[Y1 : mu]: 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(Y1)), 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind((^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind((^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.79 mbox_s4(big_p(Y1))))))(Y0, '#sk5'))))) 139.61/20.79 | '#form6'), 139.61/20.79 inference('rw_clause', [status(thm)], ['215', '90'])). 139.61/20.79 tff('217', plain, 139.61/20.79 (~ (!!((^[Y0 : mu]: 139.61/20.79 (exists_in_world(Y0, '#sk5') => 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(Y0)), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p(Y0)))), '#sk5'))))) 139.61/20.79 | '#form6'), 139.61/20.79 inference('ho_norm', [status(thm)], ['216'])). 139.61/20.79 tff('218', plain, 139.61/20.79 (~ (exists_in_world('#sk7', '#sk5') => 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p('#sk7')), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mbox_s4(big_p('#sk7')))), '#sk5')) 139.61/20.79 | '#form6'), 139.61/20.79 inference('lazy_cnf_exists', [status(thm)], ['217'])). 139.61/20.79 tff('219', plain, 139.61/20.79 (~ (exists_in_world('#sk7', '#sk5') => 139.61/20.79 mnot( 139.61/20.79 mor( 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p('#sk7')), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))))), 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mbox_s4(big_p('#sk7')))))), '#sk5')) 139.61/20.79 | '#form6'), 139.61/20.79 inference('rw', [status(thm)], ['218', '8'])). 139.61/20.79 tff('220', plain, 139.61/20.79 (~ mnot( 139.61/20.79 mor( 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p('#sk7')), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))))), 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mbox_s4(big_p('#sk7')))))), '#sk5') 139.61/20.79 | '#form6'), 139.61/20.79 inference('lazy_cnf_imply', [status(thm)], ['219'])). 139.61/20.79 tff('221', plain, 139.61/20.79 ( mor( 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p('#sk7')), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))))), 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mbox_s4(big_p('#sk7'))))), '#sk5') 139.61/20.79 | '#form6'), 139.61/20.79 inference('rw_clause', [status(thm)], ['220', '10'])). 139.61/20.79 tff('222', plain, 139.61/20.79 ( (mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mbox_s4(big_p('#sk7')))), '#sk5') | 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p('#sk7')), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 '#sk5')) 139.61/20.79 | '#form6'), 139.61/20.79 inference('rw_clause', [status(thm)], ['221', '12'])). 139.61/20.79 tff('223', plain, 139.61/20.79 ( mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mbox_s4(big_p('#sk7')))), '#sk5') 139.61/20.79 | mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p('#sk7')), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 '#sk5') 139.61/20.79 | '#form6'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['222'])). 139.61/20.79 tff('224', plain, 139.61/20.79 (~ mbox_s4( 139.61/20.79 mimplies(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mbox_s4(big_p('#sk7'))), '#sk5') 139.61/20.79 | '#form6' 139.61/20.79 | mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p('#sk7')), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 '#sk5')), 139.61/20.79 inference('rw_clause', [status(thm)], ['223', '15'])). 139.61/20.79 tff('225', plain, 139.61/20.79 (~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk5', Y0))) | 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p('#sk7')), Y0))))) 139.61/20.79 | mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p('#sk7')), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 '#sk5') 139.61/20.79 | '#form6'), 139.61/20.79 inference('rw_clause', [status(thm)], ['224', '17'])). 139.61/20.79 tff('226', plain, 139.61/20.79 (~ mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p('#sk7')), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))), '#sk5') 139.61/20.79 | '#form6' 139.61/20.79 | ~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk5', Y0))) | 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p('#sk7')), Y0)))))), 139.61/20.79 inference('rw_clause', [status(thm)], ['225', '15'])). 139.61/20.79 tff('227', plain, 139.61/20.79 (~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk5', Y0))) | 139.61/20.79 mimplies(mbox_s4(big_p('#sk7')), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), Y0))))) 139.61/20.79 | ~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk5', Y0))) | 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p('#sk7')), Y0))))) 139.61/20.79 | '#form6'), 139.61/20.79 inference('rw_clause', [status(thm)], ['226', '17'])). 139.61/20.79 tff('228', plain, 139.61/20.79 (~ mbox_s4( 139.61/20.79 mforall_ind( 139.61/20.79 (^[Y0 : mu]: 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(Y0)), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p(Y0))))))), '#sk4') 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['124'])). 139.61/20.79 tff('229', plain, 139.61/20.79 (~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk4', Y0))) | 139.61/20.79 mforall_ind( 139.61/20.79 (^[Y1 : mu]: 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(Y1)), 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind((^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind((^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.79 mbox_s4(big_p(Y1)))))), Y0))))) 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('rw_clause', [status(thm)], ['228', '17'])). 139.61/20.79 tff('230', plain, 139.61/20.79 (~ ((~(rel_s4('#sk4', '#sk5'))) | 139.61/20.79 mforall_ind( 139.61/20.79 (^[Y0 : mu]: 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(Y0)), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p(Y0)))))), '#sk5')) 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('lazy_cnf_exists', [status(thm)], ['229'])). 139.61/20.79 tff('231', plain, 139.61/20.79 (~ mforall_ind( 139.61/20.79 (^[Y0 : mu]: 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(Y0)), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p(Y0)))))), '#sk5') 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['230'])). 139.61/20.79 tff('232', plain, 139.61/20.79 (~ (!!((^[Y0 : mu]: 139.61/20.79 (exists_in_world(Y0, '#sk5') => 139.61/20.79 (^[Y1 : mu]: 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(Y1)), 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind((^[Y2 : mu]: mbox_s4(big_p(Y2))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4( 139.61/20.79 mforall_ind((^[Y2 : mu]: mbox_s4(big_p(Y2))))), 139.61/20.79 mbox_s4(big_p(Y1))))))(Y0, '#sk5'))))) 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('rw_clause', [status(thm)], ['231', '90'])). 139.61/20.79 tff('233', plain, 139.61/20.79 (~ (!!((^[Y0 : mu]: 139.61/20.79 (exists_in_world(Y0, '#sk5') => 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p(Y0)), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p(Y0)))), '#sk5'))))) 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('ho_norm', [status(thm)], ['232'])). 139.61/20.79 tff('234', plain, 139.61/20.79 (~ (exists_in_world('#sk7', '#sk5') => 139.61/20.79 mand( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p('#sk7')), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mbox_s4(big_p('#sk7')))), '#sk5')) 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('lazy_cnf_exists', [status(thm)], ['233'])). 139.61/20.79 tff('235', plain, 139.61/20.79 (~ (exists_in_world('#sk7', '#sk5') => 139.61/20.79 mnot( 139.61/20.79 mor( 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p('#sk7')), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))))), 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mbox_s4(big_p('#sk7')))))), '#sk5')) 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('rw', [status(thm)], ['234', '8'])). 139.61/20.79 tff('236', plain, 139.61/20.79 (~ mnot( 139.61/20.79 mor( 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p('#sk7')), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))))), 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mbox_s4(big_p('#sk7')))))), '#sk5') 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('lazy_cnf_imply', [status(thm)], ['235'])). 139.61/20.79 tff('237', plain, 139.61/20.79 ( mor( 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p('#sk7')), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))))), 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mbox_s4(big_p('#sk7'))))), '#sk5') 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('rw_clause', [status(thm)], ['236', '10'])). 139.61/20.79 tff('238', plain, 139.61/20.79 ( (mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mbox_s4(big_p('#sk7')))), '#sk5') | 139.61/20.79 mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p('#sk7')), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 '#sk5')) 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('rw_clause', [status(thm)], ['237', '12'])). 139.61/20.79 tff('239', plain, 139.61/20.79 ( mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mbox_s4(big_p('#sk7')))), '#sk5') 139.61/20.79 | mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p('#sk7')), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 '#sk5') 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['238'])). 139.61/20.79 tff('240', plain, 139.61/20.79 (~ mbox_s4( 139.61/20.79 mimplies(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mbox_s4(big_p('#sk7'))), '#sk5') 139.61/20.79 | rel_s4('#sk2', '#sk3') 139.61/20.79 | mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p('#sk7')), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 '#sk5')), 139.61/20.79 inference('rw_clause', [status(thm)], ['239', '15'])). 139.61/20.79 tff('241', plain, 139.61/20.79 (~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk5', Y0))) | 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p('#sk7')), Y0))))) 139.61/20.79 | mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p('#sk7')), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))))), 139.61/20.79 '#sk5') 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('rw_clause', [status(thm)], ['240', '17'])). 139.61/20.79 tff('242', plain, 139.61/20.79 (~ mbox_s4( 139.61/20.79 mimplies(mbox_s4(big_p('#sk7')), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))), '#sk5') 139.61/20.79 | rel_s4('#sk2', '#sk3') 139.61/20.79 | ~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk5', Y0))) | 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p('#sk7')), Y0)))))), 139.61/20.79 inference('rw_clause', [status(thm)], ['241', '15'])). 139.61/20.79 tff('243', plain, 139.61/20.79 (~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk5', Y0))) | 139.61/20.79 mimplies(mbox_s4(big_p('#sk7')), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), Y0))))) 139.61/20.79 | ~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk5', Y0))) | 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p('#sk7')), Y0))))) 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('rw_clause', [status(thm)], ['242', '17'])). 139.61/20.79 tff('244', plain, 139.61/20.79 (~ ((~(rel_s4('#sk5', '#sk10'))) | 139.61/20.79 mimplies(mbox_s4(big_p('#sk7')), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), '#sk10')) 139.61/20.79 | rel_s4('#sk2', '#sk3') 139.61/20.79 | ~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk5', Y0))) | 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p('#sk7')), Y0)))))), 139.61/20.79 inference('lazy_cnf_exists', [status(thm)], ['243'])). 139.61/20.79 tff('245', plain, 139.61/20.79 (~ ((~(rel_s4('#sk5', '#sk10'))) | 139.61/20.79 mor(mnot(mbox_s4(big_p('#sk7'))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), '#sk10')) 139.61/20.79 | rel_s4('#sk2', '#sk3') 139.61/20.79 | ~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk5', Y0))) | 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p('#sk7')), Y0)))))), 139.61/20.79 inference('rw', [status(thm)], ['244', '22'])). 139.61/20.79 tff('246', plain, 139.61/20.79 (~ mor(mnot(mbox_s4(big_p('#sk7'))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), '#sk10') 139.61/20.79 | ~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk5', Y0))) | 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p('#sk7')), Y0))))) 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['245'])). 139.61/20.79 tff('247', plain, 139.61/20.79 (~ (mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), '#sk10') | 139.61/20.79 mnot(mbox_s4(big_p('#sk7')), '#sk10')) 139.61/20.79 | rel_s4('#sk2', '#sk3') 139.61/20.79 | ~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk5', Y0))) | 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p('#sk7')), Y0)))))), 139.61/20.79 inference('rw_clause', [status(thm)], ['246', '25'])). 139.61/20.79 tff('248', plain, 139.61/20.79 (~ mnot(mbox_s4(big_p('#sk7')), '#sk10') 139.61/20.79 | ~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk5', Y0))) | 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p('#sk7')), Y0))))) 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['247'])). 139.61/20.79 tff('249', plain, 139.61/20.79 ( mbox_s4(big_p('#sk7'), '#sk10') 139.61/20.79 | rel_s4('#sk2', '#sk3') 139.61/20.79 | ~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk5', Y0))) | 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p('#sk7')), Y0)))))), 139.61/20.79 inference('rw_clause', [status(thm)], ['248', '10'])). 139.61/20.79 tff('250', plain, 139.61/20.79 ( (!!((^[Y0 : $i]: ((~(rel_s4('#sk10', Y0))) | big_p('#sk7', Y0))))) 139.61/20.79 | ~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk5', Y0))) | 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p('#sk7')), Y0))))) 139.61/20.79 | rel_s4('#sk2', '#sk3')), 139.61/20.79 inference('rw_clause', [status(thm)], ['249', '38'])). 139.61/20.79 tff('251', plain, 139.61/20.79 ( '#form21' 139.61/20.79 | ~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk5', Y0))) | 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p('#sk7')), Y0)))))), 139.61/20.79 define([status(thm)])). 139.61/20.79 tff('252', plain, 139.61/20.79 (~ '#form21' 139.61/20.79 | '#form6' 139.61/20.79 | ~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk5', Y0))) | 139.61/20.79 mimplies(mbox_s4(big_p('#sk7')), 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 Y0)))))), 139.61/20.79 inference('renaming', [status(thm)], ['227', '251'])). 139.61/20.79 tff('253', plain, 139.61/20.79 (~ ((~(rel_s4('#sk5', '#sk10'))) | 139.61/20.79 mimplies(mbox_s4(big_p('#sk7')), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), '#sk10')) 139.61/20.79 | '#form6' 139.61/20.79 | ~ '#form21'), 139.61/20.79 inference('lazy_cnf_exists', [status(thm)], ['252'])). 139.61/20.79 tff('254', plain, 139.61/20.79 (~ ((~(rel_s4('#sk5', '#sk10'))) | 139.61/20.79 mor(mnot(mbox_s4(big_p('#sk7'))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), '#sk10')) 139.61/20.79 | '#form6' 139.61/20.79 | ~ '#form21'), 139.61/20.79 inference('rw', [status(thm)], ['253', '22'])). 139.61/20.79 tff('255', plain, 139.61/20.79 (~ mor(mnot(mbox_s4(big_p('#sk7'))), 139.61/20.79 mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), '#sk10') 139.61/20.79 | ~ '#form21' 139.61/20.79 | '#form6'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['254'])). 139.61/20.79 tff('256', plain, 139.61/20.79 (~ (mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), '#sk10') | 139.61/20.79 mnot(mbox_s4(big_p('#sk7')), '#sk10')) 139.61/20.79 | '#form6' 139.61/20.79 | ~ '#form21'), 139.61/20.79 inference('rw_clause', [status(thm)], ['255', '25'])). 139.61/20.79 tff('257', plain, 139.61/20.79 (~ mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), '#sk10') 139.61/20.79 | ~ '#form21' 139.61/20.79 | '#form6'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['256'])). 139.61/20.79 tff('258', plain, 139.61/20.79 (~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk10', Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0))))) 139.61/20.79 | '#form6' 139.61/20.79 | ~ '#form21'), 139.61/20.79 inference('rw_clause', [status(thm)], ['257', '17'])). 139.61/20.79 tff('259', plain, 139.61/20.79 (~ mnot( 139.61/20.79 mbox_s4( 139.61/20.79 mimplies(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))), '#sk4') 139.61/20.79 | '#form6'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['190'])). 139.61/20.79 tff('260', plain, 139.61/20.79 ( mbox_s4( 139.61/20.79 mimplies(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), '#sk4') 139.61/20.79 | '#form6'), 139.61/20.79 inference('rw_clause', [status(thm)], ['259', '10'])). 139.61/20.79 tff('261', plain, 139.61/20.79 ( (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk4', Y0))) | 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mexists_ind((^[Y1 : mu]: mbox_s4(big_p(Y1)))), Y0))))) 139.61/20.79 | '#form6'), 139.61/20.79 inference('rw_clause', [status(thm)], ['260', '38'])). 139.61/20.79 tff('262', plain, 139.61/20.79 ![X1]: 139.61/20.79 ( ((~(rel_s4('#sk4', X1))) | 139.61/20.79 mimplies(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), X1)) 139.61/20.79 | '#form6'), 139.61/20.79 inference('lazy_cnf_forall', [status(thm)], ['261'])). 139.61/20.79 tff('263', plain, 139.61/20.79 ![X1]: 139.61/20.79 ( ((~(rel_s4('#sk4', X1))) | 139.61/20.79 mor(mnot(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))), 139.61/20.79 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), X1)) 139.61/20.79 | '#form6'), 139.61/20.79 inference('rw', [status(thm)], ['262', '22'])). 139.61/20.79 tff('264', plain, 139.61/20.79 ![X1]: 139.61/20.79 (~ rel_s4('#sk4', X1) 139.61/20.79 | mor(mnot(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))), 139.61/20.79 mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), X1) 139.61/20.79 | '#form6'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['263'])). 139.61/20.79 tff('265', plain, 139.61/20.79 ![X0]: 139.61/20.79 ( (mexists_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))), X0) | 139.61/20.79 mnot(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), X0)) 139.61/20.79 | '#form6' 139.61/20.79 | ~ rel_s4('#sk4', X0)), 139.61/20.79 inference('rw_clause', [status(thm)], ['264', '12'])). 139.61/20.79 tff('266', plain, 139.61/20.79 ![X0]: 139.61/20.79 ( (mnot(mforall_ind((^[Y0 : mu]: mnot(mbox_s4(big_p(Y0))))), X0) | 139.61/20.79 mnot(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), X0)) 139.61/20.79 | '#form6' 139.61/20.79 | ~ rel_s4('#sk4', X0)), 139.61/20.79 inference('rw', [status(thm)], ['265', '51'])). 139.61/20.79 tff('267', plain, 139.61/20.79 ![X0]: 139.61/20.79 ( mnot(mforall_ind((^[Y0 : mu]: mnot(mbox_s4(big_p(Y0))))), X0) 139.61/20.79 | mnot(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), X0) 139.61/20.79 | ~ rel_s4('#sk4', X0) 139.61/20.79 | '#form6'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['266'])). 139.61/20.79 tff('268', plain, 139.61/20.79 ![X0]: 139.61/20.79 (~ mforall_ind((^[Y0 : mu]: mnot(mbox_s4(big_p(Y0)))), X0) 139.61/20.79 | '#form6' 139.61/20.79 | ~ rel_s4('#sk4', X0) 139.61/20.79 | mnot(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), X0)), 139.61/20.79 inference('rw_clause', [status(thm)], ['267', '15'])). 139.61/20.79 tff('269', plain, 139.61/20.79 ![X0]: 139.61/20.79 (~ (!!((^[Y0 : mu]: 139.61/20.79 (exists_in_world(Y0, X0) => 139.61/20.79 (^[Y1 : mu]: mnot(mbox_s4(big_p(Y1))))(Y0, X0))))) 139.61/20.79 | mnot(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), X0) 139.61/20.79 | ~ rel_s4('#sk4', X0) 139.61/20.79 | '#form6'), 139.61/20.79 inference('rw_clause', [status(thm)], ['268', '90'])). 139.61/20.79 tff('270', plain, 139.61/20.79 ![X0]: 139.61/20.79 (~ (!!((^[Y0 : mu]: 139.61/20.79 (exists_in_world(Y0, X0) => mnot(mbox_s4(big_p(Y0)), X0))))) 139.61/20.79 | mnot(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), X0) 139.61/20.79 | ~ rel_s4('#sk4', X0) 139.61/20.79 | '#form6'), 139.61/20.79 inference('ho_norm', [status(thm)], ['269'])). 139.61/20.79 tff('271', plain, 139.61/20.79 ![X0]: 139.61/20.79 (~ mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), X0) 139.61/20.79 | '#form6' 139.61/20.79 | ~ rel_s4('#sk4', X0) 139.61/20.79 | ~ (!!((^[Y0 : mu]: 139.61/20.79 (exists_in_world(Y0, X0) => mnot(mbox_s4(big_p(Y0)), X0)))))), 139.61/20.79 inference('rw_clause', [status(thm)], ['270', '15'])). 139.61/20.79 tff('272', plain, 139.61/20.79 ![X0]: 139.61/20.79 (~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0))))) 139.61/20.79 | ~ (!!((^[Y0 : mu]: 139.61/20.79 (exists_in_world(Y0, X0) => mnot(mbox_s4(big_p(Y0)), X0))))) 139.61/20.79 | ~ rel_s4('#sk4', X0) 139.61/20.79 | '#form6'), 139.61/20.79 inference('rw_clause', [status(thm)], ['271', '17'])). 139.61/20.79 tff('273', plain, 139.61/20.79 ![X0]: 139.61/20.79 ( '#form22'(X0) 139.61/20.79 | ~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0)))))), 139.61/20.79 define([status(thm)])). 139.61/20.79 tff('274', plain, (~ '#form22'('#sk10') | ~ '#form21' | '#form6'), 139.61/20.79 inference('renaming', [status(thm)], ['258', '273'])). 139.61/20.79 tff('275', plain, 139.61/20.79 (~ mnot(mbox_s4(big_p('#sk7')), '#sk10') | ~ '#form21' | '#form6'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['256'])). 139.61/20.79 tff('276', plain, 139.61/20.79 ( mbox_s4(big_p('#sk7'), '#sk10') | '#form6' | ~ '#form21'), 139.61/20.79 inference('rw_clause', [status(thm)], ['275', '10'])). 139.61/20.79 tff('277', plain, 139.61/20.79 ( (!!((^[Y0 : $i]: ((~(rel_s4('#sk10', Y0))) | big_p('#sk7', Y0))))) 139.61/20.79 | ~ '#form21' 139.61/20.79 | '#form6'), 139.61/20.79 inference('rw_clause', [status(thm)], ['276', '38'])). 139.61/20.79 tff('278', plain, 139.61/20.79 ![X1]: 139.61/20.79 ( ((~(rel_s4('#sk10', X1))) | big_p('#sk7', X1)) 139.61/20.79 | '#form6' 139.61/20.79 | ~ '#form21'), 139.61/20.79 inference('lazy_cnf_forall', [status(thm)], ['277'])). 139.61/20.79 tff('279', plain, 139.61/20.79 ![X1]: 139.61/20.79 (~ rel_s4('#sk10', X1) | big_p('#sk7', X1) | ~ '#form21' | '#form6'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['278'])). 139.61/20.79 tff('280', plain, 139.61/20.79 ![X0]: 139.61/20.79 ( (!!((^[Y0 : mu]: 139.61/20.79 (exists_in_world(Y0, X0) => mnot(mbox_s4(big_p(Y0)), X0))))) 139.61/20.79 | ~ rel_s4('#sk4', X0) 139.61/20.79 | '#form6' 139.61/20.79 | (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0)))))), 139.61/20.79 inference('ho_norm', [status(thm)], ['203'])). 139.61/20.79 tff('281', plain, 139.61/20.79 ![X0]: 139.61/20.79 (~ '#form23'(X0) 139.61/20.79 | (!!((^[Y0 : mu]: 139.61/20.79 (exists_in_world(Y0, X0) => mnot(mbox_s4(big_p(Y0)), X0)))))), 139.61/20.79 define([status(thm)])). 139.61/20.79 tff('282', plain, 139.61/20.79 ![X0]: 139.61/20.79 ( '#form23'(X0) 139.61/20.79 | (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0))))) 139.61/20.79 | '#form6' 139.61/20.79 | ~ rel_s4('#sk4', X0)), 139.61/20.79 inference('renaming', [status(thm)], ['280', '281'])). 139.61/20.79 tff('283', plain, 139.61/20.79 ![X0]: 139.61/20.79 (~ '#form22'(X0) 139.61/20.79 | (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4(X0, Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0)))))), 139.61/20.79 define([status(thm)])). 139.61/20.79 tff('284', plain, 139.61/20.79 ![X0]: 139.61/20.79 ( '#form22'(X0) | ~ rel_s4('#sk4', X0) | '#form6' | '#form23'(X0)), 139.61/20.79 inference('renaming', [status(thm)], ['282', '283'])). 139.61/20.79 tff('285', plain, 139.61/20.79 ![X0, X1 : mu]: 139.61/20.79 ( rel_s4(X0, '#sk20'(X0, X1)) 139.61/20.79 | ~ exists_in_world(X1, X0) 139.61/20.79 | ~ '#form23'(X0)), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['210'])). 139.61/20.79 tff('286', plain, ( rel_s4('#sk4', '#sk5') | '#form6'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['214'])). 139.61/20.79 tff(a2, axiom, (mtransitive(rel_s4))). 139.61/20.79 tff('287', plain, mtransitive(rel_s4), inference('cnf', [status(esa)], [a2])). 139.61/20.79 tff(mtransitive, axiom, mtransitive(X22) => 139.61/20.79 ((!!((^[Y0 : $i]: 139.61/20.79 (!!((^[Y1 : $i]: 139.61/20.79 (!!((^[Y2 : $i]: 139.61/20.79 ((X22(Y1, Y2) & X22(Y0, Y1)) => X22(Y0, Y2))))))))))) 139.61/20.79 = $true)). 139.61/20.79 tff('288', plain, 139.61/20.79 ![X22:(($i * $i) > $o)]: 139.61/20.79 (mtransitive(X22) <=> 139.61/20.79 (![X27,X29,X31]: ((X22(X29,X31) & X22(X27,X29)) => X22(X27,X31)))), 139.61/20.79 inference('rw.lit', [status(esa)], [mtransitive])). 139.61/20.79 tff('289', plain, 139.61/20.79 (!!((^[Y0 : $i]: 139.61/20.79 (!!((^[Y1 : $i]: 139.61/20.79 (!!((^[Y2 : $i]: 139.61/20.79 ((rel_s4(Y1, Y2) & rel_s4(Y0, Y1)) => 139.61/20.79 rel_s4(Y0, Y2))))))))))), 139.61/20.79 inference('rw_clause', [status(thm)], ['287', '288'])). 139.61/20.79 tff('290', plain, 139.61/20.79 ![X1]: 139.61/20.79 (!!((^[Y0 : $i]: 139.61/20.79 (!!((^[Y1 : $i]: 139.61/20.79 ((rel_s4(Y0, Y1) & rel_s4(X1, Y0)) => rel_s4(X1, Y1)))))))), 139.61/20.79 inference('lazy_cnf_forall', [status(thm)], ['289'])). 139.61/20.79 tff('291', plain, 139.61/20.79 ![X1, X2]: 139.61/20.79 (!!((^[Y0 : $i]: ((rel_s4(X2, Y0) & rel_s4(X1, X2)) => rel_s4(X1, Y0))))), 139.61/20.79 inference('lazy_cnf_forall', [status(thm)], ['290'])). 139.61/20.79 tff('292', plain, 139.61/20.79 ![X1, X2, X3]: ((rel_s4(X2, X3) & rel_s4(X1, X2)) => rel_s4(X1, X3)), 139.61/20.79 inference('lazy_cnf_forall', [status(thm)], ['291'])). 139.61/20.79 tff('293', plain, 139.61/20.79 ![X1, X2, X3]: (~ (rel_s4(X2, X3) & rel_s4(X1, X2)) | rel_s4(X1, X3)), 139.61/20.79 inference('lazy_cnf_imply', [status(thm)], ['292'])). 139.61/20.79 tff('294', plain, 139.61/20.79 ![X1, X2, X3]: (~ rel_s4(X2, X3) | ~ rel_s4(X1, X2) | rel_s4(X1, X3)), 139.61/20.79 inference('lazy_cnf_and', [status(thm)], ['293'])). 139.61/20.79 tff('295', plain, ( rel_s4('#sk5', '#sk10') | ~ '#form21' | '#form6'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['254'])). 139.61/20.79 tff('296', plain, 139.61/20.79 ( '#form21' 139.61/20.79 | ~ (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk5', Y0))) | 139.61/20.79 mimplies( 139.61/20.79 mbox_s4(mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))))), 139.61/20.79 mbox_s4(big_p('#sk7')), Y0)))))), 139.61/20.79 define([status(thm)])). 139.61/20.79 tff('297', plain, 139.61/20.79 (~ ((~(rel_s4('#sk5', '#sk14'))) | 139.61/20.79 mimplies(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), 139.61/20.79 mbox_s4(big_p('#sk7')), '#sk14')) 139.61/20.79 | '#form21'), 139.61/20.79 inference('lazy_cnf_exists', [status(thm)], ['296'])). 139.61/20.79 tff('298', plain, 139.61/20.79 (~ ((~(rel_s4('#sk5', '#sk14'))) | 139.61/20.79 mor(mnot(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))), 139.61/20.79 mbox_s4(big_p('#sk7')), '#sk14')) 139.61/20.79 | '#form21'), 139.61/20.79 inference('rw', [status(thm)], ['297', '22'])). 139.61/20.79 tff('299', plain, 139.61/20.79 (~ mor(mnot(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))))), 139.61/20.79 mbox_s4(big_p('#sk7')), '#sk14') 139.61/20.79 | '#form21'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['298'])). 139.61/20.79 tff('300', plain, 139.61/20.79 (~ (mbox_s4(big_p('#sk7'), '#sk14') | 139.61/20.79 mnot(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), '#sk14')) 139.61/20.79 | '#form21'), 139.61/20.79 inference('rw_clause', [status(thm)], ['299', '25'])). 139.61/20.79 tff('301', plain, (~ mbox_s4(big_p('#sk7'), '#sk14') | '#form21'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['300'])). 139.61/20.79 tff('302', plain, 139.61/20.79 (~ (!!((^[Y0 : $i]: ((~(rel_s4('#sk14', Y0))) | big_p('#sk7', Y0))))) 139.61/20.79 | '#form21'), 139.61/20.79 inference('rw_clause', [status(thm)], ['301', '17'])). 139.61/20.79 tff('303', plain, 139.61/20.79 (~ ((~(rel_s4('#sk14', '#sk20'('#sk14', '#sk7')))) | 139.61/20.79 big_p('#sk7', '#sk20'('#sk14', '#sk7'))) 139.61/20.79 | '#form21'), 139.61/20.79 inference('lazy_cnf_exists', [status(thm)], ['302'])). 139.61/20.79 tff('304', plain, (~ big_p('#sk7', '#sk20'('#sk14', '#sk7')) | '#form21'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['303'])). 139.61/20.79 tff('305', plain, ( exists_in_world('#sk7', '#sk5') | '#form6'), 139.61/20.79 inference('lazy_cnf_imply', [status(thm)], ['219'])). 139.61/20.79 tff('306', plain, ( rel_s4('#sk14', '#sk20'('#sk14', '#sk7')) | '#form21'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['303'])). 139.61/20.79 tff('307', plain, ( rel_s4('#sk5', '#sk14') | '#form21'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['298'])). 139.61/20.79 tff(cumulative_ax, axiom, 139.61/20.79 (![X:mu,V,W]: 139.61/20.79 ((exists_in_world(X,V) & rel_s4(V,W)) => exists_in_world(X,W)))). 139.61/20.79 tff('308', plain, 139.61/20.79 (!!((^[Y0 : mu]: 139.61/20.79 (!!((^[Y1 : $i]: 139.61/20.79 (!!((^[Y2 : $i]: 139.61/20.79 ((exists_in_world(Y0, Y1) & rel_s4(Y1, Y2)) => 139.61/20.79 exists_in_world(Y0, Y2))))))))))), 139.61/20.79 inference('cnf', [status(esa)], [cumulative_ax])). 139.61/20.79 tff('309', plain, 139.61/20.79 ![X1 : mu]: 139.61/20.79 (!!((^[Y0 : $i]: 139.61/20.79 (!!((^[Y1 : $i]: 139.61/20.79 ((exists_in_world(X1, Y0) & rel_s4(Y0, Y1)) => 139.61/20.79 exists_in_world(X1, Y1)))))))), 139.61/20.79 inference('lazy_cnf_forall', [status(thm)], ['308'])). 139.61/20.79 tff('310', plain, 139.61/20.79 ![X1 : mu, X2]: 139.61/20.79 (!!((^[Y0 : $i]: 139.61/20.79 ((exists_in_world(X1, X2) & rel_s4(X2, Y0)) => 139.61/20.79 exists_in_world(X1, Y0))))), 139.61/20.79 inference('lazy_cnf_forall', [status(thm)], ['309'])). 139.61/20.79 tff('311', plain, 139.61/20.79 ![X1 : mu, X2, X3]: 139.61/20.79 ((exists_in_world(X1, X2) & rel_s4(X2, X3)) => exists_in_world(X1, X3)), 139.61/20.79 inference('lazy_cnf_forall', [status(thm)], ['310'])). 139.61/20.79 tff('312', plain, 139.61/20.79 ![X1 : mu, X2, X3]: 139.61/20.79 (~ (exists_in_world(X1, X2) & rel_s4(X2, X3)) 139.61/20.79 | exists_in_world(X1, X3)), 139.61/20.79 inference('lazy_cnf_imply', [status(thm)], ['311'])). 139.61/20.79 tff('313', plain, 139.61/20.79 ![X1 : mu, X2, X3]: 139.61/20.79 (~ exists_in_world(X1, X2) 139.61/20.79 | ~ rel_s4(X2, X3) 139.61/20.79 | exists_in_world(X1, X3)), 139.61/20.79 inference('lazy_cnf_and', [status(thm)], ['312'])). 139.61/20.79 tff(a1, axiom, (mreflexive(rel_s4))). 139.61/20.79 tff('314', plain, mreflexive(rel_s4), inference('cnf', [status(esa)], [a1])). 139.61/20.79 tff(mreflexive, axiom, mreflexive(X21) => 139.61/20.79 ((!!((^[Y0 : $i]: X21(Y0, Y0)))) = $true)). 139.61/20.79 tff('315', plain, 139.61/20.79 ![X21:(($i * $i) > $o)]: (mreflexive(X21) <=> (![X26]: X21(X26,X26))), 139.61/20.79 inference('rw.lit', [status(esa)], [mreflexive])). 139.61/20.79 tff('316', plain, (!!((^[Y0 : $i]: rel_s4(Y0, Y0)))), 139.61/20.79 inference('rw_clause', [status(thm)], ['314', '315'])). 139.61/20.79 tff('317', plain, ![X1]: rel_s4(X1, X1), 139.61/20.79 inference('lazy_cnf_forall', [status(thm)], ['316'])). 139.61/20.79 tff('318', plain, 139.61/20.79 (~ mnot(mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))))), '#sk14') 139.61/20.79 | '#form21'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['300'])). 139.61/20.79 tff('319', plain, 139.61/20.79 ( mbox_s4(mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0)))), '#sk14') 139.61/20.79 | '#form21'), 139.61/20.79 inference('rw_clause', [status(thm)], ['318', '10'])). 139.61/20.79 tff('320', plain, 139.61/20.79 ( (!!((^[Y0 : $i]: 139.61/20.79 ((~(rel_s4('#sk14', Y0))) | 139.61/20.79 mforall_ind((^[Y1 : mu]: mbox_s4(big_p(Y1))), Y0))))) 139.61/20.79 | '#form21'), 139.61/20.79 inference('rw_clause', [status(thm)], ['319', '38'])). 139.61/20.79 tff('321', plain, 139.61/20.79 ![X1]: 139.61/20.79 ( ((~(rel_s4('#sk14', X1))) | 139.61/20.79 mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))), X1)) 139.61/20.79 | '#form21'), 139.61/20.79 inference('lazy_cnf_forall', [status(thm)], ['320'])). 139.61/20.79 tff('322', plain, 139.61/20.79 ![X1]: 139.61/20.79 (~ rel_s4('#sk14', X1) 139.61/20.79 | mforall_ind((^[Y0 : mu]: mbox_s4(big_p(Y0))), X1) 139.61/20.79 | '#form21'), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['321'])). 139.61/20.79 tff('323', plain, 139.61/20.79 ![X0]: 139.61/20.79 ( (!!((^[Y0 : mu]: 139.61/20.79 (exists_in_world(Y0, X0) => 139.61/20.79 (^[Y1 : mu]: mbox_s4(big_p(Y1)))(Y0, X0))))) 139.61/20.79 | '#form21' 139.61/20.79 | ~ rel_s4('#sk14', X0)), 139.61/20.79 inference('rw_clause', [status(thm)], ['322', '42'])). 139.61/20.79 tff('324', plain, 139.61/20.79 ![X0]: 139.61/20.79 ( (!!((^[Y0 : mu]: (exists_in_world(Y0, X0) => mbox_s4(big_p(Y0), X0))))) 139.61/20.79 | '#form21' 139.61/20.79 | ~ rel_s4('#sk14', X0)), 139.61/20.79 inference('ho_norm', [status(thm)], ['323'])). 139.61/20.79 tff('325', plain, 139.61/20.79 ![X0, X1 : mu]: 139.61/20.79 ( (exists_in_world(X1, X0) => mbox_s4(big_p(X1), X0)) 139.61/20.79 | ~ rel_s4('#sk14', X0) 139.61/20.79 | '#form21'), 139.61/20.79 inference('lazy_cnf_forall', [status(thm)], ['324'])). 139.61/20.79 tff('326', plain, 139.61/20.79 ![X0, X1 : mu]: 139.61/20.79 (~ exists_in_world(X1, X0) 139.61/20.79 | mbox_s4(big_p(X1), X0) 139.61/20.79 | '#form21' 139.61/20.79 | ~ rel_s4('#sk14', X0)), 139.61/20.79 inference('lazy_cnf_imply', [status(thm)], ['325'])). 139.61/20.79 tff('327', plain, 139.61/20.79 ![X0, X1 : mu]: 139.61/20.79 ( (!!((^[Y0 : $i]: ((~(rel_s4(X0, Y0))) | big_p(X1, Y0))))) 139.61/20.79 | ~ rel_s4('#sk14', X0) 139.61/20.79 | '#form21' 139.61/20.79 | ~ exists_in_world(X1, X0)), 139.61/20.79 inference('rw_clause', [status(thm)], ['326', '38'])). 139.61/20.79 tff('328', plain, 139.61/20.79 ![X0, X1 : mu, X2]: 139.61/20.79 ( ((~(rel_s4(X0, X2))) | big_p(X1, X2)) 139.61/20.79 | ~ exists_in_world(X1, X0) 139.61/20.79 | '#form21' 139.61/20.79 | ~ rel_s4('#sk14', X0)), 139.61/20.79 inference('lazy_cnf_forall', [status(thm)], ['327'])). 139.61/20.79 tff('329', plain, 139.61/20.79 ![X0, X1 : mu, X2]: 139.61/20.79 (~ rel_s4(X0, X2) 139.61/20.79 | big_p(X1, X2) 139.61/20.79 | ~ rel_s4('#sk14', X0) 139.61/20.79 | '#form21' 139.61/20.79 | ~ exists_in_world(X1, X0)), 139.61/20.79 inference('lazy_cnf_or', [status(thm)], ['328'])). 139.61/20.79 tff('330', plain, $false, 139.61/20.79 inference('eprover', [status(thm)], 139.61/20.79 ['2', '68', '69', '80', '98', '151', '159', '163', '164', 139.61/20.79 '165', '175', '176', '177', '180', '211', '274', '279', 139.61/20.79 '284', '285', '286', '294', '295', '304', '305', '306', 139.61/20.79 '307', '313', '317', '329'])). 139.61/20.79 139.61/20.79 % SZS output end Refutation 139.61/20.79 139.61/20.79 139.61/20.79 % Terminating... 139.61/20.79 % /export/starexec/sandbox/solver/bin/lams/b.lsimp.sh running for 17s 141.05/20.92 % Runner terminated. 141.05/20.93 % Zipperpin 1.5 exiting 141.05/20.94 EOF