0.06/0.14 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.06/0.17 % Command : run_portfolio.sh /export/starexec/sandbox/benchmark/theBenchmark.p /export/starexec/sandbox/tmp/tmp.BXeqfmUpEQ 0.15/0.38 % Computer : n026.cluster.edu 0.15/0.38 % Model : x86_64 x86_64 0.15/0.38 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.15/0.38 % Memory : 8042.1875MB 0.15/0.38 % OS : Linux 3.10.0-693.el7.x86_64 0.15/0.38 % CPULimit : 960 0.15/0.38 % DateTime : Thu Jul 2 09:39:10 EDT 2020 0.15/0.39 % CPUTime : 0.15/0.39 % Running portfolio for 120 s 0.15/0.39 % File : /export/starexec/sandbox/benchmark/theBenchmark.p 0.15/0.39 % Number of cores: 8 0.15/0.39 % Python version: Python 3.6.8 0.15/0.39 % Running in HO mode 0.53/0.67 % Estimated wc time : 1286 0.53/0.67 % Estimated cpu time (7 cpus) : 183.71428571428572 0.56/0.74 % /export/starexec/sandbox/solver/bin/lams/s6.sh running for 21s 0.56/0.75 % /export/starexec/sandbox/solver/bin/lams/new_b3.2.norw.sh running for 15s 0.57/0.79 % /export/starexec/sandbox/solver/bin/lams/b.rpo.sh running for 21s 0.57/0.79 % /export/starexec/sandbox/solver/bin/lams/s6.ds.sh running for 10s 0.57/0.79 % /export/starexec/sandbox/solver/bin/lams/c.easy.sh running for 26s 0.57/0.79 % /export/starexec/sandbox/solver/bin/lams/c.s.2.sh running for 68s 0.57/0.79 % /export/starexec/sandbox/solver/bin/lams/old_b3.avatar.sh running for 15s 70.80/10.81 % /export/starexec/sandbox/solver/bin/lams/old_sp2.sh running for 17s 100.26/15.71 % /export/starexec/sandbox/solver/bin/lams/b.sh running for 31s 101.07/15.84 % /export/starexec/sandbox/solver/bin/lams/sh1.sh running for 26s 135.38/21.60 % /export/starexec/sandbox/solver/bin/lams/b.lsimp.sh running for 17s 135.38/21.63 % /export/starexec/sandbox/solver/bin/lams/b.split.sh running for 31s 166.35/26.80 % /export/starexec/sandbox/solver/bin/lams/s4.sh running for 15s 172.83/27.83 % /export/starexec/sandbox/solver/bin/lams/b.rw.sh running for 26s 237.07/38.62 % /export/starexec/sandbox/solver/bin/lams/c_ic.sh running for 42s 255.76/41.82 % /export/starexec/sandbox/solver/bin/lams/s1.f.sh running for 15s 255.76/41.86 % /export/starexec/sandbox/solver/bin/lams/c.preprocess2.sh running for 21s 285.04/46.74 % /export/starexec/sandbox/solver/bin/lams/sh.or.sh running for 26s 320.58/52.64 % /export/starexec/sandbox/solver/bin/lams/new.cs3.comb.sh running for 18s 327.66/53.87 % /export/starexec/sandbox/solver/bin/lams/old_b2.sh running for 31s 345.00/56.85 % /export/starexec/sandbox/solver/bin/lams/sp5.sh running for 36s 374.24/61.86 % Solved by lams/c.s.2.sh. 374.24/61.86 % Running : timeout 52 /export/starexec/sandbox/solver/bin/lams/eprover-ho --pos-ext=all --neg-ext=all /export/starexec/sandbox/tmp/tmp.BXeqfmUpEQ/e_input34fc03 --cpu-limit=50 --auto-schedule -s -p. 374.24/61.86 % done 709 iterations in 61.012s 374.24/61.86 % SZS status Theorem for '/export/starexec/sandbox/benchmark/theBenchmark.p' 374.24/61.86 % SZS output start Refutation 374.24/61.86 tff(def_iii, axiom, iii() = ^[X0,X1]: n_some(diffprop(X1,X0))). 374.24/61.86 tff('0', plain, iii = (^[X0,X1]: n_some(diffprop(X1,X0))), 374.24/61.86 inference('simplify_rw_rule', [status(thm)], [def_iii])). 374.24/61.86 tff('1', plain, iii = (^[V_1,V_2]: n_some(diffprop(V_2,V_1))), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff(def_nat, axiom, nat() = d_Sep(omega,^[X0]: (X0 != emptyset))). 374.24/61.86 tff('2', plain, nat = d_Sep(omega,^[X0]: (X0 != emptyset)), 374.24/61.86 inference('simplify_rw_rule', [status(thm)], [def_nat])). 374.24/61.86 tff('3', plain, nat = d_Sep(omega,^[V_1]: (V_1 != emptyset)), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff(def_all_of, axiom, all_of() = 374.24/61.86 ^[X0:(($i > $o)),X1:(($i > $o))]: (![X2]: (is_of(X2,X0) => X1(X2)))). 374.24/61.86 tff('4', plain, 374.24/61.86 all_of = 374.24/61.86 (^[X0:(($i > $o)),X1:(($i > $o))]: (![X2]: (is_of(X2,X0) => X1(X2)))), 374.24/61.86 inference('simplify_rw_rule', [status(thm)], [def_all_of])). 374.24/61.86 tff('5', plain, 374.24/61.86 all_of = 374.24/61.86 (^[V_1:(($i > $o)),V_2:(($i > $o))]: (![X4]: (is_of(X4,V_1) => V_2(X4)))), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff(satz12, axiom, 374.24/61.86 (all_of(^[X0]: in(X0,nat), 374.24/61.86 ^[X0]: 374.24/61.86 all_of(^[X1]: in(X1,nat),^[X1]: (iii(X0,X1) => d_29_ii(X1,X0)))))). 374.24/61.86 tff(zf_stmt_0, axiom, 374.24/61.86 (![X4]: 374.24/61.86 (is_of(X4,^[V_1]: in(V_1,d_Sep(omega,^[V_2]: (V_2 != emptyset)))) => 374.24/61.86 (![X6]: 374.24/61.86 (is_of(X6,^[V_3]: in(V_3,d_Sep(omega,^[V_4]: (V_4 != emptyset)))) => 374.24/61.86 (n_some(diffprop(X6,X4)) => d_29_ii(X6,X4))))))). 374.24/61.86 tff('6', plain, 374.24/61.86 ![X65, X66]: 374.24/61.86 (~ is_of(X65, 374.24/61.86 (^[Y0 : $i]: in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | d_29_ii(X65, X66) 374.24/61.86 | ~ n_some(diffprop(X65, X66)) 374.24/61.86 | ~ is_of(X66, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset))))))), 374.24/61.86 inference('cnf', [status(esa)], [zf_stmt_0])). 374.24/61.86 tff(def_n_is, axiom, n_is() = e_is(nat)). 374.24/61.86 tff('7', plain, n_is = e_is(nat), 374.24/61.86 inference('simplify_rw_rule', [status(thm)], [def_n_is, '3'])). 374.24/61.86 tff('8', plain, n_is = e_is(nat), define([status(thm)])). 374.24/61.86 tff(n_ax4, axiom, 374.24/61.86 (all_of(^[X0]: in(X0,nat), 374.24/61.86 ^[X0]: 374.24/61.86 all_of(^[X1]: in(X1,nat), 374.24/61.86 ^[X1]: (n_is(ordsucc(X0),ordsucc(X1)) => n_is(X0,X1)))))). 374.24/61.86 tff(zf_stmt_1, axiom, 374.24/61.86 (![X4]: 374.24/61.86 (is_of(X4,^[V_1]: in(V_1,d_Sep(omega,^[V_2]: (V_2 != emptyset)))) => 374.24/61.86 (![X6]: 374.24/61.86 (is_of(X6,^[V_3]: in(V_3,d_Sep(omega,^[V_4]: (V_4 != emptyset)))) => 374.24/61.86 (e_is(d_Sep(omega,^[V_5]: (V_5 != emptyset)),ordsucc(X4), 374.24/61.86 ordsucc(X6)) => 374.24/61.86 e_is(d_Sep(omega,^[V_6]: (V_6 != emptyset)),X4,X6))))))). 374.24/61.86 tff('9', plain, 374.24/61.86 ![X18, X19]: 374.24/61.86 (~ is_of(X18, 374.24/61.86 (^[Y0 : $i]: in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | e_is(d_Sep(omega, (^[Y0 : $i]: (Y0 != emptyset))), X19, X18) 374.24/61.86 | ~ e_is(d_Sep(omega, (^[Y0 : $i]: (Y0 != emptyset))), ordsucc(X19), 374.24/61.86 ordsucc(X18)) 374.24/61.86 | ~ is_of(X19, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset))))))), 374.24/61.86 inference('cnf', [status(esa)], [zf_stmt_1])). 374.24/61.86 tff('10', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('11', plain, 374.24/61.86 ![X18, X19]: 374.24/61.86 (~ is_of(X18, 374.24/61.86 (^[Y0 : $i]: in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | e_is(d_Sep(omega, (^[Y0 : $i]: (Y0 != emptyset))), X19, X18) 374.24/61.86 | ~ e_is(d_Sep(omega, (^[Y0 : $i]: (Y0 != emptyset))), ordsucc(X19), 374.24/61.86 ordsucc(X18)) 374.24/61.86 | ~ is_of(X19, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset))))))), 374.24/61.86 inference('cnf', [status(esa)], [zf_stmt_1])). 374.24/61.86 tff('12', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('13', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('14', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('15', plain, 374.24/61.86 ![X65, X66]: 374.24/61.86 (~ is_of(X65, '#l_lift10344') 374.24/61.86 | d_29_ii(X65, X66) 374.24/61.86 | ~ n_some(diffprop(X65, X66)) 374.24/61.86 | ~ is_of(X66, '#l_lift10344')), 374.24/61.86 inference('lambda_lifting', [status(thm)], ['6', '10', '12', '13', '14'])). 374.24/61.86 tff(satz11, axiom, 374.24/61.86 (all_of(^[X0]: in(X0,nat), 374.24/61.86 ^[X0]: 374.24/61.86 all_of(^[X1]: in(X1,nat),^[X1]: (d_29_ii(X0,X1) => iii(X1,X0)))))). 374.24/61.86 tff(zf_stmt_2, axiom, 374.24/61.86 (![X4]: 374.24/61.86 (is_of(X4,^[V_1]: in(V_1,d_Sep(omega,^[V_2]: (V_2 != emptyset)))) => 374.24/61.86 (![X6]: 374.24/61.86 (is_of(X6,^[V_3]: in(V_3,d_Sep(omega,^[V_4]: (V_4 != emptyset)))) => 374.24/61.86 (d_29_ii(X4,X6) => n_some(diffprop(X4,X6)))))))). 374.24/61.86 tff('16', plain, 374.24/61.86 ![X36, X37]: 374.24/61.86 (~ is_of(X36, 374.24/61.86 (^[Y0 : $i]: in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | n_some(diffprop(X37, X36)) 374.24/61.86 | ~ d_29_ii(X37, X36) 374.24/61.86 | ~ is_of(X37, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset))))))), 374.24/61.86 inference('cnf', [status(esa)], [zf_stmt_2])). 374.24/61.86 tff('17', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('18', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('19', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('20', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('21', plain, 374.24/61.86 ![X36, X37]: 374.24/61.86 (~ is_of(X36, '#l_lift10344') 374.24/61.86 | n_some(diffprop(X37, X36)) 374.24/61.86 | ~ d_29_ii(X37, X36) 374.24/61.86 | ~ is_of(X37, '#l_lift10344')), 374.24/61.86 inference('lambda_lifting', [status(thm)], ['16', '17', '18', '19', '20'])). 374.24/61.86 tff(def_n_pl, axiom, n_pl() = ^[X0]: ap(plus(X0))). 374.24/61.86 tff('22', plain, n_pl = (^[X0]: ap(plus(X0))), 374.24/61.86 inference('simplify_rw_rule', [status(thm)], [def_n_pl])). 374.24/61.86 tff('23', plain, n_pl = (^[V_1]: ap(plus(V_1))), define([status(thm)])). 374.24/61.86 tff(satz6, axiom, 374.24/61.86 (all_of(^[X0]: in(X0,nat), 374.24/61.86 ^[X0]: 374.24/61.86 all_of(^[X1]: in(X1,nat),^[X1]: n_is(n_pl(X0,X1),n_pl(X1,X0)))))). 374.24/61.86 tff(zf_stmt_3, axiom, 374.24/61.86 (![X4]: 374.24/61.86 (is_of(X4,^[V_1]: in(V_1,d_Sep(omega,^[V_2]: (V_2 != emptyset)))) => 374.24/61.86 (![X6]: 374.24/61.86 (is_of(X6,^[V_3]: in(V_3,d_Sep(omega,^[V_4]: (V_4 != emptyset)))) => 374.24/61.86 e_is(d_Sep(omega,^[V_5]: (V_5 != emptyset)),ap(plus(X4),X6), 374.24/61.86 ap(plus(X6),X4))))))). 374.24/61.86 tff('24', plain, 374.24/61.86 ![X159, X160]: 374.24/61.86 (~ is_of(X159, 374.24/61.86 (^[Y0 : $i]: in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | e_is(d_Sep(omega, (^[Y0 : $i]: (Y0 != emptyset))), 374.24/61.86 ap(plus(X160), X159), ap(plus(X159), X160)) 374.24/61.86 | ~ is_of(X160, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset))))))), 374.24/61.86 inference('cnf', [status(esa)], [zf_stmt_3])). 374.24/61.86 tff('25', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('26', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('27', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('28', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('29', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('30', plain, 374.24/61.86 ![X159, X160]: 374.24/61.86 (~ is_of(X159, '#l_lift10344') 374.24/61.86 | e_is(d_Sep(omega, '#l_lift10343'), ap(plus(X160), X159), 374.24/61.86 ap(plus(X159), X160)) 374.24/61.86 | ~ is_of(X160, '#l_lift10344')), 374.24/61.86 inference('lambda_lifting', [status(thm)], 374.24/61.86 ['24', '25', '26', '27', '28', '29'])). 374.24/61.86 tff(satz20e, axiom, 374.24/61.86 (all_of(^[X0]: in(X0,nat), 374.24/61.86 ^[X0]: 374.24/61.86 all_of(^[X1]: in(X1,nat), 374.24/61.86 ^[X1]: 374.24/61.86 all_of(^[X2]: in(X2,nat), 374.24/61.86 ^[X2]: 374.24/61.86 (n_is(n_pl(X2,X0),n_pl(X2,X1)) => n_is(X0,X1))))))). 374.24/61.86 tff(zf_stmt_4, axiom, 374.24/61.86 (![X4]: 374.24/61.86 (is_of(X4,^[V_1]: in(V_1,d_Sep(omega,^[V_2]: (V_2 != emptyset)))) => 374.24/61.86 (![X6]: 374.24/61.86 (is_of(X6,^[V_3]: in(V_3,d_Sep(omega,^[V_4]: (V_4 != emptyset)))) => 374.24/61.86 (![X8]: 374.24/61.86 (is_of(X8,^[V_5]: in(V_5,d_Sep(omega,^[V_6]: (V_6 != emptyset)))) => 374.24/61.86 (e_is(d_Sep(omega,^[V_7]: (V_7 != emptyset)),ap(plus(X8),X4), 374.24/61.86 ap(plus(X8),X6)) => 374.24/61.86 e_is(d_Sep(omega,^[V_8]: (V_8 != emptyset)),X4,X6))))))))). 374.24/61.86 tff('31', plain, 374.24/61.86 ![X164, X165, X166]: 374.24/61.86 (~ is_of(X164, 374.24/61.86 (^[Y0 : $i]: in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | e_is(d_Sep(omega, (^[Y0 : $i]: (Y0 != emptyset))), X165, X164) 374.24/61.86 | ~ e_is(d_Sep(omega, (^[Y0 : $i]: (Y0 != emptyset))), 374.24/61.86 ap(plus(X166), X165), ap(plus(X166), X164)) 374.24/61.86 | ~ is_of(X166, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | ~ is_of(X165, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset))))))), 374.24/61.86 inference('cnf', [status(esa)], [zf_stmt_4])). 374.24/61.86 tff('32', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('33', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('34', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('35', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('36', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('37', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('38', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('39', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('40', plain, 374.24/61.86 ![X164, X165, X166]: 374.24/61.86 (~ is_of(X164, '#l_lift10344') 374.24/61.86 | e_is(d_Sep(omega, '#l_lift10343'), X165, X164) 374.24/61.86 | ~ e_is(d_Sep(omega, '#l_lift10343'), ap(plus(X166), X165), 374.24/61.86 ap(plus(X166), X164)) 374.24/61.86 | ~ is_of(X166, '#l_lift10344') 374.24/61.86 | ~ is_of(X165, '#l_lift10344')), 374.24/61.86 inference('lambda_lifting', [status(thm)], 374.24/61.86 ['31', '32', '33', '34', '35', '36', '37', '38', '39'])). 374.24/61.86 tff(satz19k, axiom, 374.24/61.86 (all_of(^[X0]: in(X0,nat), 374.24/61.86 ^[X0]: 374.24/61.86 all_of(^[X1]: in(X1,nat), 374.24/61.86 ^[X1]: 374.24/61.86 all_of(^[X2]: in(X2,nat), 374.24/61.86 ^[X2]: 374.24/61.86 all_of(^[X3]: in(X3,nat), 374.24/61.86 ^[X3]: 374.24/61.86 (n_is(X0,X1) => 374.24/61.86 (iii(X2,X3) => 374.24/61.86 iii(n_pl(X2,X0),n_pl(X3,X1)))))))))). 374.24/61.86 tff(zf_stmt_5, axiom, 374.24/61.86 (![X4]: 374.24/61.86 (is_of(X4,^[V_1]: in(V_1,d_Sep(omega,^[V_2]: (V_2 != emptyset)))) => 374.24/61.86 (![X6]: 374.24/61.86 (is_of(X6,^[V_3]: in(V_3,d_Sep(omega,^[V_4]: (V_4 != emptyset)))) => 374.24/61.86 (![X8]: 374.24/61.86 (is_of(X8,^[V_5]: in(V_5,d_Sep(omega,^[V_6]: (V_6 != emptyset)))) => 374.24/61.86 (![X10]: 374.24/61.86 (is_of(X10, 374.24/61.86 ^[V_7]: 374.24/61.86 in(V_7,d_Sep(omega,^[V_8]: (V_8 != emptyset)))) => 374.24/61.86 (e_is(d_Sep(omega,^[V_9]: (V_9 != emptyset)),X4,X6) => 374.24/61.86 (n_some(diffprop(X10,X8)) => 374.24/61.86 n_some(diffprop(ap(plus(X10),X6),ap(plus(X8),X4)))))))))))))). 374.24/61.86 tff('41', plain, 374.24/61.86 ![X98, X99, X100, X101]: 374.24/61.86 (~ is_of(X98, 374.24/61.86 (^[Y0 : $i]: in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | ~ is_of(X99, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | ~ n_some(diffprop(X99, X100)) 374.24/61.86 | n_some(diffprop(ap(plus(X99), X98), ap(plus(X100), X101))) 374.24/61.86 | ~ e_is(d_Sep(omega, (^[Y0 : $i]: (Y0 != emptyset))), X101, X98) 374.24/61.86 | ~ is_of(X100, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | ~ is_of(X101, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset))))))), 374.24/61.86 inference('cnf', [status(esa)], [zf_stmt_5])). 374.24/61.86 tff('42', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('43', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('44', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('45', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('46', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('47', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('48', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('49', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('50', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('51', plain, 374.24/61.86 ![X98, X99, X100, X101]: 374.24/61.86 (~ is_of(X98, '#l_lift10344') 374.24/61.86 | ~ is_of(X99, '#l_lift10344') 374.24/61.86 | ~ n_some(diffprop(X99, X100)) 374.24/61.86 | n_some(diffprop(ap(plus(X99), X98), ap(plus(X100), X101))) 374.24/61.86 | ~ e_is(d_Sep(omega, '#l_lift10343'), X101, X98) 374.24/61.86 | ~ is_of(X100, '#l_lift10344') 374.24/61.86 | ~ is_of(X101, '#l_lift10344')), 374.24/61.86 inference('lambda_lifting', [status(thm)], 374.24/61.86 ['41', '42', '43', '44', '45', '46', '47', '48', '49', '50'])). 374.24/61.86 tff(satz15, axiom, 374.24/61.86 (all_of(^[X0]: in(X0,nat), 374.24/61.86 ^[X0]: 374.24/61.86 all_of(^[X1]: in(X1,nat), 374.24/61.86 ^[X1]: 374.24/61.86 all_of(^[X2]: in(X2,nat), 374.24/61.86 ^[X2]: (iii(X0,X1) => (iii(X1,X2) => iii(X0,X2)))))))). 374.24/61.86 tff(zf_stmt_6, axiom, 374.24/61.86 (![X4]: 374.24/61.86 (is_of(X4,^[V_1]: in(V_1,d_Sep(omega,^[V_2]: (V_2 != emptyset)))) => 374.24/61.86 (![X6]: 374.24/61.86 (is_of(X6,^[V_3]: in(V_3,d_Sep(omega,^[V_4]: (V_4 != emptyset)))) => 374.24/61.86 (![X8]: 374.24/61.86 (is_of(X8,^[V_5]: in(V_5,d_Sep(omega,^[V_6]: (V_6 != emptyset)))) => 374.24/61.86 (n_some(diffprop(X6,X4)) => 374.24/61.86 (n_some(diffprop(X8,X6)) => n_some(diffprop(X8,X4))))))))))). 374.24/61.86 tff('52', plain, 374.24/61.86 ![X60, X61, X62]: 374.24/61.86 (~ is_of(X60, 374.24/61.86 (^[Y0 : $i]: in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | ~ n_some(diffprop(X60, X61)) 374.24/61.86 | n_some(diffprop(X62, X61)) 374.24/61.86 | ~ n_some(diffprop(X62, X60)) 374.24/61.86 | ~ is_of(X62, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | ~ is_of(X61, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset))))))), 374.24/61.86 inference('cnf', [status(esa)], [zf_stmt_6])). 374.24/61.86 tff('53', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('54', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('55', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('56', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('57', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('58', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('59', plain, 374.24/61.86 ![X60, X61, X62]: 374.24/61.86 (~ is_of(X60, '#l_lift10344') 374.24/61.86 | ~ n_some(diffprop(X60, X61)) 374.24/61.86 | n_some(diffprop(X62, X61)) 374.24/61.86 | ~ n_some(diffprop(X62, X60)) 374.24/61.86 | ~ is_of(X62, '#l_lift10344') 374.24/61.86 | ~ is_of(X61, '#l_lift10344')), 374.24/61.86 inference('lambda_lifting', [status(thm)], 374.24/61.86 ['52', '53', '54', '55', '56', '57', '58'])). 374.24/61.86 tff(def_moreis, axiom, moreis() = ^[X0,X1]: l_or(d_29_ii(X0,X1),n_is(X0,X1))). 374.24/61.86 tff(def_l_or, axiom, l_or() = ^[X0:$o]: imp(d_not(X0))). 374.24/61.86 tff(def_d_not, axiom, d_not() = ^[X0:$o]: imp(X0,$false)). 374.24/61.86 tff('60', plain, d_not = (^[X0:$o]: imp(X0,$false)), 374.24/61.86 inference('simplify_rw_rule', [status(thm)], [def_d_not])). 374.24/61.86 tff('61', plain, d_not = (^[V_1:$o]: imp(V_1,$false)), define([status(thm)])). 374.24/61.86 tff('62', plain, l_or = (^[X0:$o]: imp(d_not(X0))), 374.24/61.86 inference('simplify_rw_rule', [status(thm)], [def_l_or, '61'])). 374.24/61.86 tff('63', plain, l_or = (^[V_1:$o]: imp(d_not(V_1))), define([status(thm)])). 374.24/61.86 tff('64', plain, moreis = (^[X0,X1]: l_or(d_29_ii(X0,X1),n_is(X0,X1))), 374.24/61.86 inference('simplify_rw_rule', [status(thm)], [def_moreis, '8', '3', '63'])). 374.24/61.86 tff('65', plain, moreis = (^[V_1,V_2]: l_or(d_29_ii(V_1,V_2),n_is(V_1,V_2))), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff(satz10f, axiom, 374.24/61.86 (all_of(^[X0]: in(X0,nat), 374.24/61.86 ^[X0]: 374.24/61.86 all_of(^[X1]: in(X1,nat), 374.24/61.86 ^[X1]: (d_not(iii(X0,X1)) => moreis(X0,X1)))))). 374.24/61.86 tff(zf_stmt_7, axiom, 374.24/61.86 (![X4]: 374.24/61.86 (is_of(X4,^[V_1]: in(V_1,d_Sep(omega,^[V_2]: (V_2 != emptyset)))) => 374.24/61.86 (![X6]: 374.24/61.86 (is_of(X6,^[V_3]: in(V_3,d_Sep(omega,^[V_4]: (V_4 != emptyset)))) => 374.24/61.86 (imp(n_some(diffprop(X6,X4)),$false) => 374.24/61.86 imp(imp(d_29_ii(X4,X6),$false), 374.24/61.86 e_is(d_Sep(omega,^[V_5]: (V_5 != emptyset)),X4,X6)))))))). 374.24/61.86 tff('66', plain, 374.24/61.86 ![X175, X176]: 374.24/61.86 (~ is_of(X175, 374.24/61.86 (^[Y0 : $i]: in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | imp(imp(d_29_ii(X176, X175), $false), 374.24/61.86 e_is(d_Sep(omega, (^[Y0 : $i]: (Y0 != emptyset))), X176, X175)) 374.24/61.86 | ~ imp(n_some(diffprop(X175, X176)), $false) 374.24/61.86 | ~ is_of(X176, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset))))))), 374.24/61.86 inference('cnf', [status(esa)], [zf_stmt_7])). 374.24/61.86 tff('67', plain, 374.24/61.86 ![X175, X176]: 374.24/61.86 (~ n_some(diffprop(X175, X176)) 374.24/61.86 | ~ is_of(X175, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | imp(imp(d_29_ii(X176, X175), $false), 374.24/61.86 e_is(d_Sep(omega, (^[Y0 : $i]: (Y0 != emptyset))), X176, X175)) 374.24/61.86 | ~ imp($true, $false) 374.24/61.86 | ~ is_of(X176, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset))))))), 374.24/61.86 inference('bool_simp', [status(thm)], ['66'])). 374.24/61.86 tff('68', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('69', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('70', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('71', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('72', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('73', plain, 374.24/61.86 ![X175, X176]: 374.24/61.86 (~ n_some(diffprop(X175, X176)) 374.24/61.86 | ~ is_of(X175, '#l_lift10344') 374.24/61.86 | imp(imp(d_29_ii(X176, X175), $false), 374.24/61.86 e_is(d_Sep(omega, '#l_lift10343'), X176, X175)) 374.24/61.86 | ~ imp($true, $false) 374.24/61.86 | ~ is_of(X176, '#l_lift10344')), 374.24/61.86 inference('lambda_lifting', [status(thm)], 374.24/61.86 ['67', '68', '69', '70', '71', '72'])). 374.24/61.86 tff(def_lessis, axiom, lessis() = ^[X0,X1]: l_or(iii(X0,X1),n_is(X0,X1))). 374.24/61.86 tff('74', plain, lessis = (^[X0,X1]: l_or(iii(X0,X1),n_is(X0,X1))), 374.24/61.86 inference('simplify_rw_rule', [status(thm)], 374.24/61.86 [def_lessis, '1', '8', '3', '63', '61'])). 374.24/61.86 tff('75', plain, lessis = (^[V_1,V_2]: l_or(iii(V_1,V_2),n_is(V_1,V_2))), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff(satz22d, conjecture, 374.24/61.86 (all_of(^[X0]: in(X0,nat), 374.24/61.86 ^[X0]: 374.24/61.86 all_of(^[X1]: in(X1,nat), 374.24/61.86 ^[X1]: 374.24/61.86 all_of(^[X2]: in(X2,nat), 374.24/61.86 ^[X2]: 374.24/61.86 all_of(^[X3]: in(X3,nat), 374.24/61.86 ^[X3]: 374.24/61.86 (iii(X0,X1) => 374.24/61.86 (lessis(X2,X3) => 374.24/61.86 iii(n_pl(X0,X2),n_pl(X1,X3)))))))))). 374.24/61.86 tff(zf_stmt_8, conjecture, 374.24/61.86 (![X4]: 374.24/61.86 (is_of(X4,^[V_1]: in(V_1,d_Sep(omega,^[V_2]: (V_2 != emptyset)))) => 374.24/61.86 (![X6]: 374.24/61.86 (is_of(X6,^[V_3]: in(V_3,d_Sep(omega,^[V_4]: (V_4 != emptyset)))) => 374.24/61.86 (![X8]: 374.24/61.86 (is_of(X8,^[V_5]: in(V_5,d_Sep(omega,^[V_6]: (V_6 != emptyset)))) => 374.24/61.86 (![X10]: 374.24/61.86 (is_of(X10, 374.24/61.86 ^[V_7]: 374.24/61.86 in(V_7,d_Sep(omega,^[V_8]: (V_8 != emptyset)))) => 374.24/61.86 (n_some(diffprop(X6,X4)) => 374.24/61.86 (imp(imp(n_some(diffprop(X10,X8)),$false), 374.24/61.86 e_is(d_Sep(omega,^[V_9]: (V_9 != emptyset)),X8,X10)) => 374.24/61.86 n_some(diffprop(ap(plus(X6),X10),ap(plus(X4),X8)))))))))))))). 374.24/61.86 tff(zf_stmt_9, negated_conjecture, 374.24/61.86 (~ 374.24/61.86 (![X4]: 374.24/61.86 (is_of(X4,^[V_1]: in(V_1,d_Sep(omega,^[V_2]: (V_2 != emptyset)))) => 374.24/61.86 (![X6]: 374.24/61.86 (is_of(X6,^[V_3]: in(V_3,d_Sep(omega,^[V_4]: (V_4 != emptyset)))) => 374.24/61.86 (![X8]: 374.24/61.86 (is_of(X8, 374.24/61.86 ^[V_5]: in(V_5,d_Sep(omega,^[V_6]: (V_6 != emptyset)))) => 374.24/61.86 (![X10]: 374.24/61.86 (is_of(X10, 374.24/61.86 ^[V_7]: 374.24/61.86 in(V_7,d_Sep(omega,^[V_8]: (V_8 != emptyset)))) => 374.24/61.86 (n_some(diffprop(X6,X4)) => 374.24/61.86 (imp(imp(n_some(diffprop(X10,X8)),$false), 374.24/61.86 e_is(d_Sep(omega,^[V_9]: (V_9 != emptyset)),X8,X10)) => 374.24/61.86 n_some(diffprop(ap(plus(X6),X10),ap(plus(X4),X8))))))))))))))). 374.24/61.86 tff('76', plain, 374.24/61.86 is_of(sk_X10, 374.24/61.86 (^[Y0 : $i]: in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))), 374.24/61.86 inference('cnf', [status(esa)], [zf_stmt_9])). 374.24/61.86 tff('77', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('78', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('79', plain, is_of(sk_X10, '#l_lift10344'), 374.24/61.86 inference('lambda_lifting', [status(thm)], ['76', '77', '78'])). 374.24/61.86 tff('80', plain, 374.24/61.86 is_of(sk_X8, 374.24/61.86 (^[Y0 : $i]: in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))), 374.24/61.86 inference('cnf', [status(esa)], [zf_stmt_9])). 374.24/61.86 tff('81', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('82', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('83', plain, is_of(sk_X8, '#l_lift10344'), 374.24/61.86 inference('lambda_lifting', [status(thm)], ['80', '81', '82'])). 374.24/61.86 tff('84', plain, 374.24/61.86 is_of(sk_X6, 374.24/61.86 (^[Y0 : $i]: in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))), 374.24/61.86 inference('cnf', [status(esa)], [zf_stmt_9])). 374.24/61.86 tff(satz10k, axiom, 374.24/61.86 (all_of(^[X0]: in(X0,nat), 374.24/61.86 ^[X0]: 374.24/61.86 all_of(^[X1]: in(X1,nat), 374.24/61.86 ^[X1]: (d_not(lessis(X0,X1)) => d_29_ii(X0,X1)))))). 374.24/61.86 tff(zf_stmt_10, axiom, 374.24/61.86 (![X4]: 374.24/61.86 (is_of(X4,^[V_1]: in(V_1,d_Sep(omega,^[V_2]: (V_2 != emptyset)))) => 374.24/61.86 (![X6]: 374.24/61.86 (is_of(X6,^[V_3]: in(V_3,d_Sep(omega,^[V_4]: (V_4 != emptyset)))) => 374.24/61.86 (imp(imp(imp(n_some(diffprop(X6,X4)),$false), 374.24/61.86 e_is(d_Sep(omega,^[V_5]: (V_5 != emptyset)),X4,X6)), 374.24/61.86 $false) => 374.24/61.86 d_29_ii(X4,X6))))))). 374.24/61.86 tff('85', plain, 374.24/61.86 ![X48, X49]: 374.24/61.86 (~ is_of(X48, 374.24/61.86 (^[Y0 : $i]: in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | d_29_ii(X49, X48) 374.24/61.86 | ~ imp( 374.24/61.86 imp(imp(n_some(diffprop(X48, X49)), $false), 374.24/61.86 e_is(d_Sep(omega, (^[Y0 : $i]: (Y0 != emptyset))), X49, X48)), 374.24/61.86 $false) 374.24/61.86 | ~ is_of(X49, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset))))))), 374.24/61.86 inference('cnf', [status(esa)], [zf_stmt_10])). 374.24/61.86 tff('86', plain, 374.24/61.86 ![X48, X49]: 374.24/61.86 (~ imp(imp(n_some(diffprop(X48, X49)), $false), 374.24/61.86 e_is(d_Sep(omega, (^[Y0 : $i]: (Y0 != emptyset))), X49, X48)) 374.24/61.86 | ~ is_of(X48, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | d_29_ii(X49, X48) 374.24/61.86 | ~ imp($true, $false) 374.24/61.86 | ~ is_of(X49, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset))))))), 374.24/61.86 inference('bool_simp', [status(thm)], ['85'])). 374.24/61.86 tff('87', plain, 374.24/61.86 ![X0]: 374.24/61.86 (~ is_of(X0, 374.24/61.86 (^[Y0 : $i]: in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | ~ imp($true, $false) 374.24/61.86 | d_29_ii(X0, sk_X6) 374.24/61.86 | ~ imp(imp(n_some(diffprop(sk_X6, X0)), $false), 374.24/61.86 e_is(d_Sep(omega, (^[Y0 : $i]: (Y0 != emptyset))), X0, sk_X6))), 374.24/61.86 inference('sup-', [status(thm)], ['84', '86'])). 374.24/61.86 tff('88', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('89', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('90', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('91', plain, 374.24/61.86 ![X0]: 374.24/61.86 (~ is_of(X0, '#l_lift10344') 374.24/61.86 | ~ imp($true, $false) 374.24/61.86 | d_29_ii(X0, sk_X6) 374.24/61.86 | ~ imp(imp(n_some(diffprop(sk_X6, X0)), $false), 374.24/61.86 e_is(d_Sep(omega, '#l_lift10343'), X0, sk_X6))), 374.24/61.86 inference('lambda_lifting', [status(thm)], ['87', '88', '89', '90'])). 374.24/61.86 tff(satz10c, axiom, 374.24/61.86 (all_of(^[X0]: in(X0,nat), 374.24/61.86 ^[X0]: 374.24/61.86 all_of(^[X1]: in(X1,nat), 374.24/61.86 ^[X1]: (moreis(X0,X1) => d_not(iii(X0,X1))))))). 374.24/61.86 tff(zf_stmt_11, axiom, 374.24/61.86 (![X4]: 374.24/61.86 (is_of(X4,^[V_1]: in(V_1,d_Sep(omega,^[V_2]: (V_2 != emptyset)))) => 374.24/61.86 (![X6]: 374.24/61.86 (is_of(X6,^[V_3]: in(V_3,d_Sep(omega,^[V_4]: (V_4 != emptyset)))) => 374.24/61.86 (imp(imp(d_29_ii(X4,X6),$false), 374.24/61.86 e_is(d_Sep(omega,^[V_5]: (V_5 != emptyset)),X4,X6)) => 374.24/61.86 imp(n_some(diffprop(X6,X4)),$false))))))). 374.24/61.86 tff('92', plain, 374.24/61.86 ![X46, X47]: 374.24/61.86 (~ is_of(X46, 374.24/61.86 (^[Y0 : $i]: in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | imp(n_some(diffprop(X46, X47)), $false) 374.24/61.86 | ~ imp(imp(d_29_ii(X47, X46), $false), 374.24/61.86 e_is(d_Sep(omega, (^[Y0 : $i]: (Y0 != emptyset))), X47, X46)) 374.24/61.86 | ~ is_of(X47, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset))))))), 374.24/61.86 inference('cnf', [status(esa)], [zf_stmt_11])). 374.24/61.86 tff('93', plain, 374.24/61.86 ![X46, X47]: 374.24/61.86 (~ e_is(d_Sep(omega, (^[Y0 : $i]: (Y0 != emptyset))), X47, X46) 374.24/61.86 | ~ is_of(X46, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | imp(n_some(diffprop(X46, X47)), $false) 374.24/61.86 | ~ imp(imp(d_29_ii(X47, X46), $false), $true) 374.24/61.86 | ~ is_of(X47, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset))))))), 374.24/61.86 inference('bool_simp', [status(thm)], ['92'])). 374.24/61.86 tff('94', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('95', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('96', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('97', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('98', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('99', plain, 374.24/61.86 ![X46, X47]: 374.24/61.86 (~ e_is(d_Sep(omega, '#l_lift10343'), X47, X46) 374.24/61.86 | ~ is_of(X46, '#l_lift10344') 374.24/61.86 | imp(n_some(diffprop(X46, X47)), $false) 374.24/61.86 | ~ imp(imp(d_29_ii(X47, X46), $false), $true) 374.24/61.86 | ~ is_of(X47, '#l_lift10344')), 374.24/61.86 inference('lambda_lifting', [status(thm)], 374.24/61.86 ['93', '94', '95', '96', '97', '98'])). 374.24/61.86 tff('100', plain, 374.24/61.86 is_of(sk_X6, 374.24/61.86 (^[Y0 : $i]: in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))), 374.24/61.86 inference('cnf', [status(esa)], [zf_stmt_9])). 374.24/61.86 tff('101', plain, 374.24/61.86 ![X175, X176]: 374.24/61.86 (~ n_some(diffprop(X175, X176)) 374.24/61.86 | ~ is_of(X175, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | imp(imp(d_29_ii(X176, X175), $false), 374.24/61.86 e_is(d_Sep(omega, (^[Y0 : $i]: (Y0 != emptyset))), X176, X175)) 374.24/61.86 | ~ imp($true, $false) 374.24/61.86 | ~ is_of(X176, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset))))))), 374.24/61.86 inference('bool_simp', [status(thm)], ['66'])). 374.24/61.86 tff('102', plain, 374.24/61.86 ![X0]: 374.24/61.86 (~ is_of(X0, 374.24/61.86 (^[Y0 : $i]: in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | ~ imp($true, $false) 374.24/61.86 | imp(imp(d_29_ii(X0, sk_X6), $false), 374.24/61.86 e_is(d_Sep(omega, (^[Y0 : $i]: (Y0 != emptyset))), X0, sk_X6)) 374.24/61.86 | ~ n_some(diffprop(sk_X6, X0))), 374.24/61.86 inference('sup-', [status(thm)], ['100', '101'])). 374.24/61.86 tff('103', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('104', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('105', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('106', plain, 374.24/61.86 ![X0]: 374.24/61.86 (~ is_of(X0, '#l_lift10344') 374.24/61.86 | ~ imp($true, $false) 374.24/61.86 | imp(imp(d_29_ii(X0, sk_X6), $false), 374.24/61.86 e_is(d_Sep(omega, '#l_lift10343'), X0, sk_X6)) 374.24/61.86 | ~ n_some(diffprop(sk_X6, X0))), 374.24/61.86 inference('lambda_lifting', [status(thm)], ['102', '103', '104', '105'])). 374.24/61.86 tff(satz21a, axiom, 374.24/61.86 (all_of(^[X0]: in(X0,nat), 374.24/61.86 ^[X0]: 374.24/61.86 all_of(^[X1]: in(X1,nat), 374.24/61.86 ^[X1]: 374.24/61.86 all_of(^[X2]: in(X2,nat), 374.24/61.86 ^[X2]: 374.24/61.86 all_of(^[X3]: in(X3,nat), 374.24/61.86 ^[X3]: 374.24/61.86 (iii(X0,X1) => 374.24/61.86 (iii(X2,X3) => 374.24/61.86 iii(n_pl(X0,X2),n_pl(X1,X3)))))))))). 374.24/61.86 tff(zf_stmt_12, axiom, 374.24/61.86 (![X4]: 374.24/61.86 (is_of(X4,^[V_1]: in(V_1,d_Sep(omega,^[V_2]: (V_2 != emptyset)))) => 374.24/61.86 (![X6]: 374.24/61.86 (is_of(X6,^[V_3]: in(V_3,d_Sep(omega,^[V_4]: (V_4 != emptyset)))) => 374.24/61.86 (![X8]: 374.24/61.86 (is_of(X8,^[V_5]: in(V_5,d_Sep(omega,^[V_6]: (V_6 != emptyset)))) => 374.24/61.86 (![X10]: 374.24/61.86 (is_of(X10, 374.24/61.86 ^[V_7]: 374.24/61.86 in(V_7,d_Sep(omega,^[V_8]: (V_8 != emptyset)))) => 374.24/61.86 (n_some(diffprop(X6,X4)) => 374.24/61.86 (n_some(diffprop(X10,X8)) => 374.24/61.86 n_some(diffprop(ap(plus(X6),X10),ap(plus(X4),X8)))))))))))))). 374.24/61.86 tff('107', plain, 374.24/61.86 ![X76, X77, X78, X79]: 374.24/61.86 (~ is_of(X76, 374.24/61.86 (^[Y0 : $i]: in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | ~ is_of(X77, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | ~ n_some(diffprop(X77, X78)) 374.24/61.86 | n_some(diffprop(ap(plus(X76), X77), ap(plus(X79), X78))) 374.24/61.86 | ~ n_some(diffprop(X76, X79)) 374.24/61.86 | ~ is_of(X78, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | ~ is_of(X79, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset))))))), 374.24/61.86 inference('cnf', [status(esa)], [zf_stmt_12])). 374.24/61.86 tff('108', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('109', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('110', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('111', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('112', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('113', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('114', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('115', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('116', plain, 374.24/61.86 ![X76, X77, X78, X79]: 374.24/61.86 (~ is_of(X76, '#l_lift10344') 374.24/61.86 | ~ is_of(X77, '#l_lift10344') 374.24/61.86 | ~ n_some(diffprop(X77, X78)) 374.24/61.86 | n_some(diffprop(ap(plus(X76), X77), ap(plus(X79), X78))) 374.24/61.86 | ~ n_some(diffprop(X76, X79)) 374.24/61.86 | ~ is_of(X78, '#l_lift10344') 374.24/61.86 | ~ is_of(X79, '#l_lift10344')), 374.24/61.86 inference('lambda_lifting', [status(thm)], 374.24/61.86 ['107', '108', '109', '110', '111', '112', '113', '114', '115'])). 374.24/61.86 tff('117', plain, 374.24/61.86 ~ n_some(diffprop(ap(plus(sk_X6), sk_X10), ap(plus(sk_X4), sk_X8))), 374.24/61.86 inference('cnf', [status(esa)], [zf_stmt_9])). 374.24/61.86 tff('118', plain, 374.24/61.86 is_of(sk_X4, 374.24/61.86 (^[Y0 : $i]: in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))), 374.24/61.86 inference('cnf', [status(esa)], [zf_stmt_9])). 374.24/61.86 tff('119', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('120', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('121', plain, is_of(sk_X4, '#l_lift10344'), 374.24/61.86 inference('lambda_lifting', [status(thm)], ['118', '119', '120'])). 374.24/61.86 tff('122', plain, 374.24/61.86 is_of(sk_X6, 374.24/61.86 (^[Y0 : $i]: in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))), 374.24/61.86 inference('cnf', [status(esa)], [zf_stmt_9])). 374.24/61.86 tff('123', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('124', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('125', plain, is_of(sk_X6, '#l_lift10344'), 374.24/61.86 inference('lambda_lifting', [status(thm)], ['122', '123', '124'])). 374.24/61.86 tff('126', plain, n_some(diffprop(sk_X6, sk_X4)), 374.24/61.86 inference('cnf', [status(esa)], [zf_stmt_9])). 374.24/61.86 tff('127', plain, 374.24/61.86 imp(imp(n_some(diffprop(sk_X10, sk_X8)), $false), 374.24/61.86 e_is(d_Sep(omega, (^[Y0 : $i]: (Y0 != emptyset))), sk_X8, sk_X10)), 374.24/61.86 inference('cnf', [status(esa)], [zf_stmt_9])). 374.24/61.86 tff('128', plain, 374.24/61.86 (~ e_is(d_Sep(omega, (^[Y0 : $i]: (Y0 != emptyset))), sk_X8, sk_X10) 374.24/61.86 | imp(imp(n_some(diffprop(sk_X10, sk_X8)), $false), $true)), 374.24/61.86 inference('bool_simp', [status(thm)], ['127'])). 374.24/61.86 tff('129', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('130', plain, 374.24/61.86 (~ e_is(d_Sep(omega, '#l_lift10343'), sk_X8, sk_X10) 374.24/61.86 | imp(imp(n_some(diffprop(sk_X10, sk_X8)), $false), $true)), 374.24/61.86 inference('lambda_lifting', [status(thm)], ['128', '129'])). 374.24/61.86 tff(satz16a, axiom, 374.24/61.86 (all_of(^[X0]: in(X0,nat), 374.24/61.86 ^[X0]: 374.24/61.86 all_of(^[X1]: in(X1,nat), 374.24/61.86 ^[X1]: 374.24/61.86 all_of(^[X2]: in(X2,nat), 374.24/61.86 ^[X2]: 374.24/61.86 (lessis(X0,X1) => (iii(X1,X2) => iii(X0,X2)))))))). 374.24/61.86 tff(zf_stmt_13, axiom, 374.24/61.86 (![X4]: 374.24/61.86 (is_of(X4,^[V_1]: in(V_1,d_Sep(omega,^[V_2]: (V_2 != emptyset)))) => 374.24/61.86 (![X6]: 374.24/61.86 (is_of(X6,^[V_3]: in(V_3,d_Sep(omega,^[V_4]: (V_4 != emptyset)))) => 374.24/61.86 (![X8]: 374.24/61.86 (is_of(X8,^[V_5]: in(V_5,d_Sep(omega,^[V_6]: (V_6 != emptyset)))) => 374.24/61.86 (imp(imp(n_some(diffprop(X6,X4)),$false), 374.24/61.86 e_is(d_Sep(omega,^[V_7]: (V_7 != emptyset)),X4,X6)) => 374.24/61.86 (n_some(diffprop(X8,X6)) => n_some(diffprop(X8,X4))))))))))). 374.24/61.86 tff('131', plain, 374.24/61.86 ![X105, X106, X107]: 374.24/61.86 (~ is_of(X105, 374.24/61.86 (^[Y0 : $i]: in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | ~ imp(imp(n_some(diffprop(X105, X106)), $false), 374.24/61.86 e_is(d_Sep(omega, (^[Y0 : $i]: (Y0 != emptyset))), X106, X105)) 374.24/61.86 | n_some(diffprop(X107, X106)) 374.24/61.86 | ~ n_some(diffprop(X107, X105)) 374.24/61.86 | ~ is_of(X107, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | ~ is_of(X106, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset))))))), 374.24/61.86 inference('cnf', [status(esa)], [zf_stmt_13])). 374.24/61.86 tff('132', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('133', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('134', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('135', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('136', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('137', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('138', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('139', plain, 374.24/61.86 ![X105, X106, X107]: 374.24/61.86 (~ is_of(X105, '#l_lift10344') 374.24/61.86 | ~ imp(imp(n_some(diffprop(X105, X106)), $false), 374.24/61.86 e_is(d_Sep(omega, '#l_lift10343'), X106, X105)) 374.24/61.86 | n_some(diffprop(X107, X106)) 374.24/61.86 | ~ n_some(diffprop(X107, X105)) 374.24/61.86 | ~ is_of(X107, '#l_lift10344') 374.24/61.86 | ~ is_of(X106, '#l_lift10344')), 374.24/61.86 inference('lambda_lifting', [status(thm)], 374.24/61.86 ['131', '132', '133', '134', '135', '136', '137', '138'])). 374.24/61.86 tff(satz16b, axiom, 374.24/61.86 (all_of(^[X0]: in(X0,nat), 374.24/61.86 ^[X0]: 374.24/61.86 all_of(^[X1]: in(X1,nat), 374.24/61.86 ^[X1]: 374.24/61.86 all_of(^[X2]: in(X2,nat), 374.24/61.86 ^[X2]: 374.24/61.86 (iii(X0,X1) => (lessis(X1,X2) => iii(X0,X2)))))))). 374.24/61.86 tff(zf_stmt_14, axiom, 374.24/61.86 (![X4]: 374.24/61.86 (is_of(X4,^[V_1]: in(V_1,d_Sep(omega,^[V_2]: (V_2 != emptyset)))) => 374.24/61.86 (![X6]: 374.24/61.86 (is_of(X6,^[V_3]: in(V_3,d_Sep(omega,^[V_4]: (V_4 != emptyset)))) => 374.24/61.86 (![X8]: 374.24/61.86 (is_of(X8,^[V_5]: in(V_5,d_Sep(omega,^[V_6]: (V_6 != emptyset)))) => 374.24/61.86 (n_some(diffprop(X6,X4)) => 374.24/61.86 (imp(imp(n_some(diffprop(X8,X6)),$false), 374.24/61.86 e_is(d_Sep(omega,^[V_7]: (V_7 != emptyset)),X6,X8)) => 374.24/61.86 n_some(diffprop(X8,X4))))))))))). 374.24/61.86 tff('140', plain, 374.24/61.86 ![X55, X56, X57]: 374.24/61.86 (~ is_of(X55, 374.24/61.86 (^[Y0 : $i]: in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | ~ n_some(diffprop(X55, X56)) 374.24/61.86 | n_some(diffprop(X57, X56)) 374.24/61.86 | ~ imp(imp(n_some(diffprop(X57, X55)), $false), 374.24/61.86 e_is(d_Sep(omega, (^[Y0 : $i]: (Y0 != emptyset))), X55, X57)) 374.24/61.86 | ~ is_of(X57, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | ~ is_of(X56, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset))))))), 374.24/61.86 inference('cnf', [status(esa)], [zf_stmt_14])). 374.24/61.86 tff('141', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('142', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('143', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('144', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('145', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('146', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('147', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('148', plain, 374.24/61.86 ![X55, X56, X57]: 374.24/61.86 (~ is_of(X55, '#l_lift10344') 374.24/61.86 | ~ n_some(diffprop(X55, X56)) 374.24/61.86 | n_some(diffprop(X57, X56)) 374.24/61.86 | ~ imp(imp(n_some(diffprop(X57, X55)), $false), 374.24/61.86 e_is(d_Sep(omega, '#l_lift10343'), X55, X57)) 374.24/61.86 | ~ is_of(X57, '#l_lift10344') 374.24/61.86 | ~ is_of(X56, '#l_lift10344')), 374.24/61.86 inference('lambda_lifting', [status(thm)], 374.24/61.86 ['140', '141', '142', '143', '144', '145', '146', '147'])). 374.24/61.86 tff('149', plain, 374.24/61.86 ( e_is(d_Sep(omega, (^[Y0 : $i]: (Y0 != emptyset))), sk_X8, sk_X10) 374.24/61.86 | imp(imp(n_some(diffprop(sk_X10, sk_X8)), $false), $false)), 374.24/61.86 inference('bool_simp', [status(thm)], ['127'])). 374.24/61.86 tff('150', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('151', plain, 374.24/61.86 ( e_is(d_Sep(omega, '#l_lift10343'), sk_X8, sk_X10) 374.24/61.86 | imp(imp(n_some(diffprop(sk_X10, sk_X8)), $false), $false)), 374.24/61.86 inference('lambda_lifting', [status(thm)], ['149', '150'])). 374.24/61.86 tff('152', plain, 374.24/61.86 ( imp(n_some(diffprop(sk_X10, sk_X8)), $false) 374.24/61.86 | imp($false, 374.24/61.86 e_is(d_Sep(omega, (^[Y0 : $i]: (Y0 != emptyset))), sk_X8, sk_X10))), 374.24/61.86 inference('bool_simp', [status(thm)], ['127'])). 374.24/61.86 tff('153', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('154', plain, 374.24/61.86 ( imp(n_some(diffprop(sk_X10, sk_X8)), $false) 374.24/61.86 | imp($false, e_is(d_Sep(omega, '#l_lift10343'), sk_X8, sk_X10))), 374.24/61.86 inference('lambda_lifting', [status(thm)], ['152', '153'])). 374.24/61.86 tff(satz10e, axiom, 374.24/61.86 (all_of(^[X0]: in(X0,nat), 374.24/61.86 ^[X0]: 374.24/61.86 all_of(^[X1]: in(X1,nat), 374.24/61.86 ^[X1]: (d_not(d_29_ii(X0,X1)) => lessis(X0,X1)))))). 374.24/61.86 tff(zf_stmt_15, axiom, 374.24/61.86 (![X4]: 374.24/61.86 (is_of(X4,^[V_1]: in(V_1,d_Sep(omega,^[V_2]: (V_2 != emptyset)))) => 374.24/61.86 (![X6]: 374.24/61.86 (is_of(X6,^[V_3]: in(V_3,d_Sep(omega,^[V_4]: (V_4 != emptyset)))) => 374.24/61.86 (imp(d_29_ii(X4,X6),$false) => 374.24/61.86 imp(imp(n_some(diffprop(X6,X4)),$false), 374.24/61.86 e_is(d_Sep(omega,^[V_5]: (V_5 != emptyset)),X4,X6)))))))). 374.24/61.86 tff('155', plain, 374.24/61.86 ![X38, X39]: 374.24/61.86 (~ is_of(X38, 374.24/61.86 (^[Y0 : $i]: in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | imp(imp(n_some(diffprop(X38, X39)), $false), 374.24/61.86 e_is(d_Sep(omega, (^[Y0 : $i]: (Y0 != emptyset))), X39, X38)) 374.24/61.86 | ~ imp(d_29_ii(X39, X38), $false) 374.24/61.86 | ~ is_of(X39, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset))))))), 374.24/61.86 inference('cnf', [status(esa)], [zf_stmt_15])). 374.24/61.86 tff('156', plain, 374.24/61.86 ![X38, X39]: 374.24/61.86 ( d_29_ii(X39, X38) 374.24/61.86 | ~ is_of(X38, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | imp(imp(n_some(diffprop(X38, X39)), $false), 374.24/61.86 e_is(d_Sep(omega, (^[Y0 : $i]: (Y0 != emptyset))), X39, X38)) 374.24/61.86 | ~ imp($false, $false) 374.24/61.86 | ~ is_of(X39, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset))))))), 374.24/61.86 inference('bool_simp', [status(thm)], ['155'])). 374.24/61.86 tff('157', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('158', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('159', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('160', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('161', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('162', plain, 374.24/61.86 ![X38, X39]: 374.24/61.86 ( d_29_ii(X39, X38) 374.24/61.86 | ~ is_of(X38, '#l_lift10344') 374.24/61.86 | imp(imp(n_some(diffprop(X38, X39)), $false), 374.24/61.86 e_is(d_Sep(omega, '#l_lift10343'), X39, X38)) 374.24/61.86 | ~ imp($false, $false) 374.24/61.86 | ~ is_of(X39, '#l_lift10344')), 374.24/61.86 inference('lambda_lifting', [status(thm)], 374.24/61.86 ['156', '157', '158', '159', '160', '161'])). 374.24/61.86 tff('163', plain, 374.24/61.86 ![X46, X47]: 374.24/61.86 ( imp(d_29_ii(X47, X46), $false) 374.24/61.86 | ~ is_of(X46, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | imp(n_some(diffprop(X46, X47)), $false) 374.24/61.86 | ~ imp($false, 374.24/61.86 e_is(d_Sep(omega, (^[Y0 : $i]: (Y0 != emptyset))), X47, X46)) 374.24/61.86 | ~ is_of(X47, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset))))))), 374.24/61.86 inference('bool_simp', [status(thm)], ['92'])). 374.24/61.86 tff('164', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('165', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('166', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('167', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('168', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('169', plain, 374.24/61.86 ![X46, X47]: 374.24/61.86 ( imp(d_29_ii(X47, X46), $false) 374.24/61.86 | ~ is_of(X46, '#l_lift10344') 374.24/61.86 | imp(n_some(diffprop(X46, X47)), $false) 374.24/61.86 | ~ imp($false, e_is(d_Sep(omega, '#l_lift10343'), X47, X46)) 374.24/61.86 | ~ is_of(X47, '#l_lift10344')), 374.24/61.86 inference('lambda_lifting', [status(thm)], 374.24/61.86 ['163', '164', '165', '166', '167', '168'])). 374.24/61.86 tff(satz10h, axiom, 374.24/61.86 (all_of(^[X0]: in(X0,nat), 374.24/61.86 ^[X0]: 374.24/61.86 all_of(^[X1]: in(X1,nat), 374.24/61.86 ^[X1]: (iii(X0,X1) => d_not(moreis(X0,X1))))))). 374.24/61.86 tff(zf_stmt_16, axiom, 374.24/61.86 (![X4]: 374.24/61.86 (is_of(X4,^[V_1]: in(V_1,d_Sep(omega,^[V_2]: (V_2 != emptyset)))) => 374.24/61.86 (![X6]: 374.24/61.86 (is_of(X6,^[V_3]: in(V_3,d_Sep(omega,^[V_4]: (V_4 != emptyset)))) => 374.24/61.86 (n_some(diffprop(X6,X4)) => 374.24/61.86 imp(imp(imp(d_29_ii(X4,X6),$false), 374.24/61.86 e_is(d_Sep(omega,^[V_5]: (V_5 != emptyset)),X4,X6)), 374.24/61.86 $false))))))). 374.24/61.86 tff('170', plain, 374.24/61.86 ![X103, X104]: 374.24/61.86 (~ is_of(X103, 374.24/61.86 (^[Y0 : $i]: in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset)))))) 374.24/61.86 | imp( 374.24/61.86 imp(imp(d_29_ii(X104, X103), $false), 374.24/61.86 e_is(d_Sep(omega, (^[Y0 : $i]: (Y0 != emptyset))), X104, X103)), 374.24/61.86 $false) 374.24/61.86 | ~ n_some(diffprop(X103, X104)) 374.24/61.86 | ~ is_of(X104, 374.24/61.86 (^[Y0 : $i]: 374.24/61.86 in(Y0, d_Sep(omega, (^[Y1 : $i]: (Y1 != emptyset))))))), 374.24/61.86 inference('cnf', [status(esa)], [zf_stmt_16])). 374.24/61.86 tff('171', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('172', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('173', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('174', plain, 374.24/61.86 ![X21]: '#l_lift10344'(X21) = in(X21, d_Sep(omega, '#l_lift10343')), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('175', plain, ![X20]: '#l_lift10343'(X20) = (X20 != emptyset), 374.24/61.86 define([status(thm)])). 374.24/61.86 tff('176', plain, 374.24/61.86 ![X103, X104]: 374.24/61.86 (~ is_of(X103, '#l_lift10344') 374.24/61.86 | imp( 374.24/61.86 imp(imp(d_29_ii(X104, X103), $false), 374.24/61.86 e_is(d_Sep(omega, '#l_lift10343'), X104, X103)), $false) 374.24/61.86 | ~ n_some(diffprop(X103, X104)) 374.24/61.86 | ~ is_of(X104, '#l_lift10344')), 374.24/61.86 inference('lambda_lifting', [status(thm)], 374.24/61.86 ['170', '171', '172', '173', '174', '175'])). 374.24/61.86 tff('177', plain, $false, 374.24/61.86 inference('eprover', [status(thm)], 374.24/61.86 ['15', '21', '30', '40', '51', '59', '73', '79', '83', '91', 374.24/61.86 '99', '106', '116', '117', '121', '125', '126', '130', '139', 374.24/61.86 '148', '151', '154', '162', '169', '176'])). 374.24/61.86 374.24/61.86 % SZS output end Refutation 374.24/61.86 374.24/61.86 374.24/61.86 % Terminating... 374.38/61.94 % Runner terminated. 374.38/61.95 % Zipperpin 1.5 exiting 374.38/61.96 EOF