0.00/0.04 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.03/0.04 % Command : tptp2X_and_run_prover9 %d %s 0.03/0.24 % Computer : n129.star.cs.uiowa.edu 0.03/0.24 % Model : x86_64 x86_64 0.03/0.24 % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz 0.03/0.24 % Memory : 32218.625MB 0.03/0.24 % OS : Linux 3.10.0-693.2.2.el7.x86_64 0.03/0.24 % CPULimit : 300 0.03/0.24 % DateTime : Sat Jul 14 05:40:39 CDT 2018 0.03/0.24 % CPUTime : 6.94/6.99 ============================== Prover9 =============================== 6.94/6.99 Prover9 (32) version 2009-11A, November 2009. 6.94/6.99 Process 45895 was started by sandbox2 on n129.star.cs.uiowa.edu, 6.94/6.99 Sat Jul 14 05:40:45 2018 6.94/6.99 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 300 -f /tmp/Prover9_45863_n129.star.cs.uiowa.edu". 6.94/6.99 ============================== end of head =========================== 6.94/6.99 6.94/6.99 ============================== INPUT ================================= 6.94/6.99 6.94/6.99 % Reading from file /tmp/Prover9_45863_n129.star.cs.uiowa.edu 6.94/6.99 6.94/6.99 set(prolog_style_variables). 6.94/6.99 set(auto2). 6.94/6.99 % set(auto2) -> set(auto). 6.94/6.99 % set(auto) -> set(auto_inference). 6.94/6.99 % set(auto) -> set(auto_setup). 6.94/6.99 % set(auto_setup) -> set(predicate_elim). 6.94/6.99 % set(auto_setup) -> assign(eq_defs, unfold). 6.94/6.99 % set(auto) -> set(auto_limits). 6.94/6.99 % set(auto_limits) -> assign(max_weight, "100.000"). 6.94/6.99 % set(auto_limits) -> assign(sos_limit, 20000). 6.94/6.99 % set(auto) -> set(auto_denials). 6.94/6.99 % set(auto) -> set(auto_process). 6.94/6.99 % set(auto2) -> assign(new_constants, 1). 6.94/6.99 % set(auto2) -> assign(fold_denial_max, 3). 6.94/6.99 % set(auto2) -> assign(max_weight, "200.000"). 6.94/6.99 % set(auto2) -> assign(max_hours, 1). 6.94/6.99 % assign(max_hours, 1) -> assign(max_seconds, 3600). 6.94/6.99 % set(auto2) -> assign(max_seconds, 0). 6.94/6.99 % set(auto2) -> assign(max_minutes, 5). 6.94/6.99 % assign(max_minutes, 5) -> assign(max_seconds, 300). 6.94/6.99 % set(auto2) -> set(sort_initial_sos). 6.94/6.99 % set(auto2) -> assign(sos_limit, -1). 6.94/6.99 % set(auto2) -> assign(lrs_ticks, 3000). 6.94/6.99 % set(auto2) -> assign(max_megs, 400). 6.94/6.99 % set(auto2) -> assign(stats, some). 6.94/6.99 % set(auto2) -> clear(echo_input). 6.94/6.99 % set(auto2) -> set(quiet). 6.94/6.99 % set(auto2) -> clear(print_initial_clauses). 6.94/6.99 % set(auto2) -> clear(print_given). 6.94/6.99 assign(lrs_ticks,-1). 6.94/6.99 assign(sos_limit,10000). 6.94/6.99 assign(order,kbo). 6.94/6.99 set(lex_order_vars). 6.94/6.99 clear(print_given). 6.94/6.99 6.94/6.99 % formulas(sos). % not echoed (5482 formulas) 6.94/6.99 6.94/6.99 ============================== end of input ========================== 6.94/6.99 6.94/6.99 % From the command line: assign(max_seconds, 300). 6.94/6.99 6.94/6.99 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 6.94/6.99 6.94/6.99 % Formulas that are not ordinary clauses: 6.94/6.99 1 (all X hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_1457_four__x__squared) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 2 (all X all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),hAPP_int_int(multInv(P_3),X))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),hAPP_int_int(multInv(P_3),X))),hAPP_int_int(multInv(P_3),hAPP_int_int(multInv(P_3),X)))),P_3)))))) # label(fact_2234_Int2_Oaux____1) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 3 (all C_58 all D_11 all A_136 all B_105 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_136),B_105)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_58),D_11)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_136)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_58)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_136),C_58)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_105),D_11)))))))) # label(fact_1368_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 4 (all A_109 all B_87 all C_47 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_109),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_87),C_47)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_109),B_87)),C_47)) # label(fact_1948_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 5 (all C_1 all A_3 all B_2 (B_2 = A_3 -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_1),B_2)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),C_1)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_1),A_3)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),C_1))))) # label(fact_1715_dvd_Oord__eq__less__trans) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 6 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),min)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K_1)),min)))) # label(fact_785_rel__simps_I28_J) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 7 (all A_235 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_235),A_235)),A_235) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_235),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) # label(fact_279_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 8 (all X all N hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,X)),N) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N))) # label(fact_3084_Nat__Transfer_Otransfer__int__nat__functions_I4_J) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 9 (all Y_2 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_2)) -> (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_2) = zero_zero_real <-> zero_zero_real = Y_2 & zero_zero_real = X_1)))) # label(fact_2162_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 10 (all N_1 all V hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(suc,N_1)),hAPP_int_nat(number_number_of_nat,V)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),zero_zero_nat),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,N_1),hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))))) # label(fact_5119_min__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 11 (all X all N all Y (hAPP_nat_nat(div_mod_nat(X),N) = hAPP_nat_nat(div_mod_nat(Y),N) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y),X)) -> (exists Q_4 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),Q_4)) = X)))) # label(fact_2774_nat__mod__eq__lemma) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 12 (all C all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),C)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B),A)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,C),A))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B),C)))))) # label(fact_2485_div__dvd__div) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 13 (all A_235 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_235),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_235),A_235)),A_235)) # label(fact_277_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 14 (all V_2 all W hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_2),W))) # label(fact_130_plus__numeral__code_I9_J) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 15 (all P all Q all R hAPP_f1591894897l_real(P,hAPP_r1250527377l_real(Q,R)) = hAPP_r579619957l_real(hAPP_f823702159l_real(hAPP_f1040184293l_real(cOMBB_557258983l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__036) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 16 (all W_1 all Z_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_2),one_one_int))))) # label(fact_1579_zle__diff1__eq) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 17 (all X of_real_complex(X) = hAPP_complex_complex(cnj,of_real_complex(X))) # label(fact_4080_complex__cnj__complex__of__real) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 18 (all X_1 all Y_2 (is_int(Y_2) & is_int(X_1) -> (hAPP_int_real(number267125858f_real,Y_2) = hAPP_int_real(number267125858f_real,X_1) <-> Y_2 = X_1))) # label(fact_167_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 19 (all L all U hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(L),U)))) # label(fact_5066_finite__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 20 (all N hAPP_complex_real(im,hAPP_nat_complex(semiri2020571505omplex,N)) = zero_zero_real) # label(fact_4102_complex__Im__of__nat) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 21 (all P all X (hBOOL(hAPP_f54304608l_bool(fEx_nat,P)) | -hBOOL(hAPP_nat_bool(P,X)))) # label(help_fEx_1_1_fEx_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 22 (all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),zero_zero_int)))) # label(fact_2189_double__add__less__zero__iff__single__add__less__zero) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 23 (all M all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),M))))) # label(fact_4193_prime__dvd__square) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 24 (all X all M hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,M))),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,M)),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_3695_sin__expansion__lemma) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 25 (all A_110 all B_88 all C_48 (hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_110),C_48) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_110),B_88) -> C_48 = B_88)) # label(fact_1922_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 26 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(suc,N))) # label(fact_3074_diff__Suc__eq__diff__pred) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 27 (all B_97 all A_119 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_119)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_97)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_119),B_97)))))) # label(fact_1857_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 28 (all N hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N)))) # label(fact_2985_zero__zle__int) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 29 (all N_55 all A_152 all B_114 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_152),B_114)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_152)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_152),N_55)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_114),N_55)))))) # label(fact_1237_power__mono) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 30 (all Y all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (one_one_real != A_3 -> Y = hAPP_real_real(log(A_3),hAPP_real_real(hAPP_r1250527377l_real(powr,A_3),Y))))) # label(fact_4462_log__powr__cancel) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 31 (all U_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),U_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),U_1)))) # label(fact_3364_lemma__real__divide__sqrt__less) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 32 (all X all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_nat_real(real_nat,A_3))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X)),A_3)))) # label(fact_3676_natceiling__le) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 33 (all X_9 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_9)) -> hAPP_nat_rat(semiri151668891at_rat,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_9))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(semiri151668891at_rat,hAPP_int_nat(nat,X_9))),hAPP_nat_rat(semiri151668891at_rat,hAPP_int_nat(nat,X_9))))) # label(fact_2814_of__nat__double) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 34 (all A_182 (is_int(A_182) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_182),one_one_nat) = A_182)) # label(fact_1024_power__one__right) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 35 (all Z1 all Z2 all W hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z1),W)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z2),W)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z1),Z2)),W)) # label(fact_1785_real__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 36 (all K_2 all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M)))) # label(fact_4317_dvd__gcd__D1__nat) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 37 (all B_1_1 all B_2_1 is_bool(hAPP_P1555980039t_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 38 (all B_1_1 is_int(hilbert_Eps_int(B_1_1))) # label(gsy_c_Hilbert__Choice_OEps_000tc__Int__Oint) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 39 (all A_12 all N_18 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_12),hAPP_nat_nat(suc,N_18)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_12),N_18)),A_12)) # label(fact_3005_power__Suc2) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 40 (all M all K_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2)),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)))) # label(fact_1648_add__leD1) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 41 (all P all Q (-hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)) | hBOOL(Q))) # label(help_fconj_3_1_U) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 42 (all N_1 all F (hBOOL(summable_real(F)) -> ((all M_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(F,M_1))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_f352196356l_real(suminf_real,F)))))) # label(fact_4930_series__pos__le) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 43 (all M_17 all N_30 ((zero_zero_nat != N_30 -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M_17),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_17),N_30)) & (zero_zero_nat = N_30 -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_17),N_30) = one_one_int))) # label(fact_1829_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 44 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),hAPP_int_int(bit0,K_1))))) # label(fact_114_rel__simps_I21_J) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 45 (all B_1_1 is_int(the_int(B_1_1))) # label(gsy_c_HOL_OThe_000tc__Int__Oint) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 46 (all B_120 all A_159 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_120),A_159))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_159)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_120))))) # label(fact_1166_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 47 (all P_1 P_1 = collec1347809874nt_int(P_1)) # label(fact_148_Collect__def) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 48 (all A_146 one_on1684967323de_int = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_146),zero_zero_nat)) # label(fact_1301_power__0) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 49 (all B_2 all Q_8 all R_5 all Q_2 all R_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_8)),R_5)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_3))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_5),B_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_8),Q_2))))))) # label(fact_734_unique__quotient__lemma) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 50 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> natceiling(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),one_one_real)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natceiling(X)),one_one_nat))) # label(fact_3409_natceiling__add__one) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 51 (all A_3 all B_2 all C_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B_2),C_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),C_1)) # label(fact_4311_gcd__assoc__nat) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 52 (all X_41 all Y_29 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_41),Y_29)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_41),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_29),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_41)),Y_29))) # label(fact_366_power2__sum) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 53 (all C_85 all A_193 all B_149 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_193),B_149)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_193),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_149),C_85))))) # label(fact_925_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 54 (all M hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),one_one_int) = one_one_int) # label(fact_4974_gcd__1__int) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 55 (all A_3 all R_3 all B_2 all M all C_1 all D_3 all N hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),N))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_3),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D_3),N)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_3),B_2))),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_3),D_3))),N))) # label(fact_1558_xzgcda__linear__aux1) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 56 (all N all X (hAPP_nat_real(real_nat,natfloor(X)) = X -> natfloor(hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,natfloor(X)),N))) # label(fact_3678_natfloor__power) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 57 (all Y_2 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> hBOOL(hAPP_f448129468l_bool(nat_nat_set,hAPP_i1948725293t_bool(ord_at875362053st_int(X_1),Y_2))))) # label(fact_5068_SetInterval_Otransfer__nat__int__set__function__closures) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 58 (all A_230 all C_96 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_230),C_96) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_96),A_230)) # label(fact_519_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 59 (all M_6 all N_20 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_nat_complex(semiri2020571505omplex,M_6)),N_20) = hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20))) # label(fact_2980_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 60 (all M all K_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,J_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),M)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2))),M))) # label(fact_3138_diff__Suc__diff__eq2) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 61 (all M_28 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_28),M_28) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat)),M_28)) # label(fact_692_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 62 (all P_1 ((exists X_2 hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X_2)))) <-> (exists X_2 (hBOOL(hAPP_int_bool(P_1,X_2)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) & is_int(X_2))))) # label(fact_2987_transfer__int__nat__quantifiers_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 63 (all X_1 all Y_2 (Y_2 = X_1 <-> hAPP_real_real(exp_real,Y_2) = hAPP_real_real(exp_real,X_1))) # label(fact_3827_exp__inj__iff) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 64 (all M all N (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) = M)) # label(fact_1764_add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 65 (all X all Y hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(nat_gcd(hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,X))),hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,Y)))) = hAPP_int_int(hAPP_int_fun_int_int(int_gcd,X),Y)) # label(fact_4623_Nitpick_Oint__gcd__def) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 66 (all M_12 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,M_12)))) # label(fact_2909_zero__le__imp__of__nat) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 67 (all Lx_3 all Ly_1 all Rx_3 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_3),Rx_3)),Ly_1) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_3),Ly_1)),Rx_3)) # label(fact_484_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 68 (all P all Q all R hAPP_nat_rat(hAPP_f2062659861at_rat(hAPP_f1082758869at_rat(cOMBB_rat_rat_nat,P),Q),R) = hAPP_rat_rat(P,hAPP_nat_rat(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Rat__Orat_000tc__Rat__Orat_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 69 (all X_1 all Y_2 all B (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B),X_1)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B),Y_2)))))) # label(fact_1419_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 70 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(div_mod_int(A_3),B_2)),zero_zero_int)))) # label(fact_2612_neg__mod__sign) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 71 (all W all Z1 all Z2 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z2)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z1),Z2))) # label(fact_1523_zdiff__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 72 (all K_2 all L_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),L_2)) -> (exists N_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),N_2) = L_2))) # label(fact_3953_le__Suc__ex) # label(axiom) # label(non_clause). [assumption]. 6.94/6.99 73 (all D_3 all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),A_3)))) # label(fact_4010_coprime__rmul2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 74 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),pi)) -> hAPP_real_real(arccos,hAPP_real_real(cos,X)) = X))) # label(fact_3603_arccos__cos) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 75 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (exists M_1 hAPP_nat_nat(suc,M_1) = N))) # label(fact_3265_gr0__implies__Suc) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 76 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),posDivAlg(A_3,B_2)))))) # label(fact_2799_posDivAlg__correct) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 77 (all X_1 all Y_2 all Z_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_2),X_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_2),Y_2)))))) # label(fact_1792_real__mult__le__cancel__iff2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 78 (all A_154 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_154),A_154)),zero_zero_int))) # label(fact_1200_not__square__less__zero) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 79 (all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(inv(P_3),A_3))),one_one_int),P_3)))))) # label(fact_2140_inv__is__inv) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 80 (all N_19 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),hAPP_nat_nat(suc,N_19)) = zero_z891286103de_int) # label(fact_2992_power__0__Suc) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 81 (all A_195 all B_151 hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_195),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_195),B_151)))) # label(fact_906_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 82 (all V (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V),pls)) <-> hAPP_int_nat(number_number_of_nat,V) = zero_zero_nat)) # label(fact_72_eq__number__of__0) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 83 (all B_1_1 is_int(archim856651990g_real(B_1_1))) # label(gsy_c_Archimedean__Field_Oceiling_000tc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 84 (all P all Q all R hAPP_f1344929775l_bool(P,hAPP_r632163725t_bool(Q,R)) = hAPP_r1150855451l_bool(hAPP_f381769257l_bool(hAPP_f512004221l_bool(cOMBB_188663909l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__HOL__034) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 85 (all A all C all B (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A),C)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B),C))) <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A),B)))) # label(fact_2070_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 86 (all V_18 all W_14 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_18)),hAPP_int_real(number267125858f_real,W_14)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_18),W_14))) # label(fact_88_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 87 (all C_44 all D_8 all A_102 all B_84 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_102),B_84)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_44),D_8)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_102),C_44)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_84),D_8)))))) # label(fact_2009_add__mono) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 88 (all M all N hAPP_int_int(div_mod_int(M),N) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,M),N)),N))) # label(fact_2362_zmod__zdiv__equality_H) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 89 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_nat_real(real_nat,natceiling(X))))) # label(fact_3656_real__natceiling__ge) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 90 (all N_19 zero_zero_rat = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),hAPP_nat_nat(suc,N_19))) # label(fact_2991_power__0__Suc) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 91 (all A_1 image_int_nat(nat,image_nat_int(semiri1621563631at_int,A_1)) = A_1) # label(fact_4604_transfer__int__nat__set__return__embed) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 92 (all A_3 all B_2 all C_1 hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B_2),C_1)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)),C_1)) # label(fact_4979_gcd__assoc__int) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 93 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),hAPP_int_int(bit0,K_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),K_1)))) # label(fact_350_rel__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 94 (all V (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V),pls)) <-> hAPP_int_nat(number_number_of_nat,V) = zero_zero_nat)) # label(fact_73_eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 95 (all C_69 all A_163 all B_124 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_163),B_124)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_69)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_69),A_163)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_69),B_124)))))) # label(fact_1138_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 96 (all N_26 hAPP_int_real(number267125858f_real,hAPP_nat_int(semiri1621563631at_int,N_26)) = hAPP_nat_real(semiri132038758t_real,N_26)) # label(fact_2824_number__of__int) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 97 (all N (hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3788_lemma__even__div2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 98 (all F all B all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B),X_1)) -> ((all X_2 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),hAPP_r1134773055l_bool(ord_gr788844697n_real(B),X_1))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(F,X_2))))) -> (hBOOL(hAPP_real_bool(isCont_real_real(F),X_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(F,X_1))))))) # label(fact_5174_LIM__less__bound) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 99 (all X all Y (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y)) | Y = X)) # label(help_fequal_1_1_fequal_000tc__Nat__Onat_T) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 100 (all L_2 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,pls),hAPP_int_int(bit1,L_2)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),L_2))) # label(fact_1580_diff__bin__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 101 (all A_225 all B_162 all C_91 all D_18 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_225),C_91)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_162),D_18)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_225),B_162)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_91),D_18))) # label(fact_555_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 102 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)))) # label(fact_1775_le__add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 103 (all N all D_3 (is_int(D_3) -> (zero_zero_int != D_3 -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_3),N)) -> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,N)),hAPP_int_real(real_int,D_3)) = hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N),D_3)))))) # label(fact_4450_real__of__int__div) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 104 (all N_1 hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),N_1))) = N_1) # label(fact_4764_card__Collect__less__nat) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 105 (all A_18 all C_5 all B_14 (zero_z891286103de_int != B_14 -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_5),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_18),B_14)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_18),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_5),B_14))),B_14))) # label(fact_2623_div__mult__self1) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 106 (all A all E_2 all C all B all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B),A)),E_2)),D))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),E_2)),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),E_2)),D))))) # label(fact_1566_less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 107 (all Y (Y != zero_z126310315umeral -> -(all Code_numeral Y != code_S1047413653umeral(Code_numeral)))) # label(fact_4091_code__numeral_Oexhaust) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 108 (all X_8 all Q_7 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_8),Q_7)),X_8) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_8),hAPP_nat_nat(suc,Q_7))) # label(fact_3020_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 109 (all P all Q all R hAPP_i1584592887nt_int(P,hAPP_int_int(Q,R)) = hAPP_i1584592887nt_int(hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__prod_Itc__I) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 110 (all P all Q all R hAPP_P921862901l_bool(hAPP_f365317245l_bool(hAPP_f1961001761l_bool(cOMBB_1550063917nt_int,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_P603027463t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_016) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 111 (all Lx all Rx all Ry hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx),Ry)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx),Ry))) # label(fact_507_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 112 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)))))) # label(fact_721_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 113 (all M_6 all N_20 hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(semiri132038758t_real,M_6)),N_20)) # label(fact_2981_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 114 (all Nat_5 all Nat_4 (Nat_5 = Nat_4 <-> hAPP_nat_nat(suc,Nat_5) = hAPP_nat_nat(suc,Nat_4))) # label(fact_2835_nat_Oinject) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 115 (all K_1 all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N_1))))) # label(fact_1626_nat__add__left__cancel__less) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 116 (all B_148 all A_192 all C_84 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_192),C_84)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_192),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_148),C_84))))) # label(fact_930_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 117 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) <-> hAPP_real_real(exp_real,hAPP_real_real(ln,X_1)) = X_1)) # label(fact_3847_exp__ln__iff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 118 (all M hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,M))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_3202_div2__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 119 (all P all Q all R hAPP_nat_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,P),Q),R) = hAPP_complex_complex(P,hAPP_nat_complex(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Complex__Ocomplex_000tc__Complex__Ocomplex_000tc__Na) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 120 (all Lx all Rx all Ry hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx),Ry)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx),Ry))) # label(fact_511_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 121 (all I all K_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),K_2))) # label(fact_1770_diff__add__assoc) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 122 (all P (is_bool(P) -> fTrue = P | fFalse = P)) # label(help_If_3_1_If_000tc__Nat__Onat_T) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 123 (all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A),B)) <-> (all D_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),A)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),B)) <-> one_one_nat = D_2)))) # label(fact_4028_coprime) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 124 (all Y all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X)),one_one_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,Y)),one_one_real)) -> hAPP_real_real(arctan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(arctan,X)),hAPP_real_real(arctan,Y))))) # label(fact_3408_arctan__add) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 125 (all X all Y hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)))) # label(fact_5010_gcd__ge__0__int) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 126 (all R_3 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,R_3),one_one_real)),hAPP_int_real(real_int,archim1246769320r_real(R_3))))) # label(fact_4456_real__of__int__floor__gt__diff__one) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 127 (all Y_2 all X_2 hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,minus_minus_real),hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,sin),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,plus_plus_real),Y_2)))),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),sin)),hAPP_real_real(cos,Y_2)))),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),cos)),hAPP_real_real(sin,Y_2)))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,minus_minus_real),hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,cos),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,plus_plus_real),Y_2)))),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,minus_minus_real),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),cos)),hAPP_real_real(cos,Y_2)))),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),sin)),hAPP_real_real(sin,Y_2)))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_2),zero_zero_real))) # label(fact_4801_lemma__DERIV__sin__cos__add) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 128 (all W_7 hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,W_7)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),hAPP_int_rat(number_number_of_rat,W_7))),hAPP_int_rat(number_number_of_rat,W_7))) # label(fact_374_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 129 (all A_202 zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,zero_zero_real),A_202)) # label(fact_848_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 130 (all A_29 all B_24 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_29),B_24)) -> zero_z126310315umeral = hAPP_C498520661umeral(div_mo1740067990umeral(B_24),A_29))) # label(fact_2512_dvd__imp__mod__0) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 131 (all N_1 all Ma (is_int(Ma) & is_int(N_1) -> (N_1 = Ma <-> hAPP_i1732201573de_int(quickcheck_of_int,Ma) = hAPP_i1732201573de_int(quickcheck_of_int,N_1)))) # label(fact_4554_Quickcheck__Narrowing_Oof__int__inject) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 132 (all N_16 all M_5 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_16),M_5)) -> hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_5),N_16)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_nat_rat(semiri151668891at_rat,M_5)),hAPP_nat_rat(semiri151668891at_rat,N_16)))) # label(fact_3098_of__nat__diff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 133 (all A_139 all C_61 all B_108 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_139),C_61)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_108),C_61))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_61)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_139),B_108))))) # label(fact_1350_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 134 (all X all M hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),hAPP_int_int(standardRes(M),X)),M))) # label(fact_2490_StandardRes__prop1) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 135 (all A_125 all N_36 all N_35 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_36),N_35)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_125)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_125),one_on1684967323de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_125),N_35)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_125),N_36))))))) # label(fact_1477_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 136 (all B_121 all A_160 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_160),zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_121)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_160),B_121)),zero_zero_nat))))) # label(fact_1153_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 137 (all A_16 all B_12 (zero_z891286103de_int != B_12 -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_16),B_12)),one_on1684967323de_int) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_12),A_16)),B_12))) # label(fact_2631_div__add__self1) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 138 (all B_1_1 is_bool(monoseq_real(B_1_1))) # label(gsy_c_SEQ_Omonoseq_000tc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 139 (all Y_21 all X_27 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_27)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_21)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_21),one_one_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_21),X_27)),X_27)))))) # label(fact_1386_mult__left__le__one__le) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 140 (all M all N hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,M),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N))) # label(fact_5124_min__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 141 (all P_1 all Ma all N_1 (hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) & N_1 = zero_zero_nat | (exists Q_4 (hBOOL(hAPP_nat_bool(P_1,Q_4)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),hAPP_nat_nat(suc,Q_4)))) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),Q_4)),Ma)))) <-> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,Ma),N_1))))) # label(fact_3212_split__div_H) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 142 (all Z_1 all X all Y (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),X))))) # label(fact_1727_dvd_Oless__trans) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 143 (all X_1 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)) -> zero_zero_int = hAPP_int_int(div_mod_int(X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3235_even__div__2__prop1) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 144 (all X_36 all Y_28 all Z_9 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_36),Y_28)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_36),Z_9)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_36),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,Y_28),Z_9))) # label(fact_616_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 145 (all Z_1 hAPP_complex_real(norm_norm_complex,Z_1) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_4138_cmod__def) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 146 (all A_214 all N_67 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_214),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_67))))) # label(fact_685_zero__le__even__power_H) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 147 (all A_215 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_215),one_one_rat) = A_215) # label(fact_660_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 148 (all X_1 comple124823625p_real(hAPP_r1134773055l_bool(ord_lessThan_real,X_1)) = X_1) # label(fact_4957_Sup__lessThan) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 149 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(cos,X)))))) # label(fact_3590_cos__ge__zero) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 150 (all N all M hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,quickcheck_int_of(N)),quickcheck_int_of(M))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,N),M)) # label(fact_4560_times__code__int__def) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 151 (all K_1 (is_int(K_1) -> (pls = K_1 <-> pls = hAPP_int_int(bit0,K_1)))) # label(fact_253_rel__simps_I44_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 152 (all B_2 all A_3 all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N)) -> (one_one_int = hAPP_int_int(legacy_zgcd(A_3),N) -> (exists X_2 (is_int(X_2) & (all Y_1 (is_int(Y_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),N)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Y_1)),B_2),N)) -> Y_1 = X_2))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),X_2)),B_2),N)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),N)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))))))) # label(fact_4533_zcong__lineq__unique) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 153 (all X all Y hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y)) = hAPP_real_real(dist_dist_real(X),Y)) # label(fact_4545_dist__real__def) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 154 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),N)) = zero_zero_real)) # label(fact_3752_lemma__STAR__cos) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 155 (all N_42 all A_130 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_130)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_130),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_130),N_42)))))) # label(fact_1427_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 156 (all A_90 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),A_90) = A_90) # label(fact_2114_mult__1) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 157 (all X all Y all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),Y)) = hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),Y)))) # label(fact_3912_real__root__divide) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 158 (all X hAPP_real_real(tan,hAPP_real_real(uminus_uminus_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(tan,X))) # label(fact_3596_tan__minus) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 159 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)),A_3)))) # label(fact_5013_gcd__le1__int) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 160 (all M_25 all A_185 all N_60 all B_142 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_185),N_60)),B_142)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_25),N_60)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_185),M_25)),B_142))))) # label(fact_991_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 161 (all C_80 all A_187 all B_143 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_187),B_143)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_187),C_80)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_187),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_143),C_80)))))) # label(fact_963_dvd__add) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 162 (all X_19 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_19),X_19)),one_one_int) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_19),one_one_int)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_19),one_one_int))) # label(fact_1822_real__squared__diff__one__factored) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 163 (all N hAPP_nat_nat(suc,N) != N) # label(fact_2837_n__not__Suc__n) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 164 (all A_182 A_182 = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_182),one_one_nat)) # label(fact_1026_power__one__right) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 165 (all C_34 all D_6 all A_80 all B_68 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_80),B_68)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_34),D_6)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_80),C_34)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_68),D_6)))))) # label(fact_2200_add__le__less__mono) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 166 (all A all C all D all B (is_int(D) & is_int(B) -> (B != zero_zero_int -> (D != zero_zero_int -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C),D))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),D)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),D))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),D))))))))) # label(fact_4718_le__rat) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 167 (all N all A_3 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,cis(A_3)),cis(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),A_3))) = cis(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N))),A_3))) # label(fact_4154_cis__real__of__nat__Suc__mult) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 168 (all N_66 all A_210 all B_158 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_210),B_158)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_210)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_66)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_210),N_66)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_158),N_66))))))) # label(fact_744_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 169 (all N (N != zero_zero_nat -> (exists M_1 N = hAPP_nat_nat(suc,M_1)))) # label(fact_3353_not0__implies__Suc) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 170 (all A (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),zero_zero_real)))) # label(fact_2190_double__add__less__zero__iff__single__add__less__zero) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 171 (all B_2 posDivAlg(zero_zero_int,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int)) # label(fact_2730_posDivAlg__0) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 172 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K1)),hAPP_int_int(bit0,K2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)))) # label(fact_346_less__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 173 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_complex_real(norm_norm_complex,X))),hAPP_complex_real(norm_norm_complex,X)))) # label(fact_3489_complex__mod__minus__le__complex__mod) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 174 (all P all Q all R hAPP_r50993370t_real(hAPP_f1163894827t_real(hAPP_f1998755145t_real(cOMBB_421464275l_real,P),Q),R) = hAPP_f1566856189t_real(P,hAPP_r474017924t_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__fun__029) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 175 (all R1 all A_3 all R2 all B_2 rcis(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R1),R2),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),B_2)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,rcis(R1,A_3)),rcis(R2,B_2))) # label(fact_4264_rcis__mult) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 176 (all X_1 all Y_2 (X_1 = Y_2 <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2)))) # label(fact_1699_dvd_Oeq__iff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 177 (all A all B all C (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B),A)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B)))))) # label(fact_1159_mult__less__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 178 (all A_94 hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_94),A_94) = zero_zero_rat) # label(fact_2087_diff__self) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 179 (all N_49 all A_147 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A_147)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_147),N_49))))) # label(fact_1277_one__le__power) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 180 (all B_93 all A_115 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_115),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_93),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_115),B_93)),zero_zero_rat))))) # label(fact_1886_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 181 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int)) -> hAPP_P1175774780nt_int(product_fst_int_int,hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)))) # label(fact_4228_div__pos__neg) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 182 (all B_2 all A_3 all C_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),C_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),C_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),A_3)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),B_2))))) # label(fact_1020_Euler_Oaux2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 183 (all A_145 all M_20 all N_48 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_145),M_20)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_145),N_48)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_145),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_20),N_48))) # label(fact_1304_power__add) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 184 (all B_1_1 all B_2_1 all B_3_1 is_bool(tendsto_complex_real(B_1_1,B_2_1,B_3_1))) # label(gsy_c_Limits_Otendsto_000tc__Complex__Ocomplex_000tc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 185 (all X natceiling(X) = hAPP_int_nat(nat,archim856651990g_real(X))) # label(fact_4364_natceiling__def) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 186 (all A all C all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B),C))))) # label(fact_2071_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 187 (all X_1 all K_1 (X_1 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K_1) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)))) # label(fact_3224_zEvenI) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 188 (all A all B (hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) <-> hAPP_int_int(abs_abs_int,A) = hAPP_int_int(abs_abs_int,B))) # label(fact_3458_power2__eq__iff__abs__eq) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 189 (all N_1 all Ma (hAPP_nat_real(real_nat,Ma) = hAPP_nat_real(real_nat,N_1) <-> N_1 = Ma)) # label(fact_3636_real__of__nat__inject) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 190 (all B_121 all A_160 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_160),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_121)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_160),B_121)),zero_zero_rat))))) # label(fact_1157_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 191 (all N_41 ((N_41 != zero_zero_nat -> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),N_41) = zero_zero_real) & (zero_zero_nat = N_41 -> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),N_41) = one_one_real))) # label(fact_1434_power__0__left) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 192 (all X_5 all Y_5 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_5),Y_5)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_5),Y_5)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_5),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_5),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))))) # label(fact_3191_realpow__two__diff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 193 (all N_29 all K_6 all M_16 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_6),M_16)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_6),N_29)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_6),hAPP_int_int(div_mod_int(M_16),N_29)))))) # label(fact_2451_dvd__mod) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 194 (all A_145 all M_20 all N_48 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_145),M_20)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_145),N_48)) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_145),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_20),N_48))) # label(fact_1303_power__add) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 195 (all A all B all C all D (hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),D) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C),D))))) # label(fact_2093_diff__eq__diff__less__eq) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 196 (all B_69 all A_81 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_81)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_69)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_81),B_69)))))) # label(fact_2185_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 197 (all N all X (hAPP_int_real(real_int,archim1246769320r_real(X)) = X -> archim1246769320r_real(hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,archim1246769320r_real(X)),N))) # label(fact_4425_floor__power) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 198 (all X_37 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_37),one_one_nat) = X_37) # label(fact_566_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 199 (all X_1 (one_on1684967323de_int = X_1 <-> X_1 = one_on1684967323de_int)) # label(fact_1955_one__reorient) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 200 (all C_64 all A_142 all B_111 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_64),A_142)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_64),B_111))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_64)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_142),B_111))))) # label(fact_1332_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 201 (all B_95 all C_53 all A_117 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_117)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_95),C_53)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_95),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_117),C_53)))))) # label(fact_1873_add__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 202 (all B all A (is_int(A) & is_int(B) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B),A) = B <-> A = zero_zero_int))) # label(fact_611_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 203 (all Ma all K_1 all N_1 (is_int(K_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1)) -> (hAPP_int_int(legacy_zgcd(Ma),N_1) = K_1 <-> (exists S_2 exists T (is_int(S_2) & is_int(T) & xzgcd(Ma,N_1) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,K_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,S_2),T)))))))) # label(fact_4530_xzgcd__correct) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 204 (all C_89 all D_17 all A_213 all B_159 all R_8 (R_8 != zero_zero_complex -> (B_159 = A_213 & C_89 != D_17 -> hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_159),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,R_8),D_17)) != hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_213),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,R_8),C_89))))) # label(fact_689_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 205 (all K_2 all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),K_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),K_2)),X))))) # label(fact_2594_int__div__less__self) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 206 (all X (zero_zero_real != X -> hAPP_real_real(arctan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),X)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sgn_sgn_real,X)),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_real_real(arctan,X)))) # label(fact_3558_arctan__inverse) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 207 (all A all B (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B)),zero_zero_rat)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B)))) # label(fact_2212_less__iff__diff__less__0) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 208 (all N all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),N)) -> one_one_int = hAPP_int_int(legacy_zgcd(N),P_3)))) # label(fact_4521_zprime__imp__zrelprime) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 209 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arccos,Y)),pi))))) # label(fact_3608_arccos__ubound) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 210 (all P all Q all R hAPP_i1836591008nt_int(hAPP_f617310012nt_int(hAPP_f197974549nt_int(cOMBB_638531587nt_int,P),Q),R) = hAPP_f713949180nt_int(P,hAPP_i1153444985nt_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__prod_Itc__Int__Oint) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 211 (all W_1 all X_1 (is_int(X_1) -> (hAPP_int_int(number_number_of_int,W_1) = X_1 <-> hAPP_int_int(number_number_of_int,W_1) = X_1))) # label(fact_171_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 212 (all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Y_2)))) # label(fact_100_le__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 213 (all X_45 all Y_33 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_45),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_33),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),Y_33)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,X_45),Y_33))))) # label(fact_211_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 214 (all A_106 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,zero_z891286103de_int),A_106) = A_106) # label(fact_1983_add__0) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 215 (all P all Q all R hAPP_n546711566l_real(hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,P),Q),R) = hAPP_r1250527377l_real(P,hAPP_nat_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R_008) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 216 (all X_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (one_one_int = hAPP_int_int(legacy_zgcd(X_1),Ma) -> hBOOL(hAPP_int_bool(is_RRset(hAPP_i1948725293t_bool(noXRRset(Ma),X_1)),Ma))))) # label(fact_4588_noX__is__RRset) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 217 (all B_70 all A_82 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_82),zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_70),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_82),B_70)),zero_zero_nat))))) # label(fact_2179_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 218 (all P all Q all R hAPP_i1778081398nt_int(hAPP_f2004554147nt_int(hAPP_f1228585167nt_int(cOMBB_1102189010nt_int,P),Q),R) = hAPP_f1574055885nt_int(P,hAPP_int_fun_int_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc___024) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 219 (all N (-hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)))) # label(fact_3763_odd__pos) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 220 (all X hAPP_int_int(fact_fact_int,hAPP_nat_int(semiri1621563631at_int,X)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(fact_fact_nat,X))) # label(fact_4280_transfer__int__nat__factorial) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 221 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),pi)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sin,X)))))) # label(fact_3657_sin__ge__zero) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 222 (all N_9 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,N_9)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_9)))) # label(fact_3216_of__nat__less__two__power) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 223 (all N all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_real_real(ln,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_real_real(ln,X)))) # label(fact_3697_ln__realpow) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 224 (all D all Sa exists Z all X_2 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,X_2),Z)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_2),Sa))) <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_2),Sa)))))) # label(fact_2259_minf_I9_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 225 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K_1)),pls)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),pls)))) # label(fact_345_rel__simps_I12_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 226 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,M)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N))) # label(fact_3067_mult__Suc) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 227 (all M all A_3 (is_int(A_3) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),M)) -> A_3 = zero_zero_int))))) # label(fact_1459_zcong__zless__0) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 228 (all X_36 all Y_28 all Z_9 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_36),Y_28)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_36),Z_9)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_36),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Y_28),Z_9))) # label(fact_622_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 229 (all Z_14 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_14) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Z_14),Z_14)) # label(fact_285_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 230 (all A_225 all B_162 all C_91 all D_18 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_225),B_162)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_91),D_18)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_225),C_91)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_162),D_18))) # label(fact_559_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 231 (all A_188 all B_144 all C_81 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_188),B_144)),C_81)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,B_144),C_81)))) # label(fact_959_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 232 (all P all Q all R hAPP_f429935748t_bool(hAPP_i872162435t_bool(P,R),Q) = hAPP_i1948725293t_bool(hAPP_f1791153283t_bool(hAPP_f735247205t_bool(cOMBC_983868720t_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_013) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 233 (all X_1 (zero_zero_complex = X_1 <-> zero_zero_complex = X_1)) # label(fact_1898_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 234 (all M (hAPP_nat_nat(div_mod_nat(M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) | hAPP_nat_nat(div_mod_nat(M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))) | hAPP_nat_nat(div_mod_nat(M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = one_one_nat | hAPP_nat_nat(div_mod_nat(M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_nat)) # label(fact_2771_mod__exhaust__less__4) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 235 (all N hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),pi)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),N)) # label(fact_3715_cos__npi) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 236 (all A_3 all Q_2 all R_3 (is_int(A_3) & is_int(Q_2) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,A_3),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3))) -> (A_3 != zero_zero_int -> Q_2 = one_one_int)))) # label(fact_2794_self__quotient) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 237 (all P all Q all R hAPP_r474017924t_real(hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,P),Q),R) = hAPP_r474017924t_real(P,hAPP_real_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__Nat__Onat_Mtc__RealDe) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 238 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arcsin,Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3507_arcsin__bounded) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 239 (all Y all X (hAPP_real_real(cos,X) != zero_zero_real -> (hAPP_real_real(cos,Y) != zero_zero_real -> (hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)) != zero_zero_real -> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(tan,X)),hAPP_real_real(tan,Y))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(tan,X)),hAPP_real_real(tan,Y)))) = hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)))))) # label(fact_3612_tan__add) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 240 (all C all A all B (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),B))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),B)))) # label(fact_2027_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 241 (all W_1 (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,W_1))) <-> hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,W_1)))))) # label(fact_3272_neg__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 242 (all A_108 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_108),zero_zero_complex) = A_108) # label(fact_1965_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 243 (all X hAPP_real_real(sqrt,hAPP_real_real(uminus_uminus_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sqrt,X))) # label(fact_3480_real__sqrt__minus) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 244 (all A_94 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_94),A_94) = zero_zero_complex) # label(fact_2086_diff__self) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 245 (all A_30 all B_25 all C_11 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_30),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_25),C_11))),B_25) = hAPP_nat_nat(div_mod_nat(A_30),B_25)) # label(fact_2509_mod__mult__self2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 246 (all N_1 all Ma (zero_zero_nat = N_1 & Ma = zero_zero_nat <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,N_1)),hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,Ma)))))) # label(fact_3539_int__zle__neg) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 247 (all X_6 all Q_5 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_6),hAPP_nat_nat(suc,Q_5)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_6),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_6),Q_5))) # label(fact_3032_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 248 (all M all K_2 (M != hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),K_2))))) # label(fact_3131_one__less__k) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 249 (all C_70 all A_164 all B_125 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_164),B_125)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_70)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_70),A_164)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_70),B_125)))))) # label(fact_1132_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 250 (all P all Q all R hAPP_n215258509l_bool(P,hAPP_nat_nat(Q,R)) = hAPP_n215258509l_bool(hAPP_f66927821l_bool(hAPP_f2037783381l_bool(cOMBB_1146692694ol_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HO) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 251 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)))) # label(fact_2855_Suc__less__SucD) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 252 (all K_5 all M_15 all N_28 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_5),hAPP_Q1762011733de_int(div_mo231679042de_int(M_15),N_28))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_5),N_28)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_5),M_15))))) # label(fact_2454_dvd__mod__imp__dvd) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 253 (all A_182 A_182 = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_182),one_one_nat)) # label(fact_1022_power__one__right) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 254 (all W_1 all X_1 (hAPP_int_complex(number528085621omplex,W_1) = X_1 <-> hAPP_int_complex(number528085621omplex,W_1) = X_1)) # label(fact_174_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 255 (all X_40 all N_69 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_40),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_40),N_69)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_40),N_69))) # label(fact_430_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 256 (all F all X_1 all L (hBOOL(hAPP_real_bool(deriv_real(F,X_1),L)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),L)) -> (exists D_2 ((all H_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H_1),D_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,X_1)),hAPP_real_real(F,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),H_1))))))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))))))) # label(fact_3890_DERIV__pos__inc__right) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 257 (all A_239 all B_167 all V_15 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_239),hAPP_i1732201573de_int(number1226105091de_int,V_15))),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_167),hAPP_i1732201573de_int(number1226105091de_int,V_15))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_239),B_167)),hAPP_i1732201573de_int(number1226105091de_int,V_15))) # label(fact_188_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 258 (all N_12 all R_6 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),R_6)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,R_6),one_one_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,R_6),hAPP_nat_nat(suc,N_12))),R_6))))) # label(fact_3153_realpow__Suc__le__self) # label(axiom) # label(non_clause). [assumption]. 6.94/7.00 259 (all A_200 all B_153 all C_86 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_200),B_153)),C_86) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_200),C_86)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_153),C_86))) # label(fact_874_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 260 (all B_120 all A_159 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_120),A_159))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_159)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_120))))) # label(fact_1163_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 261 (all Y all X all N (-hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) -> (X = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),N) -> Y = hAPP_real_real(hAPP_n546711566l_real(root,N),X)))) # label(fact_3899_odd__real__root__unique) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 262 (all X all Y X = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,fTrue),X),Y)) # label(help_If_1_1_If_000tc__Nat__Onat_T) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 263 (all X all Y (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Y)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y))),X)))) # label(fact_2644_zdiv__leq__prop) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 264 (all A_3 all B_2 all Qr_1 (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)),Qr_1)) -> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_P1975530577nt_int(negateSnd,Qr_1))))) # label(fact_4190_divmod__int__rel__neg) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 265 (all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3)) -> (exists P_4 (is_int(P_4) & hBOOL(hAPP_int_bool(zprime,P_4)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_4),A_3)))))) # label(fact_1508_zprime__factor__exists) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 266 (all A (zero_zero_nat = A <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A)))) # label(fact_2310_gcd__lcm__complete__lattice__nat_Otop__unique) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 267 (all X all K_2 all N all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (zero_zero_nat != N -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),K_2) -> (exists I_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),I_2) = X))))) # label(fact_4204_prime__power__exp) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 268 (all F all A_1 hBOOL(hAPP_f54304608l_bool(nat_tr160667106at_int(F),A_1))) # label(fact_2781_transfer__morphism__def) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 269 (all C_70 all A_164 all B_125 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_164),B_125)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_70)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_70),A_164)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_70),B_125)))))) # label(fact_1135_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 270 (all V_2 all W ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W))) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W)))) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W))) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_2)),one_one_int)),hAPP_int_int(number_number_of_int,W))))) # label(fact_2664_zdiv__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 271 (all D_3 all A_3 all B_2 (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)) -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),A_3) = one_one_int)) # label(fact_4997_coprime__lmult__int) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 272 (all A all C all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),C)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),C))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B),A)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),zero_zero_real)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)))) # label(fact_1198_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 273 (all K_1 hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_atMost_nat,K_1)))) # label(fact_5185_finite__atMost) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 274 (all A_3 hAPP_int_int(div_mod_int(A_3),A_3) = zero_zero_int) # label(fact_2458_zmod__self) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 275 (all M all N all K_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2)) -> hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),hAPP_int_int(legacy_zgcd(M),N)))) # label(fact_4510_zgcd__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 276 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) -> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat,X)),hAPP_int_nat(nat,Y))))) # label(fact_2743_Nat__Transfer_Otransfer__nat__int__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 277 (all A_237 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit1,pls))),A_237) = A_237) # label(fact_238_mult__numeral__1) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 278 (all A_222 zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A_222)) # label(fact_589_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 279 (all X_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),N_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X_1)) | N_1 = zero_zero_nat)) # label(fact_442_zero__less__power__nat__eq) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 280 (all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) -> -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))))) # label(fact_1728_dvd_Oless__asym) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 281 (all A_68 all B_56 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_68),B_56)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_68),B_56)),B_56)) = A_68) # label(fact_2340_semiring__div__class_Omod__div__equality_H) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 282 (all K_2 all L_2 hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),L_2)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit0,K_2)),hAPP_int_int(bit1,L_2))) # label(fact_354_add__Bit0__Bit1) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 283 (all Ma all N_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,Ma)),hAPP_int_real(number267125858f_real,N_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ma),hAPP_int_int(number_number_of_int,N_1))))) # label(fact_4442_number__of__less__real__of__int__iff2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 284 (all A_37 zero_z891286103de_int = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_37),zero_z891286103de_int)) # label(fact_2473_div__by__0) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 285 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_2)) -> hAPP_P1175774780nt_int(product_fst_int_int,posDivAlg(A_3,B_2)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)))) # label(fact_4226_div__pos__pos) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 286 (all W_1 all X_1 (hAPP_int_real(number267125858f_real,W_1) = X_1 <-> hAPP_int_real(number267125858f_real,W_1) = X_1)) # label(fact_172_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 287 (all P all Q all R hAPP_bool_bool(P,hAPP_int_bool(Q,R)) = hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 288 (all A_148 all N_51 all N_50 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_51),N_50)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),A_148)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_148),N_51)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_148),N_50)))))) # label(fact_1272_power__increasing) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 289 (all N all K_2 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)))) # label(fact_1753_Nat_Odiff__diff__eq) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 290 (all Ma all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,Ma),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(nat_case_nat(hAPP_nat_nat(suc,N_1),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,suc),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,ord_max_nat),N_1))),Ma)) # label(fact_5189_max__Suc2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 291 (all X all Y all Z_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Z_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_1))) # label(fact_1592_nat__add__left__commute) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 292 (all A_146 one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_146),zero_zero_nat)) # label(fact_1297_power__0) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 293 (all A all B all C all D (hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),D) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),D)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B))))) # label(fact_2096_diff__eq__diff__less) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 294 (all Z_1 all W hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(uminus_uminus_int,Z_1)),hAPP_int_int(uminus_uminus_int,W)) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),W))) # label(fact_3517_zminus__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 295 (all A_1 all B_1 (B_1 = A_1 <-> image_nat_int(semiri1621563631at_int,B_1) = image_nat_int(semiri1621563631at_int,A_1))) # label(fact_4602_transfer__nat__int__set__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 296 (all A_240 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_240),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),zero_zero_int))) # label(fact_16_power2__less__0) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 297 (all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),Ma)) <-> N_1 != Ma)) # label(fact_1585_nat__neq__iff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 298 (all M_18 all N_38 all A_128 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_128)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_128),M_18)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_128),N_38))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_18),N_38))))) # label(fact_1448_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 299 (all Y_6 all P_6 all N_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6))) # label(fact_3177_lemma__realpow__diff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 300 (all W_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,W_1)),zero_zero_int)))) # label(fact_348_bin__less__0__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 301 (all Z_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Z_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Z_1),Z_1)) # label(fact_569_nat__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 302 (all B_90 all A_112 all C_50 (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_90),A_112) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_50),A_112) -> B_90 = C_50)) # label(fact_1914_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 303 (all X_1 all Y_2 (hBOOL(hAPP_int_bool(even_odd_even_int,Y_2)) | hBOOL(hAPP_int_bool(even_odd_even_int,X_1)) <-> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_2))))) # label(fact_2883_Parity_Oeven__product) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 304 (all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) -> -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))))) # label(fact_1720_dvd_Oless__imp__not__less) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 305 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K1)),hAPP_int_int(bit1,K2))))) # label(fact_257_less__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 306 (all X_2 hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,X_2)) = hAPP_n1699378549t_bool(ord_less_nat,X_2)) # label(fact_4741_less__eq__Suc__le__raw) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 307 (all X_1 all Y_2 all B (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2)) <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B),X_1)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B),Y_2)))))) # label(fact_1417_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 308 (all B all A (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B),A) = B <-> zero_zero_nat = A)) # label(fact_612_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 309 (all A_113 all B_91 all C_51 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_113),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_91),C_51)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_113),B_91)),C_51)) # label(fact_1902_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 310 (all I all J_2 all K_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),K_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K_2)))) # label(fact_1632_add__lessD1) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 311 (all C_68 all A_162 all B_123 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_162),B_123)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_68)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_162),C_68)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_123),C_68)))))) # label(fact_1143_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 312 (all A_129 all N_40 all N_39 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_40),N_39)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_129)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_129),N_40)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_129),N_39)))))) # label(fact_1439_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 313 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> zero_zero_nat != N)) # label(fact_1609_gr__implies__not0) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 314 (all A_224 all N_68 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_224),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_224),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_576_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 315 (all X_8 all Q_7 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_8),hAPP_nat_nat(suc,Q_7)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_8),Q_7)),X_8)) # label(fact_3021_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 316 (all K_2 all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),K_2))))) # label(fact_2689_div__le__mono) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 317 (all K_2 hAPP_int_int(bit1,hAPP_int_int(pred,hAPP_int_int(uminus_uminus_int,K_2))) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(bit1,K_2))) # label(fact_4046_minus__Bit1) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 318 (all Z_1 (is_int(Z_1) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),Z_1) = Z_1)) # label(fact_129_zmult__1) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 319 (all L all F ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_2)),hAPP_nat_real(F,hAPP_nat_nat(suc,N_2))))) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_2)),L))) -> ((all E (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),E)) -> (exists N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,L),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(F,N_2)),E)))))) -> hBOOL(tendsto_nat_real(F,L,sequentially)))))) # label(fact_5157_increasing__LIMSEQ) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 320 (all A_227 all B_164 all C_93 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_227),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_164),C_93)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_227),B_164)),C_93)) # label(fact_545_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 321 (all A_208 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_208),zero_zero_real))) # label(fact_757_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 322 (all B_131 all A_171 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_171),zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_131)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_171),B_131)),zero_zero_nat))))) # label(fact_1093_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 323 (all C all A all B (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A),B)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A),C)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B),A)),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,C),A))) <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,B),C)))))) # label(fact_2482_div__dvd__div) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 324 (all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N))) # label(fact_1505_le0) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 325 (all A_3 all B_2 all C_1 all Q_2 all R_3 (is_int(C_1) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(B_2,C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3))) -> (zero_zero_int != C_1 -> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2),C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Q_2)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),R_3)),C_1))),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),R_3)),C_1)))))))) # label(fact_2802_zmult1__lemma) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 326 (all Sa all Ta all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B)) -> hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,A),Sa)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,B),Ta))),pair_leq)))) # label(fact_4543_pair__leqI1) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 327 (all Z_1 hBOOL(hAPP_int_bool(nat_is_nat,hAPP_nat_int(semiri1621563631at_int,Z_1)))) # label(fact_5143_Nat__Transfer_Otransfer__int__nat__function__closures_I9_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 328 (all A_61 zero_z891286103de_int = hAPP_Q1762011733de_int(div_mo231679042de_int(zero_z891286103de_int),A_61)) # label(fact_2375_mod__0) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 329 (all C_89 all D_17 all A_213 all B_159 all R_8 (R_8 != zero_zero_real -> (A_213 = B_159 & C_89 != D_17 -> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_213),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_8),C_89)) != hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_159),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_8),D_17))))) # label(fact_688_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 330 (all G all F all X_1 all D (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D)) -> ((all Z (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D)) -> Z = hAPP_real_real(G,hAPP_real_real(F,Z)))) -> ((all Z (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D)) -> hBOOL(hAPP_real_bool(isCont_real_real(F),Z)))) -> hBOOL(hAPP_real_bool(isCont_real_real(G),hAPP_real_real(F,X_1))))))) # label(fact_5176_isCont__inv__fun) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 331 (all A hBOOL(tendsto_complex_real(im,hAPP_complex_real(im,A),at_complex(A)))) # label(fact_5077_Im_Ocont) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 332 (all L_2 all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),L_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_2),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_2),M)))))) # label(fact_1741_diff__less__mono2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 333 (all M_18 all N_38 all A_128 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_128)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_128),M_18)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_128),N_38))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_18),N_38))))) # label(fact_1449_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 334 (all Z_14 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_14),Z_14) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_14)) # label(fact_284_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 335 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_complex_real(im,X)),hAPP_complex_real(im,Y)) = hAPP_complex_real(im,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),Y))) # label(fact_4101_Im_Odiff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 336 (all A all B (is_int(B) & is_int(A) -> (A = B <-> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B) = zero_zero_int))) # label(fact_2076_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 337 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(sin,X))),one_one_real))) # label(fact_3663_abs__sin__le__one) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 338 (all K_2 all A_3 all B_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),A_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),B_2)),M)))) # label(fact_1035_zcong__scalar2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 339 (all Ma all N_1 ((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) | zero_zero_nat = N_1 -> divmod_nat(Ma,N_1) = hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,zero_zero_nat),Ma)) & (-(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) | N_1 = zero_zero_nat) -> divmod_nat(Ma,N_1) = hAPP_P1586233937at_nat(produc1391996073at_nat(hAPP_f837737749at_nat(hAPP_f999082675at_nat(cOMBB_348438404at_nat,product_Pair_nat_nat),suc)),divmod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),N_1),N_1))))) # label(fact_4851_divmod__nat__if) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 340 (all A_15 all B_11 (B_11 != zero_z891286103de_int -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_15),B_11)),B_11) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_15),B_11)),one_on1684967323de_int))) # label(fact_2635_div__add__self2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 341 (all C_41 all A_99 all B_81 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_41),A_99)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_41),B_81))) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_99),B_81)))) # label(fact_2038_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 342 (all Y_2 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2)) -> ((hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven))) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat,X_1)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat,Y_2)))))) # label(fact_3242_neg__one__power__parity) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 343 (all K_2 all L_2 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,K_2),L_2) = hAPP_int_int(z3div(K_2),L_2)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2)) -> hAPP_int_int(z3div(K_2),L_2) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,K_2),hAPP_int_int(uminus_uminus_int,L_2)))))) # label(fact_3950_z3div__def) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 344 (all D int_ge_less_than(D) = collec1347809874nt_int(produc93441119t_bool(hAPP_f428220345t_bool(hAPP_f1878066172t_bool(cOMBS_1720984575t_bool,hAPP_f1315737299t_bool(hAPP_f538565589t_bool(cOMBB_1211207212ol_int,cOMBB_bool_bool_int),hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,D)))),ord_less_int)))) # label(fact_4861_int__ge__less__than__def) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 345 (all A_36 hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,zero_zero_nat),A_36) = zero_zero_nat) # label(fact_2475_div__0) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 346 (all A_26 all B_21 (B_21 != zero_z126310315umeral -> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_21),A_26)),B_21) = A_26)) # label(fact_2543_div__mult__self1__is__id) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 347 (all X_1 all A (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),hAPP_real_real(uminus_uminus_real,A)) = zero_zero_real <-> X_1 = A)) # label(fact_3483_real__add__minus__iff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 348 (all A_203 zero_zero_complex = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_203),zero_zero_complex)) # label(fact_843_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 349 (all X_7 all Q_6 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_7),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_7),Q_6)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_7),hAPP_nat_nat(suc,Q_6))) # label(fact_3027_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 350 (all K_2 all L_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),L_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),L_2)))))) # label(fact_1646_add__le__mono) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 351 (all N all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),X)),one_one_real)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),one_one_real)),N))))) # label(fact_3712_LIMSEQ__inverse__realpow__zero__lemma) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 352 (all B_2 all Q_2 all R_3 all B_5 all Q_8 all R_5 (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_3) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_8)),R_5) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_8)),R_5))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_5),B_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_5),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_2),Q_8))))))))) # label(fact_735_zdiv__mono2__lemma) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 353 (all P all Q all R hAPP_P1668131738nt_int(hAPP_f167782601nt_int(hAPP_f1769710159nt_int(cOMBC_2037038898nt_int,P),Q),R) = hAPP_f521271395nt_int(hAPP_P752050971nt_int(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc_) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 354 (all X all A_3 all B_2 hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),B_2) = hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),B_2))) # label(fact_4383_powr__powr) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 355 (all X all Y hAPP_complex_real(norm_norm_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,X),Y)) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_3954_complex__norm) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 356 (all P all Q all R hAPP_int_fun_int_int(P,hAPP_P1175774780nt_int(Q,R)) = hAPP_P2110489235nt_int(hAPP_f1855025978nt_int(hAPP_f1038672605nt_int(cOMBB_1506897658nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_012) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 357 (all A_208 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_208),zero_zero_rat))) # label(fact_761_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 358 (all W_6 hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,W_6)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_int_real(number267125858f_real,W_6))),hAPP_int_real(number267125858f_real,W_6))) # label(fact_376_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 359 (all V_2 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_2),pls)) -> zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V_2)),one_one_nat)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_2),pls)) -> hAPP_int_nat(number_number_of_nat,hAPP_int_int(pred,V_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V_2)),one_one_nat)))) # label(fact_4038_nat__number__of__diff__1) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 360 (all B all A all Ma (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(bnorRset(A),Ma))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A)))) # label(fact_4293_Bnor__mem__zle) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 361 (all X bezw(X,zero_zero_nat) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,one_one_int),zero_zero_int)) # label(fact_4155_bezw__0) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 362 (all Y_2 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),Y_2))))) # label(fact_3925_real__root__gt__1__iff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 363 (all N_11 all A_7 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_7)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_7),one_one_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_7),hAPP_nat_nat(suc,N_11))),one_one_nat))))) # label(fact_3163_power__Suc__less__one) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 364 (all A_197 all B_152 all N_64 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_197),N_64)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,B_152),N_64)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_197),B_152)),N_64)) # label(fact_898_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 365 (all A_234 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_234),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_234),A_234)) # label(fact_300_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 366 (all B all A all F all G all X_1 all D_1 (hBOOL(hAPP_real_bool(deriv_real(F,hAPP_real_real(G,X_1)),D_1)) -> (D_1 != zero_zero_real -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),B)) -> ((all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),Y_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),B)) -> Y_1 = hAPP_real_real(F,hAPP_real_real(G,Y_1)))) -> (hBOOL(hAPP_real_bool(isCont_real_real(G),X_1)) -> hBOOL(hAPP_real_bool(deriv_real(G,X_1),hAPP_real_real(inverse_inverse_real,D_1)))))))))) # label(fact_5175_DERIV__inverse__function) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 367 (all A all E_2 all C all B all D (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),E_2)),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),E_2)),D))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B),A)),E_2)),D))))) # label(fact_1565_le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 368 (all A_222 zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,zero_zero_real),A_222)) # label(fact_585_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 369 (all A_40 all B_31 all C_15 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_40),B_31)),C_15) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(div_mod_int(A_40),C_15)),B_31)),C_15)) # label(fact_2439_mod__diff__left__eq) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 370 (all N_66 all A_210 all B_158 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_210),B_158)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_210)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_66)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_210),N_66)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_158),N_66))))))) # label(fact_745_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 371 (all X all Y hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X)),Y))) # label(fact_1688_real__sum__squared__expand) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 372 (all Y_21 all X_27 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_27)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_21)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_21),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_21),X_27)),X_27)))))) # label(fact_1385_mult__left__le__one__le) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 373 (all Y_2 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y_2)))))) # label(fact_3922_real__root__ge__1__iff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 374 (all A_189 all B_145 all C_82 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_189),B_145)),C_82)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_189),C_82)))) # label(fact_953_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 375 (all N_1 all X_1 all Diff all F (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) & zero_zero_real != X_1 & (all M_1 all X_2 hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),X_2),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),X_2)))) & hAPP_n546711566l_real(Diff,zero_zero_nat) = F -> (exists T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_1))) & hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N_1),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1))) = hAPP_real_real(F,X_1) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(abs_abs_real,T))))))) # label(fact_4932_Maclaurin__all__lt__objl) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 376 (all A_103 all C_45 all B_85 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_103),C_45)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_85),C_45))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_103),B_85)))) # label(fact_2001_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 377 (all B_148 all A_192 all C_84 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_192),C_84)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_192),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_148),C_84))))) # label(fact_927_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 378 (all Z1 all Z2 all W hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),W)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),W)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z1),Z2)),W)) # label(fact_1522_zdiff__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 379 (all M hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,pi),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_real) # label(fact_3631_cos__pi__eq__zero) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 380 (all N_16 all M_5 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_16),M_5)) -> hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(semiri132038758t_real,M_5)),hAPP_nat_real(semiri132038758t_real,N_16)) = hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_5),N_16)))) # label(fact_3100_of__nat__diff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 381 (all B_69 all A_81 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_81)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_69)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_81),B_69)))))) # label(fact_2188_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 382 (all Y_2 hAPP_real_real(arctan,Y_2) = the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),tan)),Y_2))))) # label(fact_4882_arctan__def) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 383 (all B_90 all A_112 all C_50 (hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_50),A_112) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_90),A_112) -> C_50 = B_90)) # label(fact_1912_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 384 (all M hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),M)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = M) # label(fact_2749_add__self__div__2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 385 (all R_3 hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),R_3) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_3)),ii)) # label(fact_3995_complex__of__real__i) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 386 (all W_8 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,W_8)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,W_8)),hAPP_int_int(number_number_of_int,W_8))) # label(fact_362_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 387 (all C_34 all D_6 all A_80 all B_68 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_80),B_68)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_34),D_6)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_80),C_34)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_68),D_6)))))) # label(fact_2196_add__le__less__mono) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 388 (all X all Y hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Y),hAPP_real_real(uminus_uminus_real,X))) = hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(uminus_uminus_real,Y)))) # label(fact_3488_abs__minus__add__cancel) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 389 (all Ma all K_1 all N_1 (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K_1),N_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),N_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),Ma))))) # label(fact_5006_coprime__dvd__mult__iff__int) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 390 (all N archim1246769320r_real(hAPP_real_real(uminus_uminus_real,hAPP_int_real(real_int,N))) = hAPP_int_int(uminus_uminus_int,N)) # label(fact_4428_floor__minus__real__of__int) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 391 (all N all I ((zero_z126310315umeral != I -> hAPP_nat_nat(code_nat_of_aux(I),N) = hAPP_nat_nat(code_nat_of_aux(hAPP_C498520661umeral(hAPP_C1594335432umeral(minus_1690775515umeral,I),one_on1645066479umeral)),hAPP_nat_nat(suc,N))) & (I = zero_z126310315umeral -> N = hAPP_nat_nat(code_nat_of_aux(I),N)))) # label(fact_3258_nat__of__aux__code) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 392 (all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,X),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X),Y)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_4008_coprime__sos) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 393 (exists K_4 ((all X_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(cnj,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_2)),K_4)))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),K_4)))) # label(fact_4086_cnj_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 394 (all X_18 all Y_15 all A_78 all B_66 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_18),A_78)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Y_15),B_66))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_18),A_78)),B_66))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_78),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Y_15),B_66))) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_18),Y_15)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_78),B_66))) # label(fact_2282_mult_Oprod__diff__prod) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 395 (all A_198 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_198)) -> A_198 = zero_zero_nat)) # label(fact_891_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 396 (all Xa_4 all X_16 all Y_13 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_4),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_16),Y_13)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_4),X_16)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_4),Y_13))) # label(fact_2293_mult__right_Oadd) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 397 (all K_1 all N_1 all F_2 hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,F_2),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),K_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),K_1)))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),K_1)))) # label(fact_4898_sumr__offset2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 398 (all B_116 all A_155 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_155)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_116)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_155),B_116)))))) # label(fact_1186_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 399 (all X_14 all Y_11 all Ya_1 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_14),Y_11)),Ya_1) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_14),Ya_1)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Y_11),Ya_1))) # label(fact_2300_mult__left_Odiff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 400 (all X_1 all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,X_1)),hAPP_int_real(real_int,Y_2))))) # label(fact_4378_real__of__int__less__iff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 401 (all X all Y all K_2 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X),Y) -> (exists I_2 exists J_1 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),J_1) = Y & X = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),I_2)))))) # label(fact_4481_prime__power__mult) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 402 (all B all P_5 all A (is_int(B) & is_int(A) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A)) -> (-hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),one_one_int)),P_5))) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5))) -> hAPP_int_int(inv(P_5),A) = B | B = A))))) # label(fact_2227_wset__mem__imp__or) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 403 (all Ta all A_1 all D_1 (is_int(Ta) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ta),one_one_int)),A_1)) -> (all X_2 (is_int(X_2) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1)) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) != X_2)))))) -> (X_2 = Ta -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1) = Ta)))))))) # label(fact_5096_aset_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 404 (all B_95 all C_53 all A_117 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_117)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_95),C_53)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_95),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_117),C_53)))))) # label(fact_1874_add__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 405 (all P all Q all R hAPP_int_int(P,hAPP_nat_int(Q,R)) = hAPP_nat_int(hAPP_f1139079189at_int(hAPP_f1431025877at_int(cOMBB_int_int_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__Int__Oint_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 406 (all M all K_2 all N hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),N)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N)) # label(fact_4967_gcd__add__mult__int) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 407 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real)) -> hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(cos,hAPP_real_real(arcsin,X))))) # label(fact_3559_cos__arcsin) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 408 (all N_1 image_int_nat(nat,hAPP_i1948725293t_bool(ord_at875362053st_int(zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N_1))) = hAPP_n1699378549t_bool(ord_atMost_nat,N_1)) # label(fact_5183_SetInterval_Otransfer__nat__int__set__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 409 (all Z_1 all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_1))))) # label(fact_4157_termination__basic__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 410 (all X_1 all Y_2 ((hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))) <-> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_2)),zEven)))) # label(fact_2898_even__diff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 411 (all A_36 hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,zero_z126310315umeral),A_36) = zero_z126310315umeral) # label(fact_2476_div__0) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 412 (all A_239 all B_167 all V_15 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_239),B_167)),hAPP_int_rat(number_number_of_rat,V_15)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_239),hAPP_int_rat(number_number_of_rat,V_15))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_167),hAPP_int_rat(number_number_of_rat,V_15)))) # label(fact_189_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 413 (all B_1_1 is_int(archim1246769320r_real(B_1_1))) # label(gsy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_000tc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 414 (all X_1 all Y_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2)) <-> Y_2 = X_1 | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_1)))) # label(fact_1700_dvd_Ole__less) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 415 (all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1)) <-> -hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat)))))) # label(fact_3779_even__num__iff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 416 (all A_203 zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_203),zero_zero_rat)) # label(fact_845_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 417 (all Ma all N_1 all Qr (Ma = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_P1654798560at_nat(product_fst_nat_nat,Qr)),N_1)),hAPP_P1654798560at_nat(product_snd_nat_nat,Qr)) & (zero_zero_nat = N_1 -> zero_zero_nat = hAPP_P1654798560at_nat(product_fst_nat_nat,Qr)) & (zero_zero_nat != N_1 -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_P1654798560at_nat(product_snd_nat_nat,Qr))) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_P1654798560at_nat(product_snd_nat_nat,Qr)),zero_zero_nat))) & (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_P1654798560at_nat(product_snd_nat_nat,Qr))) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_P1654798560at_nat(product_snd_nat_nat,Qr)),N_1)))) <-> hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(Ma,N_1),Qr)))) # label(fact_4241_divmod__nat__rel__def) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 418 (all N -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),N))) # label(fact_2869_Suc__n__not__le__n) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 419 (all Ma all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ma),hAPP_int_int(number_number_of_int,N_1))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,Ma)),hAPP_int_real(number267125858f_real,N_1))))) # label(fact_4444_number__of__le__real__of__int__iff2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 420 (all C_79 all D_15 all A_179 all B_139 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_179),B_139)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_79),D_15)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_139)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_79)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_179),C_79)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_139),D_15)))))))) # label(fact_1058_mult__mono) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 421 (all A_50 all C_22 all B_41 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(div_mod_int(A_50),C_22)),B_41)),C_22) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_50),B_41)),C_22)) # label(fact_2409_zmod__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 422 (all P_5 hAPP_P1975530577nt_int(produc1518849193nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,hAPP_f400200275nt_int(hAPP_f1731308757nt_int(cOMBB_735549356nt_int,cOMBB_1389251822nt_int),hAPP_P621635040nt_int(hAPP_f1582586579nt_int(cOMBC_707974837nt_int,hAPP_f251264923nt_int(hAPP_f279307923nt_int(cOMBB_662120866nt_int,if_Pro1731782967nt_int),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),zero_zero_int))),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int)))),hAPP_f1574055885nt_int(hAPP_f434654626nt_int(cOMBS_2011279539nt_int,hAPP_f2094497995nt_int(hAPP_f534361363nt_int(cOMBB_1657320178nt_int,cOMBC_1964283556nt_int),hAPP_f2004554147nt_int(hAPP_f1228585167nt_int(cOMBB_1102189010nt_int,hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int)),hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,times_times_int),sgn_sgn_int)))),abs_abs_int))),quotient_of(P_5)) = quotient_of(hAPP_rat_rat(inverse_inverse_rat,P_5))) # label(fact_4839_rat__inverse__code) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 423 (all A_197 all B_152 all N_64 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_197),N_64)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_152),N_64)) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_197),B_152)),N_64)) # label(fact_892_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 424 (all C_66 all A_144 all B_113 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_66),A_144)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_66),B_113))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_66)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_144),B_113))))) # label(fact_1318_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 425 (all N all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_nat_real(real_nat,N))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,natfloor(X)),N))))) # label(fact_3698_less__natfloor) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 426 (all Z_11 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_11),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_11),Z_11)) # label(fact_295_mult__2__right) # label(axiom) # label(non_clause). [assumption]. 6.94/7.01 427 (all I_1 all K_1 (is_int(K_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),I_1)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),I_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),K_1)) | K_1 = zero_zero_int <-> zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,I_1),K_1)))) # label(fact_2607_zdiv__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 428 (all P all Q all R hAPP_int_fun_int_int(hAPP_f1760145644nt_int(hAPP_f808647893nt_int(cOMBC_736024425nt_int,P),Q),R) = hAPP_f1356864277nt_int(hAPP_i744251628nt_int(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 429 (all R_2 all X_3 (hBOOL(vanishes(X_3)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),R_2)) -> (exists K all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_2)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(X_3,N_2))),R_2))))))) # label(fact_5151_vanishesD) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 430 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hAPP_real_real(hAPP_n546711566l_real(root,N),one_one_real) = one_one_real)) # label(fact_3910_real__root__one) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 431 (all B_155 all A_204 (A_204 != zero_zero_complex -> (B_155 != zero_zero_complex -> hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_204),B_155) != zero_zero_complex))) # label(fact_832_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 432 (all M_13 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_nat_nat(semiri984289939at_nat,M_13)) = hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(suc,M_13))) # label(fact_2839_of__nat__Suc) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 433 (all Z_1 all W ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W))) -> one_one_int = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,W)))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W))) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,W))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,W)))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,W)))))) # label(fact_4789_zpower__number__of__odd) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 434 (all X all Y hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_fst_int_int,bezw(X,Y))),hAPP_nat_int(semiri1621563631at_int,X))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(X,Y))),hAPP_nat_int(semiri1621563631at_int,Y))) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y))) # label(fact_4297_bezw__aux) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 435 (all C (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,C)),one_one_real)) -> hBOOL(tendsto_nat_real(hAPP_r474017924t_real(power_power_real,C),zero_zero_real,sequentially)))) # label(fact_5045_LIMSEQ__rabs__realpow__zero2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 436 (all B_1_1 all B_2_1 is_bool(hAPP_f115998247l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__prod_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__p) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 437 (all N_1 all X_1 exists T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_1))) & hAPP_real_real(sin,X_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_r195310020l_real(hAPP_f2126667875l_real(cOMBC_746811033l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),zero_zero_real)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,div_div_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,minus_minus_nat),hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,T),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(real_nat,N_1))),pi)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1))))) # label(fact_4914_Maclaurin__sin__expansion2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 438 (all Z_14 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Z_14),Z_14) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_14)) # label(fact_282_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 439 (all Y_2 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,pls)),Y_2)))) # label(fact_413_less__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 440 (all Y_21 all X_27 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_27)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_21)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_21),one_one_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_21),X_27)),X_27)))))) # label(fact_1384_mult__left__le__one__le) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 441 (all V_3 all V_2 ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_2),V_3))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls)) -> zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3))))) # label(fact_571_mult__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 442 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2))) -> hAPP_P1975530577nt_int(adjust(B_2),negDivAlg(A_3,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))) = negDivAlg(A_3,B_2)) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2))) -> negDivAlg(A_3,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2))))) # label(fact_2669_negDivAlg__eqn) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 443 (all U (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),U)) -> hAPP_i1948725293t_bool(ord_at641636577an_int(zero_zero_int),U) = image_nat_int(semiri1621563631at_int,hAPP_n1699378549t_bool(ord_lessThan_nat,hAPP_int_nat(nat,U))))) # label(fact_4954_image__atLeastZeroLessThan__int) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 444 (all X_1 hBOOL(hAPP_real_bool(isCont_real_real(cos),X_1))) # label(fact_5161_isCont__cos) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 445 (all X_21 all N_31 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31)) -> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_21),N_31) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21))) # label(fact_1797_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 446 (all Z1 all Z2 all Z3 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),Z2)),Z3) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),Z3))) # label(fact_61_zmult__assoc) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 447 (all B_130 all A_170 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_170)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_130),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_130),A_170)),zero_zero_rat))))) # label(fact_1103_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 448 (all I all M all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),M)))) # label(fact_4981_dvd__gcd__D1__int) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 449 (all X all Y hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(powr,X),Y)))) # label(fact_4403_powr__ge__pzero) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 450 (all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5)) -> hAPP_int_int(zfact,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)) = big_co1548731110nt_int(cOMBI_int,comple219730294t_bool(setS(A,P_5))))))) # label(fact_4815_Union__SetS__setprod__prop2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 451 (all A all C all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B),C))))) # label(fact_2072_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 452 (all X_39 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_39),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_39),X_39)) # label(fact_432_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 453 (all V_13 all W_11 all Z_10 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_13),W_11))),Z_10) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_13)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,W_11)),Z_10))) # label(fact_336_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 454 (all A_3 all N cis(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),A_3)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,cis(A_3)),N)) # label(fact_4153_DeMoivre) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 455 (all W all V_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_2))) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_2))) -> hAPP_P1975530577nt_int(adjust(hAPP_int_int(number_number_of_int,V_2)),posDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(number_number_of_int,V_2)))) = posDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(number_number_of_int,V_2))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_2))) -> posDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(number_number_of_int,V_2)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),hAPP_int_int(number_number_of_int,W))))) # label(fact_2770_posDivAlg__eqn__number__of) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 456 (all A_216 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,one_one_rat),A_216) = A_216) # label(fact_653_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 457 (all A_3 all B_2 hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B_2),A_3)) # label(fact_4977_gcd__commute__int) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 458 (all X_1 all Y_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_2))))) # label(fact_2990_Nat__Transfer_Otransfer__int__nat__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 459 (all A all B (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A),B)) <-> hAPP_Q1762011733de_int(div_mo231679042de_int(B),A) = zero_z891286103de_int)) # label(fact_2517_dvd__eq__mod__eq__0) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 460 (all K_2 all L_2 hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),L_2)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(bit1,K_2)),hAPP_int_int(bit1,L_2))) # label(fact_1553_diff__bin__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 461 (all C_1 all A_3 all B_2 all Q_2 all R_3 (hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(A_3,B_2),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_2),R_3))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_1)) -> hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(A_3,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),C_1)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,Q_2),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),hAPP_nat_nat(div_mod_nat(Q_2),C_1))),R_3)))))))) # label(fact_4213_divmod__nat__rel__mult2__eq) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 462 (all A_69 all B_57 zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(div_mod_nat(A_69),B_57)),B_57)) # label(fact_2332_mod__div__trivial) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 463 (all C all D all A all B (D != C & A != B <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),D)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B),C)) != hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B),D)))) # label(fact_624_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 464 (all A all B all C (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B)))))) # label(fact_1192_mult__less__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 465 (all Z_12 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_12),Z_12) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_12),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_290_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 466 (all N hAPP_nat_int(semiri1621563631at_int,N) = archim856651990g_real(hAPP_nat_real(real_nat,N))) # label(fact_4363_ceiling__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 467 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(inverse_inverse_real,X))) # label(fact_3855_real__inv__sqrt__pow2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 468 (all K_2 all M all N hAPP_int_int(legacy_zgcd(K_2),hAPP_int_int(legacy_zgcd(M),N)) = hAPP_int_int(legacy_zgcd(M),hAPP_int_int(legacy_zgcd(K_2),N))) # label(fact_4494_zgcd__left__commute) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 469 (all A_56 all C_26 all B_47 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(div_mod_int(A_56),B_47)),C_26) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_56),C_26)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_47),C_26))) # label(fact_2393_mod__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 470 (all B_2 all A_3 (is_int(A_3) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2)) -> A_3 = hAPP_int_int(div_mod_int(A_3),B_2))))) # label(fact_2613_mod__pos__pos__trivial) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 471 (all N_1 all N_7 all F ((all N_2 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_2)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2))))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),N_7)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_1)),hAPP_nat_nat(F,N_7)))))) # label(fact_3253_dvd_Olift__Suc__mono__le) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 472 (all A_3 all B_2 all C_1 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(div_mod_int(B_2),C_1))),C_1) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)),C_1)) # label(fact_2460_zmod__zmult1__eq) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 473 (all A_19 all B_15 (is_int(B_15) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_19),B_15)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_19),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_15),A_19)) = B_15))) # label(fact_2575_dvd__mult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 474 (all X_6 all Q_5 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_6),hAPP_nat_nat(suc,Q_5)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_6),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_6),Q_5))) # label(fact_3036_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 475 (all A all C all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),A)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),C)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),C))))) # label(fact_1197_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 476 (all X all N (-hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) -> X = hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N)))) # label(fact_3901_odd__real__root__power__cancel) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 477 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),zero_zero_real)) -> zero_zero_real = hAPP_real_real(cos,arg(hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),Y))))) # label(fact_3970_cos__arg__i__mult__zero__neg) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 478 (all X all Y hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),hAPP_int_int(uminus_uminus_int,Y)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)) # label(fact_4991_gcd__neg2__int) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 479 (all X_1 (hBOOL(hAPP_int_bool(nat_is_nat,X_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)))) # label(fact_5129_is__nat__def) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 480 (all A_184 all M_23 all N_58 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_23),N_58)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_184),M_23)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_184),N_58))))) # label(fact_1005_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 481 (all C_34 all D_6 all A_80 all B_68 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_80),B_68)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_34),D_6)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_80),C_34)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_68),D_6)))))) # label(fact_2195_add__le__less__mono) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 482 (all B_74 all A_86 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_86),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_74),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_86),B_74)),zero_zero_real))))) # label(fact_2144_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 483 (all N all K_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),N)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(legacy_zgcd(M),N)))))) # label(fact_4492_zgcd__greatest) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 484 -(all T (is_int(T) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)),T) != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,s),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),one_one_int))) # label(fact_49__096_B_Bthesis_O_A_I_B_Bt_O_As_A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A_L_A1_) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 485 (all B_128 all A_168 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_168)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_128)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_168),B_128)))))) # label(fact_1111_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 486 (all Ma all N_1 all A (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A)) -> (Ma = N_1 <-> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A),Ma) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A),N_1)))) # label(fact_1285_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 487 (all A_25 all B_20 (B_20 != zero_z891286103de_int -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_25),B_20)),B_20) = A_25)) # label(fact_2548_div__mult__self2__is__id) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 488 (all Lx_2 all Ly all Rx_2 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_2),Ly)),Rx_2) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_2),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Ly),Rx_2))) # label(fact_491_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 489 (all N_1 all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),Ma)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),Ma)))) # label(fact_3064_less__eq__Suc__le) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 490 (all Y all X (hBOOL(hAPP_int_bool(nat_is_nat,X)) -> (hBOOL(hAPP_int_bool(nat_is_nat,Y)) -> hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y)))))) # label(fact_5142_Nat__Transfer_Otransfer__int__nat__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 491 (all B all C all A all D (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A),D) -> (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B),C) -> (hAPP_int_int(abs_abs_int,B) = hAPP_int_int(abs_abs_int,A) & hAPP_int_int(abs_abs_int,C) = hAPP_int_int(abs_abs_int,D) <-> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,A)),hAPP_int_int(abs_abs_int,C)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,B)),hAPP_int_int(abs_abs_int,D)))))) # label(fact_5014_coprime__crossproduct__int) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 492 (all P all Q all R hAPP_P1175774780nt_int(hAPP_f521271395nt_int(hAPP_f1914742907nt_int(cOMBS_2019229620nt_int,P),Q),R) = hAPP_int_int(hAPP_P2110489235nt_int(P,R),hAPP_P1175774780nt_int(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 493 (all A_228 all C_94 all D_19 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_228),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_94),D_19)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_228),C_94)),D_19)) # label(fact_533_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 494 (all V_5 all W_3 all C_55 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_5),W_3))),C_55) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_5)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(number267125858f_real,W_3)),C_55))) # label(fact_1575_add__number__of__diff1) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 495 (all Z_2 (is_int(Z_2) -> (zero_zero_int = Z_2 <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(abs_abs_int,Z_2)),one_one_int))))) # label(fact_3433_zabs__less__one__iff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 496 (all V_7 all B_101 all C_56 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_7)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B_101),C_56)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_7)),B_101)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_7)),C_56))) # label(fact_1547_right__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 497 (all Ma (hAPP_nat_nat(suc,zero_zero_nat) = Ma <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),hAPP_nat_nat(suc,zero_zero_nat))))) # label(fact_3050_dvd__1__iff__1) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 498 (all M all I all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,M),N)),I) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),I)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),I))) # label(fact_5121_min__diff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 499 (all Ma all K_1 all N_1 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),K_1) <-> Ma = N_1)) # label(fact_1595_nat__add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 500 (all A all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X_1)),A)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_nat_real(real_nat,A)))))) # label(fact_3689_natceiling__le__eq) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 501 (all A all B all C all D (hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C),D) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),B)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C),D))))) # label(fact_2094_diff__eq__diff__less__eq) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 502 (all X_1 all Y_2 (is_int(Y_2) & is_int(X_1) -> (hAPP_int_rat(number_number_of_rat,Y_2) = hAPP_int_rat(number_number_of_rat,X_1) <-> X_1 = Y_2))) # label(fact_169_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 503 (all X_1 all Y_2 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y_2) = hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1) <-> Y_2 = X_1))) # label(fact_3897_real__root__eq__iff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 504 (all K_2 quotient_of(hAPP_int_rat(number_number_of_rat,K_2)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,K_2)),one_one_int)) # label(fact_4664_quotient__of__number_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 505 (all Ta all B_1 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> (all X_2 (is_int(X_2) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) != X_2)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Ta)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1)),Ta)))))))) # label(fact_5094_bset_I5_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 506 (all D (D = one_one_nat <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D),zero_zero_nat)))) # label(fact_4024_coprime__0) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 507 (all C_85 all A_193 all B_149 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_193),B_149)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_193),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_149),C_85))))) # label(fact_923_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 508 (all A_240 -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_240),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),zero_zero_rat))) # label(fact_18_power2__less__0) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 509 (all A_216 A_216 = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,one_on1645066479umeral),A_216)) # label(fact_650_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 510 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X)))))) # label(fact_3818_aux4) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 511 (all P all Q all R hAPP_f330985925l_bool(P,hAPP_r2019696015l_real(Q,R)) = hAPP_r1134773055l_bool(hAPP_f1884005689l_bool(hAPP_f172451459l_bool(cOMBB_421479067l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__prod_Itc__RealDef__Orea) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 512 (all X_1 all Y_2 all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B),X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B),Y_2)))))) # label(fact_1415_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 513 (all B all A all C (hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),A) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B),A) <-> B = C)) # label(fact_1934_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 514 (all X_1 (zero_zero_rat = X_1 <-> X_1 = zero_zero_rat)) # label(fact_1900_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 515 (all B_93 all A_115 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_115),zero_z891286103de_int)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_93),zero_z891286103de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_115),B_93)),zero_z891286103de_int))))) # label(fact_1885_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 516 (all D all Sa exists Z all X_2 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,X_2),Z)) -> (-hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_2),Sa))) <-> -hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_2),Sa)))))) # label(fact_2247_minf_I10_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 517 (all X (hBOOL(hAPP_nat_bool(even_odd_even_nat,X)) -> X = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))))) # label(fact_3781_even__nat__div__two__times__two) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 518 (all A_222 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,zero_zero_complex),A_222) = zero_zero_complex) # label(fact_587_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 519 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y)))))) # label(fact_2524_Divides_Otransfer__nat__int__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 520 (all W hAPP_int_nat(number_number_of_nat,W) = hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,W))) # label(fact_2715_nat__number__of) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 521 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),hAPP_real_real(cos,X)))) # label(fact_3567_cos__ge__minus__one) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 522 (all M_9 all N_23 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_23)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_9)),hAPP_n522471361de_int(semiri1424489471de_int,N_23))))) # label(fact_2937_less__imp__of__nat__less) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 523 (all A_30 all B_25 all C_11 hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_30),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_25),C_11))),B_25) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_30),B_25)) # label(fact_2511_mod__mult__self2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 524 (all W_8 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_i1732201573de_int(number1226105091de_int,W_8)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_i1732201573de_int(number1226105091de_int,W_8)),hAPP_i1732201573de_int(number1226105091de_int,W_8))) # label(fact_360_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 525 (all B_117 all A_156 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_156)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_117),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_156),B_117)),zero_zero_int))))) # label(fact_1179_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 526 (all X all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),hAPP_int_int(multInv(P_3),X))),X),P_3)))))) # label(fact_2232_MultInv__prop4) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 527 (all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> hBOOL(hAPP_real_bool(isCont_real_real(hAPP_n546711566l_real(root,N_1)),zero_zero_real)))) # label(fact_5169_isCont__root__zero) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 528 (all D all Ta (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(uminus_uminus_int,D)),Ta)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),Ta)))) # label(fact_3520_uminus__dvd__conv_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 529 (all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(semiri984289939at_nat,Ma)),hAPP_nat_nat(semiri984289939at_nat,N_1))))) # label(fact_2949_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 530 (all B_1_1 all B_2_1 all B_3_1 is_bool(tendst1507391555omplex(B_1_1,B_2_1,B_3_1))) # label(gsy_c_Limits_Otendsto_000tc__Complex__Ocomplex_000tc__Complex__Ocomplex) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 531 (all P all Q all R hAPP_nat_int(hAPP_int_fun_nat_int(hAPP_f1673103573at_int(cOMBC_nat_int_int,P),Q),R) = hAPP_int_int(hAPP_nat_fun_int_int(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Int__Oint_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 532 (all A_9 all N_14 all B_7 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_9),hAPP_nat_nat(suc,N_14))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_7),hAPP_nat_nat(suc,N_14)))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_7)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_9),B_7))))) # label(fact_3109_power__le__imp__le__base) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 533 (all K_1 all L (is_int(L) & is_int(K_1) -> (L = K_1 <-> hAPP_int_int(bit0,L) = hAPP_int_int(bit0,K_1)))) # label(fact_178_rel__simps_I48_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 534 (all Z_1 hAPP_complex_real(re,hAPP_complex_complex(sgn_sgn_complex,Z_1)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_complex_real(re,Z_1)),hAPP_complex_real(norm_norm_complex,Z_1))) # label(fact_4129_Re__sgn) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 535 (all X hAPP_real_real(abs_abs_real,hAPP_int_real(real_int,X)) = hAPP_int_real(real_int,hAPP_int_int(abs_abs_int,X))) # label(fact_4400_real__of__int__abs) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 536 (all P_3 all Q_2 divmod_int(P_3,Q_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,P_3),Q_2)),hAPP_int_int(div_mod_int(P_3),Q_2))) # label(fact_4197_divmod__int__mod__div) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 537 (all N_33 all A_123 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_123)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_33)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_123),N_33)))))) # label(fact_1485_one__less__power) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 538 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),M)) -> zero_zero_nat = M | N = M | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)),M)))) # label(fact_1689_divides__cases) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 539 (all Ma all K_1 all N_1 ((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),K_1))))) # label(fact_1684_mult__le__cancel2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 540 (all A_61 hAPP_C498520661umeral(div_mo1740067990umeral(zero_z126310315umeral),A_61) = zero_z126310315umeral) # label(fact_2374_mod__0) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 541 (all A_3 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),zero_zero_nat))) # label(fact_2308_gcd__lcm__complete__lattice__nat_Otop__greatest) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 542 (all B all A all C (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B),A) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),A) <-> C = B)) # label(fact_1932_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 543 (all A_3 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (exists X_2 (A_3 = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_2),N) & (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1)) & hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),N) = A_3 -> Y_1 = X_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2))))))) # label(fact_1827_realpow__pos__nth__unique) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 544 (all B_1_1 all B_2_1 is_bool(hAPP_rat_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__Rat__Orat_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 545 (all A_16 all B_12 (is_int(B_12) -> (B_12 != zero_zero_int -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_12),A_16)),B_12) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_16),B_12)),one_one_int)))) # label(fact_2628_div__add__self1) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 546 (all R_3 all P_3 all Q_2 (quotient_of(R_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Q_2)))) # label(fact_4658_quotient__of__denom__pos) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 547 (all X hAPP_real_real(sin,X) = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X))) # label(fact_3734_sin__cos__eq) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 548 (all X all M all Y hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(div_mod_int(X),M)),Y)),M) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y)),M)) # label(fact_2462_zdiff__zmod__left) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 549 (all A hBOOL(tendsto_complex_real(re,hAPP_complex_real(re,A),at_complex(A)))) # label(fact_5076_Re_Ocont) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 550 (all D all Sa exists Z all X_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Z)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_2),Sa))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_2),Sa)))))) # label(fact_2264_minf_I9_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 551 (all A_236 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_236),hAPP_int_complex(number528085621omplex,hAPP_int_int(bit1,pls))) = A_236) # label(fact_241_mult__numeral__1__right) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 552 (all B_90 all A_112 all C_50 (hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_50),A_112) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_90),A_112) -> B_90 = C_50)) # label(fact_1913_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 553 (all K_1 all F all N_1 all Ma (is_int(K_1) -> ((all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),N_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),I_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),one_one_nat))),hAPP_nat_int(F,I_2)))),one_one_int)))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,Ma)),K_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),hAPP_nat_int(F,N_1))) -> (exists I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),I_2)) & K_1 = hAPP_nat_int(F,I_2) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N_1)))))))))) # label(fact_3505_nat__intermed__int__val) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 554 (all N all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,M),N)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),M))))) # label(fact_454_zdvd__not__zless) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 555 (all A_38 all C_13 all B_29 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,C_13),B_29)) -> hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(div_mo231679042de_int(A_38),B_29)),C_13) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_38),C_13))) # label(fact_2446_mod__mod__cancel) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 556 (all X all Y -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_rat))) # label(fact_9_not__sum__power2__lt__zero) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 557 (all V_11 all W_9 hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_11),W_9)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_11)),hAPP_int_complex(number528085621omplex,W_9))) # label(fact_343_number__of__add) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 558 (all A_222 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,zero_z126310315umeral),A_222) = zero_z126310315umeral) # label(fact_586_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 559 (all N_13 all A_8 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_8)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_8),hAPP_nat_nat(suc,N_13)))))) # label(fact_3117_power__gt1) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 560 (all C_73 all A_173 all B_133 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_173),B_133)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_73)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_173),C_73)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_133),C_73)))))) # label(fact_1086_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 561 (all P all Q all R hAPP_r337325687l_real(hAPP_f1732848529l_real(hAPP_f1310479593l_real(cOMBB_2092576745l_real,P),Q),R) = hAPP_f556082547l_real(P,hAPP_r1250527377l_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc_) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 562 (all A_3 all B_2 all C_1 all D_3 hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),C_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_2),D_3)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)),hAPP_real_complex(hAPP_r265291036omplex(complex,C_1),D_3))) # label(fact_3962_complex__add) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 563 (all A_3 all B_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)) # label(fact_4307_gcd__semilattice__nat_Oinf_Oleft__idem) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 564 (all X exists N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_nat_real(real_nat,N_2)))) # label(fact_3952_reals__Archimedean2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 565 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(div_mod_nat(M),N)),N)))) # label(fact_2724_mod__le__divisor) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 566 (all C_80 all A_187 all B_143 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_187),B_143)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_187),C_80)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_187),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_143),C_80)))))) # label(fact_962_dvd__add) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 567 (all A_3 all P_3 (is_int(A_3) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),P_3)) -> A_3 = hAPP_int_int(inv(P_3),hAPP_int_int(inv(P_3),A_3)))))))) # label(fact_1893_inv__inv) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 568 (all Y all X ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X)) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y) = hAPP_int_int(nat_tsub(X),Y)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X)) -> hAPP_int_int(nat_tsub(X),Y) = zero_zero_int))) # label(fact_3232_tsub__def) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 569 (all Z_1 expi(Z_1) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(hAPP_real_real(exp_real,hAPP_complex_real(re,Z_1)))),cis(hAPP_complex_real(im,Z_1)))) # label(fact_4143_expi__def) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 570 (all W_8 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,W_8)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,W_8)),hAPP_int_real(number267125858f_real,W_8))) # label(fact_364_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 571 (all A_145 all M_20 all N_48 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_145),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_20),N_48)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_145),M_20)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_145),N_48))) # label(fact_1305_power__add) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 572 (all C_57 all D_10 all A_135 all B_104 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_135),B_104)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_57),D_10)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_104)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_57)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_135),C_57)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_104),D_10)))))))) # label(fact_1375_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 573 (all X hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = X) # label(fact_3367_real__sum__of__halves) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 574 (all A_234 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_234),A_234) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_234),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_302_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 575 (all P_1 all L ((exists X_2 (hBOOL(hAPP_Q2096512830t_bool(P_1,X_2)) & hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,L),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_2),zero_z891286103de_int))))) <-> (exists X_2 hBOOL(hAPP_Q2096512830t_bool(P_1,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,L),X_2)))))) # label(fact_1678_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 576 (all P_3 normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),zero_zero_int)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int)) # label(fact_4661_normalize__denom__zero) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 577 (all Lx_6 all Ly_4 all Rx_6 all Ry_4 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_6),Rx_6)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Ly_4),Ry_4)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_6),Ly_4)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_6),Ry_4))) # label(fact_468_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 578 (all A_3 all B_2 all K_2 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),B_2)),hAPP_int_int(div_mod_int(A_3),B_2))),K_2)) # label(fact_2361_zdiv__zmod__equality2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 579 (all V_2 ((-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_nat_real(real_nat,hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_real(number267125858f_real,V_2)) & (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_nat_real(real_nat,hAPP_int_nat(number_number_of_nat,V_2)) = zero_zero_real))) # label(fact_3700_real__of__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 580 (all X_2 hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),sin)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),cos)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_2),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X_2))),hAPP_real_real(sin,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X_2))),hAPP_real_real(sin,X_2)))))) # label(fact_4802_DERIV__sin__circle__all) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 581 (all B_132 all A_172 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_172),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_132),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_172),B_132)))))) # label(fact_1090_mult__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 582 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,natfloor(X))),one_one_real)))) # label(fact_3690_real__natfloor__add__one__gt) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 583 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X))),X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) # label(fact_3477_abs__ln__one__plus__x__minus__x__bound__nonpos) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 584 -(all S1 (is_int(S1) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,S1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))))) # label(fact_764__096_B_Bthesis_O_A_I_B_Bs1_O_A_091s1_A_094_A2_A_061_A_N1_093_A_Imod_A4_) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 585 (all K_2 hAPP_int_rat(hAPP_int_fun_int_rat(fract,one_one_int),hAPP_int_int(number_number_of_int,K_2)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,K_2))) # label(fact_4667_Fract__1__number__of) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 586 (all M all N all K_2 (one_one_int = hAPP_int_int(legacy_zgcd(N),K_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M))))) # label(fact_4517_zrelprime__zdvd__zmult) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 587 (all X_1 all Y_2 (zero_zero_rat = Y_2 & zero_zero_rat = X_1 <-> zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_15_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 588 (all A_62 A_62 = hAPP_C498520661umeral(div_mo1740067990umeral(A_62),zero_z126310315umeral)) # label(fact_2370_mod__by__0) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 589 (all V_9 all U_2 all Y_19 all X_25 all A_127 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_25),A_127)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_19),A_127)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),U_2)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),V_9)) -> (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,U_2),V_9) = one_one_rat -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,U_2),X_25)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,V_9),Y_19))),A_127)))))))) # label(fact_1466_convex__bound__le) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 590 (all X X = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),X)) # label(fact_4338_gcd__0__left__nat) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 591 (all C_44 all D_8 all A_102 all B_84 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_102),B_84)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_44),D_8)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_102),C_44)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_84),D_8)))))) # label(fact_2005_add__mono) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 592 (all M all N all S_3 all S_1 all T_2 all T_1 all R_5 all R_3 all K_2 (K_2 = hAPP_int_int(legacy_zgcd(R_5),R_3) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),R_3)) -> (exists Sn exists Tn (is_int(Sn) & is_int(Tn) & hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,K_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Sn),Tn)) = xzgcda(M,N,R_5,R_3,S_3,S_1,T_2,T_1)))))) # label(fact_4535_xzgcd__correct__aux1) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 593 (all C_85 all A_193 all B_149 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_193),B_149)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_193),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_149),C_85))))) # label(fact_922_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 594 (all P_1 all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) -> (all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2))),hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),hAPP_nat_nat(suc,N_2)))))))) # label(fact_4245_Bolzano__bisect__fst__le__Suc) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 595 (all P all Q all R hAPP_int_int(P,hAPP_int_int(Q,R)) = hAPP_int_int(hAPP_f1356864277nt_int(hAPP_f747162069nt_int(cOMBB_int_int_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__Int__Oint_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 596 (all X_44 all Y_32 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_44),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_32),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_32)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_44),Y_32))))) # label(fact_218_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 597 (all N hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N)))),zero_zero_int))) # label(fact_3548_negative__zless__0) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 598 (all Z_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_2),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z_2)),Z_2)),zero_zero_int)))) # label(fact_392_odd__less__0) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 599 (all X_8 all Q_7 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_8),hAPP_nat_nat(suc,Q_7)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_8),Q_7)),X_8)) # label(fact_3018_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 600 (all M_9 all N_23 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_23)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_9)),hAPP_nat_int(semiri1621563631at_int,N_23))))) # label(fact_2940_less__imp__of__nat__less) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 601 (all C_88 all A_206 all B_157 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_206),B_157)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,B_157),C_88)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_206),C_88))))) # label(fact_802_dvd__trans) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 602 (all A_103 all C_45 all B_85 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_103),C_45)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_85),C_45))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_103),B_85)))) # label(fact_1999_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 603 (all X all A_3 all B_2 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_2)) = hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_3),B_2))) # label(fact_4413_powr__divide2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 604 (all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Y_2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,Y_2))))) # label(fact_396_less__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 605 (all X_1 all Y_2 (hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_1),Y_2))) <-> -hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)) & -hBOOL(hAPP_nat_bool(even_odd_even_nat,Y_2)) | hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)) & hBOOL(hAPP_nat_bool(even_odd_even_nat,Y_2)))) # label(fact_3757_even__sum__nat) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 606 (all L (is_int(L) -> (pls = hAPP_int_int(bit0,L) <-> L = pls))) # label(fact_254_rel__simps_I38_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 607 (all D all Sa exists Z all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z),X_2)) -> (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Sa))) <-> -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Sa)))))) # label(fact_2254_pinf_I10_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 608 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,X_1)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real)))) # label(fact_3328_real__sqrt__lt__0__iff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 609 (all A_3 all J_2 all K_2 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),zero_zero_int),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),zero_zero_int),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),K_2))),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),J_2))),P_3)))))))) # label(fact_2237_MultInv__zcong__prop2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 610 (all M_11 -hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_11)),zero_z891286103de_int))) # label(fact_2917_of__nat__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 611 (all A_231 all B_165 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_165),A_231) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_231),B_165)) # label(fact_518_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.02 612 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(cos,X)),one_one_real))) # label(fact_3562_cos__le__one) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 613 (all P all Q all R hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(P,R),Q) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(hAPP_f465821005nt_int(cOMBC_1964283556nt_int,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__prod_Itc__Int__Oin) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 614 (all A_91 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_91),one_one_complex) = A_91) # label(fact_2109_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 615 (all A_3 all B_2 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_3)),B_2))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_416_zadd__power2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 616 (all N_1 all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),Ma)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(fact_fact_nat,Ma)),hAPP_nat_nat(fact_fact_nat,N_1)) = big_co1705425894at_nat(cOMBI_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),one_one_nat)),Ma)))) # label(fact_5083_fact__div__fact) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 617 (all C_58 all D_11 all A_136 all B_105 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_136),B_105)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_58),D_11)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_136)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_58)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_136),C_58)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_105),D_11)))))))) # label(fact_1366_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 618 (all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,N_1))))) # label(fact_2946_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 619 (all A_3 all B_2 all P_3 all Q_2 (hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2) = quotient_of(hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)) -> hAPP_int_rat(hAPP_int_fun_int_rat(fract,P_3),Q_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2))) # label(fact_4669_quotient__of__eq) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 620 (all Y all X all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Y),zero_zero_int),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),X)),hAPP_int_int(multInv(P_3),Y)),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),Y),P_3)))))))) # label(fact_2233_MultInv__prop5) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 621 (all A_205 all B_156 (zero_z891286103de_int = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_205),B_156) -> A_205 = zero_z891286103de_int | B_156 = zero_z891286103de_int)) # label(fact_826_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 622 (all X_1 all Y_2 (Y_2 = X_1 <-> hAPP_nat_int(semiri1621563631at_int,X_1) = hAPP_nat_int(semiri1621563631at_int,Y_2))) # label(fact_3256_Nat__Transfer_Otransfer__int__nat__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 623 (all A_25 all B_20 (zero_zero_nat != B_20 -> A_25 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_25),B_20)),B_20))) # label(fact_2546_div__mult__self2__is__id) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 624 (all B_71 all A_83 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_83)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_71)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_83),B_71)))))) # label(fact_2166_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 625 (all X hAPP_real_real(arctan,hAPP_real_real(uminus_uminus_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(arctan,X))) # label(fact_3481_arctan__minus) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 626 (all B_96 all A_118 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_118)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_96)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_118),B_96)))))) # label(fact_1864_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 627 (all B all A all C (C = B <-> hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C),A) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B),A))) # label(fact_1929_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 628 (all P all Q all R hAPP_f617735571l_real(hAPP_r712953929l_real(P,R),Q) = hAPP_r2019696015l_real(hAPP_f1591894897l_real(hAPP_f644105375l_real(cOMBC_467724138l_real,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 629 (all A_225 all B_162 all C_91 all D_18 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_225),B_162)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_91),D_18)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_225),C_91)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_162),D_18))) # label(fact_554_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 630 (all A_12 all N_18 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_12),N_18)),A_12) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_12),hAPP_nat_nat(suc,N_18))) # label(fact_3009_power__Suc2) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 631 (all B_43 all B_42 all A_52 all C_23 all A_51 (hAPP_Q1762011733de_int(div_mo231679042de_int(A_52),C_23) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_51),C_23) -> (hAPP_Q1762011733de_int(div_mo231679042de_int(B_42),C_23) = hAPP_Q1762011733de_int(div_mo231679042de_int(B_43),C_23) -> hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_51),B_42)),C_23) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_52),B_43)),C_23)))) # label(fact_2408_mod__add__cong) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 632 (all A_3 all B_2 hAPP_int_rat(ring_1_of_int_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(sgn_sgn_int,A_3)),hAPP_int_int(sgn_sgn_int,B_2))) = hAPP_rat_rat(sgn_sgn_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2))) # label(fact_4715_sgn__rat) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 633 (all Y all X (hBOOL(hAPP_int_bool(even_odd_even_int,X)) -> (-hBOOL(hAPP_int_bool(even_odd_even_int,Y)) -> -hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y)))))) # label(fact_2889_Parity_Oeven__plus__odd) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 634 (all P all Q all R hAPP_r1230691443l_real(hAPP_f324002831l_real(hAPP_f677014621l_real(cOMBS_107495110l_real,P),Q),R) = hAPP_f1293550189l_real(hAPP_r1043989325l_real(P,R),hAPP_r2019696015l_real(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__p_042) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 635 (all B_1_1 all B_2_1 (is_int(B_2_1) -> is_int(hAPP_int_int(B_1_1,B_2_1)))) # label(gsy_c_hAPP_000tc__Int__Oint_000tc__Int__Oint) # label(hypothesis) # label(non_clause). [assumption]. 6.94/7.03 636 (all P_1 all N_1 all K_1 (is_int(K_1) & is_int(N_1) -> (hBOOL(hAPP_int_bool(P_1,hAPP_int_int(div_mod_int(N_1),K_1))) <-> (zero_zero_int = K_1 -> hBOOL(hAPP_int_bool(P_1,N_1))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1)) -> (all I_2 all J_1 (is_int(I_2) & is_int(J_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,J_1),K_1)) & N_1 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),J_1)) -> hBOOL(hAPP_int_bool(P_1,J_1)))))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),zero_zero_int)) -> (all I_2 all J_1 (is_int(I_2) & is_int(J_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),J_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_1),zero_zero_int)) & N_1 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) -> hBOOL(hAPP_int_bool(P_1,J_1))))))))) # label(fact_2660_split__zmod) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 637 (all Code_numeral_3 code_S1047413653umeral(Code_numeral_3) != zero_z126310315umeral) # label(fact_4081_code__numeral_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 638 (all Ma all N_1 (N_1 = Ma <-> hAPP_nat_complex(semiri2020571505omplex,N_1) = hAPP_nat_complex(semiri2020571505omplex,Ma))) # label(fact_2833_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 639 (all A_226 all B_163 all C_92 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_226),B_163)),C_92) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_226),C_92)),B_163)) # label(fact_553_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 640 (all I_1 all J image_nat_nat(suc,hAPP_n1699378549t_bool(ord_at4362885an_nat(I_1),J)) = hAPP_n1699378549t_bool(ord_at4362885an_nat(hAPP_nat_nat(suc,I_1)),hAPP_nat_nat(suc,J))) # label(fact_4907_image__Suc__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 641 (all F all L (hBOOL(tendsto_nat_real(F,L,sequentially)) -> hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F),hAPP_real_real(abs_abs_real,L),sequentially)))) # label(fact_5040_LIMSEQ__imp__rabs) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 642 (all C all B all A (zero_zero_nat != A -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C),A) = B <-> C = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B),A))))) # label(fact_2643_dvd__div__eq__mult) # label(axiom) # label(non_clause). [assumption]. 6.94/7.03 643 (all W_1 all Y_2 all X_1 all Z_2 (W_1 = X_1 | Z_2 = Y_2 <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,W_1),Z_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),Y_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,W_1),Y_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),Z_2)))) # label(fact_643_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 644 (all N all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,N),P_3)) -> one_one_int = hAPP_int_int(legacy_zgcd(N),P_3))))) # label(fact_4524_zless__zprime__imp__zrelprime) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 645 (all C_66 all A_144 all B_113 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_66),A_144)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_66),B_113))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_66)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_144),B_113))))) # label(fact_1322_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 646 (all N all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N)),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N)),one_one_real))) -> archim856651990g_real(X) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N),one_one_int)))) # label(fact_4474_ceiling__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 647 (all P all Q all R hAPP_r337325687l_real(hAPP_f1644353557l_real(hAPP_f1874145443l_real(cOMBB_1768400739l_real,P),Q),R) = hAPP_f203520653l_real(P,hAPP_r337325687l_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 648 (all A_3 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M)) -> (exists X_2 (is_int(X_2) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),M)) & (all Y_1 (is_int(Y_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),M)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),Y_1),M)) -> Y_1 = X_2))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),X_2),M)))))) # label(fact_1695_zcong__zless__unique) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 649 (all G all X_1 all Ma (hBOOL(hAPP_real_bool(deriv_real(G,X_1),Ma)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,cos),G),X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,hAPP_real_real(G,X_1)))),Ma))))) # label(fact_4761_DERIV__fun__cos) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 650 (all B_28 all A_34 zero_zero_nat = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_28),A_34)),B_28)) # label(fact_2492_mod__mult__self1__is__0) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 651 (all A_221 zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_221),zero_zero_int)) # label(fact_590_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 652 (all R_3 all P_3 all Q_2 (normalize(R_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Q_2)))) # label(fact_4660_normalize__denom__pos) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 653 (all B_140 all A_180 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_140),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_180),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_180)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_140)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_180),B_140))))) # label(fact_1050_split__mult__pos__le) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 654 (all M_19 all N_44 all A_132 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_132)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_132),M_19)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_132),N_44))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_19),N_44))))) # label(fact_1411_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 655 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (zero_zero_real = hAPP_real_real(sqrt,X) -> zero_zero_real = X))) # label(fact_3322_real__sqrt__eq__zero__cancel) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 656 (all A_201 all E_3 all B_154 all C_87 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_201),B_154)),E_3)),C_87) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_201),E_3)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_154),E_3)),C_87))) # label(fact_870_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 657 (all X_39 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_39),X_39) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_39),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_435_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 658 (all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),zero_zero_int)))) # label(fact_306_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 659 (all V_20 all W_16 all Z_15 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_20),W_16))),Z_15) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_20)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,W_16)),Z_15))) # label(fact_81_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 660 (all P all Q all R hAPP_r2052264067t_bool(hAPP_f77155443t_bool(hAPP_f403082607t_bool(cOMBB_984071017l_real,P),Q),R) = hAPP_f743790483t_bool(P,hAPP_r279757445t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_Mt) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 661 (all K_2 all L_2 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,K_2)),hAPP_int_int(bit1,L_2)) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),hAPP_int_int(succ,L_2)))) # label(fact_3297_add__Bit1__Bit1) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 662 (all C all B all A (A != zero_z126310315umeral -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A),B)) -> (hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B),A) = C <-> B = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C),A))))) # label(fact_2640_dvd__div__eq__mult) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 663 (all X all Y hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,X),Y)) = hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,hAPP_complex_complex(cnj,X)),hAPP_complex_complex(cnj,Y))) # label(fact_4079_complex__cnj__divide) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 664 (all A (hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially)) -> (hBOOL(monoseq_real(A)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(A,zero_zero_nat)),zero_zero_real)) -> (all N_2 hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_r1134773055l_bool(ord_at1589558736t_real(hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_2)),one_one_nat)))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_2))))))))))) # label(fact_5062_summable__Leibniz_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 665 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,Y)),hAPP_real_real(sin,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,Y)),hAPP_real_real(cos,X))) = hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y))) # label(fact_3666_sin__diff2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 666 (all A_3 all C_1 hAPP_int_rat(hAPP_int_fun_int_rat(fract,zero_zero_int),A_3) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,zero_zero_int),C_1)) # label(fact_4675_eq__rat_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 667 (all A (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),zero_zero_nat)) <-> zero_zero_nat != A)) # label(fact_2309_gcd__lcm__complete__lattice__nat_Oless__top) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 668 (all Ma all N_1 ((exists K N_1 = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),K))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)))) # label(fact_3054_less__iff__Suc__add) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 669 (all Z_2 all Y_2 all X_1 all X_3 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),X_3)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),X_1)) -> ((all X_2 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),X_3)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Z_2)))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),comple124823625p_real(X_3))))))) # label(fact_4648_SupInf_OSup__upper2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 670 (all A_3 all B_2 hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(uminus_uminus_int,A_3)),hAPP_int_int(uminus_uminus_int,B_2))) # label(fact_3523_zdiv__zminus__zminus) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 671 (all X_46 all Y_34 (is_int(X_46) & is_int(Y_34) -> (hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_34),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_46),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_46)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_34)) -> Y_34 = X_46))))) # label(fact_202_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 672 (all A_36 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,zero_z891286103de_int),A_36) = zero_z891286103de_int) # label(fact_2477_div__0) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 673 (all R_3 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,R_3),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,archim1246769320r_real(R_3))),one_one_real)))) # label(fact_4451_real__of__int__floor__add__one__ge) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 674 (all X_1 all N_1 exists T hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_r195310020l_real(hAPP_f2126667875l_real(cOMBC_746811033l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),zero_zero_real)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,div_div_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,minus_minus_nat),hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,T),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(real_nat,N_1))),pi)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1))) = hAPP_real_real(sin,X_1)) # label(fact_4923_Maclaurin__sin__expansion) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 675 (all Ma all N_1 all A (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A)) -> (Ma = N_1 <-> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A),N_1) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A),Ma)))) # label(fact_1287_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 676 (all N_1 hAPP_nat_real(cos_coeff,N_1) = hAPP_nat_real(diffs_real(sin_coeff),N_1)) # label(fact_4874_sin__fdiffs2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 677 (all X_40 all N_69 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_40),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_40),N_69)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_40),N_69))) # label(fact_425_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 678 (all J_2 all K_2 all A_3 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,J_2),K_2)),A_3),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(multInv(P_3),J_2)),J_2)),K_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(multInv(P_3),J_2)),A_3)),P_3)))) # label(fact_2222_aux______3) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 679 (all B_1_1 all B_2_1 is_bool(hAPP_complex_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__Complex__Ocomplex_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 680 (all X_1 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,X_1)),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_2))))) # label(fact_3493_real__0__less__add__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 681 (all M_17 all N_30 ((N_30 = zero_zero_nat -> one_one_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,M_17),N_30)) & (N_30 != zero_zero_nat -> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,M_17),N_30) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,M_17),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat)))))) # label(fact_1831_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 682 (all A_3 all M all B_2 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),M)),M))) # label(fact_1031_zcong__zmult__self) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 683 (all A all C all B (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),C))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B)) | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),zero_zero_rat)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B),A)))) # label(fact_1199_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 684 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,pi)),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(cos,Y)),hAPP_real_real(cos,X))))))) # label(fact_3578_cos__monotone__minus__pi__0) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 685 (all X hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3730_sin__squared__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 686 (all A_49 all B_40 all C_21 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_49),B_40)),C_21) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_49),hAPP_C498520661umeral(div_mo1740067990umeral(B_40),C_21))),C_21)) # label(fact_2415_zmod__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 687 (all K_2 all I all J_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,K_2)),I)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,K_2)),J_2)))))) # label(fact_3152_zmult__zless__mono2__lemma) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 688 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(re,X)),hAPP_complex_real(im,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(im,X)),hAPP_complex_real(re,Y))) = hAPP_complex_real(im,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X),Y))) # label(fact_4130_complex__Im__mult) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 689 (all D all B all C all A (zero_z891286103de_int != A -> (zero_z891286103de_int != C -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A),B)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,C),D)) -> (hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B),C) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A),D) <-> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,D),C) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B),A))))))) # label(fact_2637_dvd__div__div__eq__mult) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 690 (all X_1 all Y_2 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_rat)) <-> zero_zero_rat = Y_2 & X_1 = zero_zero_rat)) # label(fact_225_sum__power2__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 691 (all A_88 all B_76 hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_88),B_76)),B_76) = A_88) # label(fact_2134_add__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 692 (all Lx_4 all Ly_2 all Rx_4 all Ry_2 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_4),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Ly_2),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_4),Ry_2))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_4),Ly_2)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_4),Ry_2))) # label(fact_482_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 693 (all A_1 all K_1 (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),hAPP_n1699378549t_bool(ord_at4362885an_nat(K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),hAPP_f22106695ol_nat(finite_card_nat,A_1))))) -> A_1 = hAPP_n1699378549t_bool(ord_at4362885an_nat(K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),hAPP_f22106695ol_nat(finite_card_nat,A_1))))) # label(fact_4909_subset__card__intvl__is__intvl) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 694 (all N_43 all A_131 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_131)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_131),N_43)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_131),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_131),N_43)))))) # label(fact_1421_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 695 (all P all Q all R hAPP_r712953929l_real(hAPP_f1108539711l_real(hAPP_f1109256153l_real(cOMBB_44084907l_real,P),Q),R) = hAPP_f1730463541l_real(P,hAPP_r579619957l_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__RealDef__Oreal_044) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 696 (all C_57 all D_10 all A_135 all B_104 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_135),B_104)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_57),D_10)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_104)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_57)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_135),C_57)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_104),D_10)))))))) # label(fact_1374_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 697 (all C_63 all A_141 all B_110 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_63),A_141)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_63),B_110))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_63)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_141),B_110))))) # label(fact_1341_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 698 (all F all J all N_1 all H (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H)) -> (exists B_6 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),J)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_6),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N_1)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1))))) = hAPP_real_real(F,H)))) # label(fact_4928_Maclaurin__lemma) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 699 (all M all N hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)),hAPP_nat_nat(div_mod_nat(M),N))))) # label(fact_4220_divmod__nat__rel) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 700 (all K_2 hAPP_nat_rat(semiri151668891at_rat,K_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_nat_int(semiri1621563631at_int,K_2)),one_one_int)) # label(fact_4677_Fract__of__nat__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 701 (all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),A_3)) -> -(-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),A_3))))) # label(fact_1726_dvd_Oless__asym_H) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 702 (all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,pls)),Y_2)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,Y_2))))) # label(fact_123_le__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 703 (all C_34 all D_6 all A_80 all B_68 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_80),B_68)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_34),D_6)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_80),C_34)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_68),D_6)))))) # label(fact_2198_add__le__less__mono) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 704 (all A_110 all B_88 all C_48 (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_110),C_48) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_110),B_88) -> B_88 = C_48)) # label(fact_1928_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 705 (all P_1 all N_1 all K_1 (hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(div_mod_nat(N_1),K_1))) <-> (K_1 = zero_zero_nat -> hBOOL(hAPP_nat_bool(P_1,N_1))) & (zero_zero_nat != K_1 -> (all I_2 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),K_1)) -> (N_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),I_2)),J_1) -> hBOOL(hAPP_nat_bool(P_1,J_1)))))))) # label(fact_2735_split__mod) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 706 (all A_107 A_107 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_107),zero_z891286103de_int)) # label(fact_1973_add__0__right) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 707 (all M_11 -hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(semiri1619134803umeral,M_11)),zero_z126310315umeral))) # label(fact_2918_of__nat__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 708 (all V_3 all V_2 ((hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_nat_nat(div_mod_nat(hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = zero_zero_nat) & (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3))) -> hAPP_nat_nat(div_mod_nat(hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(number_number_of_nat,V_2)) & (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3))) -> hAPP_int_nat(nat,hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,V_3))) = hAPP_nat_nat(div_mod_nat(hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)))))) # label(fact_3285_mod__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 709 (all K_2 ((K_2 != zero_z891286103de_int -> (hAPP_Q1762011733de_int(div_mo231679042de_int(K_2),hAPP_i1732201573de_int(number1226105091de_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) != zero_z891286103de_int -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),quickcheck_int_of(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,K_2),hAPP_i1732201573de_int(number1226105091de_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),one_one_int) = quickcheck_int_of(K_2)) & (hAPP_Q1762011733de_int(div_mo231679042de_int(K_2),hAPP_i1732201573de_int(number1226105091de_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_z891286103de_int -> quickcheck_int_of(K_2) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),quickcheck_int_of(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,K_2),hAPP_i1732201573de_int(number1226105091de_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) & (zero_z891286103de_int = K_2 -> quickcheck_int_of(K_2) = zero_zero_int))) # label(fact_4250_Quickcheck__Narrowing_Oint__of__code) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 710 (all D all K_1 all P_5 (hBOOL(hAPP_nat_bool(prime,P_5)) -> ((exists I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),K_1)) & hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_5),I_2) = D)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_5),K_1)))))) # label(fact_4185_divides__primepow) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 711 (all X_1 all Y_2 (is_int(X_1) & is_int(Y_2) -> (hAPP_int_real(real_int,Y_2) = hAPP_int_real(real_int,X_1) <-> X_1 = Y_2))) # label(fact_4385_real__of__int__inject) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 712 (all K_1 all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,K_1)),hAPP_int_int(number_number_of_int,L))))) # label(fact_57_less__eq__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 713 (all V_3 hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,one_one_int),hAPP_int_int(number_number_of_int,V_3))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_int_nat(number_number_of_nat,V_3))) # label(fact_3241_one__div__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 714 (all A_3 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(legendre(A_3),P_3)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),P_3))))) # label(fact_2768_Euler__Criterion) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 715 (all A_109 all B_87 all C_47 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_109),B_87)),C_47) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_109),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_87),C_47))) # label(fact_1944_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 716 (all X_1 all Y_2 (X_1 = zero_zero_real & zero_zero_real = Y_2 <-> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),X_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Y_2)) = zero_zero_real)) # label(fact_1786_real__two__squares__add__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 717 (all I_1 all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,I_1),K_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_1))))) # label(fact_2605_pos__imp__zdiv__pos__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 718 (all A_113 all B_91 all C_51 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_113),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_91),C_51)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_113),B_91)),C_51)) # label(fact_1907_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 719 (all N archim1246769320r_real(hAPP_nat_real(real_nat,N)) = hAPP_nat_int(semiri1621563631at_int,N)) # label(fact_4020_floor__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 720 (all A_3 all P_3 (is_int(A_3) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),A_3)),one_one_int),P_3)) -> one_one_int = A_3 | hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int) = A_3)))))) # label(fact_1604_zcong__square__zless) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 721 (all B_117 all A_156 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_156)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_117),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_156),B_117)),zero_zero_real))))) # label(fact_1181_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 7.03/7.03 722 (all B all Q_1 all R_2 hAPP_P1975530577nt_int(adjust(B),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_2)) = hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,R_2),B))),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Q_1)),one_one_int)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,R_2),B))),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Q_1)),R_2))) # label(fact_4730_adjust__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 723 (all X all Y hAPP_complex_real(im,hAPP_real_complex(hAPP_r265291036omplex(complex,X),Y)) = Y) # label(fact_4092_Im) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 724 (all A_205 all B_156 (is_int(B_156) & is_int(A_205) -> (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_205),B_156) = zero_zero_int -> B_156 = zero_zero_int | A_205 = zero_zero_int))) # label(fact_821_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 725 (all R_3 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,R_3),hAPP_int_real(real_int,archim856651990g_real(R_3))))) # label(fact_4402_real__of__int__ceiling__ge) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 726 (all Diff all N_1 all X_1 (X_1 = zero_zero_real -> (zero_zero_nat != N_1 -> hAPP_real_real(hAPP_n546711566l_real(Diff,zero_zero_nat),zero_zero_real) = hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))))) # label(fact_4908_Maclaurin__zero) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 727 (all X_39 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_39),X_39) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_39),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_437_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 728 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)))) # label(fact_2875_Suc__leD) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 729 (all M all N hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N))) # label(fact_3000_int__mult) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 730 (all P all Q all R hAPP_f41603166t_bool(P,hAPP_r373117745t_bool(Q,R)) = hAPP_r481142414t_bool(hAPP_f1341793266t_bool(hAPP_f571820015t_bool(cOMBB_1159313332l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Nat__Onat_Mtc__HOL_) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 731 (all F all N_1 all R_2 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),R_2)) = hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,minus_minus_real),F)),R_2)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))) # label(fact_4897_sumr__diff__mult__const) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 732 (all X_1 all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_2))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2)))) # label(fact_3045_zdvd__int) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 733 (all X_3 all A (hBOOL(hAPP_complex_bool(sums_complex(X_3),A)) -> hBOOL(hAPP_real_bool(sums_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_3)),hAPP_complex_real(im,A))))) # label(fact_4873_Im_Osums) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 734 (all Y_2 all X_1 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_2)),zEven))))) # label(fact_2899_even__minus__even) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 735 (all X_1 (-hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)))) # label(fact_2900_not__odd__impl__even) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 736 (all A_207 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_207),A_207))) # label(fact_800_dvd__refl) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 737 (all Ma hAPP_i1948725293t_bool(bnorRset(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ma),one_one_int)),Ma) = hAPP_i1948725293t_bool(norRRset,Ma)) # label(fact_4575_norRRset__def) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 738 (all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,Ma)),hAPP_nat_rat(semiri151668891at_rat,N_1))))) # label(fact_2942_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 739 (all A_47 all B_38 all C_19 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_47),B_38)),C_19) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_47),C_19)),B_38)),C_19)) # label(fact_2422_mod__add__left__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 740 (all A_184 all M_23 all N_58 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_23),N_58)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_184),M_23)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_184),N_58))))) # label(fact_1008_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 741 (all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),pls)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,X_1)),zero_zero_real)))) # label(fact_394_less__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 742 (all X all M all Y hAPP_int_int(div_mod_int(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(div_mod_int(X),M)),Y)),M) = hAPP_int_int(div_mod_int(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Y)),M)) # label(fact_2469_zpower__zmod) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 743 (all N_65 all A_199 (is_int(A_199) -> (zero_zero_int != A_199 -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_199),N_65) != zero_zero_int))) # label(fact_881_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 744 (all A_129 all N_40 all N_39 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_40),N_39)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_129)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_129),N_40)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_129),N_39)))))) # label(fact_1442_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 745 (all N_1 all Ma hAPP_nat_nat(nat_case_nat(hAPP_nat_nat(suc,N_1),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,suc),hAPP_nat_fun_nat_nat(ord_max_nat,N_1))),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(suc,N_1)),Ma)) # label(fact_5188_max__Suc1) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 746 (all B_1_1 all B_2_1 is_int(hAPP_f1594865479ol_int(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__Int__Oint) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 747 (all P_1 all N_1 all K_1 (hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N_1),K_1))) <-> (K_1 = zero_zero_nat -> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))) & (zero_zero_nat != K_1 -> (all I_2 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),K_1)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),I_2)),J_1) = N_1 -> hBOOL(hAPP_nat_bool(P_1,I_2)))))))) # label(fact_2733_split__div) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 748 (all A_58 all B_49 all C_28 hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_58),B_49)),C_28) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_58),C_28)),hAPP_Q1762011733de_int(div_mo231679042de_int(B_49),C_28))),C_28)) # label(fact_2388_mod__mult__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 749 (all P all Q all R hAPP_f448129468l_bool(P,hAPP_r481142414t_bool(Q,R)) = hAPP_real_bool(hAPP_f101635158l_bool(hAPP_f1925507189l_bool(cOMBB_1121934354l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obo_006) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 750 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int)))) # label(fact_2207_le__iff__diff__le__0) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 751 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N) = hAPP_P1654798560at_nat(product_fst_nat_nat,divmod_nat(M,N))) # label(fact_4239_div__nat__def) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 752 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> M = hAPP_nat_nat(div_mod_nat(M),N))) # label(fact_2691_mod__less) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 753 (all X_3 (hBOOL(cauchy_complex(X_3)) -> hBOOL(cauchy_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_3))))) # label(fact_4756_cnj_OCauchy) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 754 (all N_1 all X_1 all Diff all F (hAPP_n546711566l_real(Diff,zero_zero_nat) = F -> ((all M_1 all X_2 hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),X_2),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),X_2)))) -> (exists T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_1))) & hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N_1),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1))) = hAPP_real_real(F,X_1)))))) # label(fact_4941_Maclaurin__all__le) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 755 (all A_27 all B_22 all C_9 (zero_z891286103de_int != C_9 -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_27),C_9)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_22),C_9)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_27),B_22))) # label(fact_2540_div__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 756 (all P_1 all A0 (hBOOL(hAPP_int_bool(accp_int(quickc1265749348ro_rel),A0)) -> ((all I_2 (is_int(I_2) -> (hBOOL(hAPP_int_bool(accp_int(quickc1265749348ro_rel),I_2)) -> ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),zero_zero_int)) -> (zero_zero_int != I_2 -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I_2),one_one_int))))) -> hBOOL(hAPP_int_bool(P_1,I_2)))))) -> hBOOL(hAPP_int_bool(P_1,A0))))) # label(fact_4270_around__zero_Opinduct) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 757 (all A_208 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_208),zero_zero_nat))) # label(fact_763_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 758 (all R_3 all P_3 all Q_2 (hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2) = quotient_of(R_3) -> one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,P_3),Q_2))) # label(fact_5017_quotient__of__coprime) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 759 (all K_2 all I all J_2 (one_one_int = hAPP_int_int(legacy_zgcd(I),J_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),J_2))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),K_2))))) # label(fact_4518_zrelprime__dvd__mult) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 760 (all X all Y all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(standardRes(M),X)),hAPP_int_int(standardRes(M),Y))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)),M)))) # label(fact_2320_StandardRes__prop4) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 761 (all D_3 all A_3 all B_2 (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)) -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),B_2) = one_one_int)) # label(fact_4996_coprime__rmult__int) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 762 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_3)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),A_3))) # label(fact_2324_pos__zdiv__mult__2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 763 (all A_188 all B_144 all C_81 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_188),B_144)),C_81)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,B_144),C_81)))) # label(fact_957_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 764 (all Lx_4 all Ly_2 all Rx_4 all Ry_2 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_4),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Ly_2),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx_4),Ry_2))) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_4),Ly_2)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx_4),Ry_2))) # label(fact_477_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 765 (all V_2 all W hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W)))) # label(fact_2651_zmod__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 766 (all A_149 hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_149),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_149),one_on1684967323de_int)))) # label(fact_1261_less__add__one) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 767 (all X_1 all Y_2 (Y_2 = zero_zero_real & X_1 = zero_zero_real <-> zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_14_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 768 (all P all Q all R hAPP_i627165055t_real(hAPP_f1323932638t_real(hAPP_f352226557t_real(cOMBC_824100275t_real,P),Q),R) = hAPP_f269654879t_real(hAPP_i1671359920t_real(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__RealDef__Or) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 769 (all N_56 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_int_rat(number_number_of_rat,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_56)) = one_one_rat) # label(fact_1043_power__m1__even) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 770 (all X_1 (hAPP_int_real(real_int,archim856651990g_real(X_1)) = X_1 <-> (exists N_2 (hAPP_int_real(real_int,N_2) = X_1 & is_int(N_2))))) # label(fact_4398_real__of__int__ceiling__cancel) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 771 (all X all M hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),hAPP_int_int(div_mod_int(X),M)),M))) # label(fact_2464_mod__mod__is__mod) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 772 (all A_217 all M_29 all B_160 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_217),B_160)),M_29) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_217),M_29)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_160),M_29))) # label(fact_636_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 773 (all Ma all N_1 all K_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1))))) # label(fact_1847_nat__mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 774 (all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) -> X = Y))) # label(fact_1712_dvd_Oantisym) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 775 (all A_3 all B_2 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(ring_1_of_int_rat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2))) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)),hAPP_int_rat(ring_1_of_int_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),one_one_int)))))) # label(fact_4726_rat__floor__lemma) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 776 (all B_141 all A_181 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_181)) & hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_141),zero_z891286103de_int)) | hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_181),zero_z891286103de_int)) & hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_141)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_181),B_141)),zero_z891286103de_int)))) # label(fact_1048_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 777 (all V_2 all W hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W))) # label(fact_2525_zdiv__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 778 (all K_1 all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(pred,K_1)),L)))) # label(fact_4042_le__iff__pred__less) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 779 (all X X = hAPP_real_real(ln,hAPP_real_real(exp_real,X))) # label(fact_3822_ln__exp) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 780 (all A_215 (is_int(A_215) -> A_215 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_215),one_one_int))) # label(fact_654_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 781 (all Code_numeral_5 all Code_numeral_4 (code_S1047413653umeral(Code_numeral_5) = code_S1047413653umeral(Code_numeral_4) <-> Code_numeral_5 = Code_numeral_4)) # label(fact_4074_code__numeral_Oinject) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 782 (all N_1 all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),Ma)),Ma)) <-> one_one_nat = N_1))) # label(fact_1573_dvd__mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 783 (all Lx_2 all Ly all Rx_2 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_2),Ly)),Rx_2) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ly),Rx_2))) # label(fact_492_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 784 (all A_12 all N_18 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_12),hAPP_nat_nat(suc,N_18)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_12),N_18)),A_12)) # label(fact_3004_power__Suc2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 785 (all K_2 hAPP_int_int(bit1,hAPP_int_int(pred,K_2)) = hAPP_int_int(pred,hAPP_int_int(bit0,K_2))) # label(fact_4041_pred__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 786 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arccos,Y)),pi)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(arccos,Y)))))) # label(fact_3618_arccos__lt__bounded) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 787 (all Ma all N_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,Ma)),hAPP_nat_real(semiri132038758t_real,N_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)))) # label(fact_2944_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 788 (all A_44 all B_35 hAPP_Q1762011733de_int(div_mo231679042de_int(A_44),B_35) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_44),B_35)),B_35)) # label(fact_2436_mod__add__self2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 789 (all J_2 all I -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),I)),I))) # label(fact_1625_not__add__less2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 790 (all X all N all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N)))))) # label(fact_4173_prime__divexp__n) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 791 (all X_1 all Y_2 (hBOOL(hAPP_int_bool(even_odd_even_int,Y_2)) & hBOOL(hAPP_int_bool(even_odd_even_int,X_1)) | -hBOOL(hAPP_int_bool(even_odd_even_int,X_1)) & -hBOOL(hAPP_int_bool(even_odd_even_int,Y_2)) <-> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_2))))) # label(fact_2891_even__sum) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 792 (all N_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,quickcheck_int_of(N_1)),quickcheck_int_of(Ma))) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,N_1),Ma)))) # label(fact_4252_less__eq__code__int__def) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 793 (all A_17 all C_4 all B_13 (B_13 != zero_z891286103de_int -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_4),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_17),B_13)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_17),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_13),C_4))),B_13))) # label(fact_2627_div__mult__self2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 794 (all A_134 all N_47 all B_103 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_134),N_47)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_103),N_47))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_103)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_134),B_103))))) # label(fact_1399_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 795 (all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Y_2)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_int_rat(number_number_of_rat,Y_2))))) # label(fact_398_less__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 796 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K1)),hAPP_int_int(bit1,K2))))) # label(fact_111_less__eq__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 797 (all A_27 all B_22 all C_9 (zero_zero_nat != C_9 -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_27),C_9)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_22),C_9)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_27),B_22))) # label(fact_2538_div__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 798 (all N all X hAPP_real_real(exp_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),X)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(exp_real,X)),N)) # label(fact_3846_exp__real__of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 799 (all B_128 all A_168 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_168)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_128)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_168),B_128)))))) # label(fact_1115_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 800 (all P all Q all R hAPP_f54304608l_bool(P,hAPP_i418383825t_bool(Q,R)) = hAPP_int_bool(hAPP_f41603166t_bool(hAPP_f389116883t_bool(cOMBB_1593174431ol_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obo) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 801 (all P all Q all R hAPP_f1888375463t_real(P,hAPP_r474017924t_real(Q,R)) = hAPP_r474017924t_real(hAPP_f1993765077t_real(hAPP_f895854793t_real(cOMBB_1948233853l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__fun__017) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 802 (all N hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N))))) # label(fact_3767_real__of__nat__fact__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 803 (all A_1 all B_1 (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_1)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_nat_int(semiri1621563631at_int,A_1)),image_nat_int(semiri1621563631at_int,B_1))))) # label(fact_4600_transfer__nat__int__set__relations_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 804 (all X all Y hAPP_complex_real(dist_dist_complex(X),Y) = hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),Y))) # label(fact_4546_dist__complex__def) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 805 (all I all J_2 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(legacy_zgcd(I),J_2)),J_2))) # label(fact_4484_zgcd__zdvd2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 806 (all N_27 all Y_7 all X_10 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,Y_7),X_10)) -> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_10),N_27)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_7),N_27)) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,X_10),Y_7)),N_27))) # label(fact_2581_div__power) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 807 (all U_1 all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,U_1),U_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),X)))) # label(fact_3482_real__minus__mult__self__le) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 808 (all M zero_zero_nat != hAPP_nat_nat(suc,M)) # label(fact_2848_Suc__neq__Zero) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 809 (all Ta all B_1 all D all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1)) -> (all X_2 (is_int(X_2) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) != X_2)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Ta))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1)),Ta))))))))) # label(fact_5104_bset_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 810 (all M all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N)),M))) # label(fact_4299_gcd__dvd1__nat) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 811 (all X_1 all A_1 (hBOOL(hAPP_P880235623l_bool(A_1,X_1)) <-> hBOOL(hAPP_f1469637905l_bool(hAPP_P1681069257l_bool(member308914164t_bool,X_1),A_1)))) # label(fact_144_mem__def) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 812 (all X_1 hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),X_1))) = hAPP_nat_nat(suc,X_1)) # label(fact_4746_card__bdd__nat__set__le) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 813 (all A_214 all N_67 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_214),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_67))))) # label(fact_683_zero__le__even__power_H) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 814 (all F (hBOOL(summable_real(F)) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(F,N_2)))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_f352196356l_real(suminf_real,F)))))) # label(fact_4858_suminf__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 815 (all B all P_5 all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),one_one_int)),P_5))) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5)))))) # label(fact_2224_wset__subset) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 816 (all V_12 all W_10 hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_12),W_10)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_12)),hAPP_int_real(number267125858f_real,W_10))) # label(fact_338_add__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 817 (all A_225 all B_162 all C_91 all D_18 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_225),B_162)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_91),D_18)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_225),C_91)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_162),D_18))) # label(fact_556_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 818 (all Ma all N_1 (is_int(N_1) & is_int(Ma) -> (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),N_1) = one_one_int <-> one_one_int = Ma & one_one_int = N_1 | hAPP_int_int(number_number_of_int,min) = Ma & N_1 = hAPP_int_int(number_number_of_int,min)))) # label(fact_789_zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 819 (all A_231 all B_165 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_165),A_231) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_231),B_165)) # label(fact_514_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 820 (all P all Q all R hAPP_rat_bool(P,hAPP_P1853363688nt_rat(Q,R)) = hAPP_P603027463t_bool(hAPP_f1084724746t_bool(hAPP_f374502857t_bool(cOMBB_569260360nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Rat__Orat_000tc__HOL__Obool_000tc__prod_Itc__Int__Oi) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 821 (all X hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_complex_real(re,X)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) = hAPP_complex_complex(invers1449016382omplex,X)) # label(fact_4110_complex__inverse__def) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 822 (all X hAPP_real_real(cos,X) = hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X))) # label(fact_3733_cos__sin__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 823 (all Y all X all Z_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Z_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_1))))) # label(fact_4156_termination__basic__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 824 (all B_2 all M all A_3 (is_int(B_2) & is_int(A_3) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) -> B_2 = A_3))))))) # label(fact_1460_zcong__zless__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 825 (all P all Q all R hAPP_r632163725t_bool(hAPP_f956464319t_bool(hAPP_f422375781t_bool(cOMBC_243356514t_bool,P),Q),R) = hAPP_f351634443t_bool(hAPP_r1392521531t_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Int__Oint_Mtc__fun_It) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 826 (all P all Q all R hAPP_r712953929l_real(hAPP_f1119546819l_real(hAPP_f555557763l_real(cOMBB_422423973l_real,P),Q),R) = hAPP_f800361423l_real(P,hAPP_r2019696015l_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__prod_Itc__RealDef__Orea_041) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 827 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)))))) # label(fact_4727_one__le__Fract__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 828 (all A_190 all B_146 all K_7 (A_190 = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_146),K_7) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,B_146),A_190)))) # label(fact_941_dvdI) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 829 (all X_39 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_39),X_39) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_39),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_436_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 830 (all K_1 all L (is_int(L) & is_int(K_1) -> (hAPP_int_int(bit1,L) = hAPP_int_int(bit1,K_1) <-> L = K_1))) # label(fact_177_rel__simps_I51_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 831 (all B_1_1 is_bool(cauchy_complex(B_1_1))) # label(gsy_c_SEQ_OCauchy_000tc__Complex__Ocomplex) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 832 (all X (one_one_real = hAPP_real_real(cos,X) -> zero_zero_real = hAPP_real_real(sin,X))) # label(fact_3662_cos__one__sin__zero) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 833 (all P_3 all Q_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Q_2)) -> (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,P_3),Q_2) -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2) = normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2))))) # label(fact_5022_normalize__stable) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 834 (all A_229 all C_95 all D_20 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_95),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_229),D_20)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_229),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_95),D_20))) # label(fact_531_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 835 (all X_1 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,X_1)),hAPP_real_real(sqrt,Y_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2)))) # label(fact_3313_real__sqrt__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 836 (all N_56 one_one_int = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_56))) # label(fact_1040_power__m1__even) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 837 (all A_108 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_108),zero_zero_nat) = A_108) # label(fact_1963_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 838 (all M_18 all N_38 all A_128 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_128)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_128),M_18)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_128),N_38))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_18),N_38))))) # label(fact_1450_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 839 (all K_1 all F all N_1 (is_int(K_1) -> ((all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),N_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),one_one_nat))),hAPP_nat_int(F,I_2)))),one_one_int)))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),hAPP_nat_int(F,N_1))) -> (exists I_2 (hAPP_nat_int(F,I_2) = K_1 & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N_1))))))))) # label(fact_3478_int__val__lemma) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 840 (all X_2 hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),cos_coeff)),hAPP_r474017924t_real(power_power_real,X_2))) = hAPP_real_real(cos,X_2)) # label(fact_4821_lemma__cos__ext) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 841 (all N_66 all A_210 all B_158 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_210),B_158)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_210)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_66)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_210),N_66)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_158),N_66))))))) # label(fact_749_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 842 (all A all B all A_1 all Ma (is_int(A) & is_int(B) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma)) -> (hBOOL(hAPP_int_bool(is_RRset(A_1),Ma)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),A_1)) -> A = B))))))) # label(fact_4590_RRset__zcong__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 843 (all X_1 all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,X_1)),hAPP_int_real(real_int,Y_2))))) # label(fact_4388_real__of__int__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 844 (all K_2 all L_2 hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(number_number_of_int,K_2)),hAPP_int_int(number_number_of_int,L_2)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(number_number_of_rat,K_2)),hAPP_int_rat(number_number_of_rat,L_2))) # label(fact_4710_Fract__number__of__quotient) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 845 (all X_1 all Y_2 all F ((all X_2 hBOOL(hAPP_real_bool(deriv_real(F,X_2),zero_zero_real))) -> hAPP_real_real(F,X_1) = hAPP_real_real(F,Y_2))) # label(fact_3951_DERIV__isconst__all) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 846 (all F ((all I_2 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,hAPP_nat_nat(suc,I_2))),hAPP_nat_nat(F,I_2)))) -> (exists I_2 all K hAPP_nat_nat(F,I_2) = hAPP_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),K))))) # label(fact_4571_weak__decr__stable) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 847 (all X_3 ((all R_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),R_1)) -> (exists K all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_2)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(X_3,N_2))),R_1)))))) -> hBOOL(vanishes(X_3)))) # label(fact_5152_vanishesI) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 848 (all C_43 all A_101 all B_83 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_101),B_83)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_43),A_101)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_43),B_83))))) # label(fact_2013_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 849 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sin,X)),one_one_real))) # label(fact_3644_sin__le__one) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 850 (all N (one_one_nat = hAPP_nat_nat(div_mod_nat(N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) -> hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3792_lemma__even__mod__4__div__2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 851 (all R_3 hAPP_real_complex(hAPP_r265291036omplex(complex,R_3),zero_zero_real) = of_real_complex(R_3)) # label(fact_3988_complex__of__real__def) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 852 (all C_73 all A_173 all B_133 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_173),B_133)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_73)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_173),C_73)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_133),C_73)))))) # label(fact_1083_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 853 (all X all Y hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),X)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),Y)))))) # label(fact_3335_le__real__sqrt__sumsq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 854 (all N (is_int(N) -> archim1246769320r_real(hAPP_int_real(real_int,N)) = N)) # label(fact_4381_floor__real__of__int) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 855 (all N all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N)),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N)))) -> hAPP_int_nat(nat,archim1246769320r_real(X)) = N))) # label(fact_4032_floor__eq3) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 856 (all C all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),A)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),zero_zero_int)))) # label(fact_1194_mult__less__cancel__left__disj) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 857 (all V all K_1 all A all B (B != A -> ((all X_2 hBOOL(hAPP_real_bool(deriv_real(V,X_2),K_1))) -> hAPP_real_real(V,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),B)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(V,A)),hAPP_real_real(V,B))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_3883_DERIV__const__average) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 858 (all B_2 all A_3 (zero_zero_nat != A_3 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),A_3)))) # label(fact_4345_gcd__le1__nat) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 859 (all B_132 all A_172 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_172),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_132),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_172),B_132)))))) # label(fact_1091_mult__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 860 (all Ma all K_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),N_1)) | zero_zero_nat = K_1 <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),K_1))))) # label(fact_1812_nat__mult__dvd__cancel__disj_H) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 861 (all K_1 all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K_1)),hAPP_int_int(bit1,L))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)))) # label(fact_112_rel__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 862 (all X hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X))) # label(fact_3739_minus__sin__cos__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 863 (all N all K_2 (is_int(K_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2)) -> K_2 = hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N)),K_2)))) # label(fact_4511_zgcd__zmult__eq__self2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 864 (all X_1 all P_5 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(sRStar,P_5))) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_5)))) # label(fact_2732_StandardRes__SRStar__prop1a) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 865 (all X all K_2 hAPP_real_real(sqrt,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),K_2)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,X)),K_2)) # label(fact_3319_real__sqrt__power) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 866 (all A all B all C (hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A),C) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A),B) <-> C = B)) # label(fact_1940_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 867 (all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N)) -> hAPP_int_int(fact_fact_int,N) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N),hAPP_int_int(fact_fact_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,N),one_one_int))))) # label(fact_4288_fact__reduce__int) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 868 (all N all D_3 all A_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),A_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N))))) # label(fact_4019_coprime__exp) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 869 (all A_69 all B_57 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_69),B_57)),B_57) = zero_z891286103de_int) # label(fact_2334_mod__div__trivial) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 870 (all X_1 hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),X_1))))) # label(fact_4763_bdd__nat__set__l__finite) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 871 (all B_96 all A_118 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_118)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_96)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_118),B_96)))))) # label(fact_1863_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 872 (all C all D all A all B all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),D)),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),D)),Ma))))) # label(fact_1032_zcong__zmult__prop1) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 873 (all P all Q P = hAPP_f957591787ol_nat(cOMBK_1643584600t_bool(P),Q)) # label(help_COMBK_1_1_COMBK_000tc__Nat__Onat_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 874 (all K_2 all M all N hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,K_2)),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N))) # label(fact_4994_gcd__mult__distrib__int) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 875 (all X_1 hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_int_real(real_int,X_1)),ring_1_Ints_real))) # label(fact_4760_Ints__real__of__int) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 876 (all B_95 all C_53 all A_117 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_117)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_95),C_53)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_95),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_117),C_53)))))) # label(fact_1870_add__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 877 (all A all B (hAPP_int_int(div_mod_int(B),A) = zero_zero_int <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B)))) # label(fact_2518_dvd__eq__mod__eq__0) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 878 (all P all Q all R hAPP_f1406902706l_real(hAPP_r1409264866l_real(P,R),Q) = hAPP_real_real(hAPP_f2119584768l_real(hAPP_f59594445l_real(cOMBC_2010481033l_real,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__O) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 879 (all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Y_2)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_int_rat(number_number_of_rat,Y_2))))) # label(fact_102_le__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 880 (all M all K_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) = K_2 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),K_2))))))) # label(fact_3130_prod__mn__less__k) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 881 (all P_1 all N_1 ((exists X_2 (hBOOL(hAPP_nat_bool(P_1,X_2)) & hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))))) <-> (exists M_1 (hBOOL(hAPP_nat_bool(P_1,M_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)))))) # label(fact_4901_ex__nat__less__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 882 (all X_1 all Y_2 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2))))) # label(fact_3902_real__root__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 883 (all N N != hAPP_nat_nat(suc,N)) # label(fact_2836_Suc__n__not__n) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 884 (all K_2 hAPP_int_int(bit0,K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),K_2)) # label(fact_266_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 885 (all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(sqrt,Y_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),Y_2)))) # label(fact_3332_real__sqrt__ge__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 886 (all X all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(standardRes(P_3),X))))) # label(fact_2648_StandardRes__lbound) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 887 (all Y all X (hBOOL(hAPP_int_bool(twoSqu1770640450sum2sq,X)) -> (hBOOL(hAPP_int_bool(twoSqu1770640450sum2sq,Y)) -> hBOOL(hAPP_int_bool(twoSqu1770640450sum2sq,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)))))) # label(fact_461_is__mult__sum2sq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 888 (all X all A_3 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)))) # label(fact_4404_powr__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 889 (all N_41 ((N_41 = zero_zero_nat -> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),N_41) = one_on1684967323de_int) & (N_41 != zero_zero_nat -> zero_z891286103de_int = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),N_41)))) # label(fact_1437_power__0__left) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 890 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,X_1)),one_one_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real)))) # label(fact_3841_exp__le__one__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 891 (all A_232 (is_int(A_232) -> A_232 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_232),hAPP_int_int(number_number_of_int,pls)))) # label(fact_326_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 892 (all X_1 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2)) <-> X_1 != Y_2 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2)))) # label(fact_1734_real__less__def) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 893 (all V_14 all B_166 all C_97 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_14)),B_166)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_14)),C_97)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_14)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_166),C_97))) # label(fact_196_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 894 (all A_3 all B_2 all C_1 all D_3 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)),hAPP_real_complex(hAPP_r265291036omplex(complex,C_1),D_3)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_3),C_1)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_2),D_3))) # label(fact_3964_complex__diff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 895 (all X hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3731_cos__squared__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 896 (all N_55 all A_152 all B_114 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_152),B_114)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_152)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_152),N_55)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_114),N_55)))))) # label(fact_1235_power__mono) # label(axiom) # label(non_clause). [assumption]. 7.03/7.04 897 (all N_1 all X_1 exists T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_1))) & hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(exp_real,T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1))) = hAPP_real_real(exp_real,X_1))) # label(fact_4926_Maclaurin__exp__le) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 898 (all K_2 all M all N hAPP_int_int(legacy_zgcd(K_2),hAPP_int_int(legacy_zgcd(M),N)) = hAPP_int_int(legacy_zgcd(hAPP_int_int(legacy_zgcd(K_2),M)),N)) # label(fact_4493_zgcd__assoc) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 899 (all P all Q all R hAPP_nat_fun_rat_rat(hAPP_f1830834240at_rat(hAPP_f1073644437at_rat(cOMBB_1926947at_nat,P),Q),R) = hAPP_rat_fun_rat_rat(P,hAPP_nat_rat(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Rat__Orat_000tc__fun_Itc__Rat__Orat_Mtc__Rat__Orat_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 900 (all Y all X exists R_1 exists A_2 (hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(sin,A_2)) = Y & hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(cos,A_2)) = X)) # label(fact_4544_polar__Ex) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 901 (all K_2 hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(number_number_of_int,K_2)),one_one_int) = hAPP_int_rat(number_number_of_rat,K_2)) # label(fact_4709_rat__number__collapse_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 902 (all P all Q all R hAPP_i68813070l_bool(hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_int_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 903 (all A_38 all C_13 all B_29 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,C_13),B_29)) -> hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(div_mo1740067990umeral(A_38),B_29)),C_13) = hAPP_C498520661umeral(div_mo1740067990umeral(A_38),C_13))) # label(fact_2445_mod__mod__cancel) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 904 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X))),X))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3474_abs__ln__one__plus__x__minus__x__bound__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 905 (all A_3 all B_2 zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(div_mod_int(A_3),B_2)),B_2)) # label(fact_2329_zmod__zdiv__trivial) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 906 (all B_45 all B_44 all A_54 all C_24 all A_53 (hAPP_int_int(div_mod_int(A_53),C_24) = hAPP_int_int(div_mod_int(A_54),C_24) -> (hAPP_int_int(div_mod_int(B_44),C_24) = hAPP_int_int(div_mod_int(B_45),C_24) -> hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_53),B_44)),C_24) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_54),B_45)),C_24)))) # label(fact_2401_mod__mult__cong) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 907 (all U all Ma all N_1 all I_1 all J (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),Ma) <-> Ma = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),I_1)),U)),N_1)))) # label(fact_1849_nat__eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 908 (all N_1 all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),Ma)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,Ma)),one_one_real))))) # label(fact_3686_nat__le__real__less) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 909 (all Lx_2 all Ly all Rx_2 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ly),Rx_2)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_2),Ly)),Rx_2)) # label(fact_494_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 910 (all I_1 all P_1 all K_1 (hBOOL(hAPP_int_bool(P_1,K_1)) -> ((all I_2 (is_int(I_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_2)) -> (hBOOL(hAPP_int_bool(P_1,I_2)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int))))))) -> ((all I_2 (is_int(I_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K_1)) -> (hBOOL(hAPP_int_bool(P_1,I_2)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I_2),one_one_int))))))) -> hBOOL(hAPP_int_bool(P_1,I_1)))))) # label(fact_4005_int__induct) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 911 (all B_71 all A_83 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_83)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_71)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_83),B_71)))))) # label(fact_2165_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 912 (all X_24 all N_32 (X_24 = one_on1684967323de_int | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,X_24),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_24),N_32))))) # label(fact_1494_dvd__power) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 913 (all A_225 all B_162 all C_91 all D_18 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_225),C_91)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_162),D_18)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_225),B_162)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_91),D_18))) # label(fact_557_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 914 (all X_1 all Y_2 all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),B)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B),X_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B),Y_2))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2))))) # label(fact_1416_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 915 (all Real1 all Real2 zero_zero_nat = complex_size(hAPP_real_complex(hAPP_r265291036omplex(complex,Real1),Real2))) # label(fact_3968_complex_Osize_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 916 (all Ta all D all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1)) -> (all X_2 all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D_1))),Ta))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Ta))))))) # label(fact_1531_inf__period_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 917 (all A_3 all B_2 hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),B_2)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_nat_int(semiri1621563631at_int,A_3)),hAPP_nat_int(semiri1621563631at_int,B_2))) # label(fact_3075_zdiv__int) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 918 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)))) # label(fact_2210_less__iff__diff__less__0) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 919 (all N_53 all M_21 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M_21)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),N_53)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M_21),N_53)))))) # label(fact_1245_less__1__mult) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 920 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(log(hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(exp_real,one_one_real))),hAPP_real_real(ln,X)) = hAPP_real_real(log(hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X))) # label(fact_3882_log__base__10__eq2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 921 (all B_2 all A_3 all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),B_2)) -> (exists D_2 ((all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y_1))),D_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),Y_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),B_2)))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))))))) # label(fact_3413_lemma__interval__lt) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 922 (all Y all X all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (A_3 != one_one_real -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y)) -> hAPP_real_real(log(A_3),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(log(A_3),X)),hAPP_real_real(log(A_3),Y))))))) # label(fact_3873_log__mult) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 923 (all Ma all N_1 all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A)) -> (N_1 = Ma <-> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),N_1) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),Ma)))) # label(fact_1283_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 924 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) -> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat,X)),hAPP_int_nat(nat,Y))))) # label(fact_2742_Nat__Transfer_Otransfer__nat__int__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 925 (all A_205 all B_156 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_205),B_156) = zero_zero_nat -> zero_zero_nat = A_205 | zero_zero_nat = B_156)) # label(fact_822_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 926 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(div_mod_int(A_3),B_2)),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),hAPP_int_int(div_mod_int(A_3),B_2))))) # label(fact_2611_neg__mod__conj) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 927 (all P_1 all A0 all A1 (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(norm_frac_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1))) -> ((all A_2 all B_4 (is_int(B_4) & is_int(A_2) -> (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(norm_frac_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_2),B_4))) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_4),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,hAPP_int_int(uminus_uminus_int,A_2)),hAPP_int_int(uminus_uminus_int,B_4)))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A_2),B_4)))))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1))))) # label(fact_4572_norm__frac_Opinduct) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 928 (all A_216 A_216 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),A_216)) # label(fact_648_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 929 (all X_1 all P_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(standardRes(P_5),X_1)),hAPP_i1948725293t_bool(sRStar,P_5))) <-> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_5))))) # label(fact_2764_StandardRes__SRStar__prop1) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 930 (all C all D all A all B (D != C & A != B <-> hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A),D)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B),C)) != hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A),C)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B),D)))) # label(fact_626_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 931 (all B_53 all A_65 all C_32 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_53),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_65),B_53))),hAPP_nat_nat(div_mod_nat(A_65),B_53))),C_32) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_65),C_32)) # label(fact_2351_div__mod__equality2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 932 (all A_211 all M_26 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_211),M_26)),M_26) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_211),one_one_complex)),M_26)) # label(fact_709_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 933 (all Z_1 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natfloor(Z_1)),one_one_nat)),N)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_1),hAPP_nat_real(real_nat,N))))) # label(fact_3704_ge__natfloor__plus__one__imp__gt) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 934 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),N) = M) # label(fact_1747_diff__add__inverse2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 935 (exists K_4 all X_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(re,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_2)),K_4)))) # label(fact_4142_Re_Obounded) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 936 (all X_1 all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,X_1)),hAPP_int_int(number_number_of_int,Y_2))))) # label(fact_58_le__number__of) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 937 (all C_76 all B_136 all A_176 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_136),A_176)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_76),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_176),C_76)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_136),C_76)))))) # label(fact_1070_mult__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 938 (all X_1 all Y_2 (is_int(Y_2) & is_int(X_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) <-> zero_zero_int != X_1 | Y_2 != zero_zero_int))) # label(fact_10_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 939 (all V_22 all V_21 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_22)) -> hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_21),V_22)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_21)),hAPP_int_rat(number_number_of_rat,V_22))))) # label(fact_78_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 940 (all A_105 A_105 = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,zero_zero_real),A_105)) # label(fact_1988_add__0__left) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 941 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A),B))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),zero_zero_int))))) # label(fact_2600_neg__imp__zdiv__nonneg__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 942 (all X_1 all Y_2 (hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_i769753017umeral(number1443263063umeral,Y_2)),hAPP_i769753017umeral(number1443263063umeral,X_1)) = zero_z126310315umeral <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_i769753017umeral(number1443263063umeral,X_1)),hAPP_i769753017umeral(number1443263063umeral,Y_2))))) # label(fact_2587_dvd__eq__mod__eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 943 (all A_1 all X_1 (hBOOL(hAPP_int_bool(nat_is_nat,X_1)) -> (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_int_nat(nat,X_1)),image_int_nat(nat,A_1))) <-> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_1)))))) # label(fact_5148_transfer__int__nat__set__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 944 (all C_79 all D_15 all A_179 all B_139 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_179),B_139)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_79),D_15)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_139)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_79)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_179),C_79)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_139),D_15)))))))) # label(fact_1054_mult__mono) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 945 (all X (hAPP_real_real(cos,X) != zero_zero_real -> (zero_zero_real != hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X)) -> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(tan,X))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(tan,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X))))) # label(fact_3594_tan__double) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 946 (all X_1 (zero_zero_real = hAPP_real_real(cos,X_1) <-> (exists N_2 (-hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2)) & hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = X_1)) | (exists N_2 (hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = X_1 & -hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2)))))) # label(fact_3754_cos__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 947 (all V all N_1 (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))) & (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))) -> hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))) = N_1) <-> hAPP_nat_nat(suc,N_1) = hAPP_int_nat(number_number_of_nat,V))) # label(fact_4779_eq__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 948 (all B all C all A all D (one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A),D) -> (one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B),C) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),C) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B),D) <-> C = D & B = A)))) # label(fact_4355_coprime__crossproduct__nat) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 949 (all N_43 all A_131 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_131)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_131),N_43)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_131),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_131),N_43)))))) # label(fact_1422_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 950 (all X all Y hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,fFalse),X),Y) = Y) # label(help_If_2_1_If_000tc__Nat__Onat_T) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 951 (all X_1 (zero_zero_real != hAPP_real_real(cos,X_1) -> hBOOL(hAPP_real_bool(deriv_real(tan,X_1),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3878_DERIV__tan) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 952 (all K_1 all Ma all N_1 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),N_1) <-> N_1 = Ma)) # label(fact_2876_Suc__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 953 (all B_1_1 all B_2_1 (is_bool(B_2_1) -> is_bool(hAPP_bool_bool(B_1_1,B_2_1)))) # label(gsy_c_hAPP_000tc__HOL__Obool_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 954 (all N (hAPP_nat_nat(div_mod_nat(N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))) -> -hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3791_lemma__odd__mod__4__div__2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 955 (all A all A_1 all Ma ((-(hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1)) & hBOOL(hAPP_int_bool(is_RRset(A_1),Ma)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma))) -> zero_zero_int = hAPP_int_int(rRset2norRR(A_1,Ma),A)) & (hBOOL(hAPP_int_bool(is_RRset(A_1),Ma)) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma)) -> hAPP_int_int(rRset2norRR(A_1,Ma),A) = hilbert_Eps_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,zcong(A)),Ma))),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),hAPP_i1948725293t_bool(norRRset,Ma))))))) # label(fact_4892_RRset2norRR__def) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 956 (all A_3 all B_2 all Q_2 all R_3 (A_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_3) -> ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_3))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_2)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3))) -> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3)))))) # label(fact_2801_divmod__int__relI) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 957 (all P_1 all L ((exists X_2 hBOOL(hAPP_rat_bool(P_1,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,L),X_2)))) <-> (exists X_2 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,L),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_2),zero_zero_rat))) & hBOOL(hAPP_rat_bool(P_1,X_2)))))) # label(fact_1679_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 958 (all X_37 X_37 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_37),one_one_nat)) # label(fact_565_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 959 (all P all Q all R hAPP_f766218899t_real(P,hAPP_i371647666l_real(Q,R)) = hAPP_i1671359920t_real(hAPP_f74147283t_real(hAPP_f1169087445t_real(cOMBB_2129825580al_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__031) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 960 (all D_3 all A_3 all B_2 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)) = one_one_nat -> one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),B_2))) # label(fact_4357_coprime__rmult__nat) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 961 (all C_70 all A_164 all B_125 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_164),B_125)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_70)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_70),A_164)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_70),B_125)))))) # label(fact_1133_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 962 (all K_1 all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N_1))))) # label(fact_1642_nat__add__left__cancel__le) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 963 (all X_1 (hAPP_real_real(sin,X_1) = zero_zero_real <-> (exists N_2 (hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = X_1 & hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2)))) | (exists N_2 (hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2)) & hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = X_1)))) # label(fact_3753_sin__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 964 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Y),X)) -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_int_int(abs_abs_int,Y))) # label(fact_4993_gcd__proj2__if__dvd__int) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 965 (all X_9 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_9)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(semiri984289939at_nat,hAPP_int_nat(nat,X_9))),hAPP_nat_nat(semiri984289939at_nat,hAPP_int_nat(nat,X_9))) = hAPP_nat_nat(semiri984289939at_nat,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_9))))) # label(fact_2810_of__nat__double) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 966 (all P all Q all R hAPP_i872162435t_bool(hAPP_f1251097615t_bool(hAPP_f1475822739t_bool(cOMBB_2027861294ol_int,P),Q),R) = hAPP_f1512942609t_bool(P,hAPP_i1948725293t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__025) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 967 (all N all K_2 (is_int(K_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2)) -> hAPP_int_int(legacy_zgcd(K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N)) = K_2))) # label(fact_4509_zgcd__zmult__eq__self) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 968 (all D all Sa exists Z all X_2 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_2),Z)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_2),Sa))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_2),Sa)))))) # label(fact_2262_minf_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 969 (all A all B all K_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ma)) -> (hAPP_int_int(legacy_zgcd(K_1),Ma) = one_one_int -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),K_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),K_1)),Ma)))))) # label(fact_4525_zcong__cancel) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 970 (all B all A (hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B),A) = B <-> A = zero_zero_complex)) # label(fact_614_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 971 (all Z_1 (is_int(Z_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> -(all M_1 hAPP_nat_int(semiri1621563631at_int,M_1) != Z_1)))) # label(fact_3259_nonneg__eq__int) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 972 (all I all U_1 all J_2 all K_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),U_1)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),U_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),U_1)),K_2))) # label(fact_1845_left__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 973 (all A_3 all C_1 all D_3 all B_2 (is_int(B_2) & is_int(D_3) -> (B_2 != zero_zero_int -> (zero_zero_int != D_3 -> hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),D_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),D_3)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C_1),D_3)))))) # label(fact_4719_add__rat) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 974 (all A_188 all B_144 all C_81 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_188),B_144)),C_81)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_144),C_81)))) # label(fact_961_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 975 (all X_34 all P_7 all Q_10 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_34),P_7)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_34),Q_10)) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_34),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_10))) # label(fact_668_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 976 (all A_194 all B_150 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_194),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_150),A_194)))) # label(fact_918_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 977 (all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_nat_rat(semiri151668891at_rat,N_1))))) # label(fact_3142_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 978 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (all Y_1 exists N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),X)))))) # label(fact_3751_reals__Archimedean3) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 979 (all X (hBOOL(hAPP_int_bool(nat_is_nat,X)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(fact_fact_int,X))))) # label(fact_5147_transfer__int__nat__factorial__closure) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 980 (all K_2 min != hAPP_int_int(bit0,K_2)) # label(fact_771_rel__simps_I45_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 981 (all P_1 all Y_2 all X_1 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_2))))) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_2),zero_zero_int)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),hAPP_int_int(uminus_uminus_int,Y_2)))))) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(uminus_uminus_int,X_1)),Y_2))))) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_2),zero_zero_int)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(uminus_uminus_int,X_1)),hAPP_int_int(uminus_uminus_int,Y_2)))))) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X_1),Y_2)))))))) # label(fact_5008_gcd__cases__int) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 982 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N)) # label(fact_4333_gcd__add1__nat) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 983 (all A_1 (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1)) -> hAPP_f957591787ol_nat(finite_card_int,A_1) = hAPP_f22106695ol_nat(finite_card_nat,image_int_nat(nat,A_1)))) # label(fact_4620_Nat__Transfer_Otransfer__int__nat__set__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 984 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arccos,Y)),pi)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(arccos,Y)))))) # label(fact_3615_arccos__bounded) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 985 (all N all B_2 all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,P_3),N)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,P_3),N)),B_2)))))) # label(fact_452_zprime__power__zdvd__cancel__left) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 986 (all X hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,X))) = hAPP_int_int(abs_abs_int,X)) # label(fact_3426_int__nat__abs) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 987 (all X_8 all Q_7 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_8),hAPP_nat_nat(suc,Q_7)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_8),Q_7)),X_8)) # label(fact_3019_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 988 (all A (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A)))) # label(fact_2175_zero__le__double__add__iff__zero__le__single__add) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 989 (all Y_2 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2)) -> (X_1 = Y_2 <-> hAPP_real_real(ln,Y_2) = hAPP_real_real(ln,X_1))))) # label(fact_3428_ln__inj__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 990 (all V_8 all W_4 hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_8),W_4)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_int_rat(number_number_of_rat,V_8)),hAPP_int_rat(number_number_of_rat,W_4))) # label(fact_1513_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 991 (all A_3 all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,A_3)),X)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,natfloor(X)),A_3) = natfloor(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_nat_real(real_nat,A_3))))) # label(fact_3692_natfloor__subtract) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 992 (all X_1 all Y_2 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_1)))) # label(fact_1702_dvd_Oless__le__not__le) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 993 (all A_3 rcis(one_one_real,A_3) = cis(A_3)) # label(fact_4260_cis__rcis__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 994 (all M all K_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M)),K_2))))) # label(fact_1776_le__add__diff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 995 (all X_36 all Y_28 all Z_9 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_36),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_28),Z_9)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_36),Y_28)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_36),Z_9))) # label(fact_618_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 996 (all Y_2 all X_1 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),X_1)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),Y_2)) -> (Y_2 = zero_z891286103de_int & zero_z891286103de_int = X_1 <-> zero_z891286103de_int = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_1),Y_2))))) # label(fact_2163_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 997 (all A all B (A = B <-> hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B) = zero_zero_rat)) # label(fact_2079_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 998 (all X_21 all N_31 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31)) -> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_21),N_31) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21))) # label(fact_1796_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 999 (all C_59 all D_12 all A_137 all B_106 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_137),B_106)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_59),D_12)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_137)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_59)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_137),C_59)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_106),D_12)))))))) # label(fact_1360_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1000 (all X_3 (hBOOL(cauchy_complex(X_3)) -> hBOOL(cauchy_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_3))))) # label(fact_4757_Im_OCauchy) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1001 (all X_1 (hAPP_real_real(cos,X_1) != zero_zero_real -> hBOOL(hAPP_real_bool(isCont_real_real(tan),X_1)))) # label(fact_5160_isCont__tan) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1002 (all C_46 all A_104 all B_86 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_46),A_104)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_46),B_86))) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_104),B_86)))) # label(fact_1992_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1003 (all Ma all K_1 all N_1 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,K_1),N_1) = one_one_nat -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),Ma)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N_1)))))) # label(fact_4368_coprime__dvd__mult__iff__nat) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1004 (all P_1 all K_1 all I_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_1)) -> (hBOOL(hAPP_int_bool(P_1,K_1)) -> ((all I_2 (is_int(I_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),I_2)) -> (hBOOL(hAPP_int_bool(P_1,I_2)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int))))))) -> hBOOL(hAPP_int_bool(P_1,I_1)))))) # label(fact_2220_int__ge__induct) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1005 (all Q all P (hBOOL(hAPP_bool_bool(fimplies(P),Q)) | hBOOL(P))) # label(help_fimplies_1_1_U) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1006 (all Y all X (zero_zero_real != hAPP_real_real(cos,X) -> (zero_zero_real != hAPP_real_real(cos,Y) -> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(tan,X)),hAPP_real_real(tan,Y)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y)))))) # label(fact_3713_add__tan__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1007 (all X_5 all Y_5 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_5),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_5),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_5),Y_5)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_5),Y_5))) # label(fact_3193_realpow__two__diff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1008 (all F all K_1 all A all B (A != B -> ((all X_2 hBOOL(hAPP_real_bool(deriv_real(F,X_2),K_1))) -> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B)),hAPP_real_real(F,A))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A)) = K_1))) # label(fact_3891_DERIV__const__ratio__const2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1009 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(div_mod_nat(M),N)),N)))) # label(fact_2703_mod__less__divisor) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1010 (all I all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),N))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),I)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))))) # label(fact_1603_power__dvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1011 (all N (zero_zero_nat != N -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),N) = one_one_nat)) # label(fact_4366_coprime__minus__one__nat) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1012 (all K_1 all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N_1))))) # label(fact_1669_mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1013 (all N (N != zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3234_Suc__Suc__mult__two__diff__two) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1014 (all Y_2 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),X_1)) -> X_1 = comple124823625p_real(hAPP_r1134773055l_bool(ord_gr788844697n_real(Y_2),X_1)))) # label(fact_4638_Sup__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1015 (all A_69 all B_57 zero_z126310315umeral = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_69),B_57)),B_57)) # label(fact_2333_mod__div__trivial) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1016 (all C_78 all D_14 all A_178 all B_138 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_178),B_138)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_78),D_14)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_178)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_78)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_178),C_78)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_138),D_14)))))))) # label(fact_1059_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1017 (all X all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)),X) = hAPP_real_real(hAPP_n546711566l_real(root,M),hAPP_real_real(hAPP_n546711566l_real(root,N),X)))))) # label(fact_3932_real__root__pos__mult__exp) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1018 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Y)),one_one_real)) -> Y = hAPP_real_real(cos,hAPP_real_real(arccos,Y)))) # label(fact_3604_cos__arccos__abs) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1019 (all C_70 all A_164 all B_125 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_164),B_125)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_70)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_70),A_164)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_70),B_125)))))) # label(fact_1136_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1020 (all A_184 all M_23 all N_58 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_23),N_58)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_184),M_23)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_184),N_58))))) # label(fact_1010_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1021 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A),B)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))))) # label(fact_2521_neg__imp__zdiv__neg__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1022 (all C_64 all A_142 all B_111 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_64),A_142)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_64),B_111))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_64)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_142),B_111))))) # label(fact_1330_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1023 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),pi)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sin,X)))))) # label(fact_3658_sin__gt__zero__pi) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1024 (all X all Y all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),Y)))))) # label(fact_3906_real__root__less__mono) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1025 (all P all Q all R hAPP_i1778081398nt_int(hAPP_f2094497995nt_int(hAPP_f534361363nt_int(cOMBB_1657320178nt_int,P),Q),R) = hAPP_f465821005nt_int(P,hAPP_i1778081398nt_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__prod) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1026 (all X_33 one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_33),zero_zero_nat)) # label(fact_676_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1027 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3361_real__less__half__sum) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1028 (all M all N all Q_2 all R_3 (hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_2),R_3))) -> Q_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N))) # label(fact_4218_div__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1029 (all A_225 all B_162 all C_91 all D_18 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_225),B_162)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_91),D_18)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_225),C_91)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_162),D_18))) # label(fact_558_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1030 (all X_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> hBOOL(hAPP_real_bool(isCont_real_real(hAPP_n546711566l_real(root,N_1)),X_1)))) # label(fact_5159_isCont__real__root) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1031 (all C_34 all D_6 all A_80 all B_68 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_80),B_68)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_34),D_6)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_80),C_34)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_68),D_6)))))) # label(fact_2199_add__le__less__mono) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1032 (all X hAPP_real_real(inverse_inverse_real,ratreal(X)) = ratreal(hAPP_rat_rat(inverse_inverse_rat,X))) # label(fact_4706_real__inverse__code) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1033 (all A hAPP_int_int(zfact,A) = big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(d22set,A))) # label(fact_4810_d22set__prod__zfact) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1034 (all J_2 all K_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),J_2)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),K_2)),M)) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),J_2) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),K_2)))) # label(fact_1500_neg__one__power__eq__mod__m) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1035 (all X_45 all Y_33 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_45),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_33),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_33)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_45),Y_33))))) # label(fact_209_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1036 (all W_5 ((hAPP_int_nat(number_number_of_nat,W_5) = zero_zero_nat -> one_one_rat = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),hAPP_int_nat(number_number_of_nat,W_5))) & (hAPP_int_nat(number_number_of_nat,W_5) != zero_zero_nat -> zero_zero_rat = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),hAPP_int_nat(number_number_of_nat,W_5))))) # label(fact_730_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1037 (all A all E_2 all C all B all D (is_int(D) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B)),E_2)),C) = D <-> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),E_2)),D) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),E_2)),C)))) # label(fact_1540_eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1038 (all A_148 all N_51 all N_50 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_51),N_50)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),A_148)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_148),N_51)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_148),N_50)))))) # label(fact_1276_power__increasing) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1039 (all C_69 all A_163 all B_124 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_163),B_124)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_69)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_69),A_163)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_69),B_124)))))) # label(fact_1139_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1040 (all L_2 all K_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_2),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),L_2)) -> negDivAlg(K_2,L_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,K_2),L_2)),hAPP_int_int(div_mod_int(K_2),L_2))))) # label(fact_2376_negDivAlg__div__mod) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1041 (all X all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)),X) = hAPP_real_real(hAPP_n546711566l_real(root,M),hAPP_real_real(hAPP_n546711566l_real(root,N),X))))) # label(fact_3908_real__root__mult__exp) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1042 (all M_14 all D_4 (hAPP_nat_nat(div_mod_nat(M_14),D_4) = zero_zero_nat -> (exists Q_4 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,D_4),Q_4) = M_14))) # label(fact_2788_mod__eq__0D) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1043 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(div_mod_int(A_3),B_2)),B_2)))) # label(fact_2527_pos__mod__bound) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1044 (all A_1 all B_1 all Ma (hBOOL(hAPP_f1469637905l_bool(hAPP_P1681069257l_bool(member308914164t_bool,hAPP_f538843494t_bool(hAPP_f1418982953t_bool(produc713279741t_bool,A_1),B_1)),bijR_int_int(zcongm(Ma)))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(big_co1548731110nt_int(cOMBI_int,A_1)),big_co1548731110nt_int(cOMBI_int,B_1)),Ma)))) # label(fact_5155_bijzcong__zcong__prod) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1045 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X_1)),one_one_real)))) # label(fact_3843_exp__less__one__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1046 (all A_143 all C_65 all B_112 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_143),C_65)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_112),C_65))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_65)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_143),B_112))))) # label(fact_1325_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1047 (all Y_2 all X_1 (hBOOL(hAPP_int_bool(nat_is_nat,X_1)) -> hBOOL(hAPP_f448129468l_bool(nat_nat_set,hAPP_i1948725293t_bool(ord_at875362053st_int(X_1),Y_2))))) # label(fact_5144_SetInterval_Otransfer__int__nat__set__function__closures) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1048 (all B_118 all A_157 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_157)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_118),zero_z891286103de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_118),A_157)),zero_z891286103de_int))))) # label(fact_1177_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1049 (all X_1 (hBOOL(hAPP_int_bool(even_odd_even_int,X_1)) <-> hAPP_int_int(div_mod_int(X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_int)) # label(fact_3227_even__def) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1050 (all Z_1 hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,Z_1),of_real_complex(hAPP_complex_real(norm_norm_complex,Z_1))) = hAPP_complex_complex(sgn_sgn_complex,Z_1)) # label(fact_3989_sgn__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1051 (all X_3 (hBOOL(summable_complex(X_3)) -> hBOOL(summable_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_3))))) # label(fact_4846_Re_Osummable) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1052 (all D all Sa exists Z all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Z)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Sa))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Sa)))))) # label(fact_2260_minf_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1053 (all I (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),I)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_nat(nat,I))),one_one_nat) = hAPP_list_int_nat(size_size_list_int,quickc666637781d_zero(I)))) # label(fact_4289_length__around__zero) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1054 (all P all Q all R hAPP_r1409264866l_real(hAPP_f1523861863l_real(hAPP_f1067681227l_real(cOMBB_1934313333l_real,P),Q),R) = hAPP_f1509287679l_real(P,hAPP_r474017924t_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__fun__022) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1055 (all M all V_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V_2)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(pred,V_2))))) # label(fact_4051_Suc__diff__number__of) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1056 (all P all Q all R hAPP_rat_bool(hAPP_i16292313t_bool(P,R),Q) = hAPP_int_bool(hAPP_r1900510169t_bool(hAPP_f1480506641t_bool(cOMBC_int_rat_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Rat__Orat_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1057 (all C_79 all D_15 all A_179 all B_139 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_179),B_139)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_79),D_15)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_139)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_79)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_179),C_79)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_139),D_15)))))))) # label(fact_1053_mult__mono) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1058 (all X_1 hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),cos)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_1))),hAPP_real_real(cos,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_1))),hAPP_real_real(cos,X_1)))))) # label(fact_4799_DERIV__cos__realpow2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1059 (all C_88 all A_206 all B_157 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_206),B_157)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,B_157),C_88)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_206),C_88))))) # label(fact_803_dvd__trans) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1060 (all F all L all C (hBOOL(tendsto_real_real(F,L,at_real(C))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,L),zero_zero_real)) -> (exists R_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1)) & (all X_2 (C != X_2 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),X_2))),R_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,X_2)),zero_zero_real))))))))) # label(fact_5088_LIM__fun__less__zero) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1061 (all A_3 all B_2 all Q_2 all R_3 (is_int(B_2) & is_int(Q_2) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3))) -> (B_2 != zero_zero_int -> Q_2 = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))))) # label(fact_2795_divmod__int__rel__div) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1062 (all K_5 all M_15 all N_28 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_5),hAPP_int_int(div_mod_int(M_15),N_28))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_5),N_28)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_5),M_15))))) # label(fact_2455_dvd__mod__imp__dvd) # label(axiom) # label(non_clause). [assumption]. 7.03/7.05 1063 (all X_1 all Y_2 (zero_zero_real = Y_2 & zero_zero_real = X_1 <-> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),X_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Y_2)) = zero_zero_real)) # label(fact_181_sum__squares__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1064 (all P all Q all R hAPP_r2115560837t_bool(P,hAPP_int_rat(Q,R)) = hAPP_i16292313t_bool(hAPP_f601898971t_bool(hAPP_f857296063t_bool(cOMBB_214503962ol_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Rat__Orat_000tc__fun_Itc__Rat__Orat_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1065 (all N_1 all P_1 (-hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(P_1,N_1)) -> (exists K (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),N_1)) & (all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),K)) -> -hBOOL(hAPP_nat_bool(P_1,I_2)))) & hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)))))))) # label(fact_1691_ex__least__nat__less) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1066 (all R_2 hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(times_times_real,R_2)),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(plus_plus_real,one_one_real)),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,uminus_uminus_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc))))),R_2,sequentially))) # label(fact_5048_LIMSEQ__inverse__real__of__nat__add__minus__mult) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1067 (all M hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,zero_zero_nat),M) = divmod_nat(M,zero_zero_nat)) # label(fact_4209_divmod__nat__zero) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1068 (all X_8 all Q_7 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_8),Q_7)),X_8) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_8),hAPP_nat_nat(suc,Q_7))) # label(fact_3023_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1069 (all A_237 A_237 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,pls))),A_237)) # label(fact_236_mult__numeral__1) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1070 (all X all Y all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y),Z_1))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Z_1)),Y))))) # label(fact_2608_div__prop1) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1071 (all C_63 all A_141 all B_110 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_63),A_141)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_63),B_110))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_63)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_141),B_110))))) # label(fact_1340_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1072 (all L all U hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,U),L)) = hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(ord_at641636577an_int(L),U))) # label(fact_4920_card__atLeastLessThan__int) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1073 (all A all E_2 all C all B all D (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A)),E_2)),D))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),E_2)),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),E_2)),D))))) # label(fact_1567_less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1074 (all C_59 all D_12 all A_137 all B_106 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_137),B_106)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_59),D_12)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_137)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_59)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_137),C_59)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_106),D_12)))))))) # label(fact_1365_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1075 (all A (zero_zero_complex = A <-> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_complex)) # label(fact_38_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1076 (all A all E_2 all C all B all D (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),E_2)),C) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),E_2)),D) <-> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A)),E_2)),D) = C)) # label(fact_1537_eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1077 (all C_79 all D_15 all A_179 all B_139 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_179),B_139)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_79),D_15)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_139)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_79)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_179),C_79)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_139),D_15)))))))) # label(fact_1055_mult__mono) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1078 (all N_12 all R_6 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),R_6)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,R_6),one_one_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,R_6),hAPP_nat_nat(suc,N_12))),R_6))))) # label(fact_3157_realpow__Suc__le__self) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1079 (all Nat_3 zero_zero_nat != hAPP_nat_nat(suc,Nat_3)) # label(fact_2850_nat_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1080 (all X hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_420_power2__ge__self) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1081 (all N zero_zero_real = hAPP_real_real(hAPP_n546711566l_real(root,N),zero_zero_real)) # label(fact_3895_real__root__zero) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1082 (all P all Q all R hAPP_P1860904029l_real(hAPP_r938550413l_real(P,R),hAPP_r1195171167l_real(Q,R)) = hAPP_r1195171167l_real(hAPP_f1736103445l_real(hAPP_f1908301369l_real(cOMBS_902912708l_real,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__prod_Itc__RealDef__Oreal_Mtc__) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1083 (all B_1_1 all B_2_1 is_int(hAPP_f659380387ol_int(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__Int__Oint) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1084 (all N_52 all A_150 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_150)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_150),N_52))))) # label(fact_1252_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1085 (all A (one_one_nat != A <-> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),one_one_nat)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A)))) # label(fact_2314_gcd__lcm__complete__lattice__nat_Obot__less) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1086 (all N -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N)))))) # label(fact_3549_not__zle__0__negative) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1087 (all A all B all C (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),B) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),C) <-> B = C)) # label(fact_1939_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1088 (all W_1 all Y_2 all X_1 all Z_2 (Z_2 = Y_2 | X_1 = W_1 <-> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,W_1),Y_2)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_1),Z_2)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,W_1),Z_2)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_1),Y_2)))) # label(fact_646_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1089 (all N_41 ((N_41 != zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N_41) = zero_zero_nat) & (N_41 = zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N_41) = one_one_nat))) # label(fact_1433_power__0__left) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1090 (all K_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),K_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),K_2))))) # label(fact_1597_le__trans) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1091 (all F all G all N_1 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,F),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,G),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))) = hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),F)),G)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1)))) # label(fact_4910_sum__split__even__odd) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1092 (all A_105 A_105 = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,zero_z126310315umeral),A_105)) # label(fact_1985_add__0__left) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1093 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,pi)),X)) -> hAPP_real_real(uminus_uminus_real,X) = hAPP_real_real(arccos,hAPP_real_real(cos,X))))) # label(fact_3609_arccos__cos2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1094 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(suc,N))))) # label(fact_2873_le__SucI) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1095 (all A all B all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A)),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),B)))))) # label(fact_4409_powr__less__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1096 (all C_76 all B_136 all A_176 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_136),A_176)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_76),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_176),C_76)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_136),C_76)))))) # label(fact_1068_mult__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1097 (all M all N all K_2 ((zero_zero_nat != K_2 -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N))) & (zero_zero_nat = K_2 -> zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N))))) # label(fact_2701_nat__mult__div__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1098 (all A_90 (is_int(A_90) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),A_90) = A_90)) # label(fact_2112_mult__1) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1099 (all D all Sa exists Z all X_2 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Z),X_2)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_2),Sa))) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_2),Sa)))))) # label(fact_2271_pinf_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1100 (all X all Y hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,X))),hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,Y)))) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)) # label(fact_5021_gcd__int__def) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1101 (all A_3 all B_2 (quotient_of(B_2) = quotient_of(A_3) -> B_2 = A_3)) # label(fact_4654_quotient__of__inject) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1102 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),hAPP_int_int(bit1,K_1))))) # label(fact_110_rel__simps_I22_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1103 (all A all Ma all X_1 (is_int(X_1) -> (X_1 != zero_zero_int -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),Ma)) -> big_co1548731110nt_int(cOMBI_int,image_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,times_times_int),X_1),hAPP_i1948725293t_bool(bnorRset(A),Ma))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(bnorRset(A),Ma))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(bnorRset(A),Ma)))))))) # label(fact_4812_Bnor__prod__power) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1104 (all X hAPP_real_real(sin,hAPP_real_real(uminus_uminus_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X))) # label(fact_3634_sin__minus) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1105 (all M all N (is_int(N) -> (zero_zero_int != N -> hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),N)),N) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,M),N)),one_one_rat)))) # label(fact_4723_Fract__add__one) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1106 (all X one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),one_one_nat)) # label(fact_4337_gcd__lcm__complete__lattice__nat_Oinf__bot__right) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1107 (all X_30 all Y_24 (is_int(Y_24) & is_int(X_30) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_30),Y_24)) -> (X_30 != Y_24 -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_30),Y_24)))))) # label(fact_814_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1108 (all X all Y (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)))) # label(fact_1719_dvd_Oless__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1109 (all X_1 all P_5 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(sRStar,P_5))) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(sr,P_5))))) # label(fact_2718_SRStar__SR__prop) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1110 (all N all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y),N))))) # label(fact_1803_divides__exp) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1111 (all A all B all C (is_int(B) & is_int(C) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),C) <-> B = C))) # label(fact_1937_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1112 (all P all Q all R hAPP_nat_fun_nat_nat(P,hAPP_nat_nat(Q,R)) = hAPP_nat_fun_nat_nat(hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1113 (all K_2 all L_2 hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(ring_1_of_int_rat,K_2)),hAPP_int_rat(ring_1_of_int_rat,L_2)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,K_2),L_2)) # label(fact_4680_Fract__of__int__quotient) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1114 (all A all D all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,hAPP_nat_fun_nat_nat(plus_plus_nat,A)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),D))),hAPP_n1699378549t_bool(ord_lessThan_nat,N_1))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat)),D))))) # label(fact_4953_arith__series__nat) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1115 (all N all M zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M))) # label(fact_1761_diff__add__0) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1116 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real)) -> natfloor(X) = zero_zero_nat)) # label(fact_3397_natfloor__neg) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1117 (all X_1 all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(sRStar,P_5))) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(standardRes(P_5),X_1)),hAPP_i1948725293t_bool(sRStar,P_5))))))) # label(fact_2763_StandardRes__SRStar__prop4) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1118 (all X ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> hAPP_int_int(fact_fact_int,X) = zero_zero_int) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(fact_fact_nat,hAPP_int_nat(nat,X))) = hAPP_int_int(fact_fact_int,X)))) # label(fact_4271_fact__int__def) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1119 (all N all A_3 all B_2 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N)),A_3)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N)),B_2)))))) # label(fact_4194_prime__divprod__pow) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1120 (all A_125 all N_36 all N_35 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_36),N_35)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_125)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_125),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_125),N_35)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_125),N_36))))))) # label(fact_1475_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1121 (all A_49 all B_40 all C_21 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_49),hAPP_nat_nat(div_mod_nat(B_40),C_21))),C_21) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_49),B_40)),C_21)) # label(fact_2414_zmod__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1122 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(uminus_uminus_real,X)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),X))))))) # label(fact_3504_ln__one__minus__pos__lower__bound) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1123 (all C_77 all B_137 all A_177 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_137),A_177)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_77),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_77),A_177)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_77),B_137)))))) # label(fact_1067_mult__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1124 (all Z_1 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(norm_norm_complex,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_1),hAPP_complex_complex(cnj,Z_1)))) # label(fact_4063_complex__mod__mult__cnj) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1125 (all A_190 all B_146 all K_7 (A_190 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_146),K_7) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_146),A_190)))) # label(fact_947_dvdI) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1126 (all I_1 all P_1 all P_5 ((all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),P_5)) -> (hBOOL(hAPP_nat_bool(P_1,I_2)) -> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,I_2)),P_5)))))) -> (hBOOL(hAPP_nat_bool(P_1,I_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),P_5)) -> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)))))) # label(fact_3249_mod__induct__0) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1127 (all W_1 all Y_2 all X_1 all Z_2 (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,W_1),Z_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Y_2)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,W_1),Y_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Z_2)) <-> W_1 = X_1 | Z_2 = Y_2)) # label(fact_644_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1128 (all X_1 all A_1 (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_1)) -> hBOOL(hAPP_int_bool(nat_is_nat,X_1))))) # label(fact_5139_Nat__Transfer_Otransfer__int__nat__set__function__closures_I6_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1129 (all F all G ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_nat_real(F,N_2))),hAPP_nat_real(G,N_2)))) -> (hBOOL(summable_real(G)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f352196356l_real(suminf_real,F)),hAPP_f352196356l_real(suminf_real,G))) & hBOOL(summable_real(F))))) # label(fact_4857_summable__le2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1130 (all A_27 all B_22 all C_9 (is_int(C_9) -> (zero_zero_int != C_9 -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_27),C_9)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_22),C_9)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_27),B_22)))) # label(fact_2537_div__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1131 (all A_198 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,zero_zero_rat),A_198)) -> A_198 = zero_zero_rat)) # label(fact_889_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1132 (all Lx_1 all Rx_1 all Ry_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_1),Rx_1)),Ry_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_1),Ry_1))) # label(fact_500_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1133 (all A_196 all N_63 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_196),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_196),N_63)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_196),N_63)),A_196)) # label(fact_904_power__commutes) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1134 (all P all Q all R hAPP_nat_real(hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,P),Q),R) = hAPP_P731461727l_real(P,hAPP_n332733730l_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1135 (all F all A_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (hBOOL(hAPP_f448129468l_bool(resSet(Ma),A_1)) -> ((all X_2 (is_int(X_2) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),A_1)) -> (all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),A_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(F,X_2)),hAPP_int_int(F,Xa)),Ma)) -> X_2 = Xa))))))) -> hBOOL(hAPP_f448129468l_bool(resSet(Ma),image_int_int(F,A_1))))))) # label(fact_4599_ResSet__image) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1136 (all M_14 all D_4 (is_int(M_14) -> (hAPP_int_int(div_mod_int(M_14),D_4) = zero_zero_int -> (exists Q_4 (M_14 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D_4),Q_4) & is_int(Q_4)))))) # label(fact_2773_zmod__eq__0D) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1137 (all N_43 all A_131 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_131)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_131),N_43)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_131),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_131),N_43)))))) # label(fact_1425_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1138 (all N_26 hAPP_nat_rat(semiri151668891at_rat,N_26) = hAPP_int_rat(number_number_of_rat,hAPP_nat_int(semiri1621563631at_int,N_26))) # label(fact_2826_number__of__int) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1139 (all X_40 all N_69 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_40),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_40),N_69)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_40),N_69))) # label(fact_429_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1140 (all A_182 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_182),one_one_nat) = A_182) # label(fact_1021_power__one__right) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1141 (all A_92 A_92 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_92),one_one_rat)) # label(fact_2104_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1142 (all K_1 all L ((exists N_2 L = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),L)))) # label(fact_3371_le__Suc__ex__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1143 (all W all Z1 all Z2 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z2))) # label(fact_132_zadd__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1144 (all A (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),zero_zero_real)))) # label(fact_307_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1145 (all A_29 all B_24 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_29),B_24)) -> hAPP_int_int(div_mod_int(B_24),A_29) = zero_zero_int)) # label(fact_2514_dvd__imp__mod__0) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1146 (all A_105 A_105 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A_105)) # label(fact_1987_add__0__left) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1147 (all X_1 all Y_2 (X_1 = Y_2 <-> hAPP_real_real(sqrt,Y_2) = hAPP_real_real(sqrt,X_1))) # label(fact_3311_real__sqrt__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1148 (all A_239 all B_167 all V_15 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_239),B_167)),hAPP_int_real(number267125858f_real,V_15)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_239),hAPP_int_real(number267125858f_real,V_15))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_167),hAPP_int_real(number267125858f_real,V_15)))) # label(fact_186_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1149 (all Ma (one_one_int = hAPP_int_int(abs_abs_int,Ma) <-> hAPP_int_int(legacy_zgcd(zero_zero_int),Ma) = one_one_int)) # label(fact_4519_zgcd__0__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1150 (all K_2 (is_int(K_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2)) -> -(all N_2 K_2 != hAPP_nat_int(semiri1621563631at_int,N_2))))) # label(fact_3260_nonneg__int__cases) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1151 (all X_46 all Y_34 (hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_46),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_34),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),X_46)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),Y_34)) -> Y_34 = X_46)))) # label(fact_206_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1152 (all B_94 all C_52 all A_116 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_116)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_94),C_52)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_94),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_116),C_52)))))) # label(fact_1876_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1153 (all A_103 all C_45 all B_85 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_103),C_45)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_85),C_45))) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_103),B_85)))) # label(fact_2002_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1154 (all P_3 all Y all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),Y)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y),N)))))) # label(fact_1461_zpower__zdvd__prop1) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1155 (all B_1_1 all B_2_1 is_int(hAPP_P1175774780nt_int(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oint) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1156 (all V_9 all U_2 all Y_19 all X_25 all A_127 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_25),A_127)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_19),A_127)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),U_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),V_9)) -> (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,U_2),V_9) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,U_2),X_25)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_9),Y_19))),A_127)))))))) # label(fact_1464_convex__bound__le) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1157 (all X_1 all Y_2 (Y_2 != zero_zero_rat | X_1 != zero_zero_rat <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_1),X_1)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_2),Y_2)))))) # label(fact_231_sum__squares__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1158 (all N_1 all Ma (one_one_nat = N_1 & one_one_nat = Ma <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),Ma) = one_one_nat)) # label(fact_1813_nat__mult__eq__one) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1159 (all B_129 all A_169 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_169)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_129),zero_z126310315umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_169),B_129)),zero_z126310315umeral))))) # label(fact_1107_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1160 (all Y hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_real_real(tan,Y)) = hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),Y))) # label(fact_3619_tan__inverse) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1161 (all X_1 all Y_2 all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B),X_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B),Y_2)))))) # label(fact_1453_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1162 (all N_62 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,one_on1645066479umeral),N_62) = one_on1645066479umeral) # label(fact_972_power__one) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1163 (all D all Sa exists Z all X_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z),X_2)) -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_2),Sa))) <-> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_2),Sa)))))) # label(fact_2258_pinf_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1164 (all K_2 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),min) = hAPP_int_int(succ,K_2)) # label(fact_3296_diff__bin__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1165 (all B_148 all A_192 all C_84 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_192),C_84)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_192),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_148),C_84))))) # label(fact_929_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1166 (all A_23 (is_int(A_23) -> (zero_zero_int != A_23 -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_23),A_23) = one_one_int))) # label(fact_2553_div__self) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1167 (all X_1 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat,X_1)) = hAPP_int_int(number_number_of_int,min)))) # label(fact_2752_EvenOdd_Oneg__one__odd__power) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1168 (all M_12 hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_n1108039445umeral(semiri1619134803umeral,M_12)))) # label(fact_2906_zero__le__imp__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1169 (all J_2 all A_3 all K_2 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),zero_zero_int),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,J_2),K_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),K_2))),K_2)),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,J_2),K_2)),A_3),P_3))))))) # label(fact_2238_aux______2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1170 (all Ma all N_1 (N_1 != Ma & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)))) # label(fact_1633_nat__less__le) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1171 (all A all N_1 (zero_zero_nat != N_1 & A = zero_z891286103de_int <-> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A),N_1) = zero_z891286103de_int)) # label(fact_1294_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1172 (all C_68 all A_162 all B_123 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_162),B_123)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_68)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_162),C_68)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_123),C_68)))))) # label(fact_1148_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1173 (all A_3 all B_2 hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(div_mod_nat(A_3),B_2)) = hAPP_int_int(div_mod_int(hAPP_nat_int(semiri1621563631at_int,A_3)),hAPP_nat_int(semiri1621563631at_int,B_2))) # label(fact_3085_zmod__int) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1174 (all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1)) <-> hBOOL(hAPP_int_bool(even_odd_even_int,X_1)))) # label(fact_3222_int__even__iff__2__dvd) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1175 (all M all N hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),N)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(real_nat,M)),N)) # label(fact_3655_real__of__nat__power) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1176 (all I all M hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),I))))) # label(fact_3055_less__add__Suc2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1177 (all Y hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arctan,Y)))) # label(fact_3503_arctan__lbound) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1178 (all X_1 hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_4867_Ln_Oaux2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1179 (all N_29 all K_6 all M_16 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_6),M_16)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_6),N_29)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_6),hAPP_C498520661umeral(div_mo1740067990umeral(M_16),N_29)))))) # label(fact_2449_dvd__mod) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1180 (all Ma all N_1 (-(-hBOOL(hAPP_nat_bool(even_odd_even_nat,Ma)) <-> -hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))) <-> -hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N_1))))) # label(fact_3755_odd__add) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1181 (all X (hBOOL(hAPP_nat_bool(even_odd_even_nat,X)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,X)),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))))) # label(fact_3772_even__nat__plus__one__div__two) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1182 (all A_33 all B_27 hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_33),B_27)),B_27) = zero_z891286103de_int) # label(fact_2498_mod__mult__self2__is__0) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1183 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(suc,N))) -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> M = hAPP_nat_nat(suc,N)))) # label(fact_2874_le__SucE) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1184 (all Z_7 all W_2 all Y_9 all X_12 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,Y_9),X_12)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,Z_7),W_2)) -> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_12),W_2)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Y_9),Z_7)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,X_12),Y_9)),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,W_2),Z_7))))) # label(fact_2557_div__mult__div__if__dvd) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1185 (all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),hAPP_int_int(bit1,pls))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,X_1)),one_one_real)))) # label(fact_409_less__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1186 (all C_37 all A_95 all B_77 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_95),B_77)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_95),C_37)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_77),C_37))))) # label(fact_2061_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1187 (all B_96 all A_118 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_118)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_96)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_118),B_96)))))) # label(fact_1867_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1188 (all B_121 all A_160 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_160),zero_z126310315umeral)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_121)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_160),B_121)),zero_z126310315umeral))))) # label(fact_1155_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1189 (all N_12 all R_6 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),R_6)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,R_6),one_on1645066479umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,R_6),hAPP_nat_nat(suc,N_12))),R_6))))) # label(fact_3155_realpow__Suc__le__self) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1190 (all X_1 (is_int(X_1) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)) -> -(all K (is_int(K) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K) != X_1))))) # label(fact_3248_zEvenE) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1191 (all A_55 all C_25 all B_46 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(div_mod_nat(A_55),C_25)),B_46)),C_25) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_55),B_46)),C_25)) # label(fact_2398_zmod__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1192 (all X all Y zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(cos,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(sin,Y))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(sin,Y))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3741_sin__cos__add) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1193 (all X all N hAPP_complex_complex(cnj,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X),N)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(cnj,X)),N)) # label(fact_4078_complex__cnj__power) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1194 (all Ma all N_1 image_int_nat(nat,hAPP_i1948725293t_bool(ord_at875362053st_int(hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,N_1))) = hAPP_n1699378549t_bool(ord_at238088361st_nat(Ma),N_1)) # label(fact_5071_SetInterval_Otransfer__nat__int__set__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1195 (all B_28 all A_34 zero_z891286103de_int = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_28),A_34)),B_28)) # label(fact_2494_mod__mult__self1__is__0) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1196 (all A_139 all C_61 all B_108 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_139),C_61)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_108),C_61))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_61)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_139),B_108))))) # label(fact_1353_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1197 (all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,N)),hAPP_nat_nat(fact,N)) = hAPP_nat_nat(fact,hAPP_nat_nat(suc,N))) # label(fact_4566_fact_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1198 (all C all D all A all B all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D),A)),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D),B)),Ma))))) # label(fact_1033_zcong__zmult__prop2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1199 (all G all F all X_1 all D (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D)) -> ((all Z (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D)) -> Z = hAPP_real_real(G,hAPP_real_real(F,Z)))) -> ((all Z (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D)) -> hBOOL(hAPP_real_bool(isCont_real_real(F),Z)))) -> (exists E ((all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_1),hAPP_real_real(F,X_1))))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_1),hAPP_real_real(F,X_1)))),E)) -> hAPP_real_real(F,hAPP_real_real(G,Y_1)) = Y_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),E)))))))) # label(fact_5180_isCont__inv__fun__inv) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1200 (all A_28 all B_23 all C_10 (C_10 != zero_z126310315umeral -> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_28),B_23) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_10),A_28)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_10),B_23)))) # label(fact_2535_div__mult__mult1) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1201 (all B_131 all A_171 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_171),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_131)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_171),B_131)),zero_zero_rat))))) # label(fact_1097_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1202 (all X_9 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_9)) -> hAPP_nat_real(semiri132038758t_real,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_9))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(semiri132038758t_real,hAPP_int_nat(nat,X_9))),hAPP_nat_real(semiri132038758t_real,hAPP_int_nat(nat,X_9))))) # label(fact_2811_of__nat__double) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1203 (all X_1 all A_1 (hBOOL(hAPP_int_bool(A_1,X_1)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_1)))) # label(fact_138_mem__def) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1204 (all N all M all A_3 all B_2 (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) = one_one_int -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),M)) = one_one_int)) # label(fact_4962_coprime__exp2__int) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1205 (all A_1 (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1)) -> image_nat_int(semiri1621563631at_int,image_int_nat(nat,A_1)) = A_1)) # label(fact_4618_transfer__nat__int__set__return__embed) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1206 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N) = M)) # label(fact_1772_le__add__diff__inverse2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1207 (all R_3 all A_3 rcis(R_3,A_3) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_3)),cis(A_3))) # label(fact_4266_rcis__def) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1208 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),Y)))) # label(fact_3362_real__gt__half__sum) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1209 (all Ma all W_1 (is_int(W_1) -> ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1)) -> zero_zero_nat = Ma) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1)) -> hAPP_nat_int(semiri1621563631at_int,Ma) = W_1) <-> hAPP_int_nat(nat,W_1) = Ma))) # label(fact_3166_nat__eq__iff2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1210 (all A_3 all B_2 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,expi(A_3)),expi(B_2)) = expi(hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_3),B_2))) # label(fact_4000_expi__add) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1211 (all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),zero_zero_int)) -> zero_zero_nat = hAPP_int_nat(nat,Z_1))) # label(fact_2725_nat__le__0) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1212 (all V_2 ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls)) -> hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_int_nat(number_number_of_nat,V_2))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_int_nat(number_number_of_nat,V_2)) = one_one_nat))) # label(fact_3299_nat__1__add__number__of) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1213 (all A_23 (A_23 != zero_z891286103de_int -> one_on1684967323de_int = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_23),A_23))) # label(fact_2556_div__self) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1214 (all M all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N)) -> hAPP_int_int(legacy_zgcd(M),N) = hAPP_int_int(legacy_zgcd(N),hAPP_int_int(div_mod_int(M),N)))) # label(fact_4514_zgcd__non__0) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1215 (all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(inv(P_3),A_3))))))) # label(fact_2127_inv__g__1) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1216 (all X hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = one_one_real) # label(fact_3728_sin__cos__squared__add) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1217 (all B_1_1 all B_2_1 (is_int(B_2_1) -> is_bool(hAPP_int_bool(B_1_1,B_2_1)))) # label(gsy_c_hAPP_000tc__Int__Oint_000tc__HOL__Obool) # label(hypothesis) # label(non_clause). [assumption]. 7.03/7.06 1218 (all L all K_1 (is_int(K_1) & is_int(L) -> (zero_zero_int != K_1 -> (zero_zero_int = L -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),K_1) = divmod_int(K_1,L)) & (L != zero_zero_int -> divmod_int(K_1,L) = hAPP_P1975530577nt_int(produc713050258nt_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(sgn_sgn_int,L))),hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,fdisj(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),L)),hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_1)),hAPP_bool_bool(hAPP_b589554111l_bool(fconj,hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,L),zero_zero_int)),hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),zero_zero_int)))),pdivmod(K_1,L)),hAPP_P1975530577nt_int(produc1518849193nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,hAPP_f617310012nt_int(hAPP_f197974549nt_int(cOMBB_638531587nt_int,cOMBS_1891347093nt_int),hAPP_f1148171384nt_int(hAPP_f334321603nt_int(cOMBB_1484610687nt_int,hAPP_f1582586579nt_int(cOMBC_707974837nt_int,hAPP_f251264923nt_int(hAPP_f279307923nt_int(cOMBB_662120866nt_int,if_Pro1731782967nt_int),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),zero_zero_int)))),hAPP_i1584592887nt_int(hAPP_f465821005nt_int(cOMBC_1964283556nt_int,hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),uminus_uminus_int)),zero_zero_int)))),hAPP_f1574055885nt_int(hAPP_f960630521nt_int(cOMBC_1805044090nt_int,hAPP_f142232355nt_int(hAPP_f1377453843nt_int(cOMBB_320287386nt_int,cOMBB_1846624359nt_int),hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,minus_minus_int),uminus_uminus_int)),one_one_int)))),hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(abs_abs_int,L))))),pdivmod(K_1,L)))))) & (K_1 = zero_zero_int -> divmod_int(K_1,L) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int)))) # label(fact_4854_divmod__int__pdivmod) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1219 (all Z_11 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_11),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_11),Z_11)) # label(fact_296_mult__2__right) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1220 (all B_1_1 all B_2_1 is_bool(hAPP_real_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__RealDef__Oreal_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1221 (all X_1 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,X_1)),hAPP_real_real(sqrt,Y_2))))) # label(fact_3317_real__sqrt__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1222 (all A_61 zero_zero_nat = hAPP_nat_nat(div_mod_nat(zero_zero_nat),A_61)) # label(fact_2373_mod__0) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1223 (all A_62 A_62 = hAPP_Q1762011733de_int(div_mo231679042de_int(A_62),zero_z891286103de_int)) # label(fact_2371_mod__by__0) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1224 (all N_1 all A all B all P_1 ((all A_2 all B_4 all C_3 (hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4))) -> (hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,B_4),C_3))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),B_4)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_4),C_3)) -> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),C_3)))))))) -> (-hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A),B))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) -> -hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_1))),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_1))))))))) # label(fact_4247_not__P__Bolzano__bisect) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1225 (all D all F all X_1 all L (hBOOL(hAPP_real_bool(deriv_real(F,X_1),L)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D)) -> ((all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y_1))),D)) -> hAPP_real_real(F,Y_1) = hAPP_real_real(F,X_1))) -> zero_zero_real = L)))) # label(fact_3886_DERIV__local__const) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1226 (all P_1 hBOOL(hAPP_f448129468l_bool(nat_nat_set,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),P_1))))) # label(fact_4748_Nat__Transfer_Otransfer__int__nat__set__function__closures_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1227 (all N_37 all A_126 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_126)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_126),one_one_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_126),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_126),N_37))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_126),N_37)))))) # label(fact_1467_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1228 (all X all Y hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = hAPP_nat_nat(nat_gcd(X),Y)) # label(fact_4612_gcd__eq__nitpick__gcd) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1229 (all P all Q all R hAPP_b40753821nt_int(P,hAPP_int_bool(Q,R)) = hAPP_i2104874254nt_int(hAPP_f251264923nt_int(hAPP_f279307923nt_int(cOMBB_662120866nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1230 (all K_2 all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N)))))) # label(fact_1756_dvd__diffD1) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1231 (all K_2 (is_int(K_2) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),pls) = K_2)) # label(fact_1520_diff__bin__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1232 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(ln,X_1)),zero_zero_real))))) # label(fact_3443_ln__less__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1233 (all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(suc,N_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)))) # label(fact_3063_less__Suc__eq__le) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1234 (all C_64 all A_142 all B_111 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_64),A_142)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_64),B_111))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_64)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_142),B_111))))) # label(fact_1333_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1235 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X)),hAPP_real_real(exp_real,X)))) # label(fact_3845_exp__ge__add__one__self) # label(axiom) # label(non_clause). [assumption]. 7.03/7.06 1236 (all A_140 all C_62 all B_109 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_140),C_62)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_109),C_62))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_62)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_140),B_109))))) # label(fact_1346_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1237 (all A_108 (is_int(A_108) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_108),zero_zero_int) = A_108)) # label(fact_1962_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1238 (all F (hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F),zero_zero_real,sequentially)) <-> hBOOL(tendsto_nat_real(F,zero_zero_real,sequentially)))) # label(fact_5041_LIMSEQ__rabs__zero) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1239 (all A_63 hAPP_C498520661umeral(div_mo1740067990umeral(A_63),A_63) = zero_z126310315umeral) # label(fact_2366_mod__self) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1240 (all N_65 all A_199 (zero_zero_rat != A_199 -> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_199),N_65) != zero_zero_rat)) # label(fact_884_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1241 (all Y_2 all X_1 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_2)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),Y_2))) # label(fact_3238_even__prod__div__2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1242 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hAPP_P1175774780nt_int(product_fst_int_int,negDivAlg(A_3,B_2)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)))) # label(fact_4225_div__neg__pos) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1243 (all A_48 all B_39 all C_20 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_48),C_20)),hAPP_nat_nat(div_mod_nat(B_39),C_20))),C_20) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_48),B_39)),C_20)) # label(fact_2418_mod__add__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1244 (all A_63 hAPP_Q1762011733de_int(div_mo231679042de_int(A_63),A_63) = zero_z891286103de_int) # label(fact_2367_mod__self) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1245 (all N_16 all M_5 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_16),M_5)) -> hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_5),N_16)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_nat_complex(semiri2020571505omplex,M_5)),hAPP_nat_complex(semiri2020571505omplex,N_16)))) # label(fact_3099_of__nat__diff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1246 (all A_183 all M_22 all N_57 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_183),M_22)),N_57) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_183),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_22),N_57))) # label(fact_1015_power__mult) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1247 (all B_118 all A_157 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_157)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_118),zero_z126310315umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_118),A_157)),zero_z126310315umeral))))) # label(fact_1176_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1248 (all A_15 all B_11 (is_int(B_11) -> (B_11 != zero_zero_int -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_15),B_11)),B_11) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_15),B_11)),one_one_int)))) # label(fact_2632_div__add__self2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1249 (all X all Y hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))),one_one_real))) # label(fact_3380_cos__x__y__le__one) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1250 (all A all E_2 all C all B all D (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),E_2)),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),E_2)),D))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B)),E_2)),C)),D)))) # label(fact_1561_le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1251 (all X_34 all P_7 all Q_10 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_34),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_10)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_34),P_7)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_34),Q_10))) # label(fact_669_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1252 (all X all Y all Z_1 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Z_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y),Z_1))) # label(fact_198_zadd__left__commute) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1253 (all N all A_3 all B_2 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),B_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N)),A_3)))))) # label(fact_4186_prime__power__dvd__cancel__right) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1254 (all A all B (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B)),zero_zero_rat)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),B)))) # label(fact_2209_le__iff__diff__le__0) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1255 (all X_39 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_39),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_39),X_39)) # label(fact_431_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1256 (all X_7 all Q_6 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_7),hAPP_nat_nat(suc,Q_6)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_7),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_7),Q_6))) # label(fact_3028_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1257 (all Ma all N_1 (Ma = N_1 <-> hAPP_n1108039445umeral(semiri1619134803umeral,Ma) = hAPP_n1108039445umeral(semiri1619134803umeral,N_1))) # label(fact_2829_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1258 (all A_113 all B_91 all C_51 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_113),B_91)),C_51) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_113),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_91),C_51))) # label(fact_1905_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1259 (all M_12 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_nat_rat(semiri151668891at_rat,M_12)))) # label(fact_2904_zero__le__imp__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1260 (all J_2 all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),zero_zero_int),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),A_3)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),J_2))),P_3)))))))) # label(fact_2216_MultInvPair__distinct) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1261 (all V_20 all W_16 all Z_15 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_20)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,W_16)),Z_15)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_20),W_16))),Z_15)) # label(fact_79_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1262 (all B_72 all C_35 all A_84 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_84)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_72),C_35)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_72),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_84),C_35)))))) # label(fact_2157_add__increasing) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1263 (all X all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (one_one_real != A_3 -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_real_real(log(A_3),hAPP_real_real(inverse_inverse_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(log(A_3),X)))))) # label(fact_3875_log__inverse) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1264 (all A_231 all B_165 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_231),B_165) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_165),A_231)) # label(fact_516_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1265 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sin,X)))))) # label(fact_3708_sin__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1266 (all K_2 hAPP_int_rat(hAPP_int_fun_int_rat(fract,zero_zero_int),K_2) = zero_zero_rat) # label(fact_4673_rat__number__collapse_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1267 (all Ma all N_1 all K_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1)) -> (Ma = N_1 <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma)))) # label(fact_1846_nat__mult__eq__cancel1) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1268 (all P_1 image_int_nat(nat,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_f1448486952t_bool(hAPP_f696844121t_bool(cOMBB_nat_bool_int,P_1),nat)))) = collect_nat(P_1)) # label(fact_4749_Nat__Transfer_Otransfer__nat__int__set__functions_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1269 (all Y_3 all Y_2 all X_3 all X_1 (hBOOL(tendsto_nat_real(X_3,X_1,sequentially)) -> (hBOOL(tendsto_nat_real(Y_3,Y_2,sequentially)) -> ((exists N_3 all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(X_3,N_2)),hAPP_nat_real(Y_3,N_2))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2)))))) # label(fact_5034_LIMSEQ__le) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1270 (all K_1 all I_1 all J (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),I_1) = K_1 <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),I_1) = J))) # label(fact_1771_le__imp__diff__is__add) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1271 (all A_201 all E_3 all B_154 all C_87 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_201),B_154)),E_3)),C_87) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_201),E_3)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_154),E_3)),C_87))) # label(fact_871_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1272 (all K_2 hAPP_int_int(pred,K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,min),K_2)) # label(fact_4048_add__Min) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1273 (all X all Y hAPP_int_int(int_lcm(X),Y) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(nat_lcm(hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,X))),hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,Y))))) # label(fact_4598_Nitpick_Oint__lcm__def) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1274 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),hAPP_int_int(bit1,K_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),K_1)))) # label(fact_781_rel__simps_I26_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1275 (all X_2 hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,sin),uminus_uminus_real))),sin))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,minus_minus_real),hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,cos),uminus_uminus_real))),cos))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_2),zero_zero_real))) # label(fact_4803_lemma__DERIV__sin__cos__minus) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1276 (all A all B (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(posDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A),B))) -> (-(hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B))) -> posDivAlg(A,B) = hAPP_P1975530577nt_int(adjust(B),posDivAlg(A,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B)))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),A) = posDivAlg(A,B)))) # label(fact_2784_posDivAlg_Opsimps) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1277 (all X all Y (Y != zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),hAPP_nat_nat(div_mod_nat(X),Y)))) # label(fact_4361_gcd__non__0__nat) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1278 (all Z_2 hilbert_Eps_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_f1402056515l_bool(hAPP_f824790735l_bool(cOMBB_real_bool_real,hAPP_r1134773055l_bool(fequal_real,hAPP_complex_real(re,hAPP_complex_complex(sgn_sgn_complex,Z_2)))),cos))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_f1402056515l_bool(hAPP_f824790735l_bool(cOMBB_real_bool_real,hAPP_r1134773055l_bool(fequal_real,hAPP_complex_real(im,hAPP_complex_complex(sgn_sgn_complex,Z_2)))),sin))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,pi)))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_eq_real),pi))))) = arg(Z_2)) # label(fact_4808_arg__def) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1279 (all B_5 all Q_8 all R_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_8)),R_5))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_5),B_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_5)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Q_8)))))) # label(fact_732_q__pos__lemma) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1280 (all N_55 all A_152 all B_114 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_152),B_114)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_152)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_152),N_55)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_114),N_55)))))) # label(fact_1234_power__mono) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1281 (all N all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N)),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N)))) -> N = hAPP_int_nat(nat,archim1246769320r_real(X))))) # label(fact_4035_floor__eq4) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1282 (all C_39 all D_7 all A_97 all B_79 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_97),B_79)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_39),D_7)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_97),C_39)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_79),D_7)))))) # label(fact_2050_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1283 (all N all M hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,quickcheck_int_of(N)),quickcheck_int_of(M))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,N),M)) # label(fact_4569_div__code__int__def) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1284 (all A_201 all E_3 all B_154 all C_87 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_201),E_3)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_154),E_3)),C_87)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_201),B_154)),E_3)),C_87)) # label(fact_872_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1285 (all B_71 all A_83 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_83)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_71)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_83),B_71)))))) # label(fact_2167_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1286 (all A all B all C (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A),C) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A),B) <-> B = C)) # label(fact_1938_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1287 (all X_3 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_int_int(standardRes(Ma),X_3)),collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),Ma))))))) # label(fact_4787_SR__pos) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1288 (all A_35 (is_int(A_35) -> A_35 = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_35),one_one_int))) # label(fact_2478_div__by__1) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1289 (all A_106 A_106 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A_106)) # label(fact_1980_add__0) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1290 (all P all Q all R hAPP_i418383825t_bool(hAPP_f1222178131t_bool(hAPP_f743790483t_bool(cOMBB_1647910634ol_int,P),Q),R) = hAPP_f737665877t_bool(P,hAPP_i157317923t_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__fun_) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1291 (all X_1 all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(number_number_of_int,X_1)),hAPP_int_int(number_number_of_int,Y_2))) <-> zero_zero_int = hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,Y_2)),hAPP_int_int(number_number_of_int,X_1)))) # label(fact_2589_dvd__eq__mod__eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1292 (all A_60 all B_51 all C_30 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_60),B_51)),C_30) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_60),hAPP_int_int(div_mod_int(B_51),C_30))),C_30)) # label(fact_2377_mod__mult__right__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1293 (all P all Q all R hAPP_f1375227553nt_int(P,hAPP_i1584592887nt_int(Q,R)) = hAPP_i1153444985nt_int(hAPP_f55007289nt_int(hAPP_f747475781nt_int(cOMBB_222539774nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int_038) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1294 (all Z_2 all W_1 (is_int(Z_2) & is_int(W_1) -> (Z_2 != W_1 & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_2),W_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_2),W_1))))) # label(fact_70_zless__le) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1295 (all N -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N))) # label(fact_1587_less__irrefl__nat) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1296 (all P all Q all R hAPP_i418383825t_bool(hAPP_f289318786t_bool(hAPP_f729588221t_bool(cOMBC_678732631t_bool,P),Q),R) = hAPP_f800510211t_bool(hAPP_i1796122964t_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1297 (all X_1 all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X_1)),A)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_int_real(real_int,A))))) # label(fact_4422_ceiling__le__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1298 (all D all Sa exists Z ((all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Z)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Sa))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Sa)))))) & is_int(Z))) # label(fact_2263_minf_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1299 (all B_2 all A_3 all C_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),C_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B_2),C_1))) # label(fact_4309_gcd__left__commute__nat) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1300 (all W_7 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,zero_zero_real),hAPP_int_real(number267125858f_real,W_7))),hAPP_int_real(number267125858f_real,W_7)) = hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,W_7))) # label(fact_372_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1301 (all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N))) # label(fact_1617_less__eq__nat_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1302 (all S ((exists K hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,S),hAPP_n1699378549t_bool(ord_atMost_nat,K)))) <-> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S)))) # label(fact_5195_finite__nat__iff__bounded__le) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1303 (all A_59 all B_50 all C_29 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_59),B_50)),C_29) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(div_mod_int(A_59),C_29)),B_50)),C_29)) # label(fact_2381_mod__mult__left__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1304 (all V_2 all W hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_2),W))) # label(fact_56_times__numeral__code_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1305 (all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)) <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(semiri1619134803umeral,Ma)),hAPP_n1108039445umeral(semiri1619134803umeral,N_1))))) # label(fact_2947_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1306 (all A_189 all B_145 all C_82 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_189),B_145)),C_82)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_189),C_82)))) # label(fact_954_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1307 (all A_3 all B_2 cis(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),B_2)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,cis(A_3)),cis(B_2))) # label(fact_4148_cis__mult) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1308 (all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),N))) # label(fact_4029_coprime__plus1) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1309 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arctan,Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) & hAPP_real_real(tan,hAPP_real_real(arctan,Y)) = Y & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arctan,Y))))) # label(fact_3622_arctan) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1310 (all A_3 all N all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),A_3))))) # label(fact_443_zprime__zdvd__power) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1311 (all R_3 exists N_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_2)),R_3)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_2),one_one_int)))) & is_int(N_2))) # label(fact_4479_real__lb__ub__int) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1312 (all A all N_1 (N_1 != zero_zero_nat & zero_z126310315umeral = A <-> zero_z126310315umeral = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A),N_1))) # label(fact_1292_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1313 (all A all C all B (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A),C)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B),C))) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A),B)))) # label(fact_2074_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1314 (all A_20 all C_6 all B_16 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,C_6),B_16)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_20),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_16),C_6)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_20),B_16)),C_6))) # label(fact_2571_div__mult__swap) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1315 (all A_200 all B_153 all C_86 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_200),C_86)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_153),C_86)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_200),B_153)),C_86)) # label(fact_879_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1316 (all K_2 hAPP_int_int(succ,K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),one_one_int)) # label(fact_3294_succ__def) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1317 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(exp_real,X)))) # label(fact_3833_exp__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1318 (all A_3 hAPP_int_rat(hAPP_int_fun_int_rat(fract,zero_zero_int),one_one_int) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),zero_zero_int)) # label(fact_4670_eq__rat_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1319 (all B_2 all D_3 all A_3 (one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),A_3) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),B_2) = one_one_nat -> one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2))))) # label(fact_4354_coprime__mult__nat) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1320 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3343_real__sqrt__pow2__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1321 (all A_9 all N_14 all B_7 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_9),hAPP_nat_nat(suc,N_14))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_7),hAPP_nat_nat(suc,N_14)))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_7)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_9),B_7))))) # label(fact_3113_power__le__imp__le__base) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1322 (all Xa_4 all X_16 all Y_13 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_4),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_16),Y_13)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_4),X_16)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_4),Y_13))) # label(fact_2294_mult__right_Oadd) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1323 (all B_2 all A_3 all A_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),A_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_5),B_2)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)))))) # label(fact_2595_zdiv__mono1__neg) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1324 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),Y)) -> (exists X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y),one_one_real))) & Y = hAPP_real_real(exp_real,X_2) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2)))))) # label(fact_3858_lemma__exp__total) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1325 (all M all K_2 all N (hAPP_int_int(legacy_zgcd(K_2),N) = one_one_int -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),N)),hAPP_int_int(legacy_zgcd(M),N))))) # label(fact_4516_zgcd__zmult__zdvd__zgcd) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1326 (all A_13 all B_9 (is_int(A_13) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_13),B_9)),B_9)),hAPP_int_int(div_mod_int(A_13),B_9)) = A_13)) # label(fact_2673_mod__div__decomp) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1327 (all A_149 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_149),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_149),one_one_real)))) # label(fact_1259_less__add__one) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1328 (all A_194 all B_150 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_194),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_150),A_194)))) # label(fact_919_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1329 (all N all M hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M)),N) = M) # label(fact_1746_diff__add__inverse) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1330 (all Lx_1 all Rx_1 all Ry_1 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_1),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_1),Ry_1)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_1),Rx_1)),Ry_1)) # label(fact_504_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1331 (all Y all X all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Y),zero_zero_int),P_3)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)),zero_zero_int),P_3)))))) # label(fact_1463_zcong__zmult__prop3) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1332 (all A_11 all N_17 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_11),hAPP_nat_nat(suc,N_17)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_11),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_11),N_17))) # label(fact_3012_power__Suc) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1333 (all D2 all D1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D2)) -> (exists E (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),E)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,E),D1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,E),D2))))))) # label(fact_3307_real__lbound__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1334 (all C all D all A all B (D != C & A != B <-> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),D)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),C)) != hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),D)))) # label(fact_627_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1335 (all N_37 all A_126 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_126)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_126),one_on1684967323de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_126),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_126),N_37))),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_126),N_37)))))) # label(fact_1471_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1336 (all X_1 all Y_2 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_2)),zOdd)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd)))) # label(fact_2489_odd__mult__odd__prop) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1337 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,pls)),N)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,pls)))))) # label(fact_3199_Suc__diff__eq__diff__pred) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1338 (all A_44 all B_35 hAPP_int_int(div_mod_int(A_44),B_35) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_44),B_35)),B_35)) # label(fact_2433_mod__add__self2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1339 (all W_1 all X_1 (hAPP_i1732201573de_int(number1226105091de_int,W_1) = X_1 <-> hAPP_i1732201573de_int(number1226105091de_int,W_1) = X_1)) # label(fact_175_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1340 (all B_2 all A_3 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M)) -> (one_one_int = hAPP_int_int(legacy_zgcd(A_3),M) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) -> one_one_int = hAPP_int_int(legacy_zgcd(B_2),M))))) # label(fact_4523_zgcd__zcong__zgcd) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1341 (all N_53 all M_21 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),M_21)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N_53)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_21),N_53)))))) # label(fact_1246_less__1__mult) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1342 (all A_182 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_182),one_one_nat) = A_182) # label(fact_1023_power__one__right) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1343 (all R_3 all A_3 all N hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,rcis(R_3,A_3)),N) = rcis(hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_3),N),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),A_3))) # label(fact_4251_DeMoivre2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1344 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),A_3)) -> zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)))) # label(fact_2597_div__neg__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1345 (all A hBOOL(hAPP_complex_bool(isCont_complex_real(im),A))) # label(fact_5166_Im_OisCont) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1346 (all P all Q all R hAPP_b589554111l_bool(P,hAPP_nat_bool(Q,R)) = hAPP_n1006566506l_bool(hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_005) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1347 (all X_1 all K_1 (X_1 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K_1)),one_one_int) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd)))) # label(fact_2322_zOddI) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1348 (all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A),B)) <-> (all P_4 -(hBOOL(hAPP_nat_bool(prime,P_4)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),A)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),B)))))) # label(fact_4179_coprime__prime__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1349 (all A_3 all B_2 (zero_zero_nat != hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2) -> (exists A_4 exists B_3 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_4),B_3) = one_one_nat & hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)) = B_2 & hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_4),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)) = A_3)))) # label(fact_4373_gcd__coprime__exists__nat) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1350 (all M_13 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,M_13)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,M_13))) # label(fact_2838_of__nat__Suc) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1351 (all A_3 all B_2 all C_1 all D_3 all N (hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,D_3),N) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)),C_1) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N)) -> (hAPP_int_int(legacy_zgcd(A_3),B_2) = one_one_int -> (hAPP_int_int(legacy_zgcd(B_2),C_1) = one_one_int -> (one_one_int = hAPP_int_int(legacy_zgcd(C_1),A_3) -> (exists K exists L_1 exists M_1 (is_int(M_1) & hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K),N) = hAPP_int_int(abs_abs_int,A_3) & hAPP_int_int(abs_abs_int,C_1) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_1),N) & hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,L_1),N) = hAPP_int_int(abs_abs_int,B_2) & is_int(L_1) & is_int(K))))))))) # label(fact_4538_int__triple__relprime__power__divisors) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1352 (all N all X (is_int(N) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,N)),one_one_real)),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_int_real(real_int,N))) -> N = archim856651990g_real(X))))) # label(fact_4470_ceiling__eq3) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1353 (all X all Y (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(fequal_real,X),Y)) | Y = X)) # label(help_fequal_1_1_fequal_000tc__RealDef__Oreal_T) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1354 (all W_5 ((zero_zero_nat = hAPP_int_nat(number_number_of_nat,W_5) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),hAPP_int_nat(number_number_of_nat,W_5)) = one_one_int) & (zero_zero_nat != hAPP_int_nat(number_number_of_nat,W_5) -> zero_zero_int = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),hAPP_int_nat(number_number_of_nat,W_5))))) # label(fact_724_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1355 (all M all N all R_5 all R_3 all S_3 all S_1 all T_2 all T_1 all K_2 (is_int(K_2) -> ((exists Sn exists Tn hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,K_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Sn),Tn)) = xzgcda(M,N,R_5,R_3,S_3,S_1,T_2,T_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),R_3)) -> hAPP_int_int(legacy_zgcd(R_5),R_3) = K_2)))) # label(fact_4532_xzgcd__correct__aux2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1356 (all C_2 hBOOL(hAPP_f448129468l_bool(nat_nat_set,image_nat_int(semiri1621563631at_int,C_2)))) # label(fact_4615_Nat__Transfer_Otransfer__int__nat__set__function__closures_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1357 (all R_2 hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(plus_plus_real,R_2)),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc))),R_2,sequentially))) # label(fact_5039_LIMSEQ__inverse__real__of__nat__add) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1358 (all X_3 (hBOOL(summable_complex(X_3)) -> hBOOL(summable_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_3))))) # label(fact_4845_cnj_Osummable) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1359 (all Ma all N_1 (N_1 = Ma <-> hAPP_nat_rat(semiri151668891at_rat,Ma) = hAPP_nat_rat(semiri151668891at_rat,N_1))) # label(fact_2834_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1360 (all A_226 all B_163 all C_92 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_226),B_163)),C_92) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_226),C_92)),B_163)) # label(fact_549_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1361 (all Ta all B_1 all D all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1)) -> (all X_2 (is_int(X_2) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) != X_2)))))) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Ta))) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1)),Ta))))))))) # label(fact_5103_bset_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1362 (all P (is_bool(P) -> P = fTrue | P = fFalse)) # label(help_fFalse_1_1_T) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1363 (all P all Q all R hAPP_i998473677nt_int(hAPP_f142232355nt_int(hAPP_f1377453843nt_int(cOMBB_320287386nt_int,P),Q),R) = hAPP_f2002277285nt_int(P,hAPP_i1584592887nt_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1364 (all X_41 all Y_29 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_41),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_29),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_41)),Y_29)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_41),Y_29)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_368_power2__sum) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1365 (all N_19 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),hAPP_nat_nat(suc,N_19)) = zero_zero_complex) # label(fact_2993_power__0__Suc) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1366 (all A all B all C all D (hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C),D) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C),D)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B))))) # label(fact_2092_diff__eq__diff__less__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1367 (all A_98 all C_40 all B_80 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_98),C_40)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_80),C_40))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_98),B_80)))) # label(fact_2042_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1368 (all A_107 A_107 = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_107),zero_zero_complex)) # label(fact_1972_add__0__right) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1369 (all P all Q all R hAPP_P1175774780nt_int(hAPP_f521271395nt_int(hAPP_f11151005nt_int(cOMBB_47643171nt_int,P),Q),R) = hAPP_int_int(P,hAPP_P1175774780nt_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__Int__Oint_000tc__prod_Itc__Int__Oin) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1370 (all N_61 all X_29 all Y_23 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_29),Y_23)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_29),N_61)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_23),N_61))))) # label(fact_987_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1371 (all C_41 all A_99 all B_81 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_41),A_99)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_41),B_81))) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_99),B_81)))) # label(fact_2039_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1372 (all M_8 all N_22 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_8)),hAPP_n522471361de_int(semiri1424489471de_int,N_22)) = hAPP_n522471361de_int(semiri1424489471de_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22))) # label(fact_2955_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1373 (all A (zero_zero_rat = A <-> zero_zero_rat = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_39_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1374 (all A_3 all B_2 (B_2 != zero_zero_nat -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),B_2)))) # label(fact_4346_gcd__le2__nat) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1375 (all L all U hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_gr660468384an_nat(L),U)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,U),hAPP_nat_nat(suc,L))) # label(fact_4633_card__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1376 (all X all A_3 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,archim1246769320r_real(X)),A_3) = archim1246769320r_real(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_int_real(real_int,A_3)))) # label(fact_4426_floor__subtract) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1377 (all F all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) -> ((all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),B)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X_2)) -> (exists Y_1 (hBOOL(hAPP_real_bool(deriv_real(F,X_2),Y_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,A)),hAPP_real_real(F,B)))))) # label(fact_4003_DERIV__pos__imp__increasing) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1378 (all Ta all D all D_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),D_1)) -> (all X_2 all K (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,K),D_1))),Ta))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Ta))))))) # label(fact_1528_inf__period_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1379 (all X_1 all Y_2 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_2)),zEven)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)) | hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven)))) # label(fact_2896_EvenOdd_Oeven__product) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1380 (all X all Y hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,Y),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) = hAPP_complex_complex(invers1449016382omplex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,of_real_complex(X)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),of_real_complex(Y))))) # label(fact_3986_complex__inverse__complex__split) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1381 (all A_3 all N all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),N))) -> (N != zero_zero_nat -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_3),B_2))))) # label(fact_2276_pow__divides__pow__int) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1382 (all A_219 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_219),zero_zero_complex) = A_219) # label(fact_608_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1383 (all X hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),X)) = hAPP_complex_complex(uminus473333897omplex,X)) # label(fact_3978_complex__i__mult__minus) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1384 (all C_83 all D_16 all A_191 all B_147 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_191),B_147)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_83),D_16)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_191),C_83)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_147),D_16)))))) # label(fact_940_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1385 (all P (is_bool(P) -> P = fTrue | P = fFalse)) # label(help_fTrue_1_1_T) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1386 (all A_236 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_236),hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit1,pls))) = A_236) # label(fact_242_mult__numeral__1__right) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1387 (all A_3 all B_2 all P_3 all Q_2 hAPP_P1175774780nt_int(twoSqu1907779896sum2sq,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),P_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2))),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Q_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),P_3)))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(twoSqu1907779896sum2sq,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_3),B_2))),hAPP_P1175774780nt_int(twoSqu1907779896sum2sq,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2)))) # label(fact_1843_mult__sum2sq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1388 t = zero_zero_int -> zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,s),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),one_one_int) # label(fact_4_calculation_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1389 (all B_128 all A_168 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_168)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_128)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_168),B_128)))))) # label(fact_1113_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1390 (all X all Y (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,X),Y)) | X != Y)) # label(help_fequal_2_1_fequal_000tc__Int__Oint_T) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1391 (all N all M ((M = zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) = N) & (zero_zero_nat != M -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N))))) # label(fact_3181_add__eq__if) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1392 (all X all Y ((Y != zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),hAPP_nat_nat(div_mod_nat(X),Y)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)) & (zero_zero_nat = Y -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = X))) # label(fact_4360_gcd__nat_Osimps) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1393 (all P all Q P = hAPP_nat_rat(cOMBK_rat_nat(P),Q)) # label(help_COMBK_1_1_COMBK_000tc__Rat__Orat_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1394 (all A_3 all N all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_2),N))) -> (N != zero_zero_nat -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))))) # label(fact_1815_divides__rev) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1395 (all P all Q all R hAPP_P1400155668t_bool(hAPP_f223623489t_bool(hAPP_f1815301987t_bool(cOMBB_650041899nt_int,P),Q),R) = hAPP_f1325804733t_bool(P,hAPP_P2053772734t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__f) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1396 (all A all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1)) -> hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,A)),hAPP_r474017924t_real(power_power_real,X_1)),zero_zero_real,sequentially)))) # label(fact_5043_LIMSEQ__divide__realpow__zero) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1397 (all A_229 all C_95 all D_20 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_229),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_95),D_20)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_95),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_229),D_20))) # label(fact_526_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1398 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arctan,X)),hAPP_real_real(arctan,Y))))) # label(fact_3375_arctan__monotone_H) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1399 (all X hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(sin,X))),hAPP_real_real(cos,X)) = hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X))) # label(fact_3710_sin__double) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1400 (all V_22 all V_21 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_22)) -> hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_21),V_22)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_21)),hAPP_int_nat(number_number_of_nat,V_22))))) # label(fact_75_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1401 (all X all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),hAPP_int_int(multInv(P_3),X))),hAPP_int_int(multInv(P_3),hAPP_int_int(multInv(P_3),X)))),X),P_3)))))) # label(fact_2235_Int2_Oaux____2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1402 (all Z_12 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_12),hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_12),Z_12)) # label(fact_293_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1403 (all V all V_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_int_nat(number_number_of_nat,V_1))) <-> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V),V_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V),pls))))) # label(fact_462_le__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1404 (all N hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(real_nat,N))),pi)) = zero_zero_real) # label(fact_3677_sin__2npi) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1405 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(sin,Y))) = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y))) # label(fact_3684_cos__add) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1406 (all B_1_1 is_int(archim791455193or_rat(B_1_1))) # label(gsy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_000tc__Rat__Orat) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1407 (all J_2 all A_3 all P_3 all K_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),K_2))),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,J_2),K_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),K_2))),K_2)),P_3)))) # label(fact_2221_aux______1) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1408 (all N_1 all V hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(suc,N_1)),hAPP_int_nat(number_number_of_nat,V)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),hAPP_nat_nat(suc,N_1)),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,N_1),hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))))))) # label(fact_5184_max__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1409 (all X_14 all Y_11 all Ya_1 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_14),Ya_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_11),Ya_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_14),Y_11)),Ya_1)) # label(fact_2299_mult__left_Odiff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1410 (all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,N_1))))) # label(fact_2952_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1411 (all A_3 all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> natfloor(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_nat_real(real_nat,A_3))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natfloor(X)),A_3))) # label(fact_3702_natfloor__add) # label(axiom) # label(non_clause). [assumption]. 7.03/7.07 1412 (all P_1 all N_1 ((all M_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) -> hBOOL(hAPP_nat_bool(P_1,M_1)))) <-> (all X_2 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),N_1))) -> hBOOL(hAPP_nat_bool(P_1,X_2)))))) # label(fact_5069_all__nat__less) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1413 (all Lx all Rx all Ry hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx),Ry)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx),Ry))) # label(fact_509_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1414 (all W_6 hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit1,W_6)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,W_6))),hAPP_int_rat(number_number_of_rat,W_6))) # label(fact_378_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1415 (all Y_6 all P_6 all N_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)))) # label(fact_3178_lemma__realpow__diff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1416 (all M hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),M))))) # label(fact_1653_le__cube) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1417 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(ln,X_1))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1))))) # label(fact_3453_ln__ge__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1418 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Y),X)) -> hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(abs_abs_int,X)),hAPP_int_int(abs_abs_int,Y)))) # label(fact_3430_abs__div) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1419 (all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)))) # label(fact_3147_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1420 (all X_35 all Y_27 all Q_11 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_35),Y_27)),Q_11) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_35),Q_11)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_27),Q_11))) # label(fact_667_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1421 (all C_89 all D_17 all A_213 all B_159 all R_8 (zero_zero_rat != R_8 -> (B_159 = A_213 & D_17 != C_89 -> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_159),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,R_8),D_17)) != hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_213),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,R_8),C_89))))) # label(fact_690_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1422 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(fact_fact_nat,M)),hAPP_nat_nat(fact_fact_nat,N))))) # label(fact_3802_fact__mono__nat) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1423 (all A_158 all B_119 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_158),B_119))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_158)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_119))))) # label(fact_1170_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1424 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y)) -> zero_zero_real = hAPP_real_real(cos,arg(hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),Y))))) # label(fact_3969_cos__arg__i__mult__zero__pos) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1425 (all Z_1 hAPP_complex_complex(cnj,hAPP_complex_complex(cnj,Z_1)) = Z_1) # label(fact_4073_complex__cnj__cnj) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1426 (all M all N (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) = M -> zero_zero_nat = M | N = one_one_nat)) # label(fact_1673_mult__eq__self__implies__10) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1427 (all X_1 all P_5 (is_int(X_1) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(sRStar,P_5))) -> hAPP_int_int(standardRes(P_5),X_1) = X_1))) # label(fact_2702_StandardRes__SRStar__prop3) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1428 (all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),A_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(zfact,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),P_3))))))) # label(fact_2677_zfact__prop) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1429 (all A_44 all B_35 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_44),B_35)),B_35) = hAPP_nat_nat(div_mod_nat(A_44),B_35)) # label(fact_2434_mod__add__self2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1430 (all A_3 all B_2 all C_1 all D_3 hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),D_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C_1),D_3))) # label(fact_4688_divide__rat) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1431 (all Z_2 (Z_2 = zero_zero_complex <-> hAPP_complex_complex(cnj,Z_2) = zero_zero_complex)) # label(fact_4065_complex__cnj__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1432 (all A_202 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,zero_zero_int),A_202) = zero_zero_int) # label(fact_846_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1433 (all W_1 all Z_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_2)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat,W_1)),hAPP_int_nat(nat,Z_2))))) # label(fact_2727_zless__nat__conj) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1434 (all N (zero_zero_nat != N -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),N)))) # label(fact_4031_coprime__minus1) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1435 (all N hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,N)),one_one_real)) # label(fact_3668_real__of__nat__Suc) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1436 (all N all I (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),I)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),N))))) # label(fact_3139_nat__one__le__power) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1437 (all X all N_5 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N_5)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),hAPP_real_real(hAPP_n546711566l_real(root,N_5),X)))))))) # label(fact_3937_real__root__increasing) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1438 (all A_26 all B_21 (B_21 != zero_z891286103de_int -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_21),A_26)),B_21) = A_26)) # label(fact_2544_div__mult__self1__is__id) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1439 (all A_143 all C_65 all B_112 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_143),C_65)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_112),C_65))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_65)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_143),B_112))))) # label(fact_1326_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1440 (all F all L all C (hBOOL(tendsto_real_real(F,L,at_real(C))) -> (L != zero_zero_real -> (exists R_1 ((all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),X_2))),R_1)) & C != X_2 -> hAPP_real_real(F,X_2) != zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1))))))) # label(fact_5086_LIM__fun__not__zero) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1441 (all M_17 all N_30 ((zero_zero_nat = N_30 -> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,M_17),N_30) = one_on1645066479umeral) & (zero_zero_nat != N_30 -> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,M_17),N_30) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,M_17),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat)))))) # label(fact_1832_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1442 (all A all B all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(B),A),Ma)))) # label(fact_808_zcong__sym) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1443 (all P all Q all R hAPP_f2144054103l_bool(P,hAPP_i1948725293t_bool(Q,R)) = hAPP_i1823918693l_bool(hAPP_f1344929775l_bool(hAPP_f1830214065l_bool(cOMBB_793166024ol_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__020) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1444 (all M_13 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),hAPP_nat_rat(semiri151668891at_rat,M_13)) = hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(suc,M_13))) # label(fact_2844_of__nat__Suc) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1445 (all P all Q all R hAPP_i1524277240nt_int(hAPP_f1735705551nt_int(hAPP_f151095827nt_int(cOMBB_1389251822nt_int,P),Q),R) = hAPP_P1975530577nt_int(P,hAPP_i1524277240nt_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__prod_Itc) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1446 (all A_224 all N_68 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_224),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_224),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68))) # label(fact_579_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1447 (all Y all X all U_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),U_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),U_1)))))))) # label(fact_3351_lemma__sqrt__hcomplex__capprox) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1448 (all Lx_6 all Ly_4 all Rx_6 all Ry_4 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_6),Ly_4)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_6),Ry_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_6),Rx_6)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ly_4),Ry_4))) # label(fact_465_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1449 (all C_27 all A_57 all B_48 hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_27),A_57)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_27),B_48)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_27),hAPP_Q1762011733de_int(div_mo231679042de_int(A_57),B_48))) # label(fact_2392_mod__mult__mult1) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1450 (all X_1 all Y_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(number_number_of_nat,X_1)),hAPP_int_nat(number_number_of_nat,Y_2))) <-> hAPP_nat_nat(div_mod_nat(hAPP_int_nat(number_number_of_nat,Y_2)),hAPP_int_nat(number_number_of_nat,X_1)) = zero_zero_nat)) # label(fact_2590_dvd__eq__mod__eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1451 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N)) & -hBOOL(hAPP_nat_bool(prime,N)) -> (exists M_1 exists K (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),K)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),N)) & hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),K) = N & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N)))))) # label(fact_4208_not__prime__ex__mk) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1452 (all A_208 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_208),zero_zero_int))) # label(fact_762_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1453 (all X_37 X_37 = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_37),one_one_nat)) # label(fact_563_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1454 (all Lx_5 all Ly_3 all Rx_5 all Ry_3 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_5),Ly_3)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_5),Ry_3)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_5),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_5),Ly_3)),Ry_3))) # label(fact_475_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1455 (all Z_11 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Z_11),Z_11) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Z_11),hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_298_mult__2__right) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1456 (all B_2 zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,zero_zero_int),B_2)) # label(fact_2457_zdiv__zero) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1457 (all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A)) -> hAPP_nat_int(semiri1621563631at_int,hAPP_f521865025ol_nat(finite1876863882t_bool,setS(A,P_5))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3305_SetS__card) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1458 (all A_223 all K_8 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_223),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K_8))),zero_zero_rat)) -> A_223 = zero_zero_rat)) # label(fact_582_even__power__le__0__imp__0) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1459 (all A_230 all C_96 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_96),A_230) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_230),C_96)) # label(fact_525_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1460 (all A_15 all B_11 (B_11 != zero_z126310315umeral -> hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_15),B_11)),one_on1645066479umeral) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_15),B_11)),B_11))) # label(fact_2634_div__add__self2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1461 (all C_46 all A_104 all B_86 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_46),A_104)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_46),B_86))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_104),B_86)))) # label(fact_1993_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1462 (all A_200 all B_153 all C_86 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_200),C_86)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_153),C_86)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_200),B_153)),C_86)) # label(fact_876_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1463 (all V_2 all W ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W))) -> hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W)))),one_one_int)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W))) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_2)),one_one_int)),hAPP_int_int(number_number_of_int,W)))),one_one_int) = hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,V_2))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W)))))) # label(fact_2318_zmod__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1464 (all Y exists X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) & Y = hAPP_real_real(tan,X_2))) # label(fact_3628_lemma__tan__total1) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1465 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2)) -> hAPP_P1975530577nt_int(adjust(B_2),posDivAlg(A_3,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))) = posDivAlg(A_3,B_2)) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2)) -> posDivAlg(A_3,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),A_3)))) # label(fact_2769_posDivAlg__eqn) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1466 (all N_1 all N_7 all F ((all N_2 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2))),hAPP_nat_nat(F,N_2))) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_2)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2)))))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_7)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_1)),hAPP_nat_nat(F,N_7))) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_7)),hAPP_nat_nat(F,N_1)))))) # label(fact_3263_dvd_Olift__Suc__mono__less) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1467 (all K_2 all L_2 (hAPP_int_int(div_mod_int(K_2),hAPP_int_int(uminus_uminus_int,L_2)) != zero_zero_int -> hAPP_int_int(div_mod_int(K_2),L_2) != zero_zero_int)) # label(fact_3534_zmod__zminus2__not__zero) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1468 (all V_23 hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_23),hAPP_int_int(bit1,pls))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_23)),one_one_rat)) # label(fact_47_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1469 (all P all Q all R hAPP_int_bool(hAPP_f429384717t_bool(hAPP_f1173902867t_bool(cOMBB_real_bool_int,P),Q),R) = hAPP_real_bool(P,hAPP_int_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__HOL__Obool_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1470 (all X X = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),X)) # label(fact_4304_gcd__idem__nat) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1471 (all P all Q all R hAPP_i1948725293t_bool(hAPP_f1791153283t_bool(hAPP_f1399575567t_bool(cOMBB_118231410ol_int,P),Q),R) = hAPP_i1948725293t_bool(P,hAPP_int_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1472 (all X hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(uminus_uminus_real,hAPP_complex_real(re,X))),hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X))) = hAPP_complex_complex(uminus473333897omplex,X)) # label(fact_4135_complex__minus__def) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1473 (all N all D_3 all A_3 (one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),A_3) -> one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N)))) # label(fact_4362_coprime__exp__nat) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1474 (all A all B (A = zero_zero_real & zero_zero_real = B <-> hAPP_real_complex(hAPP_r265291036omplex(complex,A),B) = zero_zero_complex)) # label(fact_3957_Complex__eq__0) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1475 (all P all Q all R hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(P,R),hAPP_i1948725293t_bool(Q,R)) = hAPP_i1948725293t_bool(hAPP_f428220345t_bool(hAPP_f1878066172t_bool(cOMBS_1720984575t_bool,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1476 (all B_155 all A_204 (zero_z126310315umeral != A_204 -> (B_155 != zero_z126310315umeral -> hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_204),B_155) != zero_z126310315umeral))) # label(fact_831_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1477 (all X hAPP_real_real(cos,X) = hAPP_real_real(cos,hAPP_real_real(uminus_uminus_real,X))) # label(fact_3561_cos__minus) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1478 (all P_1 ((all X_2 exists D_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2)) & (all A_2 all B_4 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),X_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),B_4)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_4),A_2)),D_2)) -> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4))))))) & (all A_2 all B_4 all C_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),B_4)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_4),C_3)) & hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,B_4),C_3))) & hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4))) -> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),C_3))))) -> (all A_2 all B_4 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),B_4)) -> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4))))))) # label(fact_4296_lemma__BOLZANO2) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1479 (all W_8 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_int_complex(number528085621omplex,W_8)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,W_8)),hAPP_int_complex(number528085621omplex,W_8))) # label(fact_365_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1480 (all B_141 all A_181 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_141),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_181)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_181),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_141)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_181),B_141)),zero_zero_int)))) # label(fact_1044_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1481 (all A all B all C (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),C) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),B) <-> C = B)) # label(fact_1942_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1482 (all B_95 all C_53 all A_117 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_117)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_95),C_53)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_95),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_117),C_53)))))) # label(fact_1872_add__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1483 (all A_215 A_215 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_215),one_one_complex)) # label(fact_658_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1484 (all V_4 ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4))) -> zero_zero_int = hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_4))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4))) -> hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_4)) = hAPP_int_int(number_number_of_int,V_4)))) # label(fact_3220_of__nat__number__of__lemma) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1485 (all A_3 all B_2 hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(abs_abs_int,A_3)),hAPP_int_int(abs_abs_int,B_2)) = hAPP_rat_rat(abs_abs_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2))) # label(fact_4691_abs__rat) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1486 (all C_80 all A_187 all B_143 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_187),B_143)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_187),C_80)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_187),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_143),C_80)))))) # label(fact_966_dvd__add) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1487 (all A_92 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_92),one_on1684967323de_int) = A_92) # label(fact_2103_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1488 (all A_235 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_235),A_235)),A_235) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_235),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) # label(fact_274_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1489 (all A_211 all M_26 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_211),M_26)),M_26) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_211),one_one_rat)),M_26)) # label(fact_711_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1490 (all A_3 rcis(zero_zero_real,A_3) = zero_zero_complex) # label(fact_4261_rcis__zero__mod) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1491 (exists K_4 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),K_4)) & (all X_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(im,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_2)),K_4)))))) # label(fact_4108_Im_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 7.03/7.08 1492 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(cos,X))),one_one_real))) # label(fact_3568_abs__cos__le__one) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1493 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X)) -> hAPP_int_int(nat_tsub(X),Y) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y))) # label(fact_3188_tsub__eq) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1494 (all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(semiri1424489471de_int,Ma)),hAPP_n522471361de_int(semiri1424489471de_int,N_1))))) # label(fact_2943_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1495 (all A_12 all N_18 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_12),N_18)),A_12) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_12),hAPP_nat_nat(suc,N_18))) # label(fact_3008_power__Suc2) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1496 (all P all Q all R hAPP_i744251628nt_int(hAPP_f836602773nt_int(hAPP_f888249301nt_int(cOMBB_1875755114nt_int,P),Q),R) = hAPP_f747162069nt_int(P,hAPP_int_fun_int_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc___023) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1497 (all A all B (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B)),zOdd)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),zOdd)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),zEven))))) # label(fact_3093_even__plus__odd__prop1) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1498 (all A_73 all A_72 all B_61 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_73),A_72)),B_61) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_73),B_61)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_72),B_61))) # label(fact_2302_mult_Odiff__left) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1499 (all B_1 all P_2 all P_1 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> ((exists Z all X_2 (is_int(X_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Z)) -> (hBOOL(hAPP_int_bool(P_1,X_2)) <-> hBOOL(hAPP_int_bool(P_2,X_2)))))) -> ((all X_2 (is_int(X_2) -> ((all Xa (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1)) -> X_2 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa))))) -> (hBOOL(hAPP_int_bool(P_1,X_2)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1))))))) -> ((all X_2 all K (is_int(X_2) & is_int(K) -> (hBOOL(hAPP_int_bool(P_2,X_2)) <-> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D_1))))))) -> ((exists X_2 (is_int(X_2) & (exists Xa (is_int(Xa) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),B_1)) & hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xa),X_2))))) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))))) | (exists X_2 (hBOOL(hAPP_int_bool(P_2,X_2)) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) & is_int(X_2))) <-> (exists X1 (hBOOL(hAPP_int_bool(P_1,X1)) & is_int(X1))))))))) # label(fact_5154_cpmi) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1500 (all C_42 all A_100 all B_82 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_100),B_82)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_100),C_42)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_82),C_42))))) # label(fact_2021_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1501 (all M all N all K_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)),K_2)) # label(fact_1602_nat__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1502 (all B_1 all A_1 (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1)) -> (hBOOL(hAPP_f448129468l_bool(nat_nat_set,B_1)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,image_int_nat(nat,A_1)),image_int_nat(nat,B_1))) <-> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A_1),B_1)))))) # label(fact_4611_transfer__int__nat__set__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1503 (all A_125 all N_36 all N_35 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_36),N_35)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_125)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_125),one_one_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_125),N_35)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_125),N_36))))))) # label(fact_1473_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1504 (all P_3 all Q_2 (hBOOL(hAPP_nat_bool(prime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_3),Q_2))) -> P_3 = one_one_nat | Q_2 = one_one_nat)) # label(fact_4172_prime__product) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1505 (all B_1 all A_1 (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1)) -> (hBOOL(hAPP_f448129468l_bool(nat_nat_set,B_1)) -> (image_int_nat(nat,A_1) = image_int_nat(nat,B_1) <-> B_1 = A_1)))) # label(fact_4616_transfer__int__nat__set__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1506 (all A (hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially)) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A,N_2)))) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,hAPP_nat_nat(suc,N_2))),hAPP_nat_real(A,N_2)))) -> hBOOL(tendsto_nat_real(hAPP_f292751352t_real(hAPP_f1607377819t_real(cOMBB_1533963633al_nat,hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_at4362885an_nat(zero_zero_nat)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),sequentially)))))) # label(fact_5053_summable__Leibniz_H_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1507 (all N_62 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,one_one_complex),N_62) = one_one_complex) # label(fact_973_power__one) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1508 (all A_71 all B_60 all B_59 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_71),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,B_60),B_59)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_71),B_60)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_71),B_59))) # label(fact_2306_mult_Odiff__right) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1509 (all P all Q all R hAPP_n546711566l_real(P,hAPP_nat_nat(Q,R)) = hAPP_n546711566l_real(hAPP_f1904307534l_real(hAPP_f224989205l_real(cOMBB_1419615765al_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDe) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1510 (all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),P_3)))) # label(fact_4181_prime__g__one) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1511 (all N all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y),hAPP_nat_nat(suc,N)))))) # label(fact_3088_divides__rexp) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1512 (all P all Q all R hAPP_f1676298106t_real(hAPP_r50993370t_real(P,R),Q) = hAPP_r474017924t_real(hAPP_f1501741374t_real(hAPP_f1135648319t_real(cOMBC_1329014627t_real,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Nat__Onat_Mtc__Nat__O) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1513 (all A_3 all B_2 all X (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),X)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),B_2)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),X)))) # label(fact_4313_gcd__semilattice__nat_Oless__infI2) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1514 (all N_1 all Ma (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),Ma)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,Ma))) <-> Ma = N_1))) # label(fact_2862_not__less__less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1515 (all C_39 all D_7 all A_97 all B_79 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_97),B_79)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_39),D_7)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_97),C_39)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_79),D_7)))))) # label(fact_2046_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1516 (all B_116 all A_155 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_155)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_116)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_155),B_116)))))) # label(fact_1185_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1517 (all N (zero_zero_nat != N -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)),one_one_nat)))) # label(fact_3246_Suc__mult__two__diff__one) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1518 (all X_38 all P_8 all Q_12 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_38),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_38),P_8)),Q_12)) # label(fact_444_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1519 (all P all Q all R hAPP_P1145851913nt_int(hAPP_i2104874254nt_int(P,R),hAPP_i1524277240nt_int(Q,R)) = hAPP_i1216375074nt_int(hAPP_f1375227553nt_int(hAPP_f1311564564nt_int(cOMBS_1640193733nt_int,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint__027) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1520 (all A all B (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B),zero_zero_rat)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A)) | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),zero_zero_rat)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)))) # label(fact_1118_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1521 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = X)) # label(fact_3341_real__sqrt__pow2) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1522 (all X all A_3 all B_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),B_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_2)))))) # label(fact_4410_powr__less__mono) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1523 (all C_73 all A_173 all B_133 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_173),B_133)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_73)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_173),C_73)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_133),C_73)))))) # label(fact_1088_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1524 (all N_53 all M_21 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),M_21)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),N_53)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,M_21),N_53)))))) # label(fact_1248_less__1__mult) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1525 (all B_1_1 all B_2_1 is_bool(hAPP_f526364711l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__p) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1526 (all X (zero_zero_real = hAPP_real_real(sin,X) -> hAPP_real_real(abs_abs_real,hAPP_real_real(cos,X)) = one_one_real)) # label(fact_3681_sin__zero__abs__cos__one) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1527 (all A_49 all B_40 all C_21 hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_49),B_40)),C_21) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_49),hAPP_Q1762011733de_int(div_mo231679042de_int(B_40),C_21))),C_21)) # label(fact_2416_zmod__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1528 (all B all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hBOOL(hAPP_real_bool(deriv_real(log(B),X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(ln,B)),X_1)))))) # label(fact_4791_DERIV__log) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1529 (all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1)) -> big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),N_1)) = hAPP_int_int(fact_fact_int,N_1))) # label(fact_5084_fact__altdef__int) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1530 (all B_2 all D_3 all A_3 (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),A_3) -> (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),B_2) = one_one_int -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)) = one_one_int))) # label(fact_4998_coprime__mult__int) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1531 (all B_94 all C_52 all A_116 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_116)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_94),C_52)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_94),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_116),C_52)))))) # label(fact_1878_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1532 (all Lx all Rx all Ry hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx),Ry)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx),Ry))) # label(fact_506_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1533 (all A all B all W_1 (hAPP_int_complex(number528085621omplex,W_1) = hAPP_real_complex(hAPP_r265291036omplex(complex,A),B) <-> hAPP_int_real(number267125858f_real,W_1) = A & B = zero_zero_real)) # label(fact_3959_Complex__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1534 (all Q_1 all P_1 ((all A_2 all B_4 (is_int(A_2) & is_int(B_4) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_4)) -> hBOOL(hAPP_rat_bool(P_1,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_4)))))) -> hBOOL(hAPP_rat_bool(P_1,Q_1)))) # label(fact_4729_Rat__induct__pos) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1535 (all X_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),N_1)),zEven)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven))))) # label(fact_3148_power__preserves__even) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1536 (all N all K_2 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)))))) # label(fact_1714_dvd__diff__nat) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1537 (all R_3 all X hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_complex_real(im,X)) = hAPP_complex_real(im,hAPP_complex_complex(scaleR1652505878omplex(R_3),X))) # label(fact_5108_complex__Im__scaleR) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1538 (all Ta all D all D_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),D_1)) -> (all X_2 all K (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Ta))) <-> -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,K),D_1))),Ta))))))) # label(fact_1532_inf__period_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1539 (all I (is_int(I) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),zero_zero_int)) -> hAPP_int_int(uminus_uminus_int,I) = hAPP_int_int(abs_abs_int,I)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),zero_zero_int)) -> I = hAPP_int_int(abs_abs_int,I)))) # label(fact_3542_zabs__def) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1540 (all A_18 all C_5 all B_14 (zero_z126310315umeral != B_14 -> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_18),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_5),B_14))),B_14) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_5),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_18),B_14)))) # label(fact_2622_div__mult__self1) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1541 (all X all Y -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_int))) # label(fact_7_not__sum__power2__lt__zero) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1542 (all C_63 all A_141 all B_110 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_63),A_141)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_63),B_110))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_63)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_141),B_110))))) # label(fact_1336_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1543 (all A_39 all B_30 all C_14 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_39),hAPP_int_int(div_mod_int(B_30),C_14))),C_14) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_39),B_30)),C_14)) # label(fact_2440_mod__diff__right__eq) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1544 (all W_17 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),hAPP_int_complex(number528085621omplex,W_17)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,pls)),W_17))) # label(fact_42_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1545 (all N all M hAPP_Q1762011733de_int(div_mo231679042de_int(N),M) = hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(div_mod_int(quickcheck_int_of(N)),quickcheck_int_of(M)))) # label(fact_4553_mod__code__int__def) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1546 (all M_8 all N_22 hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(semiri132038758t_real,M_8)),hAPP_nat_real(semiri132038758t_real,N_22))) # label(fact_2957_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1547 (all X_1 all N_1 all Y_2 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_2),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),hAPP_nat_nat(suc,N_1)) <-> Y_2 = X_1)) # label(fact_2880_exp__mono__eq) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1548 (all Ta all A_1 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ta),one_one_int)),A_1)) -> (all X_2 (is_int(X_2) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1)) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) != X_2)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),Ta)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1)),Ta))))))))) # label(fact_5089_aset_I6_J) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1549 (all A_89 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,one_one_rat),A_89) = A_89) # label(fact_2125_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1550 (all M_8 all N_22 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(semiri984289939at_nat,M_8)),hAPP_nat_nat(semiri984289939at_nat,N_22)) = hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22))) # label(fact_2958_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1551 (all A_111 all B_89 all C_49 (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_111),B_89) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_111),C_49) -> C_49 = B_89)) # label(fact_1921_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1552 (all K_2 all M hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),K_2),M))) # label(fact_809_zcong__refl) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1553 (all N hAPP_int_nat(number_number_of_nat,N) = natfloor(hAPP_int_real(number267125858f_real,N))) # label(fact_3387_natfloor__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1554 (all A_3 all B_2 all C_1 all D_3 hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),C_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),D_3)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C_1),D_3))) # label(fact_4690_mult__rat) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1555 (all K_2 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),K_2)),K_2) = hAPP_int_int(bit1,K_2)) # label(fact_356_Bit1__def) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1556 (all A_186 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,one_one_int),A_186))) # label(fact_981_one__dvd) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1557 (all X all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (one_one_real != A_3 -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_real_real(hAPP_r1250527377l_real(powr,A_3),hAPP_real_real(log(A_3),X)) = X)))) # label(fact_4461_powr__log__cancel) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1558 (all P all Q all R hAPP_int_rat(hAPP_f1900950721nt_rat(hAPP_f1270196437nt_rat(cOMBB_int_rat_int,P),Q),R) = hAPP_int_rat(P,hAPP_int_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__Rat__Orat_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1559 (all M all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)),M))) # label(fact_2690_div__le__dividend) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1560 (all N all K_2 all M (hAPP_int_int(legacy_zgcd(K_2),M) = one_one_int -> (one_one_int = hAPP_int_int(legacy_zgcd(N),M) -> hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N)),M) = one_one_int))) # label(fact_4486_zgcd__zgcd__zmult) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1561 (all A all B (zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B) <-> A = B)) # label(fact_2083_eq__iff__diff__eq__0) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1562 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> N = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat))))) # label(fact_3134_Suc__pred) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1563 (all A (hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially)) -> (hBOOL(monoseq_real(A)) -> hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)))))) # label(fact_5049_summable__Leibniz_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1564 (all B_131 all A_171 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_171),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_131)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_171),B_131)),zero_zero_real))))) # label(fact_1094_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1565 (all A_9 all N_14 all B_7 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_9),hAPP_nat_nat(suc,N_14))),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_7),hAPP_nat_nat(suc,N_14)))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_7)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_9),B_7))))) # label(fact_3111_power__le__imp__le__base) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1566 (all X hAPP_real_real(uminus_uminus_real,ratreal(X)) = ratreal(hAPP_rat_rat(uminus_uminus_rat,X))) # label(fact_4703_real__uminus__code) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1567 (all X_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root,N_1),X_1),hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_nat_real(real_nat,N_1))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,zero_zero_nat))))))))))) # label(fact_3893_DERIV__even__real__root) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1568 (all N all M all R_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,M)),R_3)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N)),one_one_real))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M),N))))) # label(fact_4463_lemma__floor) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1569 (all W_1 all Y_2 all X_1 all Z_2 (hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,W_1),Y_2)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_1),Z_2)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,W_1),Z_2)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_1),Y_2)) <-> Z_2 = Y_2 | W_1 = X_1)) # label(fact_645_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1570 (all Ma all N_1 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),N_1) = zero_zero_nat <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)))) # label(fact_1763_diff__is__0__eq) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1571 (all Y hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arctan,Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_3471_arctan__ubound) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1572 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hBOOL(hAPP_real_bool(deriv_real(sqrt,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X_1))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3877_DERIV__real__sqrt) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1573 (all A_226 all B_163 all C_92 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_226),B_163)),C_92) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_226),C_92)),B_163)) # label(fact_548_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1574 (all K_1 all L hAPP_int_int(legacy_zgcd(K_1),L) = hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,L),zero_zero_int)),K_1),hAPP_int_int(legacy_zgcd(L),hAPP_int_int(div_mod_int(hAPP_int_int(abs_abs_int,K_1)),hAPP_int_int(abs_abs_int,L)))))) # label(fact_4520_zgcd__code) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1575 (all L all K_1 (is_int(K_1) & is_int(L) -> (zero_zero_int != K_1 -> (L != zero_zero_int -> hAPP_P1975530577nt_int(produc713050258nt_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(sgn_sgn_int,L))),hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,hAPP_int_int(sgn_sgn_int,K_1)),hAPP_int_int(sgn_sgn_int,L))),pdivmod(K_1,L)),hAPP_P1975530577nt_int(produc1518849193nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,hAPP_f617310012nt_int(hAPP_f197974549nt_int(cOMBB_638531587nt_int,cOMBS_1891347093nt_int),hAPP_f1148171384nt_int(hAPP_f334321603nt_int(cOMBB_1484610687nt_int,hAPP_f1582586579nt_int(cOMBC_707974837nt_int,hAPP_f251264923nt_int(hAPP_f279307923nt_int(cOMBB_662120866nt_int,if_Pro1731782967nt_int),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,fequal_int),zero_zero_int)))),hAPP_i1584592887nt_int(hAPP_f465821005nt_int(cOMBC_1964283556nt_int,hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),uminus_uminus_int)),zero_zero_int)))),hAPP_f1574055885nt_int(hAPP_f960630521nt_int(cOMBC_1805044090nt_int,hAPP_f142232355nt_int(hAPP_f1377453843nt_int(cOMBB_320287386nt_int,cOMBB_1846624359nt_int),hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,minus_minus_int),uminus_uminus_int)),one_one_int)))),hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(abs_abs_int,L))))),pdivmod(K_1,L)))) = divmod_int(K_1,L)) & (L = zero_zero_int -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),K_1) = divmod_int(K_1,L))) & (zero_zero_int = K_1 -> divmod_int(K_1,L) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int)))) # label(fact_4856_divmod__int__code) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1576 (all P all Q all R hAPP_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,P),Q),R) = hAPP_int_int(hAPP_int_fun_int_int(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1577 (all N_1 all K_1 all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),Ma)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),K_1))))))) # label(fact_1766_less__diff__iff) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1578 (all B_1_1 all B_2_1 all B_3_1 is_bool(tendsto_real_real(B_1_1,B_2_1,B_3_1))) # label(gsy_c_Limits_Otendsto_000tc__RealDef__Oreal_000tc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1579 (all X all Y all Z_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),Z_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)),Z_1)) # label(fact_4312_gcd__semilattice__nat_Oinf__assoc) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1580 (all Ma all X_1 hAPP_i1948725293t_bool(noXRRset(Ma),X_1) = image_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,times_times_int),X_1),hAPP_i1948725293t_bool(norRRset,Ma))) # label(fact_4772_noXRRset__def) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1581 (all X (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))) -> X = hAPP_nat_nat(suc,zero_zero_nat) | zero_zero_nat = X)) # label(fact_3046_nat__lt__two__imp__zero__or__one) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1582 (all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,N))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,N)))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,N)))))) # label(fact_1506_number__of1) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1583 (all N_61 all X_29 all Y_23 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,X_29),Y_23)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_29),N_61)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_23),N_61))))) # label(fact_983_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1584 (all Y_3 all X_3 (hBOOL(vanishes(X_3)) -> (hBOOL(vanishes(Y_3)) -> hBOOL(vanishes(hAPP_f2062659861at_rat(hAPP_f1131305186at_rat(cOMBS_nat_rat_rat,hAPP_f1830834240at_rat(hAPP_f1073644437at_rat(cOMBB_1926947at_nat,minus_minus_rat),X_3)),Y_3)))))) # label(fact_5130_vanishes__diff) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1585 (all A_3 all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N)) -> (one_one_int = hAPP_int_int(legacy_zgcd(A_3),N) -> (exists X_2 (is_int(X_2) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),X_2)),one_one_int),N))))))) # label(fact_4534_zcong__lineq__ex) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1586 (all N_55 all A_152 all B_114 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_152),B_114)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_152)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_152),N_55)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_114),N_55)))))) # label(fact_1236_power__mono) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1587 (all Z_1 (is_int(Z_1) -> -(all M_1 all N_2 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N_2)) != Z_1))) # label(fact_3288_int__diff__cases) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1588 (all B_1_1 all B_2_1 is_bool(hAPP_f1469637905l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__prod_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__f) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1589 (all A all P_5 all X_2 (hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_2),setS(A,P_5))) -> hBOOL(hAPP_f448129468l_bool(finite_finite_int,X_2)))) # label(fact_3358_SetS__elems__finite) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1590 (all I_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),zero_zero_int)) <-> hAPP_int_nat(nat,I_1) = zero_zero_nat)) # label(fact_2726_nat__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1591 (all N_49 all A_147 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,one_on1684967323de_int),A_147)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,one_on1684967323de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_147),N_49))))) # label(fact_1281_one__le__power) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1592 (all A_106 A_106 = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,zero_zero_real),A_106)) # label(fact_1981_add__0) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1593 (all Ma all Z_2 (is_int(Z_2) -> (Z_2 = hAPP_nat_int(semiri1621563631at_int,Ma) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2)) & Ma = hAPP_int_nat(nat,Z_2)))) # label(fact_3122_int__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1594 (all X all Y (is_int(Y) -> Y = hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,fFalse),X),Y))) # label(help_If_2_1_If_000tc__Int__Oint_T) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1595 (all C (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,C)),one_one_real)) -> hBOOL(tendsto_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(abs_abs_real,C)),zero_zero_real,sequentially)))) # label(fact_5044_LIMSEQ__rabs__realpow__zero) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1596 (all K_1 all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K_1)),hAPP_int_int(bit1,L))))) # label(fact_258_rel__simps_I17_J) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1597 (all A_98 all C_40 all B_80 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_98),C_40)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_80),C_40))) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_98),B_80)))) # label(fact_2044_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1598 (all Ma all X_1 (zero_zero_int = hAPP_int_int(standardRes(Ma),X_1) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),Ma)))) # label(fact_2592_StandardRes__eq__zcong) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1599 (all X all Y (hAPP_nat_nat(suc,X) = hAPP_nat_nat(suc,Y) -> X = Y)) # label(fact_2827_Suc__inject) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1600 (all M all N hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,M)),N) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),N))) # label(fact_3083_zpower__int) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1601 (all X_15 all Y_12 all Ya_2 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_15),Ya_2)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Y_12),Ya_2)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_15),Y_12)),Ya_2)) # label(fact_2298_mult__left_Oadd) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1602 (all I all J_2 hAPP_int_int(legacy_zgcd(I),J_2) = hAPP_int_int(legacy_zgcd(I),hAPP_int_int(uminus_uminus_int,J_2))) # label(fact_4498_zgcd__zminus2) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1603 (all M all N all R_5 all S_3 all S_1 all T_2 all T_1 all Rn all Sn_1 all Tn_1 all R_3 (is_int(Rn) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),R_3)) -> (hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,Rn),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Sn_1),Tn_1)) = xzgcda(M,N,R_5,R_3,S_3,S_1,T_2,T_1) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_3),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_2),N)) = R_5 -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_1),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_1),N)) = R_3 -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Sn_1),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Tn_1),N)) = Rn)))))) # label(fact_4202_xzgcda__linear) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1604 (all P all Q (-hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)) | hBOOL(P))) # label(help_fconj_2_1_U) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1605 (all X hAPP_complex_complex(cnj,hAPP_complex_complex(invers1449016382omplex,X)) = hAPP_complex_complex(invers1449016382omplex,hAPP_complex_complex(cnj,X))) # label(fact_4064_complex__cnj__inverse) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1606 (all N_66 all A_210 all B_158 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_210),B_158)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_210)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_66)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_210),N_66)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_158),N_66))))))) # label(fact_747_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1607 (all A_91 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_91),one_on1645066479umeral) = A_91) # label(fact_2108_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1608 (all N -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N))) # label(fact_1584_less__not__refl) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1609 (all X_1 -(hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd)))) # label(fact_2903_even__odd__conj) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1610 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),negDivAlg(A_3,B_2)))))) # label(fact_2797_negDivAlg__correct) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1611 (all V_22 all V_21 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_22)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_21)),hAPP_int_int(number_number_of_int,V_22)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_21),V_22))))) # label(fact_74_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1612 (all P (is_bool(P) -> fTrue = P | fFalse = P)) # label(help_If_3_1_If_000tc__RealDef__Oreal_T) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1613 (all M all Y all X (is_int(Y) & is_int(X) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Y)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),Y),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),M)) -> X = Y)))))))) # label(fact_1312_zcong__less__eq) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1614 (all C_78 all D_14 all A_178 all B_138 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_178),B_138)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_78),D_14)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_178)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_78)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_178),C_78)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_138),D_14)))))))) # label(fact_1064_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1615 (all P all Q all R hAPP_f289318786t_bool(hAPP_r653725362t_bool(P,R),Q) = hAPP_r373117745t_bool(hAPP_f1115375056t_bool(hAPP_f271976045t_bool(cOMBC_1801450329t_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__O_021) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1616 (all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W))) -> hAPP_int_int(div_mod_int(one_one_int),hAPP_int_int(number_number_of_int,W)) = hAPP_P1175774780nt_int(product_snd_int_int,posDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W))))) # label(fact_4236_mod__pos__pos__1__number__of) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1617 (all F all Y_2 all X_1 all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) -> (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),hAPP_r1134773055l_bool(ord_gr788844697n_real(A),B))) -> (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Y_2),hAPP_r1134773055l_bool(ord_gr788844697n_real(A),B))) -> ((all X_2 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),hAPP_r1134773055l_bool(ord_gr788844697n_real(A),B))) -> hBOOL(hAPP_real_bool(deriv_real(F,X_2),zero_zero_real)))) -> hAPP_real_real(F,X_1) = hAPP_real_real(F,Y_2)))))) # label(fact_4635_DERIV__isconst3) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1618 (all X all Y ((zero_zero_nat != Y -> hAPP_nat_nat(nat_gcd(X),Y) = hAPP_nat_nat(nat_gcd(Y),hAPP_nat_nat(div_mod_nat(X),Y))) & (zero_zero_nat = Y -> hAPP_nat_nat(nat_gcd(X),Y) = X))) # label(fact_4617_nat__gcd_Osimps) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1619 (all C_73 all A_173 all B_133 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_173),B_133)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_73)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_173),C_73)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_133),C_73)))))) # label(fact_1084_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1620 (all N all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> one_one_nat = N | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),N)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,P_3),N)))) # label(fact_4188_prime__coprime) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1621 (all C_46 all A_104 all B_86 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_46),A_104)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_46),B_86))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_104),B_86)))) # label(fact_1995_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1622 (all N_19 zero_zero_int = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),hAPP_nat_nat(suc,N_19))) # label(fact_2997_power__0__Suc) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1623 (all A_215 A_215 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_215),one_on1684967323de_int)) # label(fact_659_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1624 (all A all C all B (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),C))) <-> zero_zero_rat = C | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A),B)))) # label(fact_1269_dvd__mult__cancel__right) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1625 (all Y all X all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_3)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,Y),A_3))))))) # label(fact_4430_powr__mono2) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1626 (all X (-hBOOL(hAPP_nat_bool(even_odd_even_nat,X)) -> hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(div_mod_nat(X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))))) # label(fact_3773_odd__nat__mod__two__eq__one) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1627 (all N_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(real_nat,N_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)))) # label(fact_3679_real__of__nat__gt__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1628 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,ratreal(X)),ratreal(Y)) = ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X),Y))) # label(fact_4701_real__plus__code) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1629 (all C_77 all B_137 all A_177 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_137),A_177)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_77),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_77),A_177)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_77),B_137)))))) # label(fact_1065_mult__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1630 (all N ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N),zero_zero_int)) -> one_one_int = hAPP_int_int(zfact,N)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N),zero_zero_int)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N),hAPP_int_int(zfact,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,N),one_one_int))) = hAPP_int_int(zfact,N)))) # label(fact_2213_zfact_Osimps) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1631 (all X_1 all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim1246769320r_real(X_1)),A)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,A)),one_one_real))))) # label(fact_4469_floor__le__eq) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1632 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(tan,X)),zero_zero_real))))) # label(fact_3621_tan__less__zero) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1633 (all M hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(fact_fact_int,M)))) # label(fact_4279_fact__ge__zero__int) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1634 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)))) # label(fact_2856_Suc__lessD) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1635 (all I all J_2 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),I))) # label(fact_1624_not__add__less1) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1636 (all A_3 all B_2 hAPP_complex_complex(cnj,hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)) = hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),hAPP_real_real(uminus_uminus_real,B_2))) # label(fact_4083_complex__cnj) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1637 (all M_28 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,M_28),M_28) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,one_on1645066479umeral),one_on1645066479umeral)),M_28)) # label(fact_694_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1638 (all N_49 all A_147 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,one_on1645066479umeral),A_147)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,one_on1645066479umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_147),N_49))))) # label(fact_1280_one__le__power) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1639 (all A all E_2 all C all B all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),E_2)),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),E_2)),D))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B),A)),E_2)),D))))) # label(fact_1563_le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1640 (all X_30 all Y_24 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_30),Y_24)) -> (X_30 != Y_24 -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_30),Y_24))))) # label(fact_820_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1641 (all W_1 all Ma (is_int(W_1) -> ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1)) -> Ma = zero_zero_nat) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1)) -> hAPP_nat_int(semiri1621563631at_int,Ma) = W_1) <-> Ma = hAPP_int_nat(nat,W_1)))) # label(fact_3167_nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1642 (all R_3 all S_1 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(number267125858f_real,R_3)),hAPP_int_real(number267125858f_real,S_1)) = ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(number_number_of_rat,R_3)),hAPP_int_rat(number_number_of_rat,S_1)))) # label(fact_4668_Ratreal__number__of__quotient2) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1643 (all A_46 all B_37 all C_18 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_46),B_37)),C_18) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_46),hAPP_int_int(div_mod_int(B_37),C_18))),C_18)) # label(fact_2425_mod__add__right__eq) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1644 (all A all B all N_1 (N_1 != zero_zero_nat -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A),N_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B),N_1)))))) # label(fact_2277_pow__divides__eq__nat) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1645 (all X_4 all Y_4 hAPP_real_real(sin,hAPP_real_real(arccos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_4),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_4),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_3746_sin__arccos__lemma1) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1646 (all W_1 all Z_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),one_one_int)),Z_2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)))) # label(fact_157_add1__zle__eq) # label(axiom) # label(non_clause). [assumption]. 7.24/7.26 1647 (all K_1 all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),Ma)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Ma),N_1))))) # label(fact_4328_gcd__greatest__iff__nat) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1648 (all X_1 all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,X_1)),hAPP_int_real(number267125858f_real,Y_2))))) # label(fact_331_less__number__of) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1649 (all X_6 all Q_5 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_6),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_6),Q_5)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_6),hAPP_nat_nat(suc,Q_5))) # label(fact_3034_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1650 (all V_14 all B_166 all C_97 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_14)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_166),C_97)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_14)),B_166)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_14)),C_97))) # label(fact_193_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1651 (all X hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,X)),hAPP_nat_nat(fact_fact_nat,X)) = hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(suc,X))) # label(fact_3808_Fact_Ofact__Suc) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1652 (all B_128 all A_168 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_168)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_128)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_168),B_128)))))) # label(fact_1114_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1653 (all A_146 one_one_int = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_146),zero_zero_nat)) # label(fact_1296_power__0) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1654 (all V all W_1 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_i1732201573de_int(number1226105091de_int,V)),hAPP_i1732201573de_int(number1226105091de_int,W_1))) <-> -hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_i1732201573de_int(number1226105091de_int,W_1)),hAPP_i1732201573de_int(number1226105091de_int,V))))) # label(fact_108_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1655 (all X all Y (hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = X -> zero_zero_real = Y)) # label(fact_3348_real__sqrt__sum__squares__eq__cancel) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1656 (all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),zero_zero_int)))) # label(fact_2171_double__add__le__zero__iff__single__add__le__zero) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1657 (all K_2 all A_3 all J_2 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),zero_zero_int),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(multInv(P_3),J_2)),J_2)),K_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(multInv(P_3),J_2)),A_3)),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),J_2))),P_3))))))) # label(fact_2239_aux______4) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1658 (all X_37 X_37 = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_37),one_one_nat)) # label(fact_562_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1659 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)),N)) = hAPP_nat_nat(div_mod_nat(M),N)) # label(fact_2688_Divides_Omod__div__equality_H) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1660 (all A_201 all E_3 all B_154 all C_87 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_201),E_3)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_154),E_3)),C_87)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_201),B_154)),E_3)),C_87)) # label(fact_873_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1661 (all N_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,N_1),Ma)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N_1)),one_one_real)),hAPP_int_real(real_int,Ma))))) # label(fact_4446_int__less__real__le) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1662 (all X all Y hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),X) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)) # label(fact_4306_gcd__semilattice__nat_Oinf__commute) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1663 (all M M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),zero_zero_nat)) # label(fact_1612_Nat_Oadd__0__right) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1664 (all I_1 all J all F ((all I_2 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,I_2)),hAPP_nat_nat(F,J_1))))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,I_1)),hAPP_nat_nat(F,J)))))) # label(fact_2815_less__mono__imp__le__mono) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1665 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))))) # label(fact_719_pos__zmult__pos) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1666 (all A_3 all B_2 all N hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(legacy_zgcd(A_3),B_2)),N) = hAPP_int_int(legacy_zgcd(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),N))) # label(fact_4496_zgcd__power__distrib) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1667 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sin,X)),zero_zero_real))))) # label(fact_3736_sin__less__zero) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1668 (all K_2 all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N))))) # label(fact_1557_dvd__mult__cancel) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1669 (all N hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)))) # label(fact_3711_two__realpow__gt) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1670 (all B_1_1 all B_2_1 is_bool(hAPP_f215623910l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1671 (all M_11 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_11)),zero_zero_real))) # label(fact_2919_of__nat__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1672 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A),B)))))) # label(fact_2606_pos__imp__zdiv__nonneg__iff) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1673 (all U all Ma all N_1 all I_1 all J (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),I_1)),U)),N_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N_1)))))) # label(fact_1839_nat__less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1674 (all A_9 all N_14 all B_7 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_9),hAPP_nat_nat(suc,N_14))),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_7),hAPP_nat_nat(suc,N_14)))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_7)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_9),B_7))))) # label(fact_3110_power__le__imp__le__base) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1675 (all Y_8 all Z_6 all X_11 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_6),X_11)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_6),Y_8)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_11),Z_6)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,Y_8),Z_6)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_11),Y_8)),Z_6)))) # label(fact_2580_div__add) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1676 (all C_46 all A_104 all B_86 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_46),A_104)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_46),B_86))) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_104),B_86)))) # label(fact_1997_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1677 (all A_200 all B_153 all C_86 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_200),C_86)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_153),C_86)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_200),B_153)),C_86)) # label(fact_877_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1678 (all Lx_4 all Ly_2 all Rx_4 all Ry_2 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_4),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ly_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_4),Ry_2))) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_4),Ly_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_4),Ry_2))) # label(fact_480_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1679 (all X all Y (hAPP_complex_real(re,X) = hAPP_complex_real(re,Y) -> (hAPP_complex_real(im,X) = hAPP_complex_real(im,Y) -> X = Y))) # label(fact_4117_complex__eqI) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1680 (all Z_1 all W (is_int(W) & is_int(Z_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),W)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z_1)) -> W = Z_1)))) # label(fact_63_zle__antisym) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1681 (all A all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),Ma)) -> hAPP_int_int(legacy_zgcd(big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(bnorRset(A),Ma))),Ma) = one_one_int)) # label(fact_4811_Bnor__prod__zgcd) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1682 (all A_195 all B_151 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_195),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_195),B_151)))) # label(fact_912_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1683 (all C_59 all D_12 all A_137 all B_106 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_137),B_106)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_59),D_12)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_137)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_59)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_137),C_59)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_106),D_12)))))))) # label(fact_1363_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1684 (all B all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5)))))))) # label(fact_2229_wset__mem) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1685 (all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,W))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(number_number_of_int,W)))) -> negDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(number_number_of_int,W)))) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(number_number_of_int,W)))) -> hAPP_P1975530577nt_int(adjust(hAPP_int_int(number_number_of_int,W)),negDivAlg(one_one_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(number_number_of_int,W)))) = negDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W))))) # label(fact_2321_negDivAlg__eqn__1__number__of) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1686 (all X_42 all Y_30 -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_42),X_42)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_30),Y_30))),zero_zero_rat))) # label(fact_234_not__sum__squares__lt__zero) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1687 (all A_76 all B_64 all B_63 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_76),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_64),B_63)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_76),B_64)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_76),B_63))) # label(fact_2292_mult_Oadd__right) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1688 (all C_39 all D_7 all A_97 all B_79 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_97),B_79)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_39),D_7)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_97),C_39)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_79),D_7)))))) # label(fact_2049_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1689 (all X_24 all N_32 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32)) | one_one_int = X_24 -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_24),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_24),N_32))))) # label(fact_1496_dvd__power) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1690 (all W_7 hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,W_7)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),hAPP_int_complex(number528085621omplex,W_7))),hAPP_int_complex(number528085621omplex,W_7))) # label(fact_373_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1691 (all M_7 all N_21 hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(semiri984289939at_nat,M_7)),hAPP_nat_nat(semiri984289939at_nat,N_21))) # label(fact_2965_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1692 (all X all Y (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> X != Y)) # label(fact_1717_dvd_Oless__imp__neq) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1693 (all A_194 all B_150 hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_194),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_150),A_194)))) # label(fact_913_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1694 (all A all P_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)),P_5)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int))),one_one_int),P_5)))) # label(fact_1583_inv__not__p__minus__1__aux) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1695 (all A_37 zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_37),zero_zero_int)) # label(fact_2470_div__by__0) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1696 (all B_94 all C_52 all A_116 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_116)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_94),C_52)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_94),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_116),C_52)))))) # label(fact_1880_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1697 (all Diff all F all N_1 all H (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H),zero_zero_real)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (F = hAPP_n546711566l_real(Diff,zero_zero_nat) -> ((all M_1 all T (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,H),T)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))))) -> (exists T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H),T)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,T),zero_zero_real)) & hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N_1),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N_1)))))))))) # label(fact_4936_Maclaurin__minus) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1698 (all X all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_int_real(real_int,A_3))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X)),A_3)))) # label(fact_4423_ceiling__le) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1699 (all B_129 all A_169 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_169)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_129),zero_z891286103de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_169),B_129)),zero_z891286103de_int))))) # label(fact_1108_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1700 (all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Y)))) # label(fact_4159_termination__basic__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1701 (exists X_2 (is_int(X_2) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(s1),X_2),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) & (all Y_1 (is_int(Y_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(s1),Y_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) -> Y_1 = X_2))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)))) # label(fact_439__096EX_B_As_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_A1_A_G_A_091s1) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1702 (all C all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B),A)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B))))) # label(fact_1195_mult__less__cancel__left__disj) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1703 (all K_1 all L (quickcheck_int_of(K_1) = quickcheck_int_of(L) <-> L = K_1)) # label(fact_4255_Quickcheck__Narrowing_Oint__of__inject) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1704 (all C_7 all A_22 all B_18 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_22),B_18)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_18),C_7)),A_22) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_18),A_22)),C_7))) # label(fact_2564_dvd__div__mult) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1705 (all X_1 (is_int(X_1) -> ((exists Y_1 (is_int(Y_1) & hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Y_1)),one_one_int) = X_1)) <-> -hBOOL(hAPP_int_bool(even_odd_even_int,X_1))))) # label(fact_3243_odd__equiv__def) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1706 (all A all B all P_1 ((all A_2 all B_4 all C_3 (hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4))) & hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,B_4),C_3))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_4),C_3)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),B_4)) -> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),C_3))))) -> ((all X_2 exists D_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2)) & (all A_2 all B_4 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),B_4)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_4),A_2)),D_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),X_2)) -> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4))))))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) -> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A),B))))))) # label(fact_4291_lemma__BOLZANO) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1707 (all S ((exists K hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,S),hAPP_n1699378549t_bool(ord_lessThan_nat,K)))) <-> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S)))) # label(fact_4952_finite__nat__iff__bounded) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1708 (all X_1 all P_1 ((all A_2 (is_int(A_2) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_2)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_2),one_one_int)))) -> hBOOL(hAPP_int_bool(P_1,A_2))))) -> hBOOL(hAPP_int_bool(P_1,X_1)))) # label(fact_2218_d22set__induct__old) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1709 (all Q_2 all B_2 all R_3 all C_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(div_mod_int(Q_2),C_1))),R_3))))))) # label(fact_2659_zmult2__lemma__aux3) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1710 (all C_76 all B_136 all A_176 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_136),A_176)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_76),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_176),C_76)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_136),C_76)))))) # label(fact_1069_mult__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1711 (all X_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),zero_zero_real))))) # label(fact_3917_real__root__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1712 (all K_1 (is_int(K_1) -> (min = hAPP_int_int(bit1,K_1) <-> K_1 = min))) # label(fact_766_rel__simps_I47_J) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1713 (all X_1 hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),cos)),cos),X_1),hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X_1))),hAPP_real_real(sin,X_1)))))) # label(fact_4792_DERIV__cos__cos__mult3) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1714 (all V_10 all U_3 all Y_26 all X_32 all A_209 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_32),A_209)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_26),A_209)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),U_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),V_10)) -> (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,U_3),V_10) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,U_3),X_32)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_10),Y_26))),A_209)))))))) # label(fact_750_convex__bound__lt) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1715 (all K_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),I)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),J_2))))) # label(fact_1655_mult__le__mono2) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1716 (all B_74 all A_86 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_86),zero_z891286103de_int)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_74),zero_z891286103de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_86),B_74)),zero_z891286103de_int))))) # label(fact_2145_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 7.24/7.27 1717 (all A_227 all B_164 all C_93 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_227),B_164)),C_93) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_227),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_164),C_93))) # label(fact_543_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1718 (all C_83 all D_16 all A_191 all B_147 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_191),B_147)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,C_83),D_16)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_191),C_83)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_147),D_16)))))) # label(fact_937_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1719 (all B_115 all C_67 all A_153 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_153)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_115),C_67)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_115),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_153),C_67)))))) # label(fact_1208_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1720 (all A_202 zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A_202)) # label(fact_852_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1721 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(div_mod_int(A_3),B_2))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(div_mod_int(A_3),B_2)),B_2)))) # label(fact_2614_pos__mod__conj) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1722 (all A_24 all B_19 all C_8 (is_int(C_8) -> (zero_zero_int = C_8 -> zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_8),A_24)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_8),B_19))) & (C_8 != zero_zero_int -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_24),B_19) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_8),A_24)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_8),B_19))))) # label(fact_2549_div__mult__mult1__if) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1723 (all A_1 all P_2 all P_1 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> ((exists Z all X_2 (is_int(X_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),X_2)) -> (hBOOL(hAPP_int_bool(P_1,X_2)) <-> hBOOL(hAPP_int_bool(P_2,X_2)))))) -> ((all X_2 (is_int(X_2) -> ((all Xa (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1)) -> X_2 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa))))) -> (hBOOL(hAPP_int_bool(P_1,X_2)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1))))))) -> ((all X_2 all K (is_int(X_2) & is_int(K) -> (hBOOL(hAPP_int_bool(P_2,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D_1)))) <-> hBOOL(hAPP_int_bool(P_2,X_2))))) -> ((exists X_2 ((exists Xa (hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xa),X_2))) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),A_1)) & is_int(Xa))) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) & is_int(X_2))) | (exists X_2 (is_int(X_2) & hBOOL(hAPP_int_bool(P_2,X_2)) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))))) <-> (exists X1 (hBOOL(hAPP_int_bool(P_1,X1)) & is_int(X1))))))))) # label(fact_5156_cppi) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1724 (all C_37 all A_95 all B_77 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_95),B_77)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_95),C_37)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_77),C_37))))) # label(fact_2058_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1725 (all Y all X all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,Y),A_3))))))) # label(fact_4405_powr__less__mono2) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1726 (all N_1 all Q_3 all P_1 ((all X_2 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(Q_3,X_2)),hAPP_nat_nat(P_1,X_2)))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,P_1),hAPP_n1699378549t_bool(ord_lessThan_nat,N_1))),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,Q_3),hAPP_n1699378549t_bool(ord_lessThan_nat,N_1))) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f1914919701at_nat(hAPP_f1408247010at_nat(cOMBS_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,minus_minus_nat),P_1)),Q_3)),hAPP_n1699378549t_bool(ord_lessThan_nat,N_1)))) # label(fact_4956_sum__diff__distrib) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1727 (all V_4 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4))) -> hAPP_int_real(number267125858f_real,V_4) = hAPP_nat_real(semiri132038758t_real,hAPP_int_nat(number_number_of_nat,V_4))) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4))) -> hAPP_nat_real(semiri132038758t_real,hAPP_int_nat(number_number_of_nat,V_4)) = zero_zero_real))) # label(fact_3219_of__nat__number__of__lemma) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1728 (all X_1 the_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_r481142414t_bool(hAPP_f1845081853t_bool(cOMBC_int_real_bool,hAPP_f801032407l_bool(hAPP_f499530369l_bool(cOMBB_1317819104ol_int,ord_less_eq_real),ring_1_of_int_real)),X_1))),hAPP_f429384717t_bool(hAPP_f1173902867t_bool(cOMBB_real_bool_int,hAPP_r1134773055l_bool(ord_less_real,X_1)),hAPP_f563293398t_real(hAPP_f134192053t_real(cOMBB_int_real_int,ring_1_of_int_real),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,plus_plus_int),one_one_int))))) = archim1246769320r_real(X_1)) # label(fact_4890_floor__real__def) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1729 (all A all B (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A)) | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B),zero_zero_rat)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),zero_zero_rat)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),B))))) # label(fact_1121_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1730 (all V_20 all W_16 all Z_15 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_20)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,W_16)),Z_15)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_20),W_16))),Z_15)) # label(fact_82_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1731 (all A_26 all B_21 (zero_zero_nat != B_21 -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_21),A_26)),B_21) = A_26)) # label(fact_2542_div__mult__self1__is__id) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1732 (all N hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N)) = archim1246769320r_real(hAPP_real_real(uminus_uminus_real,hAPP_nat_real(real_nat,N)))) # label(fact_4033_floor__minus__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1733 (all X_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root,N_1),X_1),hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,zero_zero_nat)))))))))) # label(fact_3942_DERIV__real__root) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1734 (all N_65 all A_199 (A_199 != zero_zero_complex -> zero_zero_complex != hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_199),N_65))) # label(fact_883_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1735 (all Y_2 all X_1 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_2)),zEven)))) # label(fact_2895_even__times__either) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1736 (all A_201 all E_3 all B_154 all C_87 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_201),E_3)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_154),E_3)),C_87)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_201),B_154)),E_3)),C_87)) # label(fact_868_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1737 (all X_36 all Y_28 all Z_9 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_36),Y_28)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_36),Z_9)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_36),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Y_28),Z_9))) # label(fact_620_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1738 (all Ma all N_1 (zero_zero_nat = Ma & zero_zero_nat != N_1 <-> zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Ma),N_1))) # label(fact_1814_nat__power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1739 (all I_1 all J (is_int(I_1) & is_int(J) -> (hAPP_int_int(legacy_zgcd(I_1),J) = zero_zero_int <-> I_1 = zero_zero_int & J = zero_zero_int))) # label(fact_4490_zgcd0) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1740 (all A_201 all E_3 all B_154 all C_87 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_201),E_3)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_154),E_3)),C_87)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_201),B_154)),E_3)),C_87)) # label(fact_869_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1741 (all A_202 zero_z891286103de_int = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,zero_z891286103de_int),A_202)) # label(fact_851_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1742 (all P all Q all R hAPP_nat_real(hAPP_r474017924t_real(P,R),Q) = hAPP_real_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__Nat__Onat_000tc__RealDef__Orea) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1743 (all P_1 all I_1 (is_int(I_1) -> ((all N_2 (hAPP_nat_int(semiri1621563631at_int,N_2) = I_1 -> hBOOL(hAPP_nat_bool(P_1,N_2)))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),zero_zero_int)) -> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))) <-> hBOOL(hAPP_nat_bool(P_1,hAPP_int_nat(nat,I_1)))))) # label(fact_3165_split__nat) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1744 (all R_3 all Q_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,R_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Q_2)) = A_3 -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),A_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Q_2)))))) # label(fact_421_self__quotient__aux1) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1745 (all N_9 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,N_9)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_9)))) # label(fact_3215_of__nat__less__two__power) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1746 (all B_95 all C_53 all A_117 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_117)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_95),C_53)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_95),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_117),C_53)))))) # label(fact_1871_add__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1747 (all M_17 all N_30 ((zero_zero_nat = N_30 -> one_one_complex = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,M_17),N_30)) & (zero_zero_nat != N_30 -> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,M_17),N_30) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,M_17),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat)))))) # label(fact_1833_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1748 (all A_207 hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_207),A_207))) # label(fact_798_dvd__refl) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1749 (all N_41 ((N_41 = zero_zero_nat -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),N_41) = one_one_int) & (N_41 != zero_zero_nat -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),N_41) = zero_zero_int))) # label(fact_1432_power__0__left) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1750 (all C_1 all A_3 all B_2 all Q_2 all R_3 (is_int(B_2) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3))) -> (B_2 != zero_zero_int -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1)) -> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1)),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Q_2),C_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(div_mod_int(Q_2),C_1))),R_3))))))))) # label(fact_2789_zmult2__lemma) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1751 (all C_39 all D_7 all A_97 all B_79 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_97),B_79)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_39),D_7)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_97),C_39)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_79),D_7)))))) # label(fact_2047_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1752 (all A_186 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,one_one_real),A_186))) # label(fact_976_one__dvd) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1753 (all Y_2 all X_1 all A (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(log(A),X_1)),hAPP_real_real(log(A),Y_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2))))))) # label(fact_3868_log__less__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1754 (all A_133 all N_46 all N_45 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_46),N_45)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_133)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_133),one_one_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_133),N_45)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_133),N_46))))))) # label(fact_1403_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1755 (all A_3 all B_2 hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(uminus_uminus_int,A_3)),hAPP_int_int(uminus_uminus_int,B_2)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)) # label(fact_4685_minus__rat__cancel) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1756 (all A (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A) = zero_zero_real <-> zero_zero_real = A)) # label(fact_247_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1757 (all N_1 ((exists M_1 hAPP_nat_nat(suc,M_1) = N_1) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)))) # label(fact_3049_gr0__conv__Suc) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1758 (all K_2 zero_zero_rat = frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,K_2),zero_zero_int))) # label(fact_4642_Frct__code__post_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1759 (all A_230 all C_96 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_96),A_230) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_230),C_96)) # label(fact_524_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1760 (all X_1 (-hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(suc,X_1))))) # label(fact_3759_even__Suc) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1761 (all L all U hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_at641636577an_int(L),U)))) # label(fact_4916_finite__atLeastLessThan__int) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1762 (all K_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),K_2))))) # label(fact_1645_add__le__mono1) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1763 (all Y_2 all X_1 (is_int(X_1) & is_int(Y_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2)) -> (zero_zero_int = X_1 & zero_zero_int = Y_2 <-> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_2) = zero_zero_int))))) # label(fact_2160_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1764 (all V_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,V_2))) -> hAPP_int_nat(number_number_of_nat,V_2) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V_2)),one_one_nat)))) # label(fact_3201_expand__Suc) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1765 (all W_1 all Z_2 (is_int(Z_2) -> ((exists N_2 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),hAPP_nat_int(semiri1621563631at_int,N_2)) = Z_2) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z_2))))) # label(fact_3003_zle__iff__zadd) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1766 (all V all N_1 ((-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),N_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_nat_nat(suc,N_1))))) # label(fact_4786_le__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1767 (all X all A_3 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,archim1246769320r_real(X)),A_3) = archim1246769320r_real(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_int_real(real_int,A_3)))) # label(fact_4420_floor__add) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1768 (all B_71 all A_83 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_83)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_71)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_83),B_71)))))) # label(fact_2170_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1769 (all A_3 all B_2 (A_3 != B_2 -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),A_3)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))))) # label(fact_1704_dvd_Oneq__le__trans) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1770 (all A_148 all N_51 all N_50 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_51),N_50)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,one_on1684967323de_int),A_148)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_148),N_51)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_148),N_50)))))) # label(fact_1275_power__increasing) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1771 (all N all M -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,N)),hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,M))))) # label(fact_3531_not__int__zless__negative) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1772 (all Y all X all Z_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Z_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_1))))) # label(fact_4158_termination__basic__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1773 (all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B)),zero_zero_real)))) # label(fact_2208_le__iff__diff__le__0) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1774 (all P all Q all R hAPP_n682674106l_real(hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,P),Q),R) = hAPP_b646336293l_real(P,hAPP_nat_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_I) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1775 (all A_60 all B_51 all C_30 hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_60),hAPP_Q1762011733de_int(div_mo231679042de_int(B_51),C_30))),C_30) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_60),B_51)),C_30)) # label(fact_2380_mod__mult__right__eq) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1776 (all B_115 all C_67 all A_153 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_153)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_115),C_67)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_115),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_153),C_67)))))) # label(fact_1205_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1777 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N))))) # label(fact_2816_Suc__mono) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1778 (all B_155 all A_204 (A_204 != zero_zero_rat -> (zero_zero_rat != B_155 -> zero_zero_rat != hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_204),B_155)))) # label(fact_834_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1779 (all A (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),zero_zero_rat)))) # label(fact_2173_double__add__le__zero__iff__single__add__le__zero) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1780 (all X_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hBOOL(hAPP_real_bool(isCont_real_real(hAPP_n546711566l_real(root,N_1)),X_1))))) # label(fact_5171_isCont__root__pos) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1781 (all A_224 all N_68 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_224),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_224),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68))) # label(fact_574_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1782 (all X all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))) -> -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),A_3)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),B_2))))) # label(fact_4315_gcd__semilattice__nat_Ole__infE) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1783 (all P all Q all R hAPP_P989584703t_bool(hAPP_f1781059817t_bool(hAPP_f326728547t_bool(cOMBB_2086486755nt_int,P),Q),R) = hAPP_f237669397t_bool(P,hAPP_P989584703t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__H) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1784 (all Y all X (hBOOL(hAPP_int_bool(nat_is_nat,X)) -> (hBOOL(hAPP_int_bool(nat_is_nat,Y)) -> hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(div_mod_int(X),Y)))))) # label(fact_5136_Divides_Otransfer__int__nat__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1785 (all K_2 (is_int(K_2) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,pls),K_2) = K_2)) # label(fact_264_add__Pls) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1786 (all D all Sa exists Z all X_2 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z),X_2)) -> (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_2),Sa))) <-> -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_2),Sa)))))) # label(fact_2256_pinf_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1787 (all K_2 all I all J_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I),K_2)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,J_2),K_2))))) # label(fact_269_zadd__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1788 (all A_198 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,zero_z126310315umeral),A_198)) -> A_198 = zero_z126310315umeral)) # label(fact_886_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1789 (all P all Q all R hAPP_f883232462nt_int(hAPP_i998473677nt_int(P,R),Q) = hAPP_i1584592887nt_int(hAPP_f1574055885nt_int(hAPP_f960630521nt_int(cOMBC_1805044090nt_int,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_019) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1790 (all X_1 all Y_2 (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_2) = zero_zero_real <-> hAPP_real_real(uminus_uminus_real,X_1) = Y_2)) # label(fact_3484_real__add__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1791 (all K_1 all F (hBOOL(summable_real(F)) -> ((all D_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),D_2)))),hAPP_nat_real(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),D_2)),one_one_nat))))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),K_1))),hAPP_f352196356l_real(suminf_real,F)))))) # label(fact_4927_sumr__pos__lt__pair) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1792 (all C_85 all A_193 all B_149 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_193),B_149)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_193),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_149),C_85))))) # label(fact_920_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1793 (all A_92 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_92),one_one_nat) = A_92) # label(fact_2099_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1794 (all B_141 all A_181 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_181),zero_z126310315umeral)) & hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_141)) | hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_141),zero_z126310315umeral)) & hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_181)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_181),B_141)),zero_z126310315umeral)))) # label(fact_1047_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1795 (all V_11 all W_9 hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_11),W_9)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_11)),hAPP_int_rat(number_number_of_rat,W_9))) # label(fact_344_number__of__add) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1796 (all Z_8 all X_23 all Y_18 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_23),Y_18)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_23),Z_8)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_23),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_18),Z_8)))))) # label(fact_1516_dvd__diff) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1797 (all X_1 all N_1 all Y_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),hAPP_nat_nat(suc,N_1))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_2),hAPP_nat_nat(suc,N_1)))))) # label(fact_3089_exp__mono__le) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1798 (all N (hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) -> (exists X_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_3797_even__square) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1799 (all A_143 all C_65 all B_112 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_143),C_65)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_112),C_65))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_65)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_143),B_112))))) # label(fact_1328_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1800 (all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),B))))) # label(fact_1120_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1801 (all Z_1 all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),Z_1)))))) # label(fact_4319_gcd__semilattice__nat_Oinf__greatest) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1802 (all A_202 zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A_202)) # label(fact_847_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1803 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X))))))) # label(fact_3473_ln__one__plus__pos__lower__bound) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1804 (all Y (Y != zero_zero_nat -> -(all Nat_1 Y != hAPP_nat_nat(suc,Nat_1)))) # label(fact_3309_nat_Oexhaust) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1805 (all A_21 all B_17 (is_int(B_17) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_21),B_17)) -> B_17 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_17),A_21)),A_21)))) # label(fact_2567_dvd__div__mult__self) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1806 (all R_3 all A_3 hAPP_complex_real(re,rcis(R_3,A_3)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_real_real(cos,A_3))) # label(fact_4262_Re__rcis) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1807 (all X_7 all Q_6 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_7),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_7),Q_6)) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_7),hAPP_nat_nat(suc,Q_6))) # label(fact_3026_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1808 (all P_5 hAPP_P1175774780nt_int(produc1298267108nt_int(div_div_int),quotient_of(P_5)) = archim791455193or_rat(P_5)) # label(fact_4833_rat__floor__code) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1809 (all B_94 all C_52 all A_116 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_116)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_94),C_52)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_94),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_116),C_52)))))) # label(fact_1877_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1810 (all X_5 all Y_5 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_5),Y_5)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_5),Y_5)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_5),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_5),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))))) # label(fact_3190_realpow__two__diff) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1811 (all B_116 all A_155 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_155)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_116)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_155),B_116)))))) # label(fact_1188_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1812 (all M all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N)),N))) # label(fact_4298_gcd__dvd2__nat) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1813 (all R_3 all S_1 all T_1 all R_5 all S_3 all M all T_2 all N (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_3),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_2),N)) = R_5 -> (R_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_1),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_1),N)) -> hAPP_int_int(div_mod_int(R_5),R_3) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,S_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,R_5),R_3)),S_1))),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,T_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,R_5),R_3)),T_1))),N))))) # label(fact_2486_xzgcda__linear__aux2) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1814 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),hAPP_int_int(div_mod_int(A_3),B_2))))) # label(fact_2526_neg__mod__bound) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1815 (all X_1 (X_1 = zero_zero_nat <-> zero_zero_nat = X_1)) # label(fact_1895_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1816 (all A all B all P_5 (hBOOL(hAPP_nat_bool(prime,P_5)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),B))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_5),A)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_5),B))))) # label(fact_4169_prime__divprod__eq) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1817 (all Y all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,one_one_real)),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,Y)),Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,X)),X)))))) # label(fact_3854_ln__x__over__x__mono) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1818 (all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> hAPP_nat_nat(suc,hAPP_int_nat(nat,Z_1)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z_1)))) # label(fact_3204_Suc__nat__eq__nat__zadd1) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1819 (all X_1 all Y_2 (hBOOL(hAPP_int_bool(even_odd_even_int,Y_2)) & hBOOL(hAPP_int_bool(even_odd_even_int,X_1)) | -hBOOL(hAPP_int_bool(even_odd_even_int,Y_2)) & -hBOOL(hAPP_int_bool(even_odd_even_int,X_1)) <-> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_2))))) # label(fact_2892_even__difference) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1820 (all R_3 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,archim1246769320r_real(R_3))),one_one_real)))) # label(fact_4452_real__of__int__floor__add__one__gt) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1821 (all N_42 all A_130 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_130)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_130),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_130),N_42)))))) # label(fact_1430_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1822 (all Z_7 all W_2 all Y_9 all X_12 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Y_9),X_12)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_7),W_2)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_12),W_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_9),Z_7)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_12),Y_9)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,W_2),Z_7))))) # label(fact_2559_div__mult__div__if__dvd) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1823 (all B all A (B = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B),A) <-> zero_zero_rat = A)) # label(fact_615_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1824 (all P all Q all R hAPP_real_real(hAPP_r1250527377l_real(P,R),hAPP_real_real(Q,R)) = hAPP_real_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__RealDef_) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1825 (all A_145 all M_20 all N_48 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_145),M_20)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_145),N_48)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_145),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_20),N_48))) # label(fact_1306_power__add) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1826 (all X_3 all A (hBOOL(tendsto_nat_complex(X_3,A,sequentially)) -> hBOOL(tendsto_nat_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_3),hAPP_complex_complex(cnj,A),sequentially)))) # label(fact_5055_cnj_OLIMSEQ) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1827 (all G all X_1 all Ma (hBOOL(hAPP_real_bool(deriv_real(G,X_1),Ma)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,sin),G),X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,hAPP_real_real(G,X_1))),Ma))))) # label(fact_4762_DERIV__fun__sin) # label(axiom) # label(non_clause). [assumption]. 7.38/7.45 1828 (all A_220 (is_int(A_220) -> A_220 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),A_220))) # label(fact_598_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1829 (all Z_1 all X all Y all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Y)),one_one_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Y),Z_1))),one_one_int),P_3)))) # label(fact_1462_zcong__zpower__zmult) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1830 (all A_110 all B_88 all C_48 (hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_110),C_48) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_110),B_88) -> C_48 = B_88)) # label(fact_1927_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1831 (all Y all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y)) -> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(ln,X)),hAPP_real_real(ln,Y)) = hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y))))) # label(fact_3454_ln__mult) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1832 (all N all M all R_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),R_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),R_3)) -> (one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)),R_3)))))) # label(fact_4367_divides__mult__nat) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1833 (all X all M hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,M))),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,M)),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3707_cos__expansion__lemma) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1834 (all A_122 all B_100 all V_6 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_122),B_100)),hAPP_int_complex(number528085621omplex,V_6)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_122),hAPP_int_complex(number528085621omplex,V_6))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_100),hAPP_int_complex(number528085621omplex,V_6)))) # label(fact_1550_left__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1835 (all L all U hAPP_i1948725293t_bool(ord_at641636577an_int(L),U) = image_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,plus_plus_int),L),hAPP_i1948725293t_bool(ord_at641636577an_int(zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,U),L)))) # label(fact_4922_image__add__int__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1836 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(ln,X))))) # label(fact_3440_ln__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1837 (all M all N hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)),hAPP_nat_nat(div_mod_nat(M),N)) = divmod_nat(M,N)) # label(fact_4211_divmod__nat__div__mod) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1838 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y)) -> (exists X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),hAPP_real_real(tan,X_2))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) # label(fact_3629_lemma__tan__total) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1839 (all A_11 all N_17 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_11),hAPP_nat_nat(suc,N_17)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_11),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_11),N_17))) # label(fact_3011_power__Suc) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1840 (all B_131 all A_171 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_171),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_131)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_171),B_131)),zero_zero_int))))) # label(fact_1092_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1841 (all A_197 all B_152 all N_64 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_197),B_152)),N_64) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_197),N_64)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_152),N_64))) # label(fact_895_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1842 (all P_1 all N_1 ((exists X_2 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),N_1))) & hBOOL(hAPP_nat_bool(P_1,X_2)))) <-> (exists M_1 (hBOOL(hAPP_nat_bool(P_1,M_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)))))) # label(fact_5070_ex__nat__less) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1843 (all M_6 all N_20 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_n1108039445umeral(semiri1619134803umeral,M_6)),N_20) = hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20))) # label(fact_2978_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1844 (all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_3)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))))))) # label(fact_2804_inv__inv__aux) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1845 (all A_190 all B_146 all K_7 (A_190 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_146),K_7) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_146),A_190)))) # label(fact_946_dvdI) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1846 (all X all N hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(abs_abs_int,X)),N)))) # label(fact_3435_zero__le__zpower__abs) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1847 (all A_64 all B_52 all C_31 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_64),C_31) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_64),B_52)),B_52)),hAPP_C498520661umeral(div_mo1740067990umeral(A_64),B_52))),C_31)) # label(fact_2356_div__mod__equality) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1848 (all N_61 all X_29 all Y_23 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_29),Y_23)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_29),N_61)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_23),N_61))))) # label(fact_988_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1849 (all C_63 all A_141 all B_110 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_63),A_141)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_63),B_110))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_63)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_141),B_110))))) # label(fact_1338_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1850 (all L all U hAPP_n1699378549t_bool(ord_at4362885an_nat(hAPP_nat_nat(suc,L)),U) = hAPP_n1699378549t_bool(ord_gr660468384an_nat(L),U)) # label(fact_4905_atLeastSucLessThan__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1851 (all N all M all R_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),R_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),R_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)),R_3)))))) # label(fact_4026_divides__mul) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1852 (all K_2 all L_2 hAPP_int_int(bit0,K_2) != hAPP_int_int(bit1,L_2)) # label(fact_252_rel__simps_I49_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1853 (all X_1 all P_5 (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_5)) <-> hAPP_int_int(standardRes(P_5),X_1) != zero_zero_int)) # label(fact_2593_StandardRes__prop3) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1854 (all N_1 all V ((-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))))) & -hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,N_1)),hAPP_int_nat(number_number_of_nat,V))))) # label(fact_4784_less__Suc__number__of) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1855 (all V_7 all B_101 all C_56 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_7)),B_101)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_7)),C_56)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_7)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,B_101),C_56))) # label(fact_1546_right__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1856 (all A_28 all B_23 all C_10 (C_10 != zero_z891286103de_int -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_10),A_28)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_10),B_23)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_28),B_23))) # label(fact_2536_div__mult__mult1) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1857 (all N_11 all A_7 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_7)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_7),one_on1684967323de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_7),hAPP_nat_nat(suc,N_11))),one_on1684967323de_int))))) # label(fact_3160_power__Suc__less__one) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1858 (all C_7 all A_22 all B_18 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_22),B_18)) -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_18),C_7)),A_22) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B_18),A_22)),C_7))) # label(fact_2562_dvd__div__mult) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1859 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat)) # label(fact_1762_diff__is__0__eq_H) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1860 (all A_20 all C_6 all B_16 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,C_6),B_16)) -> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_20),B_16)),C_6) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_20),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B_16),C_6)))) # label(fact_2569_div__mult__swap) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1861 (all N_59 all M_24 all X_28 all Y_22 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_28),Y_22)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_59),M_24)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_28),N_59)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_22),M_24)))))) # label(fact_1001_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1862 (all B_2 (is_int(B_2) -> (zero_zero_int != B_2 -> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(zero_zero_int,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int)))))) # label(fact_2793_divmod__int__rel__0) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1863 (all M all K_2 (M != hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2) -> (hAPP_nat_nat(suc,zero_zero_nat) != M -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M))))) # label(fact_3132_one__less__m) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1864 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(div_mod_nat(M),N))) # label(fact_2686_div__mod__equality_H) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1865 (all A all B all C (C != zero_zero_real -> (A = B <-> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A)))) # label(fact_1788_real__mult__left__cancel) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1866 (all B_116 all A_155 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_155)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_116)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_155),B_116)))))) # label(fact_1190_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1867 (all X_22 all Y_17 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_22),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_17),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_22)),Y_17)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_22),Y_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_1662_power2__diff) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1868 (all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,W))) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(number_number_of_int,W))) -> posDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W)) = hAPP_P1975530577nt_int(adjust(hAPP_int_int(number_number_of_int,W)),posDivAlg(one_one_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(number_number_of_int,W))))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(number_number_of_int,W))) -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int) = posDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W))))) # label(fact_2680_posDivAlg__eqn__1__number__of) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1869 (all Sa all Ta all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Sa),Ta)) -> hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,A),Sa)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,B),Ta))),pair_leq))))) # label(fact_4542_pair__leqI2) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1870 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> X = hAPP_real_real(exp_real,hAPP_real_real(ln,X)))) # label(fact_3848_exp__ln) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1871 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> (hAPP_nat_nat(suc,M) != N -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),N))))) # label(fact_2859_Suc__lessI) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1872 (all A_12 all N_18 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_12),N_18)),A_12) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_12),hAPP_nat_nat(suc,N_18))) # label(fact_3010_power__Suc2) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1873 (all B_72 all C_35 all A_84 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_84)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_72),C_35)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_72),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_84),C_35)))))) # label(fact_2153_add__increasing) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1874 (all C_83 all D_16 all A_191 all B_147 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_191),B_147)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,C_83),D_16)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_191),C_83)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_147),D_16)))))) # label(fact_935_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 7.45/7.45 1875 (all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)))) # label(fact_4195_prime__ge__2) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1876 (all N hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))))) # label(fact_3851_inv__real__of__nat__fact__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1877 (all B_97 all A_119 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_119)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_97)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_119),B_97)))))) # label(fact_1861_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1878 (all X_1 all Y_2 all Z_2 (X_1 = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_2),Z_2) <-> Y_2 = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Z_2))) # label(fact_2782_eq__diff__eq_H) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1879 (all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))) -> A_3 != hAPP_int_int(inv(P_3),A_3))))) # label(fact_2130_inv__distinct) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1880 (all A_190 all B_146 all K_7 (hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_146),K_7) = A_190 -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,B_146),A_190)))) # label(fact_943_dvdI) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1881 (all K_1 all Ma all N_1 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),N_1) <-> Ma = N_1)) # label(fact_1594_nat__add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1882 (all N all A_3 all B_2 (hAPP_int_int(legacy_zgcd(A_3),B_2) = one_one_int -> hAPP_int_int(legacy_zgcd(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N)),B_2) = one_one_int)) # label(fact_4504_zgcd__1__power__left__distrib) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1883 (all Lx_6 all Ly_4 all Rx_6 all Ry_4 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_6),Rx_6)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Ly_4),Ry_4)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_6),Ly_4)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx_6),Ry_4))) # label(fact_463_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1884 (all B_71 all A_83 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_83)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_71)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_83),B_71)))))) # label(fact_2169_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1885 (all A_3 all B_2 archim791455193or_rat(hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)) # label(fact_4683_floor__Fract) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1886 (all A_218 all B_161 all C_90 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_218),C_90)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_161),C_90)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_218),B_161)),C_90)) # label(fact_632_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1887 (all P all Q all R hAPP_r265291036omplex(P,hAPP_nat_real(Q,R)) = hAPP_n1983812447omplex(hAPP_f23851874omplex(hAPP_f406530075omplex(cOMBB_1714920583ex_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__C) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1888 (all A_124 all N_34 all B_102 (hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_102),N_34) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_124),N_34) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_124)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_102)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_34)) -> A_124 = B_102))))) # label(fact_1483_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1889 (all W_6 hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,W_6)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(number_number_of_int,W_6))),hAPP_int_int(number_number_of_int,W_6))) # label(fact_375_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1890 (all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,N),zero_zero_nat) = zero_zero_nat) # label(fact_5122_min__0R) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1891 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(real_nat,M)),hAPP_nat_real(real_nat,N)))) # label(fact_3671_real__of__nat__diff) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1892 (all X hAPP_real_real(abs_abs_real,hAPP_nat_real(real_nat,X)) = hAPP_nat_real(real_nat,X)) # label(fact_3635_abs__real__of__nat__cancel) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1893 (all N_4 ((exists M_1 all X_2 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),N_4)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),M_1)))) <-> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_4)))) # label(fact_4625_finite__nat__set__iff__bounded__le) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1894 (all N_13 all A_8 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_8)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_8),hAPP_nat_nat(suc,N_13)))))) # label(fact_3116_power__gt1) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1895 (all R_3 all X all Y hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,of_real_complex(R_3)),hAPP_real_complex(hAPP_r265291036omplex(complex,X),Y)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,R_3),X)),Y)) # label(fact_3993_complex__of__real__add__Complex) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1896 (all X_22 all Y_17 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_22),Y_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_22),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_17),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_22)),Y_17))) # label(fact_1664_power2__diff) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1897 (all A_28 all B_23 all C_10 (is_int(C_10) -> (zero_zero_int != C_10 -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_28),B_23) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_10),A_28)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_10),B_23))))) # label(fact_2533_div__mult__mult1) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1898 (all B_121 all A_160 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_160),zero_z891286103de_int)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_121)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_160),B_121)),zero_z891286103de_int))))) # label(fact_1156_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1899 (all A_21 all B_17 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_21),B_17)) -> hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B_17),A_21)),A_21) = B_17)) # label(fact_2565_dvd__div__mult__self) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1900 (all X_1 (hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_nat_int(semiri1621563631at_int,X_1))) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)))) # label(fact_3765_Parity_Otransfer__int__nat__relations) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1901 (all B_53 all A_65 all C_32 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_53),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_65),B_53))),hAPP_int_int(div_mod_int(A_65),B_53))),C_32) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_65),C_32)) # label(fact_2350_div__mod__equality2) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1902 (all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,X_1)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),pls)))) # label(fact_97_le__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1903 (all P all Q all R hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,P),Q),R) = hAPP_f448129468l_bool(hAPP_i2112223885l_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1904 (all F all A_1 hBOOL(hAPP_f448129468l_bool(nat_tr876908586nt_nat(F),A_1))) # label(fact_2780_transfer__morphism__def) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1905 (all Ma all N_1 hAPP_nat_nat(nat_case_nat(zero_zero_nat,hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,suc),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,ord_min_nat),N_1))),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Ma),hAPP_nat_nat(suc,N_1))) # label(fact_5125_min__Suc2) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1906 (all A all B all C all D (hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C),D) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),D)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B))))) # label(fact_2097_diff__eq__diff__less) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1907 (all P all Q all R hAPP_int_real(hAPP_f269654879t_real(hAPP_f766218899t_real(cOMBB_real_real_int,P),Q),R) = hAPP_real_real(P,hAPP_int_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__Int__Oin) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1908 (all Z_8 all X_23 all Y_18 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_23),Y_18)) -> (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_23),Z_8)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_23),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Y_18),Z_8)))))) # label(fact_1517_dvd__diff) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1909 (all B_118 all A_157 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_157)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_118),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_118),A_157)),zero_zero_real))))) # label(fact_1175_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1910 (all I all M all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),N)))) # label(fact_4980_dvd__gcd__D2__int) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1911 (all M hAPP_int_int(legacy_zgcd(M),one_one_int) = one_one_int) # label(fact_4488_zgcd__1) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1912 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)),N) = M)) # label(fact_2721_div__mult__self__is__m) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1913 (all A_3 all B_2 all C_1 all D_3 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)),hAPP_real_complex(hAPP_r265291036omplex(complex,C_1),D_3)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),C_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_2),D_3))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),D_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_2),C_1)))) # label(fact_3965_complex__mult) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1914 (all K_1 all I_1 all J hAPP_n1699378549t_bool(ord_at4362885an_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K_1)) = image_nat_nat(hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),K_1),hAPP_n1699378549t_bool(ord_at4362885an_nat(I_1),J))) # label(fact_4906_image__add__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1915 (all X one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,one_one_nat),X)) # label(fact_4335_gcd__lcm__complete__lattice__nat_Oinf__bot__left) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1916 (all M all N hAPP_nat_nat(div_mod_nat(M),N) = hAPP_P1654798560at_nat(product_snd_nat_nat,divmod_nat(M,N))) # label(fact_4240_mod__nat__def) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1917 (all X all A_3 all B_2 hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_2)),A_3) = hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),B_2)) # label(fact_4384_powr__powr__swap) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1918 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),M)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N))) # label(fact_4358_gcd__diff2__nat) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1919 (all X_1 all Y_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2)) <-> X_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_2))) # label(fact_4329_gcd__semilattice__nat_Ole__iff__inf) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1920 (all M zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),zero_zero_nat)) # label(fact_1620_mult__0__right) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1921 (all X all Y (is_int(X) & is_int(Y) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,X),Y)) | X = Y)) # label(help_fequal_1_1_fequal_000tc__Int__Oint_T) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1922 (all D all F all X_1 all L (hBOOL(hAPP_real_bool(deriv_real(F,X_1),L)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D)) -> ((all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y_1))),D)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(F,Y_1)),hAPP_real_real(F,X_1))))) -> L = zero_zero_real)))) # label(fact_3885_DERIV__local__max) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1923 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) -> hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),one_one_nat)))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),one_one_real))),suc))) = hAPP_real_real(ln,X_1)))) # label(fact_4826_ln__series) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1924 (all A_33 all B_27 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_33),B_27)),B_27) = zero_zero_int) # label(fact_2495_mod__mult__self2__is__0) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1925 (all A_122 all B_100 all V_6 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_122),hAPP_int_real(number267125858f_real,V_6))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_100),hAPP_int_real(number267125858f_real,V_6))) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_122),B_100)),hAPP_int_real(number267125858f_real,V_6))) # label(fact_1549_left__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1926 (all N N = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),one_one_nat)) # label(fact_1659_nat__mult__1__right) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1927 (all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(fact_fact_int,N))))) # label(fact_4281_fact__gt__zero__int) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1928 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(ln,X))))) # label(fact_3439_ln__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1929 (all A all A_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma)) -> (hBOOL(hAPP_int_bool(is_RRset(A_1),Ma)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),hAPP_int_int(rRset2norRR(A_1,Ma),A)),Ma)) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(rRset2norRR(A_1,Ma),A)),hAPP_i1948725293t_bool(norRRset,Ma))))))) # label(fact_4591_RRset2norRR__correct) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1930 (all A_109 all B_87 all C_47 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_109),B_87)),C_47) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_109),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_87),C_47))) # label(fact_1943_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1931 (all P_1 P_1 = collect_int(P_1)) # label(fact_145_Collect__def) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1932 (all A_227 all B_164 all C_93 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_227),B_164)),C_93) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_227),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_164),C_93))) # label(fact_542_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1933 (all Lx_5 all Ly_3 all Rx_5 all Ry_3 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_5),Ly_3)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_5),Ry_3)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_5),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_5),Ly_3)),Ry_3))) # label(fact_474_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1934 (all X hAPP_complex_real(im,hAPP_complex_complex(invers1449016382omplex,X)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_4112_complex__Im__inverse) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1935 (all X_3 (hBOOL(summable_complex(X_3)) -> hBOOL(summable_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_3))))) # label(fact_4847_Im_Osummable) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1936 (all C_33 all D_5 all A_79 all B_67 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_79),B_67)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_33),D_5)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_79),C_33)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_67),D_5)))))) # label(fact_2203_add__less__le__mono) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1937 (all X_19 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_19),X_19)),one_one_real) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_19),one_one_real)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_19),one_one_real))) # label(fact_1823_real__squared__diff__one__factored) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1938 (all C_38 all A_96 all B_78 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_96),B_78)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_38),A_96)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_38),B_78))))) # label(fact_2054_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1939 (all A_13 all B_9 A_13 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_13),B_9)),B_9)),hAPP_nat_nat(div_mod_nat(A_13),B_9))) # label(fact_2674_mod__div__decomp) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1940 (all K_2 all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N))) # label(fact_4334_gcd__mult__distrib__nat) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1941 (all V_9 all U_2 all Y_19 all X_25 all A_127 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_25),A_127)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_19),A_127)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),U_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),V_9)) -> (one_one_real = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,U_2),V_9) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,U_2),X_25)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,V_9),Y_19))),A_127)))))))) # label(fact_1465_convex__bound__le) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1942 (all Z_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(nat,Z_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_2)))) # label(fact_2734_zero__less__nat__eq) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1943 (all N_52 all A_150 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_150)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_150),N_52))))) # label(fact_1253_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1944 (all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))) -> hAPP_int_int(inv(P_3),A_3) != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))))) # label(fact_2128_inv__not__p__minus__1) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1945 (all A all E_2 all C all B all D (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),E_2)),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),E_2)),D))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B)),E_2)),C)),D)))) # label(fact_1570_less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1946 (all A_108 A_108 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_108),zero_z891286103de_int)) # label(fact_1966_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1947 (all A_226 all B_163 all C_92 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_226),B_163)),C_92) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_226),C_92)),B_163)) # label(fact_547_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1948 (all N_33 all A_123 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_123)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_33)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_123),N_33)))))) # label(fact_1487_one__less__power) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1949 (all A_3 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,one_one_nat),A_3))) # label(fact_4015_coprime__1_H) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1950 (all A_3 all B_2 normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_3),B_2)) = quotient_of(hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2))) # label(fact_4711_quotient__of__Fract) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1951 (all K_2 hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_nat_int(semiri1621563631at_int,K_2)),one_one_int) = hAPP_nat_rat(semiri151668891at_rat,K_2)) # label(fact_4676_of__nat__rat) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1952 (all X_47 all Y_35 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_47),X_47)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_35),Y_35))))) # label(fact_96_sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1953 (all F all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) -> ((all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),B)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X_2)) -> (exists Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_1)) & hBOOL(hAPP_real_bool(deriv_real(F,X_2),Y_1)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(F,A)),hAPP_real_real(F,B)))))) # label(fact_4550_DERIV__nonneg__imp__nonincreasing) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1954 (all N (N != one_one_nat -> (exists P_4 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),N)) & hBOOL(hAPP_nat_bool(prime,P_4)))))) # label(fact_4205_prime__factor) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1955 (all N_25 hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_n1108039445umeral(semiri1619134803umeral,N_25)))) # label(fact_2912_of__nat__0__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1956 (all Y_2 all X_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),X_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_2)) -> (X_1 = zero_zero_nat & Y_2 = zero_zero_nat <-> zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_1),Y_2))))) # label(fact_2161_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1957 (all Y all X (hAPP_real_real(exp_real,Y) = X -> Y = hAPP_real_real(ln,X))) # label(fact_3820_ln__unique) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1958 (all N hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),N) = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,pi),hAPP_nat_real(real_nat,N)))) # label(fact_3716_cos__npi2) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1959 (all F all A_1 hBOOL(hAPP_f54304608l_bool(nat_tr160667106at_int(F),A_1))) # label(fact_2779_transfer__morphismI) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1960 (all N all A_3 all B_2 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),B_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,P_3),N)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,P_3),N)),A_3)))))) # label(fact_451_zprime__power__zdvd__cancel__right) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1961 (all B_94 all C_52 all A_116 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_116)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_94),C_52)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_94),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_116),C_52)))))) # label(fact_1875_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1962 (all B all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int))) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(inv(P_5),B)),hAPP_i1948725293t_bool(wset(A),P_5))) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5)))))))))) # label(fact_2215_wset__inv__mem__mem) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1963 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(ln,X_1)))))) # label(fact_3441_ln__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1964 (all N_49 all A_147 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),A_147)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_147),N_49))))) # label(fact_1282_one__le__power) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1965 (all A_50 all C_22 all B_41 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_50),B_41)),C_22) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_50),C_22)),B_41)),C_22)) # label(fact_2411_zmod__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1966 (all A_134 all N_47 all B_103 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_134),N_47)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_103),N_47))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_103)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_134),B_103))))) # label(fact_1397_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1967 (all K_1 all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K_1)),hAPP_int_int(bit1,L))))) # label(fact_66_rel__simps_I34_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1968 (all X_1 all F ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(F,N_2)))) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_2))),X_1))) -> hBOOL(summable_real(F))))) # label(fact_4946_pos__summable) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1969 (all A_194 all B_150 hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_194),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_150),A_194)))) # label(fact_916_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1970 (all I all N hAPP_nat_nat(nat_aux(I),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat,I)),N)) # label(fact_2772_nat__aux__def) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1971 (all A_234 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_234),A_234) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_234),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_305_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1972 (all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),pls)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,X_1)),zero_zero_int)))) # label(fact_393_less__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1973 (all B_97 all A_119 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_119)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_97)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_119),B_97)))))) # label(fact_1862_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1974 (all A all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),archim856651990g_real(X_1))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,A)),X_1)))) # label(fact_4421_less__ceiling__eq) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1975 (all Lx_3 all Ly_1 all Rx_3 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_3),Rx_3)),Ly_1) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_3),Ly_1)),Rx_3)) # label(fact_488_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1976 (all N N = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),N)) # label(fact_5192_max__0L) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1977 (all N_62 one_one_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,one_one_real),N_62)) # label(fact_971_power__one) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1978 (all M zero_zero_nat = hAPP_nat_nat(div_mod_nat(M),hAPP_nat_nat(suc,zero_zero_nat))) # label(fact_3078_mod__1) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1979 (all X hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_738_quartic__square__square) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1980 (all B all A (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),zOdd)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),zOdd)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B)),zEven))))) # label(fact_3091_IntNatAux_Oodd__plus__odd) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1981 (all Z_1 zero_zero_real = hAPP_complex_real(im,of_real_complex(Z_1))) # label(fact_4094_Im__complex__of__real) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1982 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(tan,Y)),hAPP_real_real(tan,X))))))) # label(fact_3616_tan__monotone) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1983 (all W pls = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,pls),W)) # label(fact_125_mult__Pls) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1984 (all V all V_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_int_nat(number_number_of_nat,V_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),V_1)) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),V_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V_1))))) # label(fact_152_less__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1985 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K1)),hAPP_int_int(bit0,K2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)))) # label(fact_68_less__eq__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1986 (all D all A all B (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D),B) = one_one_nat & hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D),A) = one_one_nat <-> one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),B)))) # label(fact_4352_coprime__mul__eq__nat) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1987 (all Y_2 all X_1 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_2)),zEven))))) # label(fact_2897_EvenOdd_Oeven__plus__even) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1988 (all B_94 all C_52 all A_116 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_116)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_94),C_52)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_94),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_116),C_52)))))) # label(fact_1879_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1989 (all Ma all X_3 ((all Y1 all Y2 (is_int(Y1) & is_int(Y2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Y1),Y2),Ma)) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y2),X_3)) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y1),X_3)) -> Y1 = Y2))) <-> hBOOL(hAPP_f448129468l_bool(resSet(Ma),X_3)))) # label(fact_4054_ResSet__def) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1990 (all C all A all B (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A),B)) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),A)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),B))))) # label(fact_2068_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1991 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real)) -> hBOOL(monoseq_real(hAPP_r474017924t_real(power_power_real,X_1)))))) # label(fact_4806_monoseq__realpow) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1992 (all A_73 all A_72 all B_61 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_73),B_61)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_72),B_61)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_73),A_72)),B_61)) # label(fact_2301_mult_Odiff__left) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1993 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K1)),hAPP_int_int(bit0,K2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)))) # label(fact_135_less__eq__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1994 (all X_19 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_19),one_one_rat)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_19),one_one_rat)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_19),X_19)),one_one_rat)) # label(fact_1825_real__squared__diff__one__factored) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1995 (all Z_1 hAPP_complex_complex(cnj,hAPP_int_complex(ring_11397209091omplex,Z_1)) = hAPP_int_complex(ring_11397209091omplex,Z_1)) # label(fact_4580_complex__cnj__of__int) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1996 (all C_33 all D_5 all A_79 all B_67 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_79),B_67)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_33),D_5)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_79),C_33)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_67),D_5)))))) # label(fact_2201_add__less__le__mono) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1997 (all X_1 all Y_2 (quickcheck_int_of(Y_2) = quickcheck_int_of(X_1) <-> Y_2 = X_1)) # label(fact_4254_code__int_Oint__of__inject) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1998 (all X_1 all N_1 all Y_2 ((exists Q1 exists Q2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),Q2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),Q1))) <-> hAPP_nat_nat(div_mod_nat(X_1),N_1) = hAPP_nat_nat(div_mod_nat(Y_2),N_1))) # label(fact_2787_nat__mod__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 1999 (all B_55 all A_67 A_67 = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_55),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_67),B_55))),hAPP_C498520661umeral(div_mo1740067990umeral(A_67),B_55))) # label(fact_2344_mod__div__equality2) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2000 (all X_34 all P_7 all Q_10 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_34),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_10)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_34),P_7)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_34),Q_10))) # label(fact_673_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2001 (all X hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,X)),hAPP_real_real(cos,X)) = hAPP_real_real(tan,X)) # label(fact_3673_tan__def) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2002 (all A_89 (is_int(A_89) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),A_89) = A_89)) # label(fact_2119_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2003 (all A_3 ((hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),zero_zero_real)) -> hAPP_real_real(abs_abs_real,A_3) = hAPP_real_real(uminus_uminus_real,A_3)) & (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),zero_zero_real)) -> hAPP_real_real(abs_abs_real,A_3) = A_3))) # label(fact_3494_abs__real__def) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2004 (all B_96 all A_118 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_118)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_96)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_118),B_96)))))) # label(fact_1868_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2005 (all Ma all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) -> -hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(bnorRset(A),Ma))))) # label(fact_4292_Bnor__mem__zle__swap) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2006 (all A_3 all B_2 cis(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_3),B_2)) = hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,cis(A_3)),cis(B_2))) # label(fact_4151_cis__divide) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2007 (all C_79 all D_15 all A_179 all B_139 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_179),B_139)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_79),D_15)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_139)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_79)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_179),C_79)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_139),D_15)))))))) # label(fact_1057_mult__mono) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2008 (all Q_2 all R_3 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Q_2),hAPP_rat_rat(uminus_uminus_rat,R_3)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Q_2),R_3)) # label(fact_4656_diff__rat__def) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2009 (all Y all X (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) -> Y = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y))) # label(fact_4322_gcd__proj2__if__dvd__nat) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2010 (all C_88 all A_206 all B_157 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_206),B_157)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,B_157),C_88)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_206),C_88))))) # label(fact_804_dvd__trans) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2011 (all N hAPP_nat_nat(suc,N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N)) # label(fact_3071_Suc__eq__plus1__left) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2012 (all S (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S)) -> (exists K hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,S),hAPP_n1699378549t_bool(ord_lessThan_nat,K)))))) # label(fact_4955_finite__nat__bounded) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2013 (all M all N hAPP_int_int(legacy_zgcd(N),hAPP_int_int(div_mod_int(M),N)) = hAPP_int_int(legacy_zgcd(M),N)) # label(fact_4487_zgcd__eq) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2014 (all A_1 (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A_1)) <-> hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_nat_int(semiri1621563631at_int,A_1))))) # label(fact_4606_transfer__nat__int__set__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2015 (all X_39 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_39),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_39),X_39)) # label(fact_434_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2016 (all A (one_one_nat = A <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),one_one_nat)))) # label(fact_2315_gcd__lcm__complete__lattice__nat_Obot__unique) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2017 (all X_1 all N_1 all Diff all F (hAPP_n546711566l_real(Diff,zero_zero_nat) = F -> ((all M_1 all T (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_1))) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))))) -> (exists T (hAPP_real_real(F,X_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N_1),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_1)))))))) # label(fact_4933_Maclaurin__bi__le) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2018 (all K_2 all L_2 ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2)) -> hAPP_int_int(div_mod_int(K_2),hAPP_int_int(uminus_uminus_int,L_2)) = hAPP_int_int(z3mod(K_2),L_2)) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2)) -> hAPP_int_int(div_mod_int(K_2),L_2) = hAPP_int_int(z3mod(K_2),L_2)))) # label(fact_3949_z3mod__def) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2019 (all X_3 (hBOOL(vanishes(X_3)) -> hBOOL(vanishes(hAPP_f2062659861at_rat(hAPP_f1082758869at_rat(cOMBB_rat_rat_nat,uminus_uminus_rat),X_3))))) # label(fact_5133_vanishes__minus) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2020 (all C_42 all A_100 all B_82 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_100),B_82)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_100),C_42)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_82),C_42))))) # label(fact_2018_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2021 (all F all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) -> ((all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),B)) -> (exists Y_1 (hBOOL(hAPP_real_bool(deriv_real(F,X_2),Y_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),zero_zero_real)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(F,B)),hAPP_real_real(F,A)))))) # label(fact_4549_DERIV__nonpos__imp__nonincreasing) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2022 (all V_14 all B_166 all C_97 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_i1732201573de_int(number1226105091de_int,V_14)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_166),C_97)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_i1732201573de_int(number1226105091de_int,V_14)),B_166)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_i1732201573de_int(number1226105091de_int,V_14)),C_97))) # label(fact_195_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2023 (all X all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),X)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_int_int(number_number_of_int,min)),P_3))))))) # label(fact_2678_Euler__part1) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2024 (all P all Q all R hAPP_i1524277240nt_int(hAPP_f883232462nt_int(hAPP_f2002277285nt_int(cOMBB_1846624359nt_int,P),Q),R) = hAPP_i1524277240nt_int(P,hAPP_int_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2025 (all M all D_3 all R_3 (hAPP_nat_nat(div_mod_nat(M),D_3) = R_3 -> (exists Q_4 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,R_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Q_4),D_3)) = M))) # label(fact_2786_mod__eqD) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2026 (all X all N_5 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N_5)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),hAPP_real_real(hAPP_n546711566l_real(root,N_5),X)))))))) # label(fact_3938_real__root__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2027 (all X_1 all Y_2 (is_int(X_1) & is_int(Y_2) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_int <-> zero_zero_int = X_1 & zero_zero_int = Y_2))) # label(fact_13_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2028 (all M all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),M))))) # label(fact_1643_trans__le__add1) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2029 (all N_1 hAPP_n1699378549t_bool(ord_atMost_nat,N_1) = hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),N_1)) # label(fact_5186_atLeast0AtMost) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2030 (all Ta all D all D_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),D_1)) -> (all X_2 all K (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_2),Ta))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_2),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,K),D_1))),Ta))))))) # label(fact_1530_inf__period_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2031 (all A_218 all B_161 all C_90 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_218),C_90)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_161),C_90)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_218),B_161)),C_90)) # label(fact_628_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2032 (all Z_2 all W_1 (Z_2 != W_1 & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Z_2),W_1)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z_2),W_1)))) # label(fact_4657_less__rat__def) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2033 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y)) -> (exists X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2)) & Y = hAPP_real_real(tan,X_2) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) # label(fact_3626_tan__total__pos) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2034 (all B_121 all A_160 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_160),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_121)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_160),B_121)),zero_zero_real))))) # label(fact_1154_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2035 (all A_108 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_108),zero_zero_real) = A_108) # label(fact_1964_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2036 (all A_217 all M_29 all B_160 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_217),B_160)),M_29) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_217),M_29)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_160),M_29))) # label(fact_641_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2037 (all D_3 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(abs_abs_int,D_3)),D_3))) # label(fact_3419_zdvd__self__abs2) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2038 (all A_66 all B_54 A_66 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_66),B_54)),B_54)),hAPP_Q1762011733de_int(div_mo231679042de_int(A_66),B_54))) # label(fact_2349_mod__div__equality) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2039 (all X_1 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),X_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Y_2))),zero_zero_real)) <-> zero_zero_real = X_1 & Y_2 = zero_zero_real)) # label(fact_92_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2040 (all K_1 all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K_1)),hAPP_int_int(bit0,L))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)))) # label(fact_136_rel__simps_I33_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2041 (all C_41 all A_99 all B_81 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_41),A_99)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_41),B_81))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_99),B_81)))) # label(fact_2036_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2042 (all A_194 all B_150 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_194),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_150),A_194)))) # label(fact_914_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2043 (all Y_2 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2))))) # label(fact_3919_real__root__gt__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2044 (all P (-hBOOL(P) | -hBOOL(hAPP_bool_bool(fNot,P)))) # label(help_fNot_1_1_U) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2045 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),hAPP_int_int(bit1,K_1))))) # label(fact_134_rel__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2046 (all B_90 all A_112 all C_50 (is_int(B_90) & is_int(C_50) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_50),A_112) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_90),A_112) -> C_50 = B_90))) # label(fact_1909_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2047 (all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(suc,N_1))) <-> (exists J_1 (Ma = hAPP_nat_nat(suc,J_1) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),N_1)))) | Ma = zero_zero_nat)) # label(fact_3047_less__Suc__eq__0__disj) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2048 (all C_39 all D_7 all A_97 all B_79 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_97),B_79)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_39),D_7)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_97),C_39)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_79),D_7)))))) # label(fact_2051_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 7.45/7.46 2049 (all Z_1 all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_1),N)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),N))))) # label(fact_460_zdvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2050 (all X all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X)),A_3)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_int_real(real_int,A_3))))) # label(fact_4424_ceiling__le__real) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2051 (all A_189 all B_145 all C_82 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_189),B_145)),C_82)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_189),C_82)))) # label(fact_950_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2052 (all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,N_1))))) # label(fact_2989_zless__int) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2053 (all P all Q all R hAPP_r224952523l_real(hAPP_f1663701695l_real(hAPP_f774948957l_real(cOMBB_3698607l_real,P),Q),R) = hAPP_f1908301369l_real(P,hAPP_r1230691443l_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__prod_Itc__Real) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2054 (all Ma all N_1 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(semiri1424489471de_int,Ma)),hAPP_n522471361de_int(semiri1424489471de_int,N_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)))) # label(fact_2948_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2055 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K_1)),min)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),min)))) # label(fact_775_rel__simps_I13_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2056 (all X_1 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,ratreal(X_1)),ratreal(Y_2))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_1),Y_2)))) # label(fact_4693_real__less__code) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2057 (all M all N hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M),N))),N) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M)),N)) # label(fact_2879_mod__Suc__eq__Suc__mod) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2058 (all A hBOOL(hAPP_complex_bool(isCont156215680omplex(cnj),A))) # label(fact_5165_cnj_OisCont) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2059 (all A_109 all B_87 all C_47 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_109),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_87),C_47)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_109),B_87)),C_47)) # label(fact_1945_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2060 (all P_1 all X_1 all Y_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2)) -> hBOOL(P_1)))) # label(fact_1723_dvd_Oless__imp__triv) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2061 (all A all C all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B),C))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)))) # label(fact_2073_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2062 (all C_68 all A_162 all B_123 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_162),B_123)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_68)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_162),C_68)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_123),C_68)))))) # label(fact_1145_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2063 (all X all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),X)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)))),one_one_int),P_3))))) # label(fact_2753_Little__Fermat) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2064 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,X_1)),one_one_real)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_f1122023986l_real(hAPP_f1786887945l_real(cOMBB_1896127834l_real,suminf_real),hAPP_f1993765077t_real(hAPP_f895854793t_real(cOMBB_1948233853l_real,hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))))),hAPP_f1993765077t_real(hAPP_f895854793t_real(cOMBB_1948233853l_real,hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat)))))),hAPP_f1501741374t_real(hAPP_f1135648319t_real(cOMBC_1329014627t_real,hAPP_f1163894827t_real(hAPP_f1998755145t_real(cOMBB_421464275l_real,cOMBB_nat_real_nat),power_power_real)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))),X_1),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))))) # label(fact_4818_DERIV__arctan__series) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2065 (all A all B all C all X_1 (is_int(A) -> (hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,C),hAPP_int_nat(nat,X_1)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),B) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_int_nat(nat,X_1))) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd)) -> (hAPP_int_int(legacy_zgcd(A),B) = one_one_int -> (exists K (A = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K),hAPP_int_nat(nat,X_1)) & is_int(K))))))))) # label(fact_4529_int__relprime__odd__power__divisors) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2066 (all A_58 all B_49 all C_28 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_58),B_49)),C_28) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(div_mod_nat(A_58),C_28)),hAPP_nat_nat(div_mod_nat(B_49),C_28))),C_28)) # label(fact_2386_mod__mult__eq) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2067 (all A_200 all B_153 all C_86 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_200),C_86)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_153),C_86)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_200),B_153)),C_86)) # label(fact_875_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2068 (all P all Q all R hAPP_real_real(hAPP_n546711566l_real(P,R),hAPP_nat_real(Q,R)) = hAPP_nat_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__RealDef__Orea) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2069 (all A_221 zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_221),zero_zero_nat)) # label(fact_591_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2070 (all N all M all R_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),R_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),R_3)) -> (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N)),R_3)))))) # label(fact_5007_divides__mult__int) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2071 (all C all A all B (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A),B)) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),A)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),B))))) # label(fact_2026_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2072 (all N_54 all A_151 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_151)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_151),N_54))))) # label(fact_1242_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2073 (all K_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),K_1))))) # label(fact_1514_dvd__reduce) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2074 (all X hAPP_complex_complex(sgn_sgn_complex,X) = hAPP_complex_complex(scaleR1652505878omplex(hAPP_real_real(inverse_inverse_real,hAPP_complex_real(norm_norm_complex,X))),X)) # label(fact_5114_complex__sgn__def) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2075 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_3),B_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_2),A_3)) -> hAPP_int_int(abs_abs_int,A_3) = hAPP_int_int(abs_abs_int,B_2)))) # label(fact_3418_zdvd__antisym__abs) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2076 (all A_75 all A_74 all B_62 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_75),B_62)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_74),B_62)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_75),A_74)),B_62)) # label(fact_2295_mult_Oadd__left) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2077 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),zero_zero_int))))) # label(fact_2599_div__nonpos__pos__le0) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2078 (all X all Y all Z_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),Z_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Z_1))) # label(fact_4310_gcd__semilattice__nat_Oinf__left__commute) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2079 (all C_44 all D_8 all A_102 all B_84 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_102),B_84)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_44),D_8)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_102),C_44)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_84),D_8)))))) # label(fact_2007_add__mono) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2080 (all X_1 hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),sin)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X_1))),hAPP_real_real(sin,X_1))))) # label(fact_4794_DERIV__sin__realpow2a) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2081 (all Z_2 all X_1 all X_3 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),X_3)) -> ((all X_2 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),X_3)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Z_2)))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),comple124823625p_real(X_3)))))) # label(fact_4640_SupInf_OSup__upper) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2082 (all N one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),N)) # label(fact_4351_coprime__plus__one__nat) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2083 (all N_52 all A_150 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_150)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_150),N_52))))) # label(fact_1251_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2084 (all W all V_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_2))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_2)))) -> negDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(number_number_of_int,V_2)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_2)))) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_2)))) -> negDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(number_number_of_int,V_2)) = hAPP_P1975530577nt_int(adjust(hAPP_int_int(number_number_of_int,V_2)),negDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(number_number_of_int,V_2))))))) # label(fact_2668_negDivAlg__eqn__number__of) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2085 (all C_1 all D_3 all A_3 all B_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C_1),D_3),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),C_1)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_2),D_3)),M))))) # label(fact_1525_zcong__zdiff) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2086 (all P all Q all R hAPP_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,P),Q),R) = hAPP_real_real(P,hAPP_nat_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__Nat__Ona) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2087 (all X_44 all Y_32 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_44),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_32),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),Y_32)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,X_44),Y_32))))) # label(fact_220_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2088 (all M all K_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2)),N)) -> -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N))))) # label(fact_1649_add__leE) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2089 (all A_228 all C_94 all D_19 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_228),C_94)),D_19) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_228),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_94),D_19))) # label(fact_537_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2090 (all Ta all A_1 all D all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1)) -> (all X_2 (is_int(X_2) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1)) -> X_2 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa))))))) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Ta))) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1)),Ta))))))))) # label(fact_5102_aset_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2091 (all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A),B)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,B),A)))) # label(fact_4016_coprime__commute) # label(axiom) # label(non_clause). [assumption]. 7.45/7.47 2092 (all A_3 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (exists R_1 (A_3 = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_1),N) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1))))))) # label(fact_1828_realpow__pos__nth) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2093 (all Z_2 all W_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat,W_1)),hAPP_int_nat(nat,Z_2)))))) # label(fact_2739_nat__less__eq__zless) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2094 (all D all Sa exists Z all X_2 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,Z),X_2)) -> (-hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_2),Sa))) <-> -hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_2),Sa)))))) # label(fact_2253_pinf_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2095 (all Y all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)))))) # label(fact_1790_real__mult__order) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2096 (all X_1 (is_int(X_1) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd)) -> -(all K (is_int(K) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K)),one_one_int) != X_1))))) # label(fact_2671_zOddE) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2097 (all C all B all A (is_int(A) & is_int(B) & is_int(C) -> (zero_zero_int != A -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B)) -> (C = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B),A) <-> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A) = B))))) # label(fact_2642_dvd__div__eq__mult) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2098 (all B_2 all A_3 all C_1 hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B_2),C_1)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),C_1))) # label(fact_4978_gcd__int_Oleft__commute) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2099 (all A_94 zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_94),A_94)) # label(fact_2085_diff__self) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2100 (all C_68 all A_162 all B_123 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_162),B_123)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_68)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_162),C_68)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_123),C_68)))))) # label(fact_1147_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2101 (all X_8 all Q_7 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_8),hAPP_nat_nat(suc,Q_7)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_8),Q_7)),X_8)) # label(fact_3022_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2102 (all B_36 all A_45 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_36),A_45)),B_36) = hAPP_int_int(div_mod_int(A_45),B_36)) # label(fact_2429_mod__add__self1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2103 (all X_4 all Y_4 hAPP_real_real(cos,hAPP_real_real(arccos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_4),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_4),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_4),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3595_cos__arccos__lemma1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2104 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) -> -hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(d22set,A))))) # label(fact_2265_d22set__le__swap) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2105 (all C (zero_zero_rat = C <-> hBOOL(vanishes(cOMBK_rat_nat(C))))) # label(fact_5132_vanishes__const) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2106 (all A_3 (hBOOL(hAPP_nat_bool(prime,A_3)) -> hBOOL(hAPP_int_bool(zprime,hAPP_nat_int(semiri1621563631at_int,A_3))))) # label(fact_4180_prime__impl__zprime__int) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2107 (all A all C all B (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B),C))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),B)))) # label(fact_2033_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2108 (all A_10 all N_15 all B_8 (is_int(A_10) & is_int(B_8) -> (hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_10),hAPP_nat_nat(suc,N_15)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_8),hAPP_nat_nat(suc,N_15)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_10)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_8)) -> A_10 = B_8))))) # label(fact_3108_power__inject__base) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2109 (all F all F_1 all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) -> ((all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),B)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X_2)) -> hBOOL(hAPP_real_bool(deriv_real(F,X_2),hAPP_real_real(F_1,X_2))))) -> (exists Z (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),Z)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z),B)) & hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B)),hAPP_real_real(F,A)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A)),hAPP_real_real(F_1,Z))))))) # label(fact_3945_MVT2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2110 (all A_90 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,one_on1684967323de_int),A_90) = A_90) # label(fact_2117_mult__1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2111 (all B_2 all A_3 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_2),A_3))),hAPP_complex_real(norm_norm_complex,B_2))),hAPP_complex_real(norm_norm_complex,A_3)))) # label(fact_3369_complex__mod__triangle__ineq2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2112 (all N_53 all M_21 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),M_21)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),N_53)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,M_21),N_53)))))) # label(fact_1250_less__1__mult) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2113 (all D all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D),B)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D),A)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),B))))) # label(fact_4012_coprime__mul__eq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2114 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real)) -> hBOOL(hAPP_real_bool(deriv_real(arccos,X_1),hAPP_real_real(inverse_inverse_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))))))) # label(fact_3861_DERIV__arccos) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2115 (all X_1 all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(sRStar,P_5))) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(standardRes(P_5),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),hAPP_int_int(multInv(P_5),X_1)))),hAPP_i1948725293t_bool(sRStar,P_5)))))))) # label(fact_2672_SRStar__mult__prop2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2116 (all K_1 all I_1 all J hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K_1)) = image_nat_nat(hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),K_1),hAPP_n1699378549t_bool(ord_at238088361st_nat(I_1),J))) # label(fact_5073_image__add__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2117 (all A_14 all B_10 all Q_9 all Y (is_int(Y) & is_int(B_10) -> (A_14 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_10),Q_9)),Y) -> ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_10)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_10),Y)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),zero_zero_int))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_10)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),B_10)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))) -> (zero_zero_int != B_10 -> Y = hAPP_int_int(div_mod_int(A_14),B_10)))))) # label(fact_2655_divmod__int__rel__mod__eq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2118 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(cos,X)))))) # label(fact_3588_cos__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2119 (all R_3 all A_3 hAPP_real_real(abs_abs_real,R_3) = hAPP_complex_real(norm_norm_complex,rcis(R_3,A_3))) # label(fact_4259_complex__mod__rcis) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2120 (all C_60 all D_13 all A_138 all B_107 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_138),B_107)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_60),D_13)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_138)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_60)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_138),C_60)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_107),D_13)))))))) # label(fact_1357_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2121 (all N all V_2 ((-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_2))),N) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),N))) & (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N)))) # label(fact_3286_Suc__nat__number__of__add) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2122 (all M all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),M))))) # label(fact_4161_prime__dvd__power__two) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2123 (all I all K_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2)),I) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),I)),K_2))) # label(fact_1768_diff__add__assoc2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2124 (all N all J_2 all K_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_2),K_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),N)),K_2)))) # label(fact_1742_less__imp__diff__less) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2125 (all X all D_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),D_3)) -> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,X)),hAPP_nat_real(real_nat,D_3)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),D_3))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,hAPP_nat_nat(div_mod_nat(X),D_3))),hAPP_nat_real(real_nat,D_3))))) # label(fact_3714_real__of__nat__div__aux) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2126 (all B_2 all A_3 (is_int(A_3) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),A_3)) -> A_3 = hAPP_int_int(div_mod_int(A_3),B_2))))) # label(fact_2610_mod__neg__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2127 (all A_108 A_108 = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_108),zero_z126310315umeral)) # label(fact_1961_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2128 (all A_230 all C_96 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_96),A_230) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_230),C_96)) # label(fact_523_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2129 (all W_1 all Y_2 all X_1 all Z_2 (is_int(W_1) & is_int(Y_2) & is_int(X_1) & is_int(Z_2) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W_1),Z_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_2)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W_1),Y_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Z_2)) <-> W_1 = X_1 | Z_2 = Y_2))) # label(fact_642_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2130 (all I (is_int(I) -> (I != zero_zero_int -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),I)) -> hAPP_int_int(sgn_sgn_int,I) = one_one_int) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),I)) -> hAPP_int_int(sgn_sgn_int,I) = hAPP_int_int(uminus_uminus_int,one_one_int))) & (I = zero_zero_int -> hAPP_int_int(sgn_sgn_int,I) = zero_zero_int))) # label(fact_3602_zsgn__def) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2131 (all X_41 all Y_29 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_41),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_29),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_41)),Y_29)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_41),Y_29)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_370_power2__sum) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2132 (all N_1 (is_int(N_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1)) -> N_1 = hAPP_nat_int(semiri1621563631at_int,hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),N_1)))))))) # label(fact_4735_int__card__bdd__int__set__l__le) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2133 (all K_2 all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N)))) # label(fact_4316_dvd__gcd__D2__nat) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2134 (all I all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),N)) -> I = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),I)))) # label(fact_1751_diff__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2135 (all Y_2 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat,X_1)),hAPP_int_nat(nat,Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2)))))) # label(fact_2738_transfer__nat__int__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2136 (all N_1 all F (hBOOL(summable_real(F)) -> ((all M_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(F,M_1))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_f352196356l_real(suminf_real,F)))))) # label(fact_4929_series__pos__less) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2137 (all Y_2 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd)) -> (exists N_2 (is_int(N_2) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Y_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_2),X_1))))),X_1))))))) # label(fact_3475_best__odd__division__abs) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2138 (all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))) -> one_one_int != hAPP_int_int(inv(P_3),A_3))))) # label(fact_2129_inv__not__1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2139 (all Y_2 all X_1 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zOdd)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_2)),zOdd))))) # label(fact_2488_EvenOdd_Oodd__times__odd) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2140 (all X_46 all Y_34 (hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_46),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_34),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_46)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_34)) -> Y_34 = X_46)))) # label(fact_207_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2141 (all X hAPP_real_real(abs_abs_real,X) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),X))) # label(fact_3383_real__sqrt__abs2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2142 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))))) # label(fact_2280_dvd__pos__nat) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2143 (all B all A (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),zEven)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),zOdd)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B)),zOdd))))) # label(fact_3092_IntNatAux_Oeven__plus__odd) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2144 (all A (zero_zero_real != A <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_20_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2145 (all Z_13 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Z_13),Z_13) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_13)) # label(fact_289_mult__2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2146 (all B_130 all A_170 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_170)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_130),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_130),A_170)),zero_zero_nat))))) # label(fact_1099_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2147 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(exp_real,X_1))))) # label(fact_3840_one__le__exp__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2148 (all A all C all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B),C))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B)))) # label(fact_2029_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2149 (all N hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N)))))) # label(fact_3547_neg__zminus__int) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2150 (all B all A all C (B = C <-> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),A) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B),A))) # label(fact_1935_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2151 (all Ta all D all D_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),D_1)) -> (all X_2 all K (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_2),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,K),D_1))),Ta))) <-> -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_2),Ta))))))) # label(fact_1534_inf__period_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2152 (all M all N all K_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N)))) # label(fact_2720_nat__mult__div__cancel1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2153 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),hAPP_real_real(sin,X)))) # label(fact_3660_sin__ge__minus__one) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2154 (all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,N)))) # label(fact_2817_lessI) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2155 (all P_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int) = hAPP_nat_int(semiri1621563631at_int,hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(sRStar,P_5))))) # label(fact_3287_SRStar__card) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2156 (all W_8 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_i769753017umeral(number1443263063umeral,W_8)),hAPP_i769753017umeral(number1443263063umeral,W_8)) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_i769753017umeral(number1443263063umeral,W_8)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_359_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2157 (all V all W_1 (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,W_1)),hAPP_int_real(number267125858f_real,V))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,V)),hAPP_int_real(number267125858f_real,W_1))))) # label(fact_107_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2158 (all X_33 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_33),zero_zero_nat) = one_one_real) # label(fact_677_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2159 (all X_1 (one_one_rat = X_1 <-> one_one_rat = X_1)) # label(fact_1956_one__reorient) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2160 (all N all M_4 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),M_4)) -> (exists M_1 hAPP_nat_nat(suc,M_1) = M_4))) # label(fact_3290_Suc__le__D) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2161 (all A all X_3 all X_1 (hBOOL(tendsto_nat_real(X_3,X_1,sequentially)) -> ((exists N_3 all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),hAPP_nat_real(X_3,N_2))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X_1))))) # label(fact_5036_LIMSEQ__le__const) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2162 (all X_1 all Y_2 (hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X_1),Y_2))) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,Y_2)) & hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)) | -hBOOL(hAPP_nat_bool(even_odd_even_nat,Y_2)) & -hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2)))) # label(fact_3764_even__difference__nat) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2163 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real)) -> hBOOL(tendsto_nat_real(hAPP_r474017924t_real(power_power_real,X_1),zero_zero_real,sequentially))))) # label(fact_5033_LIMSEQ__realpow__zero) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2164 (all X all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),P_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,X),P_3)))))) # label(fact_4187_prime__coprime__lt) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2165 (all B_128 all A_168 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_168)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_128)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_168),B_128)))))) # label(fact_1110_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2166 (all A_1 hAPP_f22106695ol_nat(finite_card_nat,A_1) = hAPP_f957591787ol_nat(finite_card_int,image_nat_int(semiri1621563631at_int,A_1))) # label(fact_4605_Nat__Transfer_Otransfer__nat__int__set__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2167 (all A_17 all C_4 all B_13 (zero_zero_nat != B_13 -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_13),C_4))),B_13) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_4),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_17),B_13)))) # label(fact_2625_div__mult__self2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2168 (all B_90 all A_112 all C_50 (hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_90),A_112) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_50),A_112) -> C_50 = B_90)) # label(fact_1908_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2169 (all Ma all N_1 ((hBOOL(hAPP_nat_bool(even_odd_even_nat,Ma)) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N_1))))) # label(fact_3756_even__add) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2170 (all B_2 all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3)) & -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(B_2),zero_zero_int),P_3)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)),zero_zero_int),P_3)))))) # label(fact_755_zcong__zprime__prod__zero__contra) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2171 (all A_106 (is_int(A_106) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),A_106) = A_106)) # label(fact_1979_add__0) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2172 (all X hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(sin,X))) = one_one_real) # label(fact_3680_sin__cos__squared__add3) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2173 (all P all Q all R hAPP_nat_real(hAPP_f292751352t_real(hAPP_f1607377819t_real(cOMBB_1533963633al_nat,P),Q),R) = hAPP_f1406902706l_real(P,hAPP_n1699378549t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__RealDef_) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2174 (all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_n1108039445umeral(semiri1619134803umeral,N_1))))) # label(fact_3144_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2175 (all C_79 all D_15 all A_179 all B_139 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_179),B_139)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_79),D_15)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_139)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_79)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_179),C_79)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_139),D_15)))))))) # label(fact_1056_mult__mono) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2176 (all P all Q all R hAPP_i157317923t_real(hAPP_f1613846310t_real(hAPP_f1720068861t_real(cOMBC_295337211t_real,P),Q),R) = hAPP_f1888375463t_real(hAPP_i447943416t_real(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Or) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2177 (all C_60 all D_13 all A_138 all B_107 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_138),B_107)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_60),D_13)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_138)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_60)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_138),C_60)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_107),D_13)))))))) # label(fact_1354_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2178 (all Ma all K_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),K_1))))) # label(fact_1670_mult__less__cancel2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2179 (all X_1 all A_1 (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))))) # label(fact_4613_Nat__Transfer_Otransfer__nat__int__set__function__closures_I6_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2180 (all X_1 all Y_2 (is_int(X_1) & is_int(Y_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_int)) <-> zero_zero_int = X_1 & zero_zero_int = Y_2))) # label(fact_223_sum__power2__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2181 (all A_3 all C_1 all B_2 all D_3 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),C_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_2),D_3)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,C_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,D_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) # label(fact_3344_real__sqrt__sum__squares__triangle__ineq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2182 (all X all Y (is_int(X) -> X = hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,fTrue),X),Y))) # label(help_If_1_1_If_000tc__Int__Oint_T) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2183 (all A_235 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_235),A_235)),A_235) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_235),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) # label(fact_275_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2184 (all A all B all C (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B)))))) # label(fact_1382_mult__le__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2185 (all C_60 all D_13 all A_138 all B_107 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_138),B_107)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_60),D_13)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_138)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_60)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_138),C_60)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_107),D_13)))))))) # label(fact_1358_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2186 (all A_228 all C_94 all D_19 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_228),C_94)),D_19) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_228),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_94),D_19))) # label(fact_534_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2187 (all A_224 all N_68 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_224),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_224),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68))) # label(fact_578_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2188 (all A all E_2 all C all B all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B)),E_2)),C)),D)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),E_2)),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),E_2)),D))))) # label(fact_1569_less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2189 (all K_1 (hBOOL(hAPP_int_bool(iszero_int,hAPP_int_int(number_number_of_int,K_1))) <-> hBOOL(hAPP_rat_bool(iszero_rat,hAPP_int_rat(number_number_of_rat,K_1))))) # label(fact_4636_iszero__rat) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2190 (all Y_2 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_2))))) # label(fact_3916_real__root__ge__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2191 (all L_2 all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_2),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_2),M))))) # label(fact_1749_diff__le__mono2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2192 (all N_29 all K_6 all M_16 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_6),M_16)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_6),N_29)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_6),hAPP_nat_nat(div_mod_nat(M_16),N_29)))))) # label(fact_2452_dvd__mod) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2193 (all Z_1 all W hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,W))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,W))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,W)))) # label(fact_4767_zpower__number__of__even) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2194 (all F all X_1 all L (hBOOL(hAPP_real_bool(deriv_real(F,X_1),L)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,L),zero_zero_real)) -> (exists D_2 ((all H_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H_1),D_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,X_1)),hAPP_real_real(F,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),H_1))))))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))))))) # label(fact_3888_DERIV__neg__dec__left) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2195 (all A_139 all C_61 all B_108 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_139),C_61)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_108),C_61))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_61)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_139),B_108))))) # label(fact_1349_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2196 (all Y_2 the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_eq_real),pi))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),cos)),Y_2)))) = hAPP_real_real(arccos,Y_2)) # label(fact_4886_arccos__def) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2197 (all G all F all X_1 all D (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D)) -> ((all Z (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D)) -> hAPP_real_real(G,hAPP_real_real(F,Z)) = Z)) -> ((all Z (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D)) -> hBOOL(hAPP_real_bool(isCont_real_real(F),Z)))) -> (exists Z (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,X_1)),hAPP_real_real(F,Z))))))))) # label(fact_5182_lemma__isCont__inj) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2198 (all M all K_2 all N (hAPP_int_int(legacy_zgcd(K_2),N) = one_one_int -> hAPP_int_int(legacy_zgcd(M),N) = hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),N))) # label(fact_4485_zgcd__zmult__cancel) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2199 (all C_88 all A_206 all B_157 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_206),B_157)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_157),C_88)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_206),C_88))))) # label(fact_807_dvd__trans) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2200 (all N_1 all X_1 (zero_zero_real != X_1 -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (exists T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(abs_abs_real,T))) & hAPP_real_real(exp_real,X_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(exp_real,T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_1)))))))) # label(fact_4925_Maclaurin__exp__lt) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2201 (all Z_2 all X_3 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Z_2),X_3)) -> ((all X_2 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),X_3)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Z_2)))) -> comple124823625p_real(X_3) = Z_2))) # label(fact_4647_Sup__eq__maximum) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2202 (all F (hBOOL(summable_real(F)) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(F,N_2)))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_f352196356l_real(suminf_real,F)))))) # label(fact_4859_suminf__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2203 (all A_229 all C_95 all D_20 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_229),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_95),D_20)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_95),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_229),D_20))) # label(fact_532_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2204 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M)) -> M != N)) # label(fact_1588_less__not__refl2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2205 (all A_211 all M_26 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_211),M_26)),M_26) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_211),one_one_nat)),M_26)) # label(fact_706_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2206 (all Z_13 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_13),Z_13) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_13)) # label(fact_286_mult__2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2207 (all X all N (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,N))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natfloor(X)),hAPP_int_nat(number_number_of_nat,N)) = natfloor(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_int_real(number267125858f_real,N)))))) # label(fact_3381_natfloor__add__number__of) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2208 (all D all Sa exists Z ((all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),X_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Sa))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Sa)))))) & is_int(Z))) # label(fact_2273_pinf_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2209 (all M all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),hAPP_nat_nat(suc,M)))) # label(fact_3066_diff__less__Suc) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2210 (all N_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_int_real(real_int,N_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1)))) # label(fact_4440_real__of__int__ge__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2211 (all M all K_2 all N (one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,K_2),N) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M))))) # label(fact_4369_coprime__dvd__mult__nat) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2212 (all B_1 all A_1 (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1)) -> (hBOOL(hAPP_f448129468l_bool(nat_nat_set,B_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_1),B_1)) <-> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,image_int_nat(nat,A_1)),image_int_nat(nat,B_1))))))) # label(fact_4621_transfer__int__nat__set__relations_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2213 (all X_1 (zero_z891286103de_int = X_1 <-> X_1 = zero_z891286103de_int)) # label(fact_1899_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2214 (all B_1_1 all B_2_1 is_bool(hAPP_P880235623l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__prod_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__fun_Itc__I) # label(axiom) # label(non_clause). [assumption]. 7.48/7.47 2215 (all X_1 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),Y_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),X_1)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(tan,Y_2)),hAPP_real_real(tan,X_1))))))))) # label(fact_3617_tan__monotone_H) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2216 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(uminus_uminus_real,Y))) # label(fact_3487_minus__real__def) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2217 (all N_55 all A_152 all B_114 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_152),B_114)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_152)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_152),N_55)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_114),N_55)))))) # label(fact_1233_power__mono) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2218 (all X all Y hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_complex_real(re,X)),hAPP_complex_real(re,Y))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_complex_real(im,X)),hAPP_complex_real(im,Y))) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),Y)) # label(fact_4134_complex__add__def) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2219 (all K_2 all A_3 all B_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),K_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),K_2)),M)))) # label(fact_1034_zcong__scalar) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2220 (all W_1 all Z_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_2),one_one_int))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z_2)))) # label(fact_158_zle__add1__eq__le) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2221 (all W_17 hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,pls)),W_17)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(number_number_of_int,W_17))) # label(fact_40_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2222 (all X all Y all N (N != zero_zero_nat -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N)),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))))) # label(fact_1816_divides__exp2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2223 (all C_63 all A_141 all B_110 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_63),A_141)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_63),B_110))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_63)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_141),B_110))))) # label(fact_1337_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2224 (all X_43 all Y_31 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_43),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_31),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_228_sum__power2__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2225 (all A all B all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(div_mod_int(A),Ma)),hAPP_int_int(div_mod_int(B),Ma)),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma)))) # label(fact_2466_zcong__zmod) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2226 (all K_2 all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M)))))) # label(fact_1755_dvd__diffD) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2227 (all X all Y all R_3 hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),R_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),R_3)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,X),Y)),of_real_complex(R_3))) # label(fact_3990_Complex__mult__complex__of__real) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2228 (all R_3 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,R_3),one_one_real)),hAPP_int_real(real_int,archim1246769320r_real(R_3))))) # label(fact_4455_real__of__int__floor__ge__diff__one) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2229 (all N_66 all A_210 all B_158 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_210),B_158)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_210)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_66)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_210),N_66)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_158),N_66))))))) # label(fact_748_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2230 (all B_97 all A_119 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_119)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_97)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_119),B_97)))))) # label(fact_1859_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2231 (all X_22 all Y_17 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_22),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_17),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_22)),Y_17)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_22),Y_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_1661_power2__diff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2232 (all A_146 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_146),zero_zero_nat) = one_one_complex) # label(fact_1300_power__0) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2233 (all Y all U_1 all X1_1 all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X1_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X1_1),Y)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),U_1))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),hAPP_real_real(inverse_inverse_real,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,U_1),hAPP_real_real(inverse_inverse_real,X1_1)))))))) # label(fact_3838_real__mult__inverse__cancel2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2234 (all A_145 all M_20 all N_48 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_145),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_20),N_48)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_145),M_20)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_145),N_48))) # label(fact_1309_power__add) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2235 (all A_3 all B_2 all C_1 (is_int(C_1) -> (zero_zero_int != C_1 -> hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),A_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),B_2)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)))) # label(fact_4686_mult__rat__cancel) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2236 (all Y_2 all X_1 (is_int(X_1) & is_int(Y_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2)) -> (X_1 = Y_2 <-> hAPP_int_nat(nat,X_1) = hAPP_int_nat(nat,Y_2)))))) # label(fact_2707_transfer__nat__int__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2237 (all I_1 (is_int(I_1) -> (I_1 != zero_zero_int -> hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,dvd_dvd_int),I_1))))))) # label(fact_4753_finite__divisors__int) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2238 (all Y all X (-hBOOL(hAPP_int_bool(even_odd_even_int,X)) -> (-hBOOL(hAPP_int_bool(even_odd_even_int,Y)) -> -hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)))))) # label(fact_2886_Parity_Oodd__times__odd) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2239 (all C_71 all B_126 all A_165 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_126),A_165)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_71),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_165),C_71)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_126),C_71)))))) # label(fact_1128_mult__strict__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2240 (all X_43 all Y_31 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_43),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_31),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_226_sum__power2__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2241 (all A_3 all B_2 hAPP_int_int(div_mod_int(A_3),B_2) = hAPP_P1175774780nt_int(product_snd_int_int,divmod_int(A_3,B_2))) # label(fact_4233_mod__int__def) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2242 (all X_38 all P_8 all Q_12 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_38),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_38),P_8)),Q_12)) # label(fact_446_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2243 (all P all Q all R hAPP_r1134773055l_bool(P,hAPP_int_real(Q,R)) = hAPP_i688218208l_bool(hAPP_f801032407l_bool(hAPP_f499530369l_bool(cOMBB_1317819104ol_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__H) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2244 (all K_2 (is_int(K_2) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),pls) = K_2)) # label(fact_263_add__Pls__right) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2245 (all M_10 all N_24 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,M_10)),hAPP_nat_rat(semiri151668891at_rat,N_24))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_24)))) # label(fact_2930_of__nat__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2246 (all A_105 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,zero_z891286103de_int),A_105) = A_105) # label(fact_1990_add__0__left) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2247 (all C_41 all A_99 all B_81 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_41),A_99)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_41),B_81))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_99),B_81)))) # label(fact_2037_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2248 (all X_1 all N_1 (one_one_nat = X_1 | N_1 = zero_zero_nat <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),N_1) = one_one_nat)) # label(fact_1821_exp__eq__1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2249 (all Z_1 all W hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),W) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z_1)) # label(fact_54_zmult__commute) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2250 (all X_17 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_17),zero_zero_complex) = zero_zero_complex) # label(fact_2290_mult__right_Ozero) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2251 (all N_1 ((exists M_1 hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M_1)) = N_1) <-> -hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1)))) # label(fact_3787_odd__Suc__mult__two__ex) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2252 (all A_18 all C_5 all B_14 (is_int(B_14) -> (zero_zero_int != B_14 -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_18),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_5),B_14))),B_14) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_5),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_18),B_14))))) # label(fact_2620_div__mult__self1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2253 (all A_91 A_91 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_91),one_one_rat)) # label(fact_2111_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2254 (all Q_2 all B_2 all R_3 all C_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(div_mod_int(Q_2),C_1))),R_3)),zero_zero_int)))))) # label(fact_2656_zmult2__lemma__aux2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2255 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sin,X)))))) # label(fact_3726_sin__gt__zero2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2256 (all P all Q all R hAPP_i1216375074nt_int(hAPP_P621635040nt_int(hAPP_f1582586579nt_int(cOMBC_707974837nt_int,P),Q),R) = hAPP_P1145851913nt_int(hAPP_i2104874254nt_int(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2257 (all Code_numeral_2 zero_z126310315umeral != code_S1047413653umeral(Code_numeral_2)) # label(fact_4082_code__numeral_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2258 (all Z_11 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_11),Z_11) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_11),hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_297_mult__2__right) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2259 (all B_69 all A_81 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_81)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_69)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_81),B_69)))))) # label(fact_2184_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2260 (all M hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),hAPP_nat_nat(suc,zero_zero_nat)) = M) # label(fact_3077_div__1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2261 (all N zero_zero_real = hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,pi),hAPP_nat_real(real_nat,N)))) # label(fact_3640_sin__npi2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2262 (all Ma all K_1 all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),N_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(div_mod_int(Ma),N_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),Ma))))) # label(fact_2443_dvd__mod__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2263 (all Q_8 all R_5 all A_3 all B_2 all Q_2 all R_3 (is_int(Q_8) & is_int(Q_2) & is_int(B_2) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3))) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_8),R_5))) -> (B_2 != zero_zero_int -> Q_8 = Q_2))))) # label(fact_2791_unique__quotient) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2264 (all A_3 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),hAPP_nat_nat(suc,zero_zero_nat)))) # label(fact_4023_coprime__Suc0) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2265 (all D all Sa exists Z all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z),X_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Sa))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Sa)))))) # label(fact_2270_pinf_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2266 (all X all N_5 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N_5)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N_5),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),X))))))) # label(fact_3927_real__root__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2267 (all D all Sa exists Z all X_2 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,Z),X_2)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_2),Sa))) <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,D),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_2),Sa)))))) # label(fact_2269_pinf_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2268 (all A_3 all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,A_3)),X)) -> natceiling(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_nat_real(real_nat,A_3))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,natceiling(X)),A_3))) # label(fact_3694_natceiling__subtract) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2269 (all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_int_real(number267125858f_real,Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,pls)),Y_2)))) # label(fact_122_le__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2270 (all D all Sa exists Z all X_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Z)) -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_2),Sa))) <-> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_2),Sa)))))) # label(fact_2252_minf_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2271 (all X_35 all Y_27 all Q_11 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_35),Y_27)),Q_11) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_35),Q_11)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_27),Q_11))) # label(fact_661_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2272 (all A_107 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_107),zero_zero_nat) = A_107) # label(fact_1970_add__0__right) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2273 (all N_1 (one_one_nat = N_1 <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,N_1),N_1)))) # label(fact_4013_coprime__refl) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2274 (all M_7 all N_21 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,M_7)),hAPP_nat_int(semiri1621563631at_int,N_21)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21))) # label(fact_2966_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2275 (all I quickcheck_nat_of(I) = hAPP_int_nat(nat,quickcheck_int_of(I))) # label(fact_4269_nat__of__def) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2276 (all B_2 all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_3)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,archim1246769320r_real(A_3)),archim1246769320r_real(B_2))),archim1246769320r_real(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),B_2))))))) # label(fact_4034_le__mult__floor) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2277 (all K_1 all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),N_1))))) # label(fact_3069_Suc__mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2278 (all M all R_3 all S_1 all T_1 all N (is_int(R_3) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N)) -> (xzgcd(M,N) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,R_3),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,S_1),T_1)) -> R_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_1),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_1),N)))))) # label(fact_4199_xzgcd__linear) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2279 (all A_196 all N_63 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_196),N_63)),A_196) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_196),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_196),N_63))) # label(fact_905_power__commutes) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2280 (all P_1 collec1979865426at_nat(P_1) = P_1) # label(fact_150_Collect__def) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2281 (all M_18 all N_38 all A_128 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_128)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_128),M_18)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_128),N_38))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_18),N_38))))) # label(fact_1445_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2282 (all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)))) # label(fact_1667_add__gr__0) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2283 (all P all Q all R hAPP_r2019696015l_real(hAPP_f1591894897l_real(hAPP_f2114902373l_real(cOMBB_237479429l_real,P),Q),R) = hAPP_r2019696015l_real(P,hAPP_real_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__p) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2284 (all C_33 all D_5 all A_79 all B_67 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_79),B_67)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_33),D_5)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_79),C_33)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_67),D_5)))))) # label(fact_2206_add__less__le__mono) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2285 (all Z_1 (is_int(Z_1) -> ((all N_2 Z_1 != hAPP_nat_int(semiri1621563631at_int,N_2)) -> -(all N_2 Z_1 != hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_2))))))) # label(fact_4547_int__cases) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2286 (all Z_2 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_2),hAPP_nat_int(semiri1621563631at_int,Ma))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,Z_2))),Ma)))) # label(fact_3450_dvd__int__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2287 (all X_1 all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,X_1)),hAPP_int_int(number_number_of_int,Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2)))) # label(fact_330_less__number__of) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2288 (all C_88 all A_206 all B_157 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_206),B_157)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_157),C_88)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_206),C_88))))) # label(fact_806_dvd__trans) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2289 (all A_233 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,pls)),A_233) = A_233) # label(fact_324_add__numeral__0) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2290 (all L_2 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,pls),hAPP_int_int(bit0,L_2)) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,pls),L_2))) # label(fact_1554_diff__bin__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2291 (all Z_1 Z_1 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),Z_1)) # label(fact_1782_real__mult__1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2292 (all M all N (hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N)) = one_one_int -> hAPP_int_int(abs_abs_int,M) = one_one_int)) # label(fact_3421_abs__zmult__eq__1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2293 (all P all Q all R hAPP_b1482221219l_real(P,hAPP_real_bool(Q,R)) = hAPP_r931665397l_real(hAPP_f1607917947l_real(hAPP_f1076489843l_real(cOMBB_415306361l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__prod_Itc__RealDef__Oreal_) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2294 (all Lx_6 all Ly_4 all Rx_6 all Ry_4 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_6),Ly_4)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_6),Ry_4)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_6),Rx_6)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ly_4),Ry_4))) # label(fact_466_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2295 (all D all Sa exists Z all X_2 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_2),Z)) -> (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_2),Sa))) <-> -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_2),Sa)))))) # label(fact_2250_minf_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2296 (all W_6 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),hAPP_int_complex(number528085621omplex,W_6))),hAPP_int_complex(number528085621omplex,W_6)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(bit1,W_6))) # label(fact_377_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2297 (all X all Y hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3346_real__sqrt__sum__squares__ge1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2298 (all Z_1 all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),Y)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),X))))) # label(fact_1716_dvd_Ole__less__trans) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2299 (all X_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real)) -> hBOOL(hAPP_real_bool(isCont_real_real(hAPP_n546711566l_real(root,N_1)),X_1))))) # label(fact_5170_isCont__root__neg) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2300 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),hAPP_nat_nat(fact_fact_nat,N)))))) # label(fact_3812_dvd__fact__nat) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2301 (all X_3 all A (hBOOL(hAPP_complex_bool(sums_complex(X_3),A)) -> hBOOL(hAPP_complex_bool(sums_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_3)),hAPP_complex_complex(cnj,A))))) # label(fact_4871_cnj_Osums) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2302 (all A_31 all C_12 all B_26 hAPP_int_int(div_mod_int(A_31),B_26) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_31),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_12),B_26))),B_26)) # label(fact_2504_mod__mult__self1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2303 (all A all B all C (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B)))))) # label(fact_1381_mult__le__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2304 (all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(sqrt,Y_2))))) # label(fact_3334_real__sqrt__gt__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2305 (all X all Y (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X),Y)) -> hAPP_int_int(abs_abs_int,X) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y))) # label(fact_4992_gcd__proj1__if__dvd__int) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2306 (all A all C all B (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A),C)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B),C))) <-> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A),B)) | zero_zero_complex = C)) # label(fact_1268_dvd__mult__cancel__right) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2307 (all X_6 all Q_5 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_6),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_6),Q_5)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_6),hAPP_nat_nat(suc,Q_5))) # label(fact_3038_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2308 (all N_1 (N_1 = zero_zero_nat <-> hAPP_nat_int(semiri1621563631at_int,N_1) = zero_zero_int)) # label(fact_2975_int__eq__0__conv) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2309 (all P all X (-hBOOL(hAPP_int_bool(P,X)) | hBOOL(hAPP_f448129468l_bool(fEx_int,P)))) # label(help_fEx_1_1_fEx_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2310 (all N all X (hBOOL(hAPP_int_bool(nat_is_nat,X)) -> hBOOL(hAPP_int_bool(nat_is_nat,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N))))) # label(fact_5135_Nat__Transfer_Otransfer__int__nat__function__closures_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2311 (all W_1 (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,W_1))) <-> hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W_1)))))) # label(fact_3274_neg__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2312 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N) = hAPP_nat_nat(div_mod_nat(M),N))) # label(fact_2716_le__mod__geq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2313 (all N all M hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M),hAPP_int_int(div_mod_int(M),N)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,M),N))) # label(fact_2363_zmult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2314 (all C_44 all D_8 all A_102 all B_84 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_102),B_84)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_44),D_8)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_102),C_44)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_84),D_8)))))) # label(fact_2008_add__mono) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2315 (all M_12 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(semiri132038758t_real,M_12)))) # label(fact_2907_zero__le__imp__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2316 (all B_74 all A_86 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_86),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_74),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_86),B_74)),zero_zero_int))))) # label(fact_2142_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2317 (all W_1 all Z_2 (is_int(Z_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2)) <-> (exists N_2 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_1),hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_2))) = Z_2)))) # label(fact_2846_zless__iff__Suc__zadd) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2318 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,X)),hAPP_real_real(sqrt,Y))))) # label(fact_3316_real__sqrt__less__mono) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2319 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),B)),zero_zero_int)))) # label(fact_1116_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2320 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(ln,X))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X))))) # label(fact_3452_ln__ge__zero__imp__ge__one) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2321 (all X_9 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_9)) -> hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_int_nat(nat,X_9))),hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_int_nat(nat,X_9))) = hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_9))))) # label(fact_2808_of__nat__double) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2322 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(fact_fact_nat,M)),hAPP_nat_nat(fact_fact_nat,N)))))) # label(fact_3807_fact__less__mono__nat) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2323 (all A_231 all B_165 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_231),B_165) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_165),A_231)) # label(fact_515_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2324 (all N_41 ((N_41 != zero_zero_nat -> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),N_41) = zero_zero_rat) & (zero_zero_nat = N_41 -> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),N_41) = one_one_rat))) # label(fact_1438_power__0__left) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2325 (all N all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (X != zero_zero_real -> hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_nat_real(real_nat,N)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N)) & (zero_zero_real = X -> zero_zero_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N))))) # label(fact_4475_powr__realpow2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2326 (all K_1 all Xa_1 hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,cos),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,plus_plus_real),K_1)),Xa_1),hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa_1),K_1)))))) # label(fact_4740_DERIV__cos__add) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2327 (all A_148 all N_51 all N_50 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_51),N_50)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),A_148)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_148),N_51)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_148),N_50)))))) # label(fact_1273_power__increasing) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2328 (all X all Y all L_2 all M hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(uminus_uminus_real,L_2)),hAPP_real_real(uminus_uminus_real,M))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(uminus_uminus_real,L_2)))),hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Y),hAPP_real_real(uminus_uminus_real,M))))))) # label(fact_3496_abs__sum__triangle__ineq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2329 (all N_1 all K_1 all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),Ma)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),K_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),K_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)))))) # label(fact_1754_le__diff__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2330 (all X all Y hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,fFalse),X),Y) = Y) # label(help_If_2_1_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_T) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2331 (all W_12 hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,W_12)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),one_one_complex)),hAPP_int_complex(number528085621omplex,W_12))) # label(fact_272_double__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2332 (all X_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (one_one_int = hAPP_int_int(legacy_zgcd(X_1),Ma) -> hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(norRRset,Ma)) = hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(noXRRset(Ma),X_1))))) # label(fact_4587_card__nor__eq__noX) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2333 (all A (zero_zero_real = A <-> zero_zero_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_37_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2334 (all X hAPP_int_int(abs_abs_int,X) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,zero_zero_int),X)) # label(fact_5001_gcd__0__left__int) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2335 (all A all B all P_1 ((all A_2 all B_4 all C_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_4),C_3)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),B_4)) & hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,B_4),C_3))) & hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4))) -> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),C_3))))) -> (-hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A),B))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) -> (all N_2 -hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2))),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2)))))))))) # label(fact_4248_not__P__Bolzano__bisect_H) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2336 (all K_2 all I all J_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),J_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_2),K_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),K_2))))) # label(fact_62_zle__trans) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2337 (all P_1 all A all B (-((exists D_2 (A = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B),D_2) & -hBOOL(hAPP_nat_bool(P_1,D_2)))) | -hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B))) <-> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A),B))))) # label(fact_1780_nat__diff__split__asm) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2338 (all K_2 hAPP_i1732201573de_int(quickcheck_of_int,quickcheck_int_of(K_2)) = K_2) # label(fact_4556_of__int__int__of) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2339 (all N (N != zero_zero_nat -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)))) # label(fact_1610_gr0I) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2340 (all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N_1))))) # label(fact_1668_nat__0__less__mult__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2341 (all A_3 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),A_3)) -> (exists X_2 exists Y_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_3),Y_1)),one_one_nat) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),X_2))))) # label(fact_4207_bezout__prime) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2342 (all A_103 all C_45 all B_85 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_103),C_45)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_85),C_45))) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_103),B_85)))) # label(fact_1998_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2343 (all Ma all N_1 (hAPP_nat_real(semiri132038758t_real,N_1) = hAPP_nat_real(semiri132038758t_real,Ma) <-> N_1 = Ma)) # label(fact_2832_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2344 (all P all Q all R hAPP_f183342685l_real(P,hAPP_r794344869l_real(Q,R)) = hAPP_r1043989325l_real(hAPP_f1161935955l_real(hAPP_f328572565l_real(cOMBB_520243667l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__prod_Itc__Real_045) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2345 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sin,Y)),hAPP_real_real(sin,X))))))) # label(fact_3732_sin__monotone__2pi_H) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2346 (all Sa all Ta all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B)) -> hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,A),Sa)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,B),Ta))),pair_less)))) # label(fact_4541_pair__lessI1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2347 (all V_14 all B_166 all C_97 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_14)),B_166)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_14)),C_97)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_14)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_166),C_97))) # label(fact_192_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2348 (all A_184 all M_23 all N_58 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_23),N_58)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_184),M_23)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_184),N_58))))) # label(fact_1007_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2349 (all P_3 all Q_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Q_2),zero_zero_int)) -> normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(uminus_uminus_int,P_3)),hAPP_int_int(uminus_uminus_int,Q_2))) = normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2)))) # label(fact_4652_normalize__negative) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2350 (all P all Q all R hAPP_real_real(hAPP_n546711566l_real(P,R),Q) = hAPP_nat_real(hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__RealDef__Orea) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2351 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X)),hAPP_real_real(exp_real,X))))) # label(fact_3853_exp__ge__add__one__self__aux) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2352 (all A_233 A_233 = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,pls)),A_233)) # label(fact_323_add__numeral__0) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2353 (all A_196 all N_63 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_196),N_63)),A_196) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_196),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_196),N_63))) # label(fact_901_power__commutes) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2354 (all V_10 all U_3 all Y_26 all X_32 all A_209 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_32),A_209)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_26),A_209)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),U_3)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),V_10)) -> (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,U_3),V_10) = one_one_rat -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,U_3),X_32)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,V_10),Y_26))),A_209)))))))) # label(fact_752_convex__bound__lt) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2355 (all A_148 all N_51 all N_50 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_51),N_50)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,one_on1645066479umeral),A_148)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_148),N_51)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_148),N_50)))))) # label(fact_1274_power__increasing) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2356 (all C_57 all D_10 all A_135 all B_104 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_135),B_104)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_57),D_10)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_104)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_57)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_135),C_57)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_104),D_10)))))))) # label(fact_1377_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2357 (all C_38 all A_96 all B_78 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_96),B_78)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_38),A_96)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_38),B_78))))) # label(fact_2052_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2358 (all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (one_one_real != A_3 -> one_one_real = hAPP_real_real(log(A_3),A_3)))) # label(fact_3867_log__eq__one) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2359 (all A_3 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),one_one_nat))) # label(fact_4014_coprime__1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2360 (all M_27 all A_212 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_212),one_on1684967323de_int)),M_27) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,M_27),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_212),M_27))) # label(fact_703_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2361 (all N_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_1)),zero_zero_real)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,N_1),zero_zero_int)))) # label(fact_4438_real__of__int__lt__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2362 (all P all Q all R hAPP_n1699378549t_bool(P,hAPP_nat_nat(Q,R)) = hAPP_n1699378549t_bool(hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2363 (all A_25 all B_20 (B_20 != zero_z126310315umeral -> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_25),B_20)),B_20) = A_25)) # label(fact_2547_div__mult__self2__is__id) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2364 (exists K_4 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),K_4)) & (all X_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(im,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_2)),K_4)))))) # label(fact_4107_Im_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2365 (all A_1 all Ma (hBOOL(hAPP_int_bool(is_RRset(A_1),Ma)) <-> hAPP_int_nat(phi,Ma) = hAPP_f957591787ol_nat(finite_card_int,A_1) & hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,A_1),rsetR(Ma))))) # label(fact_4594_is__RRset__def) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2366 (all V_17 all V_16 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_16)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_17)) -> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_16)),hAPP_int_real(number267125858f_real,V_17)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_16),V_17))))) # label(fact_161_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2367 (all N all A_3 hAPP_complex_real(re,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,cis(A_3)),N)) = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),A_3))) # label(fact_4145_cos__n__Re__cis__pow__n) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2368 (all X_1 all Y_2 (zero_zero_real = Y_2 & zero_zero_real = X_1 <-> zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_1694_realpow__two__sum__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2369 (all P all Q all R hAPP_nat_bool(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,P),Q),R) = hAPP_nat_bool(hAPP_n1699378549t_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2370 (all A_15 all B_11 (B_11 != zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_15),B_11)),one_one_nat) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_15),B_11)),B_11))) # label(fact_2633_div__add__self2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2371 (all B_74 all A_86 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_86),zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_74),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_86),B_74)),zero_zero_nat))))) # label(fact_2143_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2372 (all P all Q all R hAPP_bool_bool(hAPP_i68813070l_bool(P,R),hAPP_int_bool(Q,R)) = hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__HOL__Obool_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2373 (all M_18 all N_38 all A_128 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_128)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_128),M_18)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_128),N_38))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_18),N_38))))) # label(fact_1447_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2374 (all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A)))) # label(fact_2192_zero__less__double__add__iff__zero__less__single__add) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2375 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,X)),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),one_one_real)))) # label(fact_3404_less__one__imp__sqr__less__one) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2376 (all A_203 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_203),zero_z891286103de_int) = zero_z891286103de_int) # label(fact_844_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2377 (all Y all X (X = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y)) -> Y = hAPP_real_real(sqrt,X)))) # label(fact_3340_real__sqrt__unique) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2378 (all F all N_1 zero_zero_real = hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),F)),hAPP_r474017924t_real(power_power_real,zero_zero_real))),hAPP_n1699378549t_bool(ord_at4362885an_nat(hAPP_nat_nat(suc,zero_zero_nat)),N_1))) # label(fact_4900_sumr__one__lb__realpow__zero) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2379 (all L all U hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_gr660468384an_nat(L),U)))) # label(fact_4631_finite__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2380 (all C_78 all D_14 all A_178 all B_138 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_178),B_138)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_78),D_14)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_178)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_78)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_178),C_78)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_138),D_14)))))))) # label(fact_1062_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2381 (all A_218 all B_161 all C_90 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_218),C_90)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_161),C_90)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_218),B_161)),C_90)) # label(fact_631_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2382 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) | N = M -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)))) # label(fact_1637_less__or__eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2383 (all A_158 all B_119 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_158),B_119))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_158)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_119))))) # label(fact_1171_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2384 (all N -hBOOL(hAPP_int_bool(nat_neg,hAPP_nat_int(semiri1621563631at_int,N)))) # label(fact_3269_not__neg__int) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2385 (all Lx_3 all Ly_1 all Rx_3 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_3),Ly_1)),Rx_3) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_3),Rx_3)),Ly_1)) # label(fact_489_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2386 (all V_17 all V_16 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_16)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_17)) -> hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_16),V_17)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_16)),hAPP_int_int(number_number_of_int,V_17))))) # label(fact_159_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2387 (all V (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,V))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V)))) # label(fact_103_less__0__number__of) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2388 (all X_1 hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),sin)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(sin,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(sin,X_1)))))) # label(fact_4795_DERIV__sin__realpow2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2389 (all K_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),K_2)))))) # label(fact_1671_mult__less__mono1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2390 (all N_54 all A_151 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_151)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_151),N_54))))) # label(fact_1241_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2391 (all A_214 all N_67 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_214),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_67))))) # label(fact_684_zero__le__even__power_H) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2392 (all Y_2 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(ln,X_1)),hAPP_real_real(ln,Y_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2)))))) # label(fact_3429_ln__less__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2393 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N))) # label(fact_3068_mult__Suc__right) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2394 (all Z_1 all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_1))))) # label(fact_4160_termination__basic__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2395 (all X hAPP_real_real(abs_abs_real,X) = hAPP_real_real(sqrt,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3402_real__sqrt__abs) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2396 (all A all A_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma)) -> (hBOOL(hAPP_int_bool(is_RRset(A_1),Ma)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hilbert_Eps_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,zcong(A)),Ma))),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),hAPP_i1948725293t_bool(norRRset,Ma))))),hAPP_i1948725293t_bool(norRRset,Ma))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),hilbert_Eps_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,zcong(A)),Ma))),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),hAPP_i1948725293t_bool(norRRset,Ma))))),Ma)))))) # label(fact_4893_aux__some) # label(axiom) # label(non_clause). [assumption]. 7.48/7.48 2397 (all A_3 all B_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B_2),A_3) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)) # label(fact_4305_gcd__commute__nat) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2398 (all K_2 hAPP_int_rat(ring_1_of_int_rat,K_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,K_2),one_one_int)) # label(fact_4679_Fract__of__int__eq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2399 (all A_87 all B_75 (is_int(A_87) -> A_87 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_87),B_75)),B_75))) # label(fact_2135_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2400 (all X all Z_1 all D_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Z_1))),one_one_int)),D_3))),Z_1)))) # label(fact_3464_decr__lemma) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2401 (all X_1 hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),cos_coeff)),hAPP_r474017924t_real(power_power_real,X_1))))) # label(fact_4832_summable__cos) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2402 (all V_7 all B_101 all C_56 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_7)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_101),C_56)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_7)),B_101)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_7)),C_56))) # label(fact_1545_right__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2403 (all N_37 all A_126 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_126)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_126),one_one_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_126),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_126),N_37))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_126),N_37)))))) # label(fact_1468_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2404 (all X_1 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_real)) <-> zero_zero_real = Y_2 & X_1 = zero_zero_real)) # label(fact_224_sum__power2__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2405 (all R_3 all X all Y hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),X)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),Y)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_3)),hAPP_real_complex(hAPP_r265291036omplex(complex,X),Y))) # label(fact_3991_complex__of__real__mult__Complex) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2406 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,X)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),one_one_real))))) # label(fact_3446_ln__le__minus__one) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2407 (all A all P_5 setS(A,P_5) = image_275383677t_bool(multInvPair(A,P_5),hAPP_i1948725293t_bool(sRStar,P_5))) # label(fact_4607_SetS__def) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2408 (all B_53 all A_65 all C_32 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_53),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_65),B_53))),hAPP_C498520661umeral(div_mo1740067990umeral(A_65),B_53))),C_32) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_65),C_32)) # label(fact_2352_div__mod__equality2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2409 (all A_10 all N_15 all B_8 (hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_8),hAPP_nat_nat(suc,N_15)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_10),hAPP_nat_nat(suc,N_15)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_10)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_8)) -> B_8 = A_10)))) # label(fact_3103_power__inject__base) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2410 (all A_133 all N_46 all N_45 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_46),N_45)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_133)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_133),one_one_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_133),N_45)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_133),N_46))))))) # label(fact_1407_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2411 (all X_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),one_one_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real))))) # label(fact_3926_real__root__lt__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2412 (all X all Y (X != Y | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(fequal_rat,X),Y)))) # label(help_fequal_2_1_fequal_000tc__Rat__Orat_T) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2413 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)))) # label(fact_1599_nat__le__linear) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2414 (all N -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N)),zero_zero_real))) # label(fact_3647_not__real__of__nat__less__zero) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2415 (all P_1 ((all X_2 (is_int(X_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) -> hBOOL(hAPP_int_bool(P_1,X_2))))) <-> (all X_2 hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X_2)))))) # label(fact_2988_transfer__int__nat__quantifiers_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2416 (all A_227 all B_164 all C_93 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_227),B_164)),C_93) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_227),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_164),C_93))) # label(fact_546_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2417 (all Ma all V (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V))) & Ma = hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,V)) <-> hAPP_int_int(number_number_of_int,V) = hAPP_nat_int(semiri1621563631at_int,Ma))) # label(fact_3168_int__eq__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2418 (all A_92 A_92 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_92),one_one_complex)) # label(fact_2102_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2419 (all A_98 all C_40 all B_80 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_98),C_40)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_80),C_40))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_98),B_80)))) # label(fact_2043_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2420 (all K_2 hAPP_int_int(bit1,K_2) != pls) # label(fact_249_rel__simps_I46_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2421 (all A all N_1 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A),N_1) = zero_zero_nat <-> N_1 != zero_zero_nat & A = zero_zero_nat)) # label(fact_1290_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2422 (all X_2 hAPP_nat_real(diffs_real(cos_coeff),X_2) = hAPP_real_real(uminus_uminus_real,hAPP_nat_real(sin_coeff,X_2))) # label(fact_4879_cos__fdiffs) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2423 (all M_19 all N_44 all A_132 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_132)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_132),M_19)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_132),N_44))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_19),N_44))))) # label(fact_1408_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2424 (all A_227 all B_164 all C_93 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_227),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_164),C_93)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_227),B_164)),C_93)) # label(fact_540_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2425 (all N_1 hAPP_nat_nat(suc,N_1) = hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),N_1)))) # label(fact_4745_card__Collect__le__nat) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2426 (all P all Q all R hAPP_f605995867t_bool(P,hAPP_r1134773055l_bool(Q,R)) = hAPP_r279757445t_bool(hAPP_f1840085295t_bool(hAPP_f1083049571t_bool(cOMBB_2050079537l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J_000tc__fun_030) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2427 (all K_2 hAPP_int_int(bit0,hAPP_int_int(uminus_uminus_int,K_2)) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(bit0,K_2))) # label(fact_3513_minus__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2428 (all A_13 all B_9 A_13 = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_13),B_9)),B_9)),hAPP_C498520661umeral(div_mo1740067990umeral(A_13),B_9))) # label(fact_2675_mod__div__decomp) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2429 (all X_36 all Y_28 all Z_9 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_36),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y_28),Z_9)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_36),Y_28)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_36),Z_9))) # label(fact_617_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2430 (all Y all X (-hBOOL(hAPP_int_bool(even_odd_even_int,X)) -> (-hBOOL(hAPP_int_bool(even_odd_even_int,Y)) -> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y)))))) # label(fact_2887_Parity_Oodd__plus__odd) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2431 (all M all N hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N))) # label(fact_3039_zadd__int) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2432 (all N exists P_4 (hBOOL(hAPP_nat_bool(prime,P_4)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_4),hAPP_nat_nat(suc,hAPP_nat_nat(fact,N)))) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),P_4)))) # label(fact_4570_euclid__bound) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2433 (all A_125 all N_36 all N_35 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_36),N_35)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_125)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_125),one_on1645066479umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_125),N_35)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_125),N_36))))))) # label(fact_1476_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2434 (all Q_2 all R_3 hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),hAPP_int_int(uminus_uminus_int,R_3)) = hAPP_P1975530577nt_int(negateSnd,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3))) # label(fact_4189_negateSnd__eq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2435 (all C_37 all A_95 all B_77 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_95),B_77)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_95),C_37)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_77),C_37))))) # label(fact_2062_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2436 (all A_3 all B_2 ((zero_zero_int = hAPP_int_int(div_mod_int(A_3),B_2) -> zero_zero_int = hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,A_3)),B_2)) & (hAPP_int_int(div_mod_int(A_3),B_2) != zero_zero_int -> hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,A_3)),B_2) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_2),hAPP_int_int(div_mod_int(A_3),B_2))))) # label(fact_3545_zmod__zminus1__eq__if) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2437 (all L all U hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,U),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,L),one_one_int))) = hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(ord_gr1297742076an_int(L),U))) # label(fact_4628_card__greaterThanLessThan__int) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2438 (all P all Q all R hAPP_r938550413l_real(hAPP_f1293550189l_real(hAPP_f183342685l_real(cOMBS_794971624l_real,P),Q),R) = hAPP_P1982799375l_real(hAPP_r931665397l_real(P,R),hAPP_r1195171167l_real(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__prod_Itc__RealDef__Oreal_Mtc___037) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2439 (all A_68 all B_56 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_68),B_56)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_68),B_56)),B_56)) = A_68) # label(fact_2341_semiring__div__class_Omod__div__equality_H) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2440 (all J all K_1 all I_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1)),I_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1))))) # label(fact_1777_le__diff__conv) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2441 (all V all N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),zero_zero_nat),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),N_1)))) # label(fact_5120_min__Suc__number__of) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2442 (all X_1 all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,X_1)),hAPP_int_rat(number_number_of_rat,Y_2))))) # label(fact_332_less__number__of) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2443 (all Lx all Rx all Ry hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx),Ry)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx),Ry))) # label(fact_508_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2444 (all A_203 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_203),zero_zero_int) = zero_zero_int) # label(fact_839_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2445 (all M all N (is_int(M) -> (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N) -> hAPP_int_int(number_number_of_int,min) = M | M = one_one_int))) # label(fact_790_pos__zmult__eq__1__iff__lemma) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2446 (all X_1 all F all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) -> ((all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),B)) -> hBOOL(hAPP_real_bool(isCont_real_real(F),X_2)))) -> ((all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),X_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),B)) -> hBOOL(hAPP_real_bool(deriv_real(F,X_2),zero_zero_real)))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),B)) -> hAPP_real_real(F,X_1) = hAPP_real_real(F,A))))))) # label(fact_5177_DERIV__isconst2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2447 (all N_66 all A_210 all B_158 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_210),B_158)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_210)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_66)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_210),N_66)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_158),N_66))))))) # label(fact_746_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2448 (all A_158 all B_119 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_158),B_119))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_158)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_119))))) # label(fact_1172_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2449 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sqrt,X)),X) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X)))) # label(fact_3850_sqrt__divide__self__eq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2450 (all X all N_5 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N_5)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N_5),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),X))))))) # label(fact_3924_real__root__decreasing) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2451 (all X_46 all Y_34 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_46),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_34),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),X_46)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_34)) -> X_46 = Y_34)))) # label(fact_203_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2452 (all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),P_3)))) # label(fact_4168_prime__g__zero) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2453 (all X_42 all Y_30 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_42),X_42)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_30),Y_30))),zero_zero_int))) # label(fact_232_not__sum__squares__lt__zero) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2454 (all X all N all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),X))))) # label(fact_4174_prime__divexp) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2455 (all A_188 all B_144 all C_81 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_188),B_144)),C_81)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_144),C_81)))) # label(fact_960_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2456 (all A_103 all C_45 all B_85 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_103),C_45)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_85),C_45))) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_103),B_85)))) # label(fact_2003_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2457 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,pi)),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(cos,Y)),hAPP_real_real(cos,X))))))) # label(fact_3573_cos__monotone__minus__pi__0_H) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2458 (all R1 all A_3 all R2 all B_2 rcis(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,R1),R2),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_3),B_2)) = hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,rcis(R1,A_3)),rcis(R2,B_2))) # label(fact_4265_rcis__divide) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2459 (all Sa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Sa)) -> hBOOL(tendsto_nat_real(hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,powr),real_nat)),hAPP_real_real(uminus_uminus_real,Sa)),zero_zero_real,sequentially)))) # label(fact_5047_LIMSEQ__neg__powr) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2460 (all X_1 all P_5 (is_int(X_1) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(sr,P_5))) -> hAPP_int_int(standardRes(P_5),X_1) = X_1))) # label(fact_2335_StandardRes__SR__prop) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2461 (all M_7 all N_21 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_7)),hAPP_n522471361de_int(semiri1424489471de_int,N_21)) = hAPP_n522471361de_int(semiri1424489471de_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21))) # label(fact_2962_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2462 (all I_1 all J image_nat_nat(suc,hAPP_n1699378549t_bool(ord_at238088361st_nat(I_1),J)) = hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_nat_nat(suc,I_1)),hAPP_nat_nat(suc,J))) # label(fact_5072_image__Suc__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2463 (all X_1 all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_2))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2)))) # label(fact_3044_Nat__Transfer_Otransfer__int__nat__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2464 (all A_200 all B_153 all C_86 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_200),B_153)),C_86) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_200),C_86)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_153),C_86))) # label(fact_880_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2465 (all P_1 all I_1 hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),P_1)),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),I_1)))))) # label(fact_4766_finite__M__bounded__by__nat) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2466 (all X_1 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),one_one_int)),zEven)))) # label(fact_3150_odd__minus__one__even) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2467 (all Ma (Ma = one_one_nat <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),one_one_nat)))) # label(fact_1515_nat__dvd__1__iff__1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2468 (all Q_2 all R_3 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Q_2),hAPP_rat_rat(inverse_inverse_rat,R_3)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,Q_2),R_3)) # label(fact_4651_divide__rat__def) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2469 (all K_2 all L_2 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(bit0,K_2)),L_2) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),L_2))) # label(fact_126_mult__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2470 (all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Y)) -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(Y,hAPP_nat_nat(div_mod_nat(X),Y)))),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_P1175774780nt_int(product_fst_int_int,bezw(Y,hAPP_nat_nat(div_mod_nat(X),Y)))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(Y,hAPP_nat_nat(div_mod_nat(X),Y)))),hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),Y))))) = bezw(X,Y))) # label(fact_4231_bezw__non__0) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2471 (all L_2 hAPP_int_int(bit1,L_2) != pls) # label(fact_250_rel__simps_I39_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2472 (all A_3 all X hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),X) = hAPP_real_real(scaleR_scaleR_real(A_3),X)) # label(fact_5107_real__scaleR__def) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2473 (all V all W_1 (-hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_i769753017umeral(number1443263063umeral,W_1)),hAPP_i769753017umeral(number1443263063umeral,V))) <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_i769753017umeral(number1443263063umeral,V)),hAPP_i769753017umeral(number1443263063umeral,W_1))))) # label(fact_104_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2474 (all M all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),M))))) # label(fact_1627_trans__less__add1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2475 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M)),N) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N))) # label(fact_2867_add__Suc) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2476 (all C_73 all A_173 all B_133 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_173),B_133)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_73)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_173),C_73)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_133),C_73)))))) # label(fact_1085_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2477 (all P all Q all R hAPP_r2019696015l_real(hAPP_f2072293513l_real(hAPP_f1763467369l_real(cOMBS_163651580l_real,P),Q),R) = hAPP_f1736103445l_real(hAPP_r224952523l_real(P,R),hAPP_r2019696015l_real(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__p) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2478 (all B_71 all A_83 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_83)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_71)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_83),B_71)))))) # label(fact_2168_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2479 (all V_3 all V_2 ((hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3))) & (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,V_3))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3))))) # label(fact_3282_div__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2480 (all B_1_1 all B_2_1 is_bool(hAPP_f448129468l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2481 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)) # label(fact_1601_nat__mult__commute) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2482 (all X hAPP_real_real(sqrt,hAPP_real_real(inverse_inverse_real,X)) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X))) # label(fact_3829_real__sqrt__inverse) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2483 (all N_56 one_one_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_56))) # label(fact_1041_power__m1__even) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2484 (all X_1 all Y_2 ((hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),zero_zero_real)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))) & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_2)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_1),Y_2))))) # label(fact_3360_real__0__le__divide__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2485 (all X_7 all Q_6 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_7),hAPP_nat_nat(suc,Q_6)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_7),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_7),Q_6))) # label(fact_3025_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2486 (all A_32 hAPP_C498520661umeral(div_mo1740067990umeral(A_32),one_on1645066479umeral) = zero_z126310315umeral) # label(fact_2501_mod__by__1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2487 (all C_66 all A_144 all B_113 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_66),A_144)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_66),B_113))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_66)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_144),B_113))))) # label(fact_1321_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2488 (all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(wset(A),P_5))),one_one_int),P_5)))))) # label(fact_4814_wset__zcong__prod__1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2489 (all A_66 all B_54 (is_int(A_66) -> A_66 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_66),B_54)),B_54)),hAPP_int_int(div_mod_int(A_66),B_54)))) # label(fact_2346_mod__div__equality) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2490 (all X hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),pi)) = hAPP_real_real(tan,X)) # label(fact_3600_tan__periodic__pi) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2491 (all A_3 all Q_2 all R_3 (is_int(R_3) & is_int(A_3) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,A_3),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3))) -> (zero_zero_int != A_3 -> R_3 = zero_zero_int)))) # label(fact_2792_self__remainder) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2492 (all X all N (-hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) -> X = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),N))) # label(fact_3900_odd__real__root__pow) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2493 (all A all B all Ma (is_int(B) -> ((exists K (is_int(K) & hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),K)) = B)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma))))) # label(fact_1314_zcong__iff__lin) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2494 (all Lx_3 all Ly_1 all Rx_3 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_3),Ly_1)),Rx_3) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_3),Rx_3)),Ly_1)) # label(fact_485_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2495 (all X_1 all Y_2 (hAPP_real_complex(hAPP_r265291036omplex(complex,X_1),Y_2) = ii <-> X_1 = zero_zero_real & one_one_real = Y_2)) # label(fact_3981_Complex__eq__i) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2496 (all X_1 all Y_2 all Z_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Z_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Z_2)))))) # label(fact_1789_real__mult__less__iff1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2497 (all X_47 all Y_35 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_47),X_47)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_35),Y_35))))) # label(fact_95_sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2498 (all D_3 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_3),hAPP_int_int(abs_abs_int,D_3)))) # label(fact_3420_zdvd__self__abs1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2499 (all A_3 (one_one_int = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) -> hAPP_int_int(abs_abs_int,A_3) = one_one_int)) # label(fact_3463_power2__eq1__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2500 (all D_3 all I (is_int(I) -> (zero_zero_int != I -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_3),I)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,D_3)),hAPP_int_int(abs_abs_int,I))))))) # label(fact_3434_dvd__imp__le__int) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2501 (all Z_8 all X_23 all Y_18 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_23),Y_18)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_23),Z_8)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_23),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Y_18),Z_8)))))) # label(fact_1519_dvd__diff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2502 (all X_24 all N_32 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32)) | one_on1645066479umeral = X_24 -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,X_24),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_24),N_32))))) # label(fact_1492_dvd__power) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2503 (all X_1 all Y_2 all Z_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y_2),Z_2))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Z_2)))) # label(fact_4327_gcd__semilattice__nat_Ole__inf__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2504 (all A_146 one_on1645066479umeral = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_146),zero_zero_nat)) # label(fact_1299_power__0) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2505 (all A_28 all B_23 all C_10 (zero_zero_nat != C_10 -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_28),B_23) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_10),A_28)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_10),B_23)))) # label(fact_2534_div__mult__mult1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2506 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_2),one_one_int))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2)))) # label(fact_1837_norR__mem__unique__aux) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2507 (all B_69 all A_81 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_81)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_69)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_81),B_69)))))) # label(fact_2183_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2508 (all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B)) <-> zero_zero_nat = hAPP_nat_nat(div_mod_nat(B),A))) # label(fact_2519_dvd__eq__mod__eq__0) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2509 (all C_88 all A_206 all B_157 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_206),B_157)) -> (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,B_157),C_88)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_206),C_88))))) # label(fact_805_dvd__trans) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2510 (all K_2 all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) # label(fact_1744_Nat_Odiff__cancel) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2511 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_2754_div__2__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2512 (all N_13 all A_8 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_8)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_8),hAPP_nat_nat(suc,N_13)))))) # label(fact_3115_power__gt1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2513 (all A_222 zero_z891286103de_int = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,zero_z891286103de_int),A_222)) # label(fact_588_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2514 (all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(number_number_of_int,Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,pls)),Y_2)))) # label(fact_411_less__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2515 (all F (hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_f352196356l_real(suminf_real,F))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F)))))) # label(fact_4830_summable__rabs) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2516 (all A_20 all C_6 all B_16 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,C_6),B_16)) -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_20),B_16)),C_6) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_20),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B_16),C_6)))) # label(fact_2570_div__mult__swap) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2517 (all A_33 all B_27 zero_zero_nat = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_33),B_27)),B_27)) # label(fact_2496_mod__mult__self2__is__0) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2518 (all C_74 all A_174 all B_134 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_174),B_134)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_74)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_74),A_174)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_74),B_134)))))) # label(fact_1080_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2519 (all A_3 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),P_3)))))) # label(fact_2766_Euler__part2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2520 (all V all W_1 (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,W_1)),hAPP_int_rat(number_number_of_rat,V))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,V)),hAPP_int_rat(number_number_of_rat,W_1))))) # label(fact_109_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2521 (all Ta all D all D_1 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D),D_1)) -> (all X_2 all K (-hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_2),Ta))) <-> -hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_2),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,K),D_1))),Ta))))))) # label(fact_1533_inf__period_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2522 (all A (is_int(A) -> (zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A) <-> A = zero_zero_int))) # label(fact_246_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2523 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),pi)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))))))) # label(fact_3566_cos__monotone__0__pi_H) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2524 (all B_55 all A_67 (is_int(A_67) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_55),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_67),B_55))),hAPP_int_int(div_mod_int(A_67),B_55)) = A_67)) # label(fact_2342_mod__div__equality2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2525 (all Y_2 hAPP_real_real(arcsin,Y_2) = the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_eq_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),sin)),Y_2))))) # label(fact_4883_arcsin__def) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2526 (all W ((hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,W))) -> hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,W)) = zero_zero_nat) & (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,W))) -> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,W)),hAPP_int_nat(number_number_of_nat,W))) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,W))))) # label(fact_4780_nat__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2527 (all A_3 hAPP_int_int(uminus_uminus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(uminus_uminus_int,A_3)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) # label(fact_3550_power3__minus) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2528 (all A_196 all N_63 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_196),N_63)),A_196) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_196),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_196),N_63))) # label(fact_903_power__commutes) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2529 (all P all Q all R hAPP_nat_fun_int_int(hAPP_f1935805932nt_int(hAPP_f125788821nt_int(cOMBB_859312247nt_nat,P),Q),R) = hAPP_int_fun_int_int(P,hAPP_nat_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_001) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2530 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X))),one_one_real)) = hAPP_real_real(tan,X)))) # label(fact_3630_tan__half) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2531 (all M all K_2 all N (is_int(K_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N)) -> (K_2 = hAPP_int_int(legacy_zgcd(M),N) -> (exists S_2 exists T (is_int(S_2) & K_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_2),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T),N)) & is_int(T))))))) # label(fact_4537_zgcd__ex__linear) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2532 (all A_18 all C_5 all B_14 (zero_zero_nat != B_14 -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_18),B_14)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_18),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_5),B_14))),B_14))) # label(fact_2621_div__mult__self1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2533 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K1)),hAPP_int_int(bit1,K2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)))) # label(fact_65_less__eq__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2534 (all X all Y hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(re,X)),hAPP_complex_real(re,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(im,X)),hAPP_complex_real(im,Y)))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(re,X)),hAPP_complex_real(im,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(im,X)),hAPP_complex_real(re,Y)))) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X),Y)) # label(fact_4136_complex__mult__def) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2535 (all X hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),natfloor(X)))) # label(fact_3385_zero__le__natfloor) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2536 (all V_22 all V_21 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_22)) -> hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_21)),hAPP_int_complex(number528085621omplex,V_22)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_21),V_22))))) # label(fact_77_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2537 (all A all B all K_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ma)) -> (hAPP_int_int(legacy_zgcd(K_1),Ma) = one_one_int -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),B)),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma)))))) # label(fact_4526_zcong__cancel2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2538 (all I_1 all M_2 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),M_2)) -> hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),M_2))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),hAPP_nat_nat(suc,I_1))))) != zero_zero_nat)) # label(fact_4769_card__less) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2539 (all X_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),N_1)),zOdd)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd))))) # label(fact_2645_power__preserves__odd) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2540 (all A_216 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,one_on1684967323de_int),A_216) = A_216) # label(fact_652_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2541 (all A all E_2 all C all B all D (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),E_2)),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),E_2)),D))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B)),E_2)),C)),D)))) # label(fact_1562_le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2542 (all B_43 all B_42 all A_52 all C_23 all A_51 (hAPP_C498520661umeral(div_mo1740067990umeral(A_52),C_23) = hAPP_C498520661umeral(div_mo1740067990umeral(A_51),C_23) -> (hAPP_C498520661umeral(div_mo1740067990umeral(B_43),C_23) = hAPP_C498520661umeral(div_mo1740067990umeral(B_42),C_23) -> hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_52),B_43)),C_23) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_51),B_42)),C_23)))) # label(fact_2407_mod__add__cong) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2543 (all A_184 all M_23 all N_58 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_23),N_58)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_184),M_23)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_184),N_58))))) # label(fact_1004_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2544 (all N_55 all A_152 all B_114 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_152),B_114)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_152)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_152),N_55)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_114),N_55)))))) # label(fact_1238_power__mono) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2545 (all P all Q all R hAPP_nat_bool(hAPP_f737665877t_bool(hAPP_f605995867t_bool(cOMBB_real_bool_nat,P),Q),R) = hAPP_real_bool(P,hAPP_nat_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__HOL__Obool_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2546 (all C_77 all B_137 all A_177 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_137),A_177)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_77),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_77),A_177)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_77),B_137)))))) # label(fact_1066_mult__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2547 (all X_18 all Y_15 all A_78 all B_66 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_18),Y_15)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_78),B_66)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_18),A_78)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_15),B_66))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_18),A_78)),B_66))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_78),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_15),B_66)))) # label(fact_2281_mult_Oprod__diff__prod) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2548 (all M_12 hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_n522471361de_int(semiri1424489471de_int,M_12)))) # label(fact_2905_zero__le__imp__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2549 (all K_3 all K_1 all F ((all N_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,K_1),hAPP_nat_real(F,N_2))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_2)),K_3)))) -> hBOOL(bseq_real(F)))) # label(fact_4948_BseqI2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2550 (all A all N_1 (zero_zero_nat != N_1 & A = zero_zero_complex <-> zero_zero_complex = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A),N_1))) # label(fact_1293_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2551 (all M all N all K_2 all L_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_2),L_2)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),L_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),N) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))))) # label(fact_1631_less__add__eq__less) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2552 (all N_59 all M_24 all X_28 all Y_22 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,X_28),Y_22)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_59),M_24)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_28),N_59)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_22),M_24)))))) # label(fact_997_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2553 (all B_70 all A_82 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_82),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_70),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_82),B_70)),zero_zero_int))))) # label(fact_2178_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2554 (all X all Y hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(re,X)),hAPP_complex_real(re,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(im,X)),hAPP_complex_real(im,Y)))) # label(fact_4131_complex__Re__mult) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2555 (all V_17 all V_16 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_16)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_17)) -> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_16)),hAPP_int_rat(number_number_of_rat,V_17)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_16),V_17))))) # label(fact_163_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2556 (all X_1 all Y_2 (is_int(X_1) & is_int(Y_2) -> (X_1 != zero_zero_int | Y_2 != zero_zero_int <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),X_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_2),Y_2))))))) # label(fact_229_sum__squares__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2557 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(ln,X))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X))))) # label(fact_3445_ln__gt__zero__imp__gt__one) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2558 (all X (is_int(X) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),zero_zero_int)) -> (exists N_2 hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_2))) = X)))) # label(fact_3557_negD) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2559 (all B_117 all A_156 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_156)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_117),zero_z126310315umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_156),B_117)),zero_z126310315umeral))))) # label(fact_1182_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2560 (all M all N hAPP_nat_nat(div_mod_nat(M),hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,N)))) = hAPP_nat_nat(div_mod_nat(M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),N))) # label(fact_3210_mod__Suc__eq__mod__add3) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2561 (all X all N hAPP_int_real(real_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(real_int,X)),N)) # label(fact_4391_real__of__int__power) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2562 (all X_1 hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),cos)),cos),X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,min))),hAPP_real_real(cos,X_1))),hAPP_real_real(sin,X_1))))) # label(fact_4782_DERIV__cos__cos__mult2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2563 (all B_1_1 all B_2_1 all B_3_1 is_bool(tendsto_nat_real(B_1_1,B_2_1,B_3_1))) # label(gsy_c_Limits_Otendsto_000tc__Nat__Onat_000tc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2564 (all A_235 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_235),A_235)),A_235) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_235),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) # label(fact_280_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2565 (all K_1 hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),K_1))))) # label(fact_4732_finite__Collect__le__nat) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2566 (all X all A_3 all B_2 hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),B_2)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_2))) # label(fact_4412_powr__add) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2567 (all M all N hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),N)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N)) # label(fact_4969_gcd__add2__int) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2568 (all A_232 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_232),hAPP_int_rat(number_number_of_rat,pls)) = A_232) # label(fact_329_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2569 (all N_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_1),Ma)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,Ma)),one_one_real))))) # label(fact_4448_int__le__real__less) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2570 (all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(semiri132038758t_real,Ma)),hAPP_nat_real(semiri132038758t_real,N_1))))) # label(fact_2951_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2571 (all A all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,A)),X_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A),natfloor(X_1)))))) # label(fact_3687_le__natfloor__eq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2572 (all X all Y hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Y))) # label(fact_3041_Nat__Transfer_Otransfer__int__nat__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2573 (all B_120 all A_159 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_120),A_159))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_159)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_120))))) # label(fact_1161_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2574 (all C_38 all A_96 all B_78 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_96),B_78)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_38),A_96)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_38),B_78))))) # label(fact_2053_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2575 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(exp_real,X))))) # label(fact_3842_exp__gt__one) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2576 (all V_20 all W_16 all Z_15 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_20),W_16))),Z_15) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_20)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,W_16)),Z_15))) # label(fact_80_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2577 (all A_6 all N_8 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_6),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_8))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_6),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_6),N_8)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3231_power__odd__eq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2578 (all A all E_2 all C all B all D (is_int(C) -> (C = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B),A)),E_2)),D) <-> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),E_2)),D) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),E_2)),C)))) # label(fact_1536_eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2579 (all N all M hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,N),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),M)),M)) # label(fact_5190_nat__minus__add__max) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2580 (all X_31 all Y_25 (X_31 != Y_25 -> (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_31),Y_25)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_25),X_31))))) # label(fact_793_linorder__neqE__linordered__idom) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2581 (all X all Y all M hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),hAPP_int_int(div_mod_int(Y),M))),M) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y)),M)) # label(fact_2463_zdiff__zmod__right) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2582 (all B all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A),B))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A))))) # label(fact_2604_nonneg1__imp__zdiv__pos__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.49 2583 (all Z_1 zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(uminus_uminus_int,Z_1)),Z_1)) # label(fact_3529_zadd__zminus__inverse2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2584 (all C all D all A all B (B != A & C != D <-> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),D)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),C)) != hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),C)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),D)))) # label(fact_625_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2585 (all A_3 all Y all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y)) -> hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),Y)),A_3) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,Y),A_3))))) # label(fact_4435_powr__divide) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2586 (all Lx_6 all Ly_4 all Rx_6 all Ry_4 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_6),Ly_4)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_6),Ry_4)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_6),Rx_6)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ly_4),Ry_4))) # label(fact_464_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2587 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K_1)),pls)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),pls)))) # label(fact_133_rel__simps_I29_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2588 (all N_1 all P_1 (hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) -> ((all N_2 (hBOOL(hAPP_nat_bool(P_1,N_2)) -> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_2))))) -> hBOOL(hAPP_nat_bool(P_1,N_1))))) # label(fact_3354_nat__induct) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2589 (all N all M hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,quickcheck_int_of(N)),quickcheck_int_of(M))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(minus_534354567de_int,N),M)) # label(fact_4562_minus__code__int__def) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2590 (all C_60 all D_13 all A_138 all B_107 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_138),B_107)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_60),D_13)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_138)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_60)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_138),C_60)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_107),D_13)))))))) # label(fact_1355_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2591 (all C_66 all A_144 all B_113 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_66),A_144)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_66),B_113))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_66)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_144),B_113))))) # label(fact_1323_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2592 (all A_26 all B_21 (is_int(A_26) & is_int(B_21) -> (zero_zero_int != B_21 -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_21),A_26)),B_21) = A_26))) # label(fact_2541_div__mult__self1__is__id) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2593 (all A_90 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,one_one_complex),A_90) = A_90) # label(fact_2116_mult__1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2594 (all V_12 all W_10 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_12)),hAPP_int_rat(number_number_of_rat,W_10)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_12),W_10))) # label(fact_340_add__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2595 (all C_83 all D_16 all A_191 all B_147 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_191),B_147)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,C_83),D_16)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_191),C_83)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_147),D_16)))))) # label(fact_939_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2596 (all B_148 all A_192 all C_84 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_192),C_84)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_192),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_148),C_84))))) # label(fact_931_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2597 (all W_17 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,W_17)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,pls)),W_17))) # label(fact_43_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2598 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real)) -> hBOOL(hAPP_real_bool(isCont_real_real(arccos),X_1))))) # label(fact_5172_isCont__arccos) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2599 (all C_75 all A_175 all B_135 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_175),B_135)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_75)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_75),A_175)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_75),B_135)))))) # label(fact_1074_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2600 (all X_38 all P_8 all Q_12 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_38),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_38),P_8)),Q_12)) # label(fact_447_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2601 (all Q_2 all B_2 all R_3 all C_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(div_mod_int(Q_2),C_1))),R_3))))))) # label(fact_2657_zmult2__lemma__aux1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2602 (all N all M (is_int(M) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),M)) -> M = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,M),N))))) # label(fact_2531_zdvd__mult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2603 (all X -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X)),zero_zero_real))) # label(fact_3835_not__exp__less__zero) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2604 (all N_1 big_co1705425894at_nat(cOMBI_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(one_one_nat),N_1)) = hAPP_nat_nat(fact_fact_nat,N_1)) # label(fact_5074_fact__altdef__nat) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2605 (all M_10 all N_24 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_10)),hAPP_n522471361de_int(semiri1424489471de_int,N_24))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_24)))) # label(fact_2931_of__nat__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2606 (all A_11 all N_17 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_11),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_11),N_17)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_11),hAPP_nat_nat(suc,N_17))) # label(fact_3016_power__Suc) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2607 (all K_2 hAPP_int_int(succ,hAPP_int_int(bit0,K_2)) = hAPP_int_int(bit1,K_2)) # label(fact_3293_succ__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2608 (all Diff all F all N_1 all H (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H),zero_zero_real)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) & F = hAPP_n546711566l_real(Diff,zero_zero_nat) & (all M_1 all T (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,H),T)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),zero_zero_real)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))))) -> (exists T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,T),zero_zero_real)) & hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N_1),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N_1))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H),T)))))) # label(fact_4938_Maclaurin__minus__objl) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2609 (all F all A_1 hBOOL(hAPP_f448129468l_bool(nat_tr876908586nt_nat(F),A_1))) # label(fact_2778_transfer__morphismI) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2610 (all X_1 (hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(uminus_uminus_int,X_1))) <-> hBOOL(hAPP_int_bool(even_odd_even_int,X_1)))) # label(fact_3527_even__neg) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2611 (all N_1 (N_1 = zero_zero_nat <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N_1)),zero_zero_real)))) # label(fact_3659_real__of__nat__le__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2612 (all P_1 all A0 all A1 (hBOOL(hAPP_P1555980039t_bool(accp_P490777396at_nat(nat_gcd_rel),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,A0),A1))) -> ((all X_2 all Y_1 (hBOOL(hAPP_P1555980039t_bool(accp_P490777396at_nat(nat_gcd_rel),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,X_2),Y_1))) -> ((Y_1 != zero_zero_nat -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Y_1),hAPP_nat_nat(div_mod_nat(X_2),Y_1)))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,X_2),Y_1))))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,A0),A1))))) # label(fact_4624_nat__gcd_Opinduct) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2613 (all F all G ((exists N_3 all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_nat_real(F,N_2))),hAPP_nat_real(G,N_2))))) -> (hBOOL(summable_real(G)) -> hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F)))))) # label(fact_4829_summable__rabs__comparison__test) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2614 (all A_154 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_154),A_154)),zero_zero_real))) # label(fact_1201_not__square__less__zero) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2615 (all B_2 all Y all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y)) -> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),hAPP_real_real(log(B_2),X)) = hAPP_real_real(log(B_2),hAPP_real_real(hAPP_r1250527377l_real(powr,X),Y))))) # label(fact_4473_log__powr) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2616 (all A_3 all B_2 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),B_2))) # label(fact_1638_zspecial__product) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2617 (all L_2 all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),L_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),L_2))))) # label(fact_1750_diff__le__mono) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2618 (all X_1 all A (hBOOL(monoseq_real(A)) -> (hBOOL(tendsto_nat_real(A,X_1,sequentially)) -> (all M_1 all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,N_2)),hAPP_nat_real(A,M_1))))) & (all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_nat_real(A,N_2)))) | (all M_1 all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,M_1)),hAPP_nat_real(A,N_2))))) & (all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,N_2)),X_1)))))) # label(fact_5037_monoseq__le) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2619 (all A all N_1 (A = zero_zero_rat & zero_zero_nat != N_1 <-> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A),N_1) = zero_zero_rat)) # label(fact_1295_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2620 (all P all Q all R hAPP_i447943416t_real(hAPP_f1295083291t_real(hAPP_f212164053t_real(cOMBB_929094628al_int,P),Q),R) = hAPP_f1690286043t_real(P,hAPP_i371647666l_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__032) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2621 (all W_5 ((hAPP_int_nat(number_number_of_nat,W_5) = zero_zero_nat -> one_one_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),hAPP_int_nat(number_number_of_nat,W_5))) & (zero_zero_nat != hAPP_int_nat(number_number_of_nat,W_5) -> zero_zero_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),hAPP_int_nat(number_number_of_nat,W_5))))) # label(fact_726_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2622 (all B_117 all A_156 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_156)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_117),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_156),B_117)),zero_zero_rat))))) # label(fact_1184_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2623 (all X hAPP_complex_real(im,hAPP_complex_complex(uminus473333897omplex,X)) = hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X))) # label(fact_4100_Im_Ominus) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2624 (all A all A_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma)) -> (hBOOL(hAPP_int_bool(is_RRset(A_1),Ma)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(rRset2norRR(A_1,Ma),A)),hAPP_i1948725293t_bool(norRRset,Ma))))))) # label(fact_4592_RRset2norRR__correct2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2625 (all B_116 all A_155 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_155)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_116)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_155),B_116)))))) # label(fact_1187_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2626 (all Z_1 all W hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_1),W) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),hAPP_int_int(uminus_uminus_int,W))) # label(fact_3537_diff__int__def__symmetric) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2627 (all A_93 A_93 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_93),zero_zero_rat)) # label(fact_2091_diff__0__right) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2628 (all A_233 A_233 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,pls)),A_233)) # label(fact_325_add__numeral__0) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2629 (all Ma all N_1 (hAPP_nat_nat(suc,zero_zero_nat) = Ma & N_1 = hAPP_nat_nat(suc,zero_zero_nat) <-> hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N_1))) # label(fact_3053_mult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2630 (all A_61 zero_zero_int = hAPP_int_int(div_mod_int(zero_zero_int),A_61)) # label(fact_2372_mod__0) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2631 (all X all Y hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),Y))) # label(fact_3076_Divides_Otransfer__int__nat__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2632 (all A all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma)) -> (one_one_int = hAPP_int_int(legacy_zgcd(A),Ma) -> (exists X_2 ((all Y_1 (is_int(Y_1) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_1),hAPP_i1948725293t_bool(norRRset,Ma))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),Y_1),Ma)) -> Y_1 = X_2))) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(norRRset,Ma))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),X_2),Ma)) & is_int(X_2)))))) # label(fact_4586_norR__mem__unique) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2633 (all X_1 all P_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(sRStar,P_5))) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(standardRes(P_5),hAPP_int_int(multInv(P_5),X_1))),hAPP_i1948725293t_bool(sRStar,P_5))))))) # label(fact_2767_StandardRes__SRStar__prop2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2634 (all X_1 hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),sin_coeff)),hAPP_r474017924t_real(power_power_real,X_1))) = hAPP_real_real(sin,X_1)) # label(fact_4820_sin__def) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2635 (all C_74 all A_174 all B_134 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_174),B_134)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_74)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_74),A_174)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_74),B_134)))))) # label(fact_1077_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2636 (all A_183 all M_22 all N_57 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_183),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_22),N_57)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_183),M_22)),N_57)) # label(fact_1018_power__mult) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2637 (all A_59 all B_50 all C_29 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_59),C_29)),B_50)),C_29) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_59),B_50)),C_29)) # label(fact_2383_mod__mult__left__eq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2638 (all Z_1 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_complex_real(im,Z_1)),hAPP_complex_real(norm_norm_complex,Z_1)) = hAPP_complex_real(im,hAPP_complex_complex(sgn_sgn_complex,Z_1))) # label(fact_4105_Im__sgn) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2639 (all X_33 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_33),zero_zero_nat) = one_on1684967323de_int) # label(fact_680_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2640 (all N_54 all A_151 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_151)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_151),N_54))))) # label(fact_1239_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2641 (all M_8 all N_22 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_n1108039445umeral(semiri1619134803umeral,M_8)),hAPP_n1108039445umeral(semiri1619134803umeral,N_22)) = hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22))) # label(fact_2953_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2642 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) -> hAPP_int_nat(nat,hAPP_int_int(div_mod_int(X),Y)) = hAPP_nat_nat(div_mod_nat(hAPP_int_nat(nat,X)),hAPP_int_nat(nat,Y))))) # label(fact_2699_Divides_Otransfer__nat__int__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2643 (all X hAPP_int_int(abs_abs_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(abs_abs_int,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) # label(fact_3448_abs__power3__distrib) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2644 (all R_3 all A_3 hAPP_real_real(abs_abs_real,R_3) = hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_3)),hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(cos,A_3)),hAPP_real_real(sin,A_3))))) # label(fact_3998_cmod__complex__polar) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2645 (all A_46 all B_37 all C_18 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_46),B_37)),C_18) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_46),hAPP_C498520661umeral(div_mo1740067990umeral(B_37),C_18))),C_18)) # label(fact_2427_mod__add__right__eq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2646 (all K_2 all I all J_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I),J_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,J_2),K_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I),K_2))))) # label(fact_1731_real__le__trans) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2647 (all P_1 all L ((exists X_2 hBOOL(hAPP_real_bool(P_1,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,L),X_2)))) <-> (exists X_2 (hBOOL(hAPP_real_bool(P_1,X_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,L),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),zero_zero_real))))))) # label(fact_1676_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2648 (all B_36 all A_45 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_36),A_45)),B_36) = hAPP_C498520661umeral(div_mo1740067990umeral(A_45),B_36)) # label(fact_2431_mod__add__self1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2649 (all X_21 all N_31 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31)) -> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_21),N_31) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21))) # label(fact_1798_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2650 (all W_1 all Z_2 (is_int(Z_2) & is_int(W_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_2),one_one_int))) <-> W_1 = Z_2 | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2))))) # label(fact_358_zless__add1__eq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2651 (all A all B all C (zero_zero_real != C -> (A = B <-> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),C) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),C)))) # label(fact_1787_real__mult__right__cancel) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2652 (all C_60 all D_13 all A_138 all B_107 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_138),B_107)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_60),D_13)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_138)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_60)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_138),C_60)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_107),D_13)))))))) # label(fact_1359_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2653 (all A_94 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_94),A_94) = zero_zero_int) # label(fact_2084_diff__self) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2654 (all A_3 all B_2 hAPP_P1175774780nt_int(product_fst_int_int,divmod_int(A_3,B_2)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)) # label(fact_4224_div__int__def) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2655 (all X_2 ((hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2)) -> zero_zero_real = hAPP_nat_real(sin_coeff,X_2)) & (-hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2)) -> hAPP_nat_real(sin_coeff,X_2) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X_2),hAPP_nat_nat(suc,zero_zero_nat))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,X_2)))))) # label(fact_3796_sin__coeff__def) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2656 (all A_3 hAPP_real_real(sin,A_3) = hAPP_complex_real(im,cis(A_3))) # label(fact_4147_Im__cis) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2657 (all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = X)) # label(fact_4323_gcd__proj1__if__dvd__nat) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2658 (all M all N hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N))) # label(fact_3001_zmult__int) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2659 (all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) = P_3 | -hBOOL(hAPP_nat_bool(even_odd_even_nat,P_3)))) # label(fact_4196_prime__odd) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2660 (all N_12 all R_6 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R_6)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,R_6),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_6),hAPP_nat_nat(suc,N_12))),R_6))))) # label(fact_3156_realpow__Suc__le__self) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2661 (all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),N))) # label(fact_1600_le__refl) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2662 (all M_25 all A_185 all N_60 all B_142 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_185),N_60)),B_142)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_25),N_60)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_185),M_25)),B_142))))) # label(fact_995_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2663 (all X all Y hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),hAPP_int_int(abs_abs_int,Y)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)) # label(fact_4988_gcd__abs2__int) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2664 (all A_48 all B_39 all C_20 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_48),B_39)),C_20) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_48),C_20)),hAPP_C498520661umeral(div_mo1740067990umeral(B_39),C_20))),C_20)) # label(fact_2419_mod__add__eq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2665 (all Lx all Rx all Ry hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx),Ry)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx),Ry))) # label(fact_510_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2666 (all B_69 all A_81 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_81)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_69)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_81),B_69)))))) # label(fact_2187_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2667 (all V_5 all W_3 all C_55 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_5)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(number_number_of_int,W_3)),C_55)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_5),W_3))),C_55)) # label(fact_1574_add__number__of__diff1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2668 (all B_2 all Bq all Br all A_3 all C_1 all Aq all Ar (is_int(C_1) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Aq),Ar))) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(B_2,C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Bq),Br))) -> (zero_zero_int != C_1 -> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2),C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aq),Bq)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ar),Br)),C_1))),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ar),Br)),C_1))))))))) # label(fact_2800_zadd1__lemma) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2669 (all X_1 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_2)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),hAPP_real_real(uminus_uminus_real,X_1))))) # label(fact_3491_real__add__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2670 (all Lx_5 all Ly_3 all Rx_5 all Ry_3 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_5),Ly_3)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_5),Ry_3)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_5),Ly_3)),Ry_3))) # label(fact_472_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2671 (all B_2 all A_3 all X (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),X)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),A_3)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),X)))) # label(fact_4314_gcd__semilattice__nat_Oless__infI1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2672 (all A_19 all B_15 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_19),B_15)) -> B_15 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_19),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B_15),A_19)))) # label(fact_2574_dvd__mult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2673 (all C_57 all D_10 all A_135 all B_104 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_135),B_104)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_57),D_10)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_104)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_57)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_135),C_57)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_104),D_10)))))))) # label(fact_1372_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2674 (all X_1 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,X_1)),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_2))))) # label(fact_3490_real__0__le__add__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2675 (all V_2 zero_zero_real = hAPP_complex_real(im,hAPP_int_complex(number528085621omplex,V_2))) # label(fact_4099_complex__Im__number__of) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2676 (all R_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),zero_zero_real)) -> (exists N_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_2)),R_3)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_2),one_one_int)))) & is_int(N_2))))) # label(fact_4478_reals__Archimedean__6c__int) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2677 (all K_2 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,K_2)),zero_zero_int))) # label(fact_2984_int__less__0__conv) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2678 (all A all C all B (is_int(C) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B)) | zero_zero_int = C <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),C)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),C)))))) # label(fact_1270_dvd__mult__cancel__right) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2679 (all Y_8 all Z_6 all X_11 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_6),X_11)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_6),Y_8)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_11),Z_6)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Y_8),Z_6)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_11),Y_8)),Z_6)))) # label(fact_2579_div__add) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2680 (all L all U hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,U),L)),one_one_int)) = hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(ord_at875362053st_int(L),U))) # label(fact_5082_card__atLeastAtMost__int) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2681 (all X_24 all N_32 (one_one_rat = X_24 | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_24),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_24),N_32))))) # label(fact_1495_dvd__power) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2682 (all R_3 all S_1 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,ratreal(hAPP_int_rat(number_number_of_rat,R_3))),ratreal(hAPP_int_rat(number_number_of_rat,S_1))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(number267125858f_real,R_3)),hAPP_int_real(number267125858f_real,S_1))) # label(fact_4714_Ratreal__number__of__quotient) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2683 (all X_1 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X_1)),hAPP_real_real(exp_real,Y_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2)))) # label(fact_3825_exp__less__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2684 (all M all N hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),N)),N)) # label(fact_4968_gcd__add1__int) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2685 (all Ma all K_1 all N_1 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_1),N_1)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_1),hAPP_C498520661umeral(div_mo1740067990umeral(Ma),N_1))) <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_1),Ma))))) # label(fact_2441_dvd__mod__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2686 (all N_43 all A_131 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_131)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_131),N_43)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_131),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_131),N_43)))))) # label(fact_1424_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2687 (all A all B (one_one_complex = hAPP_real_complex(hAPP_r265291036omplex(complex,A),B) <-> A = one_one_real & zero_zero_real = B)) # label(fact_3961_Complex__eq__1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2688 (all X -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),X)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),X)))) # label(fact_1698_dvd_Oless__irrefl) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2689 (all A_58 all B_49 all C_28 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_58),B_49)),C_28) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_58),C_28)),hAPP_C498520661umeral(div_mo1740067990umeral(B_49),C_28))),C_28)) # label(fact_2387_mod__mult__eq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2690 (all N all K_2 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N)))))) # label(fact_4321_gcd__greatest__nat) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2691 (all C all A all B (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),B))))) # label(fact_2069_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2692 (all M all K_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),K_2)) -> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M))),K_2) = one_one_nat)) # label(fact_3184_Suc__times__mod__eq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2693 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)))))) # label(fact_5011_transfer__nat__int__gcd__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2694 (all N hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N))))) # label(fact_3768_real__of__nat__fact__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2695 (all A all E_2 all C all B all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),E_2)),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),E_2)),D))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B)),E_2)),C)),D)))) # label(fact_1560_le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2696 (all A_3 all B_2 (is_int(B_2) -> (B_2 != zero_zero_int -> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),divmod_int(A_3,B_2)))))) # label(fact_4192_divmod__int__correct) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2697 (all X hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(sin,hAPP_real_real(uminus_uminus_real,X))),hAPP_real_real(sin,X))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(cos,hAPP_real_real(uminus_uminus_real,X))),hAPP_real_real(cos,X))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_real) # label(fact_3742_sin__cos__minus) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2698 (all Z_1 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Z_1),hAPP_complex_complex(cnj,Z_1)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_complex_real(im,Z_1)))),ii)) # label(fact_4090_complex__diff__cnj) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2699 (all X_1 hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),sin)),sin),X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X_1))),hAPP_real_real(sin,X_1))))) # label(fact_4790_DERIV__sin__sin__mult2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2700 (all F all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) -> ((all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),B)) -> hBOOL(hAPP_real_bool(isCont_real_real(F),X_2)))) -> ((all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),X_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),B)) -> hBOOL(hAPP_real_bool(deriv_real(F,X_2),zero_zero_real)))) -> (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),B)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X_2)) -> hAPP_real_real(F,X_2) = hAPP_real_real(F,A))))))) # label(fact_5179_DERIV__isconst1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2701 (all X all Y hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(cos,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(sin,Y)))) # label(fact_3665_sin__add) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2702 (all Lx_1 all Rx_1 all Ry_1 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_1),Rx_1)),Ry_1) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_1),Ry_1))) # label(fact_499_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2703 (all A_134 all N_47 all B_103 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_134),N_47)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_103),N_47))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_103)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_134),B_103))))) # label(fact_1400_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2704 (all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(fact,N)))) # label(fact_4564_fact__lt) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2705 (all A_111 all B_89 all C_49 (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_111),C_49) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_111),B_89) -> C_49 = B_89)) # label(fact_1918_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2706 (all X_21 all N_31 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31)) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_21),N_31) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21))) # label(fact_1799_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2707 (all P_1 all N_1 all K_1 (is_int(K_1) & is_int(N_1) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),zero_zero_int)) -> (all I_2 all J_1 (is_int(J_1) & is_int(I_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),J_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_1),zero_zero_int)) & hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) = N_1 -> hBOOL(hAPP_int_bool(P_1,I_2)))))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1)) -> (all I_2 all J_1 (is_int(I_2) & is_int(J_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),J_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,J_1),K_1)) & N_1 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) -> hBOOL(hAPP_int_bool(P_1,I_2)))))) & (zero_zero_int = K_1 -> hBOOL(hAPP_int_bool(P_1,zero_zero_int))) <-> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N_1),K_1)))))) # label(fact_2654_split__zdiv) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2708 (all Ma all N_1 (is_int(Ma) & is_int(N_1) -> (N_1 != zero_zero_int | zero_zero_int != Ma <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Ma),N_1)))))) # label(fact_4995_gcd__pos__int) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2709 (all A (zero_zero_rat = A <-> zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A))) # label(fact_248_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2710 (all W_17 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_int_real(number267125858f_real,W_17)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,pls)),W_17))) # label(fact_41_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2711 (all D all B all C all A (A != zero_zero_nat -> (C != zero_zero_nat -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C),D)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,D),C) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B),A) <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B),C) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),D))))))) # label(fact_2639_dvd__div__div__eq__mult) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2712 (all A_133 all N_46 all N_45 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_46),N_45)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_133)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_133),one_on1684967323de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_133),N_45)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_133),N_46))))))) # label(fact_1406_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 7.48/7.50 2713 (all A_218 all B_161 all C_90 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_218),B_161)),C_90) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_218),C_90)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_161),C_90))) # label(fact_629_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2714 (all M hAPP_int_int(legacy_zgcd(zero_zero_int),M) = hAPP_int_int(abs_abs_int,M)) # label(fact_4506_zgcd__0__left) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2715 (all A_63 zero_zero_nat = hAPP_nat_nat(div_mod_nat(A_63),A_63)) # label(fact_2365_mod__self) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2716 (all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_1),one_one_int)) <-> hAPP_int_int(abs_abs_int,X_1) = one_one_int)) # label(fact_3425_zdvd1__eq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2717 (all X_30 all Y_24 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_30),Y_24)) -> (X_30 != Y_24 -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_30),Y_24))))) # label(fact_816_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2718 (all B_2 all B_5 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_5),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_5)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))))))) # label(fact_2598_zdiv__mono2__neg) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2719 (all X_1 (hAPP_nat_nat(div_mod_nat(X_1),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = hAPP_nat_nat(suc,zero_zero_nat) <-> -hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)))) # label(fact_3775_odd__nat__equiv__def) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2720 (all A_3 all P_3 ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),A_3)) -> hAPP_int_int(number_number_of_int,min) = hAPP_int_int(legendre(A_3),P_3)) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),A_3)) -> hAPP_int_int(legendre(A_3),P_3) = one_one_int)) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3)) -> zero_zero_int = hAPP_int_int(legendre(A_3),P_3)))) # label(fact_1686_Legendre__def) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2721 (all X_1 all N_1 (-hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1)) -> (zero_zero_real != X_1 -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root,N_1),X_1),hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,zero_zero_nat)))))))))) # label(fact_3941_DERIV__odd__real__root) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2722 (all C_75 all A_175 all B_135 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_175),B_135)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_75)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_75),A_175)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_75),B_135)))))) # label(fact_1075_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2723 (all A_221 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_221),zero_zero_real) = zero_zero_real) # label(fact_592_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2724 (all A_87 all B_75 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_87),B_75)),B_75) = A_87) # label(fact_2137_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2725 (all B_148 all A_192 all C_84 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_192),C_84)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_192),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_148),C_84))))) # label(fact_932_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2726 (all M_28 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),one_one_complex)),M_28) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,M_28),M_28)) # label(fact_695_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2727 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N))))) # label(fact_3213_le__div__geq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2728 (all Nat hAPP_nat_nat(size_size_nat,hAPP_nat_nat(suc,Nat)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(size_size_nat,Nat)),hAPP_nat_nat(suc,zero_zero_nat))) # label(fact_4057_nat_Osize_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2729 (all B_2 all D_3 all A_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),A_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),B_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)))))) # label(fact_4011_coprime__mul) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2730 (all X_47 all Y_35 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_47),X_47)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_35),Y_35))))) # label(fact_94_sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2731 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N)) # label(fact_4332_gcd__add2__nat) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2732 (all A_47 all B_38 all C_19 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_47),B_38)),C_19) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_47),C_19)),B_38)),C_19)) # label(fact_2423_mod__add__left__eq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2733 (all A_140 all C_62 all B_109 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_140),C_62)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_109),C_62))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_62)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_140),B_109))))) # label(fact_1343_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2734 (all N all P_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),P_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_3),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(fact,N)))))) # label(fact_4552_divides__fact) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2735 (all X_1 all A_1 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_1)) <-> hBOOL(hAPP_nat_bool(A_1,X_1)))) # label(fact_141_mem__def) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2736 (all B_93 all A_115 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_115),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_93),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_115),B_93)),zero_zero_int))))) # label(fact_1882_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2737 (all B_2 negDivAlg(hAPP_int_int(number_number_of_int,min),B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_2),one_one_int))) # label(fact_2665_negDivAlg__minus1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2738 (all A_232 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_232),hAPP_int_real(number267125858f_real,pls)) = A_232) # label(fact_327_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2739 (all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(one_one_int),hAPP_int_int(number_number_of_int,min)),M)))) # label(fact_1501_one__not__neg__one__mod__m) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2740 (all Z_8 all X_23 all Y_18 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_23),Y_18)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_23),Z_8)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_23),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Y_18),Z_8)))))) # label(fact_1518_dvd__diff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2741 (all A all E_2 all C all B all D (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B),A)),E_2)),D))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),E_2)),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),E_2)),D))))) # label(fact_1568_less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2742 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(fact_fact_nat,N)),hAPP_nat_nat(fact_fact_nat,M))))) # label(fact_3809_fact__dvd) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2743 (all A_129 all N_40 all N_39 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_40),N_39)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_129)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_129),N_40)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_129),N_39)))))) # label(fact_1443_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2744 (all B_95 all C_53 all A_117 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_117)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_95),C_53)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_95),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_117),C_53)))))) # label(fact_1869_add__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2745 (all B all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int))) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B)))))) # label(fact_2225_wset__g__1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2746 (all X hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X)) = hAPP_complex_real(im,hAPP_complex_complex(cnj,X))) # label(fact_4093_complex__Im__cnj) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2747 (all N all M (is_int(M) & is_int(N) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),N)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),M)) -> M = N)))))) # label(fact_455_zdvd__antisym__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2748 (all K_2 all M all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M),N))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),N)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M))))) # label(fact_1526_zdvd__zdiffD) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2749 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arctan,Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arctan,Y))))) # label(fact_3502_arctan__bounded) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2750 (all P all Q all R hAPP_int_bool(hAPP_i1948725293t_bool(P,R),Q) = hAPP_int_bool(hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2751 (all X_1 all Y_2 (Y_2 = X_1 <-> hAPP_complex_complex(cnj,Y_2) = hAPP_complex_complex(cnj,X_1))) # label(fact_4072_complex__cnj__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2752 (all X all K_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),K_2)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),K_2))) # label(fact_3909_real__root__power) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2753 (all N_11 all A_7 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_7)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_7),one_one_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_7),hAPP_nat_nat(suc,N_11))),one_one_int))))) # label(fact_3164_power__Suc__less__one) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2754 (all V_23 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_23)),one_one_real) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_23),hAPP_int_int(bit1,pls)))) # label(fact_45_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2755 (all N hAPP_int_nat(number_number_of_nat,N) = natceiling(hAPP_int_real(number267125858f_real,N))) # label(fact_3392_natceiling__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2756 (all M all K_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(fact_fact_int,M)),hAPP_int_int(fact_fact_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),K_2)))))) # label(fact_4283_fact__mono__int__aux) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2757 (all C_7 all A_22 all B_18 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_22),B_18)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_18),A_22)),C_7) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_18),C_7)),A_22))) # label(fact_2563_dvd__div__mult) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2758 (all K_2 all L_2 hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),L_2)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,K_2)),hAPP_int_int(bit0,L_2))) # label(fact_353_add__Bit1__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2759 (all W hAPP_int_int(number_number_of_int,hAPP_int_int(uminus_uminus_int,W)) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(number_number_of_int,W))) # label(fact_3516_minus__numeral__code_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2760 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natfloor(X)),natfloor(Y))))) # label(fact_3386_natfloor__mono) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2761 (all B_5 all A_5 all A_3 all B_2 (zero_zero_int != hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) -> (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_5),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)) = A_3 -> (B_2 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)) -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_5),B_5) = one_one_int)))) # label(fact_4965_gcd__coprime__int) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2762 (all N_29 all K_6 all M_16 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_6),M_16)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_6),N_29)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_6),hAPP_Q1762011733de_int(div_mo231679042de_int(M_16),N_29)))))) # label(fact_2450_dvd__mod) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2763 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N))))) # label(fact_2860_less__SucI) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2764 (all A_239 all B_167 all V_15 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_239),B_167)),hAPP_i769753017umeral(number1443263063umeral,V_15)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_239),hAPP_i769753017umeral(number1443263063umeral,V_15))),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_167),hAPP_i769753017umeral(number1443263063umeral,V_15)))) # label(fact_183_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2765 (all X_1 (hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1)))) # label(fact_3784_nat__even__iff__2__dvd) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2766 (all X_2 hAPP_nat_real(diffs_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))),X_2) = hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,X_2)))) # label(fact_4876_exp__fdiffs) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2767 (all K_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_2),K_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,I)),K_2))))) # label(fact_2858_less__trans__Suc) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2768 (all A_122 all B_100 all V_6 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_122),B_100)),hAPP_int_rat(number_number_of_rat,V_6)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_122),hAPP_int_rat(number_number_of_rat,V_6))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_100),hAPP_int_rat(number_number_of_rat,V_6)))) # label(fact_1551_left__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2769 (all X_34 all P_7 all Q_10 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_34),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_10)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_34),P_7)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_34),Q_10))) # label(fact_671_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2770 (all B_1_1 all B_2_1 all B_3_1 is_bool(tendsto_nat_complex(B_1_1,B_2_1,B_3_1))) # label(gsy_c_Limits_Otendsto_000tc__Nat__Onat_000tc__Complex__Ocomplex) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2771 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int)) -> hAPP_P1175774780nt_int(product_snd_int_int,hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)))) = hAPP_int_int(div_mod_int(A_3),B_2)))) # label(fact_4237_mod__pos__neg) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2772 (all N N = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),N)) # label(fact_1657_nat__mult__1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2773 (all X hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),natceiling(X)))) # label(fact_3390_zero__le__natceiling) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2774 (all N_42 all A_130 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_130)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_130),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_130),N_42)))))) # label(fact_1426_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2775 (all P all Q (-hBOOL(hAPP_bool_bool(fimplies(P),Q)) | -hBOOL(P) | hBOOL(Q))) # label(help_fimplies_3_1_U) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2776 (all W hAPP_int_rat(hAPP_int_fun_int_rat(fract,W),one_one_int) = hAPP_int_rat(number_number_of_rat,W)) # label(fact_4681_rat__number__of__def) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2777 (all X_1 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),field_1210416355s_real)) -> -(all M_1 all N_2 (zero_zero_nat != N_2 -> (hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,M_1)),hAPP_nat_real(real_nat,N_2)) = hAPP_real_real(abs_abs_real,X_1) -> one_one_nat != hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M_1),N_2)))))) # label(fact_4630_Rats__abs__nat__div__natE) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2778 (all A_237 (is_int(A_237) -> A_237 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,pls))),A_237))) # label(fact_235_mult__numeral__1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2779 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,ratreal(X)),ratreal(Y)) = ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,X),Y))) # label(fact_4704_real__divide__code) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2780 (all A_149 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_149),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_149),one_one_int)))) # label(fact_1257_less__add__one) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2781 (all A_35 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_35),one_on1684967323de_int) = A_35) # label(fact_2481_div__by__1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2782 (all X_1 all N_1 (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,N_1))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,N_1)),natfloor(X_1))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,N_1)),X_1)))))) # label(fact_3410_le__natfloor__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2783 (all X_1 hBOOL(hAPP_real_bool(deriv_real(arctan,X_1),hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3879_DERIV__arctan) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2784 (all A_167 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_167),A_167)))) # label(fact_1122_zero__le__square) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2785 (all A all B (B = A <-> hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B) = zero_zero_real)) # label(fact_2077_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2786 (all Z_1 zero_zero_int != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z_1)),Z_1)) # label(fact_357_odd__nonzero) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2787 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X)),hAPP_real_real(exp_real,Y))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y)))) # label(fact_3824_exp__less__cancel) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2788 (all A_203 zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_203),zero_zero_real)) # label(fact_841_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2789 (all A_3 -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),one_one_nat)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A_3)))) # label(fact_2312_gcd__lcm__complete__lattice__nat_Onot__less__bot) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2790 (all N all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),X))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(real_nat,X))))) # label(fact_3672_real__of__nat__div4) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2791 (all P all Q all R hAPP_real_real(hAPP_f1122023986l_real(hAPP_f1786887945l_real(cOMBB_1896127834l_real,P),Q),R) = hAPP_f352196356l_real(P,hAPP_r474017924t_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__Real) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2792 (all M_19 all N_44 all A_132 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_132)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_132),M_19)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_132),N_44))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_19),N_44))))) # label(fact_1413_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2793 (all X_1 (X_1 = zero_z126310315umeral <-> X_1 = zero_z126310315umeral)) # label(fact_1897_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2794 (all Z_1 (is_int(Z_1) -> Z_1 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),Z_1))) # label(fact_268_zadd__0) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2795 (all X hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3737_cos__double) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2796 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> (exists K N = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K))))) # label(fact_3262_less__imp__Suc__add) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2797 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int)) -> hAPP_P1175774780nt_int(product_snd_int_int,hAPP_P1975530577nt_int(negateSnd,posDivAlg(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)))) = hAPP_int_int(div_mod_int(A_3),B_2)))) # label(fact_4238_mod__neg__neg) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2798 (all K_1 all P_1 all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D)) -> ((all X_2 (is_int(X_2) -> (hBOOL(hAPP_int_bool(P_1,X_2)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_1)) -> (all X_2 (hBOOL(hAPP_int_bool(P_1,X_2)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),D)))))))))) # label(fact_1690_decr__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2799 (all B_148 all A_192 all C_84 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_192),C_84)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_192),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_148),C_84))))) # label(fact_928_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2800 (all Lx_2 all Ly all Rx_2 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_2),Ly)),Rx_2) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_2),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Ly),Rx_2))) # label(fact_496_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2801 (all X_7 all Q_6 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_7),hAPP_nat_nat(suc,Q_6)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_7),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_7),Q_6))) # label(fact_3029_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2802 (all V_2 hAPP_int_real(real_int,hAPP_int_int(number_number_of_int,V_2)) = hAPP_int_real(number267125858f_real,V_2)) # label(fact_4394_real__number__of) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2803 (all A_234 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_234),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_234),A_234)) # label(fact_299_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2804 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,X)),X)))) # label(fact_3560_ln__bound) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2805 (all C_57 all D_10 all A_135 all B_104 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_135),B_104)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_57),D_10)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_104)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_57)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_135),C_57)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_104),D_10)))))))) # label(fact_1373_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2806 (all B_93 all A_115 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_115),zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_93),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_115),B_93)),zero_zero_nat))))) # label(fact_1883_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2807 (all X_1 hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),cos)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1),hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(cos,X_1))),hAPP_real_real(sin,X_1)))))) # label(fact_4798_DERIV__cos__realpow2b) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2808 (all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),hAPP_nat_nat(fact,N)))) # label(fact_4567_fact__le) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2809 (all M all V_2 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),M)),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,M)))),hAPP_int_nat(number_number_of_nat,V_2))) # label(fact_3208_Suc__mod__eq__add3__mod__number__of) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2810 (all X_21 all N_31 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31)) -> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_21),N_31) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21))) # label(fact_1802_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2811 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real)) -> hBOOL(hAPP_real_bool(isCont_real_real(arcsin),X_1))))) # label(fact_5173_isCont__arcsin) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2812 (all Z1 all Z2 all Z3 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)),Z3) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z2),Z3))) # label(fact_197_zadd__assoc) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2813 (all A_221 zero_z126310315umeral = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_221),zero_z126310315umeral)) # label(fact_593_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2814 (all X_1 all Y_2 (is_int(Y_2) & is_int(X_1) -> (hAPP_int_int(number_number_of_int,Y_2) = hAPP_int_int(number_number_of_int,X_1) <-> Y_2 = X_1))) # label(fact_166_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2815 (all Y_6 all P_6 all N_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10)) -> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6))) # label(fact_3172_lemma__realpow__diff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2816 (all A_3 all B_2 all C_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1)) -> hAPP_int_int(div_mod_int(A_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),C_1))),hAPP_int_int(div_mod_int(A_3),B_2)))) # label(fact_2503_zmod__zmult2__eq) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2817 (all A (hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially)) -> (hBOOL(monoseq_real(A)) -> hBOOL(tendsto_nat_real(hAPP_f292751352t_real(hAPP_f1607377819t_real(cOMBB_1533963633al_nat,hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_at4362885an_nat(zero_zero_nat)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),sequentially))))) # label(fact_5032_summable__Leibniz_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2818 (all A_3 hAPP_real_real(hAPP_r1250527377l_real(powr,one_one_real),A_3) = one_one_real) # label(fact_4380_powr__one__eq__one) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2819 (all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),B)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),zero_zero_real)))) # label(fact_1117_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2820 (all I all J_2 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(legacy_zgcd(I),J_2)),I))) # label(fact_4483_zgcd__zdvd1) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2821 (all K_2 all N all M hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N)),M)),N) = hAPP_nat_nat(div_mod_nat(M),N)) # label(fact_2713_mod__mult__self3) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2822 (all R_3 exists X_2 (R_3 = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_P1175774780nt_int(product_fst_int_int,X_2)),hAPP_P1175774780nt_int(product_snd_int_int,X_2)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_P1175774780nt_int(product_snd_int_int,X_2))) & (all Y_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_P1175774780nt_int(product_snd_int_int,Y_1))) & hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,Y_1)),hAPP_P1175774780nt_int(product_snd_int_int,Y_1)) = one_one_int & hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_P1175774780nt_int(product_fst_int_int,Y_1)),hAPP_P1175774780nt_int(product_snd_int_int,Y_1)) = R_3 -> X_2 = Y_1)) & hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,X_2)),hAPP_P1175774780nt_int(product_snd_int_int,X_2)) = one_one_int)) # label(fact_5024_quotient__of__unique) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2823 (all N_59 all M_24 all X_28 all Y_22 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_28),Y_22)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_59),M_24)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_28),N_59)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_22),M_24)))))) # label(fact_1002_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 7.48/7.51 2824 (all A hBOOL(hAPP_real_bool(isCont_real_real(abs_abs_real),A))) # label(fact_5168_isCont__abs) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2825 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real)) -> Y = hAPP_real_real(sin,hAPP_real_real(arcsin,Y)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arcsin,Y)),pi)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y)))))) # label(fact_3744_arcsin__pi) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2826 (all K_1 all Ma all N_1 ((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N_1))))) # label(fact_1683_mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2827 (all N_1 all Diff all F all H (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H)) -> (hAPP_n546711566l_real(Diff,zero_zero_nat) = F -> ((all M_1 all T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))))) -> (exists T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) & hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N_1),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N_1))) = hAPP_real_real(F,H) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),T)))))))) # label(fact_4937_Maclaurin2) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2828 (all C_41 all A_99 all B_81 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_41),A_99)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_41),B_81))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_99),B_81)))) # label(fact_2035_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2829 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(div_mod_int(A_3),B_2))))) # label(fact_2615_pos__mod__sign) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2830 (all N_27 all Y_7 all X_10 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,Y_7),X_10)) -> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,X_10),Y_7)),N_27) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_10),N_27)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_7),N_27)))) # label(fact_2582_div__power) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2831 (all N_1 hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),N_1)))))) # label(fact_4751_bdd__int__set__le__finite) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2832 (all I all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,I))),N)))) # label(fact_3133_diff__Suc__less) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2833 (all Ma all N_1 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,Ma))))) # label(fact_2865_not__less__eq) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2834 (all X_3 (hBOOL(cauchy_complex(X_3)) -> hBOOL(cauchy_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_3))))) # label(fact_4758_Re_OCauchy) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2835 (all X_46 all Y_34 (hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_46),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_34),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),X_46)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),Y_34)) -> X_46 = Y_34)))) # label(fact_205_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2836 (all B_141 all A_181 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_141),zero_zero_nat)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_181)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_181),zero_zero_nat)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_141)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_181),B_141)),zero_zero_nat)))) # label(fact_1045_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2837 (all Q all P (-hBOOL(P) | hBOOL(fdisj(P,Q)))) # label(help_fdisj_1_1_U) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2838 (all D_1 all X_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (X_1 != zero_zero_real -> ((hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,zero_zero_nat))))) = D_1)) -> ((hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real)) -> hAPP_real_real(uminus_uminus_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,zero_zero_nat)))))) = D_1)) -> ((-hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1)) -> hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,zero_zero_nat))))) = D_1) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root,N_1),X_1),D_1)))))))) # label(fact_3894_DERIV__real__root__generic) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2839 (all A all B all C (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B),A))))) # label(fact_1379_mult__le__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2840 (all B_73 all A_85 all C_36 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_36)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_73),A_85)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_73),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_85),C_36)))))) # label(fact_2147_add__increasing2) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2841 (all N all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,X)),one_one_real))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N),X)))) # label(fact_4447_lemma__floor2) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2842 (all N_1 ((exists M_1 N_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M_1)) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1)))) # label(fact_3785_even__mult__two__ex) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2843 (all A all B (is_int(A) & is_int(B) -> (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(norm_frac_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A),B))) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),zero_zero_int)) -> (-(A = zero_zero_int | zero_zero_int = B) -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A),hAPP_int_int(hAPP_int_fun_int_int(int_gcd,A),B))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B),hAPP_int_int(hAPP_int_fun_int_int(int_gcd,A),B))) = norm_frac(A,B)) & (zero_zero_int = B | zero_zero_int = A -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int) = norm_frac(A,B))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),zero_zero_int)) -> norm_frac(A,B) = norm_frac(hAPP_int_int(uminus_uminus_int,A),hAPP_int_int(uminus_uminus_int,B)))))) # label(fact_4804_norm__frac_Opsimps) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2844 (all R_3 all X hAPP_complex_real(re,hAPP_complex_complex(scaleR1652505878omplex(R_3),X)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_complex_real(re,X))) # label(fact_5109_complex__Re__scaleR) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2845 (all X_6 all Q_5 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_6),hAPP_nat_nat(suc,Q_5)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_6),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_6),Q_5))) # label(fact_3035_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2846 (all A_3 all B_2 (is_int(B_2) & is_int(A_3) -> (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) != zero_zero_int -> (exists A_4 exists B_3 (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_4),B_3) = one_one_int & hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_3),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)) = B_2 & A_3 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_4),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)) & is_int(B_3) & is_int(A_4)))))) # label(fact_5118_gcd__coprime__exists__int) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2847 (all C_37 all A_95 all B_77 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_95),B_77)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_95),C_37)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_77),C_37))))) # label(fact_2059_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2848 (all D int_ge_less_than2(D) = collec1347809874nt_int(produc93441119t_bool(hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,D)))),ord_less_int)))) # label(fact_4862_int__ge__less__than2__def) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2849 (all X_7 all Q_6 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_7),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_7),Q_6)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_7),hAPP_nat_nat(suc,Q_6))) # label(fact_3030_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2850 (all Z_2 (is_int(Z_2) -> (hAPP_int_int(abs_abs_int,Z_2) = one_one_int <-> Z_2 = hAPP_int_int(number_number_of_int,min) | Z_2 = one_one_int))) # label(fact_3437_abs__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2851 (all A_219 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_219),zero_z891286103de_int) = A_219) # label(fact_609_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2852 (all N hAPP_C498520661umeral(hAPP_C1594335432umeral(minus_1690775515umeral,code_S1047413653umeral(N)),one_on1645066479umeral) = N) # label(fact_4085_Suc__code__numeral__minus__one) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2853 (all V_22 all V_21 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_22)) -> hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_21),V_22)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_21)),hAPP_int_real(number267125858f_real,V_22))))) # label(fact_76_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2854 (all L all U hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_gr1297742076an_int(L),U)))) # label(fact_4629_finite__greaterThanLessThan__int) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2855 (all X all Y hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,X)),hAPP_int_real(real_int,Y))) # label(fact_4389_real__of__int__diff) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2856 (all Y_20 all X_26 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_26)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_20)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_20),one_one_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_26),Y_20)),X_26)))))) # label(fact_1387_mult__right__le__one__le) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2857 (all X_1 all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_2)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,X_1)),hAPP_int_rat(number_number_of_rat,Y_2))))) # label(fact_60_le__number__of) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2858 (all Z_1 all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z_1)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),X)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z_1))))) # label(fact_1725_dvd_Oless__le__trans) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2859 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X))),X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3414_abs__ln__one__plus__x__minus__x__bound) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2860 (all A_24 all B_19 all C_8 ((zero_z126310315umeral != C_8 -> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_24),B_19) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_8),A_24)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_8),B_19))) & (C_8 = zero_z126310315umeral -> zero_z126310315umeral = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_8),A_24)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_8),B_19))))) # label(fact_2551_div__mult__mult1__if) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2861 (all X_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (one_one_real = X_1 <-> one_one_real = hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)))) # label(fact_3911_real__root__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2862 (all C all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A),B)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),A)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),B))))) # label(fact_2024_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2863 (all X ((zero_zero_real != X -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> one_one_real = hAPP_real_real(sgn_sgn_real,X)) & (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_real_real(uminus_uminus_real,one_one_real) = hAPP_real_real(sgn_sgn_real,X))) & (zero_zero_real = X -> zero_zero_real = hAPP_real_real(sgn_sgn_real,X)))) # label(fact_3575_real__sgn__def) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2864 (all R_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R_3)) -> (exists N_2 (is_int(N_2) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_2)),R_3)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_2),one_one_int)))))))) # label(fact_4477_reals__Archimedean__6b__int) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2865 (all A_228 all C_94 all D_19 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_228),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_94),D_19)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_228),C_94)),D_19)) # label(fact_535_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2866 (all P all Q all R hAPP_real_bool(P,hAPP_real_real(Q,R)) = hAPP_real_bool(hAPP_f1402056515l_bool(hAPP_f824790735l_bool(cOMBB_real_bool_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__HOL__Obool_000tc__RealDef__Ore) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2867 (all C_80 all A_187 all B_143 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_187),B_143)) -> (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_187),C_80)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_187),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_143),C_80)))))) # label(fact_964_dvd__add) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2868 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_real_real(log(hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,hAPP_real_real(exp_real,one_one_real))),hAPP_real_real(ln,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_real_real(ln,X)))) # label(fact_3860_log__base__10__eq1) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2869 (all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_int_real(real_int,N_1))))) # label(fact_4439_real__of__int__gt__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2870 (all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1)) -> hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),N_1)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat,N_1)),one_one_nat))) # label(fact_4774_card__bdd__int__set__le) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2871 (all C all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),B))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)))) # label(fact_2067_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2872 (all N_49 all A_147 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),A_147)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_147),N_49))))) # label(fact_1279_one__le__power) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2873 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(exp_real,X)))) # label(fact_3834_exp__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2874 (all K_2 all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N))) # label(fact_1650_add__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2875 (all C all A all B (C = zero_zero_rat | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A),B)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B))))) # label(fact_1265_dvd__mult__cancel__left) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2876 (all A_121 all C_54 all B_99 all D_9 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_121),B_99)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C_54),D_9)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_121),C_54)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_99),D_9))) # label(fact_1808_add__diff__add) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2877 (all Y all X (hBOOL(hAPP_int_bool(nat_is_nat,X)) -> (hBOOL(hAPP_int_bool(nat_is_nat,Y)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)))))) # label(fact_5146_transfer__int__nat__gcd__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2878 (all X_1 all A_1 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),A_1)) <-> hBOOL(hAPP_real_bool(A_1,X_1)))) # label(fact_142_mem__def) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2879 (all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_int_real(number267125858f_real,Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Y_2)))) # label(fact_101_le__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2880 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N))) -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> N = M))) # label(fact_2857_less__SucE) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2881 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),hAPP_int_int(bit1,K_1))))) # label(fact_776_rel__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2882 (all X_1 (zero_zero_real = X_1 <-> one_one_real = hAPP_real_real(exp_real,X_1))) # label(fact_3836_exp__eq__one__iff) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2883 (all A_182 A_182 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_182),one_one_nat)) # label(fact_1025_power__one__right) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2884 (all A_198 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,zero_zero_real),A_198)) -> zero_zero_real = A_198)) # label(fact_885_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2885 (all V_2 all W hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_2),hAPP_int_int(uminus_uminus_int,W))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_int(number_number_of_int,W))) # label(fact_3543_minus__numeral__code_I6_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2886 (all A_3 all B_2 (is_int(B_2) & is_int(A_3) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int)) -> (A_3 = zero_zero_int | B_2 = zero_zero_int -> norm_frac(A_3,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int)) & (-(B_2 = zero_zero_int | zero_zero_int = A_3) -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(int_gcd,A_3),B_2))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(int_gcd,A_3),B_2))) = norm_frac(A_3,B_2))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int)) -> norm_frac(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)) = norm_frac(A_3,B_2)))) # label(fact_4807_norm__frac_Osimps) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2887 (all X_1 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2)) -> (exists X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),X_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2)) & hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),field_1210416355s_real)))))) # label(fact_4637_Rats__dense__in__real) # label(axiom) # label(non_clause). [assumption]. 7.54/7.51 2888 (all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) -> X != Y)) # label(fact_1721_dvd_Oless__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2889 (all B_1_1 all B_2_1 is_bool(hAPP_P62333565t_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__prod_Itc__) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2890 (all M_7 all N_21 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(semiri1619134803umeral,M_7)),hAPP_n1108039445umeral(semiri1619134803umeral,N_21)) = hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21))) # label(fact_2960_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2891 (all B_2 all D_3 all A_3 all C_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),C_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),D_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,C_1),D_3)))))) # label(fact_4318_gcd__semilattice__nat_Oinf__mono) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2892 (all A_3 all B_2 all C_1 hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)),C_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),C_1))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(div_mod_int(B_2),C_1))),C_1))) # label(fact_2359_zdiv__zmult1__eq) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2893 (all Ma all Ta all K_1 (is_int(K_1) -> (zero_zero_int != K_1 -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),Ta)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),Ma)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),Ta))))))) # label(fact_1674_zdvd__mono) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2894 (all X_37 X_37 = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_37),one_one_nat)) # label(fact_567_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2895 (all B_36 all A_45 hAPP_Q1762011733de_int(div_mo231679042de_int(A_45),B_36) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_36),A_45)),B_36)) # label(fact_2432_mod__add__self1) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2896 (all N all A_3 all B_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),N)) -> (one_one_int = hAPP_int_int(legacy_zgcd(M),N) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N))))))) # label(fact_4515_zcong__zgcd__zmult__zmod) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2897 (all K_2 all L_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),L_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),L_2)))))) # label(fact_1656_mult__le__mono) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2898 (all X_24 all N_32 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32)) | X_24 = one_one_real -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_24),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_24),N_32))))) # label(fact_1491_dvd__power) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2899 (all A_6 all N_8 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_6),N_8)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_6),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_8)))) # label(fact_3230_power__odd__eq) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2900 (all N all M hAPP_Q1256397320de_int(hAPP_Q952608535de_int(produc1318306967de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,N),M)),hAPP_Q1762011733de_int(div_mo231679042de_int(N),M)) = quickc495462417de_int(N,M)) # label(fact_4268_div__mod__code__int__def) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2901 (all F all K_1 all A all B (B != A -> ((all X_2 hBOOL(hAPP_real_bool(deriv_real(F,X_2),K_1))) -> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A)),K_1) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B)),hAPP_real_real(F,A))))) # label(fact_3892_DERIV__const__ratio__const) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2902 (all U all Ma all N_1 all J all I_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J)),U)),Ma)),N_1))))) # label(fact_1854_nat__le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2903 (all P_1 all I_1 all J (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J)) -> (hBOOL(hAPP_nat_bool(P_1,J)) -> ((all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J)) -> (hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,I_2))) -> hBOOL(hAPP_nat_bool(P_1,I_2))))) -> hBOOL(hAPP_nat_bool(P_1,I_1)))))) # label(fact_3254_inc__induct) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2904 (all X_1 (one_one_real = X_1 <-> hAPP_real_real(sqrt,X_1) = one_one_real)) # label(fact_3320_real__sqrt__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2905 (all Z_2 all P_1 ((all N_2 hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,N_2)))) -> ((all N_2 hBOOL(hAPP_int_bool(P_1,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_2)))))) -> hBOOL(hAPP_int_bool(P_1,Z_2))))) # label(fact_4548_int__of__nat__induct) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2906 (all A_3 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(suc,zero_zero_nat)),A_3))) # label(fact_4022_coprime__Suc0_H) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2907 (all X_30 all Y_24 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,X_30),Y_24)) -> (Y_24 != X_30 -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,X_30),Y_24))))) # label(fact_813_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2908 (all B_92 all A_114 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_114),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_92),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_114),B_92)),zero_zero_real))))) # label(fact_1890_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2909 (all Z_1 all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_1),X)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_1),Y))))) # label(fact_1732_real__add__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2910 (all A_47 all B_38 all C_19 hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_47),C_19)),B_38)),C_19) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_47),B_38)),C_19)) # label(fact_2424_mod__add__left__eq) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2911 (all B_28 all A_34 zero_zero_int = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_28),A_34)),B_28)) # label(fact_2491_mod__mult__self1__is__0) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2912 (all A all B all Q_1 all R_2 (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A,B),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_2))) -> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(hAPP_int_int(uminus_uminus_int,A),B),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,R_2),zero_zero_int)),hAPP_int_int(uminus_uminus_int,Q_1)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(uminus_uminus_int,Q_1)),one_one_int))),hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,R_2),zero_zero_int)),zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B),R_2))))))) # label(fact_3556_zminus1__lemma) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2913 (all P all Q all R hAPP_rat_bool(P,hAPP_int_rat(Q,R)) = hAPP_int_bool(hAPP_f1241350704t_bool(hAPP_f1131424617t_bool(cOMBB_rat_bool_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Rat__Orat_000tc__HOL__Obool_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2914 (all R_3 all A_3 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_real_real(cos,A_3))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_real_real(sin,A_3))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_3727_sin__cos__squared__add2__mult) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2915 (all P_2 all P_1 all X_1 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(P_1) <-> hBOOL(P_2))) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> hBOOL(P_2)) <-> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> hBOOL(P_1))))) # label(fact_1616_imp__le__cong) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2916 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),hAPP_int_int(bit0,K_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K_1)))) # label(fact_779_rel__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2917 (all B_2 all D_3 all A_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),A_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_3),B_2))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),B_2))))) # label(fact_1781_divides__add__revr) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2918 (all A (hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially)) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A,N_2)))) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,hAPP_nat_nat(suc,N_2))),hAPP_nat_real(A,N_2)))) -> hBOOL(tendsto_nat_real(hAPP_f292751352t_real(hAPP_f1607377819t_real(cOMBB_1533963633al_nat,hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_at4362885an_nat(zero_zero_nat)),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),sequentially)))))) # label(fact_5060_summable__Leibniz_H_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2919 (all A_207 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_207),A_207))) # label(fact_796_dvd__refl) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2920 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(nat,X)),hAPP_int_nat(nat,Y)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y)))) # label(fact_2696_nat__div__distrib) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2921 (all P all Q all R hAPP_real_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,P),Q),R) = hAPP_real_real(P,hAPP_real_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__RealDef_) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2922 (all X all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),hAPP_int_int(number_number_of_int,min)),P_3)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),one_one_int),P_3))))) # label(fact_812_zcong__neg__1__impl__ne__1) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2923 (all C all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),B))))) # label(fact_2065_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2924 (all B_72 all C_35 all A_84 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_84)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_72),C_35)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_72),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_84),C_35)))))) # label(fact_2154_add__increasing) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2925 (all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),hAPP_int_int(fact_fact_int,N))))) # label(fact_4282_fact__ge__one__int) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2926 (all Ma all N_1 all A (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A)) -> (hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A),N_1) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A),Ma) <-> Ma = N_1))) # label(fact_1286_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2927 (all A_219 A_219 = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_219),zero_z126310315umeral)) # label(fact_604_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2928 (all B_93 all A_115 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_115),zero_z126310315umeral)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_93),zero_z126310315umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_115),B_93)),zero_z126310315umeral))))) # label(fact_1881_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2929 (all X all Y all C_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_1),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_1),hAPP_real_real(abs_abs_real,Y)))) -> zero_zero_real = X))) # label(fact_3394_rabs__ratiotest__lemma) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2930 (all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2)) -> (exists K (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K)) & J_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K))))) # label(fact_2246_less__imp__add__positive) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2931 (all M hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,M)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,M))) # label(fact_2847_int__Suc) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2932 (all A_12 all N_18 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_12),N_18)),A_12) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_12),hAPP_nat_nat(suc,N_18))) # label(fact_3007_power__Suc2) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2933 (all X_1 (is_int(X_1) -> (zero_zero_int = X_1 <-> X_1 = zero_zero_int))) # label(fact_1894_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2934 (all X all B_2 all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (A_3 != one_one_real -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_2)) -> (one_one_real != B_2 -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,B_2)),hAPP_real_real(ln,A_3))),hAPP_real_real(log(B_2),X)) = hAPP_real_real(log(A_3),X))))))) # label(fact_3876_log__eq__div__ln__mult__log) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2935 (all B_130 all A_170 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_170)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_130),zero_z126310315umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_130),A_170)),zero_z126310315umeral))))) # label(fact_1101_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2936 (all L all U hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,U)),L) = hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(L),U))) # label(fact_5079_card__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2937 (all M hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),M)))) # label(fact_1652_le__square) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2938 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K1)),hAPP_int_int(bit1,K2))))) # label(fact_137_less__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2939 (all Ma all N_1 all A (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A),Ma) <-> Ma = N_1))) # label(fact_1284_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2940 (all P all Q all R hAPP_f729588221t_bool(P,hAPP_r1562208522t_bool(Q,R)) = hAPP_r653725362t_bool(hAPP_f1927674023t_bool(hAPP_f1261108921t_bool(cOMBB_172850899l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__fun_Itc__Nat__Onat_) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2941 (all P_1 all L ((exists X_2 (hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,L),X_2))) & is_int(X_2))) <-> (exists X_2 (is_int(X_2) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,L),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),zero_zero_int))) & hBOOL(hAPP_int_bool(P_1,X_2)))))) # label(fact_1680_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2942 (all X_3 all A (hBOOL(tendsto_nat_complex(X_3,A,sequentially)) -> hBOOL(tendsto_nat_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_3),hAPP_complex_real(re,A),sequentially)))) # label(fact_5056_Re_OLIMSEQ) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2943 (all N_1 hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,div_div_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),zero_zero_real))),hAPP_r474017924t_real(power_power_real,zero_zero_real))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(suc,N_1))) = one_one_real) # label(fact_4913_sumr__cos__zero__one) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2944 (all A_133 all N_46 all N_45 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_46),N_45)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_133)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_133),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_133),N_45)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_133),N_46))))))) # label(fact_1404_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2945 (all X hAPP_int_real(number267125858f_real,X) = hAPP_int_real(ring_1_of_int_real,X)) # label(fact_4582_number__of__real__def) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2946 (all C_80 all A_187 all B_143 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_187),B_143)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_187),C_80)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_187),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_143),C_80)))))) # label(fact_965_dvd__add) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2947 (all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(inv(P_3),A_3)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))))))) # label(fact_2126_inv__less__p__minus__1) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2948 (all N hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)))) # label(fact_1693_two__realpow__ge__one) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2949 (all P all Q all R hAPP_int_real(hAPP_f563293398t_real(hAPP_f134192053t_real(cOMBB_int_real_int,P),Q),R) = hAPP_int_real(P,hAPP_int_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__RealDef__Oreal_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2950 (all X_35 all Y_27 all Q_11 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_35),Y_27)),Q_11) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_35),Q_11)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_27),Q_11))) # label(fact_662_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2951 (all A_92 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_92),one_on1645066479umeral) = A_92) # label(fact_2101_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2952 (all K_2 all L_2 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit0,K_2)),hAPP_int_int(bit0,L_2)) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),L_2))) # label(fact_265_add__Bit0__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2953 (all Ma all N_1 (zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N_1) <-> zero_zero_nat = Ma & zero_zero_nat = N_1)) # label(fact_1613_add__is__0) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2954 (all A all B (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A),B) = one_one_nat <-> (all D_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),B)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),A)) <-> D_2 = hAPP_nat_nat(suc,zero_zero_nat))))) # label(fact_4372_coprime__Suc__0__nat) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2955 (all K_1 all N_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),N_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),Ma)))))) # label(fact_457_zdvd__reduce) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2956 (all C_63 all A_141 all B_110 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_63),A_141)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_63),B_110))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_63)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_141),B_110))))) # label(fact_1339_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2957 (all M_6 all N_20 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,M_6)),N_20) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20))) # label(fact_2983_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2958 (all N_25 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_nat_rat(semiri151668891at_rat,N_25)))) # label(fact_2910_of__nat__0__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2959 (all B adjust(B) = produc1518849193nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,hAPP_f617310012nt_int(hAPP_f197974549nt_int(cOMBB_638531587nt_int,cOMBS_1891347093nt_int),hAPP_f55007289nt_int(hAPP_f747475781nt_int(cOMBB_222539774nt_int,hAPP_f1311564564nt_int(cOMBS_1640193733nt_int,hAPP_f251264923nt_int(hAPP_f279307923nt_int(cOMBB_662120866nt_int,if_Pro1731782967nt_int),hAPP_f429935748t_bool(hAPP_f1512942609t_bool(cOMBB_int_bool_int,hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int)),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,minus_minus_int),B))))),hAPP_f1574055885nt_int(hAPP_f960630521nt_int(cOMBC_1805044090nt_int,hAPP_f142232355nt_int(hAPP_f1377453843nt_int(cOMBB_320287386nt_int,cOMBB_1846624359nt_int),hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,plus_plus_int),hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_int)))),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,minus_minus_int),B))))),hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_4841_adjust__def) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2960 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (hAPP_real_real(sin,X) = zero_zero_real -> (exists N_2 (hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2)) & X = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) # label(fact_3793_sin__zero__lemma) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2961 (all X archim1246769320r_real(ratreal(X)) = archim791455193or_rat(X)) # label(fact_4682_real__floor__code) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2962 (all X_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (zero_zero_real = hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1) <-> X_1 = zero_zero_real))) # label(fact_3904_real__root__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2963 (all A_238 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_238),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_215_zero__le__power2) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2964 (all X_1 all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_2))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2)))) # label(fact_2999_Nat__Transfer_Otransfer__int__nat__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2965 (all Z_12 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Z_12),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Z_12),Z_12)) # label(fact_291_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2966 (all Z_1 zero_zero_real = hAPP_complex_real(im,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_1),hAPP_complex_complex(cnj,Z_1)))) # label(fact_4104_complex__In__mult__cnj__zero) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2967 (all S (-hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S)) <-> (all M_1 exists N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2)) & hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,N_2),S)))))) # label(fact_4609_infinite__nat__iff__unbounded) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2968 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_2)) -> hAPP_P1175774780nt_int(product_snd_int_int,posDivAlg(A_3,B_2)) = hAPP_int_int(div_mod_int(A_3),B_2)))) # label(fact_4235_mod__pos__pos) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2969 (all C all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),B))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B)))) # label(fact_2023_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2970 (all C_41 all A_99 all B_81 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_41),A_99)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_41),B_81))) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_99),B_81)))) # label(fact_2034_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2971 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N))))) # label(fact_3061_le__imp__less__Suc) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2972 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,X)),one_one_real)) -> (exists Z (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) & X = hAPP_real_real(tan,Z) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),Z)))))) # label(fact_3625_tan__total__pi4) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2973 (all P_1 all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D)) -> ((all X_2 all K (is_int(K) & is_int(X_2) -> (hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D)))) <-> hBOOL(hAPP_int_bool(P_1,X_2))))) -> ((exists X1 (is_int(X1) & hBOOL(hAPP_int_bool(P_1,X1)))) <-> (exists X_2 (hBOOL(hAPP_int_bool(P_1,X_2)) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D))) & is_int(X_2))))))) # label(fact_5101_periodic__finite__ex) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2974 (all B_2 all Bq all Br all A_3 all C_1 all Aq all Ar (hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(A_3,C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Aq),Ar))) -> (hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(B_2,C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Bq),Br))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_1)) -> hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_3),B_2),C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Aq),Bq)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ar),Br)),C_1))),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ar),Br)),C_1)))))))) # label(fact_4214_divmod__nat__rel__add1__eq) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2975 (all N_59 all M_24 all X_28 all Y_22 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_28),Y_22)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_59),M_24)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_28),N_59)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_22),M_24)))))) # label(fact_1003_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2976 (all C_37 all A_95 all B_77 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_95),B_77)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_95),C_37)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_77),C_37))))) # label(fact_2060_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2977 (all A_88 all B_76 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_88),B_76)),B_76) = A_88) # label(fact_2132_add__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2978 (all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,min)),B_2) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_2),one_one_int))) # label(fact_2663_zmod__minus1) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2979 (all P_5 collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),P_5))) = hAPP_i1948725293t_bool(sr,P_5)) # label(fact_2317_SR__def) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2980 (all N_1 all A (hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially)) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A,N_2)))) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,hAPP_nat_nat(suc,N_2))),hAPP_nat_real(A,N_2)))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1)),one_one_nat))))))))) # label(fact_5054_summable__Leibniz_H_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2981 (all X_1 all Y_2 (hBOOL(hAPP_f526364711l_bool(hAPP_P1062890741l_bool(member2143287562nt_int,hAPP_P230595783nt_int(hAPP_P941956607nt_int(produc883642259nt_int,X_1),Y_2)),ratrel)) <-> zero_zero_int != hAPP_P1175774780nt_int(product_snd_int_int,X_1) & hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_fst_int_int,X_1)),hAPP_P1175774780nt_int(product_snd_int_int,Y_2)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_fst_int_int,Y_2)),hAPP_P1175774780nt_int(product_snd_int_int,X_1)) & hAPP_P1175774780nt_int(product_snd_int_int,Y_2) != zero_zero_int)) # label(fact_4666_ratrel__iff) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2982 (all B all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int))) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int))))))) # label(fact_2226_wset__less) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2983 (all X (is_int(X) -> X = hAPP_int_int(succ,hAPP_int_int(pred,X)))) # label(fact_4039_succ__pred) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2984 (all A_47 all B_38 all C_19 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_47),B_38)),C_19) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(div_mod_int(A_47),C_19)),B_38)),C_19)) # label(fact_2421_mod__add__left__eq) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2985 (all P all Q all R hAPP_i1948725293t_bool(P,hAPP_P1175774780nt_int(Q,R)) = hAPP_P252494982t_bool(hAPP_f291087797t_bool(hAPP_f214467655t_bool(cOMBB_182850971nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool__014) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2986 (all A all N_1 all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A),B)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A),hAPP_nat_nat(suc,N_1))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B),hAPP_nat_nat(suc,N_1)))))) # label(fact_4030_coprime__exp2) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2987 (all A_1 (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1)) -> (hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_1)) <-> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_int_nat(nat,A_1)))))) # label(fact_4619_transfer__int__nat__set__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2988 (all Ma all N_1 all A (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A)) -> (Ma = N_1 <-> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A),N_1) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A),Ma)))) # label(fact_1288_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2989 (all B_1_1 is_bool(trivial_limit_nat(B_1_1))) # label(gsy_c_Limits_Otrivial__limit_000tc__Nat__Onat) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2990 (all M_27 all A_212 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,M_27),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_212),M_27)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_212),one_on1645066479umeral)),M_27)) # label(fact_701_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2991 (all Z_1 of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_1),hAPP_complex_complex(cnj,Z_1))) # label(fact_4111_complex__mult__cnj) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2992 (all M all N all R_5 all S_3 all S_1 all T_2 all T_1 all R_3 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int)) -> hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,R_5),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,S_3),T_2)) = xzgcda(M,N,R_5,R_3,S_3,S_1,T_2,T_1)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int)) -> xzgcda(M,N,R_5,R_3,S_3,S_1,T_2,T_1) = xzgcda(M,N,R_3,hAPP_int_int(div_mod_int(R_5),R_3),S_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,S_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,R_5),R_3)),S_1)),T_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,T_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,R_5),R_3)),T_1)))))) # label(fact_4200_xzgcda_Osimps) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2993 (all X_1 all N_1 all Y_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),hAPP_nat_nat(suc,N_1))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_2),hAPP_nat_nat(suc,N_1)))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2)))) # label(fact_3087_exp__mono__lt) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2994 (all M_25 all A_185 all N_60 all B_142 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_185),N_60)),B_142)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_25),N_60)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_185),M_25)),B_142))))) # label(fact_992_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2995 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N)))) # label(fact_3060_Suc__leI) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2996 (all Y_2 all X_1 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zOdd)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_2)),zEven))))) # label(fact_3097_odd__minus__odd) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2997 (all X_3 all A (hBOOL(tendsto_nat_complex(X_3,A,sequentially)) -> hBOOL(tendsto_nat_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_3),hAPP_complex_real(im,A),sequentially)))) # label(fact_5057_Im_OLIMSEQ) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2998 (all P all Q all R hAPP_f291087797t_bool(P,hAPP_P1668131738nt_int(Q,R)) = hAPP_P2053772734t_bool(hAPP_f1198736473t_bool(hAPP_f2042071795t_bool(cOMBB_773384995nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__I) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 2999 (all N all M ((M = zero_zero_nat -> hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) = hAPP_nat_nat(fact_fact_nat,N)) & (M != zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N))) = hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N))))) # label(fact_3799_fact__add__num__eq__if2__nat) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3000 (all P_1 all L ((exists X_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,L),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_2),zero_zero_nat))) & hBOOL(hAPP_nat_bool(P_1,X_2)))) <-> (exists X_2 hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,L),X_2)))))) # label(fact_1681_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3001 (all N hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(real_nat,N))),pi)) = one_one_real) # label(fact_3735_cos__2npi) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3002 (all C all D all A all B (is_int(D) & is_int(B) & is_int(A) & is_int(C) -> (A != B & D != C <-> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),D)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),C)) != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),C)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),D))))) # label(fact_623_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3003 (all M all N (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N) = hAPP_nat_nat(div_mod_nat(M),N))) # label(fact_2711_mod__geq) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3004 (all A_3 hAPP_real_real(log(A_3),one_one_real) = zero_zero_real) # label(fact_3863_log__one) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3005 (all P_1 P_1 = collect_real(P_1)) # label(fact_149_Collect__def) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3006 (all X all Y hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(abs_abs_int,X)),Y) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)) # label(fact_4989_gcd__abs1__int) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3007 (all A_3 all B_2 hAPP_complex_complex(uminus473333897omplex,hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(uminus_uminus_real,A_3)),hAPP_real_real(uminus_uminus_real,B_2))) # label(fact_3963_complex__minus) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3008 (all X_1 (one_on1645066479umeral = X_1 <-> one_on1645066479umeral = X_1)) # label(fact_1953_one__reorient) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3009 (all A all E_2 all C all B all D (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),E_2)),D) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),E_2)),C) <-> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B)),E_2)),C) = D)) # label(fact_1543_eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3010 (all A_145 all M_20 all N_48 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_145),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_20),N_48)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_145),M_20)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_145),N_48))) # label(fact_1307_power__add) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3011 (all V (zero_zero_nat = hAPP_int_nat(number_number_of_nat,V) <-> hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))))) # label(fact_4050_neg__number__of__pred__iff__0) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3012 (all C_1 all N all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,C_1),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2) -> (exists R_1 exists S_2 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,R_1),N) = A_3 & hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,S_2),N) = B_2))))) # label(fact_4480_coprime__pow) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3013 (all N_37 all A_126 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_126)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_126),one_one_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_126),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_126),N_37))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_126),N_37)))))) # label(fact_1472_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3014 (all X_1 hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),sin)),sin),X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(sin,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X_1)),hAPP_real_real(sin,X_1)))))) # label(fact_4742_DERIV__sin__sin__mult) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3015 (all X_3 ((all J_1 exists M_3 all M_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_3),M_1)) -> (all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_3),N_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(X_3,M_1)),hAPP_nat_real(X_3,N_2)))),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,J_1))))))))) <-> hBOOL(cauchy_real(X_3)))) # label(fact_4375_Cauchy__iff2) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3016 (all I all K_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2)),I) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),I)),K_2))) # label(fact_1769_add__diff__assoc2) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3017 (all K_1 all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N_1))) <-> zero_zero_nat = K_1 | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),N_1)))) # label(fact_1848_nat__mult__dvd__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3018 (all X all Y (is_int(X) & is_int(Y) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),X)) | X = Y | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Y)))) # label(fact_179_zless__linear) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3019 (all C_66 all A_144 all B_113 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_66),A_144)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_66),B_113))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_66)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_144),B_113))))) # label(fact_1319_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3020 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(nat_tsub(X),Y)))))) # label(fact_3186_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3021 (all N_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,N_1)),Ma)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,N_1)),hAPP_int_real(real_int,Ma))))) # label(fact_4445_number__of__le__real__of__int__iff) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3022 (all U_1 all M all N all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),I)),U_1)),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),U_1)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),U_1)),N)))) # label(fact_1850_nat__diff__add__eq2) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3023 (all A_23 (A_23 != zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_23),A_23) = one_one_nat)) # label(fact_2554_div__self) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3024 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sqrt,X)),hAPP_real_real(sqrt,Y)) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),Y))) # label(fact_3356_real__sqrt__divide) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3025 (all A_221 zero_zero_complex = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_221),zero_zero_complex)) # label(fact_594_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3026 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_1)),one_one_real)) -> hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat)))))))) # label(fact_4827_summable__arctan__series) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3027 (all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_2759_Int2_Oaux__2) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3028 (all N all A_3 hAPP_complex_real(im,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,cis(A_3)),N)) = hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),A_3))) # label(fact_4144_sin__n__Im__cis__pow__n) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3029 (all P_1 all K_1 all I_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),I_1)) -> (hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_1),one_one_int))) -> ((all I_2 (is_int(I_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),I_2)) -> (hBOOL(hAPP_int_bool(P_1,I_2)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int))))))) -> hBOOL(hAPP_int_bool(P_1,I_1)))))) # label(fact_2242_int__gr__induct) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3030 (all C_58 all D_11 all A_136 all B_105 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_136),B_105)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_58),D_11)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_136)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_58)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_136),C_58)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_105),D_11)))))))) # label(fact_1370_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3031 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),one_one_real)) -> hAPP_real_real(uminus_uminus_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),X))))) = hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),X)))) # label(fact_3499_aux5) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3032 (all Real1 all Real2 zero_zero_nat = hAPP_complex_nat(size_size_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,Real1),Real2))) # label(fact_3971_complex_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3033 (all A_235 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_235),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_235),A_235)),A_235)) # label(fact_278_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3034 (all P all Q all R hAPP_int_bool(P,hAPP_int_int(Q,R)) = hAPP_int_bool(hAPP_f429935748t_bool(hAPP_f1512942609t_bool(cOMBB_int_bool_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__HOL__Obool_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3035 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,X_1)),one_one_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real)))) # label(fact_3333_real__sqrt__lt__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3036 (all Z_1 (is_int(Z_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,Z_1)) = Z_1))) # label(fact_3123_nat__0__le) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3037 (all A_17 all C_4 all B_13 (B_13 != zero_z126310315umeral -> hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_4),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_17),B_13)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_17),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_13),C_4))),B_13))) # label(fact_2626_div__mult__self2) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3038 (all D (one_one_nat = D <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,zero_zero_nat),D)))) # label(fact_4025_coprime__0_H) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3039 (all V_18 all W_14 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_18)),hAPP_int_int(number_number_of_int,W_14)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_18),W_14))) # label(fact_87_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3040 (all X_6 all Q_5 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_6),hAPP_nat_nat(suc,Q_5)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,X_6),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_6),Q_5))) # label(fact_3033_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3041 (all Y_2 all X_1 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Y_2)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Y_2),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3240_even__sum__div__2) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3042 (all P all Q all R hAPP_nat_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,P),Q),R) = hAPP_complex_real(P,hAPP_nat_complex(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Complex__Ocomplex_000tc__RealDef__Oreal_000tc__Nat__) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3043 (all P_3 (is_int(P_3) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (P_3 != hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) -> (hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))) != P_3 -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_3))))))) # label(fact_742_prime__g__5) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3044 (all X all Y hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(real_int,X)),hAPP_int_real(real_int,Y))) # label(fact_4396_real__of__int__mult) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3045 (all A_183 all M_22 all N_57 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_183),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_22),N_57)) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_183),M_22)),N_57)) # label(fact_1014_power__mult) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3046 (all X hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(arctan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) = hAPP_real_real(arctan,X)) # label(fact_3373_arctan__half) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3047 (all N_1 hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),suc)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1))) = zero_zero_real) # label(fact_4911_sumr__minus__one__realpow__zero) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3048 (all A_60 all B_51 all C_30 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_60),hAPP_C498520661umeral(div_mo1740067990umeral(B_51),C_30))),C_30) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_60),B_51)),C_30)) # label(fact_2379_mod__mult__right__eq) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3049 (all R_3 all A_3 hAPP_complex_real(im,rcis(R_3,A_3)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_real_real(sin,A_3))) # label(fact_4263_Im__rcis) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3050 (all B_2 all A_3 (B_2 != zero_zero_nat | A_3 != zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))) = one_one_nat)) # label(fact_4370_div__gcd__coprime__nat) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3051 (all M all N all Q_2 all R_3 (hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_2),R_3))) -> R_3 = hAPP_nat_nat(div_mod_nat(M),N))) # label(fact_4219_mod__eq) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3052 (all W hAPP_int_complex(number528085621omplex,W) = hAPP_complex_complex(cnj,hAPP_int_complex(number528085621omplex,W))) # label(fact_4067_complex__cnj__number__of) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3053 (all Y all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y)) -> hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(ln,X)),hAPP_real_real(ln,Y)) = hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),Y))))) # label(fact_3456_ln__div) # label(axiom) # label(non_clause). [assumption]. 7.54/7.52 3054 (all X_1 all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(sin,X_1)),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_r195310020l_real(hAPP_f2126667875l_real(cOMBC_746811033l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),zero_zero_real)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,div_div_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,minus_minus_nat),hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(abs_abs_real,X_1)),N_1))))) # label(fact_4894_Maclaurin__sin__bound) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3055 (all X all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(multInv(P_3),X)),X)),one_one_int),P_3)))))) # label(fact_2241_MultInv__prop2a) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3056 (all X (-hBOOL(hAPP_nat_bool(even_odd_even_nat,X)) -> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,X)),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))))) # label(fact_3771_odd__nat__plus__one__div__two) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3057 (all A_70 all B_58 hAPP_C498520661umeral(div_mo1740067990umeral(A_70),B_58) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(div_mo1740067990umeral(A_70),B_58)),B_58)) # label(fact_2327_mod__mod__trivial) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3058 (all M hAPP_nat_nat(div_mod_nat(M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(suc,M))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_3203_mod2__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3059 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real)) -> (exists X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_2)) & Y = hAPP_real_real(sin,X_2) & (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) & Y = hAPP_real_real(sin,Y_1) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),Y_1)) -> X_2 = Y_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))))) # label(fact_3747_sin__total) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3060 (all A all B all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma)))) # label(fact_1559_zcong__def) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3061 (all A all B (B = A <-> hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A),B) = zero_zero_complex)) # label(fact_2078_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3062 (all N all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (exists R_1 (A_3 = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_1),hAPP_nat_nat(suc,N)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1)))))) # label(fact_3250_realpow__pos__nth2) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3063 (all Y_20 all X_26 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_26)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_20)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_20),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_26),Y_20)),X_26)))))) # label(fact_1388_mult__right__le__one__le) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3064 (all K_1 all F all N_1 (is_int(K_1) -> ((all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),N_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),one_one_nat))),hAPP_nat_int(F,I_2)))),one_one_int)))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),hAPP_nat_int(F,N_1))) -> (exists I_2 (hAPP_nat_int(F,I_2) = K_1 & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N_1))))))))) # label(fact_3479_nat0__intermed__int__val) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3065 (all P_5 all N_1 (hBOOL(hAPP_nat_bool(prime,P_5)) & one_one_nat = N_1 <-> hBOOL(hAPP_nat_bool(prime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_5),N_1))))) # label(fact_4176_prime__exp) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3066 (all P all Q all R hAPP_i1948725293t_bool(hAPP_f720654810t_bool(hAPP_f908327869t_bool(cOMBB_1290694363ol_int,P),Q),R) = hAPP_f429935748t_bool(P,hAPP_int_fun_int_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3067 (all A_200 all B_153 all C_86 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_200),C_86)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_153),C_86)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_200),B_153)),C_86)) # label(fact_878_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3068 (all A_121 all C_54 all B_99 all D_9 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_121),B_99)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,C_54),D_9)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_121),C_54)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_99),D_9))) # label(fact_1809_add__diff__add) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3069 (all Z_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Z_1),Z_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_1)) # label(fact_570_nat__mult__2) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3070 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(suc,N))) # label(fact_2866_add__Suc__shift) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3071 (all M hAPP_int_int(abs_abs_int,M) = hAPP_int_int(legacy_zgcd(M),zero_zero_int)) # label(fact_4507_zgcd__0) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3072 (all S_1 all T_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S_1),T_1)) -> T_1 != S_1)) # label(fact_1589_less__not__refl3) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3073 (all X_19 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_19),one_one_complex)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_19),one_one_complex)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_19),X_19)),one_one_complex)) # label(fact_1824_real__squared__diff__one__factored) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3074 (all X_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,X_1)),zero_zero_rat)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),pls)))) # label(fact_99_le__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3075 (all M zero_zero_nat != hAPP_nat_nat(suc,M)) # label(fact_2853_Zero__not__Suc) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3076 (all A_198 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,zero_zero_complex),A_198)) -> zero_zero_complex = A_198)) # label(fact_887_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3077 (all A_197 all B_152 all N_64 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_197),B_152)),N_64) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_197),N_64)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_152),N_64))) # label(fact_897_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3078 (all N_13 all A_8 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_8)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_8),hAPP_nat_nat(suc,N_13)))))) # label(fact_3118_power__gt1) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3079 (all A_220 A_220 = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,zero_zero_real),A_220)) # label(fact_600_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3080 (all X_1 hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,times_times_real),cos)),cos),X_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_1))),hAPP_real_real(cos,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_1))),hAPP_real_real(cos,X_1)))))) # label(fact_4777_DERIV__cos__cos__mult) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3081 (all X_3 ((all R_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),R_1)) -> (exists K all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_2)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(X_3,N_2))),R_1)))))) <-> hBOOL(vanishes(X_3)))) # label(fact_5150_vanishes__def) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3082 (all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) | N_1 = Ma <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(suc,N_1))))) # label(fact_2864_less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3083 (all B_73 all A_85 all C_36 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_36)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_73),A_85)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_73),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_85),C_36)))))) # label(fact_2150_add__increasing2) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3084 (all N_62 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,one_on1684967323de_int),N_62) = one_on1684967323de_int) # label(fact_974_power__one) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3085 (all X_41 all Y_29 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_41),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_29),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_41)),Y_29)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_41),Y_29)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_367_power2__sum) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3086 (all X all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> X = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),N)))) # label(fact_3936_real__root__pow__pos) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3087 (all K_1 hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_lessThan_nat,K_1)))) # label(fact_4949_finite__lessThan) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3088 (all A_3 all X (is_int(X) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),A_3)) -> (X != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),one_one_int) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),one_one_int)))))))) # label(fact_1578_Euler_Oaux1) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3089 (all A_55 all C_25 all B_46 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_55),B_46)),C_25) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_55),C_25)),B_46)),C_25)) # label(fact_2399_zmod__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3090 (all P all Q all R hAPP_i61245889nt_int(P,hAPP_P1175774780nt_int(Q,R)) = hAPP_P1668131738nt_int(hAPP_f167782601nt_int(hAPP_f400538607nt_int(cOMBB_866192175nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__I) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3091 (all B_131 all A_171 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_171),zero_z891286103de_int)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_131)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_171),B_131)),zero_z891286103de_int))))) # label(fact_1096_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3092 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),hAPP_real_real(inverse_inverse_real,Y)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),Y)) # label(fact_3830_divide__real__def) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3093 (all A_222 zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A_222)) # label(fact_584_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3094 (all A_143 all C_65 all B_112 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_143),C_65)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_112),C_65))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_65)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_143),B_112))))) # label(fact_1329_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3095 (all V_8 all W_4 hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_8),W_4)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_int_complex(number528085621omplex,V_8)),hAPP_int_complex(number528085621omplex,W_4))) # label(fact_1512_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3096 (all K_2 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_2)) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(suc,zero_zero_nat))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K_2),M)),K_2)))) # label(fact_3237_int__power__div__base) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3097 (all Y all X all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y)) -> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),Y)) = hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)))))) # label(fact_3931_real__root__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3098 (all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> hAPP_int_nat(nat,hAPP_int_int(fact_fact_int,X)) = hAPP_nat_nat(fact_fact_nat,hAPP_int_nat(nat,X)))) # label(fact_4285_transfer__nat__int__factorial) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3099 (all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)))) # label(fact_2211_less__iff__diff__less__0) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3100 (all N hAPP_int_int(ring_1_of_int_int,hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,N)) # label(fact_4581_of__int__int__eq) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3101 (all A_239 all B_167 all V_15 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_239),B_167)),hAPP_int_int(number_number_of_int,V_15)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_239),hAPP_int_int(number_number_of_int,V_15))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_167),hAPP_int_int(number_number_of_int,V_15)))) # label(fact_184_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3102 (all Ma all K_1 all F ((all M_1 all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,M_1)),hAPP_nat_nat(F,N_2))))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(F,Ma)),K_1)),hAPP_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),K_1)))))) # label(fact_2243_mono__nat__linear__lb) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3103 (all B_2 all A_3 (is_int(A_3) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_2)) -> divmod_int(A_3,B_2) = posDivAlg(A_3,B_2)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_2)) -> (A_3 != zero_zero_int -> divmod_int(A_3,B_2) = hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)))) & (zero_zero_int = A_3 -> divmod_int(A_3,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int)))) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> divmod_int(A_3,B_2) = hAPP_P1975530577nt_int(negateSnd,posDivAlg(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> negDivAlg(A_3,B_2) = divmod_int(A_3,B_2))))) # label(fact_4162_divmod__int__def) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3104 (all X_38 all P_8 all Q_12 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_38),P_8)),Q_12) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_38),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12))) # label(fact_449_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3105 (all V_14 all B_166 all C_97 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_i769753017umeral(number1443263063umeral,V_14)),B_166)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_i769753017umeral(number1443263063umeral,V_14)),C_97)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_i769753017umeral(number1443263063umeral,V_14)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_166),C_97))) # label(fact_190_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3106 (all M_17 all N_30 ((N_30 != zero_zero_nat -> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,M_17),N_30) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,M_17),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat)))) & (N_30 = zero_zero_nat -> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,M_17),N_30) = one_on1684967323de_int))) # label(fact_1834_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3107 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) <-> hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),one_one_real) = X_1)) # label(fact_4434_powr__one__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3108 (all R_3 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,archim1246769320r_real(R_3))),R_3))) # label(fact_4401_real__of__int__floor__le) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3109 (all X (hAPP_real_real(cos,X) != zero_zero_real -> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(cos,X))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(tan,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_3857_tan__sec) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3110 (exists K_4 all X_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(im,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_2)),K_4)))) # label(fact_4109_Im_Obounded) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3111 (all P all Q all R hAPP_f1735705551nt_int(hAPP_i1836591008nt_int(P,R),hAPP_i1584592887nt_int(Q,R)) = hAPP_i1584592887nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__prod_Itc__I) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3112 (all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> hAPP_int_real(real_int,X) = hAPP_nat_real(real_nat,hAPP_int_nat(nat,X)))) # label(fact_4449_real__nat__eq__real) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3113 (all A_111 all B_89 all C_49 (hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_111),C_49) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_111),B_89) -> B_89 = C_49)) # label(fact_1915_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3114 (all D all B all C all A (is_int(A) & is_int(C) -> (zero_zero_int != A -> (zero_zero_int != C -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,C),D)) -> (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),D) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),C) <-> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B),A) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,D),C)))))))) # label(fact_2638_dvd__div__div__eq__mult) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3115 (all X_1 (hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)) <-> (exists Y_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),Y_1) = X_1))) # label(fact_3770_even__nat__equiv__def2) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3116 (all X all A_3 all B_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_2))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),B_2))))) # label(fact_4411_powr__less__cancel) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3117 (all P all Q all R hAPP_P989584703t_bool(hAPP_f1781059817t_bool(hAPP_f125837377t_bool(cOMBS_1610675060t_bool,P),Q),R) = hAPP_f237669397t_bool(hAPP_P877703507t_bool(P,R),hAPP_P989584703t_bool(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__033) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3118 (all Z_1 (is_int(Z_1) -> Z_1 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),one_one_int))) # label(fact_128_zmult__1__right) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3119 (all A_183 all M_22 all N_57 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_183),M_22)),N_57) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_183),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_22),N_57))) # label(fact_1016_power__mult) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3120 (all M all N hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),divmod_nat(M,N)))) # label(fact_4217_divmod__nat__rel__divmod__nat) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3121 (all R_3 ((hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),zero_zero_real)) -> hAPP_real_real(abs_abs_real,R_3) = hAPP_real_real(uminus_uminus_real,R_3)) & (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),zero_zero_real)) -> R_3 = hAPP_real_real(abs_abs_real,R_3)))) # label(fact_3495_real__abs__def) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3122 (all N_54 all A_151 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_151)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_151),N_54))))) # label(fact_1240_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3123 (all B_36 all A_45 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_36),A_45)),B_36) = hAPP_nat_nat(div_mod_nat(A_45),B_36)) # label(fact_2430_mod__add__self1) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3124 (all K_1 hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),K_1))))) # label(fact_4731_finite__Collect__less__nat) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3125 (all A_134 all N_47 all B_103 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_134),N_47)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_103),N_47))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_103)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_134),B_103))))) # label(fact_1396_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3126 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,M))) -> hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N)),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N))))) # label(fact_3778_fact__diff__Suc) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3127 (all Y_6 all P_6 all N_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10)) -> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)))) # label(fact_3176_lemma__realpow__diff) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3128 (all A_158 all B_119 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_158),B_119))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_158)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_119))))) # label(fact_1167_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3129 (all P all Q all R hAPP_i1584592887nt_int(hAPP_f1574055885nt_int(hAPP_f434654626nt_int(cOMBS_2011279539nt_int,P),Q),R) = hAPP_i1584592887nt_int(hAPP_i1778081398nt_int(P,R),hAPP_int_int(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__Int__Oint_000tc__fun_Itc__Int__Oint) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3130 (all A_3 all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(ln,X)),A_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(powr,A_3),A_3)),X)))))) # label(fact_4476_ln__powr__bound2) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3131 (all N hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(real_nat,N)))) # label(fact_3633_real__of__nat__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3132 (all X_2 hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),sin)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),cos)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_2),zero_zero_real))) # label(fact_4793_DERIV__sin__circle__all__zero) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3133 (all A all B all Ma (one_one_int = hAPP_int_int(legacy_zgcd(B),Ma) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(bnorRset(A),Ma))))))) # label(fact_4528_Bnor__mem__if) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3134 (all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N_1))))) # label(fact_3149_zero__less__int__conv) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3135 (all P all Q all R hAPP_i1529485324t_bool(hAPP_f1315737299t_bool(hAPP_f538565589t_bool(cOMBB_1211207212ol_int,P),Q),R) = hAPP_f627970963t_bool(P,hAPP_i68813070l_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_000tc__fun_Itc) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3136 (all A_129 all N_40 all N_39 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_40),N_39)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_129)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_129),N_40)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_129),N_39)))))) # label(fact_1444_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3137 (all A all W_1 (hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A),hAPP_int_nat(number_number_of_nat,W_1)) = zero_zero_complex <-> zero_zero_complex = A & hAPP_int_nat(number_number_of_nat,W_1) != zero_zero_nat)) # label(fact_716_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3138 (all V_19 all W_15 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_19)),hAPP_int_rat(number_number_of_rat,W_15)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_19),W_15))) # label(fact_86_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3139 (all N all M hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N)))),hAPP_nat_int(semiri1621563631at_int,M)))) # label(fact_3511_negative__zless) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3140 (all X_1 all Y_2 all P_1 ((-hBOOL(P_1) -> hAPP_nat_int(semiri1621563631at_int,Y_2) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,P_1),X_1),Y_2))) & (hBOOL(P_1) -> hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,P_1),X_1),Y_2)) = hAPP_nat_int(semiri1621563631at_int,X_1)))) # label(fact_3255_int__if__cong) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3141 (all A_38 all C_13 all B_29 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_13),B_29)) -> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(div_mod_nat(A_38),B_29)),C_13) = hAPP_nat_nat(div_mod_nat(A_38),C_13))) # label(fact_2448_mod__mod__cancel) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3142 (all A_234 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_234),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_234),A_234)) # label(fact_303_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3143 (all Ma all N_1 (hBOOL(hAPP_nat_bool(nat_case_bool(fFalse,hAPP_n1699378549t_bool(ord_less_eq_nat,Ma)),N_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Ma)),N_1)))) # label(fact_4877_less__eq__nat_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3144 (all N N = hAPP_nat_nat(size_size_nat,N)) # label(fact_4059_nat__size) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3145 (all Z_12 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_12),Z_12) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_12),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_292_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3146 (all C_69 all A_163 all B_124 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_163),B_124)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_69)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_69),A_163)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_69),B_124)))))) # label(fact_1140_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3147 (all Y_2 all X_1 all A (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(log(A),X_1)),hAPP_real_real(log(A),Y_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2))))))) # label(fact_3871_log__le__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3148 (all B_5 all A_5 all A_3 all B_2 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2) != zero_zero_nat -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)) = A_3 -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)) = B_2 -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_5),B_5) = one_one_nat)))) # label(fact_4365_gcd__coprime__nat) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3149 (all X_1 the_real(hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),exp_real)),X_1)) = hAPP_real_real(ln,X_1)) # label(fact_4884_ln__def) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3150 (all A_205 all B_156 (hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_205),B_156) = zero_zero_rat -> zero_zero_rat = A_205 | zero_zero_rat = B_156)) # label(fact_827_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3151 (all X hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),pi)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(cos,X))) # label(fact_3570_cos__periodic__pi) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3152 (all A_10 all N_15 all B_8 (hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_10),hAPP_nat_nat(suc,N_15)) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_8),hAPP_nat_nat(suc,N_15)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_10)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_8)) -> A_10 = B_8)))) # label(fact_3105_power__inject__base) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3153 (all A_113 all B_91 all C_51 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_113),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_91),C_51)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_113),B_91)),C_51)) # label(fact_1901_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3154 (all A_3 hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(cos,A_3)),hAPP_real_real(sin,A_3)) = cis(A_3)) # label(fact_4152_cis__def) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3155 (all R_3 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),R_3)) -> -(all S_2 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),S_2)) -> (all T (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),T)) -> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,S_2),T) != R_3)))))) # label(fact_4665_obtain__pos__sum) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3156 (all A_228 all C_94 all D_19 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_228),C_94)),D_19) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_228),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_94),D_19))) # label(fact_536_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3157 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A))))) # label(fact_4717_zero__less__Fract__iff) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3158 (all A_207 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_207),A_207))) # label(fact_797_dvd__refl) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3159 (all M all V_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,M)))),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),M)),hAPP_int_nat(number_number_of_nat,V_2))) # label(fact_3205_Suc__div__eq__add3__div__number__of) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3160 (all A_3 (is_int(A_3) -> (A_3 != zero_zero_int -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),A_3) = one_one_int))) # label(fact_2523_zdiv__self) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3161 (all X_1 the_Pr2103884470nt_int(hAPP_f237669397t_bool(hAPP_f461489489t_bool(cOMBS_400904860l_bool,hAPP_f365317245l_bool(hAPP_f1961001761l_bool(cOMBB_1550063917nt_int,fconj),hAPP_f1084724746t_bool(hAPP_f374502857t_bool(cOMBB_569260360nt_int,hAPP_r2115560837t_bool(fequal_rat,X_1)),hAPP_f1065357839nt_rat(hAPP_f1120137083nt_rat(cOMBS_549334880nt_rat,hAPP_f1547007718nt_rat(hAPP_f382317021nt_rat(cOMBB_170828966nt_int,fract),product_fst_int_int)),product_snd_int_int)))),hAPP_f237669397t_bool(hAPP_f461489489t_bool(cOMBS_400904860l_bool,hAPP_f365317245l_bool(hAPP_f1961001761l_bool(cOMBB_1550063917nt_int,fconj),hAPP_f892584630t_bool(hAPP_f492612297t_bool(cOMBB_798864284nt_int,hAPP_i1948725293t_bool(ord_less_int,zero_zero_int)),product_snd_int_int))),hAPP_i119638232t_bool(hAPP_f2018027565t_bool(cOMBC_33652895t_bool,hAPP_f291087797t_bool(hAPP_f214467655t_bool(cOMBB_182850971nt_int,fequal_int),hAPP_f521271395nt_int(hAPP_f1914742907nt_int(cOMBS_2019229620nt_int,hAPP_f1855025978nt_int(hAPP_f1038672605nt_int(cOMBB_1506897658nt_int,gcd_gcd_int),product_fst_int_int)),product_snd_int_int))),one_one_int)))) = quotient_of(X_1)) # label(fact_5023_quotient__of__def) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3162 (all A_133 all N_46 all N_45 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_46),N_45)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_133)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_133),one_one_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_133),N_45)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_133),N_46))))))) # label(fact_1402_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3163 (all V_2 hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,V_2)) = hAPP_int_nat(number_number_of_nat,V_2)) # label(fact_2714_nat__number__of__def) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3164 (all B_2 all Q_2 all R_3 all B_5 all Q_8 all R_5 (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_8)),R_5) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_3) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_8)),R_5)),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_5),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_8),Q_2))))))))) # label(fact_737_zdiv__mono2__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3165 (all C_80 all A_187 all B_143 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_187),B_143)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_187),C_80)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_187),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_143),C_80)))))) # label(fact_968_dvd__add) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3166 (all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(big_co1548731110nt_int(cOMBI_int,comple219730294t_bool(setS(A,P_5)))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),P_5))))))) # label(fact_4809_Union__SetS__setprod__prop1) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3167 (all B_1_1 all B_2_1 is_int(hAPP_nat_int(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__Nat__Onat_000tc__Int__Oint) # label(hypothesis) # label(non_clause). [assumption]. 7.54/7.53 3168 (all D all Sa exists Z ((all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Z)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Sa))) <-> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Sa)))))) & is_int(Z))) # label(fact_2251_minf_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3169 (all M_25 all A_185 all N_60 all B_142 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_185),N_60)),B_142)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_25),N_60)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_185),M_25)),B_142))))) # label(fact_993_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3170 (all P (is_bool(P) -> P = fTrue | fFalse = P)) # label(help_If_3_1_If_000tc__Int__Oint_T) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3171 (all K_1 all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K_1)),N_1))))) # label(fact_3065_Suc__mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3172 (all A_68 all B_56 A_68 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_68),B_56)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_68),B_56)),B_56))) # label(fact_2339_semiring__div__class_Omod__div__equality_H) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3173 (all N_33 all A_123 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_123)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_33)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_123),N_33)))))) # label(fact_1490_one__less__power) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3174 (all N_42 all A_130 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_130)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_130),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_130),N_42)))))) # label(fact_1429_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3175 (all N all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N)),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,N)),one_one_real))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat) = natceiling(X)))) # label(fact_3725_natceiling__eq) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3176 (all N_26 hAPP_int_complex(number528085621omplex,hAPP_nat_int(semiri1621563631at_int,N_26)) = hAPP_nat_complex(semiri2020571505omplex,N_26)) # label(fact_2825_number__of__int) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3177 (all A all W_1 (zero_zero_nat = A & hAPP_int_nat(number_number_of_nat,W_1) != zero_zero_nat <-> zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A),hAPP_int_nat(number_number_of_nat,W_1)))) # label(fact_713_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3178 (all N all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N)),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,N)),one_one_real))) -> N = natfloor(X)))) # label(fact_3705_natfloor__eq) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3179 (all A_217 all M_29 all B_160 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_217),M_29)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_160),M_29)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_217),B_160)),M_29)) # label(fact_639_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3180 (all Y_2 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(nat,X_1)),hAPP_int_nat(nat,Y_2))))))) # label(fact_2729_transfer__nat__int__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3181 (all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),hAPP_int_int(bit1,pls))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,X_1)),one_one_real)))) # label(fact_119_le__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3182 (all M all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),M)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),M))))) # label(fact_1311_zcong__not__zero) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3183 (all Y_2 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_1),Y_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(nat,X_1)),hAPP_int_nat(nat,Y_2))))))) # label(fact_2745_transfer__nat__int__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3184 (all K_1 all Ma all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(legacy_zgcd(Ma),N_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),N_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),Ma)))) # label(fact_4491_zgcd__greatest__iff) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3185 (all Z1 all Z2 all Z3 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z2),Z3)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z1),Z2)),Z3)) # label(fact_1784_real__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3186 (all Ta all B_1 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ta),one_one_int)),B_1)) -> (all X_2 (is_int(X_2) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1)) -> X_2 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa))))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ta),X_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ta),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1)))))))))) # label(fact_5092_bset_I8_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3187 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(cos,X)))))) # label(fact_3591_cos__gt__zero__pi) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3188 (all N zero_zero_real != hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N))) # label(fact_3762_real__of__nat__fact__not__zero) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3189 (all X_30 all Y_24 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_30),Y_24)) -> (X_30 != Y_24 -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_30),Y_24))))) # label(fact_817_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3190 (all A all E_2 all C all B all D (hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A),B)),E_2)),C) = D <-> hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A),E_2)),C) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B),E_2)),D))) # label(fact_1542_eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3191 (all P all Q all R hAPP_int_bool(hAPP_P252494982t_bool(P,R),hAPP_P1175774780nt_int(Q,R)) = hAPP_P603027463t_bool(hAPP_f892584630t_bool(hAPP_f1325804733t_bool(cOMBS_480885903t_bool,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin_004) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3192 (all C_1 all A_3 all B_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(B_2),C_1),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),C_1),M))))) # label(fact_811_zcong__trans) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3193 (all A_232 A_232 = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_232),hAPP_int_complex(number528085621omplex,pls))) # label(fact_328_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3194 (all D all Sa exists Z (is_int(Z) & (all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),X_2)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Sa))) <-> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Sa)))))))) # label(fact_2257_pinf_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3195 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),M))))) # label(fact_1527_nat__dvd__not__less) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3196 (all P all Q (is_bool(P) -> P = hAPP_nat_bool(cOMBK_bool_nat(P),Q))) # label(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3197 (all C_33 all D_5 all A_79 all B_67 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_79),B_67)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_33),D_5)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_79),C_33)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_67),D_5)))))) # label(fact_2202_add__less__le__mono) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3198 (all G all F all X_1 all D (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D)) -> ((all Z (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D)) -> Z = hAPP_real_real(G,hAPP_real_real(F,Z)))) -> ((all Z (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D)) -> hBOOL(hAPP_real_bool(isCont_real_real(F),Z)))) -> (exists Z (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Z),X_1))),D)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,Z)),hAPP_real_real(F,X_1))))))))) # label(fact_5181_lemma__isCont__inj2) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3199 (all M_10 all N_24 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(semiri1619134803umeral,M_10)),hAPP_n1108039445umeral(semiri1619134803umeral,N_24))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_24)))) # label(fact_2929_of__nat__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3200 (all K_2 all I all J_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),J_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),I)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),J_2))))) # label(fact_71_zadd__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3201 (all N_11 all A_7 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_7)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_7),one_one_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_7),hAPP_nat_nat(suc,N_11))),one_one_rat))))) # label(fact_3159_power__Suc__less__one) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3202 (all B_2 all A_3 (is_int(A_3) & is_int(B_2) -> (zero_zero_int != A_3 | zero_zero_int != B_2 -> one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)))))) # label(fact_5004_div__gcd__coprime__int) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3203 (all B_70 all A_82 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_82),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_70),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_82),B_70)),zero_zero_real))))) # label(fact_2180_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3204 (all X_39 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_39),X_39) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_39),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_433_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3205 (all X0 all F all R_4 ((all X_2 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),hAPP_r1134773055l_bool(ord_gr788844697n_real(hAPP_real_real(uminus_uminus_real,R_4)),R_4))) -> hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),F)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc)))),hAPP_r474017924t_real(power_power_real,X_2)))))) -> (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X0),hAPP_r1134773055l_bool(ord_gr788844697n_real(hAPP_real_real(uminus_uminus_real,R_4)),R_4))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_4)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_f1122023986l_real(hAPP_f1786887945l_real(cOMBB_1896127834l_real,suminf_real),hAPP_f1993765077t_real(hAPP_f895854793t_real(cOMBB_1948233853l_real,hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),F))),hAPP_f1501741374t_real(hAPP_f1135648319t_real(cOMBC_1329014627t_real,hAPP_f1163894827t_real(hAPP_f1998755145t_real(cOMBB_421464275l_real,cOMBB_nat_real_nat),power_power_real)),suc))),X0),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),F)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc)))),hAPP_r474017924t_real(power_power_real,X0))))))))) # label(fact_4842_DERIV__power__series_H) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3206 (all A_19 all B_15 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_19),B_15)) -> hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_19),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B_15),A_19)) = B_15)) # label(fact_2573_dvd__mult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3207 (all Y_2 all X_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_1)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_2)) -> (zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_1),Y_2) <-> X_1 = zero_zero_rat & Y_2 = zero_zero_rat)))) # label(fact_2164_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3208 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),one_one_real)) -> hBOOL(hAPP_real_bool(deriv_real(arcsin,X_1),hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))))) # label(fact_3881_DERIV__arcsin) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3209 (all X_1 all Y_2 (Y_2 = X_1 <-> hAPP_complex_real(re,Y_2) = hAPP_complex_real(re,X_1) & hAPP_complex_real(im,Y_2) = hAPP_complex_real(im,X_1))) # label(fact_4116_complex__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3210 (all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))) -> hAPP_int_int(inv(P_3),A_3) != zero_zero_int)))) # label(fact_2139_inv__not__0) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3211 (all Y all X (hBOOL(hAPP_int_bool(nat_is_nat,X)) -> (hBOOL(hAPP_int_bool(nat_is_nat,Y)) -> hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)))))) # label(fact_5134_Nat__Transfer_Otransfer__int__nat__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3212 (all N_1 all Ma (is_int(Ma) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),N_1)) <-> Ma = hAPP_int_int(legacy_zgcd(N_1),Ma))))) # label(fact_4512_zdvd__iff__zgcd) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3213 (all Y_6 all P_6 all N_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10)) -> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6))) # label(fact_3174_lemma__realpow__diff) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3214 (all Y_14 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,zero_zero_complex),Y_14) = zero_zero_complex) # label(fact_2286_mult__left_Ozero) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3215 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arccos,Y)),pi)) & hAPP_real_real(cos,hAPP_real_real(arccos,Y)) = Y & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(arccos,Y)))))) # label(fact_3624_arccos) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3216 (all C_58 all D_11 all A_136 all B_105 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_136),B_105)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_58),D_11)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_136)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_58)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_136),C_58)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_105),D_11)))))))) # label(fact_1371_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3217 (all M_9 all N_23 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_23)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(semiri1619134803umeral,M_9)),hAPP_n1108039445umeral(semiri1619134803umeral,N_23))))) # label(fact_2935_less__imp__of__nat__less) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3218 (all N_1 all X_1 (one_one_real != X_1 -> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1)),one_one_real)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),one_one_real)) = hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1)))) # label(fact_4904_sumr__geometric) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3219 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_complex_real(im,X))),hAPP_complex_real(norm_norm_complex,X)))) # label(fact_4103_abs__Im__le__cmod) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3220 (all A all B (-(exists P_4 (is_int(P_4) & hBOOL(hAPP_int_bool(zprime,P_4)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_4),B)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_4),A)))) <-> hAPP_int_int(legacy_zgcd(A),B) = one_one_int)) # label(fact_4522_zgcd1__iff__no__common__primedivisor) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3221 (all N all M all A_3 all B_2 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2) = one_one_nat -> one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_2),M)))) # label(fact_4301_coprime__exp2__nat) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3222 (all Z_1 exists A_2 exists R_1 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_1)),expi(A_2)) = Z_1) # label(fact_4052_complex__expi__Ex) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3223 (all P all Q all R hAPP_n1865633855at_nat(hAPP_f837737749at_nat(hAPP_f999082675at_nat(cOMBB_348438404at_nat,P),Q),R) = hAPP_n1865633855at_nat(P,hAPP_nat_nat(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__prod_Itc__N) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3224 (all A_134 all N_47 all B_103 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_134),N_47)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_103),N_47))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_103)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_134),B_103))))) # label(fact_1401_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3225 (all D all Sa exists Z all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Z)) -> (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Sa))) <-> -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_2),Sa)))))) # label(fact_2248_minf_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3226 (all M_7 all N_21 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(semiri151668891at_rat,M_7)),hAPP_nat_rat(semiri151668891at_rat,N_21)) = hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21))) # label(fact_2961_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3227 (all A_216 A_216 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,one_one_complex),A_216)) # label(fact_651_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.53 3228 (all A_50 all C_22 all B_41 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_50),B_41)),C_22) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_50),C_22)),B_41)),C_22)) # label(fact_2410_zmod__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3229 (all A all B (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(negDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A),B))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B))) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B)) = negDivAlg(A,B)) & (-(hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B)))) -> negDivAlg(A,B) = hAPP_P1975530577nt_int(adjust(B),negDivAlg(A,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B)))))) # label(fact_2776_negDivAlg_Opsimps) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3230 (all P_1 (-hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) -> ((exists X1 hBOOL(hAPP_nat_bool(P_1,X1))) -> (exists N_2 (-hBOOL(hAPP_nat_bool(P_1,N_2)) & hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_2)))))))) # label(fact_3955_exists__least__lemma) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3231 (all A_226 all B_163 all C_92 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_226),B_163)),C_92) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_226),C_92)),B_163)) # label(fact_551_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3232 (all A_3 all B_2 hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(uminus_uminus_real,B_2)),A_3) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)),ii)) # label(fact_3982_Complex__mult__i) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3233 (all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> hAPP_i1948725293t_bool(wset(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_5) = hAPP_i1948725293t_bool(d22set,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_2244_d22set__eq__wset) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3234 (all A_203 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_203),zero_zero_nat) = zero_zero_nat) # label(fact_840_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3235 (all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),Ma)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),N_1)))) # label(fact_3135_one__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3236 (all P all Q all R hAPP_nat_real(hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,P),Q),R) = hAPP_nat_real(P,hAPP_nat_nat(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3237 (all M_19 all N_44 all A_132 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_132)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_132),M_19)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_132),N_44))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_19),N_44))))) # label(fact_1412_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3238 (all B_53 all A_65 all C_32 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_53),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_65),B_53))),hAPP_Q1762011733de_int(div_mo231679042de_int(A_65),B_53))),C_32) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_65),C_32)) # label(fact_2353_div__mod__equality2) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3239 (all M_27 all A_212 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_27),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_212),M_27)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_212),one_one_nat)),M_27)) # label(fact_699_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3240 (all X_1 hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),cos_coeff)),hAPP_r474017924t_real(power_power_real,X_1))) = hAPP_real_real(cos,X_1)) # label(fact_4822_cos__def) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3241 (all A_188 all B_144 all C_81 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_188),B_144)),C_81)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,B_144),C_81)))) # label(fact_955_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3242 (all Code_numeral_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_C2129356693al_nat(size_s945831648umeral,Code_numeral_1)),hAPP_nat_nat(suc,zero_zero_nat)) = hAPP_C2129356693al_nat(size_s945831648umeral,code_S1047413653umeral(Code_numeral_1))) # label(fact_4089_code__numeral_Osize_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3243 (all A_229 all C_95 all D_20 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_95),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_229),D_20)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_229),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_95),D_20))) # label(fact_528_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3244 (all X_3 ((all R_1 exists N_3 all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_1),hAPP_nat_real(X_3,N_2))))) -> hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),X_3),zero_zero_real,sequentially)))) # label(fact_5106_LIMSEQ__inverse__zero) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3245 (all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)) -> (Ma = N_1 <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,Ma)))))) # label(fact_3059_le__less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3246 (all Ma all N_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(semiri151668891at_rat,Ma)),hAPP_nat_rat(semiri151668891at_rat,N_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)))) # label(fact_2950_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3247 (all K_1 all Ma all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),N_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Ma),N_1))))) # label(fact_4983_gcd__greatest__iff__int) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3248 (all A_63 hAPP_int_int(div_mod_int(A_63),A_63) = zero_zero_int) # label(fact_2364_mod__self) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3249 (all A_3 all B_2 all C_1 hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),C_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),C_1)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),C_1))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(div_mod_int(A_3),C_1)),hAPP_int_int(div_mod_int(B_2),C_1))),C_1))) # label(fact_2337_zdiv__zadd1__eq) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3250 (all N all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,N)),hAPP_int_real(real_int,X))),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N),X)))),one_one_real))) # label(fact_4465_real__of__int__div3) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3251 (all X all A_3 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,archim856651990g_real(X)),A_3) = archim856651990g_real(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_int_real(real_int,A_3)))) # label(fact_4427_ceiling__add) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3252 (all X all Y all R_3 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,X),Y)),of_real_complex(R_3)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),R_3)),Y)) # label(fact_3992_Complex__add__complex__of__real) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3253 (all C_74 all A_174 all B_134 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_174),B_134)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_74)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_74),A_174)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_74),B_134)))))) # label(fact_1082_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3254 (all A_35 hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_35),one_on1645066479umeral) = A_35) # label(fact_2480_div__by__1) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3255 (all N_1 hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_nat_real(real_nat,N_1)),ring_1_Ints_real))) # label(fact_4759_Ints__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3256 (all K_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N))))) # label(fact_1556_dvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3257 (all C_75 all A_175 all B_135 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_175),B_135)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_75)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_75),A_175)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_75),B_135)))))) # label(fact_1073_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3258 (all Q_8 all R_5 all A_3 all B_2 all Q_2 all R_3 (is_int(R_5) & is_int(B_2) & is_int(R_3) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3))) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_8),R_5))) -> (zero_zero_int != B_2 -> R_3 = R_5))))) # label(fact_2790_unique__remainder) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3259 (all B_55 all A_67 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_55),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_67),B_55))),hAPP_Q1762011733de_int(div_mo231679042de_int(A_67),B_55)) = A_67) # label(fact_2345_mod__div__equality2) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3260 (all X all A_3 all B_2 (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) = one_one_int -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X),B_2)) -> one_one_int = hAPP_int_int(abs_abs_int,X))))) # label(fact_5016_coprime__common__divisor__int) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3261 (all A all B all C (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B)))))) # label(fact_1378_mult__le__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3262 (all A_64 all B_52 all C_31 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_64),B_52)),B_52)),hAPP_int_int(div_mod_int(A_64),B_52))),C_31) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_64),C_31)) # label(fact_2354_div__mod__equality) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3263 (all A all B all C all D (hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C),D) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),D)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B))))) # label(fact_2095_diff__eq__diff__less) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3264 (all X all A_3 archim856651990g_real(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_int_real(real_int,A_3))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,archim856651990g_real(X)),A_3)) # label(fact_4429_ceiling__subtract) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3265 (all A_64 all B_52 all C_31 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_64),B_52)),B_52)),hAPP_nat_nat(div_mod_nat(A_64),B_52))),C_31) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_64),C_31)) # label(fact_2355_div__mod__equality) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3266 (all A_55 all C_25 all B_46 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_55),B_46)),C_25) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(div_mod_int(A_55),C_25)),B_46)),C_25)) # label(fact_2397_zmod__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3267 (all W_12 hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W_12)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),one_one_int)),hAPP_int_int(number_number_of_int,W_12))) # label(fact_270_double__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3268 (all A_70 all B_58 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(div_mod_nat(A_70),B_58)),B_58) = hAPP_nat_nat(div_mod_nat(A_70),B_58)) # label(fact_2326_mod__mod__trivial) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3269 (all P_1 all L ((exists X_2 hBOOL(hAPP_C24501650l_bool(P_1,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,L),X_2)))) <-> (exists X_2 (hBOOL(hAPP_C24501650l_bool(P_1,X_2)) & hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,L),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_2),zero_z126310315umeral))))))) # label(fact_1675_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3270 (all A all E_2 all C all B all D (D = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B)),E_2)),C) <-> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),E_2)),D) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),E_2)),C))) # label(fact_1541_eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3271 (all V_3 ((-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3))) -> hAPP_int_nat(nat,hAPP_int_int(div_mod_int(one_one_int),hAPP_int_int(number_number_of_int,V_3))) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,zero_zero_nat)),hAPP_int_nat(number_number_of_nat,V_3))) & (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3))) -> hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,zero_zero_nat)),hAPP_int_nat(number_number_of_nat,V_3))))) # label(fact_3266_one__mod__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3272 (all C_74 all A_174 all B_134 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_174),B_134)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_74)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_74),A_174)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_74),B_134)))))) # label(fact_1081_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3273 (all K_1 all L hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,L),zero_zero_int)),K_1),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,L),hAPP_int_int(div_mod_int(hAPP_int_int(abs_abs_int,K_1)),hAPP_int_int(abs_abs_int,L))))) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K_1),L)) # label(fact_5015_gcd__code__int) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3274 (all Ta all B_1 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> (all X_2 (is_int(X_2) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1)) -> X_2 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa))))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),Ta)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1)),Ta)))))))) # label(fact_5091_bset_I6_J) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3275 (all N_4 ((exists M_1 all X_2 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),N_4)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),M_1)))) <-> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_4)))) # label(fact_4626_finite__nat__set__iff__bounded) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3276 (all W hAPP_int_complex(ring_11397209091omplex,W) = hAPP_int_complex(number528085621omplex,W)) # label(fact_4584_complex__number__of__def) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3277 (all X hAPP_real_real(cos,X) = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi)))) # label(fact_3583_cos__periodic) # label(axiom) # label(non_clause). [assumption]. 7.54/7.54 3278 (all A_134 all N_47 all B_103 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_134),N_47)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_103),N_47))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_103)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_134),B_103))))) # label(fact_1398_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3279 (all V_13 all W_11 all Z_10 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_13)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,W_11)),Z_10)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_13),W_11))),Z_10)) # label(fact_334_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3280 (all X hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_complex_real(re,X)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_complex_real(re,hAPP_complex_complex(invers1449016382omplex,X))) # label(fact_4139_complex__Re__inverse) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3281 (all N_1 all G all X_1 all Ma (hBOOL(hAPP_real_bool(deriv_real(G,X_1),Ma)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),G)),N_1),X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(G,X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat)))),Ma))))) # label(fact_4773_DERIV__fun__pow) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3282 (all K_2 hAPP_int_real(number267125858f_real,K_2) = ratreal(hAPP_int_rat(number_number_of_rat,K_2))) # label(fact_4696_number__of__real__code) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3283 (all P all Q all R hAPP_f54304608l_bool(hAPP_n215258509l_bool(P,R),Q) = hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3284 (all X all A_3 hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3) = hAPP_real_real(exp_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),hAPP_real_real(ln,X)))) # label(fact_4437_powr__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3285 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arctan,X)),hAPP_real_real(arctan,Y))))) # label(fact_3377_arctan__monotone) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3286 (all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),N) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,N)))) # label(fact_3171_Suc3__eq__add__3) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3287 (all B_117 all A_156 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_156)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_117),zero_z891286103de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_156),B_117)),zero_z891286103de_int))))) # label(fact_1183_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3288 (all N natceiling(hAPP_nat_real(real_nat,N)) = N) # label(fact_3639_natceiling__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3289 (all Ta all A_1 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ta),A_1)) -> (all X_2 (is_int(X_2) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1)) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) != X_2)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Ta)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1)),Ta))))))))) # label(fact_5099_aset_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3290 (all X_1 (-hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd)))) # label(fact_2902_odd__iff__not__even) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3291 (all B all A (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B),A) = B <-> A = zero_zero_real)) # label(fact_613_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3292 (all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),pls)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,X_1)),zero_zero_real)))) # label(fact_98_le__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3293 (all P (is_int(P) -> P = hAPP_int_int(cOMBI_int,P))) # label(help_COMBI_1_1_COMBI_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3294 (all C all A all B (C = zero_zero_complex | hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A),B)) <-> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,C),A)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,C),B))))) # label(fact_1264_dvd__mult__cancel__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3295 (all M_10 all N_24 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_10)),hAPP_nat_real(semiri132038758t_real,N_24))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_24)))) # label(fact_2932_of__nat__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3296 (all B_1_1 all B_2_1 is_bool(hAPP_C24501650l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__Code____Numeral__Ocode____numeral_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3297 (all A all B all C (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A),B)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B)))))) # label(fact_1383_mult__le__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3298 (all A_3 all B_2 all M hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),M)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),M)))),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(div_mod_int(A_3),M)),hAPP_int_int(div_mod_int(B_2),M))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),B_2)) # label(fact_2487_zcong__zmod__aux) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3299 (all B_1 all K_1 all Diff all H all N_1 ((all M_1 all T (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))))) -> (N_1 = hAPP_nat_nat(suc,K_1) -> (all M_1 all T (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,minus_minus_real),hAPP_n546711566l_real(Diff,M_1))),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_f2119584768l_real(hAPP_f59594445l_real(cOMBC_2010481033l_real,hAPP_f1523861863l_real(hAPP_f1067681227l_real(cOMBB_1934313333l_real,big_co604158596t_real),hAPP_f1993765077t_real(hAPP_f895854793t_real(cOMBB_1948233853l_real,hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f1904307534l_real(hAPP_f224989205l_real(cOMBB_1419615765al_nat,Diff),hAPP_nat_fun_nat_nat(plus_plus_nat,M_1))),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))))),power_power_real))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),M_1))))),hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,hAPP_r1250527377l_real(times_times_real,B_1)),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,power_power_real),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),M_1)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),M_1))))))),T),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f1904307534l_real(hAPP_f224989205l_real(cOMBB_1419615765al_nat,Diff),hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M_1)))),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,T))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,M_1))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,T),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,M_1)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,M_1))))))))))))))) # label(fact_4924_Maclaurin__lemma2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3300 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)),zero_zero_rat)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),zero_zero_int))))) # label(fact_4716_Fract__less__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3301 (all X_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),one_one_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real))))) # label(fact_3923_real__root__le__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3302 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(tan,X)))))) # label(fact_3614_tan__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3303 (all A_231 all B_165 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_231),B_165) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_165),A_231)) # label(fact_512_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3304 (all X all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),X)),zero_zero_int),P_3)))))) # label(fact_2231_MultInv__prop3) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3305 (all A_196 all N_63 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_196),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_196),N_63)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_196),N_63)),A_196)) # label(fact_899_power__commutes) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3306 (all B_5 all Q_8 all R_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_8)),R_5)),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_5)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_8),zero_zero_int)))))) # label(fact_733_q__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3307 (all P all Q all R hAPP_bool_bool(hAPP_P921862901l_bool(P,R),hAPP_P603027463t_bool(Q,R)) = hAPP_P603027463t_bool(hAPP_f237669397t_bool(hAPP_f461489489t_bool(cOMBS_400904860l_bool,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__HOL__Obo) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3308 (all A_21 all B_17 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_21),B_17)) -> B_17 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B_17),A_21)),A_21))) # label(fact_2566_dvd__div__mult__self) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3309 (all C_68 all A_162 all B_123 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_162),B_123)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_68)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_162),C_68)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_123),C_68)))))) # label(fact_1146_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3310 (all A_91 (is_int(A_91) -> A_91 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_91),one_one_int))) # label(fact_2105_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3311 (all M zero_zero_nat != hAPP_nat_nat(suc,M)) # label(fact_2849_Zero__neq__Suc) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3312 (all A all B all C (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B),A)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B)))))) # label(fact_1160_mult__less__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3313 (all N_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(semiri132038758t_real,N_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)))) # label(fact_3145_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3314 (all A all B all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),hAPP_int_int(uminus_uminus_int,Ma))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma)))) # label(fact_3519_zcong__zminus) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3315 (all Z_1 Z_1 = hAPP_complex_real(re,of_real_complex(Z_1))) # label(fact_4114_Re__complex__of__real) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3316 (all X all Y (hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = Y -> zero_zero_real = X)) # label(fact_3347_real__sqrt__sum__squares__eq__cancel2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3317 (all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) -> zero_zero_nat = B_2 | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_3),B_2)))) # label(fact_1811_divides__ge) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3318 (all M_19 all N_44 all A_132 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_132)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_132),M_19)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_132),N_44))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_19),N_44))))) # label(fact_1410_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3319 (all C all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),C)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B),C)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B),A)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,C),A))))))) # label(fact_2484_div__dvd__div) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3320 (all A_59 all B_50 all C_29 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(div_mod_nat(A_59),C_29)),B_50)),C_29) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_59),B_50)),C_29)) # label(fact_2382_mod__mult__left__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3321 (all C_73 all A_173 all B_133 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_173),B_133)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_73)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_173),C_73)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_133),C_73)))))) # label(fact_1087_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3322 (all A_88 all B_76 (is_int(A_88) -> A_88 = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_88),B_76)),B_76))) # label(fact_2131_add__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3323 (all A_109 all B_87 all C_47 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_109),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_87),C_47)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_109),B_87)),C_47)) # label(fact_1949_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3324 (all N_37 all A_126 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_126)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_126),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_126),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_126),N_37))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_126),N_37)))))) # label(fact_1469_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3325 (all A_68 all B_56 (is_int(A_68) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(div_mod_int(A_68),B_56)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_68),B_56)),B_56)) = A_68)) # label(fact_2338_semiring__div__class_Omod__div__equality_H) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3326 (all V_13 all W_11 all Z_10 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_13)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,W_11)),Z_10)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_13),W_11))),Z_10)) # label(fact_333_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3327 (all V_2 ((hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> one_one_nat = hAPP_nat_nat(suc,hAPP_int_nat(number_number_of_nat,V_2))) & (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_nat_nat(suc,hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_2))))) # label(fact_3300_Suc__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3328 (all Lx_1 all Rx_1 all Ry_1 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_1),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx_1),Ry_1)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_1),Rx_1)),Ry_1)) # label(fact_498_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3329 (all X_45 all Y_33 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_45),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_33),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_33)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_45),Y_33))))) # label(fact_213_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3330 (all A (is_int(A) -> (zero_zero_int = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) <-> A = zero_zero_int))) # label(fact_36_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3331 (all Lx_5 all Ly_3 all Rx_5 all Ry_3 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_5),Ly_3)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx_5),Ry_3)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx_5),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx_5),Ly_3)),Ry_3))) # label(fact_470_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3332 (all A_195 all B_151 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_195),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_195),B_151)))) # label(fact_907_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3333 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X))),X)))) # label(fact_3457_ln__add__one__self__le__self2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3334 (all B_155 all A_204 (zero_zero_real != A_204 -> (zero_zero_real != B_155 -> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_204),B_155) != zero_zero_real))) # label(fact_830_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3335 (all B_140 all A_180 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_180)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_140)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_140),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_180),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_180),B_140))))) # label(fact_1051_split__mult__pos__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3336 (all V_17 all V_16 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_16)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_17)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_16)),hAPP_int_nat(number_number_of_nat,V_17)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_16),V_17))))) # label(fact_160_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3337 (all X_1 all Y_2 all B (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2)) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B),X_1)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B),Y_2)))))) # label(fact_1418_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3338 (all A_30 all B_25 all C_11 hAPP_int_int(div_mod_int(A_30),B_25) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_30),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_25),C_11))),B_25)) # label(fact_2508_mod__mult__self2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3339 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int)) -> hAPP_P1175774780nt_int(product_fst_int_int,hAPP_P1975530577nt_int(negateSnd,posDivAlg(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)))) # label(fact_4229_div__neg__neg) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3340 (all X_35 all Y_27 all Q_11 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_35),Q_11)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_27),Q_11)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_35),Y_27)),Q_11)) # label(fact_665_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3341 (all C all A all B (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C),A)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C),B))) <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A),B)))) # label(fact_2022_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3342 (all C_83 all D_16 all A_191 all B_147 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_191),B_147)) -> (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,C_83),D_16)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_191),C_83)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_147),D_16)))))) # label(fact_936_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3343 (all A_113 all B_91 all C_51 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_113),B_91)),C_51) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_113),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_91),C_51))) # label(fact_1906_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3344 (all P all Q all R hAPP_int_fun_int_int(hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,P),Q),R) = hAPP_int_fun_int_int(P,hAPP_int_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3345 (all X all Y all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),Y)))) # label(fact_3907_real__root__mult) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3346 (all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> X = Y | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)))) # label(fact_1707_dvd_Ole__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3347 (all B_73 all A_85 all C_36 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_36)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_73),A_85)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_73),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_85),C_36)))))) # label(fact_2149_add__increasing2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3348 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),M)),N) = M)) # label(fact_2722_div__mult__self1__is__m) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3349 (all Lx_1 all Rx_1 all Ry_1 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_1),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_1),Ry_1)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_1),Rx_1)),Ry_1)) # label(fact_502_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3350 (all X_1 hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,uminus_uminus_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),sin_coeff)),hAPP_r474017924t_real(power_power_real,X_1)))) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_1))) # label(fact_4823_lemma__sin__minus) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3351 (all Ma all N_1 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Ma),N_1) = zero_zero_nat <-> zero_zero_nat = Ma & zero_zero_nat = N_1)) # label(fact_4340_gcd__zero__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3352 (all A_105 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A_105) = A_105) # label(fact_1991_add__0__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3353 (all A all P_5 hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(wset(A),P_5)))) # label(fact_3357_wset__fin) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3354 (all C_75 all A_175 all B_135 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_175),B_135)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_75)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_75),A_175)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_75),B_135)))))) # label(fact_1071_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3355 (all A_190 all B_146 all K_7 (hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_146),K_7) = A_190 -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,B_146),A_190)))) # label(fact_945_dvdI) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3356 (all M_8 all N_22 hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,M_8)),hAPP_nat_int(semiri1621563631at_int,N_22))) # label(fact_2959_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3357 (all M_27 all A_212 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_212),one_one_rat)),M_27) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,M_27),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_212),M_27))) # label(fact_704_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3358 (all D_3 all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),A_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),B_2))))) # label(fact_4027_coprime__divprod) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3359 (all A_148 all N_51 all N_50 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_51),N_50)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A_148)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_148),N_51)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_148),N_50)))))) # label(fact_1271_power__increasing) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3360 (all A_229 all C_95 all D_20 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_95),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_229),D_20)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_229),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_95),D_20))) # label(fact_530_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3361 (all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(suc,N)))) # label(fact_2818_zero__less__Suc) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3362 (all A_202 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,zero_z126310315umeral),A_202) = zero_z126310315umeral) # label(fact_849_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3363 (all B_2 all A_3 (zero_zero_nat != A_3 -> (exists X_2 exists Y_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),Y_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),X_2)))) # label(fact_4374_bezout__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3364 (all A all C all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B),C))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A),B)))) # label(fact_2030_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3365 (all P hAPP_real_real(cOMBI_real,P) = P) # label(help_COMBI_1_1_COMBI_000tc__RealDef__Oreal_U) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3366 (all Z_1 all W (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_1),W)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),Z_1)) -> Z_1 = W))) # label(fact_1730_real__le__antisym) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3367 (all A hBOOL(tendst1507391555omplex(cnj,hAPP_complex_complex(cnj,A),at_complex(A)))) # label(fact_5078_cnj_Ocont) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3368 (all F (hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F))) -> hBOOL(summable_real(F)))) # label(fact_4828_summable__rabs__cancel) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3369 (all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M)) -> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M),one_one_int)) = hAPP_nat_nat(suc,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3247_inv__is__inv__aux) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3370 (all A_221 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_221),zero_z891286103de_int) = zero_z891286103de_int) # label(fact_595_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3371 (all A_3 all B_2 hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)))) # label(fact_5003_Fract__coprime) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3372 (all N_1 all X_1 ((-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real)) -> zero_zero_real = hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)) & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real)) -> hAPP_real_real(uminus_uminus_real,the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_real,zero_zero_real))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,power_power_real),N_1))),hAPP_real_real(uminus_uminus_real,X_1))))) = hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1))) & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_real,zero_zero_real))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,power_power_real),N_1))),X_1))) = hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)))) # label(fact_4885_root__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3373 (all R_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R_3)) -> (exists N_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_nat_real(real_nat,N_2))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_2),one_one_nat))),R_3)))))) # label(fact_3748_reals__Archimedean6) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3374 (all N_1 hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),N_1)))))) # label(fact_4738_bdd__int__set__l__le__finite) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3375 (all X_1 (is_int(X_1) -> ((exists A_2 exists B_4 (is_int(A_2) & X_1 = hAPP_P1175774780nt_int(twoSqu1907779896sum2sq,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_2),B_4)) & is_int(B_4))) <-> hBOOL(hAPP_int_bool(twoSqu1770640450sum2sq,X_1))))) # label(fact_1855_is__sum2sq__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3376 (all A (is_int(A) -> (zero_zero_int = A <-> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A) = zero_zero_int))) # label(fact_1975_double__zero__sym) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3377 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(ln,X)),zero_zero_real))))) # label(fact_3444_ln__less__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3378 (all C_85 all A_193 all B_149 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_193),B_149)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_193),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_149),C_85))))) # label(fact_924_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3379 (all Z_1 hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(cnj,Z_1)) = hAPP_complex_real(norm_norm_complex,Z_1)) # label(fact_4076_complex__mod__cnj) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3380 (all A_3 all B_2 hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(uminus_uminus_int,A_3)),B_2)) # label(fact_3522_zdiv__zminus2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3381 (all D all Sa exists Z all X_2 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_2),Z)) -> (-hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_2),Sa))) <-> -hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_2),Sa)))))) # label(fact_2249_minf_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3382 (all Y_3 all X_3 ((exists A_2 ((all N_2 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(X_3,N_2))),A_2))) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_2)))) -> (hBOOL(vanishes(Y_3)) -> hBOOL(vanishes(hAPP_f2062659861at_rat(hAPP_f1131305186at_rat(cOMBS_nat_rat_rat,hAPP_f1830834240at_rat(hAPP_f1073644437at_rat(cOMBB_1926947at_nat,times_times_rat),X_3)),Y_3)))))) # label(fact_5127_vanishes__mult__bounded) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3383 (all N_59 all M_24 all X_28 all Y_22 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,X_28),Y_22)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_59),M_24)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_28),N_59)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_22),M_24)))))) # label(fact_998_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.54 3384 (all K_1 all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,K_1)),hAPP_int_int(number_number_of_int,L))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)))) # label(fact_127_less__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3385 (all M_28 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M_28),M_28) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),one_one_int)),M_28)) # label(fact_691_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3386 (all K_2 hAPP_int_int(bit0,hAPP_int_int(succ,K_2)) = hAPP_int_int(succ,hAPP_int_int(bit1,K_2))) # label(fact_3292_succ__Bit1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3387 (all Z_4 all Z_1 all W_13 all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_13),W)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_4),Z_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_13),Z_4)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),Z_1)))))) # label(fact_115_zadd__zless__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3388 (all A_211 all M_26 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_211),M_26)),M_26) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_211),one_one_int)),M_26)) # label(fact_705_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3389 (all L_2 hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),L_2)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),hAPP_int_int(bit0,L_2))) # label(fact_1581_diff__bin__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3390 (all C_75 all A_175 all B_135 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_175),B_135)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_75)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_75),A_175)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_75),B_135)))))) # label(fact_1072_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3391 (all A all C all D all B (is_int(B) & is_int(D) -> (B != zero_zero_int -> (zero_zero_int != D -> (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),D) <-> hAPP_int_rat(hAPP_int_fun_int_rat(fract,C),D) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)))))) # label(fact_4687_eq__rat_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3392 (all X_21 all N_31 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_21),N_31) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21))) # label(fact_1800_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3393 (all K_2 all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,K_2),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,K_2),M)))))) # label(fact_2719_div__le__mono2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3394 (all A_69 all B_57 hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(div_mod_int(A_69),B_57)),B_57) = zero_zero_int) # label(fact_2331_mod__div__trivial) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3395 (all P_1 collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),P_1)) = image_nat_int(semiri1621563631at_int,collect_nat(hAPP_f1154658180t_bool(hAPP_f1361476881t_bool(cOMBB_int_bool_nat,P_1),semiri1621563631at_int)))) # label(fact_4734_Nat__Transfer_Otransfer__int__nat__set__functions_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3396 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natfloor(X)),one_one_nat) = natfloor(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),one_one_real)))) # label(fact_3405_natfloor__add__one) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3397 (all R_3 all Q_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,R_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Q_2)) = A_3 -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_2),one_one_int)))))) # label(fact_422_self__quotient__aux2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3398 (all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,zero_zero_nat))) <-> zero_zero_nat = N_1)) # label(fact_3048_less__Suc0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3399 (all X all Q_2 all N all R_3 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Q_2),N)),R_3) = X -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),R_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,R_3),N)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),X)))))) # label(fact_1826_divides__div__not) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3400 (all A_93 A_93 = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_93),zero_zero_complex)) # label(fact_2090_diff__0__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3401 (all W_8 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,W_8)),hAPP_int_rat(number_number_of_rat,W_8)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_int_rat(number_number_of_rat,W_8)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_361_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3402 (all P_2 all P_1 all X_1 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(P_2) <-> hBOOL(P_1))) -> (hBOOL(P_1) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) <-> hBOOL(P_2) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))))) # label(fact_1615_conj__le__cong) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3403 (all B_34 all B_33 all A_43 all C_17 all A_42 (hAPP_int_int(div_mod_int(A_42),C_17) = hAPP_int_int(div_mod_int(A_43),C_17) -> (hAPP_int_int(div_mod_int(B_33),C_17) = hAPP_int_int(div_mod_int(B_34),C_17) -> hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_42),B_33)),C_17) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_43),B_34)),C_17)))) # label(fact_2437_mod__diff__cong) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3404 (all A_216 (is_int(A_216) -> A_216 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),A_216))) # label(fact_647_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3405 (all C_58 all D_11 all A_136 all B_105 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_136),B_105)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_58),D_11)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_136)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_58)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_136),C_58)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_105),D_11)))))))) # label(fact_1369_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3406 (all A_89 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),A_89) = A_89) # label(fact_2121_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3407 (all A_66 all B_54 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_66),B_54)),B_54)),hAPP_C498520661umeral(div_mo1740067990umeral(A_66),B_54)) = A_66) # label(fact_2348_mod__div__equality) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3408 (all Y all X all U_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,X)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),U_1))))) # label(fact_3379_real__sqrt__sum__squares__less) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3409 (all B_130 all A_170 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_170)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_130),zero_z891286103de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_130),A_170)),zero_z891286103de_int))))) # label(fact_1102_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3410 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hBOOL(hAPP_real_bool(deriv_real(ln,X_1),hAPP_real_real(inverse_inverse_real,X_1))))) # label(fact_3866_DERIV__ln) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3411 (all Ma all N_1 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(semiri1619134803umeral,Ma)),hAPP_n1108039445umeral(semiri1619134803umeral,N_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)))) # label(fact_2941_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3412 (all A_91 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_91),one_on1684967323de_int) = A_91) # label(fact_2110_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3413 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real)) -> hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_real_real(sin,hAPP_real_real(arccos,X))))) # label(fact_3632_sin__arccos) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3414 (all W_5 ((zero_zero_nat = hAPP_int_nat(number_number_of_nat,W_5) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,W_5)) = one_one_nat) & (hAPP_int_nat(number_number_of_nat,W_5) != zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,W_5)) = zero_zero_nat))) # label(fact_725_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3415 (all W_12 hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,W_12)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),one_one_real)),hAPP_int_real(number267125858f_real,W_12))) # label(fact_271_double__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3416 (all N zero_zero_nat != hAPP_nat_nat(fact_fact_nat,N)) # label(fact_3801_fact__nonzero__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3417 (all Z_4 all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),Z_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat,Z_1)),hAPP_int_nat(nat,Z_4)))) # label(fact_2741_nat__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3418 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y)) -> (exists R_1 exists A_2 (hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(cos,A_2)) = X & hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(sin,A_2)) = Y)))) # label(fact_3948_polar__ex1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3419 (all X_1 hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,fact_fact_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(cos,X_1)))) # label(fact_4870_cos__paired) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3420 (all Ma all W_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),W_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat,W_1)),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),hAPP_nat_int(semiri1621563631at_int,Ma)))))) # label(fact_3189_nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3421 (all M M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),zero_zero_nat)) # label(fact_1739_minus__nat_Odiff__0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3422 (all Ma all Z_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_int_nat(nat,Z_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),Z_2)))) # label(fact_3124_zless__nat__eq__int__zless) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3423 (all N_27 all Y_7 all X_10 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Y_7),X_10)) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_10),Y_7)),N_27) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_10),N_27)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_7),N_27)))) # label(fact_2583_div__power) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3424 (all A_211 all M_26 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_211),one_on1684967323de_int)),M_26) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_211),M_26)),M_26)) # label(fact_710_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3425 (all N hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),N) = hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)))),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3709_sin__cos__npi) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3426 (all X_4 all N_6 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_6)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_4)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N_6),X_4)))))) # label(fact_3929_real__root__pos__pos) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3427 (all M all K_2 all N (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K_2),N) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M))))) # label(fact_5005_coprime__dvd__mult__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3428 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = zero_zero_int))) # label(fact_2601_div__pos__pos__trivial) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3429 (all X_1 all Y_2 (hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),Y_2))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Y_2)) & hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)))) # label(fact_3777_even__power__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3430 (all A_55 all C_25 all B_46 hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_55),C_25)),B_46)),C_25) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_55),B_46)),C_25)) # label(fact_2400_zmod__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3431 (all X all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)) = hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(uminus_uminus_real,X)))) # label(fact_3913_real__root__minus) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3432 (all A_201 all E_3 all B_154 all C_87 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_201),B_154)),E_3)),C_87) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_201),E_3)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_154),E_3)),C_87))) # label(fact_867_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3433 (all A_59 all B_50 all C_29 hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_59),B_50)),C_29) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_59),C_29)),B_50)),C_29)) # label(fact_2384_mod__mult__left__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3434 (all N_1 (zero_zero_nat = N_1 <-> hAPP_nat_real(real_nat,N_1) = zero_zero_real)) # label(fact_3645_real__of__nat__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3435 (all X all Y hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3350_real__sqrt__sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3436 (all B_1_1 is_bool(summable_real(B_1_1))) # label(gsy_c_Series_Osummable_000tc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3437 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)))))) # label(fact_3126_one__less__mult) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3438 (all D all Sa exists Z all X_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z),X_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_2),Sa))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_2),Sa)))))) # label(fact_2274_pinf_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3439 (all F all Y_2 all G all X_1 (hBOOL(hAPP_real_bool(sums_real(G),X_1)) -> (hBOOL(hAPP_real_bool(sums_real(F),Y_2)) -> hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,F),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,div_div_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,G),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,div_div_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,minus_minus_nat),one_one_nat))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_2)))))) # label(fact_4869_sums__if) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3440 (all X_1 all Y_2 all B (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2)) <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B),X_1)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B),Y_2)))))) # label(fact_1454_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3441 (all Xa_3 all X_13 all Y_10 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_3),X_13)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_3),Y_10)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa_3),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_13),Y_10))) # label(fact_2304_mult__right_Odiff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3442 (all A_3 all B_2 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),A_3)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))))) # label(fact_415_zadd__power3) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3443 (all I all J_2 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(legacy_zgcd(I),J_2)))) # label(fact_4500_zgcd__pos) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3444 (all R_3 of_real_complex(R_3) = rcis(R_3,zero_zero_real)) # label(fact_4258_rcis__zero__arg) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3445 (all N all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N))))) # label(fact_723_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3446 (all X all Y hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(legacy_zgcd(X),Y)))) # label(fact_4501_zgcd__geq__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3447 (all Z_1 (is_int(Z_1) -> Z_1 = hAPP_int_int(uminus_uminus_int,hAPP_int_int(uminus_uminus_int,Z_1)))) # label(fact_3518_zminus__zminus) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3448 (all K_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),K_2))))) # label(fact_1629_add__less__mono1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3449 (all X hAPP_complex_real(re,hAPP_complex_complex(uminus473333897omplex,X)) = hAPP_real_real(uminus_uminus_real,hAPP_complex_real(re,X))) # label(fact_4124_Re_Ominus) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3450 (all X_20 all Y_16 all A_120 all B_98 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_20),Y_16)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_120),B_98)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_20),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_16),B_98))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_20),A_120)),B_98))) # label(fact_1818_mult__diff__mult) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3451 (all N_19 zero_zero_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),hAPP_nat_nat(suc,N_19))) # label(fact_2995_power__0__Suc) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3452 (all B_118 all A_157 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_157)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_118),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_118),A_157)),zero_zero_rat))))) # label(fact_1178_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3453 (all A_1 all Ma (hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,A_1),rsetR(Ma))) -> hBOOL(hAPP_f448129468l_bool(finite_finite_int,A_1)))) # label(fact_4595_RsetR__fin) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3454 (all C_1 all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_1),A_3)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_1),B_2))))) # label(fact_1794_divides__mul__l) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3455 (all A all B all C (hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A),C) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A),B) <-> C = B)) # label(fact_1936_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3456 (all C_64 all A_142 all B_111 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_64),A_142)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_64),B_111))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_64)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_142),B_111))))) # label(fact_1335_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3457 (all A_121 all C_54 all B_99 all D_9 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_121),B_99)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C_54),D_9)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_121),C_54)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_99),D_9))) # label(fact_1810_add__diff__add) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3458 (all A_3 all B_2 all C_1 all Q_2 all R_3 (hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(B_2,C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_2),R_3))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_1)) -> hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2),C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),Q_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),R_3)),C_1))),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),R_3)),C_1))))))) # label(fact_4212_divmod__nat__rel__mult1__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3459 (all Ma all No all F ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_2)),hAPP_nat_real(F,hAPP_nat_nat(suc,N_2))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,Ma)),hAPP_nat_real(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),No)))))) # label(fact_3374_lemma__f__mono__add) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3460 (all N_1 all Ma (is_int(Ma) -> (zero_zero_int != Ma -> (one_one_int = hAPP_int_int(abs_abs_int,N_1) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),N_1)),Ma)))))) # label(fact_3449_zdvd__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3461 (all K_2 (is_int(K_2) -> hAPP_int_int(number_number_of_int,K_2) = K_2)) # label(fact_53_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3462 (all N_1 all Ma (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(real_nat,Ma))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),Ma)))) # label(fact_3648_real__of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3463 (all D all A all B (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),B)) <-> one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D),B) & one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D),A))) # label(fact_5000_coprime__mul__eq__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3464 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(hAPP_b646336293l_real(if_real,fTrue),X),Y) = X) # label(help_If_1_1_If_000tc__RealDef__Oreal_T) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3465 (all A_3 all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> natceiling(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_nat_real(real_nat,A_3))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natceiling(X)),A_3))) # label(fact_3706_natceiling__add) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3466 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,X)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) # label(fact_3856_exp__bound) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3467 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = N)) # label(fact_3180_Suc__diff__1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3468 (all K_1 all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K_1)),hAPP_int_int(bit0,L))))) # label(fact_347_rel__simps_I16_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3469 (all X_1 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,X_1)),hAPP_real_real(exp_real,Y_2))))) # label(fact_3823_exp__le__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3470 (all M all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(div_mod_nat(M),N)),M))) # label(fact_2693_mod__less__eq__dividend) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3471 (all A_76 all B_64 all B_63 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_76),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_64),B_63)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_76),B_64)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_76),B_63))) # label(fact_2291_mult_Oadd__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3472 (all X_1 hBOOL(hAPP_real_bool(isCont_real_real(sin),X_1))) # label(fact_5164_isCont__sin) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3473 (all M all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),M))) # label(fact_1748_Nat_Odiff__le__self) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3474 (all N_1 all Ma (hAPP_nat_int(semiri1621563631at_int,Ma) = hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N_1)) <-> Ma = zero_zero_nat & N_1 = zero_zero_nat)) # label(fact_3530_negative__eq__positive) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3475 (all B_128 all A_168 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_168)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_128)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_168),B_128)))))) # label(fact_1112_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3476 (all P_1 all P_2 all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D)) -> ((all X_2 all K (is_int(X_2) & is_int(K) -> (hBOOL(hAPP_int_bool(P_2,X_2)) <-> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D))))))) -> ((exists Z all X_2 (is_int(X_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),X_2)) -> (hBOOL(hAPP_int_bool(P_2,X_2)) <-> hBOOL(hAPP_int_bool(P_1,X_2)))))) -> ((exists X1 hBOOL(hAPP_int_bool(P_2,X1))) -> (exists X1 (hBOOL(hAPP_int_bool(P_1,X1)) & is_int(X1)))))))) # label(fact_4007_plusinfinity) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3477 (all A_129 all N_40 all N_39 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_40),N_39)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_129)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_129),N_40)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_129),N_39)))))) # label(fact_1441_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3478 (all A_215 A_215 = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_215),one_on1645066479umeral)) # label(fact_657_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3479 (all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),A))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A)))) # label(fact_2174_zero__le__double__add__iff__zero__le__single__add) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3480 (all K_2 all L_2 hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(abs_abs_int,K_2)),hAPP_int_int(abs_abs_int,L_2))),hAPP_int_int(div_mod_int(hAPP_int_int(abs_abs_int,K_2)),hAPP_int_int(abs_abs_int,L_2))) = pdivmod(K_2,L_2)) # label(fact_3946_pdivmod__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3481 (all W_1 all X_1 (hAPP_int_rat(number_number_of_rat,W_1) = X_1 <-> hAPP_int_rat(number_number_of_rat,W_1) = X_1)) # label(fact_176_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3482 (all C_57 all D_10 all A_135 all B_104 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_135),B_104)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_57),D_10)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_104)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_57)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_135),C_57)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_104),D_10)))))))) # label(fact_1376_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3483 (all P_3 hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3)) = frct(P_3)) # label(fact_4712_Frct__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3484 (all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,Ma)),hAPP_nat_nat(semiri984289939at_nat,N_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)))) # label(fact_2945_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3485 (all X_1 all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim856651990g_real(X_1)),A)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,A)),one_one_real))))) # label(fact_4471_ceiling__less__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3486 (all X_7 all Q_6 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_7),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_7),Q_6)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_7),hAPP_nat_nat(suc,Q_6))) # label(fact_3031_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3487 (all A_219 (is_int(A_219) -> A_219 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_219),zero_zero_int))) # label(fact_605_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3488 (all F all L all C (hBOOL(tendsto_real_real(F,L,at_real(C))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),L)) -> (exists R_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1)) & (all X_2 (X_2 != C & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),X_2))),R_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(F,X_2)))))))))) # label(fact_5087_LIM__fun__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3489 (all X_1 all Y_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_1)) <-> Y_2 != X_1 & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2)))) # label(fact_1701_dvd_Oless__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3490 (all X hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi))) = hAPP_real_real(sin,X)) # label(fact_3718_sin__periodic) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3491 (all X all Y hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)),X))) # label(fact_4331_gcd__semilattice__nat_Oinf__le1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3492 (all X_1 all Y_2 (hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)) | hBOOL(hAPP_nat_bool(even_odd_even_nat,Y_2)) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_1),Y_2))))) # label(fact_3761_even__product__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3493 (all X all Y hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(uminus_uminus_int,X)),Y)) # label(fact_4990_gcd__neg1__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3494 (all X all Y (Y = X -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)))) # label(fact_1705_dvd_Oeq__refl) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3495 (all N_1 hAPP_n1699378549t_bool(ord_lessThan_nat,N_1) = hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1)) # label(fact_4951_atLeast0LessThan) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3496 (all P all Q all R hAPP_real_complex(hAPP_n1983812447omplex(P,R),hAPP_nat_real(Q,R)) = hAPP_nat_complex(hAPP_f219720582omplex(hAPP_f1052840010omplex(cOMBS_696648398omplex,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__Complex__Ocom) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3497 (all A_216 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),A_216) = A_216) # label(fact_649_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3498 (all P all Q (-hBOOL(Q) | hBOOL(fdisj(P,Q)))) # label(help_fdisj_2_1_U) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3499 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) -> hAPP_int_nat(nat,hAPP_int_int(div_mod_int(X),Y)) = hAPP_nat_nat(div_mod_nat(hAPP_int_nat(nat,X)),hAPP_int_nat(nat,Y))))) # label(fact_2698_nat__mod__distrib) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3500 (all Y_2 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2)) -> (exists X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),X_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2)) & hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),field_1210416355s_real))))))) # label(fact_4634_Rats__dense__in__nn__real) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3501 (all A_224 all N_68 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_224),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_224),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68))) # label(fact_577_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3502 (all C all X_1 all Ta all A all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),D)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),D))),Ta))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Ta)))))) # label(fact_458_zdvd__period) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3503 (all N_1 all A (hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially)) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A,N_2)))) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,hAPP_nat_nat(suc,N_2))),hAPP_nat_real(A,N_2)))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1)))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)))))))) # label(fact_5061_summable__Leibniz_H_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3504 (all M_25 all A_185 all N_60 all B_142 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_185),N_60)),B_142)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_25),N_60)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_185),M_25)),B_142))))) # label(fact_996_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3505 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y)) -> (exists X_2 hAPP_real_real(exp_real,X_2) = Y))) # label(fact_3943_exp__total) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3506 (all Ma all N_1 (one_one_nat = Ma & N_1 = one_one_nat <-> one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N_1))) # label(fact_1658_nat__1__eq__mult__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3507 (all P_1 collect_nat(P_1) = P_1) # label(fact_146_Collect__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3508 (all X_1 all N_1 all Diff all F (hAPP_n546711566l_real(Diff,zero_zero_nat) = F -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (zero_zero_real != X_1 -> ((all M_1 all X_2 hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),X_2),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),X_2)))) -> (exists T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(abs_abs_real,T))) & hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N_1),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1))) = hAPP_real_real(F,X_1) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_1)))))))))) # label(fact_4931_Maclaurin__all__lt) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3509 (all X_34 all P_7 all Q_10 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_34),P_7)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_34),Q_10)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_34),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_10))) # label(fact_670_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3510 (all P_5 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),P_5)) & (all M_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M_1),P_5)) -> P_5 = M_1 | M_1 = one_one_nat)) <-> hBOOL(hAPP_nat_bool(prime,P_5)))) # label(fact_4184_prime__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3511 (all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_int_real(number267125858f_real,Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,pls)),Y_2)))) # label(fact_412_less__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3512 (all V_18 all W_14 hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_18),W_14)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_18)),hAPP_int_complex(number528085621omplex,W_14))) # label(fact_89_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3513 (all K_2 hAPP_int_rat(hAPP_int_fun_int_rat(fract,K_2),one_one_int) = hAPP_int_rat(ring_1_of_int_rat,K_2)) # label(fact_4678_of__int__rat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3514 (all N_1 all Diff all F all H (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H)) & hAPP_n546711566l_real(Diff,zero_zero_nat) = F & (all M_1 all T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))))) -> (exists T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),T)) & hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N_1),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N_1))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)))))) # label(fact_4935_Maclaurin2__objl) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3515 (all X all Y hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(cnj,X)),hAPP_complex_complex(cnj,Y)) = hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),Y))) # label(fact_4077_cnj_Odiff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3516 (all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Ma)),N_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)))) # label(fact_3062_Suc__le__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3517 (all A all B all C all D (hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),D) -> (A = B <-> D = C))) # label(fact_1958_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3518 (all X all Y hAPP_complex_real(im,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_complex_real(im,X)),hAPP_complex_real(im,Y))) # label(fact_4098_Im_Oadd) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3519 (all B_130 all A_170 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_170)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_130),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_130),A_170)),zero_zero_int))))) # label(fact_1098_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3520 (all A_146 one_one_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_146),zero_zero_nat)) # label(fact_1298_power__0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3521 (all P_3 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_2762_Euler_Oaux____2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3522 (all A hBOOL(hAPP_complex_bool(isCont_complex_real(re),A))) # label(fact_5167_Re_OisCont) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3523 (all X_1 all N_1 all Y_2 (hAPP_int_int(div_mod_int(X_1),N_1) = hAPP_int_int(div_mod_int(Y_2),N_1) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N_1),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_2))))) # label(fact_2532_zmod__eq__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3524 (all A_3 all B_2 hAPP_int_rat(hAPP_int_fun_int_rat(fract,B_2),A_3) = hAPP_rat_rat(inverse_inverse_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2))) # label(fact_4684_inverse__rat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3525 (all X all N hAPP_real_real(tan,X) = hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,N)),pi)))) # label(fact_3601_tan__periodic__n) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3526 (all Y_3 all X_3 (hBOOL(vanishes(X_3)) -> (hBOOL(vanishes(Y_3)) -> hBOOL(vanishes(hAPP_f2062659861at_rat(hAPP_f1131305186at_rat(cOMBS_nat_rat_rat,hAPP_f1830834240at_rat(hAPP_f1073644437at_rat(cOMBB_1926947at_nat,plus_plus_rat),X_3)),Y_3)))))) # label(fact_5131_vanishes__add) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3527 (all M hAPP_int_int(legacy_zgcd(one_one_int),M) = one_one_int) # label(fact_4489_zgcd__1__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3528 (all A all B (B = zero_zero_rat | zero_zero_rat = A <-> zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),B))) # label(fact_838_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3529 (all N_1 all Ma (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,N_1)),hAPP_int_real(real_int,Ma))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,N_1)),Ma)))) # label(fact_4443_number__of__less__real__of__int__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3530 (all A_194 all B_150 hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_194),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_150),A_194)))) # label(fact_915_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3531 (all V_2 ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),one_one_nat) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_2))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls)) -> one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),one_one_nat)))) # label(fact_3298_nat__number__of__add__1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3532 (all N (hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),N) = one_one_int | hAPP_int_int(number_number_of_int,min) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),N))) # label(fact_1685_neg__one__power) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3533 (all C_1 all D_3 all A_3 all B_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C_1),D_3),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),C_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_2),D_3)),M))))) # label(fact_1037_zcong__zadd) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3534 (all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,W)),zero_zero_int)) -> hAPP_P1175774780nt_int(product_fst_int_int,hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,one_one_int),hAPP_int_int(uminus_uminus_int,hAPP_int_int(number_number_of_int,W))))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,one_one_int),hAPP_int_int(number_number_of_int,W)))) # label(fact_4223_div__pos__neg__1__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3535 (all A_158 all B_119 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_158),B_119))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_158)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_119))))) # label(fact_1168_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3536 (all C_46 all A_104 all B_86 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_46),A_104)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_46),B_86))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_104),B_86)))) # label(fact_1994_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3537 (all A_107 A_107 = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_107),zero_zero_real)) # label(fact_1971_add__0__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3538 (all P all Q all R hAPP_P603027463t_bool(hAPP_f237669397t_bool(hAPP_f513299955t_bool(cOMBB_1344650379nt_int,P),Q),R) = hAPP_bool_bool(P,hAPP_P603027463t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__prod_Itc__Int__O) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3539 (all A_9 all N_14 all B_7 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_9),hAPP_nat_nat(suc,N_14))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_7),hAPP_nat_nat(suc,N_14)))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_7)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_9),B_7))))) # label(fact_3114_power__le__imp__le__base) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3540 (all W all V_2 ((-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_int_nat(nat,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,V_2)),hAPP_int_nat(number_number_of_nat,W))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,W))) & (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,W)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,W))))) # label(fact_3283_power__nat__number__of__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3541 (all R_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R_3)) -> (exists N_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_3),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_2)))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N_2)),R_3)))))) # label(fact_3750_reals__Archimedean6a) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3542 (all R_3 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,archim856651990g_real(R_3))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,R_3),one_one_real)))) # label(fact_4457_real__of__int__ceiling__le__add__one) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3543 (all A_3 all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,A_3)),X)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),archim1246769320r_real(X))))) # label(fact_4417_le__floor) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3544 (all A_167 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_167),A_167)))) # label(fact_1124_zero__le__square) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3545 (all X hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi))) = hAPP_real_real(tan,X)) # label(fact_3610_tan__periodic) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3546 (all A_3 all B_2 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),A_3)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),B_2))))) # label(fact_4170_prime__divprod) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3547 (all A_183 all M_22 all N_57 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_183),M_22)),N_57) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_183),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_22),N_57))) # label(fact_1017_power__mult) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3548 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,M)))),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),M)),N)) # label(fact_3206_Suc__div__eq__add3__div) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3549 (all P all Q all R hAPP_int_bool(hAPP_r481142414t_bool(hAPP_f1845081853t_bool(cOMBC_int_real_bool,P),Q),R) = hAPP_real_bool(hAPP_i688218208l_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__RealDef__Oreal_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3550 (all A (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A))))) # label(fact_2194_zero__less__double__add__iff__zero__less__single__add) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3551 (all X all Y all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),Y)))))) # label(fact_3903_real__root__le__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3552 (all X all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hAPP_real_real(hAPP_n546711566l_real(root,M),hAPP_real_real(hAPP_n546711566l_real(root,N),X)) = hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(hAPP_n546711566l_real(root,M),X))))) # label(fact_3898_real__root__commute) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3553 (all N all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(real_nat,X))),hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),X)))))) # label(fact_3699_real__of__nat__div2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3554 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X)),hAPP_real_real(exp_real,Y))))) # label(fact_3819_exp__less__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3555 (all X all Y ((Y = zero_zero_nat -> bezw(X,Y) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,one_one_int),zero_zero_int)) & (zero_zero_nat != Y -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(Y,hAPP_nat_nat(div_mod_nat(X),Y)))),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_P1175774780nt_int(product_fst_int_int,bezw(Y,hAPP_nat_nat(div_mod_nat(X),Y)))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(Y,hAPP_nat_nat(div_mod_nat(X),Y)))),hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),Y))))) = bezw(X,Y)))) # label(fact_4230_bezw_Osimps) # label(axiom) # label(non_clause). [assumption]. 7.56/7.55 3556 (all Lx_6 all Ly_4 all Rx_6 all Ry_4 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_6),Rx_6)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ly_4),Ry_4)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_6),Ly_4)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_6),Ry_4))) # label(fact_469_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3557 (all A_124 all N_34 all B_102 (hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_124),N_34) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_102),N_34) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_124)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_102)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_34)) -> B_102 = A_124))))) # label(fact_1481_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3558 (all A_208 hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_208),zero_z126310315umeral))) # label(fact_758_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3559 (all X all Y all Z_1 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Y)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Z_1)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_1))) # label(fact_64_zpower__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3560 (all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) -> (A_3 != B_2 -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),A_3)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))))) # label(fact_1708_dvd_Ole__neq__trans) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3561 (all P_1 all P1 all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D)) -> ((all X_2 all K (is_int(X_2) & is_int(K) -> (hBOOL(hAPP_int_bool(P1,X_2)) <-> hBOOL(hAPP_int_bool(P1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D))))))) -> ((exists Z all X_2 (is_int(X_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Z)) -> (hBOOL(hAPP_int_bool(P_1,X_2)) <-> hBOOL(hAPP_int_bool(P1,X_2)))))) -> ((exists X1 hBOOL(hAPP_int_bool(P1,X1))) -> (exists X1 (is_int(X1) & hBOOL(hAPP_int_bool(P_1,X1))))))))) # label(fact_4006_minusinfinity) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3562 (all V_5 all W_3 all C_55 hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_5),W_3))),C_55) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_5)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_int_rat(number_number_of_rat,W_3)),C_55))) # label(fact_1577_add__number__of__diff1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3563 (all K_2 all L_2 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),L_2))),L_2) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(bit1,K_2)),L_2)) # label(fact_244_mult__Bit1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3564 (all A_158 all B_119 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_158),B_119))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_158)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_119))))) # label(fact_1169_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3565 (all A_62 hAPP_nat_nat(div_mod_nat(A_62),zero_zero_nat) = A_62) # label(fact_2369_mod__by__0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3566 (all J all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J),zero_zero_int),P_5)) -> hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) = hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(multInvPair(A,P_5),J)))))))) # label(fact_3301_MultInvPair__card__two) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3567 (all D all Ta (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),Ta)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(uminus_uminus_int,Ta))))) # label(fact_3521_uminus__dvd__conv_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3568 (all A all X_1 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(uminus_uminus_int,A)),hAPP_int_nat(nat,X_1)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(nat,X_1))))) # label(fact_3554_neg__even__power) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3569 (all X all Y hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,fTrue),X),Y) = X) # label(help_If_1_1_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_T) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3570 (all C_42 all A_100 all B_82 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_100),B_82)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_100),C_42)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_82),C_42))))) # label(fact_2017_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3571 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)))) # label(fact_3058_Suc__le__lessD) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3572 (all X_40 all N_69 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_40),N_69)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_40),N_69)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_40),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69))) # label(fact_428_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3573 (all Y Y = hAPP_real_real(tan,hAPP_real_real(arctan,Y))) # label(fact_3598_tan__arctan) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3574 (all W_7 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),hAPP_int_int(number_number_of_int,W_7))),hAPP_int_int(number_number_of_int,W_7)) = hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W_7))) # label(fact_371_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3575 (all A all B all C all D (hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,C),D) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A),B) -> (B = A <-> D = C))) # label(fact_1959_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3576 (all P_3 quotient_of(hAPP_rat_rat(sgn_sgn_rat,P_3)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(sgn_sgn_int,hAPP_P1175774780nt_int(product_fst_int_int,quotient_of(P_3)))),one_one_int)) # label(fact_4653_rat__sgn__code) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3577 (all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),pls)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,X_1)),zero_zero_rat)))) # label(fact_395_less__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3578 (all I all K_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),J_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2)))) # label(fact_1778_diff__diff__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3579 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)),zero_zero_rat)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),zero_zero_int))))) # label(fact_4725_Fract__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3580 (all Ma all N_1 the_Pr588456374at_nat(divmod_nat_rel(Ma,N_1)) = divmod_nat(Ma,N_1)) # label(fact_4889_divmod__nat__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3581 (all A all B divmod_int_rel(A,B) = produc93441119t_bool(hAPP_f1048215610t_bool(hAPP_f472159229t_bool(cOMBC_1683390479t_bool,hAPP_f1545556668t_bool(hAPP_f85237525t_bool(cOMBB_389152643ol_int,cOMBS_int_bool_bool),hAPP_f1344929775l_bool(hAPP_f1830214065l_bool(cOMBB_793166024ol_int,hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj)),hAPP_f720654810t_bool(hAPP_f908327869t_bool(cOMBB_1290694363ol_int,hAPP_f1512942609t_bool(cOMBB_int_bool_int,hAPP_i1948725293t_bool(fequal_int,A))),hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,plus_plus_int),hAPP_int_fun_int_int(times_times_int,B)))))),hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,fimplies(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))),hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),B))))),hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,fimplies(hAPP_bool_bool(fNot,hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)))),hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,B))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),zero_zero_int))))))) # label(fact_4837_divmod__int__rel__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3582 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),zero_zero_int)) -> hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_3)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_2),one_one_int)),A_3))),one_one_int))) # label(fact_2319_neg__zmod__mult__2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3583 (all N all M all K_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N) -> N = M))) # label(fact_2279_mult__left__cancel) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3584 (all B_129 all A_169 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_169)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_129),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_169),B_129)),zero_zero_nat))))) # label(fact_1105_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3585 (all N (is_int(N) -> N = quickcheck_int_of(hAPP_i1732201573de_int(quickcheck_of_int,N)))) # label(fact_4557_int__of__of__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3586 (all X_35 all Y_27 all Q_11 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_35),Q_11)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_27),Q_11)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_35),Y_27)),Q_11)) # label(fact_666_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3587 (all V all V_1 (is_int(V) & is_int(V_1) -> ((-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V))) -> (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1))) -> V = V_1) & (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1))) -> hAPP_int_int(number_number_of_int,V) = zero_zero_int)) & (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,V_1)),zero_zero_int))) <-> hAPP_int_nat(number_number_of_nat,V) = hAPP_int_nat(number_number_of_nat,V_1)))) # label(fact_3279_eq__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3588 (all Ma all N_1 (Ma = N_1 <-> hAPP_nat_nat(semiri984289939at_nat,N_1) = hAPP_nat_nat(semiri984289939at_nat,Ma))) # label(fact_2830_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3589 (all A_133 all N_46 all N_45 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_46),N_45)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_133)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_133),one_on1645066479umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_133),N_45)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_133),N_46))))))) # label(fact_1405_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3590 (all X_31 all Y_25 (is_int(X_31) & is_int(Y_25) -> (Y_25 != X_31 -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_31),Y_25)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_25),X_31)))))) # label(fact_791_linorder__neqE__linordered__idom) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3591 (all A all W_1 (hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A),hAPP_int_nat(number_number_of_nat,W_1)) = zero_zero_real <-> A = zero_zero_real & hAPP_int_nat(number_number_of_nat,W_1) != zero_zero_nat)) # label(fact_714_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3592 (all Z_1 (hBOOL(hAPP_int_bool(nat_neg,Z_1)) -> zero_zero_nat = hAPP_int_nat(nat,Z_1))) # label(fact_3275_neg__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3593 (all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),P_5)) -> hAPP_int_nat(nat,A) = hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(bnorRset(A),P_5))))) # label(fact_4290_Bnor__prime) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3594 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) -> N = hAPP_nat_nat(suc,zero_zero_nat) | N = zero_zero_nat)) # label(fact_3225_less__2__cases) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3595 (all A_106 A_106 = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),A_106)) # label(fact_1982_add__0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3596 (all V_19 all W_15 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_19)),hAPP_int_complex(number528085621omplex,W_15)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_19),W_15))) # label(fact_85_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3597 (all K_2 hAPP_int_int(pred,K_2) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),one_one_int)) # label(fact_4045_pred__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3598 (all A_233 (is_int(A_233) -> A_233 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,pls)),A_233))) # label(fact_322_add__numeral__0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3599 (all N all P_3 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),N)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),M)))))) # label(fact_754_zprime__zdvd__zmult) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3600 (all A_3 hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),A_3) = A_3) # label(fact_4303_gcd__semilattice__nat_Oinf_Oidem) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3601 (all A_197 all B_152 all N_64 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_197),N_64)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_152),N_64)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_197),B_152)),N_64)) # label(fact_894_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3602 (all C_71 all B_126 all A_165 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_126),A_165)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_71),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_165),C_71)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_126),C_71)))))) # label(fact_1129_mult__strict__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3603 (all N_62 one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,one_one_nat),N_62)) # label(fact_970_power__one) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3604 (all X hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),zero_zero_int) = hAPP_int_int(abs_abs_int,X)) # label(fact_5002_gcd__0__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3605 (all X (-hBOOL(hAPP_nat_bool(even_odd_even_nat,X)) -> X = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))))))) # label(fact_3780_odd__nat__div__two__times__two__plus__one) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3606 (all X_1 hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_1))))) # label(fact_4835_summable__exp) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3607 (all A_231 all B_165 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_231),B_165) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_165),A_231)) # label(fact_517_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3608 (all X_5 all Y_5 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_5),Y_5)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_5),Y_5)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_5),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_5),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))))) # label(fact_3192_realpow__two__diff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3609 (all P_1 all A all B ((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B)) -> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))) & (all D_2 (A = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B),D_2) -> hBOOL(hAPP_nat_bool(P_1,D_2)))) <-> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A),B))))) # label(fact_1779_nat__diff__split) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3610 (all W_1 all X_1 (hAPP_int_nat(number_number_of_nat,W_1) = X_1 <-> hAPP_int_nat(number_number_of_nat,W_1) = X_1)) # label(fact_170_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3611 (all N exists P_4 (hBOOL(hAPP_nat_bool(prime,P_4)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),P_4)))) # label(fact_4222_euclid) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3612 (all A_24 all B_19 all C_8 ((C_8 = zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_8),A_24)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_8),B_19)) = zero_zero_nat) & (zero_zero_nat != C_8 -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_24),B_19) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_8),A_24)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_8),B_19))))) # label(fact_2550_div__mult__mult1__if) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3613 (all V_5 all W_3 all C_55 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_5),W_3))),C_55) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_5)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_int_complex(number528085621omplex,W_3)),C_55))) # label(fact_1576_add__number__of__diff1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3614 (all P_1 all N_1 ((all M_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) -> hBOOL(hAPP_nat_bool(P_1,M_1)))) <-> (all X_2 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))) -> hBOOL(hAPP_nat_bool(P_1,X_2)))))) # label(fact_4902_all__nat__less__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3615 (all A_240 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_240),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),zero_zero_real))) # label(fact_17_power2__less__0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3616 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) = M -> N = hAPP_nat_nat(suc,zero_zero_nat)))) # label(fact_3129_mn__eq__m__one) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3617 (all X_1 all F (hBOOL(summable_real(F)) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_2))),X_1))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f352196356l_real(suminf_real,F)),X_1))))) # label(fact_4944_suminf__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3618 (all X_1 all Y_2 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Y_2),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3236_even__div__2__l) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3619 (all M all N all K_2 (one_one_int = hAPP_int_int(legacy_zgcd(N),K_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M)))))) # label(fact_4527_zrelprime__zdvd__zmult__aux) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3620 (all A_189 all B_145 all C_82 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_189),B_145)),C_82)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_189),C_82)))) # label(fact_949_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3621 (all A_3 all J_2 all K_2 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),K_2),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),J_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),K_2))),P_3))))) # label(fact_2230_MultInv__zcong__prop1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3622 (all X_1 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,X_1)),hAPP_int_real(number267125858f_real,Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Y_2)))) # label(fact_59_le__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3623 (all X_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),zero_zero_real))))) # label(fact_3920_real__root__lt__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3624 (all A_182 A_182 = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_182),one_one_nat)) # label(fact_1027_power__one__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3625 (all A all B (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),B)),zOdd)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),zEven)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),zOdd))))) # label(fact_3094_even__plus__odd__prop2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3626 (all X hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),X) = hAPP_int_int(abs_abs_int,X)) # label(fact_4986_gcd__idem__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3627 (all P all Q all R hAPP_r1962736578t_bool(hAPP_f1063001928t_bool(hAPP_f2050994651t_bool(cOMBB_2138578194l_real,P),Q),R) = hAPP_f1545556668t_bool(P,hAPP_r1150855451l_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__fun_) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3628 (all V all W_1 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,W_1)),hAPP_int_nat(number_number_of_nat,V))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_int_nat(number_number_of_nat,W_1))))) # label(fact_106_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3629 (all P_5 hAPP_P1975530577nt_int(produc1518849193nt_int(hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),uminus_uminus_int)),quotient_of(P_5)) = quotient_of(hAPP_rat_rat(uminus_uminus_rat,P_5))) # label(fact_4834_rat__uminus__code) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3630 (all Y_8 all Z_6 all X_11 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,Z_6),X_11)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,Z_6),Y_8)) -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_11),Y_8)),Z_6) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,X_11),Z_6)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,Y_8),Z_6))))) # label(fact_2578_div__add) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3631 (all M_28 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,M_28),M_28) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),one_one_real)),M_28)) # label(fact_693_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3632 (all P all Q all R hAPP_f1222178131t_bool(hAPP_r2052264067t_bool(P,R),Q) = hAPP_r373117745t_bool(hAPP_f1459420095t_bool(hAPP_f693283245t_bool(cOMBC_1113900202t_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Int__Oint_Mtc__fun_It_028) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3633 (all N_1 all X_1 exists T (hAPP_real_real(cos,X_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,div_div_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),zero_zero_real))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,T),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(real_nat,N_1))),pi)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_1))))) # label(fact_4915_Maclaurin__cos__expansion) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3634 (all A_49 all B_40 all C_21 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_49),B_40)),C_21) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_49),hAPP_int_int(div_mod_int(B_40),C_21))),C_21)) # label(fact_2413_zmod__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3635 (all M all N hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,M),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N))) # label(fact_5193_max__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3636 (all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)) <-> (exists K N_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),K)))) # label(fact_1641_le__iff__add) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3637 (all N (-hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) -> (exists X_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_2)),one_one_nat) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_3795_odd__square) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3638 (all W_5 ((zero_zero_nat = hAPP_int_nat(number_number_of_nat,W_5) -> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),hAPP_int_nat(number_number_of_nat,W_5)) = one_on1684967323de_int) & (zero_zero_nat != hAPP_int_nat(number_number_of_nat,W_5) -> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),hAPP_int_nat(number_number_of_nat,W_5)) = zero_z891286103de_int))) # label(fact_729_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3639 (all X all Y all M (hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)),M) = one_one_int -> one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),M))) # label(fact_4963_invertible__coprime__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3640 (all B_115 all C_67 all A_153 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_153)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_115),C_67)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_115),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_153),C_67)))))) # label(fact_1207_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3641 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> (N != M -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))))) # label(fact_1636_le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3642 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),one_one_real)),hAPP_nat_real(real_nat,natfloor(X))))) # label(fact_3693_real__natfloor__gt__diff__one) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3643 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(fact,M)),hAPP_nat_nat(fact,N))))) # label(fact_4559_fact__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3644 (all A_11 all N_17 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_11),hAPP_nat_nat(suc,N_17)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_11),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_11),N_17))) # label(fact_3013_power__Suc) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3645 (exists K_4 all X_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(cnj,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_2)),K_4)))) # label(fact_4088_cnj_Obounded) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3646 (all K_1 all N_2 all F_2 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,F_2),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),K_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_2))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),K_1))) = hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_2),K_1)))) # label(fact_4899_sumr__offset4) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3647 (all A (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),zero_zero_real)))) # label(fact_2172_double__add__le__zero__iff__single__add__le__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3648 (all A_70 all B_58 hAPP_Q1762011733de_int(div_mo231679042de_int(A_70),B_58) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(div_mo231679042de_int(A_70),B_58)),B_58)) # label(fact_2328_mod__mod__trivial) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3649 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),zero_zero_real)) -> (exists R_1 exists A_2 (Y = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(sin,A_2)) & hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(cos,A_2)) = X)))) # label(fact_3947_polar__ex2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3650 (all X_45 all Y_33 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_45),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_33),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_33)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_45),Y_33))))) # label(fact_210_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3651 (all V_12 all W_10 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_12)),hAPP_int_complex(number528085621omplex,W_10)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_12),W_10))) # label(fact_339_add__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3652 (all A_140 all C_62 all B_109 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_140),C_62)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_109),C_62))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_62)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_140),B_109))))) # label(fact_1342_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3653 (all Ma all D (zero_zero_nat = hAPP_nat_nat(div_mod_nat(Ma),D) <-> (exists Q_4 Ma = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,D),Q_4)))) # label(fact_2706_mod__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3654 (all C_27 all A_57 all B_48 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_27),A_57)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_27),B_48)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_27),hAPP_int_int(div_mod_int(A_57),B_48))) # label(fact_2389_mod__mult__mult1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3655 (all A_3 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) # label(fact_419_cube__square) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3656 (all A_195 all B_151 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_195),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_195),B_151)))) # label(fact_911_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3657 (all X all Y hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(cnj,X)),hAPP_complex_complex(cnj,Y)) = hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X),Y))) # label(fact_4068_complex__cnj__mult) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3658 (all B_90 all A_112 all C_50 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_90),A_112) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_50),A_112) -> C_50 = B_90)) # label(fact_1910_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3659 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,natfloor(X))),X)))) # label(fact_3674_real__natfloor__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3660 (all N_25 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N_25)))) # label(fact_2915_of__nat__0__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3661 (all N_41 ((zero_zero_nat != N_41 -> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),N_41) = zero_z126310315umeral) & (N_41 = zero_zero_nat -> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),N_41) = one_on1645066479umeral))) # label(fact_1435_power__0__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3662 (all A_3 all B_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),B_2)) -> hAPP_int_int(div_mod_int(A_3),M) = hAPP_int_int(div_mod_int(hAPP_int_int(div_mod_int(A_3),B_2)),M)))) # label(fact_2617_zmod__zdvd__zmod) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3663 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real)) -> hBOOL(bseq_real(hAPP_r474017924t_real(power_power_real,X_1)))))) # label(fact_4750_Bseq__realpow) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3664 (all N hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N),one_one_int)),N) = one_one_int) # label(fact_4970_coprime__plus__one__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3665 (all X_1 all Y_2 all B (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),B)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B),X_1)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B),Y_2))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2))))) # label(fact_1456_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3666 (all X_1 all Y_2 (hBOOL(hAPP_P1555980039t_bool(accp_P490777396at_nat(nat_gcd_rel),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,X_1),Y_2))) -> (Y_2 = zero_zero_nat -> X_1 = hAPP_nat_nat(nat_gcd(X_1),Y_2)) & (Y_2 != zero_zero_nat -> hAPP_nat_nat(nat_gcd(Y_2),hAPP_nat_nat(div_mod_nat(X_1),Y_2)) = hAPP_nat_nat(nat_gcd(X_1),Y_2)))) # label(fact_4622_nat__gcd_Opsimps) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3667 (all B_2 hAPP_int_int(div_mod_int(zero_zero_int),B_2) = zero_zero_int) # label(fact_2459_zmod__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3668 (all N_43 all A_131 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_131)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_131),N_43)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_131),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_131),N_43)))))) # label(fact_1420_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3669 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> zero_zero_nat = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(fact_fact_nat,N)),hAPP_nat_nat(fact_fact_nat,M)))) # label(fact_3814_fact__mod) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3670 (all X_9 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_9)) -> hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_nat_complex(semiri2020571505omplex,hAPP_int_nat(nat,X_9))),hAPP_nat_complex(semiri2020571505omplex,hAPP_int_nat(nat,X_9))) = hAPP_nat_complex(semiri2020571505omplex,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_9))))) # label(fact_2812_of__nat__double) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3671 (all M_7 all N_21 hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(semiri132038758t_real,M_7)),hAPP_nat_real(semiri132038758t_real,N_21))) # label(fact_2964_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3672 (all C_38 all A_96 all B_78 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_96),B_78)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_38),A_96)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_38),B_78))))) # label(fact_2055_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3673 (all N_12 all R_6 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),R_6)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,R_6),one_on1684967323de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,R_6),hAPP_nat_nat(suc,N_12))),R_6))))) # label(fact_3154_realpow__Suc__le__self) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3674 (all A all B all C (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B),A)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B)))))) # label(fact_1380_mult__le__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3675 (all C_64 all A_142 all B_111 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_64),A_142)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_64),B_111))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_64)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_142),B_111))))) # label(fact_1331_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3676 (all B_96 all A_118 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_118)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_96)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_118),B_96)))))) # label(fact_1865_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3677 (all A_207 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_207),A_207))) # label(fact_799_dvd__refl) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3678 (all X_1 (zero_zero_real != hAPP_real_real(cos,X_1) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real),sin)),cos),X_1),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_4800_lemma__DERIV__tan) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3679 (all X all Y hAPP_nat_nat(nat_lcm(X),Y) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X),Y)),hAPP_nat_nat(nat_gcd(X),Y))) # label(fact_4610_Nitpick_Onat__lcm__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3680 (all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(suc,N)),N) = one_one_nat) # label(fact_4347_coprime__Suc__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3681 (all A_227 all B_164 all C_93 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_227),B_164)),C_93) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_227),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_164),C_93))) # label(fact_544_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3682 (all K_2 (is_int(K_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_2)) -> (exists N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2)) & hAPP_nat_int(semiri1621563631at_int,N_2) = K_2))))) # label(fact_3251_zero__less__imp__eq__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3683 (all P all Q all R hAPP_r285062775l_bool(hAPP_f834119977l_bool(hAPP_f848516149l_bool(cOMBB_270132509l_real,P),Q),R) = hAPP_f252743847l_bool(P,hAPP_r373117745t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Nat__Onat_Mtc__HOL__035) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3684 (all F all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) -> ((all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),B)) -> (exists Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),zero_zero_real)) & hBOOL(hAPP_real_bool(deriv_real(F,X_2),Y_1)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,B)),hAPP_real_real(F,A)))))) # label(fact_4004_DERIV__neg__imp__decreasing) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3685 (all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N)) -> hAPP_int_int(fact_fact_int,N) != zero_zero_int)) # label(fact_4278_fact__nonzero__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3686 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X))),one_one_real))))) # label(fact_3587_cos__double__less__one) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3687 (all P all Q all R hAPP_f1048215610t_bool(hAPP_r1623423338t_bool(P,R),Q) = hAPP_r632163725t_bool(hAPP_f1832844496t_bool(hAPP_f1990636069t_bool(cOMBC_412863505t_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Int__Oint_Mtc__HOL__O) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3688 (all X_1 all N_1 (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,N_1))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X_1)),hAPP_int_nat(number_number_of_nat,N_1))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_int_real(number267125858f_real,N_1))))))) # label(fact_3411_natceiling__le__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3689 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) | N = zero_zero_nat)) # label(fact_2278_divides__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3690 (all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> hAPP_int_nat(phi,P_3) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)))) # label(fact_2775_phi__prime) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3691 (all C_74 all A_174 all B_134 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_174),B_134)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_74)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_74),A_174)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_74),B_134)))))) # label(fact_1078_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3692 (all X all Y hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X),Y))) # label(fact_3002_Nat__Transfer_Otransfer__int__nat__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3693 (all X_1 (is_int(X_1) -> (one_one_int = X_1 <-> one_one_int = X_1))) # label(fact_1950_one__reorient) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3694 (all P_5 quotient_of(hAPP_rat_rat(abs_abs_rat,P_5)) = hAPP_P1975530577nt_int(produc1518849193nt_int(hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),abs_abs_int)),quotient_of(P_5))) # label(fact_4836_rat__abs__code) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3695 (all P_1 all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) -> (hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,minus_minus_real),hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,produc1935615926l_real),bolzano_bisect(P_1,A,B)))),hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,produc556554744l_real),bolzano_bisect(P_1,A,B))),zero_zero_real,sequentially)) -> (exists L_1 (hBOOL(tendsto_nat_real(hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,produc1935615926l_real),bolzano_bisect(P_1,A,B)),L_1,sequentially)) & (all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,L_1),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2))))) & hBOOL(tendsto_nat_real(hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,produc556554744l_real),bolzano_bisect(P_1,A,B)),L_1,sequentially)) & (all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2))),L_1)))))))))) # label(fact_5059_Bolzano__nest__unique) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3696 (all B all A (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(d22set,A))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B)))) # label(fact_2267_d22set__g__1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3697 (all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1)) -> hAPP_nat_int(semiri1621563631at_int,hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),N_1))))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,N_1),one_one_int))) # label(fact_4775_int__card__bdd__int__set__l__l) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3698 (all A all R_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_2)) -> (hBOOL(summable_real(A)) -> (exists N_3 all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_f352196356l_real(suminf_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,A),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),N_2))))),R_2))))))) # label(fact_4853_suminf__exist__split) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3699 (all A all B (is_int(B) & is_int(A) -> (B = A <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),zero_zero_int))))) # label(fact_1029_IntPrimes_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3700 (all L_2 min != hAPP_int_int(bit0,L_2)) # label(fact_772_rel__simps_I42_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3701 (all Y all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Y)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,natfloor(X)),Y) = natfloor(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_nat_real(real_nat,Y)))))) # label(fact_3724_natfloor__div__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3702 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),M)))))) # label(fact_3127_n__less__n__mult__m) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3703 (all N all D_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),D_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),N)) -> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(real_nat,D_3)) = hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),D_3))))) # label(fact_3696_real__of__nat__div) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3704 (all A_38 all C_13 all B_29 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,C_13),B_29)) -> hAPP_int_int(div_mod_int(hAPP_int_int(div_mod_int(A_38),B_29)),C_13) = hAPP_int_int(div_mod_int(A_38),C_13))) # label(fact_2447_mod__mod__cancel) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3705 (all X_42 all Y_30 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_42),X_42)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_30),Y_30))),zero_zero_real))) # label(fact_233_not__sum__squares__lt__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3706 (all M_17 all N_30 ((zero_zero_nat != N_30 -> hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,M_17),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat))) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,M_17),N_30)) & (N_30 = zero_zero_nat -> one_one_rat = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,M_17),N_30)))) # label(fact_1835_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3707 (all Ma hAPP_int_nat(phi,Ma) = hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(norRRset,Ma))) # label(fact_4585_phi__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3708 (all Z_1 all X all Y all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),Y),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Z_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y),Z_1)),M)))) # label(fact_1039_zcong__zpower) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3709 (all C_33 all D_5 all A_79 all B_67 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_79),B_67)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_33),D_5)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_79),C_33)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_67),D_5)))))) # label(fact_2205_add__less__le__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3710 (all Ma all N_1 (hAPP_nat_int(semiri1621563631at_int,N_1) = hAPP_nat_int(semiri1621563631at_int,Ma) <-> N_1 = Ma)) # label(fact_2831_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3711 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)),one_one_rat))))) # label(fact_4721_Fract__less__one__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3712 (all N_1 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_n522471361de_int(semiri1424489471de_int,N_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)))) # label(fact_3143_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3713 (all Lx_5 all Ly_3 all Rx_5 all Ry_3 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_5),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_5),Ly_3)),Ry_3)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_5),Ly_3)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_5),Ry_3))) # label(fact_476_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3714 (all C_60 all D_13 all A_138 all B_107 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_138),B_107)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_60),D_13)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_138)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_60)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_138),C_60)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_107),D_13)))))))) # label(fact_1356_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3715 (all X all Y all Z_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_1),X)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_1),Y)))))) # label(fact_1791_real__mult__less__mono2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3716 (all C_27 all A_57 all B_48 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_27),A_57)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_27),B_48)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_27),hAPP_nat_nat(div_mod_nat(A_57),B_48))) # label(fact_2390_mod__mult__mult1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3717 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,X_1)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),zero_zero_real)))) # label(fact_3324_real__sqrt__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3718 (all A_228 all C_94 all D_19 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_228),C_94)),D_19) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_228),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_94),D_19))) # label(fact_538_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3719 (all Q_1 all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),Q_1)),Ma)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),hAPP_nat_nat(suc,Q_1)))) <-> Q_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,Ma),N_1)))) # label(fact_3211_split__div__lemma) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3720 (all A_3 all B_2 all C_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_2),C_1))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_3),C_1)),hAPP_nat_nat(div_mod_nat(B_2),C_1))),C_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_3),B_2)),C_1)) # label(fact_2682_div__add1__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3721 (all A_19 all B_15 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_19),B_15)) -> B_15 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_19),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_15),A_19)))) # label(fact_2576_dvd__mult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3722 (all A_228 all C_94 all D_19 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_228),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_94),D_19)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_228),C_94)),D_19)) # label(fact_539_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3723 (all X_1 all X_3 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),X_3)) -> ((exists Z all X_2 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_2),X_3)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Z)))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),comple124823625p_real(X_3)))))) # label(fact_4639_Sup__upper__EX) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3724 (all Y_20 all X_26 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_26)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_20)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_20),one_one_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_26),Y_20)),X_26)))))) # label(fact_1389_mult__right__le__one__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3725 (all Y all X (hBOOL(hAPP_int_bool(nat_is_nat,X)) -> (hBOOL(hAPP_int_bool(nat_is_nat,Y)) -> hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y)))))) # label(fact_5137_Divides_Otransfer__int__nat__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3726 (all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),N) = zero_zero_nat) # label(fact_5123_min__0L) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3727 (all A all Ma hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(bnorRset(A),Ma)))) # label(fact_4294_Bnor__fin) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3728 (all X_3 (hBOOL(summable_complex(X_3)) -> hAPP_f352196356l_real(suminf_real,hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_3)) = hAPP_complex_real(im,hAPP_f213450978omplex(suminf_complex,X_3)))) # label(fact_4849_Im_Osuminf) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3729 (all X all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(inverse_inverse_real,X)) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)))) # label(fact_3915_real__root__inverse) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3730 (all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,W)),zero_zero_int)) -> hAPP_P1175774780nt_int(product_snd_int_int,hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,one_one_int),hAPP_int_int(uminus_uminus_int,hAPP_int_int(number_number_of_int,W))))) = hAPP_int_int(div_mod_int(one_one_int),hAPP_int_int(number_number_of_int,W)))) # label(fact_4232_mod__pos__neg__1__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.56 3731 (all Ma all K_1 all N_1 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_1),N_1)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_1),Ma)) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,K_1),hAPP_Q1762011733de_int(div_mo231679042de_int(Ma),N_1)))))) # label(fact_2442_dvd__mod__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3732 (all X hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,pi),X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X))) # label(fact_3669_sin__periodic__pi2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3733 (all Diff all F all N_1 all H (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hAPP_n546711566l_real(Diff,zero_zero_nat) = F -> ((all M_1 all T (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))))) -> (exists T (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N_1),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N_1))) = hAPP_real_real(F,H) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,T),H)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),T))))))))) # label(fact_4939_Maclaurin) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3734 (all N all A_3 all B_2 (hAPP_int_int(legacy_zgcd(A_3),B_2) = one_one_int -> one_one_int = hAPP_int_int(legacy_zgcd(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),N)))) # label(fact_4505_zgcd__1__power__distrib) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3735 (all N all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),N))))))) # label(fact_3817_Ln_Oaux1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3736 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(sqrt,X))))) # label(fact_3330_real__sqrt__ge__one) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3737 (all Ma all N_1 (N_1 = zero_zero_nat | zero_zero_nat = Ma <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N_1) = zero_zero_nat)) # label(fact_1621_mult__is__0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3738 (all V_4 ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4))) -> zero_zero_complex = hAPP_nat_complex(semiri2020571505omplex,hAPP_int_nat(number_number_of_nat,V_4))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4))) -> hAPP_nat_complex(semiri2020571505omplex,hAPP_int_nat(number_number_of_nat,V_4)) = hAPP_int_complex(number528085621omplex,V_4)))) # label(fact_3218_of__nat__number__of__lemma) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3739 (all R_3 all A_3 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,A_3),R_3) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),A_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),R_3))) # label(fact_3355_real__divide__square__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3740 (all A all C all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),C)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),C))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A),B)) | zero_zero_real = C)) # label(fact_1267_dvd__mult__cancel__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3741 (all X_20 all Y_16 all A_120 all B_98 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_20),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Y_16),B_98))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_20),A_120)),B_98)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_20),Y_16)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_120),B_98))) # label(fact_1817_mult__diff__mult) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3742 (all N all M (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,M))) -> N = M))) # label(fact_2861_less__antisym) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3743 (all A_98 all C_40 all B_80 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_98),C_40)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_80),C_40))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_98),B_80)))) # label(fact_2041_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3744 (all Ma all N_1 hAPP_nat_nat(nat_case_nat(zero_zero_nat,cOMBI_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),N_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),hAPP_nat_nat(suc,N_1))) # label(fact_4880_diff__Suc) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3745 (all X_44 all Y_32 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_44),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_32),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),Y_32)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_44),Y_32))))) # label(fact_221_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3746 (all A_87 all B_75 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_87),B_75)),B_75) = A_87) # label(fact_2136_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3747 (all N_26 hAPP_nat_int(semiri1621563631at_int,N_26) = hAPP_int_int(number_number_of_int,hAPP_nat_int(semiri1621563631at_int,N_26))) # label(fact_2823_number__of__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3748 (all L all U hAPP_i1948725293t_bool(ord_at641636577an_int(L),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,U),one_one_int)) = hAPP_i1948725293t_bool(ord_at875362053st_int(L),U)) # label(fact_5080_atLeastLessThanPlusOne__atLeastAtMost__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3749 (all Y (zero_zero_real != Y -> hAPP_real_real(cos,arg(hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),Y))) = zero_zero_real)) # label(fact_3973_cos__arg__i__mult__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3750 (all X_1 all N_1 (hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_1),N_1))) <-> N_1 != zero_zero_nat & hBOOL(hAPP_int_bool(even_odd_even_int,X_1)))) # label(fact_3090_even__power) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3751 (all I all K_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2)))) # label(fact_1774_add__diff__assoc) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3752 (all A_105 A_105 = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),A_105)) # label(fact_1989_add__0__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3753 (all A_106 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,zero_z126310315umeral),A_106) = A_106) # label(fact_1978_add__0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3754 (all N_11 all A_7 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_7)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_7),one_on1645066479umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_7),hAPP_nat_nat(suc,N_11))),one_on1645066479umeral))))) # label(fact_3161_power__Suc__less__one) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3755 (all N_1 hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),N_1)))))) # label(fact_4771_bdd__int__set__l__l__finite) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3756 (all Ma all N_1 (hAPP_nat_int(semiri1621563631at_int,N_1) = hAPP_nat_int(semiri1621563631at_int,Ma) <-> N_1 = Ma)) # label(fact_3257_int__int__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3757 (all N_42 all A_130 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_130)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_130),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_130),N_42)))))) # label(fact_1428_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3758 (all A_111 all B_89 all C_49 (is_int(B_89) & is_int(C_49) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_111),C_49) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_111),B_89) -> C_49 = B_89))) # label(fact_1916_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3759 (all M_27 all A_212 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_212),one_one_complex)),M_27) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,M_27),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_212),M_27))) # label(fact_702_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3760 (all X natfloor(X) = hAPP_int_nat(nat,archim1246769320r_real(X))) # label(fact_4021_natfloor__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3761 (all A_41 all B_32 all C_16 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(div_mod_int(A_41),C_16)),hAPP_int_int(div_mod_int(B_32),C_16))),C_16) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_41),B_32)),C_16)) # label(fact_2438_mod__diff__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3762 (all M_8 all N_22 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_nat_complex(semiri2020571505omplex,M_8)),hAPP_nat_complex(semiri2020571505omplex,N_22)) = hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22))) # label(fact_2956_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3763 (all B_2 all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_3)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,natfloor(A_3)),natfloor(B_2))),natfloor(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),B_2))))))) # label(fact_3403_le__mult__natfloor) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3764 (all X_43 all Y_31 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_43),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_31),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_227_sum__power2__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3765 (all A_29 all B_24 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_29),B_24)) -> zero_zero_nat = hAPP_nat_nat(div_mod_nat(B_24),A_29))) # label(fact_2515_dvd__imp__mod__0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3766 (all X_1 X_1 = hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),X_1)))) # label(fact_4765_card__bdd__nat__set__l) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3767 (all X hAPP_real_real(exp_real,X) = hAPP_real_real(abs_abs_real,hAPP_real_real(exp_real,X))) # label(fact_3828_abs__exp__cancel) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3768 (all N_62 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,one_one_rat),N_62) = one_one_rat) # label(fact_975_power__one) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3769 (all W_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,W_1)),zero_zero_int)))) # label(fact_352_bin__less__0__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3770 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)),M))))) # label(fact_2723_div__less__dividend) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3771 (all X all A_3 all B_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_3),B_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_2)))))) # label(fact_4407_powr__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3772 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(ln,X)),X)))) # label(fact_3427_ln__less__self) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3773 (all Lx_3 all Ly_1 all Rx_3 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_3),Ly_1)),Rx_3) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_3),Rx_3)),Ly_1)) # label(fact_486_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3774 (all V_2 (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_int_nat(number_number_of_nat,V_2) = zero_zero_nat)) # label(fact_3278_neg__imp__number__of__eq__0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3775 (all X_8 all Q_7 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_8),Q_7)),X_8) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_8),hAPP_nat_nat(suc,Q_7))) # label(fact_3024_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3776 (all A_225 all B_162 all C_91 all D_18 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_225),C_91)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_162),D_18)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_225),B_162)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_91),D_18))) # label(fact_560_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3777 (all M_28 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,one_on1684967323de_int),one_on1684967323de_int)),M_28) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,M_28),M_28)) # label(fact_696_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3778 (all B_122 all A_161 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_161),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_122),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_161),B_122)))))) # label(fact_1149_mult__neg__neg) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3779 (all B_2 all X all A_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),A_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),B_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)))))) # label(fact_4320_gcd__semilattice__nat_Ole__infI) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3780 (all R all S_4 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,R),S_4) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R),hAPP_real_real(inverse_inverse_real,S_4))) # label(fact_3831_real__divide__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3781 (all A all B (hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A),B) = zero_zero_complex <-> B = zero_zero_complex | A = zero_zero_complex)) # label(fact_837_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3782 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_complex_real(re,X)),hAPP_complex_real(re,Y)) = hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),Y))) # label(fact_4122_Re_Oadd) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3783 (all A_124 all N_34 all B_102 (hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_102),N_34) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_124),N_34) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_124)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_102)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_34)) -> A_124 = B_102))))) # label(fact_1484_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3784 (all N hAPP_nat_nat(suc,hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_3198_add__2__eq__Suc_H) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3785 (all A_48 all B_39 all C_20 hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_48),B_39)),C_20) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_48),C_20)),hAPP_Q1762011733de_int(div_mo231679042de_int(B_39),C_20))),C_20)) # label(fact_2420_mod__add__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3786 (all A_226 all B_163 all C_92 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_226),B_163)),C_92) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_226),C_92)),B_163)) # label(fact_550_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3787 (all X_1 all A (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),hAPP_int_real(real_int,A))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim1246769320r_real(X_1)),A)))) # label(fact_4416_floor__less__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3788 (all A_239 all B_167 all V_15 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_239),B_167)),hAPP_int_nat(number_number_of_nat,V_15)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_239),hAPP_int_nat(number_number_of_nat,V_15))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_167),hAPP_int_nat(number_number_of_nat,V_15)))) # label(fact_185_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3789 (all M all X hAPP_int_int(standardRes(M),X) = hAPP_int_int(div_mod_int(X),M)) # label(fact_2336_StandardRes__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3790 (all A_113 all B_91 all C_51 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_113),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_91),C_51)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_113),B_91)),C_51)) # label(fact_1904_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3791 (all B_1_1 is_bool(vanishes(B_1_1))) # label(gsy_c_RealDef_Ovanishes) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3792 (all X_1 hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),sin_coeff)),hAPP_r474017924t_real(power_power_real,X_1))))) # label(fact_4831_summable__sin) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3793 (all A_3 all B_2 (is_int(B_2) -> (zero_zero_int != B_2 -> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),hAPP_int_int(div_mod_int(A_3),B_2))))))) # label(fact_2798_divmod__int__rel__div__mod) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3794 (all B_97 all A_119 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_119)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_97)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_119),B_97)))))) # label(fact_1860_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3795 (all N_1 all X_1 all Diff all F ((all M_1 all X_2 hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),X_2),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),X_2)))) & F = hAPP_n546711566l_real(Diff,zero_zero_nat) -> (exists T (hAPP_real_real(F,X_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N_1),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N_1))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,X_1))))))) # label(fact_4940_Maclaurin__all__le__objl) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3796 (all P all Q all R hAPP_P877703507t_bool(hAPP_f1965248563t_bool(hAPP_f1273270095t_bool(cOMBB_595786433nt_int,P),Q),R) = hAPP_f513299955t_bool(P,hAPP_P921862901l_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_000tc__fun_Itc_040) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3797 (all X_1 hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),X_1))))) # label(fact_4744_bdd__nat__set__le__finite) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3798 (all A all W_1 (zero_zero_nat != hAPP_int_nat(number_number_of_nat,W_1) & A = zero_z126310315umeral <-> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A),hAPP_int_nat(number_number_of_nat,W_1)) = zero_z126310315umeral)) # label(fact_715_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3799 (all X_1 all W_1 (zero_zero_nat = hAPP_int_nat(number_number_of_nat,W_1) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),hAPP_int_nat(number_number_of_nat,W_1)))))) # label(fact_568_zero__less__power__nat__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3800 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,X_1)),one_one_real)))) # label(fact_3331_real__sqrt__le__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3801 (all K_2 hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(number_number_of_int,K_2)),one_one_int) = hAPP_int_rat(number_number_of_rat,K_2)) # label(fact_4708_rat__number__expand_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3802 (all A_107 A_107 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_107),zero_zero_rat)) # label(fact_1974_add__0__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3803 (all B_1_1 all B_2_1 is_bool(hAPP_Q2096512830t_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__Quickcheck____Narrowing__Ocode____int_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3804 (all W all Z_1 hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z_1))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,W))),hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,Z_1)))) # label(fact_3436_nat__abs__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3805 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),zero_zero_int))))) # label(fact_2520_div__neg__pos__less0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3806 (all A_11 all N_17 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_11),hAPP_nat_nat(suc,N_17)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_11),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_11),N_17))) # label(fact_3017_power__Suc) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3807 (all X_1 all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A)) -> (hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_1),setS(A,P_5))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(big_co1548731110nt_int(cOMBI_int,X_1)),A),P_5)))))))) # label(fact_4813_SetS__setprod__prop) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3808 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),pls)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K_1)),pls)))) # label(fact_349_rel__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3809 (all X_24 all N_32 (X_24 = one_one_nat | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_24),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_24),N_32))))) # label(fact_1497_dvd__power) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3810 (all A_189 all B_145 all C_82 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_189),B_145)),C_82)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_189),C_82)))) # label(fact_951_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3811 (all A_218 all B_161 all C_90 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_218),C_90)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_161),C_90)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_218),B_161)),C_90)) # label(fact_630_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3812 (all M_12 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(semiri984289939at_nat,M_12)))) # label(fact_2908_zero__le__imp__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3813 (all X_1 all A_1 (hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_1),A_1)) <-> hBOOL(hAPP_f448129468l_bool(A_1,X_1)))) # label(fact_140_mem__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3814 (all C_38 all A_96 all B_78 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_96),B_78)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_38),A_96)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_38),B_78))))) # label(fact_2056_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3815 (all C_64 all A_142 all B_111 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_64),A_142)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_64),B_111))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_64)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_142),B_111))))) # label(fact_1334_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3816 (all I_1 all J all K_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),J)))) # label(fact_1765_less__diff__conv) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3817 (all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,N_1))))) # label(fact_2998_zle__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3818 (all M all N (zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) -> (zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),M) -> M = N))) # label(fact_1737_diffs0__imp__equal) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3819 (all N_1 hAPP_nat_real(diffs_real(cos_coeff),N_1) = hAPP_real_real(uminus_uminus_real,hAPP_nat_real(sin_coeff,N_1))) # label(fact_4878_cos__fdiffs2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3820 (all A_198 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,zero_z891286103de_int),A_198)) -> zero_z891286103de_int = A_198)) # label(fact_888_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3821 (all N all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N)),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N)),one_one_real))) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N),one_one_int) = archim856651990g_real(X)))) # label(fact_4376_ceiling__eq2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3822 (all W_1 all Z_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat,W_1)),hAPP_int_nat(nat,Z_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_2))))) # label(fact_2728_nat__mono__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3823 (all W_5 ((hAPP_int_nat(number_number_of_nat,W_5) != zero_zero_nat -> zero_zero_complex = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),hAPP_int_nat(number_number_of_nat,W_5))) & (zero_zero_nat = hAPP_int_nat(number_number_of_nat,W_5) -> one_one_complex = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),hAPP_int_nat(number_number_of_nat,W_5))))) # label(fact_728_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3824 (all C_78 all D_14 all A_178 all B_138 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_178),B_138)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_78),D_14)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_178)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_78)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_178),C_78)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_138),D_14)))))))) # label(fact_1060_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3825 (all Ma all K_1 all N_1 (K_1 = zero_zero_nat | N_1 = Ma <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N_1),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),K_1))) # label(fact_1623_mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3826 (all M all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),J_2))))) # label(fact_1644_trans__le__add2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3827 (all A_229 all C_95 all D_20 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_229),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_95),D_20)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_95),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_229),D_20))) # label(fact_527_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3828 (all A all E_2 all C all B all D (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),E_2)),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),E_2)),D))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A)),E_2)),D))))) # label(fact_1564_le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3829 (all P all Q all R hAPP_int_bool(hAPP_f1448486952t_bool(hAPP_f696844121t_bool(cOMBB_nat_bool_int,P),Q),R) = hAPP_nat_bool(P,hAPP_int_nat(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__HOL__Obool_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3830 (all Z_14 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_14) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_14),Z_14)) # label(fact_281_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3831 (all Z_2 all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(nat,Z_2)),Ma)) <-> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_2),hAPP_nat_int(semiri1621563631at_int,Ma)))) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2)) -> zero_zero_nat = Ma))) # label(fact_3223_nat__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3832 (all B_74 all A_86 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_86),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_74),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_86),B_74)),zero_zero_rat))))) # label(fact_2146_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3833 (all A_188 all B_144 all C_81 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_188),B_144)),C_81)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,B_144),C_81)))) # label(fact_956_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3834 (all N -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat))) # label(fact_1608_less__nat__zero__code) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3835 (all U hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_at641636577an_int(zero_zero_int),U)))) # label(fact_4918_finite__atLeastZeroLessThan__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3836 (all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat) = hAPP_nat_nat(suc,N)) # label(fact_3070_Suc__eq__plus1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3837 (all B_1_1 all B_2_1 is_bool(hAPP_P603027463t_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3838 (all A all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),archim1246769320r_real(X_1))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,A)),one_one_real)),X_1)))) # label(fact_4468_less__floor__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3839 (all B_92 all A_114 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_114),zero_z891286103de_int)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_92),zero_z891286103de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_114),B_92)),zero_z891286103de_int))))) # label(fact_1891_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3840 (all A all P_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,setS(A,P_5))))) # label(fact_3370_SetS__finite) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3841 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_complex_real(re,X)),hAPP_complex_real(re,Y)) = hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),Y))) # label(fact_4126_Re_Odiff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3842 (all C all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),A)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),B))))) # label(fact_2066_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3843 (all N_1 all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma)) -> (N_1 = one_one_nat <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N_1)),Ma))))) # label(fact_1572_dvd__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3844 (all A_139 all C_61 all B_108 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_139),C_61)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_108),C_61))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_61)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_139),B_108))))) # label(fact_1351_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3845 (all A_215 A_215 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_215),one_one_real)) # label(fact_656_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3846 (all N_53 all M_21 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),M_21)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),N_53)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,M_21),N_53)))))) # label(fact_1249_less__1__mult) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3847 (all B all A (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(d22set,A))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A)))) # label(fact_2266_d22set__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3848 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),hAPP_int_int(bit0,K_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K_1)))) # label(fact_788_rel__simps_I25_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3849 (all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N)),one_one_nat) = N) # label(fact_3073_diff__Suc__1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3850 (all Y_2 all X_1 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zEven)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_2)),zOdd))))) # label(fact_3096_odd__minus__even) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3851 (all B_28 all A_34 zero_z126310315umeral = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_28),A_34)),B_28)) # label(fact_2493_mod__mult__self1__is__0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3852 (all N all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N),M)))) # label(fact_1313_zdvd__bounds) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3853 (all A_196 all N_63 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_196),N_63)),A_196) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_196),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_196),N_63))) # label(fact_900_power__commutes) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3854 (all N all M hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(div_mod_nat(M),N))) # label(fact_2687_mult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3855 (all Lx_4 all Ly_2 all Rx_4 all Ry_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_4),Ly_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_4),Ry_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_4),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ly_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_4),Ry_2)))) # label(fact_479_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3856 (all X all Y hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)),X))) # label(fact_4960_gcd__dvd1__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3857 (all C all A all B (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A),B)) <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C),A)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C),B))))) # label(fact_2064_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3858 (all Ta all B_1 all D_1 (is_int(Ta) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ta),one_one_int)),B_1)) -> (all X_2 (is_int(X_2) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1)) -> X_2 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa))))))) -> (Ta = X_2 -> Ta = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1))))))))) # label(fact_5095_bset_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3859 (all B_115 all C_67 all A_153 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_153)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_115),C_67)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_115),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_153),C_67)))))) # label(fact_1203_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3860 (all A all B all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),A)),hAPP_real_real(hAPP_r1250527377l_real(powr,X_1),B))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B))))) # label(fact_4433_powr__le__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3861 (all A_62 (is_int(A_62) -> hAPP_int_int(div_mod_int(A_62),zero_zero_int) = A_62)) # label(fact_2368_mod__by__0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3862 (all A_32 hAPP_int_int(div_mod_int(A_32),one_one_int) = zero_zero_int) # label(fact_2499_mod__by__1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3863 (all P all Q all R hAPP_int_int(hAPP_P2110489235nt_int(P,R),Q) = hAPP_P1175774780nt_int(hAPP_i61245889nt_int(hAPP_f578362681nt_int(cOMBC_1417754532nt_int,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3864 (all M hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(M),zero_zero_int),M))) # label(fact_1028_zcong__id) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3865 (all Nat_2 hAPP_nat_nat(suc,Nat_2) != zero_zero_nat) # label(fact_2852_nat_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3866 (all U_1 all X (X = hAPP_real_real(exp_real,U_1) -> hAPP_real_real(ln,X) = U_1)) # label(fact_3821_exp__ln__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3867 (all A_220 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),A_220) = A_220) # label(fact_601_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3868 (all X_38 all P_8 all Q_12 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_38),P_8)),Q_12) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_38),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12))) # label(fact_448_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3869 (all X hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = one_one_real) # label(fact_3729_sin__cos__squared__add2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3870 (all B_97 all A_119 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_119)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_97)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_119),B_97)))))) # label(fact_1858_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3871 (all A_77 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_77),zero_zero_real) = zero_zero_real) # label(fact_2287_mult_Ozero__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3872 (all A (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A) = zero_zero_rat <-> A = zero_zero_rat)) # label(fact_1977_double__zero__sym) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3873 (all A_3 hAPP_real_real(cos,A_3) = hAPP_complex_real(re,cis(A_3))) # label(fact_4146_Re__cis) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3874 (all L (is_int(L) -> (min = L <-> min = hAPP_int_int(bit1,L)))) # label(fact_767_rel__simps_I43_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3875 (all X_1 all Y_2 (X_1 = zero_zero_rat & Y_2 = zero_zero_rat <-> zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_1),X_1)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_2),Y_2)))) # label(fact_182_sum__squares__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3876 (all A_16 all B_12 (zero_zero_nat != B_12 -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_12),A_16)),B_12) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_16),B_12)),one_one_nat))) # label(fact_2629_div__add__self1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3877 (all A all X_3 all X_1 (hBOOL(tendsto_nat_real(X_3,X_1,sequentially)) -> ((exists N_3 all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_3),N_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(X_3,N_2)),A)))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),A))))) # label(fact_5035_LIMSEQ__le__const2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3878 (all A_3 zero_zero_int = hAPP_int_int(div_mod_int(A_3),hAPP_int_int(number_number_of_int,min))) # label(fact_2618_zmod__minus1__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3879 (all M all N all K_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N)),hAPP_nat_nat(suc,K_2))) # label(fact_2878_Suc__diff__diff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3880 (all N_12 all R_6 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_6)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_6),one_one_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,R_6),hAPP_nat_nat(suc,N_12))),R_6))))) # label(fact_3158_realpow__Suc__le__self) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3881 (all A_109 all B_87 all C_47 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_109),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_87),C_47)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_109),B_87)),C_47)) # label(fact_1947_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3882 (all X all Y X = hAPP_complex_real(re,hAPP_real_complex(hAPP_r265291036omplex(complex,X),Y))) # label(fact_4113_Re) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3883 (all L all U hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_at4362885an_nat(L),U)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,U),L)) # label(fact_4903_card__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3884 (all Sa all Ta all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Sa),Ta)) -> hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,hAPP_P369611207at_nat(hAPP_P402336767at_nat(produc494345619at_nat,hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,A),Sa)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,B),Ta))),pair_less))))) # label(fact_4540_pair__lessI2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3885 (all P all Q all R hAPP_r1938957037l_bool(hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_real_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_007) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3886 (all N all M hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N))),hAPP_nat_int(semiri1621563631at_int,M)))) # label(fact_3510_negative__zle) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3887 (all A_17 all C_4 all B_13 (is_int(B_13) -> (B_13 != zero_zero_int -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_17),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_13),C_4))),B_13) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_4),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_17),B_13))))) # label(fact_2624_div__mult__self2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3888 (all A_3 all B_2 all X (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),X)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),X)))) # label(fact_4324_gcd__semilattice__nat_Ole__infI2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3889 (all N_1 all Ma (hBOOL(hAPP_int_bool(nat_is_nat,Ma)) -> (hBOOL(hAPP_int_bool(nat_is_nat,N_1)) -> image_nat_int(semiri1621563631at_int,hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_int_nat(nat,Ma)),hAPP_int_nat(nat,N_1))) = hAPP_i1948725293t_bool(ord_at875362053st_int(Ma),N_1)))) # label(fact_5128_SetInterval_Otransfer__int__nat__set__functions) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3890 (all A_194 all B_150 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_194),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_150),A_194)))) # label(fact_917_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3891 (all C_83 all D_16 all A_191 all B_147 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_191),B_147)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,C_83),D_16)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_191),C_83)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_147),D_16)))))) # label(fact_934_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3892 (all M zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),M)) # label(fact_1738_diff__self__eq__0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3893 (all I_1 all P_1 all K_1 (hBOOL(hAPP_nat_bool(P_1,K_1)) -> ((all N_2 (hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_2))) -> hBOOL(hAPP_nat_bool(P_1,N_2)))) -> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,K_1),I_1)))))) # label(fact_3289_zero__induct__lemma) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3894 (all X_33 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_33),zero_zero_nat) = one_one_int) # label(fact_675_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3895 (all K_2 zero_zero_rat = hAPP_int_rat(hAPP_int_fun_int_rat(fract,K_2),zero_zero_int)) # label(fact_4674_rat__number__collapse_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3896 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N))) # label(fact_2868_add__Suc__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3897 (all N hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N))))) # label(fact_3661_real__of__nat__Suc__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3898 (all X_35 all Y_27 all Q_11 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_35),Q_11)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_27),Q_11)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_35),Y_27)),Q_11)) # label(fact_663_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3899 (all A_3 all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),archim1246769320r_real(X))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,A_3)),X)))) # label(fact_4418_real__le__floor) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3900 (all N all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N),X))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,N)),hAPP_int_real(real_int,X))))) # label(fact_4415_real__of__int__div4) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3901 (all A_3 ((A_3 = zero_zero_real -> zero_zero_real = hAPP_real_real(sgn_sgn_real,A_3)) & (zero_zero_real != A_3 -> (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> hAPP_real_real(uminus_uminus_real,one_one_real) = hAPP_real_real(sgn_sgn_real,A_3)) & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> hAPP_real_real(sgn_sgn_real,A_3) = one_one_real)))) # label(fact_3576_sgn__real__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3902 (all X_1 (X_1 = zero_zero_real <-> X_1 = zero_zero_real)) # label(fact_1896_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3903 (all P all Q all R hAPP_n546711566l_real(hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,P),Q),R) = hAPP_r1250527377l_real(hAPP_n682674106l_real(P,R),hAPP_nat_real(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__fun_Itc__Real) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3904 (all A_31 all C_12 all B_26 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_31),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_12),B_26))),B_26) = hAPP_C498520661umeral(div_mo1740067990umeral(A_31),B_26)) # label(fact_2506_mod__mult__self1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3905 (all B_72 all C_35 all A_84 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_84)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_72),C_35)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_72),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_84),C_35)))))) # label(fact_2156_add__increasing) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3906 (all Ma all D (is_int(Ma) -> (zero_zero_int = hAPP_int_int(div_mod_int(Ma),D) <-> (exists Q_4 (is_int(Q_4) & Ma = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D),Q_4)))))) # label(fact_2530_zmod__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3907 (all X all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N)) = X))) # label(fact_3933_real__root__pos2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3908 (all B_118 all A_157 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_157)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_118),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_118),A_157)),zero_zero_int))))) # label(fact_1173_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3909 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),zero_zero_int)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_3)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_2),one_one_int)),A_3))) # label(fact_2670_neg__zdiv__mult__2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3910 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),zero_zero_real)) -> hAPP_real_real(sgn_sgn_real,X) = hAPP_int_real(number267125858f_real,min))) # label(fact_3574_real__sgn__neg) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3911 (all B_73 all A_85 all C_36 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_36)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_73),A_85)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_73),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_85),C_36)))))) # label(fact_2151_add__increasing2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3912 (all D_3 all C_1 all A_3 all B_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) -> (B_2 = C_1 -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C_1),D_3),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),D_3),M)))))) # label(fact_810_zcong__eq__trans) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3913 (all A_239 all B_167 all V_15 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_239),hAPP_int_complex(number528085621omplex,V_15))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_167),hAPP_int_complex(number528085621omplex,V_15))) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_239),B_167)),hAPP_int_complex(number528085621omplex,V_15))) # label(fact_187_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3914 (all R_3 hAPP_real_real(abs_abs_real,R_3) = hAPP_real_real(norm_norm_real,R_3)) # label(fact_3382_real__norm__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3915 (all Ma all N_1 all P_1 ((all M_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,M_1),zero_zero_nat))) -> ((all M_1 all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_2),hAPP_nat_nat(div_mod_nat(M_1),N_2))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,M_1),N_2))))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Ma),N_1))))) # label(fact_4551_gcd__nat__induct) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3916 (all X all Y (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,X),Y)) -> (exists P_4 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),X)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),Y)) & hBOOL(hAPP_nat_bool(prime,P_4)))))) # label(fact_4203_coprime__prime__dvd__ex) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3917 (all A_143 all C_65 all B_112 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_143),C_65)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_112),C_65))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_65)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_143),B_112))))) # label(fact_1327_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3918 (all C_46 all A_104 all B_86 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_46),A_104)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_46),B_86))) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_104),B_86)))) # label(fact_1996_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3919 (all Z_2 all Z_3 ((-hBOOL(hAPP_int_bool(nat_neg,Z_3)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_2),Z_3))),zero_zero_nat),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_2),Z_3))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,Z_2)),hAPP_int_nat(nat,Z_3))) & (hBOOL(hAPP_int_bool(nat_neg,Z_3)) -> hAPP_int_nat(nat,Z_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,Z_2)),hAPP_int_nat(nat,Z_3))))) # label(fact_4752_diff__nat__eq__if) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3920 (all Lx all Rx all Ry hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx),Ry)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Lx),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,Rx),Ry))) # label(fact_505_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3921 (all A_140 all C_62 all B_109 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_140),C_62)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_109),C_62))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_62)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_140),B_109))))) # label(fact_1344_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3922 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),M))))) # label(fact_1759_diff__less) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3923 (all M all N all K_2 hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N),K_2))),N) = hAPP_int_int(legacy_zgcd(M),N)) # label(fact_4502_zgcd__zadd__zmult) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3924 (all A all N_1 (zero_zero_real = A & zero_zero_nat != N_1 <-> zero_zero_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A),N_1))) # label(fact_1291_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3925 (all J_2 all K_2 all A_3 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),zero_zero_int),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),zero_zero_int),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),J_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),K_2))),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),K_2),P_3))))))))) # label(fact_2236_MultInv__zcong__prop3) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3926 (all A_24 all B_19 all C_8 ((C_8 != zero_z891286103de_int -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_8),A_24)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_8),B_19)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_24),B_19)) & (C_8 = zero_z891286103de_int -> zero_z891286103de_int = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_8),A_24)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_8),B_19))))) # label(fact_2552_div__mult__mult1__if) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3927 (all Y_2 all X_1 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),zOdd)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),Y_2)),zOdd))))) # label(fact_3095_even__minus__odd) # label(axiom) # label(non_clause). [assumption]. 7.56/7.57 3928 (all K_2 all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(div_mod_nat(M),N)) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N))) # label(fact_2694_mod__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3929 (all A_10 all N_15 all B_8 (hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_10),hAPP_nat_nat(suc,N_15)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_8),hAPP_nat_nat(suc,N_15)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_10)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_8)) -> A_10 = B_8)))) # label(fact_3104_power__inject__base) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3930 (all Lx_6 all Ly_4 all Rx_6 all Ry_4 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_6),Rx_6)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Ly_4),Ry_4)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_6),Ly_4)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_6),Ry_4))) # label(fact_467_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3931 (all A_20 all C_6 all B_16 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_6),B_16)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_20),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_16),C_6)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_20),B_16)),C_6))) # label(fact_2572_div__mult__swap) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3932 (all M all N all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),N)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),M))))) # label(fact_1317_zprime__zdvd__zmult__better) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3933 (all Y_6 all P_6 all N_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10)) -> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6))) # label(fact_3173_lemma__realpow__diff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3934 (all A all B all C (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B))))) # label(fact_1191_mult__less__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3935 (all A_11 all N_17 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_11),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_11),N_17)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_11),hAPP_nat_nat(suc,N_17))) # label(fact_3014_power__Suc) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3936 (all C_88 all A_206 all B_157 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_206),B_157)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,B_157),C_88)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_206),C_88))))) # label(fact_801_dvd__trans) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3937 (all M all N all K_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(div_mod_nat(M),N)),K_2) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K_2))) # label(fact_2695_mod__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3938 (all X all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> X = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),N)))) # label(fact_3934_real__root__pow__pos2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3939 (all X all Y hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)),Y))) # label(fact_4330_gcd__semilattice__nat_Oinf__le2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3940 (all M all N all I (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),I)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),N))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))))) # label(fact_1011_nat__power__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3941 (all A_91 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_91),one_one_nat) = A_91) # label(fact_2106_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3942 (all B_92 all A_114 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_114),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_92),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_114),B_92)),zero_zero_int))))) # label(fact_1888_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3943 (all X_20 all Y_16 all A_120 all B_98 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_20),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Y_16),B_98))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_20),A_120)),B_98)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_20),Y_16)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_120),B_98))) # label(fact_1820_mult__diff__mult) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3944 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(times_times_real,ratreal(X)),ratreal(Y)) = ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X),Y))) # label(fact_4705_real__times__code) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3945 (all A_183 all M_22 all N_57 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_183),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_22),N_57)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_183),M_22)),N_57)) # label(fact_1019_power__mult) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3946 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arcsin,Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3509_arcsin__ubound) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3947 (all C_69 all A_163 all B_124 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_163),B_124)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_69)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_69),A_163)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_69),B_124)))))) # label(fact_1137_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3948 (all P_1 all A0 all A1 (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(upto_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1))) -> ((all I_2 all J_1 (is_int(J_1) & is_int(I_2) -> (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(upto_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,I_2),J_1))) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),J_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int)),J_1))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,I_2),J_1)))))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1))))) # label(fact_4573_upto_Opinduct) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3949 (all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),A_3)),one_one_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),one_one_int),P_3)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),P_3)))))) # label(fact_1605_zcong__square) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3950 (all N hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),pi)) = zero_zero_real) # label(fact_3688_tan__npi) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3951 (all Y_3 all B all X_3 all A (hBOOL(tendsto_nat_real(X_3,A,sequentially)) -> (hBOOL(tendsto_nat_real(Y_3,B,sequentially)) -> hBOOL(tendsto_nat_complex(hAPP_f219720582omplex(hAPP_f1052840010omplex(cOMBS_696648398omplex,hAPP_f23851874omplex(hAPP_f406530075omplex(cOMBB_1714920583ex_nat,complex),X_3)),Y_3),hAPP_real_complex(hAPP_r265291036omplex(complex,A),B),sequentially))))) # label(fact_5058_LIMSEQ__Complex) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3952 (all X_1 all Y_2 all P_1 ((hBOOL(P_1) -> hAPP_int_nat(nat,X_1) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,P_1),X_1),Y_2))) & (-hBOOL(P_1) -> hAPP_int_nat(nat,Y_2) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,P_1),X_1),Y_2))))) # label(fact_2683_nat__if__cong) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3953 (all C_85 all A_193 all B_149 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_193),B_149)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_193),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_149),C_85))))) # label(fact_921_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3954 (all N_1 all Ma (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,N_1),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,quickcheck_int_of(N_1)),quickcheck_int_of(Ma))))) # label(fact_4253_less__code__int__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3955 (all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(fact_fact_nat,N)))) # label(fact_3811_fact__ge__Suc__0__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3956 (all A_238 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_238),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_214_zero__le__power2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3957 (all Y all X all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),Y))))))) # label(fact_3930_real__root__less__mono__lemma) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3958 (all M all N all Z_1 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N))),Z_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,N)),Z_1))) # label(fact_3040_zadd__int__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3959 (all A_226 all B_163 all C_92 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_226),C_92)),B_163) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_226),B_163)),C_92)) # label(fact_552_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3960 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),pi)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))))))) # label(fact_3572_cos__monotone__0__pi) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3961 (all B_2 all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)),zero_zero_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(B_2),zero_zero_int),P_3)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3)))))) # label(fact_756_zcong__zprime__prod__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3962 (all B_74 all A_86 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_86),zero_z126310315umeral)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_74),zero_z126310315umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_86),B_74)),zero_z126310315umeral))))) # label(fact_2141_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3963 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_real_real(sqrt,X) != zero_zero_real)) # label(fact_3327_real__sqrt__not__eq__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3964 (all M all N all K_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),K_2)) # label(fact_1757_diff__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3965 (all X_1 hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),sin_coeff)),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_real_real(sin,X_1)))) # label(fact_4864_sin__converges) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3966 (all B all D all A (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A),B) = D <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),A)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D),B)) & (all E (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,E),A)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,E),B)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,E),D)))))) # label(fact_4326_gcd__unique__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3967 (all X_1 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y_2)),zero_zero_real)))) # label(fact_1697_real__le__eq__diff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3968 (all C_33 all D_5 all A_79 all B_67 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_79),B_67)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_33),D_5)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_79),C_33)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_67),D_5)))))) # label(fact_2204_add__less__le__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3969 (all N_54 all A_151 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_151)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_151),N_54))))) # label(fact_1243_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3970 (all X_1 all A_1 (hBOOL(hAPP_P62333565t_bool(A_1,X_1)) <-> hBOOL(hAPP_f526364711l_bool(hAPP_P1062890741l_bool(member2143287562nt_int,X_1),A_1)))) # label(fact_143_mem__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3971 (all Z_1 (is_int(Z_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,Z_1)) = Z_1) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> zero_zero_int = hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,Z_1))))) # label(fact_3121_int__nat__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3972 (all P_1 all X_1 all Y_2 (hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X_1),Y_2)))) <-> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_2),X_1)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,X_1)),hAPP_nat_int(semiri1621563631at_int,Y_2))))) & (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2)) -> hBOOL(hAPP_int_bool(P_1,zero_zero_int))))) # label(fact_3194_zdiff__int__split) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3973 (all X -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(abs_abs_real,X)),one_one_real)),X))) # label(fact_3395_abs__add__one__not__less__self) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3974 (all A_3 all B_2 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),B_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_3)),B_2))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_1665_zdiff__power2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3975 (all A_10 all N_15 all B_8 (hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_10),hAPP_nat_nat(suc,N_15)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_8),hAPP_nat_nat(suc,N_15)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_10)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_8)) -> A_10 = B_8)))) # label(fact_3106_power__inject__base) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3976 (all C_42 all A_100 all B_82 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_100),B_82)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_100),C_42)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_82),C_42))))) # label(fact_2020_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3977 (all A_205 all B_156 (hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_205),B_156) = zero_z126310315umeral -> A_205 = zero_z126310315umeral | B_156 = zero_z126310315umeral)) # label(fact_824_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3978 (all A_3 hAPP_complex_complex(invers1449016382omplex,cis(A_3)) = cis(hAPP_real_real(uminus_uminus_real,A_3))) # label(fact_4150_cis__inverse) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3979 (all Lx_5 all Ly_3 all Rx_5 all Ry_3 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_5),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_5),Ly_3)),Ry_3)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_5),Ly_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_5),Ry_3))) # label(fact_473_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3980 (all N all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M),N)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),hAPP_int_int(fact_fact_int,N)))))) # label(fact_4284_dvd__fact__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3981 (all N (is_int(N) -> N = archim856651990g_real(hAPP_int_real(real_int,N)))) # label(fact_4397_ceiling__real__of__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3982 (all A_235 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_235),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_235),A_235)),A_235)) # label(fact_276_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3983 (all P all Q all R hAPP_nat_nat(P,hAPP_nat_nat(Q,R)) = hAPP_nat_nat(hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3984 (all A_56 all C_26 all B_47 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_56),C_26)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_47),C_26)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(div_mod_nat(A_56),B_47)),C_26)) # label(fact_2394_mod__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3985 (all B_120 all A_159 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_120),A_159))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_159)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_120))))) # label(fact_1165_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3986 (all M all N all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (zero_zero_nat != N -> (N = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_3),M) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)))))) # label(fact_4183_prime__factor__lt) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3987 (all N_1 hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_nat_real(real_nat,N_1)),field_1210416355s_real))) # label(fact_4632_Rats__real__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3988 (all A_31 all C_12 all B_26 hAPP_Q1762011733de_int(div_mo231679042de_int(A_31),B_26) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_31),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_12),B_26))),B_26)) # label(fact_2507_mod__mult__self1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3989 (all X all Y hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,Y)),hAPP_real_real(cos,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,Y)),hAPP_real_real(sin,X)))) # label(fact_3682_cos__diff2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3990 (all X all A_3 hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3) != zero_zero_real) # label(fact_4379_powr__not__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3991 (all N all M hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,quickcheck_int_of(N)),quickcheck_int_of(M))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,N),M)) # label(fact_4561_plus__code__int__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3992 (all A_189 all B_145 all C_82 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_189),B_145)),C_82)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_189),C_82)))) # label(fact_952_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3993 (all D_3 all A_3 all B_2 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)) = one_one_nat -> one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),A_3))) # label(fact_4356_coprime__lmult__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3994 (all B_2 all A_3 all A_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),A_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_5),B_2)))))) # label(fact_2596_zdiv__mono1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3995 (all A_113 all B_91 all C_51 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_113),B_91)),C_51) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_113),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_91),C_51))) # label(fact_1903_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3996 (all A_110 all B_88 all C_48 (hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_110),C_48) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_110),B_88) -> B_88 = C_48)) # label(fact_1926_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3997 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K1)),hAPP_int_int(bit0,K2))))) # label(fact_260_less__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3998 (all V_19 all W_15 hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_19),W_15)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_19)),hAPP_int_int(number_number_of_int,W_15))) # label(fact_83_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 3999 (all I all J_2 hAPP_int_int(legacy_zgcd(hAPP_int_int(uminus_uminus_int,I)),J_2) = hAPP_int_int(legacy_zgcd(I),J_2)) # label(fact_4497_zgcd__zminus) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4000 (all Y_14 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,zero_zero_real),Y_14) = zero_zero_real) # label(fact_2285_mult__left_Ozero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4001 (all N_33 all A_123 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_123)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_33)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_123),N_33)))))) # label(fact_1488_one__less__power) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4002 (all C_71 all B_126 all A_165 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_126),A_165)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_71),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_165),C_71)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_126),C_71)))))) # label(fact_1130_mult__strict__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4003 (all A_219 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_219),zero_zero_rat) = A_219) # label(fact_610_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4004 (all X_1 hBOOL(hAPP_real_bool(deriv_real(sin,X_1),hAPP_real_real(cos,X_1)))) # label(fact_3862_DERIV__sin) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4005 (all N_16 all M_5 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_16),M_5)) -> hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_5),N_16)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,M_5)),hAPP_nat_int(semiri1621563631at_int,N_16)))) # label(fact_3101_of__nat__diff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4006 (all A_3 all N all B_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_2),N))) # label(fact_4302_gcd__exp__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4007 (all P all Q (-hBOOL(Q) | hBOOL(hAPP_bool_bool(fimplies(P),Q)))) # label(help_fimplies_2_1_U) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4008 (all X_44 all Y_32 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_44),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_32),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_32)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_44),Y_32))))) # label(fact_222_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4009 (all X_21 all N_31 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31)) -> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_21),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),one_one_nat))),X_21) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_21),N_31))) # label(fact_1801_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4010 (all A_139 all C_61 all B_108 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_139),C_61)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_108),C_61))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_61)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_139),B_108))))) # label(fact_1352_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4011 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(arccos,Y)))))) # label(fact_3606_arccos__lbound) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4012 (all K_2 hAPP_int_int(pred,K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),min)) # label(fact_4047_add__Min__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4013 (all A_143 all C_65 all B_112 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_143),C_65)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_112),C_65))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_65)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_143),B_112))))) # label(fact_1324_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4014 (all X_36 all Y_28 all Z_9 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_36),Y_28)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_36),Z_9)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_36),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Y_28),Z_9))) # label(fact_619_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4015 (all X all N hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),pi))) = hAPP_real_real(tan,X)) # label(fact_3691_tan__periodic__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4016 (all C all A all B (is_int(C) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B))) <-> zero_zero_int = C | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B))))) # label(fact_1266_dvd__mult__cancel__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4017 (all A_89 A_89 = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,one_on1645066479umeral),A_89)) # label(fact_2122_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4018 (all X all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N),X)))))) # label(fact_3918_real__root__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4019 (all A_208 hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_208),zero_z891286103de_int))) # label(fact_760_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4020 (all Lx_5 all Ly_3 all Rx_5 all Ry_3 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_5),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_5),Ly_3)),Ry_3)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_5),Ly_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_5),Ry_3))) # label(fact_471_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4021 (all A_88 all B_76 A_88 = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_88),B_76)),B_76)) # label(fact_2133_add__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4022 (all A_3 all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_3)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)))))) # label(fact_4432_ge__one__powr__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4023 (all X all Y (Y != X -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y),X))))) # label(fact_1586_linorder__neqE__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4024 (all R_3 all P_3 all Q_2 (normalize(R_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2) -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,P_3),Q_2) = one_one_int)) # label(fact_5018_normalize__coprime) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4025 (all A_184 all M_23 all N_58 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_23),N_58)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_184),M_23)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_184),N_58))))) # label(fact_1009_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4026 (all X all Y (hBOOL(hAPP_int_bool(even_odd_even_int,Y)) -> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y))))) # label(fact_2885_anything__times__even) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4027 (all C_59 all D_12 all A_137 all B_106 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_137),B_106)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_59),D_12)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_137)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_59)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_137),C_59)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_106),D_12)))))))) # label(fact_1364_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4028 (all X_1 hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,fact_fact_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_4840_exp__tail__after__first__two__terms__summable) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4029 (all K_1 all P_1 all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D)) -> ((all X_2 (is_int(X_2) -> (hBOOL(hAPP_int_bool(P_1,X_2)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_1)) -> (all X_2 (hBOOL(hAPP_int_bool(P_1,X_2)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),D)))))))))) # label(fact_1692_incr__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4030 (all B_2 all A_3 all X (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),X)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),X)))) # label(fact_4325_gcd__semilattice__nat_Ole__infI1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4031 (all P_1 all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) -> (all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2))),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2))))))) # label(fact_4243_Bolzano__bisect__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4032 (all P all Q all R hAPP_P1975530577nt_int(hAPP_i1216375074nt_int(P,R),hAPP_i1524277240nt_int(Q,R)) = hAPP_i1524277240nt_int(hAPP_f1735705551nt_int(hAPP_f713949180nt_int(cOMBS_1891347093nt_int,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4033 (all X_31 all Y_25 (Y_25 != X_31 -> (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_31),Y_25)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_25),X_31))))) # label(fact_792_linorder__neqE__linordered__idom) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4034 (all A_3 all B_2 ((-(hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2))) -> hAPP_P1975530577nt_int(adjust(B_2),posDivAlg(A_3,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))) = posDivAlg(A_3,B_2)) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2)) -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),A_3) = posDivAlg(A_3,B_2)))) # label(fact_2679_posDivAlg_Osimps) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4035 (all A_92 A_92 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_92),one_one_real)) # label(fact_2100_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4036 (all M all K_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2)),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N)))) # label(fact_1647_add__leD2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4037 (all Y all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Y)),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3407_real__sqrt__ge__abs2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4038 (all A_3 all B_2 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2))) -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)) = negDivAlg(A_3,B_2)) & (-(hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)))) -> negDivAlg(A_3,B_2) = hAPP_P1975530577nt_int(adjust(B_2),negDivAlg(A_3,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2)))))) # label(fact_2666_negDivAlg_Osimps) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4039 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_int_nat(nat,X)),hAPP_int_nat(nat,Y)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y))))) # label(fact_5020_transfer__nat__int__gcd_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4040 (all D all Sa exists Z all X_2 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_2),Z)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_2),Sa))) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_2),Sa)))))) # label(fact_2261_minf_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4041 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N))) # label(fact_4359_gcd__diff1__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4042 (all A_110 all B_88 all C_48 (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_110),B_88) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_110),C_48) -> C_48 = B_88)) # label(fact_1925_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4043 (all N_59 all M_24 all X_28 all Y_22 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_28),Y_22)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_59),M_24)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_28),N_59)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_22),M_24)))))) # label(fact_1000_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4044 (all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5)) -> comple219730294t_bool(setS(A,P_5)) = hAPP_i1948725293t_bool(sRStar,P_5))))) # label(fact_3306_MultInvPair__prop2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4045 (all M all N all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),M)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),N))))) # label(fact_4171_prime__dvd__mult) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4046 (all A_234 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_234),A_234) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_234),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_304_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4047 (all X_30 all Y_24 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_30),Y_24)) -> (Y_24 != X_30 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_30),Y_24))))) # label(fact_815_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4048 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)))))) # label(fact_3128_n__less__m__mult__n) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4049 (all K_1 all Ma all N_1 (N_1 = Ma | K_1 = zero_zero_nat <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma))) # label(fact_1622_mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4050 (all B_93 all A_115 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_115),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_93),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_115),B_93)),zero_zero_real))))) # label(fact_1884_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4051 (all X_1 all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(number_number_of_int,X_1)),hAPP_int_int(number_number_of_int,Y_2))) <-> hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,Y_2)),hAPP_int_int(number_number_of_int,X_1)) = zero_zero_int)) # label(fact_2619_zdvd__iff__zmod__eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4052 (all N all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_2),N))))) # label(fact_4018_coprime__exp__imp) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4053 (all X_1 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),hAPP_real_real(uminus_uminus_real,X_1))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Y_2)),zero_zero_real)))) # label(fact_3492_real__add__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4054 (all P_1 all N_1 all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) -> hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_1))),hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_1))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1)))) # label(fact_4242_Bolzano__bisect__diff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4055 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) # label(fact_2877_diff__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4056 (all X_3 (hBOOL(summable_complex(X_3)) -> hAPP_f352196356l_real(suminf_real,hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_3)) = hAPP_complex_real(re,hAPP_f213450978omplex(suminf_complex,X_3)))) # label(fact_4848_Re_Osuminf) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4057 (all C_72 all B_127 all A_166 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_127),A_166)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_72),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_72),A_166)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_72),B_127)))))) # label(fact_1125_mult__strict__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4058 (all M_8 all N_22 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(semiri151668891at_rat,M_8)),hAPP_nat_rat(semiri151668891at_rat,N_22)) = hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_8),N_22))) # label(fact_2954_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4059 (all A_195 all B_151 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_195),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_195),B_151)))) # label(fact_910_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4060 (all N_42 all A_130 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_130)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_130),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_130),N_42)))))) # label(fact_1431_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4061 (all A (zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A) <-> zero_zero_real = A)) # label(fact_1976_double__zero__sym) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4062 (all A all B (B = A <-> hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A),B) = zero_zero_real)) # label(fact_2081_eq__iff__diff__eq__0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4063 (all X all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N),X)))))) # label(fact_3921_real__root__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4064 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_real_real(sgn_sgn_real,X) = one_one_real)) # label(fact_3571_real__sgn__pos) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4065 (all A_90 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,one_on1645066479umeral),A_90) = A_90) # label(fact_2115_mult__1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4066 (all X_45 all Y_33 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_45),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_33),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_33)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_45),Y_33))))) # label(fact_208_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4067 (all A_122 all B_100 all V_6 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_122),B_100)),hAPP_int_int(number_number_of_int,V_6)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_122),hAPP_int_int(number_number_of_int,V_6))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_100),hAPP_int_int(number_number_of_int,V_6)))) # label(fact_1548_left__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4068 (all B_155 all A_204 (is_int(B_155) & is_int(A_204) -> (zero_zero_int != A_204 -> (B_155 != zero_zero_int -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_204),B_155) != zero_zero_int)))) # label(fact_828_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4069 (all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) <-> -hBOOL(hAPP_int_bool(nat_neg,X_1)))) # label(fact_3271_not__neg__eq__ge__0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4070 (all D_3 all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),B_2)))) # label(fact_4009_coprime__lmul2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4071 (all C_43 all A_101 all B_83 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_101),B_83)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_43),A_101)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_43),B_83))))) # label(fact_2012_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4072 (all M_13 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,one_on1645066479umeral),hAPP_n1108039445umeral(semiri1619134803umeral,M_13)) = hAPP_n1108039445umeral(semiri1619134803umeral,hAPP_nat_nat(suc,M_13))) # label(fact_2841_of__nat__Suc) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4073 (all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,N),zero_zero_nat) = N) # label(fact_5191_max__0R) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4074 (all X_1 hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),cos_coeff)),hAPP_r474017924t_real(power_power_real,X_1))),hAPP_real_real(cos,X_1)))) # label(fact_4865_cos__converges) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4075 (all B_155 all A_204 (zero_z891286103de_int != A_204 -> (zero_z891286103de_int != B_155 -> zero_z891286103de_int != hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_204),B_155)))) # label(fact_833_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4076 (all P_5 hBOOL(hAPP_f448129468l_bool(resSet(P_5),hAPP_i1948725293t_bool(sRStar,P_5)))) # label(fact_4055_ResSet__SRStar__prop) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4077 (all Ma all X_1 ((exists Y_1 (is_int(Y_1) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_1),Ma)))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,Ma),X_1)))) # label(fact_1687_QuadRes__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4078 (all X all Y (X != Y | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y)))) # label(help_fequal_2_1_fequal_000tc__Nat__Onat_T) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4079 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hBOOL(hAPP_real_bool(deriv_real(ln,X_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),X_1))))) # label(fact_3870_DERIV__ln__divide) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4080 (all B_140 all A_180 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_140)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_180)) | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_140),zero_zero_rat)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_180),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_180),B_140))))) # label(fact_1052_split__mult__pos__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4081 (all X_1 all Y_2 (is_int(Y_2) & is_int(X_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),X_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_2),Y_2))),zero_zero_int)) <-> X_1 = zero_zero_int & zero_zero_int = Y_2))) # label(fact_91_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4082 (all N all D_3 all A_3 (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),A_3) -> one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N)))) # label(fact_4973_coprime__exp__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4083 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),B))))) # label(fact_1119_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4084 (all N_37 all A_126 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_126)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_126),one_on1645066479umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_126),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_126),N_37))),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_126),N_37)))))) # label(fact_1470_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4085 (all X_1 (hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_nat_int(semiri1621563631at_int,X_1))) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)))) # label(fact_3766_even__nat__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4086 (all K_2 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,min),K_2) = hAPP_int_int(uminus_uminus_int,K_2)) # label(fact_3535_mult__Min) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4087 (all X all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)) = hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(abs_abs_real,X)))) # label(fact_3914_real__root__abs) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4088 (all Ta all A_1 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> (all X_2 (is_int(X_2) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1)) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa) != X_2)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ta),X_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ta),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1))))))))) # label(fact_5093_aset_I7_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4089 (all A_3 hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(number_number_of_int,min)) = hAPP_int_int(uminus_uminus_int,A_3)) # label(fact_3546_zdiv__minus1__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4090 (all A_124 all N_34 all B_102 (hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_124),N_34) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_102),N_34) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_124)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_102)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_34)) -> A_124 = B_102))))) # label(fact_1482_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4091 (all N_1 zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(hAPP_b646336293l_real(if_real,hAPP_nat_bool(even_odd_even_nat,N_1)),zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,zero_zero_nat))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),N_1))) # label(fact_3789_lemma__STAR__sin) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4092 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A),B)),zero_zero_int))))) # label(fact_2522_pos__imp__zdiv__neg__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4093 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,zero_zero_nat),M) = divmod_nat(M,N))) # label(fact_4210_divmod__nat__base) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4094 (all A_125 all N_36 all N_35 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_36),N_35)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_125)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_125),one_one_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_125),N_35)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_125),N_36))))))) # label(fact_1478_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4095 (all N (zero_zero_nat != N -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),N) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),N)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3233_lemma__Suc__Suc__4n__diff__2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4096 (all A_230 all C_96 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_230),C_96) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_96),A_230)) # label(fact_521_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4097 (all W all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),one_one_int)),Z_1)))) # label(fact_156_zless__imp__add1__zle) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4098 (all Ma all N_1 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N_1) = hAPP_nat_nat(suc,zero_zero_nat) <-> zero_zero_nat = Ma & N_1 = hAPP_nat_nat(suc,zero_zero_nat) | hAPP_nat_nat(suc,zero_zero_nat) = Ma & N_1 = zero_zero_nat)) # label(fact_3051_add__is__1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4099 (all Q_2 ((all A_2 all B_4 (is_int(A_2) & is_int(B_4) -> (Q_2 = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_4) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_4)) -> (A_2 != zero_zero_int -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_4) != one_one_int))))) -> zero_zero_rat = Q_2)) # label(fact_5025_Rat__cases__nonzero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4100 (all Ma all X_2 all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcongm(Ma),X_2),Xa)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_2),Xa),Ma)))) # label(fact_5153_zcongm__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4101 (all V_12 all W_10 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_12)),hAPP_int_int(number_number_of_int,W_10)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_12),W_10))) # label(fact_337_add__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4102 (all I all J_2 all K_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),K_2)),J_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),J_2)),K_2)) # label(fact_1842_diff__commute) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4103 (all W_12 hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,W_12)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat)),hAPP_int_rat(number_number_of_rat,W_12))) # label(fact_273_double__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4104 (all C_78 all D_14 all A_178 all B_138 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_178),B_138)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_78),D_14)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_178)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_78)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_178),C_78)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_138),D_14)))))))) # label(fact_1063_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 7.56/7.58 4105 (all K_2 all M all N (is_int(K_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N))) -> (K_2 != zero_zero_int -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),N)))))) # label(fact_456_zdvd__mult__cancel) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4106 (all C_72 all B_127 all A_166 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_127),A_166)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_72),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_72),A_166)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_72),B_127)))))) # label(fact_1127_mult__strict__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4107 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) # label(fact_1591_nat__add__commute) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4108 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),one_one_real)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X_1)),one_one_nat)))) # label(fact_3401_natceiling__le__eq__one) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4109 (all P_1 all Ma all N_1 ((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_1),Ma))) -> ((N_1 = Ma -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_1),Ma))) -> ((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),Ma)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_1),Ma))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_1),Ma)))))) # label(fact_1590_nat__less__cases) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4110 (all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(even_odd_even_int,X_1)) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_int_nat(nat,X_1)))))) # label(fact_3782_pos__int__even__equiv__nat__even) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4111 (all A (hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially)) -> (hBOOL(monoseq_real(A)) -> hBOOL(tendsto_nat_real(hAPP_f292751352t_real(hAPP_f1607377819t_real(cOMBB_1533963633al_nat,hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_at4362885an_nat(zero_zero_nat)),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),sequentially))))) # label(fact_5052_summable__Leibniz_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4112 (all Lx_2 all Ly all Rx_2 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_2),Ly)),Rx_2) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_2),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ly),Rx_2))) # label(fact_497_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4113 (all A_106 A_106 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A_106)) # label(fact_1984_add__0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4114 (all P all Q all R hAPP_real_real(hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,P),Q),R) = hAPP_real_real(hAPP_r1250527377l_real(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__RealDef_) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4115 (all M_9 all N_23 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_23)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_9)),hAPP_nat_nat(semiri984289939at_nat,N_23))))) # label(fact_2939_less__imp__of__nat__less) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4116 (all Real1_2 all Real2_2 all Real1_1 all Real2_1 (Real2_1 = Real2_2 & Real1_2 = Real1_1 <-> hAPP_real_complex(hAPP_r265291036omplex(complex,Real1_2),Real2_2) = hAPP_real_complex(hAPP_r265291036omplex(complex,Real1_1),Real2_1))) # label(fact_3956_complex_Oinject) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4117 (all Z_4 all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),zero_zero_int)) -> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),Z_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat,hAPP_int_int(uminus_uminus_int,Z_1))),hAPP_int_nat(nat,hAPP_int_int(uminus_uminus_int,Z_4))))) # label(fact_3551_nat__mult__distrib__neg) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4118 (all A_13 all B_9 A_13 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_13),B_9)),B_9)),hAPP_Q1762011733de_int(div_mo231679042de_int(A_13),B_9))) # label(fact_2676_mod__div__decomp) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4119 (all C_1 all A_3 all B_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),C_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_2),C_1)),M)))) # label(fact_1038_zcong__shift) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4120 (all A_107 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_107),zero_z126310315umeral) = A_107) # label(fact_1968_add__0__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4121 (all A_231 all B_165 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_165),A_231) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_231),B_165)) # label(fact_513_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4122 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),zero_zero_int)) -> hAPP_int_int(div_mod_int(A_3),B_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)))) # label(fact_2647_mod__pos__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4123 (all N_1 all S (hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,S)) -> ((all X_2 (hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_2),S)) -> hBOOL(hAPP_f448129468l_bool(finite_finite_int,X_2)))) -> ((all X_2 (hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_2),S)) -> hAPP_f957591787ol_nat(finite_card_int,X_2) = N_1)) -> hAPP_f521865025ol_nat(hAPP_f93439879ol_nat(big_co1758102091ol_nat,finite_card_int),S) = hAPP_f521865025ol_nat(hAPP_f93439879ol_nat(big_co1758102091ol_nat,cOMBK_1643584600t_bool(N_1)),S))))) # label(fact_5028_card__setsum__aux) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4124 (all A_121 all C_54 all B_99 all D_9 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_121),B_99)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C_54),D_9)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_121),C_54)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_99),D_9))) # label(fact_1807_add__diff__add) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4125 (all P all Q all R hAPP_P752050971nt_int(hAPP_f301246617nt_int(hAPP_f1151807883nt_int(cOMBB_563965291nt_int,P),Q),R) = hAPP_f11151005nt_int(P,hAPP_P2110489235nt_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc___039) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4126 (all F ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(F,N_2)))) -> (hBOOL(summable_real(F)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_f352196356l_real(suminf_real,F)))))) # label(fact_4860_suminf__0__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4127 (all U hAPP_int_nat(nat,U) = hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(ord_at641636577an_int(zero_zero_int),U))) # label(fact_4919_card__atLeastZeroLessThan__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4128 (exists K_4 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),K_4)) & (all X_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(cnj,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_2)),K_4)))))) # label(fact_4087_cnj_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4129 (all A (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),A))))) # label(fact_2193_zero__less__double__add__iff__zero__less__single__add) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4130 (all N all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(real_nat,X))),hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),X)))),one_one_real))) # label(fact_3701_real__of__nat__div3) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4131 (all X_1 hBOOL(hAPP_real_bool(isCont_real_real(arctan),X_1))) # label(fact_5162_isCont__arctan) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4132 (all P_1 all N_1 all K_1 (is_int(N_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),zero_zero_int)) -> ((all I_2 all J_1 (is_int(J_1) & is_int(I_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),J_1)) & hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) = N_1 & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_1),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,I_2),J_1))))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N_1),K_1)),hAPP_int_int(div_mod_int(N_1),K_1))))))) # label(fact_2585_split__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4133 (all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),hAPP_int_int(bit1,pls))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,X_1)),one_one_int)))) # label(fact_118_le__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4134 (all C_39 all D_7 all A_97 all B_79 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_97),B_79)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_39),D_7)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_97),C_39)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_79),D_7)))))) # label(fact_2048_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4135 (all F all A_1 (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1)) -> ((all X_2 (is_int(X_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(F,X_2)))))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,F),A_1)))))) # label(fact_4945_transfer__nat__int__sum__prod__closure_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4136 (all U_1 all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,U_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_3497_realpow__square__minus__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4137 (all Z_1 hAPP_real_real(sqrt,hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_1),hAPP_complex_complex(cnj,Z_1)))) = hAPP_complex_real(norm_norm_complex,Z_1)) # label(fact_4133_complex__mod__sqrt__Re__mult__cnj) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4138 (all X all Y hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),Y) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),hAPP_complex_complex(uminus473333897omplex,Y))) # label(fact_3528_complex__diff__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4139 (all X_1 all Y_2 (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,X_1),Y_2)) -> (X_1 != Y_2 -> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,X_1),Y_2))))) # label(fact_819_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4140 (all A_3 all B_2 all P_3 all Q_2 (normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_3),B_2)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2) -> hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,P_3),Q_2))) # label(fact_4692_normalize__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4141 (all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z_1))))) # label(fact_165_le__imp__0__less) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4142 (all B_132 all A_172 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_172),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_132),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_172),B_132)))))) # label(fact_1089_mult__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4143 (all X all Y ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,ratreal(X)),ratreal(Y))) # label(fact_4702_real__minus__code) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4144 (all B_1_1 is_int(code_int_of(B_1_1))) # label(gsy_c_Code__Numeral_Oint__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4145 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_1)),one_one_real)) -> hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))),zero_zero_real,sequentially)))) # label(fact_5050_zeroseq__arctan__series) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4146 (all X_20 all Y_16 all A_120 all B_98 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_20),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Y_16),B_98))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_20),A_120)),B_98)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_20),Y_16)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_120),B_98))) # label(fact_1819_mult__diff__mult) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4147 (all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A)) -> (all X_2 (hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_2),setS(A,P_5))) -> hAPP_f957591787ol_nat(finite_card_int,X_2) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) # label(fact_3302_SetS__elems__card) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4148 (all Ta all D all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1)) -> (all X_2 all K (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D_1))),Ta))) <-> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Ta))))))) # label(fact_1535_inf__period_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4149 (all B all A all C (is_int(C) & is_int(B) -> (C = B <-> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),A) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B),A)))) # label(fact_1930_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4150 (all N_1 all V (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))) & (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),hAPP_int_nat(number_number_of_nat,V))))) # label(fact_4785_le__Suc__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4151 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,X))))) # label(fact_3323_real__sqrt__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4152 (all N_49 all A_147 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),A_147)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_147),N_49))))) # label(fact_1278_one__le__power) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4153 (all Y all X (hBOOL(hAPP_int_bool(nat_is_nat,X)) -> (hBOOL(hAPP_int_bool(nat_is_nat,Y)) -> hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(nat_tsub(X),Y)))))) # label(fact_5138_Nat__Transfer_Otransfer__int__nat__function__closures_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4154 (all A_236 A_236 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_236),hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,pls)))) # label(fact_240_mult__numeral__1__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4155 (all X_38 all P_8 all Q_12 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_38),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_38),P_8)),Q_12)) # label(fact_450_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4156 (all V_23 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_23)),one_one_complex) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_23),hAPP_int_int(bit1,pls)))) # label(fact_46_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4157 (all K_2 all L_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_2),L_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),L_2)))))) # label(fact_1630_add__less__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4158 (all X all Y (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(fequal_rat,X),Y)) | X = Y)) # label(help_fequal_1_1_fequal_000tc__Rat__Orat_T) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4159 (all A all B all C (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),A))))) # label(fact_1158_mult__less__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4160 (all X all Y hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(cos,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(sin,Y)))) # label(fact_3667_sin__diff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4161 (all P P = hAPP_nat_nat(cOMBI_nat,P)) # label(help_COMBI_1_1_COMBI_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4162 (all A_217 all M_29 all B_160 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_217),B_160)),M_29) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_217),M_29)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_160),M_29))) # label(fact_635_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4163 (all X_1 hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,fact_fact_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat)))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat)))),hAPP_real_real(sin,X_1)))) # label(fact_4863_sin__paired) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4164 (all N_65 all A_199 (A_199 != zero_zero_real -> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_199),N_65) != zero_zero_real)) # label(fact_882_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4165 (all V_3 all K_2 all V_2 ((hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2))) & (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2))) & (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_3))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_2),V_3))),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2)))))) # label(fact_3280_nat__number__of__add__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4166 (all V_2 hAPP_int_real(number267125858f_real,V_2) = hAPP_complex_real(re,hAPP_int_complex(number528085621omplex,V_2))) # label(fact_4125_complex__Re__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4167 (all Z_5 (hBOOL(hAPP_int_bool(nat_neg,Z_5)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_5),zero_zero_int)))) # label(fact_3270_neg__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4168 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K_1)),pls)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),pls)))) # label(fact_113_rel__simps_I27_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4169 (all V_18 all W_14 hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_18),W_14)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_18)),hAPP_int_rat(number_number_of_rat,W_14))) # label(fact_90_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4170 (all A all B all C all D (is_int(A) & is_int(C) & is_int(D) & is_int(B) -> (hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C),D) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B) -> (C = D <-> B = A)))) # label(fact_1957_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4171 (all X all Y hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(div_mod_nat(X),Y)) = hAPP_int_int(div_mod_int(hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y))) # label(fact_3086_Divides_Otransfer__int__nat__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4172 (all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(suc,zero_zero_nat))))) # label(fact_4182_prime__nd__one) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4173 (all V_4 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4))) -> hAPP_nat_rat(semiri151668891at_rat,hAPP_int_nat(number_number_of_nat,V_4)) = hAPP_int_rat(number_number_of_rat,V_4)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_4))) -> zero_zero_rat = hAPP_nat_rat(semiri151668891at_rat,hAPP_int_nat(number_number_of_nat,V_4))))) # label(fact_3217_of__nat__number__of__lemma) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4174 (all B_2 all A_3 (is_int(A_3) & is_int(B_2) -> (zero_zero_int != B_2 | zero_zero_int != A_3 -> one_one_int = hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(legacy_zgcd(A_3),B_2))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),hAPP_int_int(legacy_zgcd(A_3),B_2)))))) # label(fact_4513_div__zgcd__relprime) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4175 (all V all V_1 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_int_nat(number_number_of_nat,V_1)) = hAPP_int_nat(number_number_of_nat,V)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_int_nat(number_number_of_nat,V_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V),hAPP_int_int(uminus_uminus_int,V_1))))),zero_zero_nat),hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V),hAPP_int_int(uminus_uminus_int,V_1)))))))) # label(fact_4796_diff__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4176 (all A_77 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_77),zero_zero_complex) = zero_zero_complex) # label(fact_2288_mult_Ozero__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4177 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hBOOL(hAPP_real_bool(isCont_real_real(ln),X_1)))) # label(fact_5158_isCont__ln) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4178 (all A_3 all B_2 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),B_2)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),A_3) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_2),A_3)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_3365_real__average__minus__first) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4179 (all X_33 one_one_rat = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_33),zero_zero_nat)) # label(fact_681_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4180 (all A (is_int(A) -> (A != zero_zero_int <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_19_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4181 (all C_42 all A_100 all B_82 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_100),B_82)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_100),C_42)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_82),C_42))))) # label(fact_2019_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4182 (all A_48 all B_39 all C_20 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(div_mod_int(A_48),C_20)),hAPP_int_int(div_mod_int(B_39),C_20))),C_20) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_48),B_39)),C_20)) # label(fact_2417_mod__add__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4183 (all M all K_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(fact_fact_int,M)),hAPP_int_int(fact_fact_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),one_one_int)),K_2))))))) # label(fact_4286_fact__less__mono__int__aux) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4184 (all C_27 all A_57 all B_48 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_27),A_57)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_27),B_48)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_27),hAPP_C498520661umeral(div_mo1740067990umeral(A_57),B_48))) # label(fact_2391_mod__mult__mult1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4185 (all A_149 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_149),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_149),one_one_nat)))) # label(fact_1258_less__add__one) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4186 (all X all A_3 hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_real_real(uminus_uminus_real,A_3)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3))) # label(fact_4436_powr__minus__divide) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4187 (all X_15 all Y_12 all Ya_2 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_15),Ya_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_12),Ya_2)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_15),Y_12)),Ya_2)) # label(fact_2297_mult__left_Oadd) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4188 (all Ta all A_1 all D all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),D_1)) -> (all X_2 (is_int(X_2) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1)) -> X_2 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa))))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),Ta))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1)),Ta))))))))) # label(fact_5105_aset_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4189 (all Q_2 -(all A_2 all B_4 (is_int(A_2) & is_int(B_4) -> (Q_2 = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_4) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_4)) -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_4) != one_one_int))))) # label(fact_5026_Rat__cases) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4190 (all A_3 all B_2 all C_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),C_1))) # label(fact_2609_zdiv__zmult2__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4191 (all X_40 all N_69 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_40),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_40),N_69)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_40),N_69))) # label(fact_424_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4192 (all B_72 all C_35 all A_84 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_84)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_72),C_35)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_72),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_84),C_35)))))) # label(fact_2155_add__increasing) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4193 (all A_205 all B_156 (zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_205),B_156) -> zero_zero_real = A_205 | zero_zero_real = B_156)) # label(fact_823_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4194 (all Y_2 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),X_1)) -> comple124823625p_real(hAPP_r1134773055l_bool(ord_at1496968948n_real(Y_2),X_1)) = X_1)) # label(fact_4917_Sup__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4195 (all A_98 all C_40 all B_80 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_98),C_40)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_80),C_40))) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_98),B_80)))) # label(fact_2040_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4196 (all P all Q all R hAPP_i1948725293t_bool(hAPP_i345030060t_bool(P,R),Q) = hAPP_i1948725293t_bool(hAPP_i345030060t_bool(hAPP_f791698359t_bool(cOMBC_241247961t_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__fun_Itc__Int__Oint) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4197 (all A_207 hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_207),A_207))) # label(fact_794_dvd__refl) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4198 (all B_2 all B_5 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_5),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_5))))))) # label(fact_2603_zdiv__mono2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4199 (all A all C all B (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A),C)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B),C))) <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A),B)))) # label(fact_2028_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4200 (all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(zfact,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))),hAPP_int_int(number_number_of_int,min)),P_3)))) # label(fact_1856_Wilson__Russ) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4201 (all Z_14 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_14) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_14),Z_14)) # label(fact_283_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4202 (all M_25 all A_185 all N_60 all B_142 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_185),N_60)),B_142)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_25),N_60)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_185),M_25)),B_142))))) # label(fact_994_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4203 (all A (hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially)) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A,N_2)))) -> ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A,hAPP_nat_nat(suc,N_2))),hAPP_nat_real(A,N_2)))) -> hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))))))) # label(fact_5065_summable) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4204 (all P all Q all R hAPP_int_bool(hAPP_f2119767738t_bool(hAPP_f423804115t_bool(cOMBB_1418110531ol_int,P),Q),R) = hAPP_f448129468l_bool(P,hAPP_i1948725293t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obo) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4205 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> M = N))) # label(fact_1596_le__antisym) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4206 (all P_1 all I_1 all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),K_1)) -> (hBOOL(hAPP_int_bool(P_1,K_1)) -> ((all I_2 (is_int(I_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K_1)) -> (hBOOL(hAPP_int_bool(P_1,I_2)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I_2),one_one_int))))))) -> hBOOL(hAPP_int_bool(P_1,I_1)))))) # label(fact_2217_int__le__induct) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4207 (all A_3 hAPP_complex_real(norm_norm_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(cos,A_3)),hAPP_real_real(sin,A_3))) = one_one_real) # label(fact_3966_cmod__unit__one) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4208 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)))) # label(fact_1635_less__imp__le__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4209 (all A_11 all N_17 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_11),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_11),N_17)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_11),hAPP_nat_nat(suc,N_17))) # label(fact_3015_power__Suc) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4210 (all X_35 all Y_27 all Q_11 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_35),Y_27)),Q_11) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_35),Q_11)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_27),Q_11))) # label(fact_664_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4211 (all A_89 A_89 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),A_89)) # label(fact_2120_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4212 (all N all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N) = hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_nat_real(real_nat,N)))) # label(fact_4459_powr__realpow) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4213 (all X_9 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_9)) -> hAPP_n522471361de_int(semiri1424489471de_int,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_9))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_n522471361de_int(semiri1424489471de_int,hAPP_int_nat(nat,X_9))),hAPP_n522471361de_int(semiri1424489471de_int,hAPP_int_nat(nat,X_9))))) # label(fact_2813_of__nat__double) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4214 (all P all Q all R hAPP_P603027463t_bool(hAPP_i119638232t_bool(hAPP_f2018027565t_bool(cOMBC_33652895t_bool,P),Q),R) = hAPP_int_bool(hAPP_P252494982t_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin_003) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4215 (all L_2 all K_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2)) -> posDivAlg(K_2,L_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,K_2),L_2)),hAPP_int_int(div_mod_int(K_2),L_2))))) # label(fact_2757_posDivAlg__div__mod) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4216 (all Lx_1 all Rx_1 all Ry_1 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_1),Rx_1)),Ry_1) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_1),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_1),Ry_1))) # label(fact_503_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4217 (all P_1 all K_1 (hBOOL(hAPP_nat_bool(P_1,K_1)) -> ((all N_2 (hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_2))) -> hBOOL(hAPP_nat_bool(P_1,N_2)))) -> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))))) # label(fact_3310_zero__induct) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4218 (all X_44 all Y_32 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_44),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_32),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_32)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_44),Y_32))))) # label(fact_217_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4219 (all M all K_2 all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),N))) # label(fact_4348_gcd__add__mult__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4220 (all X_2 hAPP_real_real(sin,X_2) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),sin_coeff)),hAPP_r474017924t_real(power_power_real,X_2)))) # label(fact_4819_lemma__sin__ext) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4221 (all M all N xzgcd(M,N) = xzgcda(M,N,M,N,one_one_int,zero_zero_int,zero_zero_int,one_one_int)) # label(fact_4201_xzgcd__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4222 (all I_1 all M_2 (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),M_2)) -> hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),M_2))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),hAPP_nat_nat(suc,I_1))))) = hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,hAPP_f66927821l_bool(hAPP_f2037783381l_bool(cOMBB_1146692694ol_nat,member_nat),suc)),M_2))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),I_1)))))) # label(fact_4768_card__less__Suc2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4223 (all R_3 all X hAPP_complex_real(im,hAPP_complex_complex(scaleR1652505878omplex(R_3),X)) = hAPP_real_real(scaleR_scaleR_real(R_3),hAPP_complex_real(im,X))) # label(fact_5112_Im_OscaleR) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4224 (all M all N hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),M)),N) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,M)))),N)) # label(fact_3209_Suc__mod__eq__add3__mod) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4225 (all A_50 all C_22 all B_41 hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_50),B_41)),C_22) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_50),C_22)),B_41)),C_22)) # label(fact_2412_zmod__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4226 (all C all A all B (C = zero_zero_real | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A),B)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B))))) # label(fact_1263_dvd__mult__cancel__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4227 (all A all X_1 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> hAPP_int_int(uminus_uminus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(nat,X_1))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(uminus_uminus_int,A)),hAPP_int_nat(nat,X_1))))) # label(fact_3555_neg__odd__power) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4228 (all Q_2 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(prime,Q_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),Q_2)) -> Q_2 = P_3)))) # label(fact_4165_primes__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4229 (all B_148 all A_192 all C_84 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_192),C_84)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_192),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_148),C_84))))) # label(fact_933_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4230 (all P_1 all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) -> (all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),hAPP_nat_nat(suc,N_2)))),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_2))))))) # label(fact_4246_Bolzano__bisect__Suc__le__snd) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4231 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K_1)),min)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),min)))) # label(fact_787_rel__simps_I11_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4232 (all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)) | hAPP_nat_nat(suc,N_1) = Ma <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),hAPP_nat_nat(suc,N_1))))) # label(fact_2871_le__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4233 (all N_1 all V (hAPP_nat_nat(suc,N_1) = hAPP_int_nat(number_number_of_nat,V) <-> -hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))) & (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))) -> hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V))) = N_1))) # label(fact_4778_Suc__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4234 (all U all Ma all N_1 all J all I_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N_1) <-> N_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J)),U)),Ma)))) # label(fact_1852_nat__eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4235 (all X all Y hAPP_P1860904029l_real(hAPP_P1982799375l_real(hAPP_b1482221219l_real(if_Pro313124157l_real,fTrue),X),Y) = X) # label(help_If_1_1_If_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_T) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4236 (all B_117 all A_156 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_156)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_117),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_156),B_117)),zero_zero_nat))))) # label(fact_1180_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4237 (all B_120 all A_159 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_120),A_159))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_159)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_120))))) # label(fact_1162_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4238 (all X (-hBOOL(hAPP_int_bool(even_odd_even_int,X)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),one_one_int) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3244_odd__plus__one__div__two) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4239 (all B_55 all A_67 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_55),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_67),B_55))),hAPP_nat_nat(div_mod_nat(A_67),B_55)) = A_67) # label(fact_2343_mod__div__equality2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4240 (all C_43 all A_101 all B_83 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_101),B_83)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_43),A_101)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_43),B_83))))) # label(fact_2011_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4241 (all V all W_1 (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,W_1)),hAPP_int_int(number_number_of_int,V))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,V)),hAPP_int_int(number_number_of_int,W_1))))) # label(fact_105_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4242 (all V_14 all B_166 all C_97 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_14)),B_166)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_14)),C_97)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_14)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_166),C_97))) # label(fact_191_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4243 (all Y all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y)) -> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),hAPP_real_real(ln,X)) = hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(powr,X),Y))))) # label(fact_4460_ln__powr) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4244 (all A_3 all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,X)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),A_3)))))) # label(fact_4377_ln__powr__bound) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4245 (all X all A_3 all B_2 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2) = one_one_nat -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),A_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),B_2)) -> X = one_one_nat)))) # label(fact_4349_coprime__common__divisor__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4246 (all X_1 all Y_2 all Z_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),Z_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Z_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2))))) # label(fact_1793_real__mult__le__cancel__iff1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4247 (all P_1 all A all B all N_1 hAPP_P1860904029l_real(produc595218619l_real(hAPP_f2072293513l_real(hAPP_f1763467369l_real(cOMBS_163651580l_real,hAPP_f1663701695l_real(hAPP_f774948957l_real(cOMBB_3698607l_real,cOMBS_902912708l_real),hAPP_f324002831l_real(hAPP_f677014621l_real(cOMBS_107495110l_real,hAPP_f1161935955l_real(hAPP_f328572565l_real(cOMBB_520243667l_real,cOMBS_794971624l_real),hAPP_f517738447l_real(hAPP_f1879982819l_real(cOMBB_731595089l_real,hAPP_f1076489843l_real(cOMBB_415306361l_real,if_Pro313124157l_real)),hAPP_f1884005689l_bool(hAPP_f172451459l_bool(cOMBB_421479067l_real,hAPP_f940061907l_bool(cOMBB_1795223523l_real,P_1)),hAPP_f320596107l_real(hAPP_f850851693l_real(cOMBS_193881978l_real,hAPP_f1119546819l_real(hAPP_f555557763l_real(cOMBB_422423973l_real,cOMBB_2107848501l_real),produc865579683l_real)),hAPP_r337325687l_real(hAPP_f1644353557l_real(cOMBC_1508019322l_real,hAPP_f1644353557l_real(hAPP_f1874145443l_real(cOMBB_1768400739l_real,cOMBC_real_real_real),hAPP_f1732848529l_real(hAPP_f1310479593l_real(cOMBB_2092576745l_real,hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real)),plus_plus_real))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))),hAPP_f1591894897l_real(hAPP_f644105375l_real(cOMBC_467724138l_real,hAPP_f1108539711l_real(hAPP_f1109256153l_real(cOMBB_44084907l_real,cOMBS_511527878l_real),hAPP_f823702159l_real(hAPP_f1040184293l_real(cOMBB_557258983l_real,hAPP_f2114902373l_real(cOMBB_237479429l_real,produc865579683l_real)),hAPP_r337325687l_real(hAPP_f1644353557l_real(cOMBC_1508019322l_real,hAPP_f1644353557l_real(hAPP_f1874145443l_real(cOMBB_1768400739l_real,cOMBC_real_real_real),hAPP_f1732848529l_real(hAPP_f1310479593l_real(cOMBB_2092576745l_real,hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real)),plus_plus_real))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),cOMBI_real)))),hAPP_f320596107l_real(hAPP_f850851693l_real(cOMBS_193881978l_real,hAPP_f1119546819l_real(hAPP_f555557763l_real(cOMBB_422423973l_real,cOMBB_2107848501l_real),produc865579683l_real)),hAPP_r337325687l_real(hAPP_f1644353557l_real(cOMBC_1508019322l_real,hAPP_f1644353557l_real(hAPP_f1874145443l_real(cOMBB_1768400739l_real,cOMBC_real_real_real),hAPP_f1732848529l_real(hAPP_f1310479593l_real(cOMBB_2092576745l_real,hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real)),plus_plus_real))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),N_1)) = hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),hAPP_nat_nat(suc,N_1))) # label(fact_4852_Bolzano__bisect_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4248 (all A_195 all B_151 hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_195),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_195),B_151)))) # label(fact_909_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4249 (all A_186 hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,one_one_complex),A_186))) # label(fact_978_one__dvd) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4250 (all Y all X (hBOOL(hAPP_int_bool(even_odd_even_int,X)) -> (hBOOL(hAPP_int_bool(even_odd_even_int,Y)) -> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y)))))) # label(fact_2890_Parity_Oeven__plus__even) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4251 (exists K_4 ((all X_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(re,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_2)),K_4)))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),K_4)))) # label(fact_4141_Re_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4252 (all D all Sa exists Z all X_2 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Z),X_2)) -> (-hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_2),Sa))) <-> -hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,D),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_2),Sa)))))) # label(fact_2255_pinf_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4253 (all P all Q all R hAPP_r1392521531t_bool(hAPP_f332806115t_bool(hAPP_f1177545951t_bool(cOMBB_1459832681l_real,P),Q),R) = hAPP_f1576199059t_bool(P,hAPP_r847664445t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__RealDef__Oreal_J_Mt) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4254 (all X_1 all Y_2 all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B),X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B),Y_2)))))) # label(fact_1452_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4255 (all B_120 all A_159 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_120),A_159))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_159)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_120))))) # label(fact_1164_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4256 (all Z_7 all W_2 all Y_9 all X_12 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_9),X_12)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_7),W_2)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_12),W_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Y_9),Z_7)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_12),Y_9)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,W_2),Z_7))))) # label(fact_2560_div__mult__div__if__dvd) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4257 (all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,N_1)),zero_zero_int)) <-> zero_zero_nat = N_1)) # label(fact_3102_int__le__0__conv) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4258 (all V_13 all W_11 all Z_10 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_13)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,W_11)),Z_10)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_13),W_11))),Z_10)) # label(fact_335_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4259 (all A all B (A = B <-> hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A),B) = zero_zero_complex)) # label(fact_2082_eq__iff__diff__eq__0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4260 (all K_2 (is_int(K_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2)) -> (exists N_2 K_2 = hAPP_nat_int(semiri1621563631at_int,N_2))))) # label(fact_3261_zero__le__imp__eq__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4261 (all P all Q all R hAPP_P603027463t_bool(hAPP_f892584630t_bool(hAPP_f492612297t_bool(cOMBB_798864284nt_int,P),Q),R) = hAPP_int_bool(P,hAPP_P1175774780nt_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__HOL__Obool_000tc__prod_Itc__Int__Oi) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4262 (all N_1 all K_1 all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),Ma)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),K_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),K_1) <-> N_1 = Ma)))) # label(fact_1752_eq__diff__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4263 (all X (is_int(X) -> (hBOOL(hAPP_int_bool(even_odd_even_int,X)) -> X = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3228_two__times__even__div__two) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4264 (all V_17 all V_16 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_16)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_17)) -> hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_16),V_17)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_16)),hAPP_int_complex(number528085621omplex,V_17))))) # label(fact_162_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4265 (all A_89 A_89 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,one_on1684967323de_int),A_89)) # label(fact_2124_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4266 (all K_2 all M all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(div_mod_int(M),N))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),N)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M))))) # label(fact_2468_zdvd__zmod__imp__zdvd) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4267 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,X)),hAPP_real_real(sqrt,Y))))) # label(fact_3312_real__sqrt__le__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4268 (all A_205 all B_156 (zero_zero_complex = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_205),B_156) -> zero_zero_complex = B_156 | A_205 = zero_zero_complex)) # label(fact_825_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4269 (all A_3 all B_2 all C_1 all N (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,C_1),N) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N)) -> (hAPP_int_int(legacy_zgcd(A_3),B_2) = one_one_int -> (exists K (is_int(K) & hAPP_int_int(abs_abs_int,A_3) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K),N))))))) # label(fact_4531_int__relprime__power__divisors) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4270 (all A_3 all C_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,zero_zero_nat)),C_1))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_3),C_1)),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,zero_zero_nat)),C_1))),C_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,A_3)),C_1)) # label(fact_3185_div__Suc) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4271 (all Z_1 all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z_1))))) # label(fact_1713_dvd_Oorder__trans) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4272 (all F all A all B hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B)),hAPP_real_real(F,A))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A))),B)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,A)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B)),hAPP_real_real(F,A))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B),A))),A))) # label(fact_3372_lemma__MVT) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4273 (all A_3 all B_2 all Q_2 all R_3 (is_int(R_3) & is_int(B_2) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),R_3))) -> (zero_zero_int != B_2 -> hAPP_int_int(div_mod_int(A_3),B_2) = R_3)))) # label(fact_2796_divmod__int__rel__mod) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4274 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_real_real(abs_abs_real,X))))) # label(fact_3398_abs__add__one__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4275 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arcsin,Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3506_arcsin__lt__bounded) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4276 (all A all A_1 all Ma (hBOOL(hAPP_int_bool(is_RRset(A_1),Ma)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1)) -> one_one_int = hAPP_int_int(legacy_zgcd(A),Ma)))) # label(fact_4589_RRset__gcd) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4277 (all X hAPP_real_real(log(hAPP_real_real(exp_real,one_one_real)),X) = hAPP_real_real(ln,X)) # label(fact_3869_log__ln) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4278 (all M_7 all N_21 hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_7),N_21)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(semiri2020571505omplex,M_7)),hAPP_nat_complex(semiri2020571505omplex,N_21))) # label(fact_2963_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 7.56/7.59 4279 (all A all E_2 all C all B all D (hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A),E_2)),C) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B),E_2)),D) <-> hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,B),A)),E_2)),D) = C)) # label(fact_1538_eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4280 (all B_92 all A_114 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_114),zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_92),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_114),B_92)),zero_zero_nat))))) # label(fact_1889_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4281 (all C_59 all D_12 all A_137 all B_106 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_137),B_106)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_59),D_12)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_137)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_59)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_137),C_59)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_106),D_12)))))))) # label(fact_1361_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4282 (all A all A_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma)) -> (hBOOL(hAPP_int_bool(is_RRset(A_1),Ma)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),A_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),hAPP_int_int(rRset2norRR(A_1,Ma),A)),Ma)))))) # label(fact_4593_RRset2norRR__correct1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4283 (all N_1 (hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1)) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(div_mod_nat(N_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3786_even__even__mod__4__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4284 (all R_3 all A_3 rcis(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),R_3),hAPP_real_real(uminus_uminus_real,A_3)) = hAPP_complex_complex(invers1449016382omplex,rcis(R_3,A_3))) # label(fact_4267_rcis__inverse) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4285 (all X all Y all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y)) -> (hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),N) = X -> hAPP_real_real(hAPP_n546711566l_real(root,N),X) = Y)))) # label(fact_3935_real__root__pos__unique) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4286 (all Q_2 all B_2 all R_3 all C_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(div_mod_int(Q_2),C_1))),R_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1))))))) # label(fact_2658_zmult2__lemma__aux4) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4287 (all M ((M != zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) = hAPP_nat_nat(fact_fact_nat,M)) & (zero_zero_nat = M -> hAPP_nat_nat(fact_fact_nat,M) = one_one_nat))) # label(fact_3815_fact__num__eq__if__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4288 (all X (hBOOL(hAPP_int_bool(even_odd_even_int,X)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3245_even__plus__one__div__two) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4289 (all X all Y hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(abs_abs_int,X)),hAPP_int_int(abs_abs_int,Y))) # label(fact_4987_gcd__abs__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4290 (all N_13 all A_8 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_8)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_8),hAPP_nat_nat(suc,N_13)))))) # label(fact_3120_power__gt1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4291 (all Y_6 all P_6 all N_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_6),N_10)) -> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N_10)),P_6)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),P_6))),Y_6))) # label(fact_3175_lemma__realpow__diff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4292 (all C all A all B (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),zero_zero_rat)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B),A)) | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B))))) # label(fact_1196_mult__less__cancel__left__disj) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4293 (all N_43 all A_131 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_131)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_131),N_43)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_131),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_131),N_43)))))) # label(fact_1423_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4294 (all A_110 all B_88 all C_48 (is_int(B_88) & is_int(C_48) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_110),C_48) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_110),B_88) -> B_88 = C_48))) # label(fact_1923_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4295 (all Z_1 exists R_1 exists A_2 Z_1 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_1)),hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(cos,A_2)),hAPP_real_real(sin,A_2)))) # label(fact_4002_complex__split__polar) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4296 (all N -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat))) # label(fact_1507_less__zeroE) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4297 (all B_2 all M all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),A_3)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M))))))) # label(fact_1310_zcong__not) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4298 (all N all X (-hBOOL(hAPP_int_bool(even_odd_even_int,X)) -> -hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N))))) # label(fact_2893_odd__pow) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4299 (all M_9 all N_23 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_23)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,M_9)),hAPP_nat_rat(semiri151668891at_rat,N_23))))) # label(fact_2936_less__imp__of__nat__less) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4300 (all Z_13 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_13) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_13),Z_13)) # label(fact_287_mult__2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4301 (all X_1 all Y_2 all B (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2)) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B),X_1)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B),Y_2)))))) # label(fact_1455_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4302 (all Z_1 Z_1 = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_complex_real(re,Z_1)),hAPP_complex_real(im,Z_1))) # label(fact_4120_complex__surj) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4303 (all Y all U_1 all X1_1 all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X1_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X1_1),Y)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),U_1))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,X)),Y)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,X1_1)),U_1))))))) # label(fact_3837_real__mult__inverse__cancel) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4304 (all A_220 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A_220) = A_220) # label(fact_599_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4305 (all N_53 all M_21 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),M_21)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),N_53)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,M_21),N_53)))))) # label(fact_1247_less__1__mult) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4306 (all X all Y hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(cnj,X)),hAPP_complex_complex(cnj,Y)) = hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),Y))) # label(fact_4071_cnj_Oadd) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4307 (all A_44 all B_35 hAPP_C498520661umeral(div_mo1740067990umeral(A_44),B_35) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_44),B_35)),B_35)) # label(fact_2435_mod__add__self2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4308 (all M hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),hAPP_nat_nat(suc,zero_zero_nat)) = hAPP_nat_nat(suc,zero_zero_nat)) # label(fact_4343_gcd__Suc__0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4309 (all P all Q all R hAPP_f1146629647l_bool(P,hAPP_i418383825t_bool(Q,R)) = hAPP_i614188481l_bool(hAPP_f252743847l_bool(hAPP_f643658017l_bool(cOMBB_2046938128ol_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc_) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4310 (all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,X_1)),one_one_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),hAPP_int_int(bit1,pls))))) # label(fact_408_less__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4311 (all A all E_2 all C all B all D (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B),A)),E_2)),D) = C <-> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),E_2)),C) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),E_2)),D))) # label(fact_1539_eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4312 (all X_1 all Y_2 all Xa_1 (X_1 = Xa_1 & Y_2 = zero_zero_real <-> of_real_complex(Xa_1) = hAPP_real_complex(hAPP_r265291036omplex(complex,X_1),Y_2))) # label(fact_3987_complex__eq__cancel__iff2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4313 (all M all N hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),N)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,M)),N)) # label(fact_3082_int__power) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4314 (all A_110 all B_88 all C_48 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_110),C_48) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_110),B_88) -> B_88 = C_48)) # label(fact_1924_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4315 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_1)),one_one_real)) -> hAPP_real_real(arctan,X_1) = hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))))) # label(fact_4817_arctan__series) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4316 (all A_108 A_108 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_108),zero_zero_rat)) # label(fact_1967_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4317 (all B_2 all A_3 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_2),A_3)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),A_3) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_2),A_3)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_3366_real__average__minus__second) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4318 (all P_1 all L ((exists X_2 (hBOOL(hAPP_complex_bool(P_1,X_2)) & hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,L),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_2),zero_zero_complex))))) <-> (exists X_2 hBOOL(hAPP_complex_bool(P_1,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,L),X_2)))))) # label(fact_1677_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4319 (all Q_2 ((Q_2 != zero_zero_rat -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),Q_2)) -> one_one_rat = hAPP_rat_rat(sgn_sgn_rat,Q_2)) & (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),Q_2)) -> hAPP_rat_rat(uminus_uminus_rat,one_one_rat) = hAPP_rat_rat(sgn_sgn_rat,Q_2))) & (zero_zero_rat = Q_2 -> hAPP_rat_rat(sgn_sgn_rat,Q_2) = zero_zero_rat))) # label(fact_4650_sgn__rat__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4320 (all P all Q all R hAPP_nat_bool(hAPP_f1154658180t_bool(hAPP_f1361476881t_bool(cOMBB_int_bool_nat,P),Q),R) = hAPP_int_bool(P,hAPP_nat_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__HOL__Obool_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4321 (all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,P_5),zOdd)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5))))) # label(fact_2667_zprime__zOdd__eq__grt__2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4322 (all P all Q all R hAPP_i345030060t_bool(hAPP_f791698359t_bool(hAPP_f847817363t_bool(cOMBB_562416518ol_int,P),Q),R) = hAPP_f428220345t_bool(P,hAPP_i345030060t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__HOL__026) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4323 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N))) # label(fact_2700_div__less) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4324 (all C_44 all D_8 all A_102 all B_84 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_102),B_84)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_44),D_8)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_102),C_44)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_84),D_8)))))) # label(fact_2006_add__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4325 (all M_13 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,one_on1684967323de_int),hAPP_n522471361de_int(semiri1424489471de_int,M_13)) = hAPP_n522471361de_int(semiri1424489471de_int,hAPP_nat_nat(suc,M_13))) # label(fact_2843_of__nat__Suc) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4326 (all Z_1 all W hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),W)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(uminus_uminus_int,Z_1)),W)) # label(fact_3515_zmult__zminus) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4327 (all A all E_2 all C all B all D (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A),E_2)),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B),E_2)),D))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B)),E_2)),C)),D)))) # label(fact_1571_less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4328 (all X_1 (is_int(X_1) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = X_1))) # label(fact_3239_even__div__2__prop2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4329 (all A_219 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_219),zero_zero_real) = A_219) # label(fact_607_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4330 (all X_6 all Q_5 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_6),hAPP_nat_nat(suc,Q_5)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_6),Q_5))) # label(fact_3037_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4331 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)),B_2)))) # label(fact_5012_gcd__le2__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4332 (all X (X != zero_zero_real -> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,X)),X) = one_one_real)) # label(fact_3839_real__mult__inverse__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4333 (all B_122 all A_161 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_161),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_122),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_161),B_122)))))) # label(fact_1150_mult__neg__neg) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4334 (all K_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(div_mod_int(M),K_2)),M)))) # label(fact_2528_zmod__le__nonneg__dividend) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4335 (all N_1 all Ma all F ((all N_2 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2))),hAPP_nat_nat(F,N_2))) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_2)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2)))))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),Ma)) <-> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,Ma)),hAPP_nat_nat(F,N_1))) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_1)),hAPP_nat_nat(F,Ma)))))) # label(fact_3264_dvd_Olift__Suc__mono__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4336 (all B_43 all B_42 all A_52 all C_23 all A_51 (hAPP_nat_nat(div_mod_nat(A_51),C_23) = hAPP_nat_nat(div_mod_nat(A_52),C_23) -> (hAPP_nat_nat(div_mod_nat(B_43),C_23) = hAPP_nat_nat(div_mod_nat(B_42),C_23) -> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_52),B_43)),C_23) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_51),B_42)),C_23)))) # label(fact_2406_mod__add__cong) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4337 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_real_real(ln,hAPP_real_real(inverse_inverse_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(ln,X)))) # label(fact_3852_ln__inverse) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4338 (all X_1 all Y_2 (is_int(X_1) & is_int(Y_2) -> (X_1 = Y_2 <-> hAPP_int_complex(number528085621omplex,X_1) = hAPP_int_complex(number528085621omplex,Y_2)))) # label(fact_168_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4339 (all X_1 all Y_2 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N_1),X_1)),hAPP_real_real(hAPP_n546711566l_real(root,N_1),Y_2)))))) # label(fact_3905_real__root__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4340 (all A_145 all M_20 all N_48 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_145),M_20)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_145),N_48)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_145),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_20),N_48))) # label(fact_1308_power__add) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4341 (all A_222 zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,zero_zero_int),A_222)) # label(fact_583_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4342 (all P all Q all R hAPP_n546711566l_real(hAPP_r195310020l_real(hAPP_f2126667875l_real(cOMBC_746811033l_real,P),Q),R) = hAPP_r1250527377l_real(hAPP_n682674106l_real(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__fun_Itc__Real) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4343 (all X all M all Y (hAPP_int_int(div_mod_int(X),M) = hAPP_int_int(div_mod_int(Y),M) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),Y),M)))) # label(fact_2465_Residues_Oaux) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4344 (all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_1),zero_zero_int)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_1)),zero_zero_real)))) # label(fact_4441_real__of__int__le__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4345 (all B_115 all C_67 all A_153 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_153)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_115),C_67)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_115),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_153),C_67)))))) # label(fact_1204_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4346 (all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N),one_one_int)),hAPP_int_int(fact_fact_int,N)) = hAPP_int_int(fact_fact_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N),one_one_int)))) # label(fact_4287_fact__plus__one__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4347 (all N_1 hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min))),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,div_div_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,zero_zero_real)))),zero_zero_real)),hAPP_n1699378549t_bool(ord_at4362885an_nat(one_one_nat),N_1)) = zero_zero_real) # label(fact_4912_lemma__STAR__cos2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4348 (all A all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),A)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),zero_zero_int),Ma)))) # label(fact_1316_zcong__zero__equiv__div) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4349 (all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),hAPP_int_int(bit1,pls))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,X_1)),one_one_rat)))) # label(fact_410_less__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4350 (all P all Q all R hAPP_i1529485324t_bool(hAPP_f1545556668t_bool(hAPP_f85237525t_bool(cOMBB_389152643ol_int,P),Q),R) = hAPP_f727283836t_bool(P,hAPP_i1823918693l_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__HOL__Obool_Mtc__HOL) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4351 (all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,M),zero_zero_int)) -> zero_zero_int = hAPP_int_int(fact_fact_int,M))) # label(fact_4275_fact__neg__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4352 (all P_1 all A0 all A1 (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(lazy_small_lazy_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1))) -> ((all D_2 all I_2 (is_int(I_2) & is_int(D_2) -> (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(lazy_small_lazy_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,D_2),I_2))) -> ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,D_2),I_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,D_2),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int)))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,D_2),I_2)))))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1))))) # label(fact_4574_small__lazy_H_Opinduct) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4353 (all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),hAPP_nat_nat(fact_fact_nat,N)))) # label(fact_3810_fact__ge__one__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4354 (all M all N all Qr_1 (hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),Qr_1)) -> divmod_nat(M,N) = Qr_1)) # label(fact_4216_divmod__nat__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4355 (all K_2 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(suc,zero_zero_nat)),K_2))) # label(fact_2819_dvd__1__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4356 (all E_1 all B_2 all D_3 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_3),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,E_1),B_2)) -> (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) = one_one_int -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),E_1) = one_one_int)))) # label(fact_4972_coprime__divisors__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4357 (all M_27 all A_212 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_212),one_one_int)),M_27) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M_27),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_212),M_27))) # label(fact_698_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4358 (all M_28 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat)),M_28) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,M_28),M_28)) # label(fact_697_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4359 (all X_1 all Y_2 all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B),X_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B),Y_2))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2))))) # label(fact_1451_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4360 (all A_3 all B_2 (is_int(B_2) -> (zero_zero_int != B_2 -> (hAPP_int_int(div_mod_int(A_3),B_2) != zero_zero_int -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))),one_one_int) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(uminus_uminus_int,B_2))) & (zero_zero_int = hAPP_int_int(div_mod_int(A_3),B_2) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)))))) # label(fact_3552_zdiv__zminus2__eq__if) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4361 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (exists N_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_2))),Y))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),Y)),X))))))) # label(fact_3749_reals__Archimedean4) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4362 (all P_1 all A all B hAPP_n332733730l_real(bolzano_bisect(P_1,A,B),zero_zero_nat) = hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A),B)) # label(fact_4244_Bolzano__bisect_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4363 (all A all B (hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A),B) = zero_zero_real <-> zero_zero_real = A | zero_zero_real = B)) # label(fact_836_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4364 (all N hAPP_nat_complex(semiri2020571505omplex,N) = hAPP_complex_complex(cnj,hAPP_nat_complex(semiri2020571505omplex,N))) # label(fact_4075_complex__cnj__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4365 (all A_93 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_93),zero_zero_real) = A_93) # label(fact_2089_diff__0__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4366 (all Ma all Z_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,Z_2)))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,Ma)),Z_2)))) # label(fact_3451_int__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4367 (all C_66 all A_144 all B_113 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_66),A_144)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_66),B_113))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_66)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_144),B_113))))) # label(fact_1320_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4368 (all X_1 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_2)) | Y_2 = X_1 <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2)))) # label(fact_1733_less__eq__real__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4369 (all X all D_3 (is_int(D_3) -> (D_3 != zero_zero_int -> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),D_3))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,hAPP_int_int(div_mod_int(X),D_3))),hAPP_int_real(real_int,D_3))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,X)),hAPP_int_real(real_int,D_3))))) # label(fact_4466_real__of__int__div__aux) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4370 (all A_3 -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_3)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),zero_zero_nat)))) # label(fact_2307_gcd__lcm__complete__lattice__nat_Onot__top__less) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4371 (all A all P_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> hBOOL(hAPP_f448129468l_bool(finite_finite_int,comple219730294t_bool(setS(A,P_5)))))) # label(fact_3352_Union__SetS__finite) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4372 (all A_89 A_89 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,one_one_complex),A_89)) # label(fact_2123_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4373 (all U_1 all M all N all J_2 all I (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),I)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),J_2)),U_1)),M)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),U_1)),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),U_1)),N)))) # label(fact_1853_nat__diff__add__eq1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4374 (all X_2 ((-hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2)) -> zero_zero_real = hAPP_nat_real(cos_coeff,X_2)) & (hBOOL(hAPP_nat_bool(even_odd_even_nat,X_2)) -> hAPP_nat_real(cos_coeff,X_2) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,X_2)))))) # label(fact_3798_cos__coeff__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4375 (all Lx_2 all Ly all Rx_2 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_2),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Ly),Rx_2)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_2),Ly)),Rx_2)) # label(fact_495_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4376 (all A all W_1 (zero_zero_rat = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A),hAPP_int_nat(number_number_of_nat,W_1)) <-> zero_zero_rat = A & hAPP_int_nat(number_number_of_nat,W_1) != zero_zero_nat)) # label(fact_718_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4377 (all N_13 all A_8 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_8)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_8),hAPP_nat_nat(suc,N_13)))))) # label(fact_3119_power__gt1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4378 (all A_32 hAPP_Q1762011733de_int(div_mo231679042de_int(A_32),one_on1684967323de_int) = zero_z891286103de_int) # label(fact_2502_mod__by__1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4379 (all Lx_1 all Rx_1 all Ry_1 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_1),Ry_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_1),Rx_1)),Ry_1)) # label(fact_501_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4380 (all R_3 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,archim856651990g_real(R_3))),one_one_real)),R_3))) # label(fact_4458_real__of__int__ceiling__diff__one__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4381 (all X all Y hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Y),hAPP_int_int(div_mod_int(X),Y))) # label(fact_4984_gcd__red__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4382 (all A_92 (is_int(A_92) -> A_92 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_92),one_one_int))) # label(fact_2098_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4383 (all G all X_1 (hBOOL(hAPP_real_bool(sums_real(G),X_1)) -> hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_r195310020l_real(hAPP_f2126667875l_real(cOMBC_746811033l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),zero_zero_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,G),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,div_div_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,minus_minus_nat),one_one_nat))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),X_1)))) # label(fact_4868_sums__if_H) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4384 (all Y_2 all X_1 all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(sRStar,P_5))) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_2),hAPP_i1948725293t_bool(sRStar,P_5))) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(standardRes(P_5),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),Y_2))),hAPP_i1948725293t_bool(sRStar,P_5)))))))) # label(fact_2765_SRStar__mult__prop1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4385 (all I all J_2 all K_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),J_2)),K_2)) # label(fact_1745_diff__diff__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4386 (all A_3 all B_2 ((zero_zero_int = hAPP_int_int(div_mod_int(A_3),B_2) -> zero_zero_int = hAPP_int_int(div_mod_int(A_3),hAPP_int_int(uminus_uminus_int,B_2))) & (hAPP_int_int(div_mod_int(A_3),B_2) != zero_zero_int -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(div_mod_int(A_3),B_2)),B_2) = hAPP_int_int(div_mod_int(A_3),hAPP_int_int(uminus_uminus_int,B_2))))) # label(fact_3544_zmod__zminus2__eq__if) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4387 (all X all Y hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X)),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3406_real__sqrt__ge__abs1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4388 (all X hAPP_complex_complex(uminus473333897omplex,hAPP_complex_complex(cnj,X)) = hAPP_complex_complex(cnj,hAPP_complex_complex(uminus473333897omplex,X))) # label(fact_4070_cnj_Ominus) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4389 (all A_111 all B_89 all C_49 (hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_111),B_89) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_111),C_49) -> B_89 = C_49)) # label(fact_1919_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4390 (all B_129 all A_169 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_169)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_129),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_169),B_129)),zero_zero_real))))) # label(fact_1106_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4391 (all V all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,V)),hAPP_nat_nat(suc,N_1))) <-> (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,V)))),N_1))))) # label(fact_4783_less__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4392 (all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),min)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K_1)),min)))) # label(fact_782_rel__simps_I30_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4393 (all X_1 all Y_2 (zero_zero_rat = Y_2 & zero_zero_rat = X_1 <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_1),X_1)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_2),Y_2))),zero_zero_rat)))) # label(fact_93_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4394 (all R_3 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),of_real_complex(R_3)) = hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),R_3)) # label(fact_3996_i__complex__of__real) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4395 (all P all Q all R hAPP_int_int(hAPP_nat_fun_int_int(hAPP_f2105620693nt_int(cOMBC_int_nat_int,P),Q),R) = hAPP_nat_int(hAPP_int_fun_nat_int(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Nat__Onat_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4396 (all Q_2 ((-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Q_2),zero_zero_rat)) -> hAPP_rat_rat(abs_abs_rat,Q_2) = Q_2) & (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Q_2),zero_zero_rat)) -> hAPP_rat_rat(abs_abs_rat,Q_2) = hAPP_rat_rat(uminus_uminus_rat,Q_2)))) # label(fact_4649_abs__rat__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4397 (all A_183 all M_22 all N_57 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_183),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_22),N_57)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_183),M_22)),N_57)) # label(fact_1013_power__mult) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4398 (all B_90 all A_112 all C_50 (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_50),A_112) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_90),A_112) -> B_90 = C_50)) # label(fact_1911_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4399 (all N_61 all X_29 all Y_23 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_29),Y_23)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_29),N_61)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_23),N_61))))) # label(fact_985_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4400 (all N_1 (zero_zero_nat != N_1 <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)))) # label(fact_1607_neq0__conv) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4401 (all B_131 all A_171 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_171),zero_z126310315umeral)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_131)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_171),B_131)),zero_z126310315umeral))))) # label(fact_1095_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4402 (all M_19 all N_44 all A_132 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_132)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_132),M_19)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_132),N_44))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_19),N_44))))) # label(fact_1409_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4403 (all M_11 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_11)),zero_zero_int))) # label(fact_2921_of__nat__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4404 (all A_229 all C_95 all D_20 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_229),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_95),D_20)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_95),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_229),D_20))) # label(fact_529_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4405 (all C_37 all A_95 all B_77 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_95),B_77)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_95),C_37)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_77),C_37))))) # label(fact_2063_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4406 (all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),zero_zero_nat)) <-> zero_zero_nat = N_1)) # label(fact_1618_le__0__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4407 (all D_1 all X_1 (zero_zero_real != X_1 -> ((hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X_1))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = D_1) -> ((hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),zero_zero_real)) -> D_1 = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X_1)))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) -> hBOOL(hAPP_real_bool(deriv_real(sqrt,X_1),D_1)))))) # label(fact_3880_DERIV__real__sqrt__generic) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4408 (all A all C all B (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A),B)) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A),C)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B),C))))) # label(fact_2032_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4409 (all N N = hAPP_int_nat(nat,hAPP_nat_int(semiri1621563631at_int,N))) # label(fact_2854_nat__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4410 (all V_8 all W_4 hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_8),W_4)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(number_number_of_int,V_8)),hAPP_int_int(number_number_of_int,W_4))) # label(fact_1510_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4411 (all K_1 all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K_1)),hAPP_int_int(bit1,L))))) # label(fact_151_rel__simps_I15_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4412 (all M_6 all N_20 hAPP_n522471361de_int(semiri1424489471de_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_6)),N_20)) # label(fact_2977_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4413 (all P all Q all R hAPP_bool_bool(hAPP_r1938957037l_bool(P,R),hAPP_real_bool(Q,R)) = hAPP_real_bool(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__HOL__Obool_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4414 (all N all F_3 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,F_3),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,F_3),N)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,F_3),hAPP_int_int(div_mod_int(M),N)))))) # label(fact_2467_zdvd__zmod) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4415 (all C_1 all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),C_1))))) # label(fact_1795_divides__mul__r) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4416 (all V_3 all V_2 ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_3),pls)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)) = hAPP_int_nat(number_number_of_nat,V_2)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_3),pls)) -> hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_2),V_3)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3)))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls)) -> hAPP_int_nat(number_number_of_nat,V_3) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_int_nat(number_number_of_nat,V_3))))) # label(fact_164_add__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4417 (all A_234 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_234),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_234),A_234)) # label(fact_301_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4418 (all A_190 all B_146 all K_7 (A_190 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_146),K_7) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,B_146),A_190)))) # label(fact_944_dvdI) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4419 (all Y_2 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_2),X_1)) -> comple124823625p_real(hAPP_r1134773055l_bool(ord_at1589558736t_real(Y_2),X_1)) = X_1)) # label(fact_5064_Sup__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4420 (all N all X (is_int(N) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N)),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N)),one_one_real))) -> N = archim1246769320r_real(X))))) # label(fact_4467_floor__eq2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4421 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X_1)) -> hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_r474017924t_real(power_power_real,X_1)),zero_zero_real,sequentially)))) # label(fact_5038_LIMSEQ__inverse__realpow__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4422 (all X all Y hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)) # label(fact_4308_gcd__semilattice__nat_Oinf__left__idem) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4423 (all A_140 all C_62 all B_109 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_140),C_62)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_109),C_62))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_62)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_140),B_109))))) # label(fact_1345_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4424 (all K_1 all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_1),L)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)))) # label(fact_1555_less__bin__lemma) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4425 (all B_65 zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,zero_zero_real),B_65)) # label(fact_2283_mult_Ozero__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4426 (all P_1 collec50511176nt_int(P_1) = P_1) # label(fact_147_Collect__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4427 (all P_1 all I_1 all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),K_1)) -> (hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_1),one_one_int))) -> ((all I_2 (is_int(I_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),K_1)) -> (hBOOL(hAPP_int_bool(P_1,I_2)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I_2),one_one_int))))))) -> hBOOL(hAPP_int_bool(P_1,I_1)))))) # label(fact_2219_int__less__induct) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4428 (all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hAPP_int_int(number_number_of_int,min) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,min)),B_2))) # label(fact_2646_div__eq__minus1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4429 (all X (is_int(X) -> (-hBOOL(hAPP_int_bool(even_odd_even_int,X)) -> X = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_int)))) # label(fact_2805_two__times__odd__div__two__plus__one) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4430 (all Y_2 all X_1 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),X_1)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),Y_2)) -> (hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_1),Y_2) = zero_z126310315umeral <-> Y_2 = zero_z126310315umeral & X_1 = zero_z126310315umeral)))) # label(fact_2159_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4431 (all B all A all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int))) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(wset(A),P_5))) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(inv(P_5),B)),hAPP_i1948725293t_bool(wset(A),P_5)))))))) # label(fact_2214_wset__mem__inv__mem) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4432 (all X_34 all P_7 all Q_10 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_34),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_10)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_34),P_7)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_34),Q_10))) # label(fact_674_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4433 (all Ma all N_1 (is_int(N_1) & is_int(Ma) -> (zero_zero_int = N_1 & zero_zero_int = Ma <-> zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Ma),N_1)))) # label(fact_4975_gcd__zero__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4434 (all Xa_3 all X_13 all Y_10 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_3),X_13)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_3),Y_10)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa_3),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_13),Y_10))) # label(fact_2303_mult__right_Odiff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4435 (all B all A all D (is_int(D) -> (D = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A),B) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),D)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),A)) & (all E (is_int(E) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,E),B)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,E),A)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,E),D))))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D),B))))) # label(fact_5009_gcd__unique__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4436 (all Q all P (-hBOOL(Q) | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)) | -hBOOL(P))) # label(help_fconj_1_1_U) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4437 (all X_1 (hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)) <-> hAPP_nat_nat(div_mod_nat(X_1),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = zero_zero_nat)) # label(fact_3776_even__nat__equiv__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4438 (all A_31 all C_12 all B_26 hAPP_nat_nat(div_mod_nat(A_31),B_26) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_31),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_12),B_26))),B_26)) # label(fact_2505_mod__mult__self1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4439 (all Q_2 all R_3 all B_2 all C_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,R_3),B_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),hAPP_nat_nat(div_mod_nat(Q_2),C_1))),R_3)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),C_1)))))) # label(fact_2736_mod__lemma) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4440 (all P all Q all R hAPP_real_bool(hAPP_r1134773055l_bool(P,R),Q) = hAPP_real_bool(hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__HOL__Obo) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4441 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hAPP_real_real(ln,X) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),one_one_real) -> one_one_real = X))) # label(fact_3432_ln__eq__minus__one) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4442 (all A_3 all B_2 (is_int(B_2) -> (B_2 != zero_zero_int -> (hAPP_int_int(div_mod_int(A_3),B_2) != zero_zero_int -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))),one_one_int) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(uminus_uminus_int,A_3)),B_2)) & (zero_zero_int = hAPP_int_int(div_mod_int(A_3),B_2) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(uminus_uminus_int,A_3)),B_2) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)))))) # label(fact_3553_zdiv__zminus1__eq__if) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4443 (all M_27 all A_212 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,M_27),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_212),M_27)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_212),one_one_real)),M_27)) # label(fact_700_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4444 (all Y_2 all X_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_1)) -> (X_1 = Y_2 <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2))))) # label(fact_1706_dvd_Oantisym__conv) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4445 (all A_3 all B_2 all C_1 (is_int(A_3) -> (C_1 = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),B_2) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_1),B_2) = A_3))) # label(fact_1524_Int2_Oaux1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4446 (all N_33 all A_123 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_123)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_33)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_123),N_33)))))) # label(fact_1486_one__less__power) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4447 (all L all U hAPP_i1948725293t_bool(ord_gr1297742076an_int(L),U) = hAPP_i1948725293t_bool(ord_at641636577an_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,L),one_one_int)),U)) # label(fact_4921_atLeastPlusOneLessThan__greaterThanLessThan__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4448 (all C all B all A (zero_z891286103de_int != A -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A),B)) -> (hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B),A) = C <-> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C),A) = B)))) # label(fact_2641_dvd__div__eq__mult) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4449 (all A (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),zero_zero_rat)))) # label(fact_2191_double__add__less__zero__iff__single__add__less__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4450 (all P_1 all A0 all A1 (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(negDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1))) -> ((all A_2 all B_4 (is_int(B_4) & is_int(A_2) -> (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(negDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_2),B_4))) -> ((-(hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_4),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_2),B_4)))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_4)))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A_2),B_4)))))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1))))) # label(fact_2783_negDivAlg_Opinduct) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4451 (all N hAPP_nat_nat(suc,hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)) # label(fact_3197_add__2__eq__Suc) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4452 (all A_3 all B_2 all C_1 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)),C_1) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(div_mod_int(B_2),C_1))),C_1)) # label(fact_2461_zmod__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4453 (all X all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M)) -> (hAPP_int_int(legacy_zgcd(X),M) = one_one_int -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(phi,M))),one_one_int),M))))) # label(fact_4482_Euler__Fermat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4454 (all Z_1 hAPP_complex_real(re,hAPP_int_complex(ring_11397209091omplex,Z_1)) = hAPP_int_real(ring_1_of_int_real,Z_1)) # label(fact_4579_complex__Re__of__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4455 (all A_223 all K_8 (is_int(A_223) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_223),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K_8))),zero_zero_int)) -> zero_zero_int = A_223))) # label(fact_580_even__power__le__0__imp__0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4456 (all A_33 all B_27 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_33),B_27)),B_27) = zero_z126310315umeral) # label(fact_2497_mod__mult__self2__is__0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.60 4457 (all V_11 all W_9 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_11)),hAPP_int_real(number267125858f_real,W_9)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_11),W_9))) # label(fact_342_number__of__add) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4458 (all A_32 hAPP_nat_nat(div_mod_nat(A_32),one_one_nat) = zero_zero_nat) # label(fact_2500_mod__by__1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4459 (all A_21 all B_17 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_21),B_17)) -> B_17 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_17),A_21)),A_21))) # label(fact_2568_dvd__div__mult__self) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4460 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_2758_Suc__n__div__2__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4461 (all X_1 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd)) | hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)))) # label(fact_2901_even__odd__disj) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4462 (all Y all X (zero_zero_real != hAPP_real_real(cos,X) -> (zero_zero_real != hAPP_real_real(cos,Y) -> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(tan,X)),hAPP_real_real(tan,Y)))))) # label(fact_3613_lemma__tan__add1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4463 (all C_85 all A_193 all B_149 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_193),B_149)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_193),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_149),C_85))))) # label(fact_926_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4464 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_complex_real(re,X)),hAPP_complex_real(norm_norm_complex,X)))) # label(fact_4118_complex__Re__le__cmod) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4465 (all A (A != zero_zero_rat <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_21_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4466 (all Code_numeral_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,code_c271388182l_size(Code_numeral_1)),hAPP_nat_nat(suc,zero_zero_nat)) = code_c271388182l_size(code_S1047413653umeral(Code_numeral_1))) # label(fact_4062_code__numeral_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4467 (all D all F all X_1 all L (hBOOL(hAPP_real_bool(deriv_real(F,X_1),L)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D)) -> ((all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),Y_1))),D)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(F,X_1)),hAPP_real_real(F,Y_1))))) -> zero_zero_real = L)))) # label(fact_3884_DERIV__local__min) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4468 (all X all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,X)),A_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),natfloor(A_3))))) # label(fact_3675_le__natfloor) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4469 (all P_1 ((all X_2 (is_int(X_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) -> hBOOL(hAPP_nat_bool(P_1,hAPP_int_nat(nat,X_2)))))) <-> (all X1 hBOOL(hAPP_nat_bool(P_1,X1))))) # label(fact_2708_all__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4470 (all X_34 all P_7 all Q_10 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_34),P_7)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_34),Q_10)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_34),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_10))) # label(fact_672_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4471 (all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sqrt,Y_2))))) # label(fact_3329_real__sqrt__gt__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4472 (all K_1 all Ma all N_1 (zero_zero_nat = K_1 | N_1 = Ma <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma))) # label(fact_1844_nat__mult__eq__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4473 (all P all Q all R hAPP_r1195171167l_real(hAPP_f617735571l_real(hAPP_f1730463541l_real(cOMBS_511527878l_real,P),Q),R) = hAPP_r1195171167l_real(hAPP_r2019696015l_real(P,R),hAPP_real_real(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__prod_Itc) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4474 (all B_2 all A_3 (is_int(B_2) & is_int(A_3) -> (B_2 != zero_zero_int | A_3 != zero_zero_int -> (exists C_3 exists D_2 (hAPP_int_int(legacy_zgcd(C_3),D_2) = one_one_int & hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(legacy_zgcd(A_3),B_2)),D_2) = B_2 & A_3 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(legacy_zgcd(A_3),B_2)),C_3) & is_int(D_2) & is_int(C_3)))))) # label(fact_4539_make__zrelprime) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4475 (all A_146 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_146),zero_zero_nat) = one_one_rat) # label(fact_1302_power__0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4476 (all Z_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_int_nat(nat,Z_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Z_2)))) # label(fact_3200_one__less__nat__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4477 (all B all A all C (hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C),A) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B),A) <-> B = C)) # label(fact_1933_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4478 (all K_2 all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N))) # label(fact_1758_diff__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4479 (all B_1_1 all B_2_1 is_bool(hAPP_P1463840893t_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__prod_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__prod_Itc__) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4480 (all C_42 all A_100 all B_82 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_100),B_82)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_100),C_42)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_82),C_42))))) # label(fact_2016_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4481 (all A_188 all B_144 all C_81 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_188),B_144)),C_81)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,B_144),C_81)))) # label(fact_958_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4482 (all Z_1 hAPP_complex_complex(cnj,Z_1) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_complex_real(re,Z_1)),hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,Z_1)))) # label(fact_4132_cnj__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4483 (all B_2 all A_3 all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),B_2)) -> (exists D_2 ((all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y_1))),D_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),B_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_3),Y_1)))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))))))) # label(fact_3412_lemma__interval) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4484 (all X_1 (is_int(X_1) -> ((exists Y_1 (X_1 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Y_1) & is_int(Y_1))) <-> hBOOL(hAPP_int_bool(even_odd_even_int,X_1))))) # label(fact_3221_even__equiv__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4485 (all C_59 all D_12 all A_137 all B_106 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_137),B_106)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_59),D_12)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_137)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_59)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_137),C_59)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_106),D_12)))))))) # label(fact_1362_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4486 (all B_141 all A_181 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_141),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_181)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_181),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_141)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_181),B_141)),zero_zero_real)))) # label(fact_1046_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4487 (all A (hBOOL(tendsto_nat_real(A,zero_zero_real,sequentially)) -> (hBOOL(monoseq_real(A)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(A,zero_zero_nat))) -> (all N_2 hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A))),hAPP_r1134773055l_bool(ord_at1589558736t_real(hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_2)))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_2)),one_one_nat))))))))))) # label(fact_5063_summable__Leibniz_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4488 (all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_1)) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1)))) # label(fact_3783_even__dvd) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4489 (all X_1 all Y_2 all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B),X_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B),Y_2))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2))))) # label(fact_1414_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4490 (all A all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,A)),one_one_real)),X_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),archim856651990g_real(X_1))))) # label(fact_4472_le__ceiling__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4491 (all R_3 all S_1 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,R_3),hAPP_real_real(uminus_uminus_real,S_1)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,R_3),S_1)) # label(fact_3486_real__diff__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4492 (all N_1 hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),N_1)))))) # label(fact_4739_bdd__int__set__l__finite) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4493 (all X_1 all R_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_1)),R_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),R_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,R_2)),X_1)))) # label(fact_3485_abs__le__interval__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4494 (all A_37 zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_37),zero_zero_nat)) # label(fact_2471_div__by__0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4495 (all F all A_1 (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A_1)) -> ((all X_2 (is_int(X_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(F,X_2)))))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),big_co1548731110nt_int(F,A_1)))))) # label(fact_4816_transfer__nat__int__sum__prod__closure_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4496 (all P all Q all R hAPP_i1836591008nt_int(hAPP_f400200275nt_int(hAPP_f1731308757nt_int(cOMBB_735549356nt_int,P),Q),R) = hAPP_f151095827nt_int(P,hAPP_i1216375074nt_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__p) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4497 (all P_1 all A0 all A1 (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(posDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1))) -> ((all A_2 all B_4 (is_int(A_2) & is_int(B_4) -> (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(posDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_2),B_4))) -> ((-(hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_4),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),B_4))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_4)))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A_2),B_4)))))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1))))) # label(fact_2785_posDivAlg_Opinduct) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4498 (all N_25 hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_n522471361de_int(semiri1424489471de_int,N_25)))) # label(fact_2911_of__nat__0__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4499 (all K_1 all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K_1),L)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K_1)),hAPP_int_int(bit0,L))))) # label(fact_69_rel__simps_I31_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4500 (all A_3 all B_2 all C_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),C_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),B_2)),C_1)) # label(fact_2692_div__mult2__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4501 (all N one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,N),one_one_int)),N)) # label(fact_4971_coprime__minus__one__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4502 (all V_7 all B_101 all C_56 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_7)),B_101)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_7)),C_56)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_7)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_101),C_56))) # label(fact_1544_right__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4503 (all K_1 hAPP_n1699378549t_bool(ord_lessThan_nat,hAPP_nat_nat(suc,K_1)) = hAPP_n1699378549t_bool(ord_atMost_nat,K_1)) # label(fact_5187_lessThan__Suc__atMost) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4504 (all X all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(standardRes(P_3),X)),P_3)))) # label(fact_2591_StandardRes__ubound) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4505 (all X_3 (hBOOL(summable_complex(X_3)) -> hAPP_f213450978omplex(suminf_complex,hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_3)) = hAPP_complex_complex(cnj,hAPP_f213450978omplex(suminf_complex,X_3)))) # label(fact_4844_cnj_Osuminf) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4506 (all B_141 all A_181 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_181),zero_zero_rat)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_141)) | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_141),zero_zero_rat)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_181)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_181),B_141)),zero_zero_rat)))) # label(fact_1049_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4507 (all X all Y all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y),Z_1)),Z_1))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Z_1)),Y))))) # label(fact_2652_div__prop2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4508 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),A)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)))))) # label(fact_4722_one__less__Fract__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4509 (all A_3 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(abs_abs_int,A_3)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(abs_abs_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3459_abs__power2__distrib) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4510 (all B_1_1 is_bool(summable_complex(B_1_1))) # label(gsy_c_Series_Osummable_000tc__Complex__Ocomplex) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4511 (all K_2 quickcheck_int_of(hAPP_i1732201573de_int(number1226105091de_int,K_2)) = hAPP_int_int(number_number_of_int,K_2)) # label(fact_4256_int__of__number) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4512 (all X_46 all Y_34 (hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_46),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_34),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_46)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_34)) -> Y_34 = X_46)))) # label(fact_204_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4513 (all P all Q all R hAPP_f472159229t_bool(P,hAPP_r1962736578t_bool(Q,R)) = hAPP_r1623423338t_bool(hAPP_f704188951t_bool(hAPP_f1456711209t_bool(cOMBB_690122963l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__fun_Itc__Int__Oint_) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4514 (all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,P_3)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_2760_Int2_Oaux__1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4515 (all A_14 all B_10 all Y all R_7 (is_int(B_10) & is_int(Y) -> (A_14 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_10),Y)),R_7) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_10)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_7),B_10)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_7))) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_10)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_10),R_7)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_7),zero_zero_int))) -> (B_10 != zero_zero_int -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_14),B_10) = Y))))) # label(fact_2653_divmod__int__rel__div__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4516 (all M_18 all N_38 all A_128 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_128)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_128),M_18)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_128),N_38))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_18),N_38))))) # label(fact_1446_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4517 (all X (is_int(X) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) -> one_one_int = X | zero_zero_int = X)))) # label(fact_417_int__pos__lt__two__imp__zero__or__one) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4518 (all A_154 -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_154),A_154)),zero_zero_rat))) # label(fact_1202_not__square__less__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4519 (all Diff all F all N_1 all H (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) & F = hAPP_n546711566l_real(Diff,zero_zero_nat) & (all M_1 all T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))))) -> (exists T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),T)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,T),H)) & hAPP_real_real(F,H) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,Diff),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,H))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(Diff,N_1),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N_1)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),N_1))))))) # label(fact_4934_Maclaurin__objl) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4520 (all A_215 A_215 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_215),one_one_nat)) # label(fact_655_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4521 (all A_125 all N_36 all N_35 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_36),N_35)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_125)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_125),one_one_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_125),N_35)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_125),N_36))))))) # label(fact_1474_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4522 (all A all B all C (C = B <-> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A),B) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A),C))) # label(fact_1941_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4523 (all Ma all N_1 (hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N_1) <-> N_1 = hAPP_nat_nat(suc,zero_zero_nat) & zero_zero_nat = Ma | Ma = hAPP_nat_nat(suc,zero_zero_nat) & zero_zero_nat = N_1)) # label(fact_3052_one__is__add) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4524 (all C_80 all A_187 all B_143 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_187),B_143)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_187),C_80)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_187),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_143),C_80)))))) # label(fact_967_dvd__add) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4525 (all R_3 all X hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_complex_real(re,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),hAPP_complex_real(im,X))) = hAPP_complex_complex(scaleR1652505878omplex(R_3),X)) # label(fact_5115_complex__scaleR__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4526 (all K_2 hAPP_int_rat(number_number_of_rat,K_2) = frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,K_2)),one_one_int))) # label(fact_4641_Frct__code__post_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4527 -(all S_2 (is_int(S_2) -> -(hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),S_2)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,S_2),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(s1),S_2),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)))))) # label(fact_440__096_B_Bthesis_O_A_I_B_Bs_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4528 (all B_69 all A_81 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_81)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_69)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_81),B_69)))))) # label(fact_2186_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4529 (all A_220 A_220 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A_220)) # label(fact_603_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4530 (all N_1 all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),Ma)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),hAPP_nat_nat(suc,Ma))))) # label(fact_2872_Suc__le__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4531 (all A_149 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_149),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_149),one_one_rat)))) # label(fact_1262_less__add__one) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4532 (all A_3 all B_2 (is_int(A_3) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))),hAPP_int_int(div_mod_int(A_3),B_2)) = A_3)) # label(fact_2358_zmod__zdiv__equality) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4533 (all N -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat))) # label(fact_1606_not__less0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4534 (all K_2 all N all M hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N)),M))),N) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M)),N)) # label(fact_3140_mod__mult__self4) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4535 (all S (-hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S)) <-> (all M_1 exists N_2 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,N_2),S)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_2)))))) # label(fact_4608_infinite__nat__iff__unbounded__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4536 (all A_30 all B_25 all C_11 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_30),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_25),C_11))),B_25) = hAPP_C498520661umeral(div_mo1740067990umeral(A_30),B_25)) # label(fact_2510_mod__mult__self2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4537 (all V_3 all K_2 all V_2 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls)) -> zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2))) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_2),V_3))),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_3)),K_2))))) # label(fact_572_nat__number__of__mult__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4538 (all A all B (quotient_of(B) = quotient_of(A) <-> A = B)) # label(fact_4655_quotient__of__inject__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4539 (all N hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),N)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3723_fact__lemma) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4540 (all B_130 all A_170 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_170)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_130),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_130),A_170)),zero_zero_real))))) # label(fact_1100_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4541 (all A_197 all B_152 all N_64 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_197),B_152)),N_64) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_197),N_64)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_152),N_64))) # label(fact_896_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4542 (all X_1 all P_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),zero_zero_int),P_5)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_5),X_1)))) # label(fact_1315_zcong__eq__zdvd__prop) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4543 (all X_1 the_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_r1900510169t_bool(hAPP_f1480506641t_bool(cOMBC_int_rat_bool,hAPP_f601898971t_bool(hAPP_f857296063t_bool(cOMBB_214503962ol_int,ord_less_eq_rat),ring_1_of_int_rat)),X_1))),hAPP_f1241350704t_bool(hAPP_f1131424617t_bool(cOMBB_rat_bool_int,hAPP_r2115560837t_bool(ord_less_rat,X_1)),hAPP_f1900950721nt_rat(hAPP_f1270196437nt_rat(cOMBB_int_rat_int,ring_1_of_int_rat),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,plus_plus_int),one_one_int))))) = archim791455193or_rat(X_1)) # label(fact_4891_floor__rat__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4544 (all X_1 all A_1 (hBOOL(hAPP_P1463840893t_bool(A_1,X_1)) <-> hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,X_1),A_1)))) # label(fact_139_mem__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4545 (all R_3 all A_3 all B_2 hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),A_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_3),B_2)) = hAPP_complex_complex(scaleR1652505878omplex(R_3),hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2))) # label(fact_5110_complex__scaleR) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4546 (all X all Y hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),X)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),Y)))))) # label(fact_3336_real__sqrt__mult__self__sum__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4547 (all W hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),W))) # label(fact_52_zle__refl) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4548 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(nat,X)),hAPP_int_nat(nat,Y)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y))))) # label(fact_2697_Divides_Otransfer__nat__int__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4549 (exists K_4 ((all X_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(re,X_2))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_2)),K_4)))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),K_4)))) # label(fact_4140_Re_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4550 (all A_3 all B_2 hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(uminus_uminus_int,A_3)),B_2) = hAPP_rat_rat(uminus_uminus_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2))) # label(fact_4689_minus__rat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4551 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hAPP_nat_nat(fact_fact_nat,N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))))) # label(fact_3816_fact__reduce__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4552 (all X_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),natfloor(X_1))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X_1)))) # label(fact_3400_le__natfloor__eq__one) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4553 (all A_129 all N_40 all N_39 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_40),N_39)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_129)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_129),N_40)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_129),N_39)))))) # label(fact_1440_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4554 (all X_33 one_on1645066479umeral = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_33),zero_zero_nat)) # label(fact_678_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4555 (all B_122 all A_161 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_161),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_122),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_161),B_122)))))) # label(fact_1151_mult__neg__neg) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4556 (all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Ma),N_1))) <-> zero_zero_nat != Ma | N_1 != zero_zero_nat)) # label(fact_4344_gcd__pos__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4557 (all Z1 all Z2 all W hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),W)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),W)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)),W)) # label(fact_131_zadd__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4558 (all X_1 hBOOL(hAPP_real_bool(deriv_real(cos,X_1),hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X_1))))) # label(fact_3865_DERIV__cos) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4559 (all A_12 all N_18 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_12),hAPP_nat_nat(suc,N_18)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_12),N_18)),A_12)) # label(fact_3006_power__Suc2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4560 (all Z_1 all W (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_1),W)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),Z_1)))) # label(fact_1735_real__le__linear) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4561 (all Z_1 (is_int(Z_1) -> (-hBOOL(hAPP_int_bool(nat_neg,Z_1)) -> Z_1 = hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,Z_1))))) # label(fact_3277_not__neg__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4562 (all P all Q all R hAPP_f892584630t_bool(hAPP_P1400155668t_bool(P,R),hAPP_P1668131738nt_int(Q,R)) = hAPP_P989584703t_bool(hAPP_f1187443546t_bool(hAPP_f11709033t_bool(cOMBS_1676841879t_bool,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc_) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4563 (all A all W_1 (is_int(A) -> (zero_zero_int = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),hAPP_int_nat(number_number_of_nat,W_1)) <-> hAPP_int_nat(number_number_of_nat,W_1) != zero_zero_nat & zero_zero_int = A))) # label(fact_712_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4564 (all X hAPP_real_real(sgn_sgn_real,X) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_real_real(abs_abs_real,X))) # label(fact_3565_real__sgn__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4565 (all K_2 all L_2 frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,K_2)),hAPP_int_int(number_number_of_int,L_2))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(number_number_of_rat,K_2)),hAPP_int_rat(number_number_of_rat,L_2))) # label(fact_4645_Frct__code__post_I6_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4566 (all A_58 all B_49 all C_28 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(div_mod_int(A_58),C_28)),hAPP_int_int(div_mod_int(B_49),C_28))),C_28) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_58),B_49)),C_28)) # label(fact_2385_mod__mult__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4567 (all A_224 all N_68 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_224),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_224),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68))) # label(fact_573_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4568 (all P all Q all R hAPP_bool_bool(hAPP_n1006566506l_bool(P,R),hAPP_nat_bool(Q,R)) = hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__HOL__Obool_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4569 (all B_70 all A_82 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_82),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_70),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_82),B_70)),zero_zero_rat))))) # label(fact_2182_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4570 (all A_3 all B_2 hAPP_int_int(uminus_uminus_int,hAPP_int_int(div_mod_int(A_3),B_2)) = hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,A_3)),hAPP_int_int(uminus_uminus_int,B_2))) # label(fact_3525_zmod__zminus__zminus) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4571 (all A_186 hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,one_on1645066479umeral),A_186))) # label(fact_977_one__dvd) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4572 (all A all N_1 (is_int(A) -> (A = zero_zero_int & zero_zero_nat != N_1 <-> zero_zero_int = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),N_1)))) # label(fact_1289_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4573 (all X_1 (one_one_complex = X_1 <-> X_1 = one_one_complex)) # label(fact_1954_one__reorient) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4574 (all A_56 all C_26 all B_47 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(div_mo231679042de_int(A_56),B_47)),C_26) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_56),C_26)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_47),C_26))) # label(fact_2396_mod__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4575 (all M all N (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) = M -> zero_zero_nat = N)) # label(fact_1614_add__eq__self__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4576 (all A_91 A_91 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_91),one_one_real)) # label(fact_2107_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4577 (all U all F ((all N_2 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),hAPP_nat_nat(F,N_2)))) -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,ord_less_eq_nat),F)),U)))))) # label(fact_4895_finite__less__ub) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4578 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3)) -> hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_3)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(div_mod_int(B_2),A_3))))) # label(fact_2323_pos__zmod__mult__2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4579 (all A all B ((exists X_2 exists Y_1 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),X_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B),Y_1)) = one_one_nat | one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B),X_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A),Y_1)))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A),B)))) # label(fact_4036_coprime__bezout) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4580 (all N all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,N)),hAPP_int_real(real_int,X))),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N),X)))))) # label(fact_4464_real__of__int__div2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4581 (all Lx_3 all Ly_1 all Rx_3 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_3),Ly_1)),Rx_3) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_3),Rx_3)),Ly_1)) # label(fact_490_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4582 (all X_1 ((exists N_2 (is_int(N_2) & hAPP_int_real(real_int,N_2) = X_1)) <-> hAPP_int_real(real_int,archim1246769320r_real(X_1)) = X_1)) # label(fact_4382_real__of__int__floor__cancel) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4583 (all C_70 all A_164 all B_125 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_164),B_125)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_70)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_70),A_164)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_70),B_125)))))) # label(fact_1131_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4584 (all A all B (is_int(B) & is_int(A) -> (zero_zero_int = A | zero_zero_int = B <-> zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),B)))) # label(fact_835_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4585 (all X all Y all Xa_2 all Ya hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Ya),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Ya),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_3349_real__sqrt__sum__squares__mult__squared__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4586 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Y)),one_one_real)) -> hAPP_real_real(sin,hAPP_real_real(arccos,Y)) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3743_sin__arccos__abs) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4587 (all Ta all A_1 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> (all X_2 (is_int(X_2) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1)) -> X_2 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa))))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ta),X_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ta),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1))))))))) # label(fact_5090_aset_I8_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4588 (all X_1 all A_1 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(semiri1621563631at_int,X_1)),image_nat_int(semiri1621563631at_int,A_1))) <-> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A_1)))) # label(fact_4603_transfer__nat__int__set__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4589 (all N zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),N)) # label(fact_1740_diff__0__eq__0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4590 (all X_1 all Y_2 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_1),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,ratreal(X_1)),ratreal(Y_2))))) # label(fact_4700_real__less__eq__code) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4591 (all C all A all B (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A),B)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A),C)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,B),A)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,C),A))) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,B),C)))))) # label(fact_2483_div__dvd__div) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4592 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)))) # label(fact_3125_zdiff__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4593 (all A_75 all A_74 all B_62 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_75),A_74)),B_62) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_75),B_62)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_74),B_62))) # label(fact_2296_mult_Oadd__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4594 (all Q_1 all P_1 ((all A_2 all B_4 (is_int(B_4) & is_int(A_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_4)) -> (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_4) -> hBOOL(hAPP_rat_bool(P_1,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_4))))))) -> hBOOL(hAPP_rat_bool(P_1,Q_1)))) # label(fact_5027_Rat__induct) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4595 (all L_2 hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),L_2)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),hAPP_int_int(bit1,L_2))) # label(fact_1582_diff__bin__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4596 (all M all K_2 all N (one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,K_2),N) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N))) # label(fact_4353_gcd__mult__cancel__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4597 (all K_2 all L_2 (zero_zero_int != hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,K_2)),L_2) -> hAPP_int_int(div_mod_int(K_2),L_2) != zero_zero_int)) # label(fact_3533_zmod__zminus1__not__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4598 (all X one_one_real = hAPP_real_real(hAPP_r1250527377l_real(powr,X),zero_zero_real)) # label(fact_4408_powr__zero__eq__one) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4599 (all M hAPP_int_int(abs_abs_int,hAPP_nat_int(semiri1621563631at_int,M)) = hAPP_nat_int(semiri1621563631at_int,M)) # label(fact_3417_abs__int__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4600 (all B_2 all Q_8 all R_5 all Q_2 all R_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_8)),R_5)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_2)),R_3))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_5)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_2),Q_8))))))) # label(fact_736_unique__quotient__lemma__neg) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4601 (all A_223 all K_8 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_223),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K_8))),zero_zero_real)) -> zero_zero_real = A_223)) # label(fact_581_even__power__le__0__imp__0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4602 (all A_189 all B_145 all C_82 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_189),B_145)),C_82)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_189),C_82)))) # label(fact_948_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4603 (all P all Q all R hAPP_real_bool(hAPP_f330985925l_bool(hAPP_f940061907l_bool(cOMBB_1795223523l_real,P),Q),R) = hAPP_P1333854989l_bool(P,hAPP_r1195171167l_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc_010) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4604 (all I_1 all K_1 all J (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_1)),J)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K_1)))))) # label(fact_1773_le__diff__conv2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4605 (all N_54 all A_151 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_151)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_151),N_54))))) # label(fact_1244_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4606 (all A_3 all B_2 all C_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)),C_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_2),C_1))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),hAPP_nat_nat(div_mod_nat(B_2),C_1))),C_1))) # label(fact_2685_div__mult1__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4607 (all Lx_4 all Ly_2 all Rx_4 all Ry_2 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_4),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ly_2),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_4),Ry_2))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_4),Ly_2)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_4),Ry_2))) # label(fact_483_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4608 (all X all Y hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,X),Y) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X),hAPP_complex_complex(invers1449016382omplex,Y))) # label(fact_3972_complex__divide__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4609 (all U hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_lessThan_nat,U)) = U) # label(fact_4950_card__lessThan) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4610 (all A_230 all C_96 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_230),C_96) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_96),A_230)) # label(fact_522_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4611 (all X_1 (X_1 = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,X_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)))) # label(fact_3342_real__sqrt__pow2__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4612 (all R_3 all X hAPP_complex_real(re,hAPP_complex_complex(scaleR1652505878omplex(R_3),X)) = hAPP_real_real(scaleR_scaleR_real(R_3),hAPP_complex_real(re,X))) # label(fact_5111_Re_OscaleR) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4613 (all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2)) -> (B_2 != one_one_nat -> (exists X_2 exists Y_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),X_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),Y_1)),one_one_nat))))) # label(fact_4037_coprime__bezout__strong) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4614 (all X all Y hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(sin,Y)))) # label(fact_3683_cos__diff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4615 (all X_36 all Y_28 all Z_9 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_36),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,Y_28),Z_9)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_36),Y_28)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_36),Z_9))) # label(fact_621_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4616 (all M all N (N = M -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)))) # label(fact_1598_eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4617 (all X all Y hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y))) # label(fact_4966_transfer__int__nat__gcd_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4618 (all L all U hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_at875362053st_int(L),U)))) # label(fact_5067_finite__atLeastAtMost__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4619 (all A_27 all B_22 all C_9 (C_9 != zero_z126310315umeral -> hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_27),B_22) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_27),C_9)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_22),C_9)))) # label(fact_2539_div__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4620 (all A_196 all N_63 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_196),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_196),N_63)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_196),N_63)),A_196)) # label(fact_902_power__commutes) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4621 (all X_1 hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,power_power_real),cos)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,min))),hAPP_real_real(cos,X_1))),hAPP_real_real(sin,X_1))))) # label(fact_4797_DERIV__cos__realpow2a) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4622 (all A_124 all N_34 all B_102 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_124),N_34) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_102),N_34) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_124)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_102)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_34)) -> A_124 = B_102))))) # label(fact_1480_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4623 (all X hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),zero_zero_nat) = X) # label(fact_4339_gcd__0__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4624 (all A_3 all B_2 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),B_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),A_3)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))))) # label(fact_1666_zdiff__power3) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4625 (all N all K_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),N)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N)))))) # label(fact_4982_gcd__greatest__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4626 (all N_1 all P_1 (-hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(P_1,N_1)) -> (exists K (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N_1)) & hBOOL(hAPP_nat_bool(P_1,K)) & (all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),K)) -> -hBOOL(hAPP_nat_bool(P_1,I_2))))))))) # label(fact_2245_ex__least__nat__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4627 (all N_59 all M_24 all X_28 all Y_22 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_28),Y_22)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_59),M_24)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_28),N_59)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_22),M_24)))))) # label(fact_999_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4628 (all B_115 all C_67 all A_153 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_153)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_115),C_67)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_115),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_153),C_67)))))) # label(fact_1206_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 7.56/7.61 4629 (all N_41 ((N_41 = zero_zero_nat -> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),N_41) = one_one_complex) & (N_41 != zero_zero_nat -> zero_zero_complex = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),N_41)))) # label(fact_1436_power__0__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4630 (all P_3 all X hAPP_int_int(multInv(P_3),X) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_2756_MultInv__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4631 (all L all U hAPP_n1699378549t_bool(ord_at4362885an_nat(L),hAPP_nat_nat(suc,U)) = hAPP_n1699378549t_bool(ord_at238088361st_nat(L),U)) # label(fact_5075_atLeastLessThanSuc__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4632 (all A_124 all N_34 all B_102 (is_int(B_102) & is_int(A_124) -> (hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_102),N_34) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_124),N_34) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_124)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_102)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_34)) -> A_124 = B_102)))))) # label(fact_1479_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4633 (all I all K_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,I)),K_2)) -> -(all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_1)) -> hAPP_nat_nat(suc,J_1) != K_2)))) # label(fact_3304_Suc__lessE) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4634 (all X1_2 all X2 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (hAPP_int_int(standardRes(Ma),X2) = hAPP_int_int(standardRes(Ma),X1_2) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X1_2),X2),Ma))))) # label(fact_2650_StandardRes__prop2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4635 (all A_56 all C_26 all B_47 hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_56),C_26)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_47),C_26)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(div_mo1740067990umeral(A_56),B_47)),C_26)) # label(fact_2395_mod__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4636 (all M all N hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,M)),hAPP_nat_real(real_nat,N))) # label(fact_3652_real__of__nat__add) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4637 (all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),hAPP_int_int(number_number_of_int,Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,pls)),Y_2)))) # label(fact_121_le__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4638 (all N hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),hAPP_nat_nat(fact_fact_nat,N))) # label(fact_3813_fact__plus__one__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4639 (all P all Q all R hAPP_f1902329768t_bool(P,hAPP_r285062775l_bool(Q,R)) = hAPP_r1562208522t_bool(hAPP_f2084827628t_bool(hAPP_f1121609875t_bool(cOMBB_2125215742l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Nat__Onat_Mtc__fun_) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4640 (all Ma all N_1 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),N_1) = one_one_nat <-> one_one_nat = N_1 & Ma = one_one_nat)) # label(fact_1660_nat__mult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4641 (all P_3 all A_3 hAPP_int_int(inv(P_3),A_3) = hAPP_int_int(div_mod_int(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),P_3)) # label(fact_2761_WilsonRuss_Oinv__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4642 (all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) | Ma = N_1)) # label(fact_1634_le__eq__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4643 (all C_70 all A_164 all B_125 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_164),B_125)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_70)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_70),A_164)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_70),B_125)))))) # label(fact_1134_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4644 (all P_5 hAPP_i1948725293t_bool(sRStar,P_5) = collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),P_5)))) # label(fact_2731_SRStar__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4645 (all M all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),J_2))))) # label(fact_1628_trans__less__add2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4646 (all M all N ((hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M),N)) != N -> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M)),N) = hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M),N))) & (N = hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M),N)) -> zero_zero_nat = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M)),N)))) # label(fact_3079_mod__Suc) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4647 (all R_3 all X hAPP_complex_complex(scaleR1652505878omplex(R_3),hAPP_complex_complex(cnj,X)) = hAPP_complex_complex(cnj,hAPP_complex_complex(scaleR1652505878omplex(R_3),X))) # label(fact_5113_cnj_OscaleR) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4648 (all K_2 hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,K_2)) = frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,one_one_int),hAPP_int_int(number_number_of_int,K_2)))) # label(fact_4646_Frct__code__post_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4649 (all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1)) -> hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),N_1)))) = hAPP_int_nat(nat,N_1))) # label(fact_4737_card__bdd__int__set__l__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4650 (all C_1 all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) -> (C_1 = B_2 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),C_1))))) # label(fact_1710_dvd_Oord__le__eq__trans) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4651 (all M_13 hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(suc,M_13)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_nat_real(semiri132038758t_real,M_13))) # label(fact_2840_of__nat__Suc) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4652 (all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N_1)) -> hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),N_1)))) = hAPP_int_nat(nat,N_1))) # label(fact_4736_card__bdd__int__set__l) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4653 (all X all Y all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),Y),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),X)),hAPP_int_int(multInv(P_3),Y)),P_3))))) # label(fact_2228_MultInv__prop1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4654 (all A all B ((all D_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),B)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),A)) <-> D_2 = one_one_nat)) <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A),B) = one_one_nat)) # label(fact_4350_coprime__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4655 (all D all Sa exists Z all X_2 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z),X_2)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_2),Sa))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_2),Sa)))))) # label(fact_2272_pinf_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4656 (all B_45 all B_44 all A_54 all C_24 all A_53 (hAPP_Q1762011733de_int(div_mo231679042de_int(A_53),C_24) = hAPP_Q1762011733de_int(div_mo231679042de_int(A_54),C_24) -> (hAPP_Q1762011733de_int(div_mo231679042de_int(B_44),C_24) = hAPP_Q1762011733de_int(div_mo231679042de_int(B_45),C_24) -> hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_53),B_44)),C_24) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_54),B_45)),C_24)))) # label(fact_2404_mod__mult__cong) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4657 (all P all Q all R hAPP_r2019696015l_real(hAPP_f320596107l_real(hAPP_f850851693l_real(cOMBS_193881978l_real,P),Q),R) = hAPP_f617735571l_real(hAPP_r712953929l_real(P,R),hAPP_r1250527377l_real(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4658 (all M all N hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,M)),hAPP_nat_real(real_nat,N)) = hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N))) # label(fact_3651_real__of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4659 (all V_19 all W_15 hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_19),W_15)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_19)),hAPP_int_real(number267125858f_real,W_15))) # label(fact_84_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4660 (all N all V_2 ((-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,V_2)),N) = hAPP_int_nat(nat,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,V_2)),N))) & (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,V_2)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N)))) # label(fact_3284_power__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4661 (all M zero_zero_nat != hAPP_nat_nat(suc,M)) # label(fact_2851_Suc__not__Zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4662 (all K_2 all M all N hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,K_2)),hAPP_int_int(legacy_zgcd(M),N)) = hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N))) # label(fact_4503_zgcd__zmult__distrib2__abs) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4663 (all K_2 zero_zero_rat = frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),K_2))) # label(fact_4643_Frct__code__post_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4664 (all X -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,X)),zero_zero_real))) # label(fact_3832_not__exp__le__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4665 (all A_186 hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,one_on1684967323de_int),A_186))) # label(fact_979_one__dvd) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4666 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N))))) # label(fact_3182_div__geq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4667 (all N hAPP_complex_real(re,hAPP_nat_complex(semiri2020571505omplex,N)) = hAPP_nat_real(semiri132038758t_real,N)) # label(fact_4127_complex__Re__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4668 (all Z_1 all W hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),W) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),Z_1)) # label(fact_199_zadd__commute) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4669 (all Z_1 hAPP_complex_real(im,hAPP_int_complex(ring_11397209091omplex,Z_1)) = zero_zero_real) # label(fact_4578_complex__Im__of__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4670 (all A_208 hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_208),zero_zero_complex))) # label(fact_759_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4671 (all K_2 all I all J_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),I)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),J_2)))))) # label(fact_201_zmult__zless__mono2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4672 (all X all Y hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y))) # label(fact_4985_abs__gcd__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4673 (all P all Q all R hAPP_r1250527377l_real(P,hAPP_int_real(Q,R)) = hAPP_i371647666l_real(hAPP_f1352333353l_real(hAPP_f318350629l_real(cOMBB_1946221070al_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4674 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (zero_zero_real = hAPP_real_real(ln,X_1) <-> X_1 = one_one_real))) # label(fact_3442_ln__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4675 (all A_227 all B_164 all C_93 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_227),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_164),C_93)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_227),B_164)),C_93)) # label(fact_541_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4676 (all X all Y hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),hAPP_nat_nat(div_mod_nat(X),Y)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)) # label(fact_4342_gcd__red__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4677 (all Ta all B_1 all D_1 (is_int(Ta) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ta),B_1)) -> (all X_2 (is_int(X_2) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa) != X_2)))))) -> (Ta != X_2 -> Ta != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1))))))))) # label(fact_5098_bset_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4678 (all X_1 all Y_2 (X_1 = Y_2 <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_2)))) # label(fact_1703_divides__antisym) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4679 (all B_1_1 is_int(quickcheck_int_of(B_1_1))) # label(gsy_c_Quickcheck__Narrowing_Oint__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4680 (all X all Y all Xa_2 all Ya hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Ya),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) # label(fact_3308_real__sqrt__sum__squares__mult__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4681 (all X all Y -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_real))) # label(fact_8_not__sum__power2__lt__zero) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4682 (all M all N -(all Q_4 all R_1 -hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_4),R_1))))) # label(fact_4221_divmod__nat__rel__ex) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4683 (all Y all X all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (A_3 != one_one_real -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y)) -> hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(log(A_3),X)),hAPP_real_real(log(A_3),Y)) = hAPP_real_real(log(A_3),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),Y))))))) # label(fact_3874_log__divide) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4684 (all A_93 (is_int(A_93) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_93),zero_zero_int) = A_93)) # label(fact_2088_diff__0__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4685 (all Z_1 (is_int(Z_1) -> Z_1 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),zero_zero_int))) # label(fact_267_zadd__0__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4686 (all X all Y hAPP_P1860904029l_real(hAPP_P1982799375l_real(hAPP_b1482221219l_real(if_Pro313124157l_real,fFalse),X),Y) = Y) # label(help_If_2_1_If_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_T) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4687 (all Z_7 all W_2 all Y_9 all X_12 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,Y_9),X_12)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,Z_7),W_2)) -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_12),W_2)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Y_9),Z_7)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,X_12),Y_9)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,W_2),Z_7))))) # label(fact_2558_div__mult__div__if__dvd) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4688 (all N_61 all X_29 all Y_23 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_29),Y_23)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_29),N_61)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_23),N_61))))) # label(fact_989_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4689 (all B_43 all B_42 all A_52 all C_23 all A_51 (hAPP_int_int(div_mod_int(A_51),C_23) = hAPP_int_int(div_mod_int(A_52),C_23) -> (hAPP_int_int(div_mod_int(B_43),C_23) = hAPP_int_int(div_mod_int(B_42),C_23) -> hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_52),B_43)),C_23) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_51),B_42)),C_23)))) # label(fact_2405_mod__add__cong) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4690 (all Y_8 all Z_6 all X_11 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,Z_6),X_11)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,Z_6),Y_8)) -> hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,X_11),Z_6)),hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,Y_8),Z_6)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_11),Y_8)),Z_6)))) # label(fact_2577_div__add) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4691 (all P all Q all R hAPP_r1134773055l_bool(P,hAPP_real_real(Q,R)) = hAPP_r1134773055l_bool(hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__H_009) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4692 (all N all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,P_3),N)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),N)))) # label(fact_4178_prime__coprime__strong) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4693 (all A_202 zero_zero_complex = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,zero_zero_complex),A_202)) # label(fact_850_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4694 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,N)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),N))) # label(fact_3207_div__Suc__eq__div__add3) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4695 (all P all Q all R hAPP_P507092031nt_rat(hAPP_f1547007718nt_rat(hAPP_f382317021nt_rat(cOMBB_170828966nt_int,P),Q),R) = hAPP_int_fun_int_rat(P,hAPP_P1175774780nt_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Rat__Orat_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4696 (all B_1_1 all B_2_1 is_bool(hAPP_nat_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__Nat__Onat_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4697 (all D all B all C all A (A != zero_z126310315umeral -> (C != zero_z126310315umeral -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A),B)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,C),D)) -> (hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B),A) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,D),C) <-> hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A),D) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B),C))))))) # label(fact_2636_dvd__div__div__eq__mult) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4698 (all Y exists X_2 ((all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),Y_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) & hAPP_real_real(tan,Y_1) = Y -> X_2 = Y_1)) & hAPP_real_real(tan,X_2) = Y & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X_2)))) # label(fact_3627_tan__total) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4699 (all A_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma)) -> (hBOOL(hAPP_int_bool(is_RRset(A_1),Ma)) -> image_int_int(rRset2norRR(A_1,Ma),A_1) = hAPP_i1948725293t_bool(norRRset,Ma)))) # label(fact_4597_RRset2norRR__eq__norR) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4700 (all N_56 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_int_complex(number528085621omplex,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_56)) = one_one_complex) # label(fact_1042_power__m1__even) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4701 (all N N = natfloor(hAPP_nat_real(real_nat,N))) # label(fact_3638_natfloor__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4702 (all B_65 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,zero_zero_complex),B_65) = zero_zero_complex) # label(fact_2284_mult_Ozero__left) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4703 (all X all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),hAPP_int_int(multInv(P_3),X))),one_one_int),P_3)))))) # label(fact_2240_MultInv__prop2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4704 (all A_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_3)) -> zero_zero_nat = A_3)) # label(fact_2311_gcd__lcm__complete__lattice__nat_Otop__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4705 (all X all Y (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Y)) -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Y),hAPP_int_int(div_mod_int(X),Y)))) # label(fact_4964_gcd__non__0__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4706 (all A_139 all C_61 all B_108 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_139),C_61)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_108),C_61))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_61)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_139),B_108))))) # label(fact_1348_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4707 (all A_64 all B_52 all C_31 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_64),C_31) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,A_64),B_52)),B_52)),hAPP_Q1762011733de_int(div_mo231679042de_int(A_64),B_52))),C_31)) # label(fact_2357_div__mod__equality) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4708 (all V_10 all U_3 all Y_26 all X_32 all A_209 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_32),A_209)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_26),A_209)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),U_3)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),V_10)) -> (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,U_3),V_10) = one_one_real -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,U_3),X_32)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,V_10),Y_26))),A_209)))))))) # label(fact_751_convex__bound__lt) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4709 (all M_17 all N_30 ((N_30 = zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_17),N_30) = one_one_nat) & (N_30 != zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_17),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_30),one_one_nat))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_17),N_30)))) # label(fact_1830_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4710 (all P_5 all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A),hAPP_i1948725293t_bool(wset(A),P_5))))) # label(fact_2223_wset__mem__mem) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4711 (all X all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(inverse_inverse_real,X)) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X))))) # label(fact_3940_real__root__inverse__lemma) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4712 (all A_230 all C_96 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_230),C_96) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_96),A_230)) # label(fact_520_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4713 (all B_72 all C_35 all A_84 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_84)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_72),C_35)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_72),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_84),C_35)))))) # label(fact_2158_add__increasing) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4714 (all X all N (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,N))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natceiling(X)),hAPP_int_nat(number_number_of_nat,N)) = natceiling(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_int_real(number267125858f_real,N)))))) # label(fact_3378_natceiling__add__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4715 (all A_23 (A_23 != zero_z126310315umeral -> one_on1645066479umeral = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_23),A_23))) # label(fact_2555_div__self) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4716 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real)) -> Y = hAPP_real_real(cos,hAPP_real_real(arccos,Y))))) # label(fact_3607_cos__arccos) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4717 (all C_44 all D_8 all A_102 all B_84 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_102),B_84)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_44),D_8)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_102),C_44)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_84),D_8)))))) # label(fact_2004_add__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4718 (all B_2 all N all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_real_real(log(B_2),X)) = hAPP_real_real(log(B_2),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N)))) # label(fact_3872_log__nat__power) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4719 (all Lx_4 all Ly_2 all Rx_4 all Ry_2 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_4),Ly_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_4),Ry_2)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_4),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ly_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_4),Ry_2)))) # label(fact_478_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4720 (all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) -> X != Y)) # label(fact_1722_dvd_Oless__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4721 (all C_7 all A_22 all B_18 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_22),B_18)) -> hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,B_18),A_22)),C_7) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_18),C_7)),A_22))) # label(fact_2561_dvd__div__mult) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4722 (all N zero_zero_real = hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),pi))) # label(fact_3641_sin__npi) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4723 (all A_109 all B_87 all C_47 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_109),B_87)),C_47) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_109),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_87),C_47))) # label(fact_1946_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4724 (all X hAPP_complex_real(re,hAPP_complex_complex(cnj,X)) = hAPP_complex_real(re,X)) # label(fact_4115_complex__Re__cnj) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4725 (all N_25 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(semiri984289939at_nat,N_25)))) # label(fact_2914_of__nat__0__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4726 (all X hAPP_i1732201573de_int(quickcheck_of_int,quickcheck_int_of(X)) = X) # label(fact_4555_int__of__inverse) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4727 (all X_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X_1)) | N_1 = zero_zero_nat <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),N_1))))) # label(fact_1012_nat__zero__less__power__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4728 (all Z_1 all W hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_1),W) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,W),Z_1)) # label(fact_1783_real__mult__commute) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4729 (all P all Q all R hAPP_nat_nat(hAPP_f1914919701at_nat(hAPP_f1408247010at_nat(cOMBS_nat_nat_nat,P),Q),R) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(P,R),hAPP_nat_nat(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4730 (all A_90 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),A_90) = A_90) # label(fact_2113_mult__1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4731 (all A_98 all C_40 all B_80 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_98),C_40)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_80),C_40))) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_98),B_80)))) # label(fact_2045_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4732 (all B_1_1 all B_2_1 is_bool(hAPP_f54304608l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4733 (all A all B all N_1 (N_1 != zero_zero_nat -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A),N_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B),N_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B))))) # label(fact_2275_pow__divides__eq__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4734 (all A_16 all B_12 (B_12 != zero_z126310315umeral -> hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_16),B_12)),one_on1645066479umeral) = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_12),A_16)),B_12))) # label(fact_2630_div__add__self1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4735 (all P all Q all R hAPP_r1250527377l_real(hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,P),Q),R) = hAPP_r1250527377l_real(P,hAPP_real_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R_011) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4736 (all A_1 all B_1 (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,image_nat_int(semiri1621563631at_int,A_1)),image_nat_int(semiri1621563631at_int,B_1))) <-> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A_1),B_1)))) # label(fact_4601_transfer__nat__int__set__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4737 (all Ta all B_1 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ta),B_1)) -> (all X_2 (is_int(X_2) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B_1)) -> X_2 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa))))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ta),X_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ta),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_2),D_1)))))))))) # label(fact_5100_bset_I7_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4738 (all A_220 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,zero_z126310315umeral),A_220) = A_220) # label(fact_597_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4739 (all W hAPP_int_int(ring_1_of_int_int,W) = hAPP_int_int(number_number_of_int,W)) # label(fact_4583_int__number__of__def) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4740 (all Y all X (hBOOL(hAPP_int_bool(even_odd_even_int,X)) -> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y))))) # label(fact_2884_even__times__anything) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4741 (all P all Q all R hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,P),Q),R) = hAPP_bool_bool(P,hAPP_nat_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4742 (all N_27 all Y_7 all X_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_7),X_10)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_10),N_27)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_7),N_27)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X_10),Y_7)),N_27))) # label(fact_2584_div__power) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4743 (all N_1 all X_1 hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,power_power_real),N_1),X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_nat_nat(suc,zero_zero_nat))))))) # label(fact_4781_DERIV__pow) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4744 (all W hAPP_int_complex(number528085621omplex,W) != ii) # label(fact_3977_complex__i__not__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4745 (all Ma all N_1 (Ma = N_1 <-> hAPP_n522471361de_int(semiri1424489471de_int,N_1) = hAPP_n522471361de_int(semiri1424489471de_int,Ma))) # label(fact_2828_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4746 (all W_8 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,W_8)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,W_8)),hAPP_int_nat(number_number_of_nat,W_8))) # label(fact_363_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4747 (all X_1 all N_1 (is_int(X_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(abs_abs_int,X_1)),N_1))) <-> X_1 != zero_zero_int | N_1 = zero_zero_nat))) # label(fact_3447_zero__less__zpower__abs__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4748 (all B_45 all B_44 all A_54 all C_24 all A_53 (hAPP_nat_nat(div_mod_nat(A_53),C_24) = hAPP_nat_nat(div_mod_nat(A_54),C_24) -> (hAPP_nat_nat(div_mod_nat(B_44),C_24) = hAPP_nat_nat(div_mod_nat(B_45),C_24) -> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_54),B_45)),C_24) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_53),B_44)),C_24)))) # label(fact_2402_mod__mult__cong) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4749 (all P all Q all R hAPP_i1948725293t_bool(hAPP_f351634443t_bool(hAPP_f1576199059t_bool(cOMBB_95956018ol_int,P),Q),R) = hAPP_f429384717t_bool(P,hAPP_i627165055t_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__RealDef__Oreal_J_000tc__fun_) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4750 (all P (hBOOL(P) | hBOOL(hAPP_bool_bool(fNot,P)))) # label(help_fNot_2_1_U) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4751 (all A all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),Ma)) -> hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,hAPP_i1948725293t_bool(bnorRset(A),Ma)),rsetR(Ma))))) # label(fact_4596_Bnor__in__RsetR) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4752 (all Z_1 all X all D_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Z_1))),one_one_int)),D_3)))))) # label(fact_3465_incr__lemma) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4753 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X)),natceiling(Y))))) # label(fact_3391_natceiling__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4754 (all N_1 all Ma (is_int(Ma) & is_int(N_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),N_1) <-> one_one_int = N_1 & one_one_int = Ma)))) # label(fact_245_pos__zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4755 (all X_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,X_1)),one_one_rat)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),hAPP_int_int(bit1,pls))))) # label(fact_120_le__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4756 (all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(semiri984289939at_nat,N_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)))) # label(fact_3146_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4757 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(fequal_real,X),Y)) | X != Y)) # label(help_fequal_2_1_fequal_000tc__RealDef__Oreal_T) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4758 (all A_236 (is_int(A_236) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_236),hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,pls))) = A_236)) # label(fact_239_mult__numeral__1__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4759 (all A_71 all B_60 all B_59 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_71),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_60),B_59)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_71),B_60)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_71),B_59))) # label(fact_2305_mult_Odiff__right) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4760 (all C_1 all A_3 all B_2 (A_3 = B_2 -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),C_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),C_1))))) # label(fact_1709_dvd_Oord__eq__le__trans) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4761 (all A_111 all B_89 all C_49 (hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_111),B_89) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_111),C_49) -> B_89 = C_49)) # label(fact_1920_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4762 (all A all B (zero_z126310315umeral = hAPP_C498520661umeral(div_mo1740067990umeral(B),A) <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A),B)))) # label(fact_2516_dvd__eq__mod__eq__0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4763 (all K_2 hAPP_int_int(pred,hAPP_int_int(bit1,K_2)) = hAPP_int_int(bit0,K_2)) # label(fact_4040_pred__Bit1) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4764 (all B_116 all A_155 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_155)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_116)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_155),B_116)))))) # label(fact_1189_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4765 (all X all N hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(real_int,X)),N) = hAPP_int_real(real_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N))) # label(fact_4390_power__real__of__int) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4766 (all A_211 all M_26 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_211),one_one_real)),M_26) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_211),M_26)),M_26)) # label(fact_707_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4767 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hAPP_real_real(arcsin,hAPP_real_real(sin,X)) = X))) # label(fact_3740_arcsin__sin) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4768 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hAPP_P1175774780nt_int(product_snd_int_int,negDivAlg(A_3,B_2)) = hAPP_int_int(div_mod_int(A_3),B_2)))) # label(fact_4234_mod__neg__pos) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4769 (all F all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),B)) -> ((all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),B)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),X_2)) -> hBOOL(hAPP_real_bool(isCont_real_real(F),X_2)))) -> ((all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A),X_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),B)) -> hBOOL(hAPP_real_bool(deriv_real(F,X_2),zero_zero_real)))) -> hAPP_real_real(F,A) = hAPP_real_real(F,B))))) # label(fact_5178_DERIV__isconst__end) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4770 (all X_1 all A_1 all Ma (hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,A_1),rsetR(Ma))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (hAPP_int_int(legacy_zgcd(X_1),Ma) = one_one_int -> hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,image_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,times_times_int),X_1),A_1)),rsetR(Ma))))))) # label(fact_4788_RsetR__zmult__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4771 (all N_19 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),hAPP_nat_nat(suc,N_19)) = zero_zero_nat) # label(fact_2996_power__0__Suc) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4772 (all Ta all A_1 all D_1 (is_int(Ta) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ta),A_1)) -> (all X_2 (is_int(X_2) -> ((all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A_1)) -> X_2 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa))))))) -> (X_2 != Ta -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_2),D_1) != Ta)))))))) # label(fact_5097_aset_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4773 (all A_111 all B_89 all C_49 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_111),C_49) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_111),B_89) -> B_89 = C_49)) # label(fact_1917_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4774 (all M_10 all N_24 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_10)),hAPP_nat_int(semiri1621563631at_int,N_24))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_24)))) # label(fact_2934_of__nat__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4775 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real)) -> (exists X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),pi)) & (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),pi)) & Y = hAPP_real_real(cos,Y_1) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_1)) -> Y_1 = X_2)) & Y = hAPP_real_real(cos,X_2)))))) # label(fact_3592_cos__total) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4776 (all N_1 all Ma (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,N_1)),one_one_real)),hAPP_nat_real(real_nat,Ma))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),Ma)))) # label(fact_3685_nat__less__real__le) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4777 (all Z_1 all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),W)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z_1)))) # label(fact_55_zle__linear) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4778 (all X_40 all N_69 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_40),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_40),N_69)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_40),N_69))) # label(fact_427_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4779 (all N zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,zero_zero_nat),N)) # label(fact_1619_mult__0) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4780 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N))),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) = divmod_nat(M,N)))) # label(fact_4206_divmod__nat__step) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4781 (all C_69 all A_163 all B_124 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_163),B_124)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_69)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_69),A_163)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_69),B_124)))))) # label(fact_1141_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4782 (all N (-hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) -> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3790_lemma__not__even__div2) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4783 (all Ma (Ma != zero_zero_nat -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,dvd_dvd_nat),Ma)))))) # label(fact_4754_finite__divisors__nat) # label(axiom) # label(non_clause). [assumption]. 7.56/7.62 4784 (all M_6 all N_20 hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(semiri984289939at_nat,M_6)),N_20)) # label(fact_2982_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4785 (all P all Q all R hAPP_r1195171167l_real(P,hAPP_real_real(Q,R)) = hAPP_r1195171167l_real(hAPP_f617735571l_real(hAPP_f800361423l_real(cOMBB_2107848501l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__prod_Itc__RealDef__Oreal_Mtc__) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4786 (all P all Q all R hAPP_P1853363688nt_rat(hAPP_f1065357839nt_rat(hAPP_f1120137083nt_rat(cOMBS_549334880nt_rat,P),Q),R) = hAPP_int_rat(hAPP_P507092031nt_rat(P,R),hAPP_P1175774780nt_int(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin_002) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4787 (all U_1 all V_2 all X all Y (Y = X -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,U_1)),V_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),U_1)),Y))),V_2))))) # label(fact_3396_sin__bound__lemma) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4788 (all M all K_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2)),hAPP_nat_nat(suc,J_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2))))) # label(fact_3137_diff__Suc__diff__eq1) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4789 (all A_25 all B_20 (is_int(A_25) & is_int(B_20) -> (B_20 != zero_zero_int -> A_25 = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_25),B_20)),B_20)))) # label(fact_2545_div__mult__self2__is__id) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4790 (all K_2 all L_2 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(bit0,K_2)),hAPP_int_int(bit0,L_2)) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),L_2))) # label(fact_1521_diff__bin__simps_I7_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4791 (all C_43 all A_101 all B_83 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_101),B_83)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_43),A_101)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_43),B_83))))) # label(fact_2014_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4792 (all A all C all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B),C))))) # label(fact_2031_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4793 (all P all Q all R hAPP_i1948725293t_bool(hAPP_f1048215610t_bool(hAPP_f472159229t_bool(cOMBC_1683390479t_bool,P),Q),R) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool__015) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4794 (all B_96 all A_118 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_118)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_96)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_118),B_96)))))) # label(fact_1866_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4795 (all A_66 all B_54 A_66 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_66),B_54)),B_54)),hAPP_nat_nat(div_mod_nat(A_66),B_54))) # label(fact_2347_mod__div__equality) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4796 (all M_9 all N_23 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_23)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_9)),hAPP_nat_real(semiri132038758t_real,N_23))))) # label(fact_2938_less__imp__of__nat__less) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4797 (all Nat hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(nat_size,Nat)),hAPP_nat_nat(suc,zero_zero_nat)) = hAPP_nat_nat(nat_size,hAPP_nat_nat(suc,Nat))) # label(fact_4056_nat_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4798 (all X_44 all Y_32 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_44),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_32),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_32)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_44),Y_32))))) # label(fact_219_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4799 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> N = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))) # label(fact_3179_Suc__pred_H) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4800 (all X_45 all Y_33 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_45),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_33),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),Y_33)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_45),Y_33))))) # label(fact_212_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4801 (all X_1 all Y_2 (is_int(X_1) & is_int(Y_2) -> (X_1 = zero_zero_int & Y_2 = zero_zero_int <-> zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_1),X_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_2),Y_2))))) # label(fact_180_sum__squares__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4802 (all A_218 all B_161 all C_90 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_218),C_90)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_161),C_90)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_218),B_161)),C_90)) # label(fact_634_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4803 (all X all Y Y = hAPP_real_real(hAPP_r1250527377l_real(hAPP_b646336293l_real(if_real,fFalse),X),Y)) # label(help_If_2_1_If_000tc__RealDef__Oreal_T) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4804 (all B_1_1 all B_2_1 is_int(big_co1548731110nt_int(B_1_1,B_2_1))) # label(gsy_c_Big__Operators_Ocomm__monoid__mult__class_Osetprod_000tc__Int__Oint_000tc_) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4805 (all C_43 all A_101 all B_83 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_101),B_83)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_43),A_101)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,C_43),B_83))))) # label(fact_2010_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4806 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real)) -> zero_zero_nat = natceiling(X))) # label(fact_3399_natceiling__neg) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4807 (all X_24 all N_32 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32)) | one_one_complex = X_24 -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_24),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_24),N_32))))) # label(fact_1493_dvd__power) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4808 (all N all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,M),N)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(fact_fact_int,M)),hAPP_int_int(fact_fact_int,N)))))) # label(fact_4274_fact__less__mono__int) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4809 (all Lx_4 all Ly_2 all Rx_4 all Ry_2 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_4),Ly_2)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_4),Ry_2)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_4),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Ly_2),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_4),Ry_2)))) # label(fact_481_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4810 (all A_220 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,zero_z891286103de_int),A_220) = A_220) # label(fact_602_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4811 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),X))),hAPP_real_real(uminus_uminus_real,X)))))) # label(fact_3498_ln__one__minus__pos__upper__bound) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4812 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arcsin,Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) & hAPP_real_real(sin,hAPP_real_real(arcsin,Y)) = Y))) # label(fact_3745_arcsin) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4813 (all P (is_bool(P) -> P = fFalse | P = fTrue)) # label(help_If_3_1_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_T) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4814 (all N all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(nat,X)),N) = hAPP_int_nat(nat,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N)))) # label(fact_2747_Nat__Transfer_Otransfer__nat__int__functions_I4_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4815 (all W_1 all X_1 (X_1 = hAPP_i769753017umeral(number1443263063umeral,W_1) <-> hAPP_i769753017umeral(number1443263063umeral,W_1) = X_1)) # label(fact_173_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4816 (all Q_2 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(prime,Q_2)) -> (P_3 != Q_2 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,P_3),Q_2)))))) # label(fact_4166_distinct__prime__coprime) # label(axiom) # label(non_clause). [assumption]. 7.65/7.62 4817 (all X_1 all Ma (hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_1),Ma) <-> zero_zero_nat = Ma | X_1 = hAPP_nat_nat(suc,zero_zero_nat))) # label(fact_3080_nat__power__eq__Suc__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4818 (all Ma all K_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),hAPP_nat_nat(div_mod_nat(Ma),N_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),Ma))))) # label(fact_2444_dvd__mod__iff) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4819 (all B_129 all A_169 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_169)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_129),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_169),B_129)),zero_zero_rat))))) # label(fact_1109_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4820 (all Z_3 all Z_2 (is_int(Z_2) & is_int(Z_3) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_3)) -> (Z_2 = Z_3 <-> hAPP_int_nat(nat,Z_3) = hAPP_int_nat(nat,Z_2)))))) # label(fact_2710_eq__nat__nat__iff) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4821 (all M all X (is_int(X) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),M)) -> X = zero_zero_int))))) # label(fact_1458_Int2_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4822 (all K_1 all F (hAPP_nat_nat(F,zero_zero_nat) = zero_zero_nat -> hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_nat_nat(suc,zero_zero_nat)),K_1)) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),K_1)))) # label(fact_5081_setsum__shift__lb__Suc0__0) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4823 (all A all B ((all D_2 (is_int(D_2) -> (D_2 = one_one_int <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_2),B)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_2),A)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),D_2))))) <-> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A),B) = one_one_int)) # label(fact_5019_coprime__int) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4824 (all N_11 all A_7 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_7)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_7),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_7),hAPP_nat_nat(suc,N_11))),one_one_real))))) # label(fact_3162_power__Suc__less__one) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4825 (all A (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),zero_zero_rat)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat)))) # label(fact_308_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4826 (all Y_2 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,X_1)),hAPP_real_real(ln,Y_2))))))) # label(fact_3438_ln__le__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4827 (all A_46 all B_37 all C_18 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_46),B_37)),C_18) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_46),hAPP_nat_nat(div_mod_nat(B_37),C_18))),C_18)) # label(fact_2426_mod__add__right__eq) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4828 (all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),Ma)) <-> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1)))) # label(fact_2870_not__less__eq__eq) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4829 (all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W))) -> hAPP_P1175774780nt_int(product_fst_int_int,posDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,one_one_int),hAPP_int_int(number_number_of_int,W)))) # label(fact_4227_div__pos__pos__1__number__of) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4830 (all A_238 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_238),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_216_zero__le__power2) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4831 (all N_25 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(semiri132038758t_real,N_25)))) # label(fact_2913_of__nat__0__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4832 (all Ma all N_1 all K_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N_1))))) # label(fact_1840_nat__mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4833 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,X)),hAPP_int_real(real_int,Y)) = hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y))) # label(fact_4393_real__of__int__add) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4834 (all K_1 all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K_1)),hAPP_int_int(bit0,L))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K_1),L)))) # label(fact_261_rel__simps_I14_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4835 (all X hAPP_real_real(uminus_uminus_real,hAPP_int_real(real_int,X)) = hAPP_int_real(real_int,hAPP_int_int(uminus_uminus_int,X))) # label(fact_4399_real__of__int__minus) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4836 (all B_1_1 is_bool(cauchy_real(B_1_1))) # label(gsy_c_SEQ_OCauchy_000tc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4837 (all Z_2 all W_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),W_1)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(nat,W_1)),hAPP_int_nat(nat,Z_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z_2))))) # label(fact_2740_nat__le__eq__zle) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4838 (all P_3 all R_3 all S_1 all Q_2 (is_int(S_1) & is_int(Q_2) -> (zero_zero_int != Q_2 -> (zero_zero_int != S_1 -> (normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_2)) = normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,R_3),S_1)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_3),Q_2) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,P_3),S_1)))))) # label(fact_4662_normalize__crossproduct) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4839 (all Ta all D all D_1 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D),D_1)) -> (all X_2 all K (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_2),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,K),D_1))),Ta))) <-> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_2),Ta))))))) # label(fact_1529_inf__period_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4840 (all X_1 (-hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)) <-> (exists Y_1 X_1 = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),Y_1))))) # label(fact_3769_odd__nat__equiv__def2) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4841 (all Z_12 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Z_12),hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Z_12),Z_12)) # label(fact_294_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4842 (all A_103 all C_45 all B_85 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_103),C_45)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_85),C_45))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_103),B_85)))) # label(fact_2000_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4843 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(d22set,A)))))) # label(fact_2268_d22set__mem) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4844 (all A_3 all B_2 hAPP_int_int(uminus_uminus_int,hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,A_3)),B_2)) = hAPP_int_int(div_mod_int(A_3),hAPP_int_int(uminus_uminus_int,B_2))) # label(fact_3524_zmod__zminus2) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4845 (all A_218 all B_161 all C_90 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_218),C_90)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_161),C_90)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_218),B_161)),C_90)) # label(fact_633_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4846 (all A_3 all B_2 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),one_one_int))) # label(fact_1030_zcong__1) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4847 (all X all Y hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)),Y))) # label(fact_4961_gcd__dvd2__int) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4848 (all P_1 ((exists X1 hBOOL(hAPP_nat_bool(P_1,X1))) <-> (exists X_2 (is_int(X_2) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) & hBOOL(hAPP_nat_bool(P_1,hAPP_int_nat(nat,X_2))))))) # label(fact_2709_ex__nat) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4849 (all N_1 all N_4 ((all X_2 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),N_4)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),N_1)))) -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_4)))) # label(fact_4627_bounded__nat__set__is__finite) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4850 (all K_5 all M_15 all N_28 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_5),hAPP_C498520661umeral(div_mo1740067990umeral(M_15),N_28))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_5),N_28)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,K_5),M_15))))) # label(fact_2453_dvd__mod__imp__dvd) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4851 (all X_9 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_9)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,X_9))),hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,X_9))) = hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_9))))) # label(fact_2809_of__nat__double) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4852 (all W hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,W)),hAPP_int_nat(number_number_of_nat,W)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,W))) # label(fact_4747_nat__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4853 (all I all K_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K_2)) -> (hAPP_nat_nat(suc,I) != K_2 -> -(all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_1)) -> K_2 != hAPP_nat_nat(suc,J_1)))))) # label(fact_3303_lessE) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4854 (all P all Q (hBOOL(P) | hBOOL(Q) | -hBOOL(fdisj(P,Q)))) # label(help_fdisj_3_1_U) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4855 (all A_70 all B_58 hAPP_int_int(div_mod_int(A_70),B_58) = hAPP_int_int(div_mod_int(hAPP_int_int(div_mod_int(A_70),B_58)),B_58)) # label(fact_2325_mod__mod__trivial) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4856 (all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,Y_2))))) # label(fact_3325_real__sqrt__ge__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4857 (all N hAPP_nat_real(real_nat,N) = hAPP_int_real(real_int,hAPP_nat_int(semiri1621563631at_int,N))) # label(fact_4392_real__of__int__of__nat__eq) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4858 (all X_1 all Y_2 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) <-> zero_zero_rat != X_1 | Y_2 != zero_zero_rat)) # label(fact_12_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4859 (all B_1_1 all B_2_1 (is_bool(B_1_1) & is_bool(B_2_1) -> is_bool(fdisj(B_1_1,B_2_1)))) # label(gsy_c_fdisj) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4860 (all A_190 all B_146 all K_7 (hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_146),K_7) = A_190 -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,B_146),A_190)))) # label(fact_942_dvdI) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4861 (all X_1 (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),X_1))) <-> zero_zero_real = X_1)) # label(fact_1509_not__real__square__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4862 (all B all A all C (C = B <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),A) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B),A))) # label(fact_1931_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4863 (all N_9 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,N_9)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_9)))) # label(fact_3214_of__nat__less__two__power) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4864 (all A_197 all B_152 all N_64 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_197),B_152)),N_64) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_197),N_64)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_152),N_64))) # label(fact_893_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4865 (all C_89 all D_17 all A_213 all B_159 all R_8 (zero_zero_nat != R_8 -> (A_213 = B_159 & D_17 != C_89 -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_159),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,R_8),D_17)) != hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_213),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,R_8),C_89))))) # label(fact_687_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4866 (all A_36 hAPP_int_int(hAPP_int_fun_int_int(div_div_int,zero_zero_int),A_36) = zero_zero_int) # label(fact_2474_div__0) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4867 (all Z_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Z_2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_2)))) # label(fact_153_int__one__le__iff__zero__less) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4868 (all A_3 all N all B_2 hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),N)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)),N)) # label(fact_4976_gcd__exp__int) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4869 (all B_45 all B_44 all A_54 all C_24 all A_53 (hAPP_C498520661umeral(div_mo1740067990umeral(A_54),C_24) = hAPP_C498520661umeral(div_mo1740067990umeral(A_53),C_24) -> (hAPP_C498520661umeral(div_mo1740067990umeral(B_44),C_24) = hAPP_C498520661umeral(div_mo1740067990umeral(B_45),C_24) -> hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_54),B_45)),C_24) = hAPP_C498520661umeral(div_mo1740067990umeral(hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_53),B_44)),C_24)))) # label(fact_2403_mod__mult__cong) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4870 (all A_46 all B_37 all C_18 hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_46),B_37)),C_18) = hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_46),hAPP_Q1762011733de_int(div_mo231679042de_int(B_37),C_18))),C_18)) # label(fact_2428_mod__add__right__eq) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4871 (all B_2 all A_3 all K_2 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))),hAPP_int_int(div_mod_int(A_3),B_2))),K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),K_2)) # label(fact_2360_zdiv__zmod__equality) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4872 (all Y all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3345_real__sqrt__sum__squares__ge2) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4873 (all M all N all K_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K_2))) # label(fact_1651_add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4874 (all A_90 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,one_one_rat),A_90) = A_90) # label(fact_2118_mult__1) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4875 (all A_35 hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_35),one_one_nat) = A_35) # label(fact_2479_div__by__1) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4876 (all K_1 all Xa_1 hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,sin),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,plus_plus_real),K_1)),Xa_1),hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa_1),K_1))))) # label(fact_4743_DERIV__sin__add) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4877 (all Qr_2 all M all N all Qr_1 (hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),Qr_1)) -> (hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),Qr_2)) -> Qr_2 = Qr_1))) # label(fact_4215_divmod__nat__rel__unique) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4878 (all N_61 all X_29 all Y_23 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,X_29),Y_23)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_29),N_61)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_23),N_61))))) # label(fact_984_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4879 (all B_121 all A_160 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_160),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_121)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_160),B_121)),zero_zero_int))))) # label(fact_1152_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4880 (all B all A all Ma (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B),hAPP_i1948725293t_bool(bnorRset(A),Ma))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)))) # label(fact_4295_Bnor__mem__zg) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4881 (all N all X (is_int(N) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N)),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N)),one_one_real))) -> archim1246769320r_real(X) = N)))) # label(fact_4453_floor__eq) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4882 (all A all W_1 (hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A),hAPP_int_nat(number_number_of_nat,W_1)) = zero_z891286103de_int <-> hAPP_int_nat(number_number_of_nat,W_1) != zero_zero_nat & A = zero_z891286103de_int)) # label(fact_717_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4883 (all A_3 all X hAPP_real_real(log(A_3),X) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,X)),hAPP_real_real(ln,A_3))) # label(fact_3864_Log_Olog__def) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4884 (all N all M hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M)))) # label(fact_1640_le__add1) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4885 (all X_17 zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_17),zero_zero_real)) # label(fact_2289_mult__right_Ozero) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4886 (all Y all X all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,Y),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3))))))) # label(fact_4406_powr__less__mono2__neg) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4887 (all N_1 all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),Ma)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N_1)),hAPP_nat_real(real_nat,Ma))))) # label(fact_3650_real__of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4888 (all X all Y all M (hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X),Y)),M) = one_one_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),M) = one_one_nat)) # label(fact_4371_invertible__coprime__nat) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4889 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X_1)),one_one_real)) -> hBOOL(monoseq_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))))) # label(fact_4805_monoseq__arctan__series) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4890 (all B_1_1 all B_2_1 is_bool(hAPP_P1333854989l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__HOL__Obo) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4891 (all A_10 all N_15 all B_8 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_10),hAPP_nat_nat(suc,N_15)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_8),hAPP_nat_nat(suc,N_15)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_10)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_8)) -> B_8 = A_10)))) # label(fact_3107_power__inject__base) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4892 (all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(suc,zero_zero_nat)),N) = hAPP_nat_nat(suc,zero_zero_nat)) # label(fact_3081_power__Suc__0) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4893 (all C_78 all D_14 all A_178 all B_138 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_178),B_138)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_78),D_14)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_178)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_78)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_178),C_78)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_138),D_14)))))))) # label(fact_1061_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4894 (all A_221 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_221),zero_zero_rat) = zero_zero_rat) # label(fact_596_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4895 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y)))))) # label(fact_722_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4896 (all Z_1 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Z_1)))) # label(fact_2986_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4897 (all J all I_1 all P_1 all P_5 ((all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),P_5)) -> (hBOOL(hAPP_nat_bool(P_1,I_2)) -> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,I_2)),P_5)))))) -> (hBOOL(hAPP_nat_bool(P_1,I_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),P_5)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J),P_5)) -> hBOOL(hAPP_nat_bool(P_1,J))))))) # label(fact_3252_mod__induct) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4898 (all N_19 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),hAPP_nat_nat(suc,N_19)) = zero_z126310315umeral) # label(fact_2994_power__0__Suc) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4899 (all F all K_3 all N_1 ((all P_4 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,P_4),N_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,P_4)),K_3)))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),K_3))))) # label(fact_4942_real__setsum__nat__ivl__bounded) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4900 (all C_1 all D_3 all A_3 all B_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C_1),D_3),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),C_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),D_3)),M))))) # label(fact_1036_zcong__zmult) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4901 (all R_2 hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(plus_plus_real,R_2)),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,uminus_uminus_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),suc)))),R_2,sequentially))) # label(fact_5046_LIMSEQ__inverse__real__of__nat__add__minus) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4902 (all M all N ((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> hAPP_nat_nat(div_mod_nat(M),N) = M) & (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N) = hAPP_nat_nat(div_mod_nat(M),N)))) # label(fact_2712_mod__if) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4903 (all P_1 all N_1 all K_1 (is_int(N_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1)) -> ((all I_2 all J_1 (is_int(I_2) & is_int(J_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),J_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,J_1),K_1)) & hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),I_2)),J_1) = N_1 -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,I_2),J_1))))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N_1),K_1)),hAPP_int_int(div_mod_int(N_1),K_1))))))) # label(fact_2586_split__pos__lemma) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4904 (all A_3 all B_2 all C_1 hAPP_nat_nat(div_mod_nat(A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),C_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),B_2)),C_1))),hAPP_nat_nat(div_mod_nat(A_3),B_2))) # label(fact_2684_mod__mult2__eq) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4905 (all P_5 (is_int(P_5) -> (hBOOL(hAPP_int_bool(zprime,P_5)) <-> (all M_1 (is_int(M_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M_1),P_5)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M_1)) -> M_1 = P_5 | one_one_int = M_1))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),P_5))))) # label(fact_741_zprime__def) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4906 (all A_140 all C_62 all B_109 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_140),C_62)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_109),C_62))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_62)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_140),B_109))))) # label(fact_1347_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4907 (all V_14 all B_166 all C_97 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_14)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_166),C_97)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_14)),B_166)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_14)),C_97))) # label(fact_194_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4908 (all C_38 all A_96 all B_78 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_96),B_78)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_38),A_96)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_38),B_78))))) # label(fact_2057_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4909 (all A_195 all B_151 hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_195),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_195),B_151)))) # label(fact_908_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4910 (all A all C all D all B (is_int(B) & is_int(D) -> (B != zero_zero_int -> (zero_zero_int != D -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C),D))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),D)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),D))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B),D))))))))) # label(fact_4713_less__rat) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4911 (all M all N ((hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) != zero_zero_nat -> hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),one_one_nat)))) & (zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) -> hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) = one_one_nat))) # label(fact_3800_fact__add__num__eq__if__nat) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4912 (all A_3 all C_1 all D_3 all B_2 (is_int(D_3) & is_int(B_2) -> (zero_zero_int != B_2 -> (zero_zero_int != D_3 -> hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C_1),D_3)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),D_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),D_3)))))) # label(fact_4720_diff__rat) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4913 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (exists N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_2)))),X))))) # label(fact_3859_reals__Archimedean) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4914 (all X_1 all Y_2 (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_1),Y_2)) -> (X_1 != Y_2 -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,X_1),Y_2))))) # label(fact_818_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4915 (all A hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(d22set,A)))) # label(fact_3359_d22set__fin) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4916 (all A_207 hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_207),A_207))) # label(fact_795_dvd__refl) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4917 (all X_1 (X_1 = one_one_real <-> X_1 = one_one_real)) # label(fact_1952_one__reorient) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4918 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),M)) -> M = N))) # label(fact_1711_dvd__antisym) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4919 (all N N = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),N)) # label(fact_1611_plus__nat_Oadd__0) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4920 (all P all Q all R hAPP_r1250527377l_real(hAPP_r337325687l_real(hAPP_f1644353557l_real(cOMBC_1508019322l_real,P),Q),R) = hAPP_r1250527377l_real(hAPP_r337325687l_real(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__fun_Itc_) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4921 (all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(fact_fact_nat,N)))) # label(fact_3806_fact__gt__zero__nat) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4922 (all X_41 all Y_29 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_41),Y_29)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_41),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_29),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_41)),Y_29))) # label(fact_369_power2__sum) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4923 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sqrt,X)),hAPP_real_real(sqrt,Y)) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y))) # label(fact_3318_real__sqrt__mult) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4924 (all M all K_2 all N (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K_2),N) -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),N) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N))) # label(fact_4999_gcd__mult__cancel__int) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4925 (all A_3 all B_2 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(uminus_uminus_real,B_2)),A_3)) # label(fact_3983_i__mult__Complex) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4926 (all X_1 all Y_2 (zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X_1),Y_2) <-> zero_zero_nat = X_1 & zero_zero_nat = Y_2)) # label(fact_4341_gcd__lcm__complete__lattice__nat_Oinf__eq__top__iff) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4927 (all X_1 all Y_2 (zero_zero_real != X_1 | zero_zero_real != Y_2 <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_11_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4928 (all N all M code_d418564891umeral(N,M) = hAPP_C2132160860umeral(hAPP_C1614127039umeral(produc2136830103umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,N),M)),hAPP_C498520661umeral(div_mo1740067990umeral(N),M))) # label(fact_4198_div__mod__code__numeral__def) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4929 (all C_58 all D_11 all A_136 all B_105 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_136),B_105)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_58),D_11)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_136)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_58)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_136),C_58)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_105),D_11)))))))) # label(fact_1367_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4930 (all B_129 all A_169 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_169)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_129),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_169),B_129)),zero_zero_int))))) # label(fact_1104_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4931 (all F all X_1 all F_1 all N_1 all Ma ((all R_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),R_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,R_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),N_1))) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(F,R_1),X_1),hAPP_real_real(hAPP_n546711566l_real(F_1,R_1),X_1))))) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_f2119584768l_real(hAPP_f59594445l_real(cOMBC_2010481033l_real,hAPP_f1523861863l_real(hAPP_f1067681227l_real(cOMBB_1934313333l_real,big_co604158596t_real),hAPP_f42270917t_real(cOMBC_nat_real_real,F))),hAPP_n1699378549t_bool(ord_at4362885an_nat(Ma),N_1)),X_1),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,F_1),X_1)),hAPP_n1699378549t_bool(ord_at4362885an_nat(Ma),N_1)))))) # label(fact_4943_DERIV__sumr) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4932 (all X all Y all Z_1 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Y)),Z_1) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Y),Z_1))) # label(fact_453_zpower__zpower) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4933 (all A all C all B (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B),C))))) # label(fact_2075_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4934 (all P all Q all R hAPP_i345030060t_bool(hAPP_f725770969t_bool(hAPP_f983569339t_bool(cOMBB_550562460ol_int,P),Q),R) = hAPP_f1791153283t_bool(P,hAPP_int_fun_int_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc___018) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4935 (all K_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),K_2))))) # label(fact_1654_mult__le__mono1) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4936 (all N_62 one_one_int = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,one_one_int),N_62)) # label(fact_969_power__one) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4937 (all X_3 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (hBOOL(hAPP_f448129468l_bool(resSet(Ma),X_3)) -> hBOOL(hAPP_f448129468l_bool(finite_finite_int,X_3))))) # label(fact_4053_ResSet__finite) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4938 (all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_int_real(number267125858f_real,Y_2))))) # label(fact_397_less__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4939 (all N all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(suc,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_nat_nat(suc,N))) = X)) # label(fact_3928_real__root__pos) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4940 (all X all Y hAPP_int_int(nat_tsub(hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X),Y))) # label(fact_2820_Nat__Transfer_Otransfer__int__nat__functions_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4941 (all P_3 all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2)) -> -(hBOOL(hAPP_nat_bool(prime,P_3)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),B_2)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),A_3))))) # label(fact_4177_coprime__prime) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4942 (all A all D all N_1 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_f659380387ol_int(hAPP_f1599440987ol_int(big_co1024481617at_int,hAPP_f1139079189at_int(hAPP_f1431025877at_int(cOMBB_int_int_nat,hAPP_int_fun_int_int(plus_plus_int,A)),hAPP_int_fun_nat_int(hAPP_f1673103573at_int(cOMBC_nat_int_int,hAPP_f1935805932nt_int(hAPP_f125788821nt_int(cOMBB_859312247nt_nat,times_times_int),semiri1621563631at_int)),D))),hAPP_n1699378549t_bool(ord_lessThan_nat,N_1))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,N_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat))),D))))) # label(fact_4947_arith__series__int) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4943 (all A_186 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A_186))) # label(fact_982_one__dvd) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4944 (all N_1 all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),Ma))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)))) # label(fact_1760_zero__less__diff) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4945 (all A_224 all N_68 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_224),N_68)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_224),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_68))) # label(fact_575_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4946 (all M_13 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),hAPP_nat_complex(semiri2020571505omplex,M_13)) = hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(suc,M_13))) # label(fact_2842_of__nat__Suc) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4947 (all K_2 ((K_2 = zero_z126310315umeral -> zero_zero_int = code_int_of(K_2)) & (K_2 != zero_z126310315umeral -> (zero_z126310315umeral != hAPP_C498520661umeral(div_mo1740067990umeral(K_2),hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) -> code_int_of(K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),code_int_of(hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,K_2),hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),one_one_int)) & (hAPP_C498520661umeral(div_mo1740067990umeral(K_2),hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_z126310315umeral -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),code_int_of(hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,K_2),hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = code_int_of(K_2))))) # label(fact_2806_Code__Numeral_Oint__of__code) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4948 (all A_29 all B_24 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_29),B_24)) -> zero_z891286103de_int = hAPP_Q1762011733de_int(div_mo231679042de_int(B_24),A_29))) # label(fact_2513_dvd__imp__mod__0) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4949 (all E_1 all B_2 all D_3 all A_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),A_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,E_1),B_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),E_1)))))) # label(fact_4017_coprime__divisors) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4950 (all P all Q all R hAPP_f2119767738t_bool(P,hAPP_r632163725t_bool(Q,R)) = hAPP_r481142414t_bool(hAPP_f1625314454t_bool(hAPP_f340152631t_bool(cOMBB_1453147152l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__HOL_) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4951 (all M all N all K_2 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N)))) # label(fact_4300_gcd__dvd__prod__nat) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4952 (all F all X_1 all L (hBOOL(hAPP_real_bool(deriv_real(F,X_1),L)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),L)) -> (exists D_2 ((all H_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H_1),D_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),H_1))),hAPP_real_real(F,X_1)))))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))))))) # label(fact_3887_DERIV__pos__inc__left) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4953 (all X_3 all A (hBOOL(hAPP_complex_bool(sums_complex(X_3),A)) -> hBOOL(hAPP_real_bool(sums_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_3)),hAPP_complex_real(re,A))))) # label(fact_4872_Re_Osums) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4954 (all F all X_1 all L (hBOOL(hAPP_real_bool(deriv_real(F,X_1),L)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,L),zero_zero_real)) -> (exists D_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2)) & (all H_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H_1),D_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),H_1))),hAPP_real_real(F,X_1))))))))))) # label(fact_3889_DERIV__neg__dec__right) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4955 (all V_8 all W_4 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(number267125858f_real,V_8)),hAPP_int_real(number267125858f_real,W_4)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_8),W_4))) # label(fact_1511_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4956 (all V_23 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_23)),one_one_int) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_23),hAPP_int_int(bit1,pls)))) # label(fact_44_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4957 (all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(fact_fact_int,X))))) # label(fact_4277_transfer__nat__int__factorial__closure) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4958 (all X_1 all Y_2 all Z_2 (hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,Y_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_2)) = X_1 <-> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_1) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,Y_2),Z_2))) # label(fact_3368_eq__divide__2__times__iff) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4959 (all B_118 all A_157 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_157)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_118),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_118),A_157)),zero_zero_nat))))) # label(fact_1174_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4960 (all C_69 all A_163 all B_124 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_163),B_124)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_69)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_69),A_163)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_69),B_124)))))) # label(fact_1142_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4961 (all X_38 all P_8 all Q_12 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_38),P_8)),Q_12) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_38),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_8),Q_12))) # label(fact_445_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4962 (all Lx_3 all Ly_1 all Rx_3 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_3),Ly_1)),Rx_3) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_3),Rx_3)),Ly_1)) # label(fact_487_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4963 (all C_89 all D_17 all A_213 all B_159 all R_8 (is_int(C_89) & is_int(R_8) & is_int(D_17) -> (zero_zero_int != R_8 -> (C_89 != D_17 & A_213 = B_159 -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_159),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_8),D_17)) != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_213),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_8),C_89)))))) # label(fact_686_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4964 (all B_155 all A_204 (zero_zero_nat != A_204 -> (B_155 != zero_zero_nat -> zero_zero_nat != hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_204),B_155)))) # label(fact_829_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4965 (all X hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(suc,zero_zero_nat)),X) = X) # label(fact_3896_real__root__Suc__0) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4966 (all Y all X (-hBOOL(hAPP_int_bool(even_odd_even_int,X)) -> (hBOOL(hAPP_int_bool(even_odd_even_int,Y)) -> -hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y)))))) # label(fact_2888_odd__plus__even) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4967 (all P all Q all R hAPP_f561022312t_bool(P,hAPP_i614188481l_bool(Q,R)) = hAPP_i1796122964t_bool(hAPP_f1902329768t_bool(hAPP_f1182713365t_bool(cOMBB_2087181719ol_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__HOL__Obool_Mtc__HOL) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4968 (all M_25 all A_185 all N_60 all B_142 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_185),N_60)),B_142)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_25),N_60)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_185),M_25)),B_142))))) # label(fact_990_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4969 (all A all B all C all D all X_1 (is_int(C) & is_int(B) & is_int(A) -> (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A),B)),C) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,D),hAPP_int_nat(nat,X_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_int_nat(nat,X_1))) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zOdd)) -> (one_one_int = hAPP_int_int(legacy_zgcd(A),B) -> (one_one_int = hAPP_int_int(legacy_zgcd(B),C) -> (one_one_int = hAPP_int_int(legacy_zgcd(C),A) -> (exists K exists L_1 exists M_1 (is_int(K) & B = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,L_1),hAPP_int_nat(nat,X_1)) & hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_1),hAPP_int_nat(nat,X_1)) = C & A = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K),hAPP_int_nat(nat,X_1)) & is_int(M_1) & is_int(L_1))))))))))) # label(fact_4536_int__triple__relprime__odd__power__divisors) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4970 (all P (is_bool(P) -> fTrue = P | P = fFalse)) # label(help_If_3_1_If_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_T) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4971 (all A all B all C (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A),B)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B)))))) # label(fact_1193_mult__less__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4972 (all N_33 all A_123 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_123)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_33)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_123),N_33)))))) # label(fact_1489_one__less__power) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4973 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y)))))) # label(fact_3508_arcsin__lbound) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4974 (all K_2 all L_2 hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(pred,K_2)),L_2)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(bit0,K_2)),hAPP_int_int(bit1,L_2))) # label(fact_4049_diff__bin__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4975 (all Z_13 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_13) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_13),Z_13)) # label(fact_288_mult__2) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4976 (all S (hBOOL(hAPP_f448129468l_bool(nat_nat_set,S)) <-> (all X_2 (is_int(X_2) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_2),S)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2))))))) # label(fact_4614_nat__set__def) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4977 (all X zero_zero_real != hAPP_real_real(cos,hAPP_real_real(arctan,X))) # label(fact_3564_cos__arctan__not__zero) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4978 (all N_61 all X_29 all Y_23 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_29),Y_23)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_29),N_61)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_23),N_61))))) # label(fact_986_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4979 (all C_72 all B_127 all A_166 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_127),A_166)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_72),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_72),A_166)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_72),B_127)))))) # label(fact_1126_mult__strict__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4980 (all B_92 all A_114 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_114),zero_z126310315umeral)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_92),zero_z126310315umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_114),B_92)),zero_z126310315umeral))))) # label(fact_1887_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4981 (all X all Y hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Y)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X),Y))) # label(fact_1806_diff__square) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4982 (all F all G ((all N_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_2)),hAPP_nat_real(G,N_2)))) -> (hBOOL(summable_real(F)) -> (hBOOL(summable_real(G)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f352196356l_real(suminf_real,F)),hAPP_f352196356l_real(suminf_real,G))))))) # label(fact_4881_summable__le) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4983 (all A_217 all M_29 all B_160 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_217),B_160)),M_29) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_217),M_29)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_160),M_29))) # label(fact_640_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4984 (all M all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M),N)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(fact_fact_int,M)),hAPP_int_int(fact_fact_int,N))))) # label(fact_4272_fact__mono__int) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4985 (all X_1 all Y_2 (hAPP_Q1762011733de_int(div_mo231679042de_int(hAPP_i1732201573de_int(number1226105091de_int,Y_2)),hAPP_i1732201573de_int(number1226105091de_int,X_1)) = zero_z891286103de_int <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_i1732201573de_int(number1226105091de_int,X_1)),hAPP_i1732201573de_int(number1226105091de_int,Y_2))))) # label(fact_2588_dvd__eq__mod__eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4986 (all P all Q all R hAPP_i1153444985nt_int(hAPP_f1148171384nt_int(hAPP_f334321603nt_int(cOMBB_1484610687nt_int,P),Q),R) = hAPP_P621635040nt_int(P,hAPP_i1524277240nt_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc_) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4987 (all X all I hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(real_int,I)),pi))) = hAPP_real_real(tan,X)) # label(fact_4454_tan__periodic__int) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4988 (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(exp_real,X_1))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)))) # label(fact_3844_one__less__exp__iff) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4989 (all A_3 all B_2 hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,A_3),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,B_2)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) = hAPP_complex_complex(invers1449016382omplex,hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2))) # label(fact_3967_complex__inverse) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4990 (all A_87 all B_75 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_87),B_75)),B_75) = A_87) # label(fact_2138_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4991 (all V_11 all W_9 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_11)),hAPP_int_int(number_number_of_int,W_9)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_11),W_9))) # label(fact_341_number__of__add) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4992 (all B_73 all A_85 all C_36 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_36)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_73),A_85)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_73),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_85),C_36)))))) # label(fact_2152_add__increasing2) # label(axiom) # label(non_clause). [assumption]. 7.65/7.63 4993 (all N_52 all A_150 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_150)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_150),N_52))))) # label(fact_1254_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 4994 (all X_37 (is_int(X_37) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_37),one_one_nat) = X_37)) # label(fact_564_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 4995 (all X_1 hBOOL(hAPP_real_bool(isCont_real_real(sqrt),X_1))) # label(fact_5163_isCont__real__sqrt) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 4996 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)) & (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N))))) # label(fact_3183_div__if) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 4997 (all N_52 all A_150 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_150)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_150),N_52))))) # label(fact_1255_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 4998 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sqrt,X))))) # label(fact_3326_real__sqrt__gt__zero) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 4999 (all X_1 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),zEven)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat,X_1)) = one_one_int))) # label(fact_2807_EvenOdd_Oneg__one__even__power) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5000 (all A_37 zero_z126310315umeral = hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,A_37),zero_z126310315umeral)) # label(fact_2472_div__by__0) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5001 (all P_3 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_P1175774780nt_int(product_snd_int_int,P_3))) -> normalize(P_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_P1175774780nt_int(product_snd_int_int,P_3)),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3))))) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_P1175774780nt_int(product_snd_int_int,P_3))) -> (zero_zero_int = hAPP_P1175774780nt_int(product_snd_int_int,P_3) -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int) = normalize(P_3)) & (zero_zero_int != hAPP_P1175774780nt_int(product_snd_int_int,P_3) -> normalize(P_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3))))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_P1175774780nt_int(product_snd_int_int,P_3)),hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3))))))))) # label(fact_4959_normalize__def) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5002 (all K_5 all M_15 all N_28 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_5),hAPP_nat_nat(div_mod_nat(M_15),N_28))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_5),N_28)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_5),M_15))))) # label(fact_2456_dvd__mod__imp__dvd) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5003 (all A_3 (is_int(A_3) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),zero_zero_int) = zero_zero_int & A_3 = hAPP_int_int(div_mod_int(A_3),zero_zero_int))) # label(fact_2330_DIVISION__BY__ZERO) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5004 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)),hAPP_int_int(number_number_of_int,min))) -> one_one_int != hAPP_int_int(legendre(hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)) # label(fact_1503__096_126_AQuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_A_061_061_062_ALegendre_A_) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5005 (all A_105 (is_int(A_105) -> A_105 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),A_105))) # label(fact_1986_add__0__left) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5006 (all Lx_2 all Ly all Rx_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_2),Ly)),Rx_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ly),Rx_2))) # label(fact_493_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5007 (all B_70 all A_82 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_82),zero_z891286103de_int)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_70),zero_z891286103de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_82),B_70)),zero_z891286103de_int))))) # label(fact_2181_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5008 (all P_2 all P_1 ((all X_2 (is_int(X_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) -> (hBOOL(hAPP_int_bool(P_1,X_2)) <-> hBOOL(hAPP_int_bool(P_2,X_2)))))) -> collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),P_1)) = collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),P_2)))) # label(fact_4958_transfer__nat__int__set__cong) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5009 (all W hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),W))) # label(fact_1736_real__le__refl) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5010 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,X)),hAPP_int_nat(nat,Y)) = hAPP_int_nat(nat,hAPP_int_int(nat_tsub(X),Y))))) # label(fact_2803_Nat__Transfer_Otransfer__nat__int__functions_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5011 (all U all Ma all N_1 all I_1 all J (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),I_1)),U)),N_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N_1)))))) # label(fact_1851_nat__le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5012 (all A_237 A_237 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(bit1,pls))),A_237)) # label(fact_237_mult__numeral__1) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5013 (all K_2 hAPP_int_real(number267125858f_real,K_2) = ratreal(hAPP_int_rat(number_number_of_rat,K_2))) # label(fact_4697_Ratreal__number__collapse_I3_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5014 (all I_1 all M_2 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),M_2)) -> hAPP_nat_nat(suc,hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,hAPP_f66927821l_bool(hAPP_f2037783381l_bool(cOMBB_1146692694ol_nat,member_nat),suc)),M_2))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),I_1))))) = hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),M_2))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),hAPP_nat_nat(suc,I_1))))))) # label(fact_4770_card__less__Suc) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5015 (all Z_4 all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_4)) -> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),Z_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat,Z_1)),hAPP_int_nat(nat,Z_4))))) # label(fact_2744_nat__add__distrib) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5016 (all Y all X all P_3 (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X),P_3)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),Y))))) # label(fact_740_Euler_Oaux____1) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5017 (all U all Ma all N_1 all J all I_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J)),U)),Ma)),N_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J),U)),N_1)))))) # label(fact_1838_nat__less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5018 (all C_74 all A_174 all B_134 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_174),B_134)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_74)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_74),A_174)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_74),B_134)))))) # label(fact_1079_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5019 (all A_9 all N_14 all B_7 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_9),hAPP_nat_nat(suc,N_14))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_7),hAPP_nat_nat(suc,N_14)))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_7)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_9),B_7))))) # label(fact_3112_power__le__imp__le__base) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5020 (all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,N_1)),one_one_nat) = hAPP_f957591787ol_nat(finite_card_int,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),N_1)))))) # label(fact_4776_card__bdd__int__set__l__l) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5021 (all B_1_1 all B_2_1 is_bool(hAPP_f1501066425l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5022 (all M_10 all N_24 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_10)),hAPP_nat_nat(semiri984289939at_nat,N_24))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_24)))) # label(fact_2933_of__nat__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5023 (all C_43 all A_101 all B_83 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_101),B_83)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_43),A_101)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_43),B_83))))) # label(fact_2015_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5024 (all C all A all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),A)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),B))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A),B)))) # label(fact_2025_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5025 (all B_92 all A_114 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_114),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_92),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_114),B_92)),zero_zero_rat))))) # label(fact_1892_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5026 (all C_75 all A_175 all B_135 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_175),B_135)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_75)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_75),A_175)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_75),B_135)))))) # label(fact_1076_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5027 (all M (hBOOL(hAPP_int_bool(zprime,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),M)),one_one_int))) -> hAPP_int_int(legendre(hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),M)),one_one_int)) = one_one_int)) # label(fact_1504_Legendre__1mod4) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5028 (all B_70 all A_82 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_82),zero_z126310315umeral)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_70),zero_z126310315umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_82),B_70)),zero_z126310315umeral))))) # label(fact_2177_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5029 (all M_6 all N_20 hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_20)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_nat_rat(semiri151668891at_rat,M_6)),N_20)) # label(fact_2979_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5030 (all M_11 -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,M_11)),zero_zero_rat))) # label(fact_2916_of__nat__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5031 (all N all M ((M != zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N))) & (M = zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) = zero_zero_nat))) # label(fact_1804_mult__eq__if) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5032 (all X_37 X_37 = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_37),one_one_nat)) # label(fact_561_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5033 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X)) -> (exists N_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Y),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_2),X))))),X)) & is_int(N_2))))) # label(fact_3476_best__division__abs) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5034 (all W_5 ((hAPP_int_nat(number_number_of_nat,W_5) != zero_zero_nat -> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),hAPP_int_nat(number_number_of_nat,W_5)) = zero_z126310315umeral) & (hAPP_int_nat(number_number_of_nat,W_5) = zero_zero_nat -> one_on1645066479umeral = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),hAPP_int_nat(number_number_of_nat,W_5))))) # label(fact_727_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5035 (all N hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N))),zero_zero_int))) # label(fact_3540_negative__zle__0) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5036 (all I all J_2 hAPP_int_int(legacy_zgcd(J_2),I) = hAPP_int_int(legacy_zgcd(I),J_2)) # label(fact_4495_zgcd__commute) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5037 (all A_219 A_219 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_219),zero_zero_nat)) # label(fact_606_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5038 (all A_3 all N all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),A_3))))) # label(fact_4175_prime__dvd__power) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5039 (all A_3 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A_3))) # label(fact_2313_gcd__lcm__complete__lattice__nat_Obot__least) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5040 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N))) # label(fact_3072_Suc__diff__le) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5041 (all Z_1 of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_complex_real(re,Z_1))) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_1),hAPP_complex_complex(cnj,Z_1))) # label(fact_4137_complex__add__cnj) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5042 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hAPP_real_real(arctan,hAPP_real_real(tan,X)) = X))) # label(fact_3623_arctan__tan) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5043 (all L all U hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_at4362885an_nat(L),U)))) # label(fact_4896_finite__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5044 (exists X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) & hAPP_real_real(cos,X_2) = zero_zero_real & (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_1)) & zero_zero_real = hAPP_real_real(cos,Y_1) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) -> X_2 = Y_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_2)))) # label(fact_3593_cos__is__zero) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5045 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_complex_real(re,X))),hAPP_complex_real(norm_norm_complex,X)))) # label(fact_4128_abs__Re__le__cmod) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5046 (all X (hBOOL(hAPP_nat_bool(even_odd_even_nat,X)) -> zero_zero_nat = hAPP_nat_nat(div_mod_nat(X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))))) # label(fact_3774_even__nat__mod__two__eq__zero) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5047 (all M all N hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(real_nat,M)),N) = hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),N))) # label(fact_3654_power__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5048 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A))))) # label(fact_4724_zero__le__Fract__iff) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5049 (all N hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))))) # label(fact_3849_inv__real__of__nat__fact__ge__zero) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5050 (all C_34 all D_6 all A_80 all B_68 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_80),B_68)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_34),D_6)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_80),C_34)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_68),D_6)))))) # label(fact_2197_add__le__less__mono) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5051 (all A_211 all M_26 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_211),M_26)),M_26) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_211),one_on1645066479umeral)),M_26)) # label(fact_708_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5052 (all P_3 all M ((zero_zero_nat = M -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),M) = one_one_nat) & (zero_zero_nat != M -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),M)))) # label(fact_1805_power__eq__if) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5053 (all A_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),one_one_nat)) -> A_3 = one_one_nat)) # label(fact_2316_gcd__lcm__complete__lattice__nat_Ole__bot) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5054 (all X_1 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X_1)),hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,fact_fact_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,X_1)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) = hAPP_real_real(exp_real,X_1)) # label(fact_4824_exp__first__two__terms) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5055 (all Z_1 all W hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_1),W) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),hAPP_int_int(uminus_uminus_int,W))) # label(fact_3536_diff__int__def) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5056 (all P all Q all R hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,P),Q),R) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5057 (all A_217 all M_29 all B_160 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_217),M_29)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_160),M_29)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_217),B_160)),M_29)) # label(fact_638_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5058 (all Ma (one_one_nat = hAPP_nat_nat(div_mod_nat(Ma),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(div_mod_nat(Ma),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_2755_mod2__gr__0) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5059 (all Z_1 all Z_4 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_4)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_4),Z_1)) -> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_1),Z_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,Z_1)),hAPP_int_nat(nat,Z_4))))) # label(fact_2746_nat__diff__distrib) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5060 (all K_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),I)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),J_2)))))) # label(fact_1672_mult__less__mono2) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5061 (all P all Q all R hAPP_f1607917947l_real(P,hAPP_r1134773055l_bool(Q,R)) = hAPP_r794344869l_real(hAPP_f517738447l_real(hAPP_f1879982819l_real(cOMBB_731595089l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J_000tc__fun_043) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5062 (all K_2 all L_2 hAPP_int_int(bit0,L_2) != hAPP_int_int(bit1,K_2)) # label(fact_251_rel__simps_I50_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5063 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real)) -> Y = hAPP_real_real(sin,hAPP_real_real(arcsin,Y))))) # label(fact_3703_sin__arcsin) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5064 (all X_40 all N_69 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_40),N_69)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_40),N_69)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_40),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_69))) # label(fact_426_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5065 (all Y all N all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y),N))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),Y)))))) # label(fact_753_zpower__zdvd__prop2) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5066 (all N all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(nat,Z_1)),N) = hAPP_int_nat(nat,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),N)))) # label(fact_2748_nat__power__eq) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5067 (all M all N all K_2 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(legacy_zgcd(M),N)),hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),N)))) # label(fact_4508_zgcd__zdvd__zgcd__zmult) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5068 (all X_1 all Y_2 (zero_zero_real != X_1 | zero_zero_real != Y_2 <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_1),X_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_2),Y_2)))))) # label(fact_230_sum__squares__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5069 (all N hAPP_int_nat(nat,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N))) = zero_zero_nat) # label(fact_3541_nat__zminus__int) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5070 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (hAPP_real_real(cos,X) = zero_zero_real -> (exists N_2 (-hBOOL(hAPP_nat_bool(even_odd_even_nat,N_2)) & hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_2)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = X))))) # label(fact_3794_cos__zero__lemma) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5071 (all A all B (is_int(B) & is_int(A) -> (B = A <-> zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A),B)))) # label(fact_2080_eq__iff__diff__eq__0) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5072 (all N all M hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)))) # label(fact_1639_le__add2) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5073 (all K_2 all L_2 hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),L_2)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(bit1,K_2)),hAPP_int_int(bit0,L_2))) # label(fact_1552_diff__bin__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5074 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(div_mod_int(X),Y)))))) # label(fact_2529_Divides_Otransfer__nat__int__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5075 (all A_167 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_167),A_167)))) # label(fact_1123_zero__le__square) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5076 (all X all A_3 hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)) = hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_real_real(uminus_uminus_real,A_3))) # label(fact_4414_powr__minus) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5077 (all M (is_int(M) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M)) -> M = hAPP_int_int(legacy_zgcd(M),M)))) # label(fact_4499_zgcd__self) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5078 (all N_1 all Ma hAPP_nat_nat(nat_case_nat(zero_zero_nat,hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,suc),hAPP_nat_fun_nat_nat(ord_min_nat,N_1))),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(suc,N_1)),Ma)) # label(fact_5126_min__Suc1) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5079 (all C_1 all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_3),B_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_1),A_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A_3),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,B_2),C_1)))))) # label(fact_1767_diff__less__mono) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5080 (all A_184 all M_23 all N_58 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_23),N_58)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_184),M_23)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_184),N_58))))) # label(fact_1006_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5081 (all M one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),one_one_nat)) # label(fact_4336_gcd__1__nat) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5082 (all K_2 all L_2 (is_int(L_2) -> (zero_zero_int = L_2 -> pdivmod(K_2,L_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),hAPP_int_int(abs_abs_int,K_2))) & (L_2 != zero_zero_int -> pdivmod(K_2,L_2) = posDivAlg(hAPP_int_int(abs_abs_int,K_2),hAPP_int_int(abs_abs_int,L_2))))) # label(fact_3944_pdivmod__posDivAlg) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5083 (all M all K_2 all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),K_2))) # label(fact_1743_diff__cancel2) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5084 (all X_1 (hAPP_int_int(div_mod_int(X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_one_int <-> hAPP_int_int(div_mod_int(X_1),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) != zero_zero_int)) # label(fact_2661_neq__one__mod__two) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5085 (all X_1 (is_int(X_1) -> (zero_zero_real = hAPP_int_real(real_int,X_1) <-> zero_zero_int = X_1))) # label(fact_4387_real__of__int__zero__cancel) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5086 (all X all M hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,hAPP_int_int(div_mod_int(X),M))),M) = hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,X)),M)) # label(fact_3526_zminus__zmod) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5087 (all X hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),X))) # label(fact_1696_dvd_Oorder__refl) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5088 (all N_26 hAPP_nat_nat(semiri984289939at_nat,N_26) = hAPP_int_nat(number_number_of_nat,hAPP_nat_int(semiri1621563631at_int,N_26))) # label(fact_2822_number__of__int) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5089 (all A all B all C all D (hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C),D) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A),B) -> (A = B <-> C = D))) # label(fact_1960_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5090 (all A_3 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),P_3)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,P_3)),one_one_nat))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(nat,P_3)))) # label(fact_2751_Euler_Oaux__1) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5091 (all C_1 all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),A_3)) -> (C_1 = B_2 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),C_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_1),A_3))))) # label(fact_1724_dvd_Oord__less__eq__trans) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5092 (all A_203 zero_z126310315umeral = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_203),zero_z126310315umeral)) # label(fact_842_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5093 (all X_22 all Y_17 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_22),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_17),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_22)),Y_17)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_22),Y_17)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_1663_power2__diff) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5094 (all C_83 all D_16 all A_191 all B_147 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_191),B_147)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,C_83),D_16)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_191),C_83)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_147),D_16)))))) # label(fact_938_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5095 (all M all N all K_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),K_2)) # label(fact_1593_nat__add__assoc) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5096 (all Ma all N_1 all K_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),N_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_1),N_1)))))) # label(fact_1841_nat__mult__dvd__cancel1) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5097 (all P_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,P_5),zOdd)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_2649_Euler_Oaux__2) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5098 (all A (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A),A))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A)))) # label(fact_2176_zero__le__double__add__iff__zero__le__single__add) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5099 (all A all B all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A),B),Ma)) <-> hAPP_int_int(div_mod_int(B),Ma) = hAPP_int_int(div_mod_int(A),Ma)))) # label(fact_2616_zcong__zmod__eq) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5100 (all P all Q all R hAPP_nat_rat(hAPP_f2062659861at_rat(hAPP_f1131305186at_rat(cOMBS_nat_rat_rat,P),Q),R) = hAPP_rat_rat(hAPP_nat_fun_rat_rat(P,R),hAPP_nat_rat(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__Rat__Orat_000tc__Rat__Orat_U) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5101 (all A_3 all Y all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y)) -> hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)),A_3) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,Y),A_3))))) # label(fact_4431_powr__mult) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5102 (all A_186 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,one_one_rat),A_186))) # label(fact_980_one__dvd) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5103 (all X all Y (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> -(-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X))))) # label(fact_1718_dvd_Oless__not__sym) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5104 (all A_149 hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_149),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_149),one_on1645066479umeral)))) # label(fact_1260_less__add__one) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5105 (all A_107 (is_int(A_107) -> A_107 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_107),zero_zero_int))) # label(fact_1969_add__0__right) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5106 (all V_2 ((hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_2)) = zero_zero_int) & (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_int_int(number_number_of_int,V_2) = hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_2))))) # label(fact_3281_int__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5107 (all P_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(sRStar,P_5))))) # label(fact_3363_SRStar__finite) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5108 (all X_1 (hAPP_real_real(sqrt,X_1) = zero_zero_real <-> X_1 = zero_zero_real)) # label(fact_3314_real__sqrt__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5109 (all Ma all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,Ma)),hAPP_nat_nat(suc,N_1))))) # label(fact_2863_Suc__less__eq) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5110 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X))),X)))) # label(fact_3455_ln__add__one__self__le__self) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5111 (all A_60 all B_51 all C_30 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_60),hAPP_nat_nat(div_mod_nat(B_51),C_30))),C_30) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_60),B_51)),C_30)) # label(fact_2378_mod__mult__right__eq) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5112 (all B_73 all A_85 all C_36 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_36)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_73),A_85)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_73),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_85),C_36)))))) # label(fact_2148_add__increasing2) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5113 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),zero_zero_int)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = hAPP_int_int(number_number_of_int,min)))) # label(fact_2662_div__pos__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5114 (all C_68 all A_162 all B_123 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_162),B_123)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_68)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_162),C_68)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_123),C_68)))))) # label(fact_1144_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5115 (all P all Q all R hAPP_f1173902867t_bool(P,hAPP_r1134773055l_bool(Q,R)) = hAPP_r847664445t_bool(hAPP_f1920544743t_bool(hAPP_f866672739t_bool(cOMBB_1247074793l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J_000tc__fun) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5116 (all A all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),archim1246769320r_real(X_1))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,A)),X_1)))) # label(fact_4419_le__floor__eq) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5117 (all M_11 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_11)),zero_zero_nat))) # label(fact_2920_of__nat__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5118 (all B_1_1 is_bool(bseq_real(B_1_1))) # label(gsy_c_SEQ_OBseq_000tc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 7.65/7.64 5119 (all U hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_atMost_nat,U)) = hAPP_nat_nat(suc,U)) # label(fact_5194_card__atMost) # label(axiom) # label(non_clause). [assumption]. 7.87/7.88 5120 (all N_52 all A_150 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_150)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_150),N_52))))) # label(fact_1256_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 7.87/7.88 5121 (all X hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),pi)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X))) # label(fact_3670_sin__periodic__pi) # label(axiom) # label(non_clause). [assumption]. 7.87/7.88 5122 (all I all M hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),M))))) # label(fact_3056_less__add__Suc1) # label(axiom) # label(non_clause). [assumption]. 7.87/7.88 5123 (all X_33 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_33),zero_zero_nat) = one_one_complex) # label(fact_679_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 7.87/7.88 5124 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A),B)),one_one_rat)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B))))) # label(fact_4728_Fract__le__one__iff) # label(axiom) # label(non_clause). [assumption]. 7.87/7.88 5125 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),zero_zero_int))))) # label(fact_2602_div__nonneg__neg__le0) # label(axiom) # label(non_clause). [assumption]. 7.87/7.88 5126 (all X_1 (one_one_nat = X_1 <-> X_1 = one_one_nat)) # label(fact_1951_one__reorient) # label(axiom) # label(non_clause). [assumption]. 7.87/7.88 5127 (all A_217 all M_29 all B_160 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_217),B_160)),M_29) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_217),M_29)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_160),M_29))) # label(fact_637_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 7.87/7.88 5128 (all A_198 (is_int(A_198) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,zero_zero_int),A_198)) -> zero_zero_int = A_198))) # label(fact_890_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 7.87/7.88 5129 (all P all Q all R hAPP_i1948725293t_bool(hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,P),Q),R) = hAPP_f1805168059t_bool(P,hAPP_i1948725293t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc_) # label(axiom) # label(non_clause). [assumption]. 7.87/7.88 5130 (all X all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),X)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),one_one_int),P_3))))))) # label(fact_2681_Euler__part3) # label(axiom) # label(non_clause). [assumption]. 7.87/7.88 7.87/7.88 ============================== end of process non-clausal formulas === 7.87/7.88 7.87/7.88 ============================== PROCESS INITIAL CLAUSES =============== 7.87/7.88 7.87/7.88 ============================== PREDICATE ELIMINATION ================= 7.87/7.88 7.87/7.88 ============================== end predicate elimination ============= 16.87/16.89 16.87/16.89 Auto_denials: (non-Horn, no changes). 16.87/16.89 16.87/16.89 Term ordering decisions: 16.87/16.89 Function symbol KB weights: ord_less_int=1. zero_zero_int=1. member_int=1. one_one_int=1. minus_minus_int=1. times_times_int=1. plus_plus_int=1. zero_zero_real=1. zero_zero_nat=1. bit1=1. pls=1. ord_less_real=1. ord_less_eq_real=1. bit0=1. ord_less_eq_int=1. ord_less_nat=1. times_times_real=1. number_number_of_nat=1. times_times_nat=1. suc=1. number_number_of_int=1. plus_plus_nat=1. ord_less_eq_nat=1. plus_plus_real=1. power_power_real=1. number267125858f_real=1. dvd_dvd_nat=1. one_one_real=1. zero_zero_rat=1. one_one_nat=1. times_times_rat=1. dvd_dvd_int=1. inverse_divide_real=1. power_power_int=1. div_div_int=1. nat=1. real_nat=1. semiri1621563631at_int=1. minus_minus_nat=1. ord_less_rat=1. ord_less_eq_rat=1. minus_minus_real=1. plus_plus_rat=1. times_times_complex=1. power_power_nat=1. zero_z126310315umeral=1. times_123202395de_int=1. times_1655362735umeral=1. zero_z891286103de_int=1. min=1. cos=1. div_div_nat=1. gcd_gcd_nat=1. plus_p1446045655de_int=1. plus_p1627245867umeral=1. plus_plus_complex=1. gcd_gcd_int=1. cOMBS_nat_real_real=1. power_power_rat=1. abs_abs_real=1. cOMBB_1308947378al_nat=1. pi=1. product_Pair_int_int=1. sin=1. fact_fact_nat=1. real_int=1. ord_le1860547276de_int=1. uminus_uminus_real=1. ord_le1304079648umeral=1. uminus_uminus_int=1. ord_le258702272de_int=1. ord_le565307924umeral=1. even_odd_even_nat=1. power_2100829034umeral=1. power_881366806de_int=1. number_number_of_rat=1. abs_abs_int=1. sqrt=1. root=1. produc865579683l_real=1. one_one_rat=1. cOMBB_nat_real_nat=1. zero_zero_complex=1. number528085621omplex=1. power_power_complex=1. fract=1. big_co604158596t_real=1. nat_neg=1. even_odd_even_int=1. ln=1. complex=1. re=1. dvd_dv174992974umeral=1. dvd_dv1760642554de_int=1. im=1. zprime=1. coprime=1. fconj=1. minus_minus_rat=1. one_on1645066479umeral=1. inverse_inverse_real=1. minus_minus_complex=1. one_on1684967323de_int=1. sequentially=1. powr=1. cnj=1. dvd_dvd_complex=1. dvd_dvd_rat=1. dvd_dvd_real=1. exp_real=1. cOMBC_nat_nat_nat=1. div_di1430059507de_int=1. div_di1218280263umeral=1. tan=1. prime=1. zEven=1. cOMBC_nat_real_real=1. one_one_complex=1. pred=1. zOdd=1. semiri132038758t_real=1. semiri151668891at_rat=1. product_snd_int_int=1. suminf_real=1. cOMBB_1652995168ol_int=1. cOMBB_882763271l_real=1. cOMBS_int_bool_bool=1. cOMBB_real_real_nat=1. semiri984289939at_nat=1. norm_norm_complex=1. semiri1424489471de_int=1. semiri1619134803umeral=1. nat_nat_set=1. finite_finite_nat=1. nat_is_nat=1. member_real=1. product_Pair_nat_nat=1. member_nat=1. arctan=1. product_fst_int_int=1. number1226105091de_int=1. product_snd_nat_nat=1. semiri2020571505omplex=1. finite_card_int=1. m=1. ii=1. number1443263063umeral=1. cOMBS_real_real_real=1. fact_fact_int=1. cOMBC_int_int_bool=1. cOMBB_963856155at_nat=1. finite_finite_int=1. cOMBC_real_nat_real=1. arccos=1. produc1935615926l_real=1. produc556554744l_real=1. fequal_int=1. arcsin=1. cOMBC_nat_nat_bool=1. sRStar=1. finite_card_nat=1. cOMBB_624415285l_real=1. cOMBC_real_real_real=1. cOMBS_real_bool_bool=1. quadRes=1. cOMBB_599996122l_real=1. cOMBC_real_real_bool=1. fFalse=1. fTrue=1. member_fun_int_bool=1. quickcheck_of_int=1. fequal_real=1. ord_min_nat=1. product_fst_nat_nat=1. s=1. cOMBB_real_real_real=1. cOMBC_int_int_int=1. norRRset=1. ord_lessThan_nat=1. succ=1. cOMBI_int=1. if_real=1. uminus473333897omplex=1. cOMBB_64450052al_nat=1. cOMBB_722610049al_nat=1. invers1449016382omplex=1. negateSnd=1. ord_le1568362934t_bool=1. ord_max_nat=1. sgn_sgn_int=1. sin_coeff=1. cOMBB_1480366270al_nat=1. cOMBB_782625832nt_int=1. cOMBB_nat_nat_nat=1. cos_coeff=1. norm_frac_rel=1. sgn_sgn_real=1. abs_abs_rat=1. cOMBB_1419615765al_nat=1. cOMBB_1948233853l_real=1. cOMBB_430461877l_real=1. fact=1. if_Pro1731782967nt_int=1. if_nat=1. inverse_divide_rat=1. negDivAlg_rel=1. posDivAlg_rel=1. ring_1_of_int_rat=1. t=1. big_co1758102091ol_nat=1. cOMBB_1496585939nt_int=1. cOMBB_1934313333l_real=1. cOMBC_2010481033l_real=1. if_int=1. twoSqu1770640450sum2sq=1. cOMBB_1015721476ol_nat=1. cOMBB_800536526ol_nat=1. cOMBS_nat_bool_bool=1. field_1210416355s_real=1. norm_norm_real=1. ord_atMost_nat=1. produc282740534nt_int=1. s1=1. twoSqu1907779896sum2sq=1. uminus_uminus_rat=1. zfact=1. big_co387207925at_nat=1. cOMBB_117013006ex_nat=1. cOMBS_337378985l_real=1. d22set=1. fEx_int=1. fNot=1. int_gcd=1. invers1025623611omplex=1. lazy_small_lazy_rel=1. member180897546at_nat=1. nat_gcd_rel=1. ord_le951220754t_bool=1. ring_1_of_int_real=1. sgn_sgn_complex=1. upto_rel=1. cOMBB_1533963633al_nat=1. cOMBB_1550063917nt_int=1. cOMBB_182850971nt_int=1. cOMBB_662120866nt_int=1. cOMBB_bool_bool_int=1. cOMBB_int_bool_int=1. cOMBC_226598744l_bool=1. cOMBC_746811033l_real=1. cOMBS_1470252563nt_int=1. finite1395289673t_bool=1. ord_le1912455174t_bool=1. ord_le382113706t_bool=1. phi=1. quickc1265749348ro_rel=1. ring_11397209091omplex=1. sgn_sgn_rat=1. cOMBB_1344650379nt_int=1. cOMBB_1418110531ol_int=1. cOMBB_1506897658nt_int=1. cOMBB_1768400739l_real=1. cOMBB_1846624359nt_int=1. cOMBB_1896127834l_real=1. cOMBB_1926947at_nat=1. cOMBB_2092576745l_real=1. cOMBB_320287386nt_int=1. cOMBB_421464275l_real=1. cOMBB_638531587nt_int=1. cOMBC_1329014627t_real=1. cOMBC_1508019322l_real=1. cOMBC_1805044090nt_int=1. cOMBC_1964283556nt_int=1. cOMBC_33652895t_bool=1. cOMBC_707974837nt_int=1. cOMBC_94739984l_bool=1. cOMBI_nat=1. cOMBS_1891347093nt_int=1. cOMBS_400904860l_bool=1. cOMBS_nat_rat_rat=1. fequal_nat=1. inverse_inverse_rat=1. member2143287562nt_int=1. produc494345619at_nat=1. size_size_nat=1. suminf_complex=1. big_co230513141nt_int=1. cOMBB_1121934354l_real=1. cOMBB_1146692694ol_nat=1. cOMBB_1484610687nt_int=1. cOMBB_1946221070al_int=1. cOMBB_2027861294ol_int=1. cOMBB_2107848501l_real=1. cOMBB_389152643ol_int=1. cOMBB_422423973l_real=1. cOMBB_591320580ol_int=1. cOMBB_793166024ol_int=1. cOMBB_real_bool_int=1. cOMBB_real_bool_real=1. cOMBC_1683390479t_bool=1. cOMBC_983868720t_bool=1. cOMBC_int_nat_int=1. cOMBS_193881978l_real=1. fequal_rat=1. if_Pro313124157l_real=1. member308914164t_bool=1. minus_1690775515umeral=1. nat_size=1. ratrel=1. size_s945831648umeral=1. sr=1. cOMBB_1102189010nt_int=1. cOMBB_1159313332l_real=1. cOMBB_118231410ol_int=1. cOMBB_1211207212ol_int=1. cOMBB_1247074793l_real=1. cOMBB_1290694363ol_int=1. cOMBB_1317819104ol_int=1. cOMBB_1389251822nt_int=1. cOMBB_1453147152l_real=1. cOMBB_1459832681l_real=1. cOMBB_1593174431ol_int=1. cOMBB_1647910634ol_int=1. cOMBB_1657320178nt_int=1. cOMBB_170828966nt_int=1. cOMBB_1714920583ex_nat=1. cOMBB_172850899l_real=1. cOMBB_1795223523l_real=1. cOMBB_1875755114nt_int=1. cOMBB_188663909l_real=1. cOMBB_2046938128ol_int=1. cOMBB_2050079537l_real=1. cOMBB_2086486755nt_int=1. cOMBB_2087181719ol_int=1. cOMBB_2125215742l_real=1. cOMBB_2129825580al_int=1. cOMBB_2138578194l_real=1. cOMBB_214503962ol_int=1. cOMBB_222539774nt_int=1. cOMBB_237479429l_real=1. cOMBB_270132509l_real=1. cOMBB_348438404at_nat=1. cOMBB_3698607l_real=1. cOMBB_415306361l_real=1. cOMBB_421479067l_real=1. cOMBB_44084907l_real=1. cOMBB_47643171nt_int=1. cOMBB_520243667l_real=1. cOMBB_550562460ol_int=1. cOMBB_557258983l_real=1. cOMBB_562416518ol_int=1. cOMBB_563965291nt_int=1. cOMBB_569260360nt_int=1. cOMBB_595786433nt_int=1. cOMBB_650041899nt_int=1. cOMBB_690122963l_real=1. cOMBB_731595089l_real=1. cOMBB_735549356nt_int=1. cOMBB_773384995nt_int=1. cOMBB_798864284nt_int=1. cOMBB_859312247nt_nat=1. cOMBB_866192175nt_int=1. cOMBB_929094628al_int=1. cOMBB_95956018ol_int=1. cOMBB_984071017l_real=1. cOMBB_bool_bool_nat=1. cOMBB_int_bool_nat=1. cOMBB_int_int_int=1. cOMBB_int_int_nat=1. cOMBB_int_rat_int=1. cOMBB_int_real_int=1. cOMBB_nat_bool_int=1. cOMBB_rat_bool_int=1. cOMBB_rat_rat_nat=1. cOMBB_real_bool_nat=1. cOMBB_real_real_int=1. cOMBC_1113900202t_bool=1. cOMBC_1417754532nt_int=1. cOMBC_1801450329t_bool=1. cOMBC_2037038898nt_int=1. cOMBC_241247961t_bool=1. cOMBC_243356514t_bool=1. cOMBC_295337211t_real=1. cOMBC_412863505t_bool=1. cOMBC_467724138l_real=1. cOMBC_678732631t_bool=1. cOMBC_736024425nt_int=1. cOMBC_824100275t_real=1. cOMBC_int_rat_bool=1. cOMBC_int_real_bool=1. cOMBC_nat_int_int=1. cOMBI_real=1. cOMBS_107495110l_real=1. cOMBS_1610675060t_bool=1. cOMBS_163651580l_real=1. cOMBS_1640193733nt_int=1. cOMBS_1676841879t_bool=1. cOMBS_1720984575t_bool=1. cOMBS_2011279539nt_int=1. cOMBS_2019229620nt_int=1. cOMBS_480885903t_bool=1. cOMBS_511527878l_real=1. cOMBS_549334880nt_rat=1. cOMBS_696648398omplex=1. cOMBS_794971624l_real=1. cOMBS_902912708l_real=1. cOMBS_nat_nat_nat=1. fEx_nat=1. iszero_int=1. iszero_rat=1. pair_leq=1. pair_less=1. produc883642259nt_int=1. ring_1_Ints_real=1. ring_1_of_int_int=1. big_co1024481617at_int=1. finite1876863882t_bool=1. frac=1. int=1. minus_534354567de_int=1. ord_lessThan_real=1. pred_nat=1. produc1318306967de_int=1. produc2136830103umeral=1. produc713279741t_bool=1. size_size_complex=1. size_size_list_int=1. c1=1. c2=1. c3=1. c4=1. c5=1. c6=1. c7=1. c8=1. c9=1. c10=1. c11=1. c12=1. c13=1. c14=1. hAPP_int_bool=1. hAPP_int_int=1. hAPP_i1948725293t_bool=1. hAPP_int_fun_int_int=1. hAPP_f448129468l_bool=1. hAPP_i2112223885l_bool=1. hAPP_real_real=1. hAPP_nat_nat=1. hAPP_real_bool=1. hAPP_r1134773055l_bool=1. hAPP_nat_bool=1. hAPP_nat_fun_nat_nat=1. hAPP_n1699378549t_bool=1. hAPP_r1250527377l_real=1. hAPP_int_nat=1. hAPP_nat_real=1. hAPP_rat_rat=1. hAPP_rat_fun_rat_rat=1. hAPP_nat_int=1. hAPP_rat_bool=1. hAPP_r2115560837t_bool=1. hAPP_int_real=1. hAPP_complex_complex=1. hAPP_r474017924t_real=1. hAPP_Q1762011733de_int=1. hAPP_C498520661umeral=1. hAPP_c172932010omplex=1. hAPP_Q1729056540de_int=1. hAPP_C1594335432umeral=1. hAPP_Q2096512830t_bool=1. hAPP_C24501650l_bool=1. hAPP_Q1010094925t_bool=1. hAPP_C10902773l_bool=1. hAPP_int_fun_nat_int=1. hAPP_n546711566l_real=1. hAPP_int_rat=1. hAPP_nat_rat=1. hAPP_f1888375463t_real=1. hAPP_complex_real=1. hAPP_f451358433l_real=1. hAPP_n1108039445umeral=1. hAPP_n522471361de_int=1. hAPP_f197615720t_real=1. hAPP_rat_fun_nat_rat=1. hAPP_f1092742621l_real=1. hAPP_i1584592887nt_int=1. hAPP_i1524277240nt_int=1. hAPP_C471253896umeral=1. hAPP_Q125967240de_int=1. deriv_real=1. hAPP_nat_complex=1. hAPP_r2019696015l_real=1. hAPP_r1195171167l_real=1. hAPP_P1333854989l_bool=1. hAPP_int_complex=1. hAPP_f1566856189t_real=1. hAPP_f1676298106t_real=1. hAPP_c1088319240omplex=1. hAPP_int_fun_int_rat=1. hAPP_f54304608l_bool=1. hAPP_P1175774780nt_int=1. hAPP_f1509287679l_real=1. hAPP_f1406902706l_real=1. hAPP_P603027463t_bool=1. hAPP_r265291036omplex=1. hAPP_real_complex=1. hAPP_complex_bool=1. hAPP_c1403687985x_bool=1. hAPP_f416620757at_nat=1. hAPP_f42270917t_real=1. hAPP_f1805168059t_bool=1. hAPP_P1654798560at_nat=1. hAPP_i1732201573de_int=1. hAPP_f1950183573l_real=1. hAPP_f352196356l_real=1. hAPP_f1734373249l_bool=1. hAPP_f1960414057l_real=1. hAPP_f2144054103l_bool=1. hAPP_f1690286043t_real=1. hAPP_f727283836t_bool=1. hAPP_P1555980039t_bool=1. hAPP_P1975530577nt_int=1. hAPP_f556082547l_real=1. hAPP_n332733730l_real=1. hAPP_f1501066425l_bool=1. hAPP_n1865633855at_nat=1. hAPP_r1130899993l_bool=1. hAPP_P731461727l_real=1. hAPP_n1289843868at_nat=1. hAPP_f428220345t_bool=1. hAPP_n215258509l_bool=1. divmod_nat_rel=1. posDivAlg=1. divmod_int_rel=1. negDivAlg=1. hAPP_i769753017umeral=1. hAPP_f2069516469l_real=1. hAPP_f957591787ol_nat=1. hAPP_f1639111240at_nat=1. hAPP_f1731313045at_nat=1. hAPP_f22106695ol_nat=1. image_nat_int=1. hAPP_f248847589l_real=1. image_int_nat=1. hAPP_f215623910l_bool=1. hAPP_bool_bool=1. hAPP_f229349961t_bool=1. rcis=1. hAPP_f1010710745l_bool=1. hAPP_f103356543l_bool=1. hAPP_f1485493409l_bool=1. hAPP_f1574055885nt_int=1. hAPP_f1861673493l_bool=1. hAPP_f60716669l_bool=1. big_co1548731110nt_int=1. divmod_int=1. hAPP_f1407095736t_real=1. hAPP_f1796904975t_real=1. hAPP_f628503027l_bool=1. hAPP_f695504873l_bool=1. hAPP_f203520653l_real=1. hAPP_f800510211t_bool=1. divmod_nat=1. hAPP_f1992866895l_real=1. hAPP_f2135977429nt_int=1. bezw=1. hAPP_f1914919701at_nat=1. image_int_int=1. hAPP_f1290670424t_real=1. hAPP_f284875647l_bool=1. hAPP_f327026105t_real=1. hAPP_f73570213t_real=1. hAPP_f918606171t_real=1. hAPP_f1129044203l_real=1. hAPP_f1585078997at_nat=1. hAPP_f1760145644nt_int=1. hAPP_f2054677283nt_int=1. hAPP_f264196187l_real=1. norm_frac=1. setS=1. hAPP_b589554111l_bool=1. hAPP_b992065680at_nat=1. hAPP_f1393328161l_bool=1. hAPP_f1900646853l_bool=1. hAPP_f1904307534l_real=1. hAPP_f1993765077t_real=1. hAPP_f224989205l_real=1. hAPP_f465821005nt_int=1. hAPP_f521865025ol_nat=1. hAPP_f895854793t_real=1. rRset2norRR=1. hAPP_b1463609396nt_int=1. hAPP_f1067681227l_real=1. hAPP_f1523861863l_real=1. hAPP_f1629352853nt_int=1. hAPP_f1644353557l_real=1. hAPP_f2119584768l_real=1. hAPP_f237669397t_bool=1. hAPP_f59594445l_real=1. hAPP_f93439879ol_nat=1. hAPP_r337325687l_real=1. hAPP_P1145851913nt_int=1. hAPP_P408881810nt_int=1. hAPP_f1080886329l_bool=1. hAPP_f1146629647l_bool=1. hAPP_f115998247l_bool=1. hAPP_f1505651103t_bool=1. hAPP_f561022312t_bool=1. hAPP_f618557131t_bool=1. hAPP_i1730167831nt_int=1. pdivmod=1. hAPP_P1876494581l_bool=1. hAPP_b40753821nt_int=1. hAPP_f1457396693omplex=1. hAPP_f1646117269omplex=1. hAPP_f2062659861at_rat=1. hAPP_f335634176l_real=1. hAPP_f782000547ol_nat=1. hAPP_i345030060t_bool=1. fdisj=1. hAPP_P989584703t_bool=1. hAPP_f1081447804nt_int=1. hAPP_f1607377819t_real=1. hAPP_f1722879237t_bool=1. hAPP_f1791153283t_bool=1. hAPP_f1961001761l_bool=1. hAPP_f2126667875l_real=1. hAPP_f214467655t_bool=1. hAPP_f251264923nt_int=1. hAPP_f279307923nt_int=1. hAPP_f291087797t_bool=1. hAPP_f292751352t_real=1. hAPP_f365317245l_bool=1. hAPP_f526364711l_bool=1. hAPP_f627970963t_bool=1. hAPP_nat_fun_int_int=1. hAPP_r195310020l_real=1. hAPP_r481142414t_bool=1. nat_case_nat=1. hAPP_P1062890741l_bool=1. hAPP_P1668131738nt_int=1. hAPP_P1860904029l_real=1. hAPP_P2110489235nt_int=1. hAPP_P369611207at_nat=1. hAPP_P402336767at_nat=1. hAPP_b646336293l_real=1. hAPP_f1038672605nt_int=1. hAPP_f1073644437at_rat=1. hAPP_f1122023986l_real=1. hAPP_f1131305186at_rat=1. hAPP_f1135648319t_real=1. hAPP_f1163894827t_real=1. hAPP_f1310479593l_real=1. hAPP_f1377453843nt_int=1. hAPP_f142232355nt_int=1. hAPP_f1469637905l_bool=1. hAPP_f1501741374t_real=1. hAPP_f1512942609t_bool=1. hAPP_f1582586579nt_int=1. hAPP_f1591894897l_real=1. hAPP_f1594865479ol_int=1. hAPP_f167782601nt_int=1. hAPP_f1732848529l_real=1. hAPP_f1781059817t_bool=1. hAPP_f1786887945l_real=1. hAPP_f1830834240at_rat=1. hAPP_f1855025978nt_int=1. hAPP_f1874145443l_real=1. hAPP_f197974549nt_int=1. hAPP_f1998755145t_real=1. hAPP_f2018027565t_bool=1. hAPP_f202917053t_bool=1. hAPP_f2119767738t_bool=1. hAPP_f213450978omplex=1. hAPP_f423804115t_bool=1. hAPP_f429935748t_bool=1. hAPP_f461489489t_bool=1. hAPP_f513299955t_bool=1. hAPP_f521271395nt_int=1. hAPP_f617310012nt_int=1. hAPP_f617735571l_real=1. hAPP_f791698359t_bool=1. hAPP_f892584630t_bool=1. hAPP_f960630521nt_int=1. hAPP_i119638232t_bool=1. hAPP_i1216375074nt_int=1. hAPP_i1529485324t_bool=1. hAPP_i1778081398nt_int=1. hAPP_i418383825t_bool=1. hAPP_r373117745t_bool=1. hAPP_r632163725t_bool=1. hAPP_r712953929l_real=1. image_nat_nat=1. xzgcd=1. hAPP_C2129356693al_nat=1. hAPP_P1463840893t_bool=1. hAPP_P1681069257l_bool=1. hAPP_P1982799375l_real=1. hAPP_P252494982t_bool=1. hAPP_P621635040nt_int=1. hAPP_P62333565t_bool=1. hAPP_P880235623l_bool=1. hAPP_P921862901l_bool=1. hAPP_b1482221219l_real=1. hAPP_f101635158l_bool=1. hAPP_f1048215610t_bool=1. hAPP_f1119546819l_real=1. hAPP_f1148171384nt_int=1. hAPP_f1173902867t_bool=1. hAPP_f1251097615t_bool=1. hAPP_f1344929775l_bool=1. hAPP_f1352333353l_real=1. hAPP_f1402056515l_bool=1. hAPP_f1475822739t_bool=1. hAPP_f1545556668t_bool=1. hAPP_f1735705551nt_int=1. hAPP_f1830214065l_bool=1. hAPP_f1925507189l_bool=1. hAPP_f1926459811ol_int=1. hAPP_f2037783381l_bool=1. hAPP_f2105620693nt_int=1. hAPP_f318350629l_real=1. hAPP_f320596107l_real=1. hAPP_f334321603nt_int=1. hAPP_f429384717t_bool=1. hAPP_f472159229t_bool=1. hAPP_f555557763l_real=1. hAPP_f654702867t_bool=1. hAPP_f66927821l_bool=1. hAPP_f735247205t_bool=1. hAPP_f824790735l_bool=1. hAPP_f850851693l_real=1. hAPP_f85237525t_bool=1. hAPP_i1153444985nt_int=1. hAPP_i1836591008nt_int=1. hAPP_i2104874254nt_int=1. hAPP_i371647666l_real=1. hAPP_i68813070l_bool=1. hAPP_n682674106l_real=1. big_co1705425894at_nat=1. hAPP_P1400155668t_bool=1. hAPP_P1853363688nt_rat=1. hAPP_P2053772734t_bool=1. hAPP_P230595783nt_int=1. hAPP_P507092031nt_rat=1. hAPP_P752050971nt_int=1. hAPP_P877703507t_bool=1. hAPP_P941956607nt_int=1. hAPP_f1040184293l_real=1. hAPP_f1052840010omplex=1. hAPP_f1063001928t_bool=1. hAPP_f1065357839nt_rat=1. hAPP_f1076489843l_real=1. hAPP_f1082758869at_rat=1. hAPP_f1083049571t_bool=1. hAPP_f1084724746t_bool=1. hAPP_f1108539711l_real=1. hAPP_f1109256153l_real=1. hAPP_f11151005nt_int=1. hAPP_f1115375056t_bool=1. hAPP_f1120137083nt_rat=1. hAPP_f1121609875t_bool=1. hAPP_f1131424617t_bool=1. hAPP_f1139079189at_int=1. hAPP_f1151807883nt_int=1. hAPP_f1154658180t_bool=1. hAPP_f1161935955l_real=1. hAPP_f1169087445t_real=1. hAPP_f11709033t_bool=1. hAPP_f1177545951t_bool=1. hAPP_f1182713365t_bool=1. hAPP_f1187443546t_bool=1. hAPP_f1198736473t_bool=1. hAPP_f1222178131t_bool=1. hAPP_f1228585167nt_int=1. hAPP_f1241350704t_bool=1. hAPP_f125788821nt_int=1. hAPP_f125837377t_bool=1. hAPP_f1261108921t_bool=1. hAPP_f1270196437nt_rat=1. hAPP_f1273270095t_bool=1. hAPP_f1293550189l_real=1. hAPP_f1295083291t_real=1. hAPP_f1311564564nt_int=1. hAPP_f1315737299t_bool=1. hAPP_f1323932638t_real=1. hAPP_f1325804733t_bool=1. hAPP_f1341793266t_bool=1. hAPP_f134192053t_real=1. hAPP_f1356864277nt_int=1. hAPP_f1361476881t_bool=1. hAPP_f1375227553nt_int=1. hAPP_f1399575567t_bool=1. hAPP_f1408247010at_nat=1. hAPP_f1431025877at_int=1. hAPP_f1448486952t_bool=1. hAPP_f1456711209t_bool=1. hAPP_f1459420095t_bool=1. hAPP_f1480506641t_bool=1. hAPP_f151095827nt_int=1. hAPP_f1547007718nt_rat=1. hAPP_f1576199059t_bool=1. hAPP_f1607917947l_real=1. hAPP_f1613846310t_real=1. hAPP_f1625314454t_bool=1. hAPP_f1663701695l_real=1. hAPP_f1673103573at_int=1. hAPP_f1720068861t_real=1. hAPP_f172451459l_bool=1. hAPP_f1730463541l_real=1. hAPP_f1731308757nt_int=1. hAPP_f1736103445l_real=1. hAPP_f1763467369l_real=1. hAPP_f1769710159nt_int=1. hAPP_f1815301987t_bool=1. hAPP_f1832844496t_bool=1. hAPP_f183342685l_real=1. hAPP_f1840085295t_bool=1. hAPP_f1845081853t_bool=1. hAPP_f1878066172t_bool=1. hAPP_f1879982819l_real=1. hAPP_f1884005689l_bool=1. hAPP_f1900950721nt_rat=1. hAPP_f1902329768t_bool=1. hAPP_f1908301369l_real=1. hAPP_f1914742907nt_int=1. hAPP_f1920544743t_bool=1. hAPP_f1927674023t_bool=1. hAPP_f1935805932nt_int=1. hAPP_f1965248563t_bool=1. hAPP_f1990636069t_bool=1. hAPP_f2002277285nt_int=1. hAPP_f2004554147nt_int=1. hAPP_f2042071795t_bool=1. hAPP_f2050994651t_bool=1. hAPP_f2072293513l_real=1. hAPP_f2084827628t_bool=1. hAPP_f2094497995nt_int=1. hAPP_f2114902373l_real=1. hAPP_f212164053t_real=1. hAPP_f219720582omplex=1. hAPP_f223623489t_bool=1. hAPP_f23851874omplex=1. hAPP_f252743847l_bool=1. hAPP_f269654879t_real=1. hAPP_f271976045t_bool=1. hAPP_f289318786t_bool=1. hAPP_f301246617nt_int=1. hAPP_f324002831l_real=1. hAPP_f326728547t_bool=1. hAPP_f328572565l_real=1. hAPP_f330985925l_bool=1. hAPP_f332806115t_bool=1. hAPP_f340152631t_bool=1. hAPP_f351634443t_bool=1. hAPP_f352226557t_real=1. hAPP_f374502857t_bool=1. hAPP_f381769257l_bool=1. hAPP_f382317021nt_rat=1. hAPP_f389116883t_bool=1. hAPP_f400200275nt_int=1. hAPP_f400538607nt_int=1. hAPP_f403082607t_bool=1. hAPP_f406530075omplex=1. hAPP_f41603166t_bool=1. hAPP_f422375781t_bool=1. hAPP_f434654626nt_int=1. hAPP_f492612297t_bool=1. hAPP_f499530369l_bool=1. hAPP_f512004221l_bool=1. hAPP_f517738447l_real=1. hAPP_f534361363nt_int=1. hAPP_f538565589t_bool=1. hAPP_f55007289nt_int=1. hAPP_f563293398t_real=1. hAPP_f571820015t_bool=1. hAPP_f578362681nt_int=1. hAPP_f601898971t_bool=1. hAPP_f605995867t_bool=1. hAPP_f643658017l_bool=1. hAPP_f644105375l_real=1. hAPP_f659380387ol_int=1. hAPP_f677014621l_real=1. hAPP_f693283245t_bool=1. hAPP_f696844121t_bool=1. hAPP_f704188951t_bool=1. hAPP_f713949180nt_int=1. hAPP_f720654810t_bool=1. hAPP_f725770969t_bool=1. hAPP_f729588221t_bool=1. hAPP_f737665877t_bool=1. hAPP_f74147283t_real=1. hAPP_f743790483t_bool=1. hAPP_f747162069nt_int=1. hAPP_f747475781nt_int=1. hAPP_f766218899t_real=1. hAPP_f77155443t_bool=1. hAPP_f774948957l_real=1. hAPP_f800361423l_real=1. hAPP_f801032407l_bool=1. hAPP_f808647893nt_int=1. hAPP_f823702159l_real=1. hAPP_f834119977l_bool=1. hAPP_f836602773nt_int=1. hAPP_f837737749at_nat=1. hAPP_f847817363t_bool=1. hAPP_f848516149l_bool=1. hAPP_f857296063t_bool=1. hAPP_f866672739t_bool=1. hAPP_f883232462nt_int=1. hAPP_f888249301nt_int=1. hAPP_f894608603t_bool=1. hAPP_f908327869t_bool=1. hAPP_f940061907l_bool=1. hAPP_f956464319t_bool=1. hAPP_f983569339t_bool=1. hAPP_f999082675at_nat=1. hAPP_i157317923t_real=1. hAPP_i16292313t_bool=1. hAPP_i1671359920t_real=1. hAPP_i1796122964t_bool=1. hAPP_i1823918693l_bool=1. hAPP_i447943416t_real=1. hAPP_i61245889nt_int=1. hAPP_i614188481l_bool=1. hAPP_i627165055t_real=1. hAPP_i688218208l_bool=1. hAPP_i744251628nt_int=1. hAPP_i872162435t_bool=1. hAPP_i998473677nt_int=1. hAPP_n1006566506l_bool=1. hAPP_n1983812447omplex=1. hAPP_nat_fun_rat_rat=1. hAPP_r1043989325l_real=1. hAPP_r1150855451l_bool=1. hAPP_r1230691443l_real=1. hAPP_r1392521531t_bool=1. hAPP_r1409264866l_real=1. hAPP_r1562208522t_bool=1. hAPP_r1623423338t_bool=1. hAPP_r1900510169t_bool=1. hAPP_r1938957037l_bool=1. hAPP_r1962736578t_bool=1. hAPP_r2052264067t_bool=1. hAPP_r224952523l_real=1. hAPP_r279757445t_bool=1. hAPP_r285062775l_bool=1. hAPP_r50993370t_real=1. hAPP_r579619957l_real=1. hAPP_r653725362t_bool=1. hAPP_r794344869l_real=1. hAPP_r847664445t_bool=1. hAPP_r931665397l_real=1. hAPP_r938550413l_real=1. multInvPair=1. nat_case_bool=1. code_d418564891umeral=1. hAPP_C1614127039umeral=1. hAPP_C2132160860umeral=1. hAPP_P1586233937at_nat=1. hAPP_Q1256397320de_int=1. hAPP_Q952608535de_int=1. hAPP_complex_nat=1. hAPP_f1418982953t_bool=1. hAPP_f1599440987ol_int=1. hAPP_f538843494t_bool=1. hAPP_list_int_nat=1. image_275383677t_bool=1. quickc495462417de_int=1. f2=1. f5=1. f9=1. f19=1. f20=1. f21=1. f32=1. f33=1. f38=1. f43=1. f44=1. f52=1. f53=1. f55=1. f65=1. f76=1. f78=1. f79=1. f80=1. f81=1. f82=1. f90=1. f91=1. f96=1. f99=1. f102=1. f105=1. f108=1. f109=1. f121=1. f125=1. f126=1. f127=1. f128=1. f135=1. f136=1. f143=1. f144=1. f158=1. f159=1. f162=1. f179=1. f189=1. f191=1. f193=1. f197=1. f198=1. f199=1. f204=1. f205=1. f206=1. f207=1. f217=1. f222=1. f227=1. f228=1. f229=1. f235=1. f236=1. f238=1. f246=1. f251=1. f252=1. f263=1. f270=1. f271=1. f272=1. f282=1. f289=1. f290=1. f292=1. f295=1. f296=1. f299=1. f300=1. f301=1. f302=1. f304=1. f307=1. f308=1. f309=1. f310=1. f311=1. f312=1. f313=1. f314=1. f318=1. f319=1. f320=1. f323=1. f327=1. f328=1. f334=1. f337=1. f342=1. f348=1. f349=1. f350=1. f353=1. f359=1. f360=1. f366=1. f367=1. f375=1. f376=1. f377=1. f393=1. f394=1. f395=1. f396=1. f397=1. f399=1. f400=1. f401=1. f405=1. f417=1. f421=1. f427=1. f432=1. f433=1. f437=1. f445=1. f451=1. f452=1. f456=1. f464=1. f465=1. f468=1. f474=1. f476=1. f477=1. f478=1. f482=1. f483=1. f484=1. f485=1. f486=1. f487=1. f488=1. f492=1. f493=1. f502=1. f506=1. f507=1. f521=1. f522=1. f523=1. ord_at875362053st_int=1. div_mod_int=1. zcong=1. div_mod_nat=1. legacy_zgcd=1. ord_at4362885an_nat=1. div_mo231679042de_int=1. div_mo1740067990umeral=1. isCont_real_real=1. accp_P2006205492nt_int=1. archim1246769320r_real=1. natfloor=1. quickcheck_int_of=1. multInv=1. ratreal=1. summable_real=1. log=1. of_real_complex=1. collect_int=1. archim856651990g_real=1. natceiling=1. quotient_of=1. cis=1. standardRes=1. resSet=1. vanishes=1. ord_at238088361st_nat=1. collect_nat=1. is_RRset=1. wset=1. normalize=1. sums_real=1. adjust=1. comple124823625p_real=1. monoseq_real=1. bnorRset=1. ord_gr788844697n_real=1. inv=1. nat_gcd=1. ord_at641636577an_int=1. scaleR1652505878omplex=1. code_S1047413653umeral=1. summable_complex=1. the_real=1. at_real=1. cauchy_real=1. expi=1. frct=1. legendre=1. nat_tsub=1. accp_P490777396at_nat=1. code_int_of=1. produc1518849193nt_int=1. rsetR=1. accp_int=1. archim791455193or_rat=1. cOMBK_1643584600t_bool=1. cauchy_complex=1. diffs_real=1. fimplies=1. arg=1. comple219730294t_bool=1. hilbert_Eps_int=1. produc93441119t_bool=1. sums_complex=1. at_complex=1. bseq_real=1. cOMBK_rat_nat=1. code_c271388182l_size=1. code_nat_of_aux=1. collec1347809874nt_int=1. collect_real=1. nat_tr160667106at_int=1. nat_tr876908586nt_nat=1. noXRRset=1. ord_at1589558736t_real=1. ord_gr1297742076an_int=1. ord_gr660468384an_nat=1. produc713050258nt_int=1. scaleR_scaleR_real=1. the_int=1. zcongm=1. cOMBK_bool_nat=1. collec1979865426at_nat=1. collec50511176nt_int=1. isCont_complex_real=1. nat_lcm=1. produc1298267108nt_int=1. z3div=1. z3mod=1. bijR_int_int=1. complex_size=1. dist_dist_complex=1. dist_dist_real=1. hilbert_Eps_real=1. int_ge_less_than=1. int_ge_less_than2=1. int_lcm=1. isCont156215680omplex=1. nat_aux=1. ord_at1496968948n_real=1. produc1391996073at_nat=1. produc1922581855t_bool=1. produc595218619l_real=1. produc973645403t_bool=1. quickc666637781d_zero=1. quickcheck_nat_of=1. the_Pr2103884470nt_int=1. the_Pr588456374at_nat=1. trivial_limit_nat=1. undefined_int=1. f3=1. f4=1. f6=1. f8=1. f12=1. f17=1. f42=1. f66=1. f69=1. f73=1. f74=1. f75=1. f88=1. f89=1. f92=1. f93=1. f95=1. f110=1. f112=1. f122=1. f123=1. f124=1. f138=1. f139=1. f140=1. f142=1. f145=1. f146=1. f147=1. f156=1. f157=1. f163=1. f164=1. f171=1. f178=1. f190=1. f194=1. f195=1. f196=1. f200=1. f203=1. f210=1. f216=1. f224=1. f237=1. f239=1. f243=1. f244=1. f253=1. f254=1. f255=1. f262=1. f273=1. f280=1. f281=1. f284=1. f288=1. f291=1. f297=1. f298=1. f303=1. f305=1. f306=1. f317=1. f322=1. f326=1. f331=1. f332=1. f333=1. f343=1. f344=1. f346=1. f347=1. f354=1. f361=1. f362=1. f363=1. f378=1. f382=1. f386=1. f392=1. f398=1. f402=1. f430=1. f431=1. f434=1. f440=1. f441=1. f442=1. f443=1. f447=1. f449=1. f450=1. f462=1. f463=1. f467=1. f475=1. f479=1. f494=1. f501=1. f503=1. f504=1. f505=1. f512=1. f513=1. f520=1. f524=1. tendsto_nat_real=1. bolzano_bisect=1. tendsto_real_real=1. tendsto_nat_complex=1. tendsto_complex_real=1. tendst1507391555omplex=1. f1=1. f7=1. f10=1. f11=1. f13=1. f14=1. f16=1. f34=1. f48=1. f49=1. f50=1. f51=1. f57=1. f58=1. f59=1. f60=1. f61=1. f70=1. f71=1. f72=1. f83=1. f84=1. f85=1. f86=1. f87=1. f97=1. f101=1. f104=1. f106=1. f107=1. f134=1. f137=1. f141=1. f166=1. f167=1. f172=1. f173=1. f174=1. f175=1. f188=1. f202=1. f208=1. f209=1. f214=1. f215=1. f219=1. f231=1. f247=1. f248=1. f249=1. f250=1. f258=1. f259=1. f260=1. f261=1. f267=1. f268=1. f269=1. f274=1. f275=1. f276=1. f277=1. f278=1. f279=1. f283=1. f294=1. f321=1. f324=1. f325=1. f329=1. f330=1. f341=1. f365=1. f369=1. f370=1. f371=1. f373=1. f374=1. f387=1. f388=1. f390=1. f391=1. f403=1. f404=1. f406=1. f416=1. f418=1. f419=1. f420=1. f422=1. f423=1. f424=1. f425=1. f435=1. f436=1. f444=1. f453=1. f454=1. f455=1. f457=1. f458=1. f459=1. f460=1. f466=1. f469=1. f470=1. f497=1. f498=1. f509=1. f510=1. f511=1. f515=1. f516=1. f18=1. f22=1. f23=1. f25=1. f26=1. f27=1. f28=1. f29=1. f30=1. f31=1. f35=1. f36=1. f40=1. f41=1. f54=1. f56=1. f62=1. f63=1. f64=1. f77=1. f98=1. f113=1. f114=1. f115=1. f116=1. f117=1. f118=1. f119=1. f149=1. f150=1. f151=1. f152=1. f153=1. f154=1. f155=1. f160=1. f161=1. f168=1. f169=1. f170=1. f176=1. f177=1. f181=1. f182=1. f183=1. f184=1. f185=1. f186=1. f187=1. f201=1. f211=1. f212=1. f213=1. f218=1. f225=1. f226=1. f230=1. f232=1. f233=1. f234=1. f256=1. f257=1. f264=1. f265=1. f266=1. f285=1. f286=1. f287=1. f293=1. f315=1. f316=1. f335=1. f336=1. f338=1. f339=1. f340=1. f351=1. f352=1. f355=1. f356=1. f372=1. f379=1. f380=1. f381=1. f383=1. f384=1. f385=1. f389=1. f407=1. f408=1. f409=1. f410=1. f411=1. f412=1. f413=1. f414=1. f415=1. f428=1. f429=1. f448=1. f461=1. f471=1. f472=1. f473=1. f480=1. f481=1. f490=1. f491=1. f495=1. f496=1. f499=1. f500=1. f508=1. f67=1. f68=1. f120=1. f129=1. f130=1. f131=1. f132=1. f133=1. f148=1. f165=1. f180=1. f220=1. f221=1. f357=1. f358=1. f438=1. f439=1. f514=1. f517=1. f518=1. f519=1. f24=1. xzgcda=1. f45=1. f46=1. 16.87/16.89 16.87/16.89 ============================== STATISTICS ============================ 16.87/16.89 16.87/16.89 Given=0. Generated=9465. Kept=8433. proofs=0. 16.87/16.89 Usable=0. Sos=0. Demods=1193. Limbo=8432, Disabled=9465. Hints=0. 16.87/16.89 Kept_by_rule=0, Deleted_by_rule=0. 16.87/16.89 Forward_subsumed=1032. Back_subsumed=0. 16.87/16.89 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 16.87/16.89 New_demodulators=1193 (16 lex), Back_demodulated=0. Back_unit_deleted=0. 16.87/16.89 Demod_attempts=404579. Demod_rewrites=2778. 16.87/16.89 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 16.87/16.89 Nonunit_fsub_feature_tests=6153. Nonunit_bsub_feature_tests=0. 16.87/16.90 Megabytes=419.43. 16.87/16.90 User_CPU=10.53, System_CPU=0.13, Wall_clock=11. 16.87/16.90 16.87/16.90 Megs malloced by palloc(): 400. 16.87/16.90 type (bytes each) gets frees in use bytes 16.87/16.90 chunk ( 104) 228157 228157 0 0.0 K 16.87/16.90 string_buf ( 8) 191802 191802 0 0.0 K 16.87/16.90 token ( 20) 500586 500586 0 0.0 K 16.87/16.90 pterm ( 16) 307559 307559 0 0.0 K 16.87/16.90 hashtab ( 8) 5130 5130 0 0.0 K 16.87/16.90 hashnode ( 8) 18502 18502 0 0.0 K 16.87/16.90 term ( 20) 2641950 2005516 636434 12430.4 K 16.87/16.90 term arg arrays: 3077.8 K 16.87/16.90 attribute ( 12) 48748 1034 47714 559.1 K 16.87/16.90 ilist ( 8) 100490347 100481247 9100 71.1 K 16.87/16.90 plist ( 8) 150029 76370 73659 575.5 K 16.87/16.90 i2list ( 12) 904782 904782 0 0.0 K 16.87/16.90 just ( 12) 34514 8096 26418 309.6 K 16.87/16.90 parajust ( 16) 0 0 0 0.0 K 16.87/16.90 instancejust ( 8) 0 0 0 0.0 K 16.87/16.90 ivyjust ( 24) 0 0 0 0.0 K 16.87/16.90 formula ( 28) 196240 133890 62350 1704.9 K 16.87/16.90 formula arg arrays: 203.5 K 16.87/16.90 topform ( 52) 24757 1032 23725 1204.8 K 16.87/16.90 clist_pos ( 20) 29252 9465 19787 386.5 K 16.87/16.90 clist ( 16) 8 0 8 0.1 K 16.87/16.90 context ( 808) 170552 170552 0 0.0 K 16.87/16.90 trail ( 12) 27395 27395 0 0.0 K 16.87/16.90 ac_match_pos (70044) 0 0 0 0.0 K 16.87/16.90 ac_match_free_vars_pos (20020) 16.87/16.90 0 0 0 0.0 K 16.87/16.90 btm_state ( 60) 0 0 0 0.0 K 16.87/16.90 btu_state ( 60) 0 0 0 0.0 K 16.87/16.90 ac_position (285432) 0 0 0 0.0 K 16.87/16.90 fpa_trie ( 20) 35954 0 35954 702.2 K 16.87/16.90 fpa_state ( 28) 1217 1217 0 0.0 K 16.87/16.90 fpa_index ( 12) 10 0 10 0.1 K 16.87/16.90 fpa_chunk ( 20) 31769 10984 20785 406.0 K 16.87/16.90 fpa_list ( 16) 13987 0 13987 218.5 K 16.87/16.90 fpa_list chunks: 830.4 K 16.87/16.90 discrim ( 12) 27612 0 27612 323.6 K 16.87/16.90 discrim_pos ( 16) 7447 7447 0 0.0 K 16.87/16.90 flat2 ( 32) 556317 556317 0 0.0 K 16.87/16.90 flat ( 48) 0 0 0 0.0 K 16.87/16.90 flatterm ( 32) 717102 717102 0 0.0 K 16.87/16.90 mindex ( 28) 13 0 13 0.4 K 16.87/16.90 mindex_pos ( 56) 137431 137431 0 0.0 K 16.87/16.90 lindex ( 12) 5 0 5 0.1 K 16.87/16.90 clash ( 40) 0 0 0 0.0 K 16.87/16.90 di_tree ( 12) 32877885 0 32877885 385287.7 K 16.87/16.90 avl_node ( 20) 0 0 0 0.0 K 16.87/16.90 16.87/16.90 Memory report, 20 @ 20 = 400 megs (400.00 megs used). 16.87/16.90 List 1, length 9, 0.0 K 16.87/16.90 List 5, length 2, 0.0 K 16.87/16.90 List 7, length 10, 0.3 K 16.87/16.90 List 8, length 7329, 229.0 K 16.87/16.90 List 11, length 180, 7.7 K 16.87/16.90 List 12, length 100, 4.7 K 16.87/16.90 List 14, length 4, 0.2 K 16.87/16.90 List 15, length 2, 0.1 K 16.87/16.90 List 16, length 505, 31.6 K 16.87/16.90 List 19, length 2, 0.1 K 16.87/16.90 List 20, length 2, 0.2 K 16.87/16.90 List 26, length 320, 32.5 K 16.87/16.90 List 27, length 1, 0.1 K 16.87/16.90 List 32, length 31, 3.9 K 16.87/16.90 List 49, length 2, 0.4 K 16.87/16.90 List 76, length 2, 0.6 K 16.87/16.90 List 98, length 2, 0.8 K 16.87/16.90 List 101, length 2, 0.8 K 16.87/16.90 List 202, length 2, 1.6 K 16.87/16.90 16.87/16.90 ============================== SELECTOR REPORT ======================= 16.87/16.90 Sos_deleted=0, Sos_displaced=0, Sos_size=0 16.87/16.90 SELECTOR PART PRIORITY ORDER SIZE SELECTED 16.87/16.90 I 2147483647 high age 0 0 16.87/16.90 H 1 high weight 0 0 16.87/16.90 A 1 low age 0 0 16.87/16.90 F 4 low weight 0 0 16.87/16.90 T 4 low weight 0 0 16.87/16.90 ============================== end of selector report ================ 16.87/16.90 16.87/16.90 ============================== end of statistics ===================== 16.87/16.90 16.87/16.90 Exiting with failure. 16.87/16.90 16.87/16.90 Process 45895 exit (max_megs) Sat Jul 14 05:40:56 2018 16.87/16.90 Prover9 interrupted 16.87/16.91 EOF