0.00/0.07 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.06/0.07 % Command : tptp2X_and_run_prover9 %d %s 0.06/0.26 % Computer : n012.cluster.edu 0.06/0.26 % Model : x86_64 x86_64 0.06/0.26 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.06/0.26 % Memory : 8042.1875MB 0.06/0.26 % OS : Linux 3.10.0-693.el7.x86_64 0.06/0.26 % CPULimit : 960 0.06/0.26 % DateTime : Thu Jul 2 08:24:52 EDT 2020 0.06/0.26 % CPUTime : 4.65/4.75 ============================== Prover9 =============================== 4.65/4.75 Prover9 (32) version 2009-11A, November 2009. 4.65/4.75 Process 28384 was started by sandbox2 on n012.cluster.edu, 4.65/4.75 Thu Jul 2 08:24:56 2020 4.65/4.75 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 960 -f /tmp/Prover9_28225_n012.cluster.edu". 4.65/4.75 ============================== end of head =========================== 4.65/4.75 4.65/4.75 ============================== INPUT ================================= 4.65/4.75 4.65/4.75 % Reading from file /tmp/Prover9_28225_n012.cluster.edu 4.65/4.75 4.65/4.75 set(prolog_style_variables). 4.65/4.75 set(auto2). 4.65/4.75 % set(auto2) -> set(auto). 4.65/4.75 % set(auto) -> set(auto_inference). 4.65/4.75 % set(auto) -> set(auto_setup). 4.65/4.75 % set(auto_setup) -> set(predicate_elim). 4.65/4.75 % set(auto_setup) -> assign(eq_defs, unfold). 4.65/4.75 % set(auto) -> set(auto_limits). 4.65/4.75 % set(auto_limits) -> assign(max_weight, "100.000"). 4.65/4.75 % set(auto_limits) -> assign(sos_limit, 20000). 4.65/4.75 % set(auto) -> set(auto_denials). 4.65/4.75 % set(auto) -> set(auto_process). 4.65/4.75 % set(auto2) -> assign(new_constants, 1). 4.65/4.75 % set(auto2) -> assign(fold_denial_max, 3). 4.65/4.75 % set(auto2) -> assign(max_weight, "200.000"). 4.65/4.75 % set(auto2) -> assign(max_hours, 1). 4.65/4.75 % assign(max_hours, 1) -> assign(max_seconds, 3600). 4.65/4.75 % set(auto2) -> assign(max_seconds, 0). 4.65/4.75 % set(auto2) -> assign(max_minutes, 5). 4.65/4.75 % assign(max_minutes, 5) -> assign(max_seconds, 300). 4.65/4.75 % set(auto2) -> set(sort_initial_sos). 4.65/4.75 % set(auto2) -> assign(sos_limit, -1). 4.65/4.75 % set(auto2) -> assign(lrs_ticks, 3000). 4.65/4.75 % set(auto2) -> assign(max_megs, 400). 4.65/4.75 % set(auto2) -> assign(stats, some). 4.65/4.75 % set(auto2) -> clear(echo_input). 4.65/4.75 % set(auto2) -> set(quiet). 4.65/4.75 % set(auto2) -> clear(print_initial_clauses). 4.65/4.75 % set(auto2) -> clear(print_given). 4.65/4.75 assign(lrs_ticks,-1). 4.65/4.75 assign(sos_limit,10000). 4.65/4.75 assign(order,kbo). 4.65/4.75 set(lex_order_vars). 4.65/4.75 clear(print_given). 4.65/4.75 4.65/4.75 % formulas(sos). % not echoed (5492 formulas) 4.65/4.75 4.65/4.75 ============================== end of input ========================== 4.65/4.75 4.65/4.75 % From the command line: assign(max_seconds, 960). 4.65/4.75 4.65/4.75 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 4.65/4.75 4.65/4.75 % Formulas that are not ordinary clauses: 4.65/4.75 1 (all P_3 all Q_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Q_1)) -> (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,P_3),Q_1) -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_1) = normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_1))))) # label(fact_5021_normalize__stable) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 2 (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_3004_Suc__less__SucD) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 3 (all X all M hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),hAPP_int_int(standardRes(M),X)),M))) # label(fact_3029_StandardRes__prop1) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 4 (all P_1 all A_1 all B_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1)) -> (all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A_1,B_1),N_1))),hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A_1,B_1),hAPP_nat_nat(suc,N_1)))))))) # label(fact_4244_Bolzano__bisect__fst__le__Suc) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 5 (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_3413_exp__ge__add__one__self__aux) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 6 (all Xa (Xa = hAPP_real_real(exp_real,hAPP_real_real(ln,Xa)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa)))) # label(fact_3412_exp__ln__iff) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 7 (all A all B (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A),B)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,image_nat_int(semiri1621563631at_int,A)),image_nat_int(semiri1621563631at_int,B))))) # label(fact_4603_transfer__nat__int__set__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 8 (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_4496_zgcd__assoc) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 9 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)) | Ya = Xa <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)))) # label(fact_2018_order__le__less) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 10 (all X all Y (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X),Y)) -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_int_int(abs_abs_int,X))) # label(fact_4973_gcd__proj1__if__dvd__int) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 11 (all A_1 all B_1 all C (hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_1),B_1) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_1),C) <-> B_1 = C)) # label(fact_810_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 12 (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 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),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,Y_1),B_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),Y_1))))))))) # label(fact_3287_lemma__interval__lt) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 13 (all Xa hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),Xa))))) # label(fact_4761_bdd__nat__set__l__finite) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 14 (all X_2 (hBOOL(cauchy_complex(X_2)) -> hBOOL(cauchy_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_2))))) # label(fact_4754_Im_OCauchy) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 15 (all A_126 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,one_on1684967323de_int),A_126) = A_126) # label(fact_1207_mult__1) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 16 (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),N)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),M))))) # label(fact_4170_prime__dvd__mult) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 17 (all Ya all Xa (hBOOL(hAPP_int_bool(nat_is_nat,Xa)) -> hBOOL(hAPP_f448129468l_bool(nat_nat_set,hAPP_i1948725293t_bool(ord_at875362053st_int(Xa),Ya))))) # label(fact_5143_SetInterval_Otransfer__int__nat__set__function__closures) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 18 (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]. 4.65/4.75 19 (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_2482_diff__add__inverse2) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 20 (all N -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),N))) # label(fact_3018_Suc__n__not__le__n) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 21 (all F all J all Na 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),Na))),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),Na)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,Na))))) = hAPP_real_real(F,H)))) # label(fact_4927_Maclaurin__lemma) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 22 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(exp_real,X)))) # label(fact_3399_exp__ge__zero) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 23 (all Xa all Ya (is_int(Ya) & is_int(Xa) -> (hAPP_int_int(number_number_of_int,Xa) = hAPP_int_int(number_number_of_int,Ya) <-> Xa = Ya))) # label(fact_87_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 24 (all A_59 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_59),A_59) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_59),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_1902_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 25 (all W hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),W))) # label(fact_506_zle__refl) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 26 -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))) -> 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_2195__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]. 4.65/4.75 27 (all N -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N))) # label(fact_276_less__not__refl) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 28 (all A_131 all E_3 all B_93 all C_49 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_131),E_3)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_93),E_3)),C_49)) = 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_131),B_93)),E_3)),C_49)) # label(fact_1159_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 29 (all K_2 all L_2 hAPP_int_int(bit0,K_2) != hAPP_int_int(bit1,L_2)) # label(fact_134_rel__simps_I49_J) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 30 (all Va all Na ((-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va)))) -> 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,Va)))),Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,Va)),hAPP_nat_nat(suc,Na))))) # label(fact_4786_le__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 31 (all N_24 all A_77 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_77)) -> 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_77),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_77),N_24)))))) # label(fact_1603_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 32 (all V_11 all B_62 all C_34 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_11)),B_62)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_11)),C_34)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_11)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_62),C_34))) # label(fact_1456_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 33 (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_4117_complex__Re__le__cmod) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 34 (all M hAPP_int_int(abs_abs_int,M) = hAPP_int_int(legacy_zgcd(zero_zero_int),M)) # label(fact_4506_zgcd__0__left) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 35 (all A_1 all Na (is_int(A_1) -> (hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_1),Na) = zero_zero_int <-> Na != zero_zero_nat & A_1 = zero_zero_int))) # label(fact_451_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 36 (all V_11 all B_62 all C_34 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_i1732201573de_int(number1226105091de_int,V_11)),B_62)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_i1732201573de_int(number1226105091de_int,V_11)),C_34)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_i1732201573de_int(number1226105091de_int,V_11)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_62),C_34))) # label(fact_1457_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 37 (all M_14 all A_90 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_90),one_one_nat)),M_14) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_14),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_90),M_14))) # label(fact_1482_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 38 (all X hAPP_int_real(number267125858f_real,X) = hAPP_int_real(ring_1_of_int_real,X)) # label(fact_4575_number__of__real__def) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 39 (all X ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> zero_zero_int = hAPP_int_int(fact_fact_int,X)) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> hAPP_int_int(fact_fact_int,X) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(fact_fact_nat,hAPP_int_nat(nat,X)))))) # label(fact_4270_fact__int__def) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 40 (all M_15 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,M_15),M_15) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),one_one_complex)),M_15)) # label(fact_1472_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 41 (all X_35 all Y_29 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_35),Y_29)) -> Y_29 != X_35)) # label(fact_1801_order__less__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 42 (all Ya all Xa (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ya),zOdd)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xa),Ya)),zOdd))))) # label(fact_3354_even__minus__odd) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 43 (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_4194_prime__ge__2) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 44 (all A_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),A_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),zero_zero_int),Ma)))) # label(fact_2593_zcong__zero__equiv__div) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 45 (all Na all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),Ma))))) # label(fact_2503_zero__less__diff) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 46 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,ratreal(X)),ratreal(Y)) = ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X),Y))) # label(fact_4701_real__minus__code) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 47 (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_2893_nat__mult__div__cancel1) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 48 (all B_133 all A_190 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_190)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_133)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_190),B_133)))))) # label(fact_490_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 49 (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_4200_xzgcd__def) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 50 (all V_1 hAPP_complex_real(re,hAPP_int_complex(number528085621omplex,V_1)) = hAPP_int_real(number267125858f_real,V_1)) # label(fact_4124_complex__Re__number__of) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 51 (all A_1 all B_1 (is_int(A_1) & is_int(B_1) -> (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(norm_frac_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_1),B_1))) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),zero_zero_int)) -> (-(A_1 = zero_zero_int | B_1 = zero_zero_int) -> norm_frac(A_1,B_1) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_1),hAPP_int_int(hAPP_int_fun_int_int(int_gcd,A_1),B_1))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_1),hAPP_int_int(hAPP_int_fun_int_int(int_gcd,A_1),B_1)))) & (zero_zero_int = A_1 | B_1 = zero_zero_int -> norm_frac(A_1,B_1) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),zero_zero_int)) -> norm_frac(A_1,B_1) = norm_frac(hAPP_int_int(uminus_uminus_int,A_1),hAPP_int_int(uminus_uminus_int,B_1)))))) # label(fact_4803_norm__frac_Opsimps) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 52 (all C all A_1 all B_1 (A_1 = B_1 -> (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,B_1),C)) -> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A_1),C))))) # label(fact_1763_ord__eq__less__trans) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 53 (all B_70 all A_99 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_99),zero_z891286103de_int)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_70)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_99),B_70)),zero_z891286103de_int))))) # label(fact_1410_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 54 (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_3675_arccos__ubound) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 55 (all K_2 (is_int(K_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_2)) -> (exists N_1 (K_2 = hAPP_nat_int(semiri1621563631at_int,N_1) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))))))) # label(fact_473_zero__less__imp__eq__int) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 56 (all A_20 all N_9 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(abs_abs_int,A_20)),N_9) = hAPP_int_int(abs_abs_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_20),N_9))) # label(fact_2689_power__abs) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 57 (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_4689_mult__rat) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 58 (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_292_trans__less__add2) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 59 (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_3756_natceiling__eq) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 60 (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]. 4.65/4.75 61 (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_870_le__imp__0__less) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 62 (all Q_2 all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Na),hAPP_nat_nat(suc,Q_2)))) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Na),Q_2)),Ma)) <-> Q_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,Ma),Na)))) # label(fact_3146_split__div__lemma) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 63 (all L all U hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(ord_at641636577an_int(L),U)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,U),L))) # label(fact_4919_card__atLeastLessThan__int) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 64 (all 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))) = 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_4634_Frct__code__post_I5_J) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 65 (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]. 4.65/4.75 66 (all A_86 all C_32 all B_59 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_86),C_32)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_59),C_32))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_32)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_86),B_59))))) # label(fact_1523_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 67 (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_2811_Int2_Oaux__2) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 68 (all B_42 all A_65 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_42),A_65)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_65),B_42)))) # label(fact_1775_xt1_I9_J) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 69 (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_2778_zero__le__zpower__abs) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 70 (all X_5 all Y_5 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_5),Y_5)) -> (X_5 != Y_5 -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_5),Y_5))))) # label(fact_2566_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 71 (all X_8 all N_12 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_12)) | X_8 = one_one_real -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_8),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_8),N_12))))) # label(fact_2454_dvd__power) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 72 (all A_1 all B_1 (zero_zero_complex = hAPP_real_complex(hAPP_r265291036omplex(complex,A_1),B_1) <-> A_1 = zero_zero_real & zero_zero_real = B_1)) # label(fact_3951_Complex__eq__0) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 73 (all Ma all Na (one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),Na) <-> Ma = one_one_nat & Na = one_one_nat)) # label(fact_2107_nat__mult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 74 (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_2888_aux______3) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 75 (all X_47 all Y_41 (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_47),Y_41)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_41),X_47)))) # label(fact_1624_linorder__le__cases) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 76 (all A_132 all B_94 all C_50 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_132),B_94)),C_50) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_132),C_50)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_94),C_50))) # label(fact_1156_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 77 (all B_42 all A_65 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_42),A_65)) -> -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_65),B_42)))) # label(fact_1773_xt1_I9_J) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 78 (exists K_4 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),K_4)) & (all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(cnj,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_1)),K_4)))))) # label(fact_4085_cnj_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 79 (all X_1 hAPP_n1699378549t_bool(ord_less_nat,X_1) = hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,X_1))) # label(fact_4744_less__eq__Suc__le__raw) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 80 (all R_2 all A_3 all N hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,rcis(R_2,A_3)),N) = rcis(hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_2),N),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),A_3))) # label(fact_4250_DeMoivre2) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 81 (all Y 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)) -> (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_3451_log__mult) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 82 (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_4414_real__of__int__div4) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 83 (all Na all Ma (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),Ma)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),hAPP_nat_nat(suc,Ma))) <-> Ma = Na))) # label(fact_3011_not__less__less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 84 (all C_24 all D_9 all A_78 all B_51 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_78),B_51)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_24),D_9)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_51)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_24)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_78),C_24)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_51),D_9)))))))) # label(fact_1569_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 85 (all Ma all Na (zero_zero_nat = Ma & Na = zero_zero_nat <-> zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),Na))) # label(fact_315_add__is__0) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 86 (all F all Ya all Xa all A_1 all B_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B_1)) -> (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Xa),hAPP_r1134773055l_bool(ord_gr788844697n_real(A_1),B_1))) -> (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Ya),hAPP_r1134773055l_bool(ord_gr788844697n_real(A_1),B_1))) -> ((all X_1 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),hAPP_r1134773055l_bool(ord_gr788844697n_real(A_1),B_1))) -> hBOOL(hAPP_real_bool(deriv_real(F,X_1),zero_zero_real)))) -> hAPP_real_real(F,Ya) = hAPP_real_real(F,Xa)))))) # label(fact_4628_DERIV__isconst3) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 87 (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)) -> (hAPP_int_int(legacy_zgcd(M),N) = one_one_int -> 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_4513_zcong__zgcd__zmult__zmod) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 88 (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,M),hAPP_real_real(hAPP_n546711566l_real(root,N),X)) = hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)),X))))) # label(fact_3926_real__root__pos__mult__exp) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 89 (all Wa all Z_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Wa),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_2),one_one_int))))) # label(fact_2442_zle__diff1__eq) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 90 (all Xa all Na all Ya (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Xa),hAPP_nat_nat(suc,Na))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Ya),hAPP_nat_nat(suc,Na)))))) # label(fact_3073_exp__mono__le) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 91 (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]. 4.65/4.75 92 (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_2469_zcong__square) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 93 (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]. 4.65/4.75 94 (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_4520_zprime__imp__zrelprime) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 95 (all A_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_int_bool(hAPP_i1948725293t_bool(zcong(A_1),zero_zero_int),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(big_co1548731110nt_int(cOMBI_int,comple219730294t_bool(setS(A_1,P_5)))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_1),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_4808_Union__SetS__setprod__prop1) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 96 (all A_82 all C_28 all B_55 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_82),C_28)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_55),C_28))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_28)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_82),B_55))))) # label(fact_1550_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 97 (all V_7 all W_3 hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_7),W_3)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_int_complex(number528085621omplex,V_7)),hAPP_int_complex(number528085621omplex,W_3))) # label(fact_2229_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 98 (all V_8 all U_2 all Y_22 all X_27 all A_60 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_27),A_60)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_22),A_60)) -> (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_8)) -> (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,U_2),V_8) = 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_2),X_27)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,V_8),Y_22))),A_60)))))))) # label(fact_1877_convex__bound__lt) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 99 (all A_1 (is_int(A_1) -> (zero_zero_int = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) <-> A_1 = zero_zero_int))) # label(fact_22_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 100 (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_3237_odd__plus__even) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 101 (all X_60 all Y_50 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_60),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_50),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_50)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,X_60),Y_50))))) # label(fact_892_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 102 (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_3080_neg__mod__sign) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 103 (all A_137 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_137),zero_zero_rat) = zero_zero_rat) # label(fact_1113_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 104 (all V_22 all W_12 hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_22),W_12)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_22)),hAPP_int_int(number_number_of_int,W_12))) # label(fact_198_number__of__add) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 105 (all Z_15 all X_37 all Y_31 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_37),Y_31)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_31),Z_15)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_37),Z_15))))) # label(fact_1734_order__less__trans) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 106 (all L_2 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),hAPP_int_int(bit1,L_2)) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),L_2))) # label(fact_2444_diff__bin__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 107 (all Xa all Na 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,Xa)),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,Xa))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na))))),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,Na)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(abs_abs_real,Xa)),Na))))) # label(fact_4893_Maclaurin__sin__bound) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 108 (all A_183 all N_42 all N_41 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_42),N_41)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),A_183)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_183),N_42)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_183),N_41)))))) # label(fact_550_power__increasing) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 109 (all N_31 all A_153 all B_104 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_153),B_104)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_153)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_153),N_31)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_104),N_31))))))) # label(fact_880_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 110 (all Xa all Na all Ya (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Xa),hAPP_nat_nat(suc,Na))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Ya),hAPP_nat_nat(suc,Na)))))) # label(fact_3071_exp__mono__lt) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 111 (all B_70 all A_99 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_99),zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_70)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_99),B_70)),zero_zero_nat))))) # label(fact_1413_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 112 (all B_71 all A_100 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_71),A_100))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_100)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_71))))) # label(fact_1402_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 113 (all A_1 all B_1 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,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B_1))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_1),A_1))))) # label(fact_1407_mult__less__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 114 (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_2113_zpower__zpower) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 115 (all R_2 of_real_complex(R_2) = rcis(R_2,zero_zero_real)) # label(fact_4257_rcis__zero__arg) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 116 (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_3348_ln__less__self) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 117 (all B_31 all A_42 all C_11 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_42),C_11)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_42),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_31),C_11))))) # label(fact_2277_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 118 (all X_30 all Y_24 (X_30 = Y_24 | hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_24),X_30)) | hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_30),Y_24)))) # label(fact_1836_linorder__less__linear) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 119 (all B_123 all A_179 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_179)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_123)) -> 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_179),B_123)))))) # label(fact_589_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 120 (all M_22 all N_45 all A_193 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_193)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_193),M_22)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_193),N_45))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_22),N_45))))) # label(fact_438_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 121 (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_3673_arccos__lbound) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 122 (all C all B_1 all A_1 (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B_1),A_1)) -> (C = B_1 -> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C),A_1))))) # label(fact_1658_xt1_I4_J) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 123 (all P_1 all A_1 all B_1 hAPP_n332733730l_real(bolzano_bisect(P_1,A_1,B_1),zero_zero_nat) = hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_1),B_1)) # label(fact_4243_Bolzano__bisect_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 124 (all Xa (one_one_real = Xa <-> one_one_real = Xa)) # label(fact_854_one__reorient) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 125 (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]. 4.65/4.75 126 (all F all A all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (hBOOL(hAPP_f448129468l_bool(resSet(Ma),A)) -> ((all X_1 (is_int(X_1) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A)) -> (all Xa_1 (is_int(Xa_1) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),A)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(F,X_1)),hAPP_int_int(F,Xa_1)),Ma)) -> X_1 = Xa_1))))))) -> hBOOL(hAPP_f448129468l_bool(resSet(Ma),image_int_int(F,A))))))) # label(fact_4597_ResSet__image) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 127 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sqrt,X)),X))) # label(fact_3859_sqrt__divide__self__eq) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 128 (all A_13 all B_13 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(abs_abs_real,A_13)),hAPP_real_real(abs_abs_real,B_13))),hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_13),A_13))))) # label(fact_2730_abs__triangle__ineq2__sym) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 129 (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_3165_realpow__pos__nth2) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 130 (all N_35 all A_166 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),A_166)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_166),N_35))))) # label(fact_710_one__le__power) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 131 (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_4682_of__int__rat) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 132 (all N_27 all M_16 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M_16)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),N_27)) -> 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_16),N_27)))))) # label(fact_1455_less__1__mult) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 133 (all Z_10 all Y_21 all X_25 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_21),X_25)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Z_10),Y_21)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Z_10),X_25))))) # label(fact_1922_xt1_I8_J) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 134 (all M all N all S_3 all S_1 all T_2 all T_1 all R_4 all R_2 all K_2 (K_2 = hAPP_int_int(legacy_zgcd(R_4),R_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),R_2)) -> (exists Sn exists Tn (is_int(Sn) & 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_4,R_2,S_3,S_1,T_2,T_1) & is_int(Tn)))))) # label(fact_4534_xzgcd__correct__aux1) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 135 (all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,Xa)),one_one_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),hAPP_int_int(bit1,pls))))) # label(fact_123_less__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 136 (all A_1 (hBOOL(tendsto_nat_real(A_1,zero_zero_real,sequentially)) -> (hBOOL(monoseq_real(A_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(A_1,zero_zero_nat))) -> (all N_1 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_1))),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_1)),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_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_1)),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_5061_summable__Leibniz_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 137 (all X hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),natfloor(X)))) # label(fact_3304_zero__le__natfloor) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 138 (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_3023_le__SucE) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 139 (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]. 4.65/4.75 140 (all A_191 one_on1684967323de_int = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_191),zero_zero_nat)) # label(fact_459_power__0) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 141 (all A_11 all B_11 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(abs_abs_real,A_11)),hAPP_real_real(abs_abs_real,B_11)))),hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_11),B_11))))) # label(fact_2736_abs__triangle__ineq3) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 142 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(uminus_uminus_real,Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y)) # label(fact_3433_minus__real__def) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 143 (all C_12 all D_8 all A_43 all B_32 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_43),B_32)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,C_12),D_8)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_43),C_12)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_32),D_8)))))) # label(fact_2273_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 144 (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_2447_zprime__power__zdvd__cancel__right) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 145 (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_3443_abs__sum__triangle__ineq) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 146 (all X_30 all Y_24 (is_int(Y_24) & is_int(X_30) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_24),X_30)) | X_30 = Y_24 | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_30),Y_24)))) # label(fact_1839_linorder__less__linear) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 147 (all A_199 all N_54 all N_53 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_54),N_53)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_199)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_199),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_199),N_53)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_199),N_54))))))) # label(fact_263_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 148 (all M all N all Q_1 all R_2 (hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_1),R_2))) -> R_2 = hAPP_nat_nat(div_mod_nat(M),N))) # label(fact_4218_mod__eq) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 149 (all Xa 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,Xa),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,Xa),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_4866_Ln_Oaux2) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 150 (all A_146 (is_int(A_146) -> A_146 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),A_146))) # label(fact_951_add__0__left) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 151 (all Xa all Ya (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Ya)) -> (Ya != Xa -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya))))) # label(fact_2564_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 152 (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_3493_neg__zminus__int) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 153 (all Xa 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,Xa),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),Xa))),hAPP_i1948725293t_bool(sRStar,P_5))))))) # label(fact_3212_StandardRes__SRStar__prop2) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 154 (all G all F all Xa all D_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_1)) -> ((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),Xa))),D_1)) -> 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),Xa))),D_1)) -> 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),Xa))),D_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,Z)),hAPP_real_real(F,Xa))))))))) # label(fact_5179_lemma__isCont__inj2) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 155 (all Y_43 all X_49 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_49)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_43)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_43),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_49),Y_43)),X_49)))))) # label(fact_1593_mult__right__le__one__le) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 156 (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_2484_diff__diff__left) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 157 (all B_2 (is_int(B_2) -> (B_2 != zero_zero_int -> 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_3257_divmod__int__rel__0) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 158 (all Xa (hAPP_int_int(div_mod_int(Xa),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(Xa),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) != zero_zero_int)) # label(fact_3149_neq__one__mod__two) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 159 (all P all Q all R hAPP_n1865633855at_nat(P,hAPP_nat_nat(Q,R)) = hAPP_n1865633855at_nat(hAPP_f837737749at_nat(hAPP_f999082675at_nat(cOMBB_348438404at_nat,P),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]. 4.65/4.75 160 (all Y_39 all X_45 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_39),X_45)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_45),Y_39)) -> Y_39 = X_45))) # label(fact_1639_xt1_I5_J) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 161 (all M_15 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,M_15),M_15) = 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_15)) # label(fact_1470_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 162 (all C_73 all A_184 all B_127 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_184),B_127)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_184),C_73)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_127),C_73))))) # label(fact_536_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 163 (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_3560_cos__x__y__le__one) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 164 (all F all G ((all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_1)),hAPP_nat_real(G,N_1)))) -> (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_4880_summable__le) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 165 (all J all K all 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))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K)),I_1)))) # label(fact_2517_le__diff__conv) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 166 (all K all L ((exists N_1 L = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),N_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),L)))) # label(fact_2933_le__Suc__ex__iff) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 167 (all Xa (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Xa),field_1210416355s_real)) -> -(all M_1 all N_1 (N_1 != zero_zero_nat -> (hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,M_1)),hAPP_nat_real(real_nat,N_1)) = hAPP_real_real(abs_abs_real,Xa) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M_1),N_1) != one_one_nat))))) # label(fact_4622_Rats__abs__nat__div__natE) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 168 (all B_117 all C_67 all A_171 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_171)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_117),C_67)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_117),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_171),C_67)))))) # label(fact_662_add__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 169 (all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),one_one_nat)) <-> one_one_nat = Ma)) # label(fact_2500_nat__dvd__1__iff__1) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 170 (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]. 4.65/4.75 171 (all B_118 all C_68 all A_172 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_172)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_118),C_68)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_118),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_172),C_68)))))) # label(fact_657_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption]. 4.65/4.75 172 (all A_1 all E_2 all C all B_1 all D_1 (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_1),E_2)),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_1),E_2)),D_1))) <-> 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_1),A_1)),E_2)),D_1))))) # label(fact_2422_le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 173 (all R_2 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),R_2)) -> -(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)) -> R_2 != hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,S_2),T))))))) # label(fact_4647_obtain__pos__sum) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 174 (all R_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,R_2),one_one_real)),hAPP_int_real(real_int,archim1246769320r_real(R_2))))) # label(fact_4455_real__of__int__floor__gt__diff__one) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 175 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,Ma)),hAPP_nat_real(semiri132038758t_real,Na))))) # label(fact_357_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 176 (all Z_18 all Y_40 all X_46 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_40),X_46)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_18),Y_40)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_18),X_46))))) # label(fact_1633_xt1_I6_J) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 177 (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_3007_less__trans__Suc) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 178 (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_2461_zdiff__power2) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 179 (all M_23 all N_47 hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_23),N_47)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_nat_complex(semiri2020571505omplex,M_23)),N_47)) # label(fact_383_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 180 (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_4297_gcd__dvd1__nat) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 181 (all A_30 all M_7 all N_13 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_7),N_13)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_30),M_7)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_30),N_13))))) # label(fact_2373_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 182 (all Va all V (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,Va)),hAPP_int_nat(number_number_of_nat,V))) <-> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Va),V)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Va),pls))))) # label(fact_1621_le__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 183 (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_3397_ln__x__over__x__mono) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 184 (all C_26 all D_11 all A_80 all B_53 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_80),B_53)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_26),D_11)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_80)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_26)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_80),C_26)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_53),D_11)))))))) # label(fact_1561_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 185 (all P_3 all Q_1 hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,P_3),Q_1)),hAPP_int_int(div_mod_int(P_3),Q_1)) = divmod_int(P_3,Q_1)) # label(fact_4196_divmod__int__mod__div) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 186 (all A_1 all B_1 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,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B_1))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1))))) # label(fact_1579_mult__le__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 187 (all A_37 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,one_one_rat),A_37))) # label(fact_2308_one__dvd) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 188 (all A_1 all C all D_1 all B_1 (is_int(D_1) & is_int(B_1) -> (zero_zero_int != B_1 -> (zero_zero_int != D_1 -> (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_1),D_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),D_1))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),D_1)))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_1)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C),D_1)))))))) # label(fact_4712_less__rat) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 189 (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_4230_bezw__non__0) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 190 (all V_17 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_17))) -> hAPP_int_real(number267125858f_real,V_17) = hAPP_nat_real(semiri132038758t_real,hAPP_int_nat(number_number_of_nat,V_17))) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_17))) -> hAPP_nat_real(semiri132038758t_real,hAPP_int_nat(number_number_of_nat,V_17)) = zero_zero_real))) # label(fact_901_of__nat__number__of__lemma) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 191 (all A_127 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,one_on1645066479umeral),A_127) = A_127) # label(fact_1202_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 192 (all A_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),Ma)) -> hAPP_int_int(legacy_zgcd(big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(bnorRset(A_1),Ma))),Ma) = one_one_int)) # label(fact_4810_Bnor__prod__zgcd) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 193 (all Z_13 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Z_13),Z_13) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Z_13),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_1887_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 194 (all C_56 all A_145 all B_103 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_56),A_145)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_56),B_103))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_145),B_103)))) # label(fact_957_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 195 (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_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat,Z_1)),hAPP_int_nat(nat,Z_4)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),Z_4))))) # label(fact_2794_nat__add__distrib) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 196 (all A_149 A_149 = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_149),zero_zero_real)) # label(fact_925_add__0__right) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 197 (all M_6 all N_10 ((N_10 = zero_zero_nat -> one_on1645066479umeral = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,M_6),N_10)) & (zero_zero_nat != N_10 -> hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,M_6),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,M_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),one_one_nat))) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,M_6),N_10)))) # label(fact_2610_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 198 (all K 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)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K)) = image_nat_nat(hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),K),hAPP_n1699378549t_bool(ord_at238088361st_nat(I_1),J))) # label(fact_5073_image__add__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 199 (all M_24 all N_48 hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_24),N_48)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,M_24)),hAPP_nat_int(semiri1621563631at_int,N_48))) # label(fact_375_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 200 (all A_40 all B_29 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_40),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_29),A_40)))) # label(fact_2288_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 201 (all X_62 one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_62),zero_zero_nat)) # label(fact_481_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 202 (all B_116 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_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(plus_plus_rat,A_170),B_116)))))) # label(fact_665_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 203 (all V_24 all W_14 all Z_21 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_24)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,W_14)),Z_21)) = 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_24),W_14))),Z_21)) # label(fact_189_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 204 (all Na (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Na),zero_zero_int)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,Na)),zero_zero_real)))) # label(fact_4437_real__of__int__lt__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 205 (all X hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(suc,zero_zero_nat)),X) = X) # label(fact_3890_real__root__Suc__0) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 206 (all Xa all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ya),Xa)) <-> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)))) # label(fact_2044_linorder__not__le) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 207 (all Xa all Ya (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) | Ya = Xa <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)))) # label(fact_2019_order__le__less) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 208 (all R_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,R_2),hAPP_int_real(real_int,archim856651990g_real(R_2))))) # label(fact_4401_real__of__int__ceiling__ge) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 209 (all X_41 all Y_35 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_41),Y_35)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_35),X_41)))) # label(fact_1710_linorder__linear) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 210 (all X_35 all Y_29 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_35),Y_29)) -> Y_29 != X_35)) # label(fact_1800_order__less__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 211 (all K all Na 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))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na)) = 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,Na),K)))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),K)))) # label(fact_4897_sumr__offset2) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 212 (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_4267_div__mod__code__int__def) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 213 (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_2494_diff__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 4.65/4.76 214 (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_5129_Nat__Transfer_Otransfer__int__nat__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 215 (all A_3 all B_2 expi(hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_3),B_2)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,expi(A_3)),expi(B_2))) # label(fact_3996_expi__add) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 216 (all Xa all Ya (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Xa),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,ratreal(Xa)),ratreal(Ya))))) # label(fact_4692_real__less__code) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 217 (all A_1 hBOOL(hAPP_complex_bool(isCont_complex_real(re),A_1))) # label(fact_5165_Re_OisCont) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 218 (all X_59 all Y_49 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_59),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_49),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_49)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_59),Y_49))))) # label(fact_988_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 219 (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_4970_gcd__mult__distrib__int) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 220 (all A_162 all B_110 all C_64 (hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_162),B_110) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_162),C_64) -> B_110 = C_64)) # label(fact_791_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 221 (all A_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_int_bool(hAPP_i1948725293t_bool(zcong(A_1),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_1,P_5))))))) # label(fact_4814_Union__SetS__setprod__prop2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 222 (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_3139_one__less__nat__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 223 (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_3386_aux4) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 224 (all Lx_4 all Ly_4 all Rx_4 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_4),Ly_4)),Rx_4) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_4),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Ly_4),Rx_4))) # label(fact_1057_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 225 (all B_96 all A_136 (zero_z891286103de_int != A_136 -> (zero_z891286103de_int != B_96 -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_136),B_96) != zero_z891286103de_int))) # label(fact_1125_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 226 (all Xa comple124823625p_real(hAPP_r1134773055l_bool(ord_atMost_real,Xa)) = Xa) # label(fact_5190_Sup__atMost) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 227 (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_3421_exp__bound) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 228 (all Y (zero_z126310315umeral != Y -> -(all Code_numeral Y != code_S1047413653umeral(Code_numeral)))) # label(fact_4090_code__numeral_Oexhaust) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 229 (all D_1 all F all Xa all L (hBOOL(hAPP_real_bool(deriv_real(F,Xa),L)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_1)) -> ((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,Xa),Y_1))),D_1)) -> hAPP_real_real(F,Y_1) = hAPP_real_real(F,Xa))) -> zero_zero_real = L)))) # label(fact_3878_DERIV__local__const) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 230 (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),Y)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),X)) & hBOOL(hAPP_nat_bool(prime,P_4)))))) # label(fact_4202_coprime__prime__dvd__ex) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 231 (all Xa all Ya (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ya),Xa)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)))) # label(fact_2025_less__le__not__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 232 (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)))))) -> X = hAPP_real_real(arctan,hAPP_real_real(tan,X))))) # label(fact_3770_arctan__tan) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 233 (all A_1 (A_1 = zero_zero_complex <-> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_complex)) # label(fact_20_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 234 (all Ra hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(plus_plus_real,Ra)),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))),Ra,sequentially))) # label(fact_5039_LIMSEQ__inverse__real__of__nat__add) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 235 (all Xa all Ya all B_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_1),Xa)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_1),Ya)))))) # label(fact_432_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 236 (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_3108_diff__Suc__diff__eq1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 237 (all N_52 all A_197 (A_197 != zero_zero_complex -> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_197),N_52) != zero_zero_complex)) # label(fact_345_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 238 (all P_1 all L ((exists X_1 hBOOL(hAPP_Q2096512830t_bool(P_1,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,L),X_1)))) <-> (exists X_1 (hBOOL(hAPP_Q2096512830t_bool(P_1,X_1)) & hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,L),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_1),zero_z891286103de_int))))))) # label(fact_2417_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 239 (all N_22 all A_62 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_62)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_62),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_62),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_62),N_22))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_62),N_22)))))) # label(fact_1863_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 240 (all B_1 all A_1 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_1),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_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),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_1)),hAPP_i1948725293t_bool(wset(A_1),P_5))) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),hAPP_i1948725293t_bool(wset(A_1),P_5)))))))))) # label(fact_2924_wset__inv__mem__mem) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 241 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> divmod_nat(M,N) = hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,zero_zero_nat),M))) # label(fact_4209_divmod__nat__base) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 242 (all N_56 all A_200 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_200)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_56)) -> 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_200),N_56)))))) # label(fact_246_one__less__power) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 243 (all B_1 all A_1 (is_int(A_1) & is_int(B_1) -> (zero_zero_int = A_1 <-> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_1),A_1) = B_1))) # label(fact_907_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 244 (all A_74 (is_int(A_74) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,pls))),A_74) = A_74)) # label(fact_1618_mult__numeral__1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 245 (all W_11 hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,W_11)) = 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_11))),hAPP_int_real(number267125858f_real,W_11))) # label(fact_222_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 246 (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_4574_norRRset__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 247 (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_3769_arctan) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 248 (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_4710_quotient__of__Fract) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 249 (all M_6 all N_10 ((zero_zero_nat != N_10 -> hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,M_6),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,M_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),one_one_nat))) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,M_6),N_10)) & (N_10 = zero_zero_nat -> one_one_complex = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,M_6),N_10)))) # label(fact_2609_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 250 (all X all Y all N (zero_zero_nat != N -> (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_2528_divides__exp2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 251 (all V_16 all W_8 all Z_20 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_16)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,W_8)),Z_20)) = 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_16),W_8))),Z_20)) # label(fact_1091_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 252 (all Va (hAPP_int_nat(number_number_of_nat,Va) = zero_zero_nat <-> hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va)))))) # label(fact_4049_neg__number__of__pred__iff__0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 253 (all Z_1 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(uminus_uminus_int,Z_1)),Z_1) = zero_zero_int) # label(fact_3475_zadd__zminus__inverse2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 254 (all A_130 all B_92 all C_48 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_130),C_48)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_92),C_48)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_130),B_92)),C_48)) # label(fact_1175_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 255 (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_775_zless__imp__add1__zle) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 256 (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_4111_complex__Im__inverse) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 257 (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_2847_zdiv__leq__prop) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 258 (all V_2 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_2))) = 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_2))) # label(fact_3159_one__div__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 259 (all P (is_bool(P) -> fTrue = P | P = fFalse)) # label(help_fTrue_1_1_T) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 260 (all X_2 (hBOOL(vanishes(X_2)) <-> (all R_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),R_1)) -> (exists K_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(X_2,N_1))),R_1)))))))) # label(fact_5150_vanishes__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 261 (all A_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_1),A_1)),zero_zero_real)))) # label(fact_487_double__add__less__zero__iff__single__add__less__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 262 (all K 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),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),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))),hAPP_f352196356l_real(suminf_real,F)))))) # label(fact_4926_sumr__pos__lt__pair) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 263 (all V_15 all W_7 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_15)),hAPP_int_rat(number_number_of_rat,W_7)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_15),W_7))) # label(fact_1092_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 264 (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_4271_fact__mono__int) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 265 (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_2915_aux______2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 266 (all A_3 A_3 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),A_3)) # label(fact_4311_gcd__semilattice__nat_Oinf_Oidem) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 267 (all Xa all Ya (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)) <-> Ya = Xa))) # label(fact_2001_linorder__antisym__conv1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 268 (all A_1 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_1),one_one_nat)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A_1)) <-> one_one_nat != A_1)) # label(fact_2966_gcd__lcm__complete__lattice__nat_Obot__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 269 (all I_1 all J (is_int(J) & is_int(I_1) -> (J = zero_zero_int & zero_zero_int = I_1 <-> zero_zero_int = hAPP_int_int(legacy_zgcd(I_1),J)))) # label(fact_4489_zgcd0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 270 (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_3904_real__root__one) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 271 (all Xa all 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) = Xa -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd)))) # label(fact_2951_zOddI) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 272 (all K_2 pls != hAPP_int_int(bit1,K_2)) # label(fact_131_rel__simps_I46_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 273 (all B_118 all C_68 all A_172 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_172)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_118),C_68)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_118),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_172),C_68)))))) # label(fact_653_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 274 (all B_132 all A_189 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_189),zero_z891286103de_int)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_132),zero_z891286103de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_189),B_132)),zero_z891286103de_int))))) # label(fact_496_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 275 (all Va (hAPP_int_nat(number_number_of_nat,Va) = zero_zero_nat <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Va),pls)))) # label(fact_755_eq__number__of__0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 276 (all W_15 ((zero_zero_nat = hAPP_int_nat(number_number_of_nat,W_15) -> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),hAPP_int_nat(number_number_of_nat,W_15)) = one_one_rat) & (zero_zero_nat != hAPP_int_nat(number_number_of_nat,W_15) -> 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_15))))) # label(fact_41_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 277 (all Wa all Z_2 (is_int(Z_2) -> ((exists N_1 Z_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Wa),hAPP_nat_int(semiri1621563631at_int,N_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Wa),Z_2))))) # label(fact_738_zle__iff__zadd) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 278 (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_3685_sin__cos__npi) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 279 (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),X)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z_1))))) # label(fact_2626_dvd_Oless__le__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 280 (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_4448_real__nat__eq__real) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 281 (all A_59 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_59),A_59) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_59),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_1903_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 282 (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_3973_complex__i__mult__minus) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 283 (all K all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K),Na)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K),Ma)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Ma),Na))))) # label(fact_4315_gcd__greatest__iff__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 284 (all Xa hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Xa))) # label(fact_1017_order__refl) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 285 (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_4323_gcd__semilattice__nat_Ole__infI) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 286 (all Na (hAPP_nat_int(semiri1621563631at_int,Na) = zero_zero_int <-> Na = zero_zero_nat)) # label(fact_71_int__eq__0__conv) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 287 (all A_48 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_48),A_48))) # label(fact_2222_dvd__refl) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 288 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,Ma)),hAPP_nat_nat(semiri984289939at_nat,Na))))) # label(fact_358_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 289 (all C_31 all A_85 all B_58 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_31),A_85)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_31),B_58))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_31)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_85),B_58))))) # label(fact_1527_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 290 (all Z_2 all Xa all Ya (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Ya),Z_2)) -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Z_2))))) # label(fact_1736_order__less__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 291 (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_4423_ceiling__le__real) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 292 (all N_22 all A_62 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_62)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_62),one_on1684967323de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_62),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_62),N_22))),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_62),N_22)))))) # label(fact_1861_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 293 (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_3745_natfloor__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 294 (all U U = hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_lessThan_nat,U))) # label(fact_4949_card__lessThan) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 295 (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,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_2636_dvd_Oord__eq__less__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 296 (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_2988_zmod__zmult1__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 297 (all N_35 all A_166 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),A_166)) -> 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_166),N_35))))) # label(fact_707_one__le__power) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 298 (all X_34 all Y_28 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_34),Y_28)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_28),X_34)))) # label(fact_1809_order__less__imp__not__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 299 (all Z_11 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_11) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_11),Z_11)) # label(fact_1894_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 300 (all A_183 all N_42 all N_41 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_42),N_41)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),A_183)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_183),N_42)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_183),N_41)))))) # label(fact_553_power__increasing) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 301 (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_4187_prime__coprime) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 302 (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_2352_diff__bin__simps_I7_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 303 (all Ma all K (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(abs_abs_rat,Ma)),K)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,Ma),K)))) # label(fact_2694_abs__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 304 (all B_69 all A_98 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_98),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_69),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_98),B_69)))))) # label(fact_1417_mult__neg__neg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 305 (all Lx all Ly all Rx all Ry hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx),Rx)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Ly),Ry)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx),Ly)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx),Ry))) # label(fact_1082_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 306 (all X_56 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_56),X_56))) # label(fact_1021_order__refl) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 307 (all Na one_one_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,zero_zero_real))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(suc,Na)))) # label(fact_4912_sumr__cos__zero__one) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 308 (all A_3 all B_2 hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = archim791455193or_rat(hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2))) # label(fact_4680_floor__Fract) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 309 (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_2840_div__nonneg__neg__le0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 310 (all A_1 (A_1 = zero_zero_rat <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(abs_abs_rat,A_1)),zero_zero_rat)))) # label(fact_2708_abs__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 311 (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_4829_summable__rabs) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 312 (all B_134 all C_78 all A_196 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_196)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_134),C_78)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_134),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_196),C_78)))))) # label(fact_392_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 313 (all V_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))) -> 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_1)),one_one_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_1))),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_1))),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_1)),hAPP_int_int(number_number_of_int,W))))) # label(fact_2854_zdiv__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 314 (all N all M (is_int(N) & is_int(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,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_2404_zdvd__antisym__nonneg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 315 (all Na all N_6 all F ((all N_1 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,hAPP_nat_nat(suc,N_1))),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,hAPP_nat_nat(suc,N_1)))))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),N_6)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,Na)),hAPP_nat_nat(F,N_6))) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_6)),hAPP_nat_nat(F,Na)))))) # label(fact_3215_dvd_Olift__Suc__mono__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 316 (all M_19 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(semiri132038758t_real,M_19)))) # label(fact_722_zero__le__imp__of__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 317 (all Xa all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) -> (one_one_real = Xa <-> hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa) = one_one_real))) # label(fact_3905_real__root__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 318 (all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) -> (hBOOL(hAPP_int_bool(even_odd_even_int,Xa)) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_int_nat(nat,Xa)))))) # label(fact_3812_pos__int__even__equiv__nat__even) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 319 (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_4833_rat__uminus__code) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 320 (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_4993_gcd__assoc__int) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 321 (all W_5 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_5)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,W_5))) # label(fact_1866_double__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 322 (all R_2 all X hAPP_real_real(scaleR_scaleR_real(R_2),hAPP_complex_real(im,X)) = hAPP_complex_real(im,hAPP_complex_complex(scaleR1652505878omplex(R_2),X))) # label(fact_5107_Im_OscaleR) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 323 (all Z_13 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_13),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_13),Z_13)) # label(fact_1886_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 324 (all A_44 all B_33 all K_6 (A_44 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_33),K_6) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,B_33),A_44)))) # label(fact_2269_dvdI) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 325 (all N_27 all M_16 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),M_16)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),N_27)) -> 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_16),N_27)))))) # label(fact_1450_less__1__mult) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 326 (all A_16 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(abs_abs_int,A_16)),zero_zero_int))) # label(fact_2722_abs__not__less__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 327 (all M all N (M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) -> zero_zero_nat = M | N = one_one_nat)) # label(fact_2125_mult__eq__self__implies__10) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 328 (all A_158 all B_106 all C_60 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_158),B_106)),C_60) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_158),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_106),C_60))) # label(fact_827_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 329 (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_64_zpower__int) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 330 (all C all A_1 all B_1 (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A_1),B_1)) -> (B_1 = C -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A_1),C))))) # label(fact_1750_ord__less__eq__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 331 (all C_45 all A_112 all B_82 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_112),B_82)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_45)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_45),A_112)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_45),B_82)))))) # label(fact_1333_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 332 (all F all B_1 all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_1),Xa)) -> ((all X_1 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),hAPP_r1134773055l_bool(ord_gr788844697n_real(B_1),Xa))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(F,X_1))))) -> (hBOOL(hAPP_real_bool(isCont_real_real(F),Xa)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(F,Xa))))))) # label(fact_5171_LIM__less__bound) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 333 (all W hAPP_int_int(number_number_of_int,W) = hAPP_int_int(ring_1_of_int_int,W)) # label(fact_4576_int__number__of__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 334 (all Sb all Ta all A_1 all B_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),B_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Sb),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_1),Sb)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,B_1),Ta))),pair_less))))) # label(fact_4539_pair__lessI2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 335 (all Xa (hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_nat_int(semiri1621563631at_int,Xa))) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa)))) # label(fact_3795_even__nat__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 336 (all A_1 all B_1 (hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_1),B_1) = zero_zero_rat <-> B_1 = A_1)) # label(fact_2317_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 337 (all C_1 all A_3 all B_2 all Q_1 all R_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_1),R_2))) -> (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_1),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_1),C_1))),R_2))))))))) # label(fact_3223_zmult2__lemma) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 338 (all R_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,archim856651990g_real(R_2))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,R_2),one_one_real)))) # label(fact_4456_real__of__int__ceiling__le__add__one) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 339 (all Q_1 ((all A_2 all B_4 (is_int(B_4) & is_int(A_2) -> (Q_1 = 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 -> one_one_int != hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_4)))))) -> Q_1 = zero_zero_rat)) # label(fact_5024_Rat__cases__nonzero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 340 (all M_21 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_21)),zero_zero_real))) # label(fact_455_of__nat__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 341 (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) & 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)) = K_2 & is_int(T))))))) # label(fact_4536_zgcd__ex__linear) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 342 (all C_70 all D_21 all A_178 all B_122 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_178),B_122)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_70),D_21)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_178),C_70)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_122),D_21)))))) # label(fact_604_add__le__less__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 343 (all A_3 all X hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,X)),hAPP_real_real(ln,A_3)) = hAPP_real_real(log(A_3),X)) # label(fact_3436_Log_Olog__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 344 (all Ma all Na all K (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K)) -> (Na = Ma <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Na) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Ma)))) # label(fact_2194_nat__mult__eq__cancel1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 345 (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(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(hAPP_nat_fun_nat_nat(div_div_nat,M),N)))) # label(fact_3148_le__div__geq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 346 (all C_20 all A_70 all B_47 (B_47 = A_70 -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_47),C_20)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_70),C_20))))) # label(fact_1681_ord__eq__le__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 347 (all A_3 all C_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,A_3)),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))) # label(fact_3200_div__Suc) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 348 (all A_176 all N_40 all N_39 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_40),N_39)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_176)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_176),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_176),N_39)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_176),N_40))))))) # label(fact_622_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 349 (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_2998_zpower__zmod) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 350 (all A_1 all B_1 (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_1),B_1))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_1),zero_zero_rat)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_1),zero_zero_rat)) | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_1)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_1)))) # label(fact_1286_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 351 (all X_60 all Y_50 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_60),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_50),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_50)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_60),Y_50))))) # label(fact_895_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 352 (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_4988_dvd__gcd__D1__int) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 353 (all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(sqrt,Ya))))) # label(fact_3534_real__sqrt__ge__1__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 354 (all Xa hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),Xa))))) # label(fact_4751_bdd__nat__set__le__finite) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 355 (all Xa all Ya (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)))))),Ya)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ya),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)))))),Xa)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),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,Ya)),hAPP_real_real(tan,Xa))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ya),Xa)))))))) # label(fact_3764_tan__monotone_H) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 356 (all Xa all Ya all B_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_1),Xa)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_1),Ya))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya))))) # label(fact_634_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 357 (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_3372_ln__add__one__self__le__self) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 358 (all N hAPP_nat_complex(semiri2020571505omplex,N) = hAPP_complex_complex(cnj,hAPP_nat_complex(semiri2020571505omplex,N))) # label(fact_4071_complex__cnj__of__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 359 (all Xa all Ya (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Xa),Ya)) -> (Ya = Xa <-> -hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Xa),Ya))))) # label(fact_1969_linorder__antisym__conv2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 360 (all K all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) <-> 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)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),Na))))) # label(fact_3062_Suc__mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 361 (all Na (Na = zero_zero_nat <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),hAPP_nat_nat(suc,zero_zero_nat))))) # label(fact_3040_less__Suc0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 362 (all N 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) = one_one_int) # label(fact_4981_coprime__minus__one__int) # label(axiom) # label(non_clause). [assumption]. 4.69/4.76 363 (all C all B_1 all A_1 (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,B_1),A_1)) -> (B_1 = C -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,C),A_1))))) # label(fact_1743_xt1_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 364 (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_3026_diff__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 365 (all B_134 all C_78 all A_196 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_196)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_134),C_78)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_134),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_196),C_78)))))) # label(fact_395_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 366 (all 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)) = hAPP_real_real(cos,X)) # label(fact_3625_cos__sin__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 367 (all A_134 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A_134) = zero_zero_rat) # label(fact_1138_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 368 (all A_36 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_36),A_36) = zero_zero_complex) # label(fact_2326_diff__self) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 369 (all Xa 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)),Xa))),hAPP_f1241350704t_bool(hAPP_f1131424617t_bool(cOMBB_rat_bool_int,hAPP_r2115560837t_bool(ord_less_rat,Xa)),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(Xa)) # label(fact_4890_floor__rat__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 370 (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_1275_add__leE) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 371 (all Xa (Xa = hAPP_int_real(real_int,archim856651990g_real(Xa)) <-> (exists N_1 (is_int(N_1) & hAPP_int_real(real_int,N_1) = Xa)))) # label(fact_4396_real__of__int__ceiling__cancel) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 372 (all C_21 all A_71 all B_48 (A_71 = B_48 -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_21),B_48)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_21),A_71))))) # label(fact_1675_xt1_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 373 (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_4150_cis__divide) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 374 (all A_1 all B_1 (zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_1),B_1) <-> A_1 = B_1)) # label(fact_2319_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 375 (all Xa all A_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_int_bool(hAPP_i1948725293t_bool(zcong(A_1),zero_zero_int),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A_1)) -> (hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,Xa),setS(A_1,P_5))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(big_co1548731110nt_int(cOMBI_int,Xa)),A_1),P_5)))))))) # label(fact_4812_SetS__setprod__prop) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 376 (all Xa all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(number_number_of_int,Xa)),hAPP_int_int(number_number_of_int,Ya))) <-> zero_zero_int = hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,Ya)),hAPP_int_int(number_number_of_int,Xa)))) # label(fact_3086_zdvd__iff__zmod__eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 377 (all Lx_3 all Ly_3 all Rx_3 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_3),Ly_3)),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_3)) # label(fact_1063_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 378 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(div_mod_nat(M),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)) # label(fact_3193_div__mod__equality_H) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 379 (all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),N) = N) # label(fact_317_plus__nat_Oadd__0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 380 (all X_1 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_1) = hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,X_1)))) # label(fact_4875_exp__fdiffs) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 381 (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_4300_coprime__exp2__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 382 (all K all Na all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),Na)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Na),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),Ma)))))) # label(fact_2406_zdvd__reduce) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 383 (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_3939_less__mono__imp__le__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 384 (all X_60 all Y_50 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_60),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_50),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_50)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_60),Y_50))))) # label(fact_894_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 385 (all A_57 all B_39 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_57),B_39)) -> (A_57 != B_39 -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_57),B_39))))) # label(fact_1957_order__le__neq__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 386 (all B_123 all A_179 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_179)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_123)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_179),B_123)))))) # label(fact_591_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 387 (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]. 4.69/4.77 388 (all Xa all Ya all Z_2 (Xa = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,Ya),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)) <-> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,Ya),Z_2) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Xa))) # label(fact_3415_eq__divide__2__times__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 389 (all Xa (Xa = zero_zero_rat <-> Xa = zero_zero_rat)) # label(fact_763_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 390 (all L_2 pls != hAPP_int_int(bit1,L_2)) # label(fact_132_rel__simps_I39_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 391 (all C_46 all A_113 all B_83 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_113),B_83)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_46)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_46),A_113)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_46),B_83)))))) # label(fact_1325_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 392 (all A_163 all B_111 all C_65 (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_163),C_65) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_163),B_111) -> B_111 = C_65)) # label(fact_784_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 393 (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_3461_minus__Bit0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 394 (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) -> zero_zero_int != hAPP_int_int(div_mod_int(K_2),L_2))) # label(fact_3482_zmod__zminus1__not__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 395 (all M all N all R_4 all S_3 all S_1 all T_2 all T_1 all R_2 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_2),zero_zero_int)) -> hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,R_4),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,S_3),T_2)) = xzgcda(M,N,R_4,R_2,S_3,S_1,T_2,T_1)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_2),zero_zero_int)) -> xzgcda(M,N,R_2,hAPP_int_int(div_mod_int(R_4),R_2),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_4),R_2)),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_4),R_2)),T_1))) = xzgcda(M,N,R_4,R_2,S_3,S_1,T_2,T_1)))) # label(fact_4199_xzgcda_Osimps) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 396 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa),Ya))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,Xa)),Ya)))) # label(fact_3439_real__0__less__add__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 397 (all Xa (Xa = zero_zero_nat <-> zero_zero_nat = Xa)) # label(fact_768_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 398 (all Y_15 all X_19 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_15),X_19)) -> -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_19),Y_15)))) # label(fact_1982_leD) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 399 (all B_125 all A_181 all C_72 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_72)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_125),A_181)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_125),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_181),C_72)))))) # label(fact_575_add__increasing2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 400 (all Y all X (X = hAPP_real_real(exp_real,Y) -> Y = hAPP_real_real(ln,X))) # label(fact_3396_ln__unique) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 401 (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_3944_SRStar__finite) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 402 (all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),hAPP_nat_nat(fact,N)))) # label(fact_4566_fact__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 403 (all Real1 all Real2 zero_zero_nat = complex_size(hAPP_real_complex(hAPP_r265291036omplex(complex,Real1),Real2))) # label(fact_3963_complex_Osize_I1_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 404 (all X_42 all Y_36 (X_42 = Y_36 -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_42),Y_36)))) # label(fact_1698_order__eq__refl) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 405 (all F (hBOOL(tendsto_nat_real(F,zero_zero_real,sequentially)) <-> hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F),zero_zero_real,sequentially)))) # label(fact_5040_LIMSEQ__rabs__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 406 (all M hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,zero_zero_nat),M) = divmod_nat(M,zero_zero_nat)) # label(fact_4208_divmod__nat__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 407 (all Xa all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root,Na),Xa),hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,Na)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),hAPP_nat_nat(suc,zero_zero_nat)))))))))) # label(fact_3887_DERIV__real__root) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 408 (all V_20 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_20)) -> hAPP_int_real(number267125858f_real,hAPP_int_int(succ,V_20)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_int_real(number267125858f_real,V_20)))) # label(fact_746_semiring__1__add__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 409 (all X (is_int(X) -> hAPP_int_int(succ,hAPP_int_int(pred,X)) = X)) # label(fact_4038_succ__pred) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 410 (all Xa 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)))),Xa),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,Xa))),hAPP_real_real(sin,Xa)))))) # label(fact_4798_DERIV__cos__realpow2b) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 411 (all A_89 all M_13 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_89),one_on1645066479umeral)),M_13) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_89),M_13)),M_13)) # label(fact_1487_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 412 (all C all A_1 all B_1 (A_1 = B_1 -> (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C),B_1)) -> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C),A_1))))) # label(fact_1672_xt1_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 413 (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_3387_exp__less__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 414 (all V_1 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_1))),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_1)),hAPP_int_int(number_number_of_int,W))) # label(fact_2827_zdiv__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 415 (all A_1 (A_1 != zero_zero_real <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(abs_abs_real,A_1))))) # label(fact_2718_zero__less__abs__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 416 (all B_89 all A_119 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_119)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_89)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_119),B_89)))))) # label(fact_1293_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 417 (all Lx all Ly all Rx all Ry hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx),Rx)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Ly),Ry)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx),Ly)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx),Ry))) # label(fact_1081_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 418 (all C_12 all D_8 all A_43 all B_32 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_43),B_32)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,C_12),D_8)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_43),C_12)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_32),D_8)))))) # label(fact_2271_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 419 (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_4966_Fract__coprime) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 420 (all C_46 all A_113 all B_83 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_113),B_83)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_46)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_46),A_113)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_46),B_83)))))) # label(fact_1326_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 421 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)) -> (Xa = Ya <-> -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya))))) # label(fact_1970_linorder__antisym__conv2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 422 (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_3664_arcsin__sin) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 423 (all R_2 hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),R_2) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),of_real_complex(R_2))) # label(fact_3991_i__complex__of__real) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 424 (all B_126 all A_182 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_182),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_126),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_182),B_126)),zero_zero_real))))) # label(fact_567_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 425 (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,zero_zero_nat),hAPP_int_nat(nat,Z_2))))) # label(fact_2782_zero__less__nat__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 426 (all Xa (one_one_complex = Xa <-> one_one_complex = Xa)) # label(fact_852_one__reorient) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 427 (all Y exists X_1 (Y = hAPP_real_real(tan,X_1) & (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)) & Y = hAPP_real_real(tan,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)))))) -> X_1 = Y_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,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)))) # label(fact_3776_tan__total) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 428 (all Z_7 all X_22 all Y_18 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_22),Y_18)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_18),Z_7)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_22),Z_7))))) # label(fact_1946_order__less__le__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 429 (all A_1 all B_1 (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_1)) -> (B_1 != A_1 -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A_1),B_1))))) # label(fact_1956_order__le__neq__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 430 (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_3578_arctan__ubound) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 431 (all P_3 all Q_1 (hBOOL(hAPP_nat_bool(prime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_3),Q_1))) -> P_3 = one_one_nat | Q_1 = one_one_nat)) # label(fact_4171_prime__product) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 432 (all M_15 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,M_15),M_15) = 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_15)) # label(fact_1471_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 433 (all Xa all Ya (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya)) -> -hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Ya),Xa)))) # label(fact_1813_order__less__not__sym) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 434 (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_3457_negative__zless) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 435 (all Lx_6 all Rx_6 all Ry_4 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_6),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_6),Ry_4)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_6),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_6),Ry_4))) # label(fact_1039_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 436 (all K_3 all K all F ((all N_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_1)),K_3)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,K),hAPP_nat_real(F,N_1))))) -> hBOOL(bseq_real(F)))) # label(fact_4947_BseqI2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 437 (all A_1 all D_1 all Na hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,Na)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),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,Na),one_one_nat))),D_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_1)),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_1))),hAPP_n1699378549t_bool(ord_lessThan_nat,Na)))) # label(fact_4946_arith__series__int) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 438 (all M all N (is_int(M) -> (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N) = one_one_int -> M = one_one_int | hAPP_int_int(number_number_of_int,min) = M))) # label(fact_2185_pos__zmult__eq__1__iff__lemma) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 439 (all A_127 A_127 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,one_on1684967323de_int),A_127)) # label(fact_1200_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 440 (all V_13 all V_12 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_12)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_13)) -> hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_12)),hAPP_int_complex(number528085621omplex,V_13)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_12),V_13))))) # label(fact_1279_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 441 (all 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) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,N))) # label(fact_3136_add__2__eq__Suc) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 442 (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_65_int__power) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 443 (all A_1 all A all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma)) -> (hBOOL(hAPP_int_bool(is_RRset(A),Ma)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),A)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(rRset2norRR(A,Ma),A_1)),hAPP_i1948725293t_bool(norRRset,Ma))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),hAPP_int_int(rRset2norRR(A,Ma),A_1)),Ma)))))) # label(fact_4590_RRset2norRR__correct) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 444 (all A_56 all B_38 (B_38 != A_56 -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_38),A_56)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_38),A_56))))) # label(fact_1988_xt1_I12_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 445 (all B_134 all C_78 all A_196 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_196)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_134),C_78)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_134),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_196),C_78)))))) # label(fact_394_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 446 (all A_160 all B_108 all C_62 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_160),C_62)),B_108) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_160),B_108)),C_62)) # label(fact_817_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 447 (all N_37 all A_168 all B_114 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_168),B_114)) -> (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,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_168),N_37)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_114),N_37)))))) # label(fact_695_power__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 448 (all X_2 (hBOOL(vanishes(X_2)) -> hBOOL(vanishes(hAPP_f2062659861at_rat(hAPP_f1082758869at_rat(cOMBB_rat_rat_nat,uminus_uminus_rat),X_2))))) # label(fact_5139_vanishes__minus) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 449 (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 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1)) & A_3 = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_1),N)))))) # label(fact_1009_realpow__pos__nth) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 450 (all A_1 all E_2 all C all B_1 all D_1 (is_int(D_1) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),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_1),E_2)),D_1) <-> D_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(hAPP_int_fun_int_int(minus_minus_int,A_1),B_1)),E_2)),C)))) # label(fact_2384_eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 451 (all C_75 all D_22 all A_186 all B_129 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_186),B_129)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_75),D_22)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_186),C_75)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_129),D_22)))))) # label(fact_529_add__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 452 (all M_25 all N_49 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_25)),hAPP_nat_int(semiri1621563631at_int,N_49))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_25),N_49)))) # label(fact_369_of__nat__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 453 (all X_32 all Y_26 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_32),Y_26)) -> Y_26 != X_32)) # label(fact_1822_less__imp__neq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 454 (all C all B_1 all A_1 (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_1),A_1)) -> (C = B_1 -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C),A_1))))) # label(fact_1659_xt1_I4_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 455 (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(hAPP_int_fun_int_int(times_times_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_4509_zgcd__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 456 (all X all Y ((Y != zero_zero_nat -> hAPP_nat_nat(nat_gcd(Y),hAPP_nat_nat(div_mod_nat(X),Y)) = hAPP_nat_nat(nat_gcd(X),Y)) & (zero_zero_nat = Y -> hAPP_nat_nat(nat_gcd(X),Y) = X))) # label(fact_4658_nat__gcd_Osimps) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 457 (all N_16 all X_12 all Y_9 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_12),Y_9)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_12),N_16)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_9),N_16))))) # label(fact_2315_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 458 (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(div_div_nat,A_3),B_2)),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))) # label(fact_2883_div__mult2__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 459 (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_2581_zcong__zmult__self) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 460 (all B_115 all A_169 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_169)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_115)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_169),B_115)))))) # label(fact_676_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 461 (all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) <-> (exists M_1 hAPP_nat_nat(suc,M_1) = Na))) # label(fact_3041_gr0__conv__Suc) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 462 (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_3448_realpow__square__minus__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 463 (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_4756_zpower__number__of__even) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 464 (all W hAPP_int_int(uminus_uminus_int,hAPP_int_int(number_number_of_int,W)) = hAPP_int_int(number_number_of_int,hAPP_int_int(uminus_uminus_int,W))) # label(fact_3465_minus__numeral__code_I5_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 465 (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]. 4.69/4.77 466 (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_504_zle__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 467 (all A_33 all B_25 (is_int(A_33) -> A_33 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_33),B_25)),B_25))) # label(fact_2346_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 468 (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_2543_zcong__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 469 (all X all N_4 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_4)) -> (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_4),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),X))))))) # label(fact_3918_real__root__decreasing) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 470 (all S_1 all T_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S_1),T_1)) -> S_1 != T_1)) # label(fact_290_less__not__refl3) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 471 (all P all Q all R hAPP_i1584592887nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,P),Q),R) = hAPP_f1735705551nt_int(hAPP_i1836591008nt_int(P,R),hAPP_i1584592887nt_int(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]. 4.69/4.77 472 (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_2907_inv__not__0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 473 (all W_4 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,W_4)),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_4)),hAPP_int_int(number_number_of_int,W_4))) # label(fact_1915_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 474 (all A_35 A_35 = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_35),zero_zero_complex)) # label(fact_2330_diff__0__right) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 475 (all P all Q all R hAPP_r1043989325l_real(hAPP_f1161935955l_real(hAPP_f328572565l_real(cOMBB_520243667l_real,P),Q),R) = hAPP_f183342685l_real(P,hAPP_r794344869l_real(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]. 4.69/4.77 476 (all X_21 all Y_17 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_21),Y_17)) -> Y_17 = X_21 | hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_21),Y_17)))) # label(fact_1964_order__le__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 477 (all C_25 all D_10 all A_79 all B_52 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_79),B_52)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_25),D_10)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_79)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_25)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_79),C_25)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_52),D_10)))))))) # label(fact_1568_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 478 (all A_1 all B_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),B_1)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),zero_zero_int)))) # label(fact_1291_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 479 (all A_148 A_148 = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),A_148)) # label(fact_934_add__0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 480 (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]. 4.69/4.77 481 (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]. 4.69/4.77 482 (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]. 4.69/4.77 483 (all N_57 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,N_57)),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_57)))) # label(fact_236_of__nat__less__two__power) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 484 (all X_53 all P_7 all Q_8 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_53),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_8)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_53),P_7)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_53),Q_8))) # label(fact_1499_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 485 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Xa)) <-> Xa = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,Xa)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3545_real__sqrt__pow2__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 486 (all Na 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),Na)))))) # label(fact_4740_bdd__int__set__l__finite) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 487 (all B_120 all A_174 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_174),zero_z891286103de_int)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_120),zero_z891286103de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_174),B_120)),zero_z891286103de_int))))) # label(fact_642_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 488 (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_2850_div__prop2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 489 (all X_61 all Y_51 (hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_51),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,X_61),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_61)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),Y_51)) -> Y_51 = X_61)))) # label(fact_885_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 490 (all A_165 all N_33 all B_113 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_165),N_33)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_113),N_33))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_113)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_165),B_113))))) # label(fact_753_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 491 (all A_124 A_124 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_124),one_one_real)) # label(fact_1224_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 492 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),Xa)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(ln,Xa)))))) # label(fact_3364_ln__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 493 (all A_149 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_149),zero_zero_rat) = A_149) # label(fact_922_add__0__right) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 494 (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_3617_sin__periodic__pi) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 495 (all A_132 all B_94 all C_50 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_132),B_94)),C_50) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_132),C_50)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_94),C_50))) # label(fact_1155_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 496 (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_3450_aux5) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 497 (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_3473_zmod__zminus2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 498 (all A_57 all B_39 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_57),B_39)) -> (A_57 != B_39 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_57),B_39))))) # label(fact_1959_order__le__neq__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 499 (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_2914_aux______4) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 500 (all X_9 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_9),one_one_rat)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_9),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_9),X_9)),one_one_rat)) # label(fact_2433_real__squared__diff__one__factored) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 501 (all X_59 all Y_49 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_59),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_49),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_49)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,X_59),Y_49))))) # label(fact_990_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 502 (all A_1 all A all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma)) -> (hBOOL(hAPP_int_bool(is_RRset(A),Ma)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),A)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),hAPP_int_int(rRset2norRR(A,Ma),A_1)),Ma)))))) # label(fact_4592_RRset2norRR__correct1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 503 (all P_1 ((exists X1 hBOOL(hAPP_nat_bool(P_1,X1))) <-> (exists X_1 (hBOOL(hAPP_nat_bool(P_1,hAPP_int_nat(nat,X_1))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) & is_int(X_1))))) # label(fact_2744_ex__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 504 (all A_146 A_146 = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,zero_zero_real),A_146)) # label(fact_949_add__0__left) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 505 (all A_1 all B_1 (-(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_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_4),B_1)))) <-> one_one_int = hAPP_int_int(legacy_zgcd(A_1),B_1))) # label(fact_4521_zgcd1__iff__no__common__primedivisor) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 506 (all A_1 all B_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_1),B_1)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),zero_zero_int))))) # label(fact_2823_pos__imp__zdiv__neg__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 507 (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,M)),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = 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))))))) # label(fact_3771_sin__expansion__lemma) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 508 (all B_124 all C_71 all A_180 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_180)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_124),C_71)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_124),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_180),C_71)))))) # label(fact_578_add__increasing) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 509 (all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N))) # label(fact_1024_le0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 510 (all A_35 (is_int(A_35) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_35),zero_zero_int) = A_35)) # label(fact_2332_diff__0__right) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 511 (all A_201 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_201),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),zero_zero_real))) # label(fact_212_power2__less__0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 512 (all X all Y ((zero_zero_nat = Y -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = X) & (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)))) # label(fact_4359_gcd__nat_Osimps) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 513 (all C all A_1 all B_1 (A_1 = B_1 -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,B_1),C)) -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A_1),C))))) # label(fact_1764_ord__eq__less__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 514 (all Xa all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,Xa)),hAPP_int_real(real_int,Ya))))) # label(fact_4388_real__of__int__le__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 515 (all C_52 all A_141 all B_99 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_141),B_99)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_141),C_52)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_99),C_52))))) # label(fact_973_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 516 (all A all Xa (hBOOL(hAPP_int_bool(nat_is_nat,Xa)) -> (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),A)) <-> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_int_nat(nat,Xa)),image_int_nat(nat,A))))))) # label(fact_5147_transfer__int__nat__set__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 517 (all Xa all Ya (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Ya),Xa)) <-> -hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Xa),Ya)))) # label(fact_2046_linorder__not__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 518 (all W_15 ((hAPP_int_nat(number_number_of_nat,W_15) != zero_zero_nat -> zero_z126310315umeral = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),hAPP_int_nat(number_number_of_nat,W_15))) & (hAPP_int_nat(number_number_of_nat,W_15) = zero_zero_nat -> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),hAPP_int_nat(number_number_of_nat,W_15)) = one_on1645066479umeral))) # label(fact_44_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 519 (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_3486_negative__zle__0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 520 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real)) -> zero_zero_nat = natfloor(X))) # label(fact_3313_natfloor__neg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 521 (all Xa all Ya (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya)) -> -hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Ya),Xa)))) # label(fact_1722_order__less__asym) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 522 (all Na all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,Na)),Ma)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,Na)),hAPP_int_real(real_int,Ma))))) # label(fact_4443_number__of__le__real__of__int__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 523 (all A_131 all E_3 all B_93 all C_49 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_131),E_3)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_93),E_3)),C_49)) = 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_131),B_93)),E_3)),C_49)) # label(fact_1161_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 524 (all Xa all Ya all B_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),B_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_1),Xa)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_1),Ya))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya))))) # label(fact_633_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 525 (all X_16 all Y_12 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_12),X_16)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_16),Y_12)))) # label(fact_2037_linorder__le__less__linear) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 526 (all A_131 all E_3 all B_93 all C_49 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_131),B_93)),E_3)),C_49) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_131),E_3)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_93),E_3)),C_49))) # label(fact_1160_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 527 (all Na (zero_zero_nat = Na <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,Na)),zero_zero_real)))) # label(fact_3712_real__of__nat__le__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 528 (all N_14 all M_8 all X_11 all Y_8 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_11),Y_8)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_14),M_8)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_11),N_14)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_8),M_8)))))) # label(fact_2367_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 529 (all A_201 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_201),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),zero_zero_int))) # label(fact_213_power2__less__0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 530 (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_4522_zgcd__zcong__zgcd) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 531 (all X_8 all N_12 (X_8 = one_on1645066479umeral | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_12)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,X_8),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_8),N_12))))) # label(fact_2453_dvd__power) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 532 (all K_2 hAPP_int_int(succ,hAPP_int_int(bit1,K_2)) = hAPP_int_int(bit0,hAPP_int_int(succ,K_2))) # label(fact_773_succ__Bit1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 533 (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_2855_less__one__imp__sqr__less__one) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 534 (all Xa hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Xa))) # label(fact_1016_order__refl) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 535 (all A_1 all B_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_1),B_1)),zero_zero_rat)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_1)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_1),zero_zero_rat)) | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_1),zero_zero_rat)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_1)))) # label(fact_1289_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 536 (all M_14 all A_90 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_90),one_on1684967323de_int)),M_14) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,M_14),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_90),M_14))) # label(fact_1478_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 537 (all Xa all Ya all B_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),B_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_1),Xa)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_1),Ya))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya))))) # label(fact_632_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 538 (all A_158 all B_106 all C_60 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_158),B_106)),C_60) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_158),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_106),C_60))) # label(fact_829_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 539 (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_3586_arcsin__ubound) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 540 (all X_33 all Y_27 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_33),Y_27)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_27),X_33)))) # label(fact_1817_order__less__not__sym) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 541 (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_2973_StandardRes__prop4) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 542 (all B_85 all A_115 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_115),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_85),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_115),B_85)))))) # label(fact_1317_mult__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 543 (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_4223_div__int__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 544 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(suc,Na))) <-> (exists J_1 (Ma = hAPP_nat_nat(suc,J_1) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),Na)))) | zero_zero_nat = Ma)) # label(fact_3039_less__Suc__eq__0__disj) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 545 (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_3548_real__sqrt__pow2__gt__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 546 (all C_27 all D_12 all A_81 all B_54 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_81),B_54)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_27),D_12)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_81)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_27)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_81),C_27)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_54),D_12)))))))) # label(fact_1555_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 547 (all L_2 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),hAPP_int_int(bit0,L_2)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),L_2))) # label(fact_2445_diff__bin__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 548 (all A_21 all B_16 hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_21),B_16)) = hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_16),A_21))) # label(fact_2685_abs__minus__commute) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 549 (all G all F all Xa all D_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_1)) -> ((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),Xa))),D_1)) -> 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),Xa))),D_1)) -> hBOOL(hAPP_real_bool(isCont_real_real(F),Z)))) -> hBOOL(hAPP_real_bool(isCont_real_real(G),hAPP_real_real(F,Xa))))))) # label(fact_5173_isCont__inv__fun) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 550 (all P all Q all R hAPP_rat_fun_rat_rat(P,hAPP_nat_rat(Q,R)) = hAPP_nat_fun_rat_rat(hAPP_f1830834240at_rat(hAPP_f1073644437at_rat(cOMBB_1926947at_nat,P),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]. 4.69/4.77 551 (all Xa (hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa)) <-> (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) = Xa))) # label(fact_3799_even__nat__equiv__def2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 552 (all C_43 all B_80 all A_110 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_80),A_110)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_43),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_43),A_110)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_43),B_80)))))) # label(fact_1340_mult__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 553 (all C_74 all A_185 all B_128 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_185),B_128)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_74),A_185)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_74),B_128))))) # label(fact_534_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 554 (all A_156 all C_58 all D_17 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_156),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_58),D_17)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_58),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_156),D_17))) # label(fact_839_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 555 (all A_3 all B_2 hAPP_complex_complex(invers1449016382omplex,hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),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)))))))) # label(fact_3962_complex__inverse) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 556 (all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(fact,N)))) # label(fact_4563_fact__lt) # label(axiom) # label(non_clause). [assumption]. 4.69/4.77 557 (all F all A_1 all B_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B_1)) -> ((all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),X_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),B_1)) -> hBOOL(hAPP_real_bool(isCont_real_real(F),X_1)))) -> ((all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),B_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),X_1)) -> hBOOL(hAPP_real_bool(deriv_real(F,X_1),zero_zero_real)))) -> (all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),X_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),B_1)) -> hAPP_real_real(F,A_1) = hAPP_real_real(F,X_1))))))) # label(fact_5174_DERIV__isconst1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 558 (all C_42 all D_15 all A_109 all B_79 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_109),B_79)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_42),D_15)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_109)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_42)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_109),C_42)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_79),D_15)))))))) # label(fact_1344_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 559 (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]. 4.69/4.78 560 (all W hAPP_int_int(hAPP_int_fun_int_int(times_times_int,pls),W) = pls) # label(fact_1254_mult__Pls) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 561 (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]. 4.69/4.78 562 (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_3416_real__sum__of__halves) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 563 (all Wa all Xa (is_int(Xa) -> (hAPP_int_int(number_number_of_int,Wa) = Xa <-> hAPP_int_int(number_number_of_int,Wa) = Xa))) # label(fact_93_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 564 (all N_31 all A_153 all B_104 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_153),B_104)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_153)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_153),N_31)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_104),N_31))))))) # label(fact_878_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 565 (all Va all Wa (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,Va)),hAPP_int_nat(number_number_of_nat,Wa))) <-> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,Wa)),hAPP_int_nat(number_number_of_nat,Va))))) # label(fact_692_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 566 (all Xa all Ya (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)))) # label(fact_2647_dvd_Oless__le__not__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 567 (all N_34 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(semiri984289939at_nat,N_34)))) # label(fact_717_of__nat__0__le__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 568 (all P all Q all R hAPP_r1562208522t_bool(hAPP_f2084827628t_bool(hAPP_f1121609875t_bool(cOMBB_2125215742l_real,P),Q),R) = hAPP_f1902329768t_bool(P,hAPP_r285062775l_bool(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]. 4.69/4.78 569 (all Xa all Ya all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa)),hAPP_real_real(hAPP_n546711566l_real(root,Na),Ya)))))) # label(fact_3899_real__root__less__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 570 (all C_30 all A_84 all B_57 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_30),A_84)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_30),B_57))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_30)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_84),B_57))))) # label(fact_1534_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 571 (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_2913_inv__is__inv) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 572 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(exp_real,X)))) # label(fact_3401_exp__gt__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 573 (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]. 4.69/4.78 574 (all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> hAPP_nat_nat(fact_fact_nat,hAPP_int_nat(nat,X)) = hAPP_int_nat(nat,hAPP_int_int(fact_fact_int,X)))) # label(fact_4284_transfer__nat__int__factorial) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 575 (all A_28 all B_20 all V_5 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_28),hAPP_int_int(number_number_of_int,V_5))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_20),hAPP_int_int(number_number_of_int,V_5))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_28),B_20)),hAPP_int_int(number_number_of_int,V_5))) # label(fact_2400_left__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 576 (all Wa all Ya all Xa all Z_2 (is_int(Z_2) & is_int(Xa) & is_int(Ya) & is_int(Wa) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Wa),Z_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Xa),Ya)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Wa),Ya)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Xa),Z_2)) <-> Z_2 = Ya | Wa = Xa))) # label(fact_1191_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 577 (all M hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),one_one_int) = one_one_int) # label(fact_4982_gcd__1__int) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 578 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y)) -> (exists R_1 exists A_2 (X = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(cos,A_2)) & hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(sin,A_2)) = Y)))) # label(fact_3850_polar__ex1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 579 (all C_19 all B_46 all A_69 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_46),A_69)) -> (C_19 = B_46 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_19),A_69))))) # label(fact_1746_xt1_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 580 (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))) -> (zero_zero_nat != N -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))))) # label(fact_2527_divides__rev) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 581 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),min)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K)),min)))) # label(fact_2183_rel__simps_I11_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 582 (all B_76 all A_106 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_106)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_76),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_106),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_76)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_106),B_76)),zero_zero_int)))) # label(fact_1363_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 583 (all Ma all D_1 (is_int(Ma) -> ((exists Q_4 (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D_1),Q_4) = Ma & is_int(Q_4))) <-> hAPP_int_int(div_mod_int(Ma),D_1) = zero_zero_int))) # label(fact_3032_zmod__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 584 (all Xa all Ya (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) & Xa != Ya <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)))) # label(fact_2033_order__less__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 585 (all X_47 all Y_41 (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_47),Y_41)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_41),X_47)))) # label(fact_1626_linorder__le__cases) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 586 (all P all Q all R hAPP_n215258509l_bool(hAPP_f66927821l_bool(hAPP_f2037783381l_bool(cOMBB_1146692694ol_nat,P),Q),R) = hAPP_n215258509l_bool(P,hAPP_nat_nat(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]. 4.69/4.78 587 (all Z_16 all Y_32 all X_38 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_32),X_38)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_16),Y_32)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_16),X_38))))) # label(fact_1731_xt1_I10_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 588 (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)))) -> N = hAPP_int_nat(nat,archim1246769320r_real(X))))) # label(fact_4031_floor__eq3) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 589 (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_4220_divmod__nat__rel__ex) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 590 (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_4143_sin__n__Im__cis__pow__n) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 591 (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_2605_zprime__zdvd__zmult) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 592 (all A_3 all B_2 all C_1 all Q_1 all R_2 (hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(B_2,C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_1),R_2))) -> (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_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),R_2)),C_1))),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),R_2)),C_1))))))) # label(fact_4212_divmod__nat__rel__mult1__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 593 (all B_117 all C_67 all A_171 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_171)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_117),C_67)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_117),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_171),C_67)))))) # label(fact_661_add__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 594 (all A_187 all C_76 all B_130 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_187),C_76)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_130),C_76))) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_187),B_130)))) # label(fact_520_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 595 (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]. 4.69/4.78 596 (all X_58 all Y_48 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_58),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_48),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_998_sum__power2__ge__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 597 (all Na 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)))),Na))) = zero_zero_real) # label(fact_4910_sumr__minus__one__realpow__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 598 (all N_16 all X_12 all Y_9 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_12),Y_9)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_12),N_16)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_9),N_16))))) # label(fact_2313_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 599 (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]. 4.69/4.78 600 (all D_1 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_1)))),ord_less_int))) = int_ge_less_than2(D_1)) # label(fact_4860_int__ge__less__than2__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 601 (all A_1 all B_1 all C all Xa (is_int(A_1) -> (hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,C),hAPP_int_nat(nat,Xa)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),B_1) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_int_nat(nat,Xa))) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd)) -> (hAPP_int_int(legacy_zgcd(A_1),B_1) = one_one_int -> (exists K_1 (is_int(K_1) & A_1 = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K_1),hAPP_int_nat(nat,Xa)))))))))) # label(fact_4528_int__relprime__odd__power__divisors) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 602 (all Xa all Ya all B_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),B_1)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_1),Xa)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_1),Ya))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya))))) # label(fact_427_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 603 (all B_119 all A_173 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_173),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_119),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_173),B_119)),zero_zero_real))))) # label(fact_649_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 604 (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_2604_zcong__zprime__prod__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 605 (all A_22 all B_17 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(abs_abs_int,A_22)),hAPP_int_int(abs_abs_int,B_17)) = hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(abs_abs_int,A_22)),hAPP_int_int(abs_abs_int,B_17)))) # label(fact_2680_abs__add__abs) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 606 (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_120_int__less__0__conv) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 607 (all A_151 A_151 = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_151),zero_z126310315umeral)) # label(fact_914_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 608 (all A_191 one_one_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_191),zero_zero_nat)) # label(fact_462_power__0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 609 (all Ma all Xa hAPP_i1948725293t_bool(noXRRset(Ma),Xa) = image_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,times_times_int),Xa),hAPP_i1948725293t_bool(norRRset,Ma))) # label(fact_4772_noXRRset__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 610 (all Ya (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_int_rat(number_number_of_rat,Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Ya)))) # label(fact_107_less__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 611 (exists K_4 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),K_4)) & (all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(im,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_1)),K_4)))))) # label(fact_4106_Im_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 612 (all B_115 all A_169 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_169)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_115)) -> 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_169),B_115)))))) # label(fact_671_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 613 (all X_16 all Y_12 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_12),X_16)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_16),Y_12)))) # label(fact_2039_linorder__le__less__linear) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 614 (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_2410_diff__bin__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 615 (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_2639_dvd__antisym) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 616 (all C all A_1 all B_1 (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_1)) -> (C = B_1 -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),C))))) # label(fact_1666_ord__le__eq__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 617 (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_3743_ge__natfloor__plus__one__imp__gt) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 618 (all B_1 all A_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_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,A_1),B_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),A_1))))) # label(fact_2842_nonneg1__imp__zdiv__pos__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 619 (all X_2 (hBOOL(cauchy_complex(X_2)) -> hBOOL(cauchy_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_2))))) # label(fact_4753_Re_OCauchy) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 620 (all Lx all Ly all Rx all Ry hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx),Ly)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx),Ry)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx),Rx)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ly),Ry))) # label(fact_1085_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 621 (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_4167_prime__g__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 622 (all A_1 all C all B_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_1),C)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_1),C))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_1),A_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),zero_zero_real)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C)))) # label(fact_1368_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 623 (all A_8 all B_9 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_8),B_9))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(abs_abs_rat,A_8)),hAPP_rat_rat(abs_abs_rat,B_9))))) # label(fact_2760_abs__triangle__ineq4) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 624 (all U hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_at641636577an_int(zero_zero_int),U)))) # label(fact_4916_finite__atLeastZeroLessThan__int) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 625 (all A_1 all B_1 ((all D_2 (is_int(D_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_2),A_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_2),B_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),D_2)) <-> D_2 = one_one_int))) <-> one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_1),B_1))) # label(fact_5018_coprime__int) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 626 (all N_34 hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_n1108039445umeral(semiri1619134803umeral,N_34)))) # label(fact_715_of__nat__0__le__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 627 (all N zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,N),zero_zero_nat)) # label(fact_5120_min__0R) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 628 (all Z_8 all Y_19 all X_23 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_19),X_23)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_8),Y_19)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_8),X_23))))) # label(fact_1938_xt1_I7_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 629 (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_3116_Suc__diff__1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 630 (all Code_numeral_2 zero_z126310315umeral != code_S1047413653umeral(Code_numeral_2)) # label(fact_4081_code__numeral_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 631 (all X_17 all Y_13 (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_17),Y_13)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_13),X_17)))) # label(fact_2011_leI) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 632 (all A_199 all N_54 all N_53 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_54),N_53)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_199)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_199),one_on1645066479umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_199),N_53)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_199),N_54))))))) # label(fact_261_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 633 (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_3801_odd__nat__plus__one__div__two) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 634 (all C_38 all A_95 all B_66 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_95),B_66)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_38)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_38),A_95)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_38),B_66)))))) # label(fact_1434_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 635 (all D_3 all A_3 all B_2 (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 -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),A_3) = one_one_int)) # label(fact_5001_coprime__lmult__int) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 636 (all L all U hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(L),U)))) # label(fact_5065_finite__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 637 (all N_22 all A_62 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_62)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_62),one_on1645066479umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_62),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_62),N_22))),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_62),N_22)))))) # label(fact_1862_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 638 (all Z_1 hAPP_int_complex(ring_11397209091omplex,Z_1) = hAPP_complex_complex(cnj,hAPP_int_complex(ring_11397209091omplex,Z_1))) # label(fact_4579_complex__cnj__of__int) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 639 (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_3434_abs__minus__add__cancel) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 640 (all N_14 all M_8 all X_11 all Y_8 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_11),Y_8)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_14),M_8)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_11),N_14)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_8),M_8)))))) # label(fact_2369_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 641 (all A_148 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,zero_z126310315umeral),A_148) = A_148) # label(fact_938_add__0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 642 (all Ma all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,Ma),Xa)) <-> (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))))),Xa),Ma)))))) # label(fact_2467_QuadRes__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 643 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,Xa)),one_one_real)))) # label(fact_3405_exp__le__one__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 644 (all A_150 A_150 = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_150),zero_zero_real)) # label(fact_918_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 645 (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]. 4.69/4.78 646 (all C_41 all D_14 all A_108 all B_78 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_108),B_78)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_41),D_14)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_78)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_41)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_108),C_41)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_78),D_14)))))))) # label(fact_1353_mult__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 647 (all C all D_1 all A_1 all B_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),B_1),Ma)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),D_1)),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),D_1)),Ma))))) # label(fact_2580_zcong__zmult__prop1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 648 (all Na all Ma hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(suc,Na)),Ma) = hAPP_nat_nat(nat_case_nat(hAPP_nat_nat(suc,Na),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,suc),hAPP_nat_fun_nat_nat(ord_max_nat,Na))),Ma)) # label(fact_5192_max__Suc1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 649 (all Xa all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa)),zero_zero_real))))) # label(fact_3914_real__root__lt__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 650 (all A_149 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_149),zero_z126310315umeral) = A_149) # label(fact_928_add__0__right) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 651 (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_4064_complex__cnj__divide) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 652 (all W_15 ((hAPP_int_nat(number_number_of_nat,W_15) = zero_zero_nat -> one_on1684967323de_int = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),hAPP_int_nat(number_number_of_nat,W_15))) & (zero_zero_nat != hAPP_int_nat(number_number_of_nat,W_15) -> zero_z891286103de_int = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),hAPP_int_nat(number_number_of_nat,W_15))))) # label(fact_42_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 653 (all Z_1 all W hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),Z_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),W)) # label(fact_102_zadd__commute) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 654 (all Ma all Na hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,Ma),hAPP_nat_nat(suc,Na)) = hAPP_nat_nat(nat_case_nat(hAPP_nat_nat(suc,Na),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),Na))),Ma)) # label(fact_5191_max__Suc2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 655 (all Xa (one_one_nat = Xa <-> one_one_nat = Xa)) # label(fact_855_one__reorient) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 656 (all X_30 all Y_24 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_24),X_30)) | X_30 = Y_24 | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_30),Y_24)))) # label(fact_1838_linorder__less__linear) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 657 (all F all Xa all L (hBOOL(hAPP_real_bool(deriv_real(F,Xa),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,Xa)),hAPP_real_real(F,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Xa),H_1)))))))))))) # label(fact_3879_DERIV__neg__dec__left) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 658 (all C_21 all A_71 all B_48 (A_71 = B_48 -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_21),B_48)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_21),A_71))))) # label(fact_1677_xt1_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 659 (all A_101 all B_72 (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_101),B_72))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_101)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_72))))) # label(fact_1398_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 660 (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_3683_tan__half) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 661 (all A (hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_nat_int(semiri1621563631at_int,A))) <-> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)))) # label(fact_4598_transfer__nat__int__set__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 662 (all N_35 all A_166 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A_166)) -> 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_166),N_35))))) # label(fact_712_one__le__power) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 663 (all Na all Ma (is_int(Ma) & is_int(Na) -> (hAPP_i1732201573de_int(quickcheck_of_int,Na) = hAPP_i1732201573de_int(quickcheck_of_int,Ma) <-> Na = Ma))) # label(fact_4557_Quickcheck__Narrowing_Oof__int__inject) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 664 (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_4518_zgcd__0__1__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 665 (all V_14 all W_6 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_14)),hAPP_int_int(number_number_of_int,W_6)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_14),W_6))) # label(fact_1099_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 666 (all A_12 all B_12 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(abs_abs_rat,A_12)),hAPP_rat_rat(abs_abs_rat,B_12))),hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_12),B_12))))) # label(fact_2732_abs__triangle__ineq2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 667 (all K_2 all M all 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)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(div_mod_nat(M),N))) # label(fact_3173_mod__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 668 (all Xa all Na all Ya (hAPP_nat_nat(div_mod_nat(Ya),Na) = hAPP_nat_nat(div_mod_nat(Xa),Na) <-> (exists Q1 exists Q2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Xa),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Na),Q1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ya),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Na),Q2))))) # label(fact_3219_nat__mod__eq__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 669 (all P all Q all R hAPP_r279757445t_bool(hAPP_f1840085295t_bool(hAPP_f1083049571t_bool(cOMBB_2050079537l_real,P),Q),R) = hAPP_f605995867t_bool(P,hAPP_r1134773055l_bool(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]. 4.69/4.78 670 (all W_15 ((zero_zero_nat = hAPP_int_nat(number_number_of_nat,W_15) -> one_one_complex = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),hAPP_int_nat(number_number_of_nat,W_15))) & (zero_zero_nat != hAPP_int_nat(number_number_of_nat,W_15) -> zero_zero_complex = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),hAPP_int_nat(number_number_of_nat,W_15))))) # label(fact_43_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 671 (all V_17 ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_17))) -> zero_zero_int = hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_17))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_17))) -> hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_17)) = hAPP_int_int(number_number_of_int,V_17)))) # label(fact_902_of__nat__number__of__lemma) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 672 (all A_1 all B_1 (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A_1),B_1)) -> -hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,B_1),A_1)))) # label(fact_1778_order__less__asym_H) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 673 (all N -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat))) # label(fact_305_not__less0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 674 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),pls)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K)),pls)))) # label(fact_113_rel__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 675 (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_3464_zminus__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 676 (all Xa 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,Xa))),hAPP_real_real(sin,Xa)))) # label(fact_4863_sin__converges) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 677 (all Z_2 all Ya all Xa all X_2 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Xa),X_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ya),Xa)) -> ((all X_1 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),X_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Z_2)))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ya),comple124823625p_real(X_2))))))) # label(fact_4643_SupInf_OSup__upper2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 678 (all A_1 (A_1 = zero_zero_rat <-> zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_1),A_1))) # label(fact_128_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 679 (all R_2 all S_1 all T_1 all R_4 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_4 -> (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_2 -> hAPP_int_int(div_mod_int(R_4),R_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(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_4),R_2)),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_4),R_2)),T_1))),N))))) # label(fact_3111_xzgcda__linear__aux2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 680 (all X_59 all Y_49 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_59),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_49),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_49)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_59),Y_49))))) # label(fact_989_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 681 (all K_2 hAPP_int_rat(number_number_of_rat,K_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(number_number_of_int,K_2)),one_one_int)) # label(fact_4708_rat__number__expand_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 682 (all Na big_co1705425894at_nat(cOMBI_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(one_one_nat),Na)) = hAPP_nat_nat(fact_fact_nat,Na)) # label(fact_5072_fact__altdef__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 683 (all M_19 hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_n522471361de_int(semiri1424489471de_int,M_19)))) # label(fact_720_zero__le__imp__of__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 684 (all V_9 all U_3 all Y_23 all X_28 all A_63 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_28),A_63)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_23),A_63)) -> (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_9)) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,U_3),V_9) = one_one_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,U_3),X_28)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_9),Y_23))),A_63)))))))) # label(fact_1859_convex__bound__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 685 (all I hAPP_int_nat(nat,quickcheck_int_of(I)) = quickcheck_nat_of(I)) # label(fact_4268_nat__of__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 686 (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_2516_diff__diff__right) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 687 (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_4129_complex__Im__mult) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 688 (all Z_2 all Wa (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Z_2),Wa)) & Wa != Z_2 <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z_2),Wa)))) # label(fact_4646_less__rat__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 689 (all Va all Na (hAPP_nat_nat(suc,Na) = hAPP_int_nat(number_number_of_nat,Va) <-> (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va)))) -> hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va))) = Na) & -hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va)))))) # label(fact_4778_eq__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 690 (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_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))) = 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)))))) # label(fact_4789_zpower__number__of__odd) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 691 (all M_24 all N_48 hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_24),N_48)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(semiri151668891at_rat,M_24)),hAPP_nat_rat(semiri151668891at_rat,N_48))) # label(fact_370_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 692 (all N -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N))) # label(fact_288_less__irrefl__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 693 (all X_3 all Y_3 hAPP_real_real(sin,hAPP_real_real(arccos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_3),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_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,Y_3),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_3),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_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,Y_3),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_3682_sin__arccos__lemma1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 694 (all C_39 all A_96 all B_67 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_96),B_67)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_39)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_39),A_96)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_39),B_67)))))) # label(fact_1425_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 695 (all W_15 ((hAPP_int_nat(number_number_of_nat,W_15) = 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_15))) & (zero_zero_nat != hAPP_int_nat(number_number_of_nat,W_15) -> zero_zero_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),hAPP_int_nat(number_number_of_nat,W_15))))) # label(fact_45_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 696 (all M_14 all A_90 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_90),one_on1645066479umeral)),M_14) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,M_14),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_90),M_14))) # label(fact_1480_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 697 (all A_158 all B_106 all C_60 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_158),B_106)),C_60) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_158),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_106),C_60))) # label(fact_831_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 698 (all Ya all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Xa)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Ya)) -> (zero_zero_real = Ya & Xa = zero_zero_real <-> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa),Ya) = zero_zero_real)))) # label(fact_585_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 699 (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_1 (Y = hAPP_real_real(cos,X_1) & (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),pi)) & Y = hAPP_real_real(cos,Y_1) -> X_1 = Y_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),pi)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1))))))) # label(fact_3667_cos__total) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 700 (all B_1 all A_1 all C (C = B_1 <-> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),A_1) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_1),A_1))) # label(fact_802_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 701 (all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,pls)),Ya)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),hAPP_int_int(number_number_of_int,Ya))))) # label(fact_640_le__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 702 (all V_11 all B_62 all C_34 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_i769753017umeral(number1443263063umeral,V_11)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_62),C_34)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_i769753017umeral(number1443263063umeral,V_11)),B_62)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_i769753017umeral(number1443263063umeral,V_11)),C_34))) # label(fact_1462_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 703 (all Xa all Ya (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Xa),Ya)) & Xa != Ya <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Xa),Ya)))) # label(fact_2031_order__less__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 704 (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_4419_floor__add) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 705 (all Ya (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,pls)),Ya)))) # label(fact_124_less__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 706 (all A_3 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),zero_zero_nat))) # label(fact_2962_gcd__lcm__complete__lattice__nat_Otop__greatest) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 707 (all P_1 collect_real(P_1) = P_1) # label(fact_146_Collect__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 708 (all B_118 all C_68 all A_172 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_172)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_118),C_68)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_118),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_172),C_68)))))) # label(fact_654_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 709 (all R_2 all P_3 all Q_1 (hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_1) = normalize(R_2) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Q_1)))) # label(fact_4660_normalize__denom__pos) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 710 (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_2130_is__mult__sum2sq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 711 (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_731_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 712 (all N_17 all M_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_17),M_10)) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,M_10)),hAPP_nat_int(semiri1621563631at_int,N_17)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_10),N_17)))) # label(fact_2244_of__nat__diff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 713 (all A_120 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_120),A_120)))) # label(fact_1285_zero__le__square) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 714 (all M_20 all N_38 all A_175 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_175)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_175),M_20)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_175),N_38))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_20),N_38))))) # label(fact_624_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 715 (all Xa 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,Xa)))) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,Xa))) # label(fact_4822_lemma__sin__minus) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 716 (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_2475_dvd__diffD1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 717 (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_156_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 718 (all Ya all Xa (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Ya),Xa)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Xa),Ya)) <-> Xa = Ya))) # label(fact_1688_order__antisym__conv) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 719 (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_4172_prime__dvd__power) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 720 (all Xa all Ya (zero_zero_real = Ya & Xa = zero_zero_real <-> zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa),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_3_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 721 (all A_1 all C all B_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_1),C))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B_1)))) # label(fact_549_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 722 (all P_1 P_1 = collect_int(P_1)) # label(fact_150_Collect__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 723 (all B_2 all A_3 (zero_zero_nat != A_3 | zero_zero_nat != B_2 -> 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_4369_div__gcd__coprime__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 724 (all A_3 hAPP_real_real(cos,A_3) = hAPP_complex_real(re,cis(A_3))) # label(fact_4145_Re__cis) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 725 (all N_17 all M_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_17),M_10)) -> hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_10),N_17)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(semiri132038758t_real,M_10)),hAPP_nat_real(semiri132038758t_real,N_17)))) # label(fact_2243_of__nat__diff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 726 (all P all Q all R hAPP_i1671359920t_real(hAPP_f74147283t_real(hAPP_f1169087445t_real(cOMBB_2129825580al_int,P),Q),R) = hAPP_f766218899t_real(P,hAPP_i371647666l_real(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]. 4.69/4.78 727 (all X_54 all Y_46 all Q_9 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_54),Y_46)),Q_9) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_54),Q_9)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_46),Q_9))) # label(fact_1251_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 728 (all N_17 all M_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_17),M_10)) -> hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_10),N_17)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_nat_rat(semiri151668891at_rat,M_10)),hAPP_nat_rat(semiri151668891at_rat,N_17)))) # label(fact_2241_of__nat__diff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 729 (all P all Q all R hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,P),Q),R) = hAPP_bool_bool(hAPP_n1006566506l_bool(P,R),hAPP_nat_bool(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__HOL__Obool_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 730 (all P (is_bool(P) -> P = fTrue | P = fFalse)) # label(help_If_3_1_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_T) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 731 (all Xa (Xa = one_on1645066479umeral <-> Xa = one_on1645066479umeral)) # label(fact_853_one__reorient) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 732 (all X_7 all Y_6 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_7),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_6),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_7)),Y_6)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_7),Y_6)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_2457_power2__diff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 733 (all Xa all Ya all Z_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Ya),Z_2))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Z_2)))) # label(fact_4316_gcd__semilattice__nat_Ole__inf__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 734 (all P all Q all R hAPP_real_bool(P,hAPP_nat_real(Q,R)) = hAPP_nat_bool(hAPP_f737665877t_bool(hAPP_f605995867t_bool(cOMBB_real_bool_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__HOL__Obool_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 735 (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_4495_zgcd__left__commute) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 736 (all Xa all Ya (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ya),Xa)) <-> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)))) # label(fact_2048_linorder__not__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 737 (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_2920_MultInv__prop2a) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 738 (all N_22 all A_62 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_62)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_62),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_62),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_62),N_22))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_62),N_22)))))) # label(fact_1860_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 739 (all A_121 all B_90 all N_28 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_121),N_28)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_90),N_28)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_121),B_90)),N_28)) # label(fact_1243_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 740 (all B_115 all A_169 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_169)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_115)) -> 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_169),B_115)))))) # label(fact_675_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 741 (all Ma all Wa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),hAPP_nat_int(semiri1621563631at_int,Ma))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat,Wa)),Ma))))) # label(fact_2805_nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 742 (all Xa (is_int(Xa) -> (zero_zero_int = Xa <-> Xa = zero_zero_int))) # label(fact_769_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 743 (all X_43 all Y_37 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_43),Y_37)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_37),X_43)) -> Y_37 = X_43))) # label(fact_1655_order__antisym) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 744 (all Ma all Na (Na = Ma <-> hAPP_nat_complex(semiri2020571505omplex,Na) = hAPP_nat_complex(semiri2020571505omplex,Ma))) # label(fact_323_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 745 (all C_75 all D_22 all A_186 all B_129 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_186),B_129)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_75),D_22)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_186),C_75)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_129),D_22)))))) # label(fact_526_add__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 746 (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_3626_sin__cos__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 747 (all A_1 all E_2 all C all B_1 all D_1 (hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_1),E_2)),C) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_1),E_2)),D_1) <-> 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_1),A_1)),E_2)),D_1) = C)) # label(fact_2386_eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 748 (all K_2 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),min) = hAPP_int_int(pred,K_2)) # label(fact_4046_add__Min__right) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 749 (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_3660_sin__less__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 750 (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_3556_real__sqrt__sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 751 (all C_38 all A_95 all B_66 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_95),B_66)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_38)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_38),A_95)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_38),B_66)))))) # label(fact_1432_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 752 (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]. 4.69/4.78 753 (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_5079_atLeastLessThanPlusOne__atLeastAtMost__int) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 754 (all A_125 A_125 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_125),one_one_nat)) # label(fact_1218_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 755 (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_3037_zdiv__zadd1__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 756 (all A_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_1),A_1)),zero_zero_real)))) # label(fact_98_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 757 (all M_21 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_21)),zero_zero_nat))) # label(fact_456_of__nat__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 758 (all N_36 all A_167 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_167)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_167),N_36))))) # label(fact_703_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 759 (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_3940_Suc__le__D) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 760 (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_4326_dvd__gcd__D1__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 761 (all X_42 all Y_36 (X_42 = Y_36 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_42),Y_36)))) # label(fact_1699_order__eq__refl) # label(axiom) # label(non_clause). [assumption]. 4.69/4.78 762 (all Ma all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),K)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),hAPP_int_int(abs_abs_int,K))))) # label(fact_2696_dvd__abs__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 763 (all Na all Xa all Diff all F (F = hAPP_n546711566l_real(Diff,zero_zero_nat) & (all M_1 all X_1 hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),X_1),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),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,Xa))) & hAPP_real_real(F,Xa) = 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,Xa))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na))),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,Na),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,Na)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa),Na))))))) # label(fact_4940_Maclaurin__all__le__objl) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 764 (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_3621_sin__cos__squared__add) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 765 (all K_2 hAPP_int_int(number_number_of_int,K_2) = quickcheck_int_of(hAPP_i1732201573de_int(number1226105091de_int,K_2))) # label(fact_4252_int__of__number) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 766 (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_1264_zadd__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 767 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(nat,X)),hAPP_int_nat(nat,Y)))) # label(fact_2897_nat__div__distrib) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 768 (all B_132 all A_189 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_189),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_132),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_189),B_132)),zero_zero_real))))) # label(fact_497_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 769 (all A_134 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,zero_zero_complex),A_134) = zero_zero_complex) # label(fact_1140_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 770 (all Z_6 all X_13 all Y_10 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_13),Y_10)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_13),Z_6)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_13),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Y_10),Z_6)))))) # label(fact_2216_dvd__diff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 771 (all B_15 all D_6 all A_15 all C_5 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,A_15)),C_5)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,B_15)),D_6)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(abs_abs_rat,A_15)),hAPP_rat_rat(abs_abs_rat,B_15))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_5),D_6)))))) # label(fact_2723_abs__mult__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 772 (all X all A_3 zero_zero_real != hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)) # label(fact_4380_powr__not__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 773 (all C_33 all A_87 all B_60 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_33),A_87)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_33),B_60))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_33)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_87),B_60))))) # label(fact_1518_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 774 (all C all A_1 all B_1 (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A_1),B_1)) -> (B_1 = C -> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A_1),C))))) # label(fact_1749_ord__less__eq__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 775 (all L all U hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(ord_gr1297742076an_int(L),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)))) # label(fact_4620_card__greaterThanLessThan__int) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 776 (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_1023_zadd__power2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 777 (all X_20 all Y_16 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_20),Y_16)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_20),Y_16)))) # label(fact_1978_order__less__imp__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 778 (all A_152 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_152),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_896_zero__le__power2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 779 (all X_56 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_56),X_56))) # label(fact_1020_order__refl) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 780 (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_4687_Fract__of__nat__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 781 (all C_73 all A_184 all B_127 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_184),B_127)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_184),C_73)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_127),C_73))))) # label(fact_539_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 782 (all Xa all Ya (-hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Xa),Ya)) -> (Xa = Ya <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Xa),Ya))))) # label(fact_2000_linorder__antisym__conv1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 783 (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_3640_sin__gt__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 784 (all A_144 all C_55 all B_102 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_144),C_55)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_102),C_55))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_144),B_102)))) # label(fact_960_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 785 (all A_50 hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_50),zero_zero_complex))) # label(fact_2208_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 786 (all Xa all A_1 (hBOOL(monoseq_real(A_1)) -> (hBOOL(tendsto_nat_real(A_1,Xa,sequentially)) -> (all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A_1,N_1)),Xa))) & (all M_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A_1,M_1)),hAPP_nat_real(A_1,N_1))))) | (all M_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A_1,N_1)),hAPP_nat_real(A_1,M_1))))) & (all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),hAPP_nat_real(A_1,N_1))))))) # label(fact_5042_monoseq__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 787 (all P_1 all A_1 all B_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1)) -> (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_1,B_1)))),hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,produc556554744l_real),bolzano_bisect(P_1,A_1,B_1))),zero_zero_real,sequentially)) -> (exists L_1 (hBOOL(tendsto_nat_real(hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,produc556554744l_real),bolzano_bisect(P_1,A_1,B_1)),L_1,sequentially)) & (all N_1 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_1,B_1),N_1))))) & hBOOL(tendsto_nat_real(hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,produc1935615926l_real),bolzano_bisect(P_1,A_1,B_1)),L_1,sequentially)) & (all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A_1,B_1),N_1))),L_1)))))))))) # label(fact_5058_Bolzano__nest__unique) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 788 (all Xa (zero_zero_real = Xa <-> -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa),Xa))))) # label(fact_2119_not__real__square__gt__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 789 (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(hAPP_int_int(div_mod_int(A_3),B_2)),M) = hAPP_int_int(div_mod_int(A_3),M)))) # label(fact_3084_zmod__zdvd__zmod) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 790 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Xa)),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,Xa)),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_5049_zeroseq__arctan__series) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 791 (all C_47 all A_114 all B_84 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_114),B_84)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_47)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_114),C_47)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_84),C_47)))))) # label(fact_1323_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 792 (all A_3 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A_3))) # label(fact_2965_gcd__lcm__complete__lattice__nat_Obot__least) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 793 (all Ma all Na (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),Na) = zero_zero_nat <-> Ma = zero_zero_nat | Na = zero_zero_nat)) # label(fact_2086_mult__is__0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 794 (all B_1 all A_1 (B_1 = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_1),A_1) <-> zero_zero_complex = A_1)) # label(fact_904_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 795 (all Z_8 all Y_19 all X_23 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_19),X_23)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Z_8),Y_19)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Z_8),X_23))))) # label(fact_1936_xt1_I7_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 796 (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_4453_tan__periodic__int) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 797 (all A_3 all B_2 (quotient_of(B_2) = quotient_of(A_3) -> B_2 = A_3)) # label(fact_4650_quotient__of__inject) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 798 (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_3590_sin__diff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 799 (all M_23 all N_47 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,M_23)),N_47) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_23),N_47))) # label(fact_386_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 800 (all P all Q all R hAPP_int_int(hAPP_f1356864277nt_int(hAPP_f747162069nt_int(cOMBB_int_int_int,P),Q),R) = hAPP_int_int(P,hAPP_int_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__Int__Oint_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 801 (all A_140 all B_98 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_98),A_140) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_140),B_98)) # label(fact_1037_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 802 (all X_5 all Y_5 (is_int(X_5) & is_int(Y_5) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_5),Y_5)) -> (Y_5 != X_5 -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_5),Y_5)))))) # label(fact_2568_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 803 (all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(sqrt,Ya))))) # label(fact_3536_real__sqrt__gt__1__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 804 (all X all Y (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,X),Y)) | Y != X)) # label(help_fequal_2_1_fequal_000tc__Int__Oint_T) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 805 (all X_50 all Y_44 -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_50),X_50)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_44),Y_44))),zero_zero_real))) # label(fact_1588_not__sum__squares__lt__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 806 (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_3507_real__sqrt__divide) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 807 (all N zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,zero_zero_nat),N)) # label(fact_2084_mult__0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 808 (all A_44 all B_33 all K_6 (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_33),K_6) = A_44 -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_33),A_44)))) # label(fact_2265_dvdI) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 809 (all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_int_real(number267125858f_real,Ya))))) # label(fact_615_le__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 810 (all B_74 all A_103 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_103)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_74),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_103),B_74)),zero_zero_nat))))) # label(fact_1386_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 811 (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(I_2) & is_int(J_1) -> (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_4572_upto_Opinduct) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 812 (all X_39 all Y_33 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_39),Y_33)) -> -hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_33),X_39)))) # label(fact_1723_order__less__asym) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 813 (all A_1 all Na (zero_zero_rat = A_1 & zero_zero_nat != Na <-> zero_zero_rat = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_1),Na))) # label(fact_445_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 814 (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_4568_div__code__int__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 815 (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_2777_int__nat__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 816 (all C_53 all A_142 all B_100 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_142),B_100)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_53),A_142)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_53),B_100))))) # label(fact_968_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 817 (all A_1 all B_1 (A_1 != B_1 -> (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B_1),A_1)) -> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,B_1),A_1))))) # label(fact_1986_xt1_I12_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 818 (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_2472_dvd__diff__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 819 (all X_33 all Y_27 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_33),Y_27)) -> -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_27),X_33)))) # label(fact_1815_order__less__not__sym) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 820 (all Xa all Ya (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Xa),Ya)) | Ya = Xa <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Xa),Ya)))) # label(fact_2017_order__le__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 821 (all K_8 hAPP_int_real(number267125858f_real,hAPP_int_int(succ,K_8)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_int_real(number267125858f_real,K_8))) # label(fact_873_number__of__succ) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 822 (all V_6 all B_21 all C_7 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_6)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,B_21),C_7)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_6)),B_21)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_6)),C_7))) # label(fact_2394_right__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 823 (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_3731_real__natfloor__add__one__gt) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 824 (all X_47 all Y_41 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_47),Y_41)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_41),X_47)))) # label(fact_1625_linorder__le__cases) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 825 (all A_9 all N_7 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,A_9)),N_7)))) # label(fact_2759_zero__le__power__abs) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 826 (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_4155_termination__basic__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 827 (all P_1 all K all I_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),I_1)) -> (hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K),one_one_int))) -> ((all I_2 (is_int(I_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),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_271_int__gr__induct) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 828 (all X_34 all Y_28 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_34),Y_28)) -> -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_28),X_34)))) # label(fact_1808_order__less__imp__not__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 829 (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_5003_transfer__int__nat__gcd_I1_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 830 (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_3538_le__real__sqrt__sumsq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 831 (all Na (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(semiri132038758t_real,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)))) # label(fact_243_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 832 (all Z_15 all X_37 all Y_31 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_37),Y_31)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_31),Z_15)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_37),Z_15))))) # label(fact_1739_order__less__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 833 (all X_2 ((all R_1 exists N_2 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),N_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_1),hAPP_nat_real(X_2,N_1))))) -> hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),X_2),zero_zero_real,sequentially)))) # label(fact_5105_LIMSEQ__inverse__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 834 (all Y all X (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = Y)) # label(fact_4321_gcd__proj2__if__dvd__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 835 (all A_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),A_1)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),zero_zero_int)))) # label(fact_99_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 836 (all Wa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,Wa)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),zero_zero_int)))) # label(fact_117_bin__less__0__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 837 (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_3750_cos__npi) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 838 (all X hAPP_complex_real(re,hAPP_complex_complex(invers1449016382omplex,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))))))) # label(fact_4138_complex__Re__inverse) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 839 (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_5195_max__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 840 (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_3 = Z_2 <-> hAPP_int_nat(nat,Z_3) = hAPP_int_nat(nat,Z_2)))))) # label(fact_2741_eq__nat__nat__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 841 (all B_133 all A_190 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_190)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_133)) -> 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_190),B_133)))))) # label(fact_492_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 842 (all A_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_int_bool(hAPP_i1948725293t_bool(zcong(A_1),zero_zero_int),P_5)) -> hAPP_i1948725293t_bool(sRStar,P_5) = comple219730294t_bool(setS(A_1,P_5)))))) # label(fact_3331_MultInvPair__prop2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 843 (all Xa all Ya all P_1 ((-hBOOL(P_1) -> hAPP_nat_int(semiri1621563631at_int,Ya) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,P_1),Xa),Ya))) & (hBOOL(P_1) -> hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,P_1),Xa),Ya)) = hAPP_nat_int(semiri1621563631at_int,Xa)))) # label(fact_327_int__if__cong) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 844 (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,N),hAPP_real_real(hAPP_n546711566l_real(root,M),X)) = hAPP_real_real(hAPP_n546711566l_real(root,M),hAPP_real_real(hAPP_n546711566l_real(root,N),X))))) # label(fact_3892_real__root__commute) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 845 (all C_46 all A_113 all B_83 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_113),B_83)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_46)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_46),A_113)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_46),B_83)))))) # label(fact_1330_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 846 (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)) -> divmod_nat(M,N) = 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))))) # label(fact_4205_divmod__nat__step) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 847 (all Va all Wa (-hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_i769753017umeral(number1443263063umeral,Wa)),hAPP_i769753017umeral(number1443263063umeral,Va))) <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_i769753017umeral(number1443263063umeral,Va)),hAPP_i769753017umeral(number1443263063umeral,Wa))))) # label(fact_694_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 848 (all P all Q all R hAPP_n1006566506l_bool(hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,P),Q),R) = hAPP_b589554111l_bool(P,hAPP_nat_bool(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]. 4.69/4.79 849 (all B_74 all A_103 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_103)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_74),zero_z126310315umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_103),B_74)),zero_z126310315umeral))))) # label(fact_1384_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 850 (all P_2 all P_1 ((all X_1 (is_int(X_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(P_1,X_1)) <-> hBOOL(hAPP_int_bool(P_2,X_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)) = 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_4957_transfer__nat__int__set__cong) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 851 (all V_2 all V_1 ((-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1))) -> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,V_1)),hAPP_int_int(number_number_of_int,V_2))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(number_number_of_nat,V_1)),hAPP_int_nat(number_number_of_nat,V_2))) & (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(number_number_of_nat,V_1)),hAPP_int_nat(number_number_of_nat,V_2)) = zero_zero_nat))) # label(fact_3274_div__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 852 (all A_47 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,zero_zero_complex),A_47)) -> A_47 = zero_zero_complex)) # label(fact_2249_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 853 (all X all Y ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,ratreal(X)),ratreal(Y))) # label(fact_4700_real__plus__code) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 854 (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]. 4.69/4.79 855 (all X_17 all Y_13 (-hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_17),Y_13)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_13),X_17)))) # label(fact_2010_leI) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 856 (all Ma all Na (Ma = Na <-> hAPP_nat_int(semiri1621563631at_int,Ma) = hAPP_nat_int(semiri1621563631at_int,Na))) # label(fact_103_int__int__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 857 (all A_194 hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_194),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_194),one_on1684967323de_int)))) # label(fact_416_less__add__one) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 858 (all Xa all Ya (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,Xa)),hAPP_nat_int(semiri1621563631at_int,Ya))))) # label(fact_2530_Nat__Transfer_Otransfer__int__nat__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 859 (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_3811_even__nat__div__two__times__two) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 860 (all Lx_5 all Rx_5 all Ry_3 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_5),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_5),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),Rx_5)),Ry_3)) # label(fact_1049_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 861 (all A_33 all B_25 A_33 = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_33),B_25)),B_25)) # label(fact_2345_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 862 (all Ya all Xa 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,Xa),hAPP_i1948725293t_bool(sRStar,P_5))) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ya),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,Xa),Ya))),hAPP_i1948725293t_bool(sRStar,P_5)))))))) # label(fact_3211_SRStar__mult__prop1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 863 (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,Z_13),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_1888_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 864 (all A_1 all B_1 all K all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ma)) -> (one_one_int = hAPP_int_int(legacy_zgcd(K),Ma) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),K)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),K)),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),B_1),Ma)))))) # label(fact_4525_zcong__cancel) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 865 (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_5011_gcd__ge__0__int) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 866 (all W all V_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_1))) -> (-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_1))) -> posDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(number_number_of_int,V_1)) = hAPP_P1975530577nt_int(adjust(hAPP_int_int(number_number_of_int,V_1)),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_1))))) & (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_1))) -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),hAPP_int_int(number_number_of_int,W)) = posDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(number_number_of_int,V_1))))) # label(fact_3290_posDivAlg__eqn__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 867 (all C_56 all A_145 all B_103 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_56),A_145)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_56),B_103))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_145),B_103)))) # label(fact_956_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 868 (all X all Y all 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)) = 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))) # label(fact_1508_zpower__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 869 (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_3216_mod__induct__0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 870 (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 & B_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,S_2),N)))))) # label(fact_4479_coprime__pow) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 871 (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_3074_StandardRes__ubound) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 872 (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_2474_dvd__diffD) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 873 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),Y) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),hAPP_real_real(inverse_inverse_real,Y))) # label(fact_3854_divide__real__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 874 (all B_132 all A_189 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_189),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_132),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_189),B_132)),zero_zero_int))))) # label(fact_499_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 875 (all B_15 all D_6 all A_15 all C_5 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,A_15)),C_5)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,B_15)),D_6)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(abs_abs_real,A_15)),hAPP_real_real(abs_abs_real,B_15))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_5),D_6)))))) # label(fact_2724_abs__mult__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 876 (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]. 4.69/4.79 877 (all Z_6 all X_13 all Y_10 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_13),Y_10)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_13),Z_6)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_13),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_10),Z_6)))))) # label(fact_2218_dvd__diff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 878 (all C_36 all B_64 all A_93 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_64),A_93)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_36),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_36),A_93)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_36),B_64)))))) # label(fact_1441_mult__strict__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 879 (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_3366_ln__le__minus__one) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 880 (all A all B (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_nat_int(semiri1621563631at_int,A)),image_nat_int(semiri1621563631at_int,B))) <-> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)))) # label(fact_4604_transfer__nat__int__set__relations_I5_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 881 (all X hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(suc,X)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,X)),hAPP_nat_nat(fact_fact_nat,X))) # label(fact_3838_Fact_Ofact__Suc) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 882 (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]. 4.69/4.79 883 (all Xa all Ya (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Xa),Ya)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Ya),Xa)))) # label(fact_2040_linorder__not__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 884 (all X all 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_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y))) # label(fact_3596_sin__add) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 885 (all N all M all R_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),R_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),R_2)) -> (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_2)))))) # label(fact_4025_divides__mul) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 886 (all Na all A_1 (hBOOL(tendsto_nat_real(A_1,zero_zero_real,sequentially)) -> ((all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A_1,N_1)))) -> ((all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A_1,hAPP_nat_nat(suc,N_1))),hAPP_nat_real(A_1,N_1)))) -> 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_1)),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)))),Na)))),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_1)))))))) # label(fact_5060_summable__Leibniz_H_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 887 (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_2828_zdvd__mult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 888 (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_3117_Suc__pred_H) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 889 (all X (is_int(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(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) = X))) # label(fact_3225_two__times__odd__div__two__plus__one) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 890 (all Xa all Ya (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Xa),Ya)),zOdd)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd)))) # label(fact_2941_odd__mult__odd__prop) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 891 (all A_154 all N_32 all B_105 (hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_154),N_32) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_105),N_32) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_154)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_105)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32)) -> A_154 = B_105))))) # label(fact_859_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 892 (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_3708_real__of__nat__power) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 893 (all Ma all Na all A_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_1)) -> (hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_1),Na) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_1),Ma) <-> Ma = Na))) # label(fact_424_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 894 (all Xa all Ya (hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa)) & hBOOL(hAPP_nat_bool(even_odd_even_nat,Ya)) | -hBOOL(hAPP_nat_bool(even_odd_even_nat,Ya)) & -hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa)) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Xa),Ya))))) # label(fact_3787_even__sum__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 895 (all Wa all Z_2 (is_int(Z_2) -> ((exists N_1 Z_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Wa),hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_1)))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_2))))) # label(fact_3106_zless__iff__Suc__zadd) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 896 (all X_14 all P_6 all Q_7 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_14),P_6)),Q_7) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_14),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_6),Q_7))) # label(fact_2092_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 897 (all N_46 all A_195 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_195)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_195),N_46))))) # label(fact_411_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 898 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),hAPP_int_int(bit0,K))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K)))) # label(fact_2173_rel__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 899 (all N N = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),N)) # label(fact_5194_max__0L) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 900 (all Xa all Ya (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya)) -> Ya != Xa)) # label(fact_1792_order__less__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 901 (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]. 4.69/4.79 902 (all C_26 all D_11 all A_80 all B_53 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_80),B_53)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_26),D_11)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_80)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_26)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_80),C_26)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_53),D_11)))))))) # label(fact_1562_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 903 (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)) -> posDivAlg(A_3,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),A_3)) & (-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)))) # label(fact_3291_posDivAlg__eqn) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 904 (all K all Xa 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)),Xa),hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa),K))))) # label(fact_4748_DERIV__sin__add) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 905 (all X_43 all Y_37 (is_int(X_43) & is_int(Y_37) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_43),Y_37)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_37),X_43)) -> Y_37 = X_43)))) # label(fact_1656_order__antisym) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 906 (all A_126 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,one_one_complex),A_126) = A_126) # label(fact_1208_mult__1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 907 (all L all U hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_gr660468384an_nat(L),U)))) # label(fact_4624_finite__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 908 (all A_21 all B_16 hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B_16),A_21)) = hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_21),B_16))) # label(fact_2684_abs__minus__commute) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 909 (all A_3 all B_2 (is_int(B_2) -> (zero_zero_int != B_2 -> (zero_zero_int != hAPP_int_int(div_mod_int(A_3),B_2) -> 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_3498_zdiv__zminus2__eq__if) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 910 (all P_1 all L ((exists X_1 (hBOOL(hAPP_nat_bool(P_1,X_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,L),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_1),zero_zero_nat))))) <-> (exists X_1 hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,L),X_1)))))) # label(fact_2414_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 911 (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]. 4.69/4.79 912 (all A_191 one_one_int = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_191),zero_zero_nat)) # label(fact_464_power__0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 913 (all M_14 all A_90 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_90),one_one_rat)),M_14) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,M_14),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_90),M_14))) # label(fact_1477_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 914 (all Xa all A (hBOOL(hAPP_real_bool(A,Xa)) <-> hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Xa),A)))) # label(fact_140_mem__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 915 (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),X)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))))) # label(fact_2623_dvd_Oless__asym) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 916 (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_4012_coprime__lmul2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 917 (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_2493_Nat_Odiff__le__self) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 918 (all C_31 all A_85 all B_58 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_31),A_85)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_31),B_58))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_31)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_85),B_58))))) # label(fact_1532_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 919 (all P_1 all L ((exists X_1 (hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,L),X_1))) & is_int(X_1))) <-> (exists X_1 (is_int(X_1) & hBOOL(hAPP_int_bool(P_1,X_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,L),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),zero_zero_int))))))) # label(fact_2415_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 920 (all Ma all Na (hBOOL(hAPP_nat_bool(nat_case_bool(fFalse,hAPP_n1699378549t_bool(ord_less_eq_nat,Ma)),Na)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Ma)),Na)))) # label(fact_4876_less__eq__nat_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 921 (all B_75 all A_104 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_104)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_75)) -> 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_104),B_75)))))) # label(fact_1376_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 922 (all B_112 all A_164 all C_66 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_112),A_164) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_66),A_164) -> C_66 = B_112)) # label(fact_782_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 923 (all A_123 A_123 = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_123),one_on1645066479umeral)) # label(fact_1230_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 924 (all A_1 all B_1 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_1),B_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B_1)))))) # label(fact_1580_mult__le__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 925 (all F all A_1 all B_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1)) -> ((all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),B_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),X_1)) -> (exists Y_1 (hBOOL(hAPP_real_bool(deriv_real(F,X_1),Y_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_1)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(F,A_1)),hAPP_real_real(F,B_1)))))) # label(fact_4546_DERIV__nonneg__imp__nonincreasing) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 926 (all X hAPP_real_real(exp_real,X) = hAPP_real_real(abs_abs_real,hAPP_real_real(exp_real,X))) # label(fact_3393_abs__exp__cancel) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 927 (all A_91 all B_61 all V_10 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_91),hAPP_int_nat(number_number_of_nat,V_10))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_61),hAPP_int_nat(number_number_of_nat,V_10))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_91),B_61)),hAPP_int_nat(number_number_of_nat,V_10))) # label(fact_1467_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 928 (all A_1 all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),archim856651990g_real(Xa))) <-> 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_1)),one_one_real)),Xa)))) # label(fact_4471_le__ceiling__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 929 (all X_61 all Y_51 (hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_51),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_61),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_61)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),Y_51)) -> X_61 = Y_51)))) # label(fact_886_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 930 (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_2189_left__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 931 (all A_126 A_126 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,one_one_rat),A_126)) # label(fact_1206_mult__1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 932 (all X_16 all Y_12 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_16),Y_12)) | hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_12),X_16)))) # label(fact_2036_linorder__le__less__linear) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 933 (all A_1 all B_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_1),B_1)) <-> (exists X_1 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_1),X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_1),Y_1)) = 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_1),X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_1),Y_1)) = one_one_nat)))) # label(fact_4035_coprime__bezout) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 934 (all B_2 all A_3 (A_3 != 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)),A_3)))) # label(fact_4345_gcd__le1__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 935 (all V_1 all W hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_1),W)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_1)),hAPP_int_int(number_number_of_int,W))) # label(fact_59_plus__numeral__code_I9_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 936 (all Xa all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ya),Xa)) <-> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)))) # label(fact_2049_linorder__not__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 937 (all F (hBOOL(summable_real(F)) -> ((all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(F,N_1)))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_f352196356l_real(suminf_real,F)))))) # label(fact_4857_suminf__gt__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 938 (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_1270_trans__le__add2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 939 (all A_124 A_124 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_124),one_one_nat)) # label(fact_1225_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 940 (all B_71 all A_100 (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_71),A_100))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_100)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_71))))) # label(fact_1400_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 941 (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_2595_zprime__zdvd__zmult__better) # label(axiom) # label(non_clause). [assumption]. 4.69/4.79 942 (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_1619_mult__Bit1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 943 (all Na all Ma (hAPP_nat_real(real_nat,Na) = hAPP_nat_real(real_nat,Ma) <-> Na = Ma)) # label(fact_3690_real__of__nat__inject) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 944 (all Xa all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,Xa)),hAPP_int_real(number267125858f_real,Ya))))) # label(fact_508_le__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 945 (all A_64 all B_41 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_64),B_41)) -> -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_41),A_64)))) # label(fact_1780_order__less__asym_H) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 946 (all Ma all Na divmod_nat(Ma,Na) = the_Pr588456374at_nat(divmod_nat_rel(Ma,Na))) # label(fact_4888_divmod__nat__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 947 (all Xa all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) -> hBOOL(hAPP_real_bool(isCont_real_real(hAPP_n546711566l_real(root,Na)),Xa)))) # label(fact_5155_isCont__real__root) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 948 (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_3042_dvd__1__iff__1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 949 (all Z_9 all X_24 all Y_20 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_24),Y_20)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_20),Z_9)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_24),Z_9))))) # label(fact_1929_order__le__less__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 950 (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(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_3901_real__root__mult) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 951 (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_3656_cos__double__less__one) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 952 (all A_86 all C_32 all B_59 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_86),C_32)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_59),C_32))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_32)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_86),B_59))))) # label(fact_1521_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 953 (all P hAPP_nat_nat(cOMBI_nat,P) = P) # label(help_COMBI_1_1_COMBI_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 954 (all F all G ((all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_nat_real(F,N_1))),hAPP_nat_real(G,N_1)))) -> (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_4856_summable__le2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 955 (all X_2 (hBOOL(cauchy_complex(X_2)) -> hBOOL(cauchy_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_2))))) # label(fact_4755_cnj_OCauchy) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 956 (all B_40 all A_58 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_40),A_58)) -> (B_40 != A_58 -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_40),A_58))))) # label(fact_1950_xt1_I11_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 957 (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_2504_diff__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 958 (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_3558_real__sqrt__ge__abs2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 959 (all A_1 all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),archim1246769320r_real(Xa))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,A_1)),Xa)))) # label(fact_4418_le__floor__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 960 (all Na 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,Na)),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,Na),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,Na))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),Na))) # label(fact_3819_lemma__STAR__sin) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 961 (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]. 4.69/4.80 962 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,Xa)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),zero_zero_real)))) # label(fact_3529_real__sqrt__lt__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 963 (all Xa (Xa = one_on1684967323de_int <-> one_on1684967323de_int = Xa)) # label(fact_851_one__reorient) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 964 (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_4464_real__of__int__div3) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 965 (all A_9 all N_7 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(abs_abs_real,A_9)),N_7)))) # label(fact_2758_zero__le__power__abs) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 966 (all Na all Ma (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,Na)),hAPP_int_real(real_int,Ma))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,Na)),Ma)))) # label(fact_4442_number__of__less__real__of__int__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 967 (all Va all Na hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_int_nat(number_number_of_nat,Va)),hAPP_nat_nat(suc,Na)) = 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,Va)))),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,Va)))),Na)))) # label(fact_5119_min__Suc__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 968 (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_3133_Nat__Transfer_Otransfer__int__nat__functions_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 969 (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_4398_real__of__int__minus) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 970 (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_3676_arccos__cos2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 971 (all C_20 all A_70 all B_47 (B_47 = A_70 -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_47),C_20)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_70),C_20))))) # label(fact_1678_ord__eq__le__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 972 (all P all Q all R hAPP_f800510211t_bool(hAPP_i1796122964t_bool(P,R),Q) = hAPP_i418383825t_bool(hAPP_f289318786t_bool(hAPP_f729588221t_bool(cOMBC_678732631t_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 973 (all C all A_1 all B_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),A_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),B_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B_1)))) # label(fact_981_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 974 (all X all Y hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),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))))) # label(fact_5020_gcd__int__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 975 (all Xa all A (hBOOL(hAPP_int_bool(A,Xa)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),A)))) # label(fact_144_mem__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 976 (all B_71 all A_100 (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_71),A_100))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_100)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_71))))) # label(fact_1404_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 977 (all Ma all Na all K (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na))))) # label(fact_2152_nat__mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 978 (all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,pls)),Ya)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,Ya))))) # label(fact_638_le__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 979 (all X_29 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_29),X_29))) # label(fact_1854_order__less__irrefl) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 980 (all Ta all B all D (is_int(Ta) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ta),B)) -> (all X_1 (is_int(X_1) -> ((all Xa_1 (is_int(Xa_1) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa_1) != X_1)))))) -> (X_1 != Ta -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),D) != Ta)))))))) # label(fact_5099_bset_I4_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 981 (all A all B all Ma (hBOOL(hAPP_f1469637905l_bool(hAPP_P1681069257l_bool(member308914164t_bool,hAPP_f538843494t_bool(hAPP_f1418982953t_bool(produc713279741t_bool,A),B)),bijR_int_int(zcongm(Ma)))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(big_co1548731110nt_int(cOMBI_int,A)),big_co1548731110nt_int(cOMBI_int,B)),Ma)))) # label(fact_5154_bijzcong__zcong__prod) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 982 (all M all N (M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) -> zero_zero_nat = N)) # label(fact_314_add__eq__self__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 983 (all A_1 all B_1 all A all Ma (is_int(B_1) & is_int(A_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma)) -> (hBOOL(hAPP_int_bool(is_RRset(A),Ma)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),B_1),Ma)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),A)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),A)) -> B_1 = A_1))))))) # label(fact_4589_RRset__zcong__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 984 (all A_155 all C_57 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_57),A_155) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_155),C_57)) # label(fact_844_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 985 (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_3748_LIMSEQ__inverse__realpow__zero__lemma) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 986 (all Ma all Na 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),Na))),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Ma),hAPP_nat_nat(suc,Na))) # label(fact_5124_min__Suc2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 987 (all M_22 all N_45 all A_193 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_193)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_193),M_22)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_193),N_45))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_22),N_45))))) # label(fact_436_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 988 (all A_147 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),A_147) = A_147) # label(fact_941_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 989 (all Xa all Ya (is_int(Ya) & is_int(Xa) -> (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,Xa),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,Ya),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) <-> zero_zero_int != Xa | Ya != zero_zero_int))) # label(fact_216_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 990 (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]. 4.69/4.80 991 (all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,pls)),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_int_real(number267125858f_real,Ya))))) # label(fact_639_le__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 992 (all A_82 all C_28 all B_55 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_82),C_28)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_55),C_28))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_28)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_82),B_55))))) # label(fact_1547_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 993 (all Xa all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),one_one_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa)),one_one_real))))) # label(fact_3920_real__root__lt__1__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 994 (all A_165 all N_33 all B_113 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_165),N_33)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_113),N_33))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_113)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_165),B_113))))) # label(fact_754_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 995 (all P_2 all P_1 all Xa ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) -> (hBOOL(P_1) <-> hBOOL(P_2))) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) -> hBOOL(P_1)) <-> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) -> hBOOL(P_2))))) # label(fact_2051_imp__le__cong) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 996 (all N_58 hAPP_int_complex(number528085621omplex,hAPP_nat_int(semiri1621563631at_int,N_58)) = hAPP_nat_complex(semiri2020571505omplex,N_58)) # label(fact_204_number__of__int) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 997 (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_3927_real__root__pos2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 998 (all A_159 all B_107 all C_61 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_159),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_107),C_61)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_159),B_107)),C_61)) # label(fact_825_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 999 (all Ta all B all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D)) -> (all X_1 (is_int(X_1) -> ((all Xa_1 (is_int(Xa_1) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B)) -> X_1 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa_1))))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Ta)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),D)),Ta)))))))) # label(fact_5095_bset_I5_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1000 (all X_8 all N_12 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_12)) | X_8 = one_one_complex -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_8),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_8),N_12))))) # label(fact_2452_dvd__power) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1001 (all W_4 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_int_complex(number528085621omplex,W_4)),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_4)),hAPP_int_complex(number528085621omplex,W_4))) # label(fact_1912_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1002 (all Z_4 all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),zero_zero_int)) -> 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))) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),Z_4)))) # label(fact_3497_nat__mult__distrib__neg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1003 (all P all Q all R hAPP_f1805168059t_bool(P,hAPP_i1948725293t_bool(Q,R)) = hAPP_i1948725293t_bool(hAPP_f428220345t_bool(hAPP_f654702867t_bool(cOMBB_591320580ol_int,P),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]. 4.69/4.80 1004 (all B_1 all A_1 all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),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_1),hAPP_i1948725293t_bool(wset(A_1),P_5))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int))))))) # label(fact_2931_wset__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1005 (all Na all Xa hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,power_power_real),Na),Xa),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,Na)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),hAPP_nat_nat(suc,zero_zero_nat))))))) # label(fact_4781_DERIV__pow) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1006 (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_3915_real__root__gt__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1007 (all B_5 all Q_5 all R_4 (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_5)),R_4)),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_4)) -> (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_5),zero_zero_int)))))) # label(fact_2063_q__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1008 (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]. 4.69/4.80 1009 (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_4186_prime__coprime__lt) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1010 (all A_1 all B_1 all C all D_1 (hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),D_1) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_1),B_1) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C),D_1))))) # label(fact_2334_diff__eq__diff__less__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1011 (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_3035_zmod__zdiv__trivial) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1012 (all K all Ma all Na (Ma = Na <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),Na))) # label(fact_3025_Suc__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1013 (all X_2 all A_1 (hBOOL(hAPP_complex_bool(sums_complex(X_2),A_1)) -> hBOOL(hAPP_complex_bool(sums_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_2)),hAPP_complex_complex(cnj,A_1))))) # label(fact_4870_cnj_Osums) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1014 (all K_2 hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,K_2)),one_one_int) = quotient_of(hAPP_int_rat(number_number_of_rat,K_2))) # label(fact_4654_quotient__of__number_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1015 (all A_3 all Q_1 all R_2 (is_int(A_3) & is_int(R_2) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,A_3),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_2))) -> (zero_zero_int != A_3 -> R_2 = zero_zero_int)))) # label(fact_3256_self__remainder) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1016 (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_390_Nat__Transfer_Otransfer__int__nat__functions_I4_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1017 (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_1273_add__leD2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1018 (all A_1 hBOOL(tendst1507391555omplex(cnj,hAPP_complex_complex(cnj,A_1),at_complex(A_1)))) # label(fact_5078_cnj_Ocont) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1019 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ya),Xa)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)) <-> Xa != Ya)) # label(fact_1847_linorder__neq__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1020 (all Na all Va (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,Na)),hAPP_int_nat(number_number_of_nat,Va))) <-> -hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va)))) & (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va)))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va)))))))) # label(fact_4783_less__Suc__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1021 (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_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),D_3)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(real_nat,D_3))))) # label(fact_3736_real__of__nat__div) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1022 (all K_2 (is_int(K_2) -> K_2 = hAPP_int_int(number_number_of_int,K_2))) # label(fact_279_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1023 (all Ma all Va (hAPP_int_int(number_number_of_int,Va) = hAPP_nat_int(semiri1621563631at_int,Ma) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,Va))) & hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,Va)) = Ma)) # label(fact_2797_int__eq__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1024 (all C_19 all B_46 all A_69 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_46),A_69)) -> (B_46 = C_19 -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_19),A_69))))) # label(fact_1747_xt1_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1025 (all Na image_int_nat(nat,hAPP_i1948725293t_bool(ord_at875362053st_int(zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Na))) = hAPP_n1699378549t_bool(ord_atMost_nat,Na)) # label(fact_5182_SetInterval_Otransfer__nat__int__set__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1026 (all X_8 all N_12 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_12)) | one_on1684967323de_int = X_8 -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,X_8),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_8),N_12))))) # label(fact_2451_dvd__power) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1027 (all F all Xa all L (hBOOL(hAPP_real_bool(deriv_real(F,Xa),L)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),L)) -> (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(minus_minus_real,Xa),H_1))),hAPP_real_real(F,Xa))))))))))) # label(fact_3880_DERIV__pos__inc__left) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1028 (all N hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(suc,zero_zero_nat)),N)) # label(fact_3069_power__Suc__0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1029 (all A_140 all B_98 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_140),B_98) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_98),A_140)) # label(fact_1032_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1030 (all Xa all Ya (Xa != Ya & hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Ya)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Ya)))) # label(fact_2029_order__less__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1031 (all W_5 hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,W_5)) = 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_5))) # label(fact_1867_double__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1032 (all X_2 all A_1 (hBOOL(hAPP_complex_bool(sums_complex(X_2),A_1)) -> hBOOL(hAPP_real_bool(sums_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_2)),hAPP_complex_real(im,A_1))))) # label(fact_4872_Im_Osums) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1033 (all Na (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Na),zero_zero_int)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,Na)),zero_zero_real)))) # label(fact_4440_real__of__int__le__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1034 (all Ya all Xa (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ya),Xa)) -> (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)) <-> Xa = Ya))) # label(fact_1832_linorder__antisym__conv3) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1035 (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_2912_MultInv__prop3) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1036 (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_3803_odd__nat__mod__two__eq__one) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1037 (all Xa all Na all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Na),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xa),Ya))) <-> hAPP_int_int(div_mod_int(Xa),Na) = hAPP_int_int(div_mod_int(Ya),Na))) # label(fact_3043_zmod__eq__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1038 (all K_2 K_2 = hAPP_i1732201573de_int(quickcheck_of_int,quickcheck_int_of(K_2))) # label(fact_4555_of__int__int__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1039 (all D_1 all A_1 all B_1 (one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_1),A_1) & hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_1),B_1) = one_one_nat <-> one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_1),B_1)))) # label(fact_4351_coprime__mul__eq__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1040 (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]. 4.69/4.80 1041 (all A_187 all C_76 all B_130 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_187),C_76)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_130),C_76))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_187),B_130)))) # label(fact_522_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1042 (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_3009_less__SucI) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1043 (all X 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(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)))))) # label(fact_3633_sin__cos__minus) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1044 (all A_1 all B_1 (hAPP_int_int(abs_abs_int,B_1) = hAPP_int_int(abs_abs_int,A_1) <-> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_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,B_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_2803_power2__eq__iff__abs__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1045 (all K_2 hAPP_int_int(succ,hAPP_int_int(bit0,K_2)) = hAPP_int_int(bit1,K_2)) # label(fact_774_succ__Bit0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1046 (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_3384_nat__of__aux__code) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1047 (all Ya all Xa (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ya),zEven)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xa),Ya)),zEven))))) # label(fact_3341_even__minus__even) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1048 (all A_89 all M_13 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_89),M_13)),M_13) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_89),one_one_rat)),M_13)) # label(fact_1484_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1049 (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]. 4.69/4.80 1050 (all N_56 all A_200 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_200)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_56)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_200),N_56)))))) # label(fact_247_one__less__power) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1051 (all A_1 all C all B_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),C)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),C))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),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,B_1),A_1)))) # label(fact_1369_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1052 (all A_157 all C_59 all D_18 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_157),C_59)),D_18) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_157),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_59),D_18))) # label(fact_833_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1053 (all Ta all D_1 all D (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D_1),D)) -> (all X_1 all K_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,K_1),D))),Ta))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Ta))))))) # label(fact_2240_inf__period_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1054 (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)) -> X = hAPP_real_real(hAPP_r1250527377l_real(powr,A_3),hAPP_real_real(log(A_3),X)))))) # label(fact_4460_powr__log__cancel) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1055 (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)) -> W = Z_1))) # label(fact_2078_real__le__antisym) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1056 (all B_42 all A_65 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_42),A_65)) -> -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_65),B_42)))) # label(fact_1769_xt1_I9_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1057 (all A_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_1)) <-> 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_1),A_1))))) # label(fact_600_zero__le__double__add__iff__zero__le__single__add) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1058 (all C_52 all A_141 all B_99 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_141),B_99)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_141),C_52)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_99),C_52))))) # label(fact_977_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1059 (all A_161 all B_109 all C_63 all D_19 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_161),C_63)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_109),D_19)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_161),B_109)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_63),D_19))) # label(fact_798_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1060 (all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),zero_zero_int)) -> hAPP_int_nat(nat,Z_1) = zero_zero_nat)) # label(fact_2770_nat__le__0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1061 (all Xa (is_int(Xa) -> (zero_zero_int = Xa <-> zero_zero_real = hAPP_int_real(real_int,Xa)))) # label(fact_4378_real__of__int__zero__cancel) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1062 (all N_51 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,one_one_rat),N_51) = one_one_rat) # label(fact_348_power__one) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1063 (all A_1 all B_1 (hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_1),B_1) = zero_zero_complex <-> B_1 = A_1)) # label(fact_2318_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1064 (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_4007_coprime__sos) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1065 (all A_135 all B_95 (zero_z126310315umeral = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_135),B_95) -> B_95 = zero_z126310315umeral | A_135 = zero_z126310315umeral)) # label(fact_1134_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1066 (all A_1 all C all B_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_1),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_1),C))))) # label(fact_986_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1067 (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))) -> 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)))) & (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_3286_negDivAlg__eqn) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1068 (all X X = hAPP_i1732201573de_int(quickcheck_of_int,quickcheck_int_of(X))) # label(fact_4556_int__of__inverse) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1069 (all Xa (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)))),Xa)) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa)))) # label(fact_3813_nat__even__iff__2__dvd) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1070 (all A_1 all E_2 all C all B_1 all D_1 (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_1),B_1)),E_2)),C) = D_1 <-> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_1),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_1),E_2)),D_1))) # label(fact_2381_eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1071 (all N_36 all A_167 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_167)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_167),N_36))))) # label(fact_702_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1072 (all C_23 all B_50 all A_73 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_50),A_73)) -> (C_23 = B_50 -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_23),A_73))))) # label(fact_1663_xt1_I4_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1073 (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_4683_Fract__of__int__quotient) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1074 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(fact_fact_nat,N)),hAPP_nat_nat(fact_fact_nat,M)) = zero_zero_nat)) # label(fact_3844_fact__mod) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1075 (all A_157 all C_59 all D_18 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_157),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_59),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_157),C_59)),D_18)) # label(fact_832_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1076 (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_2520_add__diff__assoc) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1077 (all C_16 all A_66 all B_43 (A_66 = B_43 -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_43),C_16)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_66),C_16))))) # label(fact_1766_ord__eq__less__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1078 (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_3409_exp__gt__one) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1079 (all Na hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na) = hAPP_n1699378549t_bool(ord_lessThan_nat,Na)) # label(fact_4951_atLeast0LessThan) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1080 (all F all A_1 all B_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1)) -> ((all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),B_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),X_1)) -> (exists Y_1 (hBOOL(hAPP_real_bool(deriv_real(F,X_1),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_1)),hAPP_real_real(F,A_1)))))) # label(fact_4547_DERIV__nonpos__imp__nonincreasing) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1081 (all N zero_zero_real = hAPP_complex_real(im,hAPP_nat_complex(semiri2020571505omplex,N))) # label(fact_4101_complex__Im__of__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1082 (all Z_15 all X_37 all Y_31 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_37),Y_31)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_31),Z_15)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_37),Z_15))))) # label(fact_1738_order__less__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1083 (all Xa all Ya (Ya = Xa <-> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Ya)) & hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ya),Xa)))) # label(fact_1703_order__eq__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1084 (all Z_9 all X_24 all Y_20 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_24),Y_20)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_20),Z_9)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_24),Z_9))))) # label(fact_1930_order__le__less__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1085 (all V_1 all W hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_1)),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_1),W))) # label(fact_1276_times__numeral__code_I5_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1086 (all K_2 frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),K_2)) = zero_zero_rat) # label(fact_4638_Frct__code__post_I1_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1087 (all M all 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)) = hAPP_int_int(div_mod_int(M),N)) # label(fact_3092_zmod__zdiv__equality_H) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1088 (all A_123 A_123 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_123),one_one_real)) # label(fact_1231_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1089 (all A_18 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_18)) -> hAPP_rat_rat(abs_abs_rat,A_18) = A_18)) # label(fact_2711_abs__of__nonneg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1090 (all X_20 all Y_16 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_20),Y_16)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_20),Y_16)))) # label(fact_1979_order__less__imp__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1091 (all N_36 all A_167 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_167)) -> 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_167),N_36))))) # label(fact_706_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1092 (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_3335_abs__ln__one__plus__x__minus__x__bound__nonneg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1093 (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]. 4.69/4.80 1094 (all Xa all Ya (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Ya)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ya),Xa)) -> Xa = Ya))) # label(fact_1651_order__antisym) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1095 (all Ra all X_2 (hBOOL(vanishes(X_2)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),Ra)) -> (exists K_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(X_2,N_1))),Ra))))))) # label(fact_5149_vanishesD) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1096 (all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),pls)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,Xa)),zero_zero_rat)))) # label(fact_104_less__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1097 (all Ta all B all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ta),B)) -> (all X_1 (is_int(X_1) -> ((all Xa_1 (is_int(Xa_1) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa_1) != X_1)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ta),X_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ta),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),D)))))))))) # label(fact_5097_bset_I7_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1098 (all M all N all K_2 ((K_2 = zero_zero_nat -> 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))) & (zero_zero_nat != K_2 -> 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)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)))) # label(fact_2885_nat__mult__div__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1099 (all K all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,K)),hAPP_int_int(number_number_of_int,L))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L)))) # label(fact_564_less__eq__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1100 (all X all 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)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)) # label(fact_4977_gcd__abs__int) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1101 (all Wa 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,Wa)),hAPP_int_nat(nat,Z_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_2))))) # label(fact_2773_nat__mono__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1102 (all X hAPP_real_real(uminus_uminus_real,hAPP_real_real(tan,X)) = hAPP_real_real(tan,hAPP_real_real(uminus_uminus_real,X))) # label(fact_3688_tan__minus) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1103 (all P all Q hAPP_nat_rat(cOMBK_rat_nat(P),Q) = P) # label(help_COMBK_1_1_COMBK_000tc__Rat__Orat_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1104 (all N_37 all A_168 all B_114 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_168),B_114)) -> (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,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_168),N_37)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_114),N_37)))))) # label(fact_696_power__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1105 (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_2105_mult__le__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1106 (all N_46 all A_195 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_195)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_195),N_46))))) # label(fact_412_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1107 (all A_1 all B_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_1)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),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_1),B_1))))) # label(fact_1288_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1108 (all Z_18 all Y_40 all X_46 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_40),X_46)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_18),Y_40)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_18),X_46))))) # label(fact_1634_xt1_I6_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1109 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,Xa),Ya))) <-> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Ya)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),zero_zero_real))) & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Xa)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ya),zero_zero_real))))) # label(fact_3402_real__0__le__divide__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1110 (all Ma all Na hAPP_n1699378549t_bool(ord_at238088361st_nat(Ma),Na) = image_int_nat(nat,hAPP_i1948725293t_bool(ord_at875362053st_int(hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,Na)))) # label(fact_5074_SetInterval_Otransfer__nat__int__set__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1111 (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_2535_dvd__mult__cancel) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1112 (all A_61 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_61),A_61)),A_61) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_61),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) # label(fact_1873_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1113 (all A_192 all N_44 all N_43 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_44),N_43)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_192)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_192),N_44)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_192),N_43)))))) # label(fact_439_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1114 (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)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = zero_zero_int))) # label(fact_2835_div__neg__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1115 (all V_4 all W_2 all C_6 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_4)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(number267125858f_real,W_2)),C_6)) = 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_4),W_2))),C_6)) # label(fact_2439_add__number__of__diff1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1116 (all C_23 all B_50 all A_73 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_50),A_73)) -> (B_50 = C_23 -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_23),A_73))))) # label(fact_1657_xt1_I4_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1117 (all P all Q all R hAPP_f1908301369l_real(P,hAPP_r1230691443l_real(Q,R)) = hAPP_r224952523l_real(hAPP_f1663701695l_real(hAPP_f774948957l_real(cOMBB_3698607l_real,P),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]. 4.69/4.80 1118 (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_3615_abs__cos__le__one) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1119 (all X hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),X))) # label(fact_2621_dvd_Oorder__refl) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1120 (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 (X = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),I_2) & Y = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),J_1)))))) # label(fact_4480_prime__power__mult) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1121 (all X_1 hAPP_real_real(cos,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)))) # label(fact_4821_lemma__cos__ext) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1122 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,Xa)),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))))),Xa),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,Xa)),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_4817_DERIV__arctan__series) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1123 (all A_1 (is_int(A_1) -> (zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),A_1) <-> A_1 = zero_zero_int))) # label(fact_931_double__zero__sym) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1124 (all A_32 all C_8 all B_24 all D_7 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_32),B_24)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,C_8),D_7)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_32),C_8)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_24),D_7))) # label(fact_2348_add__diff__add) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1125 (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_3611_sin__ge__minus__one) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1126 (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_2881_div__le__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1127 (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,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)) = 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)))) # label(fact_2537_nat__diff__add__eq2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1128 (all C all A_1 all B_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),A_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),B_1))))) # label(fact_542_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1129 (all Ma all Na (one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),Na) <-> Na = one_one_nat & Ma = one_one_nat)) # label(fact_2109_nat__1__eq__mult__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1130 (all K_2 all 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)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),L_2))) # label(fact_2409_diff__bin__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1131 (all A_1 all B_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_1),B_1)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B_1)))) # label(fact_2377_le__iff__diff__le__0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1132 (all A_194 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_194),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_194),one_one_real)))) # label(fact_418_less__add__one) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1133 (all A_1 all C all D_1 all B_1 (is_int(B_1) & is_int(D_1) -> (B_1 != zero_zero_int -> (zero_zero_int != D_1 -> (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_1),D_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),D_1))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),D_1)))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_1)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C),D_1)))))))) # label(fact_4717_le__rat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1134 (all X_33 all Y_27 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_33),Y_27)) -> -hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_27),X_33)))) # label(fact_1814_order__less__not__sym) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1135 (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_4436_powr__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1136 (all A_1 all B_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_1)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_1)))))) # label(fact_4715_zero__less__Fract__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.80 1137 (all X all M 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))))))) = 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))))))) # label(fact_3686_cos__expansion__lemma) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1138 (all A_1 all B_1 all C all D_1 (hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,C),D_1) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_1),B_1) -> (C = D_1 <-> A_1 = B_1))) # label(fact_2225_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1139 (all A_1 all Wa (hAPP_int_nat(number_number_of_nat,Wa) != zero_zero_nat & zero_zero_rat = A_1 <-> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_1),hAPP_int_nat(number_number_of_nat,Wa)) = zero_zero_rat)) # label(fact_180_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1140 (all A_138 zero_z891286103de_int = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,zero_z891286103de_int),A_138)) # label(fact_1107_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1141 (all N_56 all A_200 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_200)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_56)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_200),N_56)))))) # label(fact_248_one__less__power) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1142 (all Ya all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ya)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(nat,Xa)),hAPP_int_nat(nat,Ya))))))) # label(fact_2774_transfer__nat__int__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1143 (all X_58 all Y_48 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_58),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_48),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_999_sum__power2__ge__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1144 (all P_1 all P_2 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> ((all X_1 all K_1 (is_int(K_1) & is_int(X_1) -> (hBOOL(hAPP_int_bool(P_2,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),D_1)))) <-> hBOOL(hAPP_int_bool(P_2,X_1))))) -> ((exists Z all X_1 (is_int(X_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),X_1)) -> (hBOOL(hAPP_int_bool(P_1,X_1)) <-> hBOOL(hAPP_int_bool(P_2,X_1)))))) -> ((exists X1 hBOOL(hAPP_int_bool(P_2,X1))) -> (exists X1 (is_int(X1) & hBOOL(hAPP_int_bool(P_1,X1))))))))) # label(fact_4006_plusinfinity) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1145 (all C_45 all A_112 all B_82 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_112),B_82)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_45)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_45),A_112)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_45),B_82)))))) # label(fact_1331_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1146 (all F all G (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,F),G)) <-> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G)) & -hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,G),F)))) # label(fact_2077_less__fun__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1147 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y)) -> (exists X_1 (Y = hAPP_real_real(tan,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_eq_real,zero_zero_real),X_1)))))) # label(fact_3774_tan__total__pos) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1148 (all R_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,archim1246769320r_real(R_2))),R_2))) # label(fact_4400_real__of__int__floor__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1149 (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_2488_Nat_Odiff__diff__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1150 (all K_2 ratreal(hAPP_int_rat(number_number_of_rat,K_2)) = hAPP_int_real(number267125858f_real,K_2)) # label(fact_4696_Ratreal__number__collapse_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1151 (all Xa 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)))),Xa),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,Xa))),hAPP_real_real(cos,Xa))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,Xa))),hAPP_real_real(cos,Xa)))))) # label(fact_4799_DERIV__cos__realpow2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1152 (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_3552_real__sqrt__sum__squares__triangle__ineq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1153 (all M_26 all N_50 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_26),N_50)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_26)),hAPP_nat_real(semiri132038758t_real,N_50))))) # label(fact_362_less__imp__of__nat__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1154 (all A_52 all K_7 (is_int(A_52) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_52),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_7))),zero_zero_int)) -> zero_zero_int = A_52))) # label(fact_2144_even__power__le__0__imp__0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1155 (all C_9 all A_38 all B_27 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_38),B_27)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_38),C_9)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_38),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_27),C_9)))))) # label(fact_2301_dvd__add) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1156 (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_2121_real__mult__order) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1157 (all V_15 all W_7 hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_15),W_7)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_15)),hAPP_int_int(number_number_of_int,W_7))) # label(fact_1095_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1158 (all A_1 all P_5 all X_1 (hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_1),setS(A_1,P_5))) -> hBOOL(hAPP_f448129468l_bool(finite_finite_int,X_1)))) # label(fact_3943_SetS__elems__finite) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1159 (all Z_1 (is_int(Z_1) -> ((all N_1 Z_1 != hAPP_nat_int(semiri1621563631at_int,N_1)) -> -(all N_1 Z_1 != hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_1))))))) # label(fact_4548_int__cases) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1160 (all Xa (-hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven)))) # label(fact_3342_not__odd__impl__even) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1161 (all S (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S)) <-> (exists K_1 hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,S),hAPP_n1699378549t_bool(ord_atMost_nat,K_1)))))) # label(fact_5184_finite__nat__iff__bounded__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1162 (all Z_1 all W hAPP_real_real(hAPP_r1250527377l_real(times_times_real,W),Z_1) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_1),W)) # label(fact_2158_real__mult__commute) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1163 (all A_30 all M_7 all N_13 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_7),N_13)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_30),M_7)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_30),N_13))))) # label(fact_2372_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1164 (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_3924_real__root__less__mono__lemma) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1165 (all P all Q all R hAPP_real_bool(hAPP_f101635158l_bool(hAPP_f1925507189l_bool(cOMBB_1121934354l_real,P),Q),R) = hAPP_f448129468l_bool(P,hAPP_r481142414t_bool(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]. 4.69/4.81 1166 (all Xa (-hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa)) <-> (exists Y_1 Xa = 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_3800_odd__nat__equiv__def2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1167 (all A_1 all Xa (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) -> hAPP_int_int(uminus_uminus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_1),hAPP_int_nat(nat,Xa))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(uminus_uminus_int,A_1)),hAPP_int_nat(nat,Xa))))) # label(fact_3501_neg__odd__power) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1168 (all M all N (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> 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_2510_add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1169 (all A_125 A_125 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_125),one_one_rat)) # label(fact_1213_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1170 (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]. 4.69/4.81 1171 (all K_2 all M all 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)) = 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))) # label(fact_4505_zgcd__zmult__distrib2__abs) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1172 (all V_23 all W_13 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_23)),hAPP_int_rat(number_number_of_rat,W_13)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_23),W_13))) # label(fact_191_add__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1173 (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)) -> zero_zero_nat = 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)))))) # label(fact_2806_nat__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1174 (all A_1 hBOOL(hAPP_complex_bool(isCont_complex_real(im),A_1))) # label(fact_5164_Im_OisCont) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1175 (all Na all F (hBOOL(summable_real(F)) -> ((all M_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),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),Na))),hAPP_f352196356l_real(suminf_real,F)))))) # label(fact_4928_series__pos__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1176 (all Z_15 all X_37 all Y_31 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_37),Y_31)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_31),Z_15)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_37),Z_15))))) # label(fact_1737_order__less__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1177 (all A_37 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A_37))) # label(fact_2306_one__dvd) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1178 (all Va all Wa (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,Wa)),hAPP_int_real(number267125858f_real,Va))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,Va)),hAPP_int_real(number267125858f_real,Wa))))) # label(fact_691_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1179 (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]. 4.69/4.81 1180 (all A_11 all B_11 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(abs_abs_rat,A_11)),hAPP_rat_rat(abs_abs_rat,B_11)))),hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_11),B_11))))) # label(fact_2735_abs__triangle__ineq3) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1181 (all A_138 zero_zero_complex = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,zero_zero_complex),A_138)) # label(fact_1108_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1182 (all W_10 hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,W_10)) = 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_10))),hAPP_int_int(number_number_of_int,W_10))) # label(fact_227_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1183 (all B_71 all A_100 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_71),A_100))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_100)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_71))))) # label(fact_1403_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1184 (all A_1 all B_1 all C all D_1 (hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_1),B_1) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),D_1) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B_1)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),D_1))))) # label(fact_2337_diff__eq__diff__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1185 (all Z_1 (is_int(Z_1) -> hAPP_int_int(uminus_uminus_int,hAPP_int_int(uminus_uminus_int,Z_1)) = Z_1)) # label(fact_3459_zminus__zminus) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1186 (all R_2 all A_3 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_2),A_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_2),R_2)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,A_3),R_2)) # label(fact_3392_real__divide__square__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1187 (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_4008_coprime__exp) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1188 (all Xa all Ya (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Ya)) -> -hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Ya),Xa)))) # label(fact_1805_order__less__imp__not__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1189 (all K hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_lessThan_nat,K)))) # label(fact_4948_finite__lessThan) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1190 (all X all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)) = hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(inverse_inverse_real,X)))) # label(fact_3909_real__root__inverse) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1191 (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_3638_cos__monotone__minus__pi__0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1192 (all X_57 X_57 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_57),one_one_nat)) # label(fact_1002_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1193 (all M_26 all N_50 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_26),N_50)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,M_26)),hAPP_nat_rat(semiri151668891at_rat,N_50))))) # label(fact_360_less__imp__of__nat__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1194 (all Na all Xa exists T (hAPP_real_real(exp_real,Xa) = 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,Xa))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na))),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,Na)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa),Na))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,Xa))))) # label(fact_4925_Maclaurin__exp__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1195 (all A_1 all B_1 all C all D_1 (hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_1),B_1) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C),D_1) -> (D_1 = C <-> B_1 = A_1))) # label(fact_2224_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1196 (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,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_1),zero_zero_real))) # label(fact_4732_lemma__DERIV__sin__cos__minus) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1197 (all X_56 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_56),X_56))) # label(fact_1019_order__refl) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1198 (all Ma hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(norRRset,Ma)) = phi(Ma)) # label(fact_4584_phi__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1199 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ya),Xa)) | Ya = Xa <-> -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)))) # label(fact_1842_not__less__iff__gr__or__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1200 (all A_34 all B_26 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_34),B_26)),B_26) = A_34) # label(fact_2340_add__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1201 (all Z_1 (is_int(Z_1) -> (-hBOOL(hAPP_int_bool(nat_neg,Z_1)) -> hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,Z_1)) = Z_1))) # label(fact_3253_not__neg__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1202 (all A_1 all B_1 all Na (Na != 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_1),Na)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_1),Na))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_1),B_1))))) # label(fact_2955_pow__divides__eq__int) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1203 (all N hAPP_nat_nat(suc,N) != N) # label(fact_2985_n__not__Suc__n) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1204 (all C_12 all D_8 all A_43 all B_32 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_43),B_32)) -> (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,C_12),D_8)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_43),C_12)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_32),D_8)))))) # label(fact_2274_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1205 (all A_89 all M_13 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_89),one_on1684967323de_int)),M_13) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_89),M_13)),M_13)) # label(fact_1485_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1206 (all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) <-> -hBOOL(hAPP_int_bool(nat_neg,Xa)))) # label(fact_3247_not__neg__eq__ge__0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1207 (all K all L (quickcheck_int_of(K) = quickcheck_int_of(L) <-> K = L)) # label(fact_4254_Quickcheck__Narrowing_Oint__of__inject) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1208 (all A_1 all B_1 ((all P_4 -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),B_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),A_1)) & hBOOL(hAPP_nat_bool(prime,P_4)))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_1),B_1)))) # label(fact_4179_coprime__prime__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1209 (all Na all Ma (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Na),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,quickcheck_int_of(Na)),quickcheck_int_of(Ma))))) # label(fact_4253_less__code__int__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1210 (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)) -> hAPP_real_real(sin,hAPP_real_real(arcsin,Y)) = 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_3665_arcsin__pi) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1211 (all P_1 all Xa all Ya (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Xa),Ya)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Ya),Xa)) -> hBOOL(P_1)))) # label(fact_1783_order__less__imp__triv) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1212 (all B_77 all A_107 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_107)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_77)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_107),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_77),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_107),B_77))))) # label(fact_1357_split__mult__pos__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1213 (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_3522_arctan__monotone) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1214 (all B_76 all A_106 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_76),zero_zero_rat)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_106)) | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_76)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_106),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_106),B_76)),zero_zero_rat)))) # label(fact_1358_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1215 (all R_2 ((-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_2),zero_zero_real)) -> R_2 = hAPP_real_real(abs_abs_real,R_2)) & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_2),zero_zero_real)) -> hAPP_real_real(uminus_uminus_real,R_2) = hAPP_real_real(abs_abs_real,R_2)))) # label(fact_3442_real__abs__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1216 (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_3016_add__Suc) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1217 (all X one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),one_one_nat)) # label(fact_4335_gcd__lcm__complete__lattice__nat_Oinf__bot__right) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1218 (all N_46 all A_195 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_195)) -> 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_195),N_46))))) # label(fact_414_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1219 (all B_112 all A_164 all C_66 (is_int(C_66) & is_int(B_112) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_66),A_164) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_112),A_164) -> B_112 = C_66))) # label(fact_783_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1220 (all C_39 all A_96 all B_67 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_96),B_67)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_39)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_39),A_96)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_39),B_67)))))) # label(fact_1424_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1221 (all A_150 (is_int(A_150) -> A_150 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_150),zero_zero_int))) # label(fact_920_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1222 (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]. 4.69/4.81 1223 (all V_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))) -> hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,V_1))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,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_1)),one_one_int)),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(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_1)),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_1))),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W)))))) # label(fact_2970_zmod__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1224 (all P all Q all R hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,P),Q),R) = hAPP_bool_bool(P,hAPP_int_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1225 (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_3192_div__mult1__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1226 (all C_23 all B_50 all A_73 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_50),A_73)) -> (C_23 = B_50 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_23),A_73))))) # label(fact_1662_xt1_I4_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1227 (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_4103_complex__In__mult__cnj__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1228 (all P_1 all Xa all Ya (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Ya),Xa)) -> hBOOL(P_1)))) # label(fact_1785_order__less__imp__triv) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1229 (all Xa all Ya (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Xa),Ya)) & -hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Ya),Xa)) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Xa),Ya)))) # label(fact_2024_less__le__not__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1230 (all Xa all Ya (Ya = Xa <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Xa),Ya)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Ya),Xa)))) # label(fact_1701_order__eq__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1231 (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)) -> hAPP_int_int(legacy_zgcd(N),P_3) = one_one_int)))) # label(fact_4523_zless__zprime__imp__zrelprime) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1232 (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_4156_termination__basic__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1233 (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_3140_div2__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1234 (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_3121_div__geq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1235 (all N_22 all A_62 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_62)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_62),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_62),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_62),N_22))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_62),N_22)))))) # label(fact_1864_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1236 (all P_1 collect_nat(P_1) = P_1) # label(fact_149_Collect__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1237 (all N_31 all A_153 all B_104 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_153),B_104)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_153)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_153),N_31)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_104),N_31))))))) # label(fact_881_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1238 (all I_1 all M_2 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),M_2)) -> zero_zero_nat != 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_4767_card__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1239 (all A_1 all C all B_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_1),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_1),C))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),B_1)))) # label(fact_548_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1240 (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)) & (zero_zero_nat != M -> 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_3830_fact__add__num__eq__if2__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1241 (all A_122 all N_29 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_122),N_29)),A_122) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_122),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_122),N_29))) # label(fact_1238_power__commutes) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1242 (all A_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_1),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_1),A_1)),zero_zero_rat)))) # label(fact_97_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1243 (all X_36 all Y_30 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_36),Y_30)) -> X_36 != Y_30)) # label(fact_1796_order__less__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1244 (all B_1 all A_1 (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B_1),A_1)) -> (B_1 != A_1 -> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,B_1),A_1))))) # label(fact_1948_xt1_I11_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1245 (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_3318_natfloor__add__one) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1246 (all Q_5 all R_4 all A_3 all B_2 all Q_1 all R_2 (is_int(R_4) & is_int(R_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_1),R_2))) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_5),R_4))) -> (zero_zero_int != B_2 -> R_2 = R_4))))) # label(fact_3254_unique__remainder) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1247 (all C_47 all A_114 all B_84 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_114),B_84)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_47)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_114),C_47)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_84),C_47)))))) # label(fact_1320_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1248 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,Na))))) # label(fact_1507_zle__int) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1249 (all N all A_3 cis(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N))),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)))) # label(fact_4153_cis__real__of__nat__Suc__mult) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1250 (all Xa all Na exists T hAPP_real_real(sin,Xa) = 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,Xa))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na))),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,Na))),pi)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,Na)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa),Na)))) # label(fact_4922_Maclaurin__sin__expansion) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1251 (all Z_2 all Ya all Xa (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Ya),Xa)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Z_2),Ya)) -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Z_2),Xa))))) # label(fact_1729_xt1_I10_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1252 (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_2606_zpower__zdvd__prop2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1253 (all B_126 all A_182 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_182),zero_z891286103de_int)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_126),zero_z891286103de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_182),B_126)),zero_z891286103de_int))))) # label(fact_566_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1254 (all X all Y (hAPP_complex_real(re,X) = hAPP_complex_real(re,Y) -> (hAPP_complex_real(im,X) = hAPP_complex_real(im,Y) -> Y = X))) # label(fact_4115_complex__eqI) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1255 (all X_2 all A_1 (hBOOL(tendsto_nat_complex(X_2,A_1,sequentially)) -> hBOOL(tendsto_nat_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_2),hAPP_complex_complex(cnj,A_1),sequentially)))) # label(fact_5054_cnj_OLIMSEQ) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1256 (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_3551_real__sqrt__sum__squares__ge2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1257 (all N_22 all A_62 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_62)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_62),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_62),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_62),N_22))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_62),N_22)))))) # label(fact_1865_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1258 (all N_51 one_on1684967323de_int = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,one_on1684967323de_int),N_51)) # label(fact_349_power__one) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1259 (all D_1 all F all Xa all L (hBOOL(hAPP_real_bool(deriv_real(F,Xa),L)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_1)) -> ((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,Xa),Y_1))),D_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(F,Xa)),hAPP_real_real(F,Y_1))))) -> L = zero_zero_real)))) # label(fact_3877_DERIV__local__min) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1260 (all Ya all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Xa)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd)) -> (exists N_1 (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,Ya),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_1),Xa))))),Xa)) & is_int(N_1)))))) # label(fact_2953_best__odd__division__abs) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1261 (all N_55 ((zero_zero_nat != N_55 -> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),N_55) = zero_zero_rat) & (zero_zero_nat = N_55 -> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),N_55) = one_one_rat))) # label(fact_252_power__0__left) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1262 (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)) -> (K_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),K_2))))))) # label(fact_3097_prod__mn__less__k) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1263 (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_1086_zmult__commute) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1264 (all Va all Na 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,Va)))),hAPP_nat_nat(suc,Na)),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va)))),Na))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_int_nat(number_number_of_nat,Va)),hAPP_nat_nat(suc,Na))) # label(fact_5189_max__Suc__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1265 (all A_1 all B_1 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),B_1)),zOdd)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),zEven)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),zOdd))))) # label(fact_3350_even__plus__odd__prop2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1266 (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_3655_sin__gt__zero2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1267 (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_2891_div__mult__self1__is__m) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1268 (all R_2 all X hAPP_complex_complex(scaleR1652505878omplex(R_2),hAPP_complex_complex(cnj,X)) = hAPP_complex_complex(cnj,hAPP_complex_complex(scaleR1652505878omplex(R_2),X))) # label(fact_5108_cnj_OscaleR) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1269 (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_4127_abs__Re__le__cmod) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1270 (all C_25 all D_10 all A_79 all B_52 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_79),B_52)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_25),D_10)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_79)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_25)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_79),C_25)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_52),D_10)))))))) # label(fact_1565_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1271 (all A_1 all B_1 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_1),A_1)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A_1)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B_1)))))) # label(fact_1575_mult__le__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1272 (all N_35 all A_166 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,one_on1645066479umeral),A_166)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,one_on1645066479umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_166),N_35))))) # label(fact_709_one__le__power) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1273 (all A_125 (is_int(A_125) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_125),one_one_int) = A_125)) # label(fact_1219_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1274 (all P (is_bool(P) -> fTrue = P | fFalse = P)) # label(help_If_3_1_If_000tc__Int__Oint_T) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1275 (all P_5 all Na (hBOOL(hAPP_nat_bool(prime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_5),Na))) <-> hBOOL(hAPP_nat_bool(prime,P_5)) & one_one_nat = Na)) # label(fact_4175_prime__exp) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1276 (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]. 4.69/4.81 1277 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_real_real(uminus_uminus_real,hAPP_real_real(ln,X)) = hAPP_real_real(ln,hAPP_real_real(inverse_inverse_real,X)))) # label(fact_3861_ln__inverse) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1278 (all A_57 all B_39 (is_int(B_39) & is_int(A_57) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_57),B_39)) -> (A_57 != B_39 -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_57),B_39)))))) # label(fact_1960_order__le__neq__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1279 (all C all D_1 all A_1 all B_1 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_1),D_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_1),C)) != hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_1),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_1),D_1)) <-> C != D_1 & B_1 != A_1)) # label(fact_1173_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1280 (all Ma all Na (is_int(Na) & is_int(Ma) -> (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Ma),Na) = zero_zero_int <-> Na = zero_zero_int & zero_zero_int = Ma))) # label(fact_4996_gcd__zero__int) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1281 (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_2900_inv__less__p__minus__1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1282 (all A_3 all B_2 hAPP_rat_rat(inverse_inverse_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,B_2),A_3)) # label(fact_4676_inverse__rat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1283 (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]. 4.69/4.81 1284 (all Ma all Na (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,Ma)),hAPP_nat_rat(semiri151668891at_rat,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)))) # label(fact_355_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1285 (all M all N all K_2 hAPP_int_int(legacy_zgcd(M),N) = 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)) # label(fact_4499_zgcd__zadd__zmult) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1286 (all X all Y all M hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y)),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)) # label(fact_2991_zdiff__zmod__right) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1287 (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_2492_diff__le__mono2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1288 (all Xa all P_5 (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(sRStar,P_5))) -> hAPP_int_int(standardRes(P_5),Xa) = Xa))) # label(fact_3175_StandardRes__SRStar__prop3) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1289 (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_5146_transfer__int__nat__factorial__closure) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1290 (all X one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,one_one_nat),X)) # label(fact_4337_gcd__lcm__complete__lattice__nat_Oinf__bot__left) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1291 (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)) -> (I_2 != zero_zero_int -> 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_4269_around__zero_Opinduct) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1292 (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_2810_Euler_Oaux__1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1293 (all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),N) = N) # label(fact_2110_nat__mult__1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1294 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),hAPP_int_int(bit1,K))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),K)))) # label(fact_2175_rel__simps_I26_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1295 (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_3622_sin__cos__squared__add2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1296 (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_2969_SR__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1297 (all C_45 all A_112 all B_82 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_112),B_82)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_45)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_45),A_112)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_45),B_82)))))) # label(fact_1335_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1298 (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_3719_le__natfloor) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1299 (all A_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(abs_abs_rat,A_1))) <-> zero_zero_rat != A_1)) # label(fact_2717_zero__less__abs__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1300 (all K hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_atMost_nat,K)))) # label(fact_5183_finite__atMost) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1301 (all P all Q all R hAPP_nat_nat(hAPP_nat_fun_nat_nat(P,R),Q) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1302 (all X hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),pi)) = hAPP_real_real(tan,X)) # label(fact_3711_tan__periodic__pi) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1303 (all X_61 all Y_51 (hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_51),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_61),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_61)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_51)) -> X_61 = Y_51)))) # label(fact_887_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1304 (all A_91 all B_61 all V_10 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_91),B_61)),hAPP_int_rat(number_number_of_rat,V_10)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_91),hAPP_int_rat(number_number_of_rat,V_10))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_61),hAPP_int_rat(number_number_of_rat,V_10)))) # label(fact_1463_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1305 (all D all Xa all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) -> (Xa != zero_zero_real -> ((hBOOL(hAPP_nat_bool(even_odd_even_nat,Na)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa)) -> hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,Na)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),hAPP_nat_nat(suc,zero_zero_nat))))) = D)) -> ((hBOOL(hAPP_nat_bool(even_odd_even_nat,Na)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),zero_zero_real)) -> D = 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,Na)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),hAPP_nat_nat(suc,zero_zero_nat)))))))) -> ((-hBOOL(hAPP_nat_bool(even_odd_even_nat,Na)) -> D = hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,Na)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),hAPP_nat_nat(suc,zero_zero_nat)))))) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root,Na),Xa),D)))))))) # label(fact_3885_DERIV__real__root__generic) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1306 (all Na (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Na)) -> hAPP_int_nat(nat,Na) = 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),Na)))))) # label(fact_4742_card__bdd__int__set__l) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1307 (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]. 4.69/4.81 1308 (all Q_1 ((Q_1 != zero_zero_rat -> (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),Q_1)) -> hAPP_rat_rat(sgn_sgn_rat,Q_1) = hAPP_rat_rat(uminus_uminus_rat,one_one_rat)) & (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),Q_1)) -> one_one_rat = hAPP_rat_rat(sgn_sgn_rat,Q_1))) & (Q_1 = zero_zero_rat -> hAPP_rat_rat(sgn_sgn_rat,Q_1) = zero_zero_rat))) # label(fact_4635_sgn__rat__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1309 (all C_31 all A_85 all B_58 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_31),A_85)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_31),B_58))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_31)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_85),B_58))))) # label(fact_1528_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1310 (all P_1 all L ((exists X_1 hBOOL(hAPP_rat_bool(P_1,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,L),X_1)))) <-> (exists X_1 (hBOOL(hAPP_rat_bool(P_1,X_1)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,L),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_1),zero_zero_rat))))))) # label(fact_2416_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1311 (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]. 4.69/4.81 1312 (all N (hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) -> 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_3818_lemma__even__div2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1313 (all B_86 all A_116 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_116),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_86)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_116),B_86)),zero_zero_int))))) # label(fact_1315_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1314 (all P all Q all R hAPP_i1823918693l_bool(hAPP_f1344929775l_bool(hAPP_f1830214065l_bool(cOMBB_793166024ol_int,P),Q),R) = hAPP_f2144054103l_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__020) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1315 (all Z_1 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_1),hAPP_complex_complex(cnj,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)))) # label(fact_4136_complex__add__cnj) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1316 (all A_27 hAPP_real_real(abs_abs_real,hAPP_real_real(abs_abs_real,A_27)) = hAPP_real_real(abs_abs_real,A_27)) # label(fact_2655_abs__idempotent) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1317 (all V_25 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_25)),one_one_int) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_25),hAPP_int_int(bit1,pls)))) # label(fact_30_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1318 (all A_1 (A_1 != zero_zero_real <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_209_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1319 (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_2801_Little__Fermat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1320 (all Na all A_1 (hBOOL(tendsto_nat_real(A_1,zero_zero_real,sequentially)) -> ((all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A_1,N_1)))) -> ((all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A_1,hAPP_nat_nat(suc,N_1))),hAPP_nat_real(A_1,N_1)))) -> 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_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(power_power_real,hAPP_int_real(number267125858f_real,min)))),A_1)),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)))),Na)),one_one_nat))))))))) # label(fact_5052_summable__Leibniz_H_I4_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.81 1321 (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_2584_zcong__zdiff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1322 (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]. 4.69/4.82 1323 (all P all Q all R hAPP_f556082547l_real(P,hAPP_r1250527377l_real(Q,R)) = hAPP_r337325687l_real(hAPP_f1732848529l_real(hAPP_f1310479593l_real(cOMBB_2092576745l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc_) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1324 (all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_nat_rat(semiri151668891at_rat,Na))))) # label(fact_240_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1325 (all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,Ya))))) # label(fact_3527_real__sqrt__ge__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1326 (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_2499_divides__mul__r) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1327 (all Ya all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ya),Xa)) -> (Xa = Ya <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya))))) # label(fact_1689_order__antisym__conv) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1328 (all Xa (hBOOL(hAPP_int_bool(even_odd_even_int,Xa)) <-> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(uminus_uminus_int,Xa))))) # label(fact_3474_even__neg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1329 (all A_154 all N_32 all B_105 (hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_154),N_32) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_105),N_32) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_154)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_105)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32)) -> A_154 = B_105))))) # label(fact_860_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1330 (all A_150 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_150),zero_zero_nat) = A_150) # label(fact_919_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1331 (all I_1 all K all J (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),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))) <-> 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)),J))))) # label(fact_2521_le__diff__conv2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1332 (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_2586_zcong__less__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1333 (all M_6 all N_10 ((zero_zero_nat = N_10 -> one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_10)) & (zero_zero_nat != N_10 -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_10) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),one_one_nat)))))) # label(fact_2612_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1334 (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_3634_real__sgn__neg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1335 (all Xa (zero_zero_int = hAPP_int_int(div_mod_int(Xa),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) <-> hBOOL(hAPP_int_bool(even_odd_even_int,Xa)))) # label(fact_3270_even__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1336 (all A_122 all N_29 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_122),N_29)),A_122) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_122),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_122),N_29))) # label(fact_1234_power__commutes) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1337 (all V_14 all W_6 hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_14),W_6)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_14)),hAPP_int_rat(number_number_of_rat,W_6))) # label(fact_1096_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1338 (all Ma all Na (-hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),Na))) <-> -(-hBOOL(hAPP_nat_bool(even_odd_even_nat,Na)) <-> -hBOOL(hAPP_nat_bool(even_odd_even_nat,Ma))))) # label(fact_3785_odd__add) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1339 (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,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_11)) # label(fact_1893_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1340 (all A_1 all B_1 all C (C = B_1 <-> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_1),B_1) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_1),C))) # label(fact_808_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1341 (all A_1 all E_2 all C all B_1 all D_1 (D_1 = 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_1),B_1)),E_2)),C) <-> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_1),E_2)),D_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_1),E_2)),C))) # label(fact_2383_eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1342 (all A_1 all B_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_1),B_1)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B_1)))) # label(fact_2379_less__iff__diff__less__0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1343 (all B_10 all A_10 ((hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_10),zero_zero_rat)) | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_10))) & (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_10),zero_zero_rat)) | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_10))) -> hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(abs_abs_rat,A_10)),hAPP_rat_rat(abs_abs_rat,B_10)) = hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_10),B_10)))) # label(fact_2754_abs__eq__mult) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1344 (all X_50 all Y_44 -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_50),X_50)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_44),Y_44))),zero_zero_int))) # label(fact_1589_not__sum__squares__lt__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1345 (all P all Q all R hAPP_P603027463t_bool(hAPP_f237669397t_bool(hAPP_f461489489t_bool(cOMBS_400904860l_bool,P),Q),R) = hAPP_bool_bool(hAPP_P921862901l_bool(P,R),hAPP_P603027463t_bool(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]. 4.69/4.82 1346 (all N zero_zero_real = hAPP_real_real(hAPP_n546711566l_real(root,N),zero_zero_real)) # label(fact_3889_real__root__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1347 (all M_5 all D_4 (zero_zero_nat = hAPP_nat_nat(div_mod_nat(M_5),D_4) -> (exists Q_4 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,D_4),Q_4) = M_5))) # label(fact_3220_mod__eq__0D) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1348 (all X_5 all Y_5 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_5),Y_5)) -> (X_5 != Y_5 -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_5),Y_5))))) # label(fact_2565_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1349 (all N_18 one_one_rat = 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_18))) # label(fact_2190_power__m1__even) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1350 (all Z_2 arg(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)))))) # label(fact_4807_arg__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1351 (all B_88 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_le258702272de_int,B_88),zero_z891286103de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_118),B_88)),zero_z891286103de_int))))) # label(fact_1299_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1352 (all Lx_5 all Rx_5 all 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),Rx_5)),Ry_3) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_5),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_5),Ry_3))) # label(fact_1044_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1353 (all P all Q all R hAPP_f1406902706l_real(P,hAPP_n1699378549t_bool(Q,R)) = hAPP_nat_real(hAPP_f292751352t_real(hAPP_f1607377819t_real(cOMBB_1533963633al_nat,P),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]. 4.69/4.82 1354 (all Y exists X_1 (hAPP_real_real(tan,X_1) = Y & 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,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)))) # label(fact_3775_lemma__tan__total1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1355 (all A_1 all B_1 (A_1 = B_1 <-> quotient_of(A_1) = quotient_of(B_1))) # label(fact_4649_quotient__of__inject__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1356 (all B_69 all A_98 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_98),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_69),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_98),B_69)))))) # label(fact_1415_mult__neg__neg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1357 (all A_30 all M_7 all N_13 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_7),N_13)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_30),M_7)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_30),N_13))))) # label(fact_2371_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1358 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(hAPP_b646336293l_real(if_real,fFalse),X),Y) = Y) # label(help_If_2_1_If_000tc__RealDef__Oreal_T) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1359 (all C_77 all A_188 all B_131 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_77),A_188)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_77),B_131))) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_188),B_131)))) # label(fact_515_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1360 (all Xa all Ya (-hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Xa),Ya)) <-> Xa = Ya | hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Ya),Xa)))) # label(fact_1841_not__less__iff__gr__or__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1361 (all Xa all Ya (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Xa),Ya)) <-> -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Ya),Xa)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Xa),Ya)))) # label(fact_2021_less__le__not__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1362 (exists X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) & zero_zero_real = hAPP_real_real(cos,X_1) & (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))))) -> Y_1 = X_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)))) # label(fact_3668_cos__is__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1363 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),Xa)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),one_one_real)) -> hBOOL(hAPP_real_bool(deriv_real(arcsin,Xa),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,Xa),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))))) # label(fact_3866_DERIV__arcsin) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1364 (all X (hBOOL(hAPP_nat_bool(even_odd_even_nat,X)) -> 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_3802_even__nat__plus__one__div__two) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1365 (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_3138_Suc__diff__eq__diff__pred) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1366 (all N_23 all A_76 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_76)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_76),N_23)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_76),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_76),N_23)))))) # label(fact_1606_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1367 (all Z_17 all X_44 all Y_38 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_44),Y_38)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_38),Z_17)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_44),Z_17))))) # label(fact_1643_order__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1368 (all V_23 all W_13 hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_23),W_13)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_23)),hAPP_int_real(number267125858f_real,W_13))) # label(fact_193_add__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1369 (all A_149 (is_int(A_149) -> A_149 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_149),zero_zero_int))) # label(fact_927_add__0__right) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1370 (all C_74 all A_185 all B_128 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_185),B_128)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_74),A_185)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_74),B_128))))) # label(fact_533_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1371 (all B_74 all A_103 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_103)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_74),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_103),B_74)),zero_zero_real))))) # label(fact_1385_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1372 (all X_34 all Y_28 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_34),Y_28)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_28),X_34)))) # label(fact_1810_order__less__imp__not__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1373 (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_3359_ln__ge__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1374 (all P all Q all R hAPP_real_real(P,hAPP_real_real(Q,R)) = hAPP_real_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__RealDef_) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1375 (all C_33 all A_87 all B_60 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_33),A_87)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_33),B_60))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_33)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_87),B_60))))) # label(fact_1516_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1376 (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_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) & (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1)) & A_3 = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),N) -> Y_1 = X_1)) & A_3 = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_1),N)))))) # label(fact_1008_realpow__pos__nth__unique) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1377 (all Z_9 all X_24 all Y_20 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_24),Y_20)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_20),Z_9)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_24),Z_9))))) # label(fact_1931_order__le__less__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1378 (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_2901_inv__g__1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1379 (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_3470_zdiv__zminus2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1380 (all P_3 all Q_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Q_1),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_1))) = normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_1)))) # label(fact_4656_normalize__negative) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1381 (all Xa all Wa (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Xa),hAPP_int_nat(number_number_of_nat,Wa)))) <-> zero_zero_nat = hAPP_int_nat(number_number_of_nat,Wa) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Xa)))) # label(fact_308_zero__less__power__nat__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1382 (all B_75 all A_104 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_104)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_75)) -> 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_104),B_75)))))) # label(fact_1381_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1383 (all P all Q all R hAPP_r1195171167l_real(hAPP_f617735571l_real(hAPP_f800361423l_real(cOMBB_2107848501l_real,P),Q),R) = hAPP_r1195171167l_real(P,hAPP_real_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__prod_Itc__RealDef__Oreal_Mtc__) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1384 (all N N = natceiling(hAPP_nat_real(real_nat,N))) # label(fact_3689_natceiling__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1385 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)) -> (exists X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Ya)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),X_1)) & hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),field_1210416355s_real)))))) # label(fact_4630_Rats__dense__in__real) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1386 -(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_2213__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]. 4.69/4.82 1387 (all Xa all Ya (is_int(Ya) & is_int(Xa) -> (hAPP_int_real(real_int,Xa) = hAPP_int_real(real_int,Ya) <-> Xa = Ya))) # label(fact_4384_real__of__int__inject) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1388 (all P_1 all Na all K (is_int(Na) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Na),K)),hAPP_int_int(div_mod_int(Na),K))) <-> (all I_2 all J_1 (is_int(J_1) & is_int(I_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),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),I_2)),J_1) = Na & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,J_1),K)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,I_2),J_1))))))))) # label(fact_3150_split__pos__lemma) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1389 (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]. 4.69/4.82 1390 (all A_16 -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,A_16)),zero_zero_rat))) # label(fact_2720_abs__not__less__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1391 (all Xa (-hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd)))) # label(fact_3344_odd__iff__not__even) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1392 (all Na hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_nat_real(real_nat,Na)),ring_1_Ints_real))) # label(fact_4729_Ints__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1393 (all A_123 A_123 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_123),one_one_nat)) # label(fact_1232_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1394 (all A_159 all B_107 all C_61 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_159),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_107),C_61)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_159),B_107)),C_61)) # label(fact_824_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1395 (all Xa hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),Xa))) = Xa) # label(fact_4758_card__bdd__nat__set__l) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1396 (all A_131 all E_3 all B_93 all C_49 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_131),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_93),E_3)),C_49)) = 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_131),B_93)),E_3)),C_49)) # label(fact_1158_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1397 (all A_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),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_1),A_1)),zero_zero_int)))) # label(fact_488_double__add__less__zero__iff__single__add__less__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1398 (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_3365_ln__gt__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1399 (all X all Y Y = hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,fFalse),X),Y)) # label(help_If_2_1_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_T) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1400 (all A_101 all B_72 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_101),B_72))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_101)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_72))))) # label(fact_1397_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1401 (all A_22 all B_17 hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(abs_abs_rat,A_22)),hAPP_rat_rat(abs_abs_rat,B_17))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(abs_abs_rat,A_22)),hAPP_rat_rat(abs_abs_rat,B_17))) # label(fact_2678_abs__add__abs) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1402 (all Xa all Ya (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Ya)) -> Ya = Xa | hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya)))) # label(fact_1963_order__le__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1403 (all K all F all Na all Ma (is_int(K) -> ((all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),Na)) & 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),Na)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,Ma)),K)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),hAPP_nat_int(F,Na))) -> (exists I_2 (hAPP_nat_int(F,I_2) = K & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),Na)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),I_2)))))))))) # label(fact_2817_nat__intermed__int__val) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1404 (all Q_1 all B_2 all R_2 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_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_2),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_1),C_1))),R_2))))))) # label(fact_3128_zmult2__lemma__aux1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1405 (all A_126 (is_int(A_126) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),A_126) = A_126)) # label(fact_1212_mult__1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1406 (all Z_8 all Y_19 all X_23 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_19),X_23)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_8),Y_19)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_8),X_23))))) # label(fact_1937_xt1_I7_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1407 (all M_6 all N_10 ((N_10 != zero_zero_nat -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,M_6),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,M_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),one_one_nat))) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,M_6),N_10)) & (N_10 = zero_zero_nat -> one_on1684967323de_int = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,M_6),N_10)))) # label(fact_2608_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1408 (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_3492_zdiv__minus1__right) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1409 (all R_2 all X hAPP_complex_real(re,hAPP_complex_complex(scaleR1652505878omplex(R_2),X)) = hAPP_real_real(scaleR_scaleR_real(R_2),hAPP_complex_real(re,X))) # label(fact_5106_Re_OscaleR) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1410 (all A_3 all B_2 all C_1 all 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)) = 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))) # label(fact_3957_complex__add) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1411 (all A_1 all C all B_1 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_1),B_1)) | zero_zero_complex = C <-> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_1),C)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_1),C))))) # label(fact_2357_dvd__mult__cancel__right) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1412 -(all R_1 all V_3 (is_int(R_1) & is_int(V_3) -> -(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,V_3))),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n)))) & hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,x),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n)))) = V_3))) # label(fact_2614__096_B_Bthesis_O_A_I_B_Br_Av_O_Av_A_061_Ax_A_N_Ar_A_K_A_I1_A_L_Aint_An) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1413 (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_2544_zcong__eq__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1414 (all V_22 all W_12 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_22)),hAPP_int_rat(number_number_of_rat,W_12)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_22),W_12))) # label(fact_195_number__of__add) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1415 (all P_1 collec50511176nt_int(P_1) = P_1) # label(fact_148_Collect__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1416 (all Va all Na ((-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va)))) -> 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,Va)))),Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,Va)),hAPP_nat_nat(suc,Na))))) # label(fact_4784_less__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1417 (all A_47 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_47)) -> A_47 = zero_zero_nat)) # label(fact_2245_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1418 (all M_18 all N_30 hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_18),N_30)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(semiri151668891at_rat,M_18)),hAPP_nat_rat(semiri151668891at_rat,N_30))) # label(fact_1100_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1419 (all F all G all Na 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)))),Na))) = 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),Na))),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),Na)))) # label(fact_4909_sum__split__even__odd) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1420 (all C_46 all A_113 all B_83 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_113),B_83)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_46)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_46),A_113)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_46),B_83)))))) # label(fact_1327_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1421 (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_2831_div__prop1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1422 (all C_9 all A_38 all B_27 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_38),B_27)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_38),C_9)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_38),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_27),C_9)))))) # label(fact_2300_dvd__add) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1423 (all P all Q all R hAPP_P252494982t_bool(hAPP_f291087797t_bool(hAPP_f214467655t_bool(cOMBB_182850971nt_int,P),Q),R) = hAPP_i1948725293t_bool(P,hAPP_P1175774780nt_int(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]. 4.69/4.82 1424 (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_1827_linorder__neqE) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1425 (all M hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),hAPP_nat_nat(suc,zero_zero_nat))) # label(fact_4342_gcd__Suc__0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1426 (all C_33 all A_87 all B_60 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_33),A_87)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_33),B_60))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_33)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_87),B_60))))) # label(fact_1519_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1427 (all C_23 all B_50 all A_73 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_50),A_73)) -> (C_23 = B_50 -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_23),A_73))))) # label(fact_1660_xt1_I4_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1428 (all P all Q all R hAPP_P1175774780nt_int(hAPP_i61245889nt_int(hAPP_f578362681nt_int(cOMBC_1417754532nt_int,P),Q),R) = hAPP_int_int(hAPP_P2110489235nt_int(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1429 (all A_124 A_124 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_124),one_one_complex)) # label(fact_1222_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1430 (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_4313_gcd__semilattice__nat_Oinf__le2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1431 (all X all Y (is_int(X) & is_int(Y) -> X = Y | -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,X),Y)))) # label(help_fequal_1_1_fequal_000tc__Int__Oint_T) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1432 (all B_123 all A_179 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_179)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_123)) -> 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_179),B_123)))))) # label(fact_592_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1433 (all A_47 (is_int(A_47) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,zero_zero_int),A_47)) -> zero_zero_int = A_47))) # label(fact_2246_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1434 (all A_1 hBOOL(tendsto_complex_real(re,hAPP_complex_real(re,A_1),at_complex(A_1)))) # label(fact_5076_Re_Ocont) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1435 (all A_27 hAPP_rat_rat(abs_abs_rat,A_27) = hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(abs_abs_rat,A_27))) # label(fact_2654_abs__idempotent) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1436 (all B_73 all A_102 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_102)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_73),zero_z891286103de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_73),A_102)),zero_z891286103de_int))))) # label(fact_1389_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1437 (all A_45 all B_34 all C_13 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_45),B_34)),C_13)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_45),C_13)))) # label(fact_2262_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1438 (all A_37 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,one_one_int),A_37))) # label(fact_2307_one__dvd) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1439 (all X_20 all Y_16 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_20),Y_16)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_20),Y_16)))) # label(fact_1976_order__less__imp__le) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1440 (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_3985_sgn__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1441 (all Lx all Ly all Rx all Ry hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx),Ly)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx),Ry)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx),Rx)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ly),Ry))) # label(fact_1084_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1442 (all A_1 all B_1 (A_1 = zero_zero_rat | zero_zero_rat = B_1 <-> hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_1),B_1) = zero_zero_rat)) # label(fact_1120_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1443 (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_4028_coprime__plus1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1444 (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_2630_dvd_Oless__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1445 (all X all Y all 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)) = 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))) # label(fact_101_zadd__left__commute) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1446 (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_3144_Suc__div__eq__add3__div) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1447 (all D_1 all Ta (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),hAPP_int_int(uminus_uminus_int,Ta))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),Ta)))) # label(fact_3468_uminus__dvd__conv_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1448 (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_770_int__one__le__iff__zero__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1449 (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_3582_arctan__bounded) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1450 (all K hAPP_n1699378549t_bool(ord_atMost_nat,K) = hAPP_n1699378549t_bool(ord_lessThan_nat,hAPP_nat_nat(suc,K))) # label(fact_5185_lessThan__Suc__atMost) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1451 (all A all Ma (hBOOL(hAPP_int_bool(is_RRset(A),Ma)) <-> phi(Ma) = hAPP_f957591787ol_nat(finite_card_int,A) & hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,A),rsetR(Ma))))) # label(fact_4593_is__RRset__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1452 (all X_1 ((hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)) -> hAPP_nat_real(cos_coeff,X_1) = 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_1),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_1)))) & (-hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)) -> zero_zero_real = hAPP_nat_real(cos_coeff,X_1)))) # label(fact_3828_cos__coeff__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1453 (all Ra hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(plus_plus_real,Ra)),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)))),Ra,sequentially))) # label(fact_5032_LIMSEQ__inverse__real__of__nat__add__minus) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1454 (all X_8 all N_12 (one_one_rat = X_8 | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_12)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_8),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_8),N_12))))) # label(fact_2450_dvd__power) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1455 (all P_1 (-hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) -> ((exists X1 hBOOL(hAPP_nat_bool(P_1,X1))) -> (exists N_1 (-hBOOL(hAPP_nat_bool(P_1,N_1)) & hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_1)))))))) # label(fact_3950_exists__least__lemma) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1456 (all Q_1 all R_2 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_2),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_1),C_1))),R_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),C_1)))))) # label(fact_3198_mod__lemma) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1457 (all B_1_1 is_bool(vanishes(B_1_1))) # label(gsy_c_RealDef_Ovanishes) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1458 (all A all Ma (hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,A),rsetR(Ma))) -> hBOOL(hAPP_f448129468l_bool(finite_finite_int,A)))) # label(fact_4595_RsetR__fin) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1459 (all U all Ma all Na 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)),Na))) <-> 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)),Na)))))) # label(fact_2550_nat__less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1460 (all Ya 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,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),Ya)))),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,Ya)))),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,Ya)))))),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),Ya)))),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,Ya)))),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,Ya)))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_1),zero_zero_real))) # label(fact_4801_lemma__DERIV__sin__cos__add) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1461 (all X_61 all Y_51 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_51),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,X_61),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_61)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_51)) -> Y_51 = X_61)))) # label(fact_888_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1462 (all C_27 all D_12 all A_81 all B_54 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_81),B_54)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_27),D_12)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_81)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_27)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_81),C_27)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_54),D_12)))))))) # label(fact_1553_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1463 (all X all M all Y (hAPP_int_int(div_mod_int(Y),M) = hAPP_int_int(div_mod_int(X),M) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),Y),M)))) # label(fact_2999_Residues_Oaux) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1464 (all C_20 all A_70 all B_47 (B_47 = A_70 -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_47),C_20)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_70),C_20))))) # label(fact_1684_ord__eq__le__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1465 (all M all V_1 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_1)) = 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_1))) # label(fact_3143_Suc__div__eq__add3__div__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1466 (all M all N hAPP_int_int(legacy_zgcd(M),N) = hAPP_int_int(legacy_zgcd(N),hAPP_int_int(div_mod_int(M),N))) # label(fact_4490_zgcd__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1467 (all Xa all Ya (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Xa)),hAPP_nat_int(semiri1621563631at_int,Ya))))) # label(fact_1506_Nat__Transfer_Otransfer__int__nat__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1468 (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_3841_fact__ge__Suc__0__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1469 (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_4221_euclid) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1470 (all B_87 all A_117 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_117)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_87),zero_z891286103de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_87),A_117)),zero_z891286103de_int))))) # label(fact_1305_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1471 (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_3478_diff__int__def__symmetric) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1472 (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,P_5),zOdd))))) # label(fact_2949_zprime__zOdd__eq__grt__2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1473 (all A_3 all B_2 all Q_1 all R_2 (is_int(Q_1) & 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_1),R_2))) -> (B_2 != zero_zero_int -> Q_1 = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))))) # label(fact_3260_divmod__int__rel__div) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1474 (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_3929_real__root__pos__unique) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1475 (all P all Q all R hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,P),Q),R) = hAPP_f54304608l_bool(hAPP_n215258509l_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1476 (all Ya all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Ya)) -> (hAPP_real_real(ln,Xa) = hAPP_real_real(ln,Ya) <-> Xa = Ya)))) # label(fact_3347_ln__inj__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1477 (all A_1 all Na (zero_zero_nat != Na & A_1 = zero_zero_complex <-> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_1),Na) = zero_zero_complex)) # label(fact_447_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1478 (all Lx_6 all Rx_6 all Ry_4 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_6),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_6),Ry_4)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_6),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_6),Ry_4))) # label(fact_1038_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1479 (all B_87 all A_117 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_117)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_87),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_87),A_117)),zero_zero_int))))) # label(fact_1309_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1480 (all A_161 all B_109 all C_63 all 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_161),C_63)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_109),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_161),B_109)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_63),D_19))) # label(fact_796_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1481 (all X (is_int(X) -> (hBOOL(hAPP_int_bool(even_odd_even_int,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))))) = X))) # label(fact_3271_two__times__even__div__two) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1482 (all A_61 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_61),A_61)),A_61) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_61),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) # label(fact_1872_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1483 (all X_6 all N_11 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_11)) -> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_6),N_11) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_11),one_one_nat))),X_6))) # label(fact_2556_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1484 (all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sqrt,Ya))))) # label(fact_3528_real__sqrt__gt__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1485 (all B_115 all A_169 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_169)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_115)) -> 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_169),B_115)))))) # label(fact_674_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1486 (all N_23 all A_76 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_76)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_76),N_23)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_76),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_76),N_23)))))) # label(fact_1607_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1487 (all Xa all Ya (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Ya)) <-> Ya = Xa | hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Ya)))) # label(fact_2015_order__le__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1488 (all Xa all Ya ((hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ya),zEven))) <-> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xa),Ya)),zEven)))) # label(fact_3340_even__diff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1489 (all A_1 hBOOL(tendsto_complex_real(im,hAPP_complex_real(im,A_1),at_complex(A_1)))) # label(fact_5077_Im_Ocont) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1490 (all Ma all Na (zero_zero_nat = Ma & hAPP_nat_nat(suc,zero_zero_nat) = Na | zero_zero_nat = Na & Ma = hAPP_nat_nat(suc,zero_zero_nat) <-> hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),Na))) # label(fact_3045_add__is__1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1491 (all X_53 all P_7 all Q_8 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_53),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_8)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_53),P_7)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_53),Q_8))) # label(fact_1498_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1492 (all W_16 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,W_16)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,pls)),W_16))) # label(fact_23_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1493 (all C_40 all A_97 all B_68 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_97),B_68)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_40)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_97),C_40)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_68),C_40)))))) # label(fact_1422_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1494 (all M_20 all N_38 all A_175 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_175)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_175),M_20)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_175),N_38))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_20),N_38))))) # label(fact_627_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1495 (all C all A_1 all B_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),B_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),A_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),B_1))))) # label(fact_543_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1496 (all Xa all Ya (hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ya)) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Xa),Ya))))) # label(fact_3807_even__power__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1497 (all N_55 ((N_55 != zero_zero_nat -> zero_zero_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),N_55)) & (zero_zero_nat = N_55 -> one_one_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),N_55)))) # label(fact_256_power__0__left) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1498 (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_2524_diff__add__assoc) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1499 (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)) -> natfloor(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_int_real(number267125858f_real,N))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natfloor(X)),hAPP_int_nat(number_number_of_nat,N))))) # label(fact_3299_natfloor__add__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1500 (all B_123 all A_179 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_179)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_123)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_179),B_123)))))) # label(fact_594_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1501 (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_4322_gcd__greatest__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1502 (all B_126 all A_182 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_182),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_126),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_182),B_126)),zero_zero_rat))))) # label(fact_565_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1503 (all N hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),pi)) = zero_zero_real) # label(fact_3727_sin__npi) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1504 (all Y_14 all X_18 (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_14),X_18)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_18),Y_14)))) # label(fact_2006_not__leE) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1505 (all X_2 all A_1 (hBOOL(tendsto_nat_complex(X_2,A_1,sequentially)) -> hBOOL(tendsto_nat_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_2),hAPP_complex_real(im,A_1),sequentially)))) # label(fact_5056_Im_OLIMSEQ) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1506 (all X (hAPP_real_real(cos,X) = one_one_real -> hAPP_real_real(sin,X) = zero_zero_real)) # label(fact_3595_cos__one__sin__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1507 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) | hAPP_nat_nat(suc,Na) = Ma <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),hAPP_nat_nat(suc,Na))))) # label(fact_3020_le__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1508 (all X all Y all R_2 hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),R_2)),Y) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,X),Y)),of_real_complex(R_2))) # label(fact_3988_Complex__add__complex__of__real) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1509 (all Ta all A all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ta),A)) -> (all X_1 (is_int(X_1) -> ((all Xa_1 (is_int(Xa_1) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A)) -> X_1 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa_1))))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Ta)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),D)),Ta))))))))) # label(fact_5096_aset_I5_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1510 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,M))) -> 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))) = hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N)))) # label(fact_3808_fact__diff__Suc) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1511 (all Na 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),Na)) -> 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),Na))),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,Na),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,Na)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),Na))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)))))) # label(fact_4934_Maclaurin2__objl) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1512 (all B_96 all A_136 (A_136 != zero_zero_nat -> (zero_zero_nat != B_96 -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_136),B_96) != zero_zero_nat))) # label(fact_1129_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1513 (all A_128 A_128 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),A_128)) # label(fact_1196_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1514 (all Ta all D_1 all D (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D_1),D)) -> (all X_1 all K_1 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D_1),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_1),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,K_1),D))),Ta))) <-> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D_1),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_1),Ta))))))) # label(fact_2239_inf__period_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1515 (all Xa (hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa)) <-> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_nat_int(semiri1621563631at_int,Xa))))) # label(fact_3796_Parity_Otransfer__int__nat__relations) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1516 (all Ma all X_1 all Xa_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),Xa_1),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcongm(Ma),X_1),Xa_1)))) # label(fact_5152_zcongm__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.82 1517 (all A_157 all C_59 all D_18 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_157),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_59),D_18)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_157),C_59)),D_18)) # label(fact_834_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1518 (all C all A_1 all B_1 (A_1 = B_1 -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,C),B_1)) -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,C),A_1))))) # label(fact_1757_xt1_I1_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1519 (all P (is_bool(P) -> fTrue = P | fFalse = P)) # label(help_fFalse_1_1_T) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1520 (all A_1 all B_1 (zero_zero_complex = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_1),B_1) <-> B_1 = A_1)) # label(fact_2322_eq__iff__diff__eq__0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1521 (all X_16 all Y_12 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_12),X_16)) | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_16),Y_12)))) # label(fact_2035_linorder__le__less__linear) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1522 (all Y_39 all X_45 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_39),X_45)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_45),Y_39)) -> X_45 = Y_39))) # label(fact_1641_xt1_I5_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1523 (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_1 K_2 = hAPP_nat_int(semiri1621563631at_int,N_1))))) # label(fact_1013_zero__le__imp__eq__int) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1524 (exists K_4 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),K_4)) & (all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(re,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_1)),K_4)))))) # label(fact_4139_Re_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1525 (all G all Xa all Ma (hBOOL(hAPP_real_bool(deriv_real(G,Xa),Ma)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,sin),G),Xa),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,hAPP_real_real(G,Xa))),Ma))))) # label(fact_4757_DERIV__fun__sin) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1526 (all A_3 (hBOOL(hAPP_nat_bool(prime,A_3)) -> hBOOL(hAPP_int_bool(zprime,hAPP_nat_int(semiri1621563631at_int,A_3))))) # label(fact_4176_prime__impl__zprime__int) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1527 (all V_13 all V_12 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_12)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_13)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_12)),hAPP_int_int(number_number_of_int,V_13)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_12),V_13))))) # label(fact_1282_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1528 (all M hAPP_nat_nat(suc,M) != zero_zero_nat) # label(fact_2996_Zero__neq__Suc) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1529 (all C_46 all A_113 all B_83 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_113),B_83)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_46)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_46),A_113)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_46),B_83)))))) # label(fact_1329_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1530 (all A_1 all X_2 all Xa (hBOOL(tendsto_nat_real(X_2,Xa,sequentially)) -> ((exists N_2 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),N_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),hAPP_nat_real(X_2,N_1))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),Xa))))) # label(fact_5043_LIMSEQ__le__const) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1531 (all N (hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) -> (exists X_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(bit0,hAPP_int_int(bit1,pls))))),X_1) = 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_3827_even__square) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1532 (all A_156 all C_58 all D_17 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_58),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_156),D_17)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_156),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_58),D_17))) # label(fact_838_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1533 (all Na all Ma (is_int(Ma) -> (zero_zero_int != Ma -> (one_one_int = hAPP_int_int(abs_abs_int,Na) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),Na)),Ma)))))) # label(fact_2796_zdvd__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1534 (all P all Q all R hAPP_f737665877t_bool(P,hAPP_i157317923t_real(Q,R)) = hAPP_i418383825t_bool(hAPP_f1222178131t_bool(hAPP_f743790483t_bool(cOMBB_1647910634ol_int,P),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]. 4.69/4.83 1535 (all M all N hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),divmod_nat(M,N)))) # label(fact_4215_divmod__nat__rel__divmod__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1536 (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_3900_real__root__less__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1537 (all N (N != zero_zero_nat -> 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_4365_coprime__minus__one__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1538 (all Z_1 (is_int(Z_1) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),Z_1) = Z_1)) # label(fact_158_zadd__0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1539 (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))) -> (hAPP_P1175774780nt_int(product_snd_int_int,P_3) = zero_zero_int -> 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) -> 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))))) = normalize(P_3))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_P1175774780nt_int(product_snd_int_int,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)))) = normalize(P_3)))) # label(fact_4958_normalize__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1540 (all A_176 all N_40 all N_39 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_40),N_39)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_176)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_176),one_on1645066479umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_176),N_39)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_176),N_40))))))) # label(fact_619_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1541 (all Va all V ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls)) -> 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,Va),hAPP_int_int(uminus_uminus_int,V))))),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,Va),hAPP_int_int(uminus_uminus_int,V))))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,Va)),hAPP_int_nat(number_number_of_nat,V))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls)) -> hAPP_int_nat(number_number_of_nat,Va) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,Va)),hAPP_int_nat(number_number_of_nat,V))))) # label(fact_4795_diff__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1542 (all Xa all Ya (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) <-> Xa = Ya))) # label(fact_2002_linorder__antisym__conv1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1543 (all N all V_1 ((hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_1)),N))) & (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_1))),N) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_1)),N))))) # label(fact_3276_Suc__nat__number__of__add) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1544 (all A_198 A_198 = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_198),one_one_nat)) # label(fact_318_power__one__right) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1545 (all A_1 all Xa (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(uminus_uminus_int,A_1)),hAPP_int_nat(nat,Xa)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_1),hAPP_int_nat(nat,Xa))))) # label(fact_3500_neg__even__power) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1546 (all Y all X (zero_zero_real != hAPP_real_real(cos,X) -> (zero_zero_real != hAPP_real_real(cos,Y) -> (hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)) != zero_zero_real -> hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)) = 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)))))))) # label(fact_3759_tan__add) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1547 (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_1266_le__add1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1548 (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_3112_zcong__zmod__aux) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1549 (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_4285_fact__less__mono__int__aux) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1550 (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]. 4.69/4.83 1551 (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_4409_powr__less__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1552 (all N_34 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N_34)))) # label(fact_718_of__nat__0__le__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1553 (all N_27 all M_16 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),M_16)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),N_27)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,M_16),N_27)))))) # label(fact_1452_less__1__mult) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1554 (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_2963_gcd__lcm__complete__lattice__nat_Onot__top__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1555 (all A_46 all B_35 all C_14 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_46),B_35)),C_14)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,B_35),C_14)))) # label(fact_2256_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1556 (all A_1 (is_int(A_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,A_1)),zero_zero_int)) <-> zero_zero_int = A_1))) # label(fact_2710_abs__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1557 (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_4964_coprime__dvd__mult__int) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1558 (all A_125 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_125),one_on1645066479umeral) = A_125) # label(fact_1216_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1559 (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_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)))))),X_1)) & (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)))))) & hAPP_real_real(sin,Y_1) = 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_1)) -> Y_1 = X_1)) & Y = hAPP_real_real(sin,X_1) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_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))))))))))) # label(fact_3666_sin__total) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1560 (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_3540_arctan__half) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1561 (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_4768_card__less__Suc) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1562 (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_3424_log__base__10__eq1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1563 (all W_4 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,W_4)),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_4)),hAPP_int_real(number267125858f_real,W_4))) # label(fact_1913_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1564 (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_2599_zcong__zless__imp__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1565 (all C_54 all D_16 all A_143 all B_101 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_143),B_101)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_54),D_16)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_143),C_54)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_101),D_16)))))) # label(fact_967_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1566 (all M_25 all N_49 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,M_25)),hAPP_nat_rat(semiri151668891at_rat,N_49))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_25),N_49)))) # label(fact_365_of__nat__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1567 (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_2858_zfact__prop) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1568 (all Y_4 all X_4 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_4)) -> hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(abs_abs_rat,Y_4)),X_4) = hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_4),X_4)))) # label(fact_2751_abs__mult__pos) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1569 (all A_48 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_48),A_48))) # label(fact_2221_dvd__refl) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1570 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_nat_real(real_nat,natceiling(X))))) # label(fact_3709_real__natceiling__ge) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1571 (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_3310_natceiling__mono) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1572 (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) -> (A_3 = 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)) -> (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)) = B_2 -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_5),B_5) = one_one_int)))) # label(fact_5004_gcd__coprime__int) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1573 (all 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)) = negDivAlg(hAPP_int_int(number_number_of_int,min),B_2)) # label(fact_3277_negDivAlg__minus1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1574 (all X_5 all Y_5 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_5),Y_5)) -> (X_5 != Y_5 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_5),Y_5))))) # label(fact_2567_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1575 (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_3671_arccos__cos) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1576 (all P all Q all R hAPP_f1545556668t_bool(P,hAPP_r1150855451l_bool(Q,R)) = hAPP_r1962736578t_bool(hAPP_f1063001928t_bool(hAPP_f2050994651t_bool(cOMBB_2138578194l_real,P),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]. 4.69/4.83 1577 (all C all A_1 all B_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),A_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),B_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B_1)))) # label(fact_982_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1578 (all Lx_6 all Rx_6 all Ry_4 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_6),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,Rx_6),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_6),Ry_4))) # label(fact_1043_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1579 (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]. 4.69/4.83 1580 (all A_54 all M_11 all N_21 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_54),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_11),N_21)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_54),M_11)),N_21)) # label(fact_2094_power__mult) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1581 (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,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)) = 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))) # label(fact_2540_nat__diff__add__eq1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1582 (all C_10 all A_41 all B_30 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_41),B_30)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_41),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_30),C_10))))) # label(fact_2282_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1583 (all K all L hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K),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),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)),hAPP_int_int(abs_abs_int,L)))))) # label(fact_5014_gcd__code__int) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1584 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa)) -> hBOOL(hAPP_real_bool(deriv_real(sqrt,Xa),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,Xa))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3871_DERIV__real__sqrt) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1585 (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_3847_Ln_Oaux1) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1586 (all U_1 all V_1 all X all Y (X = Y -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,U_1)),V_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(hAPP_r1250527377l_real(plus_plus_real,X),U_1)),Y))),V_1))))) # label(fact_2922_sin__bound__lemma) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1587 (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_202_odd__nonzero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1588 (all X -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,X)),zero_zero_real))) # label(fact_3398_not__exp__le__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1589 (all A_6 all B_7 all Q_6 all Y (is_int(Y) & is_int(B_7) -> (A_6 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_7),Q_6)),Y) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_7)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),B_7)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_7)) -> 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,B_7),Y))) -> (zero_zero_int != B_7 -> hAPP_int_int(div_mod_int(A_6),B_7) = Y))))) # label(fact_3130_divmod__int__rel__mod__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1590 (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_4238_div__nat__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1591 (all Xa -hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Xa))) # label(fact_1852_order__less__irrefl) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1592 (all N 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) = one_one_nat) # label(fact_4350_coprime__plus__one__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1593 (all M all 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)) = hAPP_nat_nat(div_mod_nat(M),hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,N))))) # label(fact_3204_mod__Suc__eq__mod__add3) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1594 (all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),N) = zero_zero_nat) # label(fact_2479_diff__0__eq__0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1595 (all C_18 all A_68 all B_45 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_68),B_45)) -> (B_45 = C_18 -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_68),C_18))))) # label(fact_1751_ord__less__eq__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1596 (all P_1 all Xa all Ya (hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Xa),Ya)))) <-> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ya),Xa)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,Xa)),hAPP_nat_int(semiri1621563631at_int,Ya))))) & (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) -> hBOOL(hAPP_int_bool(P_1,zero_zero_int))))) # label(fact_2561_zdiff__int__split) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1597 (all M one_one_int = hAPP_int_int(legacy_zgcd(one_one_int),M)) # label(fact_4488_zgcd__1__left) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1598 (all K all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K)),hAPP_int_int(bit0,L))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L)))) # label(fact_112_rel__simps_I16_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1599 (all A_1 all B_1 (zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_1),B_1) <-> zero_zero_real = B_1 | zero_zero_real = A_1)) # label(fact_1122_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1600 (all X_36 all Y_30 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_36),Y_30)) -> Y_30 != X_36)) # label(fact_1795_order__less__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1601 (all N_36 all A_167 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_167)) -> 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_167),N_36))))) # label(fact_705_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1602 (all B_116 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_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(plus_plus_real,A_170),B_116)))))) # label(fact_667_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1603 (all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)) = phi(P_3))) # label(fact_2937_phi__prime) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1604 (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(abs_abs_real,X)) = hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)))) # label(fact_3908_real__root__abs) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1605 (all Xa 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,Xa)),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_4839_exp__tail__after__first__two__terms__summable) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1606 (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_4709_Fract__number__of__quotient) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1607 (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_2460_zdiff__power3) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1608 (all M_19 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_nat_rat(semiri151668891at_rat,M_19)))) # label(fact_719_zero__le__imp__of__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1609 (all X all B_2 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),B_2)) -> (B_2 != one_one_real -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_real_real(log(A_3),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)))))))) # label(fact_3453_log__eq__div__ln__mult__log) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1610 (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_2916_MultInv__zcong__prop2) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1611 (all M all X hAPP_int_int(div_mod_int(X),M) = hAPP_int_int(standardRes(M),X)) # label(fact_2981_StandardRes__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1612 (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]. 4.69/4.83 1613 (all Na all Xa (Xa != zero_zero_real -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) -> (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,Xa) = 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,Xa))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na))),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,Na)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa),Na))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,Xa)))))))) # label(fact_4924_Maclaurin__exp__lt) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1614 (all Xa all Ya all B_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),B_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_1),Xa)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_1),Ya))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya))))) # label(fact_431_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1615 (all A_134 zero_z126310315umeral = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,zero_z126310315umeral),A_134)) # label(fact_1141_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1616 (all B_96 all A_136 (A_136 != zero_zero_rat -> (B_96 != zero_zero_rat -> hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_136),B_96) != zero_zero_rat))) # label(fact_1124_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1617 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> N != zero_zero_nat)) # label(fact_299_gr__implies__not0) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1618 (all I_1 all J all K (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))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K)),J)))) # label(fact_2509_less__diff__conv) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1619 (all N natfloor(hAPP_nat_real(real_nat,N)) = N) # label(fact_3693_natfloor__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1620 (all A_3 cis(A_3) = rcis(one_one_real,A_3)) # label(fact_4259_cis__rcis__eq) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1621 (all X all Y all R_2 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,X),Y)),of_real_complex(R_2)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),R_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),R_2))) # label(fact_3987_Complex__mult__complex__of__real) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1622 (all C_77 all A_188 all B_131 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_77),A_188)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_77),B_131))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_188),B_131)))) # label(fact_519_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1623 (all Y all 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)))) = X -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y)) -> Y = hAPP_real_real(sqrt,X)))) # label(fact_3547_real__sqrt__unique) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1624 (all Z_1 all W (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),Z_1)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_1),W)))) # label(fact_2080_real__le__linear) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1625 (all V_11 all B_62 all C_34 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_11)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_62),C_34)) = 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_11)),B_62)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_11)),C_34))) # label(fact_1461_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1626 (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)) -> hAPP_int_int(div_mod_int(A_3),B_2) = A_3)))) # label(fact_3079_mod__pos__pos__trivial) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1627 (all Xa all Ya (quickcheck_int_of(Ya) = quickcheck_int_of(Xa) <-> Ya = Xa)) # label(fact_4255_code__int_Oint__of__inject) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1628 (all A_61 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_61),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_61),A_61)),A_61)) # label(fact_1871_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1629 (all N_52 all A_197 (zero_zero_rat != A_197 -> zero_zero_rat != hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_197),N_52))) # label(fact_344_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1630 (all A_3 hAPP_int_int(div_mod_int(A_3),A_3) = zero_zero_int) # label(fact_2986_zmod__self) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1631 (all A_89 all M_13 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_89),one_one_nat)),M_13) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_89),M_13)),M_13)) # label(fact_1489_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1632 (all B_74 all A_103 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_103)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_74),zero_z891286103de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_103),B_74)),zero_z891286103de_int))))) # label(fact_1383_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1633 (all Xa 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,Xa)))) # label(fact_2927_d22set__induct__old) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1634 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,Xa)),hAPP_int_real(number267125858f_real,Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)))) # label(fact_61_less__number__of) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1635 (all M_6 all N_10 ((zero_zero_nat != N_10 -> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,M_6),N_10) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,M_6),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,M_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),one_one_nat)))) & (zero_zero_nat = N_10 -> one_one_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,M_6),N_10)))) # label(fact_2611_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1636 (all B_1 all A_1 all C (hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C),A_1) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_1),A_1) <-> B_1 = C)) # label(fact_804_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1637 (all B_1 all A_1 all C (C = B_1 <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_1),A_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),A_1))) # label(fact_806_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1638 (all N (-hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) -> (exists X_1 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_1)),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_3825_odd__square) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1639 (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_5132_Nat__Transfer_Otransfer__int__nat__function__closures_I4_J) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1640 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,Xa)),zero_zero_real)))) # label(fact_3526_real__sqrt__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1641 (all Z_2 all Xa all Ya (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Ya)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Ya),Z_2)) -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Z_2))))) # label(fact_1928_order__le__less__trans) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1642 (all F all K all A_1 all B_1 (A_1 != B_1 -> ((all X_1 hBOOL(hAPP_real_bool(deriv_real(F,X_1),K))) -> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_1),A_1)),K) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B_1)),hAPP_real_real(F,A_1))))) # label(fact_3884_DERIV__const__ratio__const) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1643 (all P_1 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> ((all X_1 all K_1 (is_int(K_1) & is_int(X_1) -> (hBOOL(hAPP_int_bool(P_1,X_1)) <-> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),D_1))))))) -> ((exists X_1 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1))) & hBOOL(hAPP_int_bool(P_1,X_1)) & is_int(X_1))) <-> (exists X1 (hBOOL(hAPP_int_bool(P_1,X1)) & is_int(X1))))))) # label(fact_5100_periodic__finite__ex) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1644 (all Ya all Xa (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ya),zOdd)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xa),Ya)),zEven))))) # label(fact_3356_odd__minus__odd) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1645 (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_4544_dist__real__def) # label(axiom) # label(non_clause). [assumption]. 4.69/4.83 1646 (all Ya all Xa (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),Xa)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),Ya)) -> (Xa = zero_z126310315umeral & Ya = zero_z126310315umeral <-> hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,Xa),Ya) = zero_z126310315umeral)))) # label(fact_588_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1647 (all L_2 min != hAPP_int_int(bit0,L_2)) # label(fact_2165_rel__simps_I42_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1648 (all Y_14 all X_18 (-hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_14),X_18)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_18),Y_14)))) # label(fact_2005_not__leE) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1649 (all D_1 all F all Xa all L (hBOOL(hAPP_real_bool(deriv_real(F,Xa),L)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_1)) -> ((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,Xa),Y_1))),D_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(F,Y_1)),hAPP_real_real(F,Xa))))) -> L = zero_zero_real)))) # label(fact_3876_DERIV__local__max) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1650 (all P all Q all R hAPP_int_bool(hAPP_f429935748t_bool(hAPP_f1512942609t_bool(cOMBB_int_bool_int,P),Q),R) = hAPP_int_bool(P,hAPP_int_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__HOL__Obool_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1651 (all W hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,W)) = hAPP_int_nat(number_number_of_nat,W)) # label(fact_2746_nat__number__of) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1652 (all Xa hAPP_real_real(sin,Xa) = 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,Xa)))) # label(fact_4818_sin__def) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1653 (all V_2 ((hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> 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_2))) & (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_int_nat(nat,hAPP_int_int(div_mod_int(one_one_int),hAPP_int_int(number_number_of_int,V_2))) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,zero_zero_nat)),hAPP_int_nat(number_number_of_nat,V_2))))) # label(fact_3224_one__mod__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1654 (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_3449_ln__one__minus__pos__upper__bound) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1655 (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_3657_sin__monotone__2pi_H) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1656 (all K all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(pred,K)),L)))) # label(fact_4041_le__iff__pred__less) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1657 (all B_88 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_eq_real,B_88),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_118),B_88)),zero_zero_real))))) # label(fact_1301_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1658 (all A_150 A_150 = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_150),zero_zero_complex)) # label(fact_917_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1659 (all P_1 all A_1 all B_1 (hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A_1),B_1))) <-> -(-hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B_1)) | (exists D_2 (-hBOOL(hAPP_nat_bool(P_1,D_2)) & A_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_1),D_2)))))) # label(fact_2531_nat__diff__split__asm) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1660 (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]. 4.77/4.83 1661 (all A_86 all C_32 all B_59 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_86),C_32)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_59),C_32))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_32)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_86),B_59))))) # label(fact_1525_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1662 (all K all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Na))))) # label(fact_2117_mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1663 (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_4580_complex__Re__of__int) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1664 (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_3720_exp__real__of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1665 (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]. 4.77/4.83 1666 (all Z_1 exists A_2 exists R_1 Z_1 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_1)),expi(A_2))) # label(fact_4051_complex__expi__Ex) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1667 (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]. 4.77/4.83 1668 (all Xa (zero_zero_real != hAPP_real_real(cos,Xa) -> 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),Xa),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,Xa)),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]. 4.77/4.83 1669 (all C_70 all D_21 all A_178 all B_122 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_178),B_122)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_70),D_21)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_178),C_70)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_122),D_21)))))) # label(fact_603_add__le__less__mono) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1670 (all Na 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),Na)) <-> Ma = hAPP_int_int(legacy_zgcd(Na),Ma))))) # label(fact_4511_zdvd__iff__zgcd) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1671 (all U all F ((all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_nat_nat(F,N_1)))) -> 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_4894_finite__less__ub) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1672 (all R_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R_2)) -> (exists N_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_2),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_1)))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N_1)),R_2)))))) # label(fact_3780_reals__Archimedean6a) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1673 (all Z_1 hBOOL(hAPP_int_bool(nat_is_nat,hAPP_nat_int(semiri1621563631at_int,Z_1)))) # label(fact_5138_Nat__Transfer_Otransfer__int__nat__function__closures_I9_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1674 (all V_6 all B_21 all C_7 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_6)),B_21)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_6)),C_7)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_6)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_21),C_7))) # label(fact_2396_right__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1675 (all A_183 all N_42 all N_41 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_42),N_41)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,one_on1645066479umeral),A_183)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_183),N_42)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_183),N_41)))))) # label(fact_552_power__increasing) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1676 (all Ma all Na (Ma = zero_zero_nat & hAPP_nat_nat(suc,zero_zero_nat) = Na | Na = zero_zero_nat & Ma = hAPP_nat_nat(suc,zero_zero_nat) <-> hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),Na))) # label(fact_3044_one__is__add) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1677 (all Na all N_6 all F ((all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_1)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_1))))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_6)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,Na)),hAPP_nat_nat(F,N_6)))))) # label(fact_3167_dvd_Olift__Suc__mono__le) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1678 (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_5145_transfer__int__nat__gcd__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1679 (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_3060_mult__Suc) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1680 (all Xa all F (hBOOL(summable_real(F)) -> ((all N_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))),Xa))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f352196356l_real(suminf_real,F)),Xa))))) # label(fact_4943_suminf__le) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1681 (all P all Q all R hAPP_i1948725293t_bool(hAPP_f428220345t_bool(hAPP_f1878066172t_bool(cOMBS_1720984575t_bool,P),Q),R) = hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(P,R),hAPP_i1948725293t_bool(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]. 4.77/4.83 1682 (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_3024_Suc__leD) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1683 (all P all Q all R hAPP_int_bool(hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,P),Q),R) = hAPP_int_bool(hAPP_i1948725293t_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1684 (all A_146 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A_146) = A_146) # label(fact_950_add__0__left) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1685 (all M all N (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> hAPP_nat_nat(div_mod_nat(M),N) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N))) # label(fact_3180_mod__geq) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1686 (all Ya all Xa (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ya),zEven)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Xa),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,Ya),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,Xa),Ya)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_3379_even__sum__div__2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1687 (all X all Y Y = hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,fFalse),X),Y)) # label(help_If_2_1_If_000tc__Nat__Onat_T) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1688 (all N all M hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(div_mod_int(quickcheck_int_of(N)),quickcheck_int_of(M))) = hAPP_Q1762011733de_int(div_mo231679042de_int(N),M)) # label(fact_4552_mod__code__int__def) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1689 (all P_1 all K all I_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),I_1)) -> (hBOOL(hAPP_int_bool(P_1,K)) -> ((all I_2 (is_int(I_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),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_1014_int__ge__induct) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1690 (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),divmod_int(A_3,B_2)))))) # label(fact_4191_divmod__int__correct) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1691 (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_3519_real__sqrt__minus) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1692 (all N all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> 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))) # label(fact_4458_powr__realpow) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1693 (all Na all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Na),Ma)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,Na)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,Ma)),one_one_real))))) # label(fact_4446_int__le__real__less) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1694 (all A_91 all B_61 all V_10 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_91),hAPP_i1732201573de_int(number1226105091de_int,V_10))),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_61),hAPP_i1732201573de_int(number1226105091de_int,V_10))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_91),B_61)),hAPP_i1732201573de_int(number1226105091de_int,V_10))) # label(fact_1464_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1695 (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),phi(M))),one_one_int),M))))) # label(fact_4481_Euler__Fermat) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1696 (all Wa all Xa (Xa = hAPP_i1732201573de_int(number1226105091de_int,Wa) <-> Xa = hAPP_i1732201573de_int(number1226105091de_int,Wa))) # label(fact_89_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1697 (all P_1 all L ((exists X_1 hBOOL(hAPP_complex_bool(P_1,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,L),X_1)))) <-> (exists X_1 (hBOOL(hAPP_complex_bool(P_1,X_1)) & hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,L),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_1),zero_zero_complex))))))) # label(fact_2418_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1698 (all A_83 all C_29 all B_56 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_83),C_29)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_56),C_29))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_29)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_83),B_56))))) # label(fact_1539_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1699 (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_2833_zdiv__mono1__neg) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1700 (all Xa all A (hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,Xa),A)) <-> hBOOL(hAPP_P1463840893t_bool(A,Xa)))) # label(fact_143_mem__def) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1701 (all Xa all P_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_5),Xa)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Xa),zero_zero_int),P_5)))) # label(fact_2592_zcong__eq__zdvd__prop) # label(axiom) # label(non_clause). [assumption]. 4.77/4.83 1702 (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))) & 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_3081_neg__mod__conj) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1703 (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_3317_le__mult__natfloor) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1704 (all N_35 all A_166 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),A_166)) -> 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_166),N_35))))) # label(fact_711_one__le__power) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1705 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)) & Ya != Xa)) # label(fact_2032_order__less__le) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1706 (all A_156 all C_58 all D_17 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_58),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_156),D_17)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_156),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_58),D_17))) # label(fact_840_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1707 (all A_44 all B_33 all K_6 (hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_33),K_6) = A_44 -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,B_33),A_44)))) # label(fact_2267_dvdI) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1708 (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)) -> 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)) = negDivAlg(K_2,L_2)))) # label(fact_3278_negDivAlg__div__mod) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1709 (all A_18 (is_int(A_18) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_18)) -> hAPP_int_int(abs_abs_int,A_18) = A_18))) # label(fact_2713_abs__of__nonneg) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1710 (all A_123 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_123),one_one_complex) = A_123) # label(fact_1229_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1711 (all L all K (is_int(L) & is_int(K) -> (zero_zero_int = K -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int) = divmod_int(K,L)) & (zero_zero_int != K -> (zero_zero_int != 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,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,hAPP_int_int(sgn_sgn_int,K)),hAPP_int_int(sgn_sgn_int,L))),pdivmod(K,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,L)))) = divmod_int(K,L)) & (zero_zero_int = L -> divmod_int(K,L) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),K))))) # label(fact_4855_divmod__int__code) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1712 (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_3677_arccos__bounded) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1713 (all C_46 all A_113 all B_83 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_113),B_83)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_46)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_46),A_113)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_46),B_83)))))) # label(fact_1328_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1714 (all C_42 all D_15 all A_109 all B_79 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_109),B_79)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_42),D_15)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_109)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_42)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_109),C_42)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_79),D_15)))))))) # label(fact_1346_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1715 (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_4237_mod__neg__neg) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1716 (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_2491_diff__le__mono) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1717 (all N_3 (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_3)) <-> (exists M_1 all X_1 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),N_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),M_1)))))) # label(fact_4609_finite__nat__set__iff__bounded__le) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1718 (all Ta all A all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D)) -> (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)) -> (all X_1 (is_int(X_1) -> ((all Xa_1 (is_int(Xa_1) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A)) -> X_1 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa_1))))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Ta)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),D)),Ta))))))))) # label(fact_5089_aset_I6_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1719 (all Ya 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)),Ya)))) = hAPP_real_real(arccos,Ya)) # label(fact_4885_arccos__def) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1720 (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]. 4.77/4.84 1721 (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]. 4.77/4.84 1722 (all A_132 all B_94 all C_50 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_132),B_94)),C_50) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_132),C_50)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_94),C_50))) # label(fact_1154_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1723 (all B_89 all A_119 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_119)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_89)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_119),B_89)))))) # label(fact_1295_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1724 (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_3094_n__less__n__mult__m) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1725 (all P (is_bool(P) -> P = fTrue | fFalse = P)) # label(help_If_3_1_If_000tc__RealDef__Oreal_T) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1726 (all V_1 all W hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_1),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_1)),hAPP_int_int(number_number_of_int,W))) # label(fact_3488_minus__numeral__code_I6_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1727 (all Z_1 (hBOOL(hAPP_int_bool(nat_neg,Z_1)) -> zero_zero_nat = hAPP_int_nat(nat,Z_1))) # label(fact_3250_neg__nat) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1728 (all A_32 all C_8 all B_24 all D_7 hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_32),C_8)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_24),D_7)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_32),B_24)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C_8),D_7))) # label(fact_2347_add__diff__add) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1729 (all N hAPP_nat_nat(suc,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(bit1,hAPP_int_int(bit1,pls)))),N)) # label(fact_3115_Suc3__eq__add__3) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1730 (all Ya all Xa (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ya),Xa)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Ya)) <-> Xa = Ya))) # label(fact_1687_order__antisym__conv) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1731 (all P all Q all R hAPP_r1623423338t_bool(hAPP_f704188951t_bool(hAPP_f1456711209t_bool(cOMBB_690122963l_real,P),Q),R) = hAPP_f472159229t_bool(P,hAPP_r1962736578t_bool(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]. 4.77/4.84 1732 (all M_26 all N_50 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_26),N_50)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_26)),hAPP_nat_nat(semiri984289939at_nat,N_50))))) # label(fact_363_less__imp__of__nat__less) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1733 (all Wa all Z_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),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,Wa)),hAPP_int_nat(nat,Z_2))))) # label(fact_2772_zless__nat__conj) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1734 (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_3226_lemma__interval) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1735 (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]. 4.77/4.84 1736 (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]. 4.77/4.84 1737 (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_3390_exp__less__cancel) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1738 (all X all Y ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,ratreal(X)),ratreal(Y))) # label(fact_4703_real__divide__code) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1739 (all K_2 (is_int(K_2) -> K_2 = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),pls))) # label(fact_2351_diff__bin__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1740 (all B_70 all A_99 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_99),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_70)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_99),B_70)),zero_zero_int))))) # label(fact_1414_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1741 (all Na all 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,Na)),one_one_real)),hAPP_int_real(real_int,Ma))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Na),Ma)))) # label(fact_4445_int__less__real__le) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1742 (all N_55 ((N_55 != zero_zero_nat -> zero_z891286103de_int = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),N_55)) & (N_55 = zero_zero_nat -> one_on1684967323de_int = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),N_55)))) # label(fact_253_power__0__left) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1743 (all B_124 all C_71 all A_180 (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,B_124),C_71)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_124),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_180),C_71)))))) # label(fact_581_add__increasing) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1744 (all X_8 all N_12 (one_one_int = X_8 | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_12)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_8),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_8),N_12))))) # label(fact_2449_dvd__power) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1745 (all N_34 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_nat_rat(semiri151668891at_rat,N_34)))) # label(fact_713_of__nat__0__le__iff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1746 (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_3623_sin__squared__eq) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1747 (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) != one_one_int)))) # label(fact_2903_inv__not__1) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1748 (all A_1 (hBOOL(tendsto_nat_real(A_1,zero_zero_real,sequentially)) -> (hBOOL(monoseq_real(A_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(A_1,zero_zero_nat)),zero_zero_real)) -> (all N_1 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_1))),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_1)),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)))),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_1)),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_5062_summable__Leibniz_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1749 (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_3860_inv__real__of__nat__fact__gt__zero) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1750 (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_4688_rat__number__of__def) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1751 (all A_1 all B_1 (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(posDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_1),B_1))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B_1)) -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),A_1) = posDivAlg(A_1,B_1)) & (-(hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B_1)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),zero_zero_int))) -> posDivAlg(A_1,B_1) = hAPP_P1975530577nt_int(adjust(B_1),posDivAlg(A_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)))),B_1)))))) # label(fact_3301_posDivAlg_Opsimps) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1752 (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))) -> archim1246769320r_real(X) = N)))) # label(fact_4466_floor__eq2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1753 (all X_29 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_29),X_29))) # label(fact_1855_order__less__irrefl) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1754 (all A_133 zero_z891286103de_int = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_133),zero_z891286103de_int)) # label(fact_1146_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1755 (all Lx_3 all Ly_3 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_3)),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_3)) # label(fact_1066_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1756 (all Xa all F all G (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G)) -> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,Xa)),hAPP_nat_bool(G,Xa))))) # label(fact_1693_le__funD) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1757 (all X all Y hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,X)),hAPP_int_real(real_int,Y))) # label(fact_4395_real__of__int__add) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1758 (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_nat_real(real_nat,N)),pi)))) # label(fact_3696_tan__periodic__nat) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1759 (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(times_times_nat,hAPP_int_nat(nat,X)),hAPP_int_nat(nat,Y)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y))))) # label(fact_2792_Nat__Transfer_Otransfer__nat__int__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1760 (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_2223_zdvd__zdiffD) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1761 (all K_2 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),one_one_int) = hAPP_int_int(succ,K_2)) # label(fact_857_succ__def) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1762 (all Y_2 all Ya all X_2 all Xa (hBOOL(tendsto_nat_real(X_2,Xa,sequentially)) -> (hBOOL(tendsto_nat_real(Y_2,Ya,sequentially)) -> ((exists N_2 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),N_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(X_2,N_1)),hAPP_nat_real(Y_2,N_1))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)))))) # label(fact_5045_LIMSEQ__le) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1763 (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_2861_Euler__Criterion) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1764 (all Y_42 all X_48 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_48)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_42)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_42),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_42),X_48)),X_48)))))) # label(fact_1598_mult__left__le__one__le) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1765 (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_4274_fact__neg__int) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1766 (all A_1 (hBOOL(tendsto_nat_real(A_1,zero_zero_real,sequentially)) -> ((all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A_1,N_1)))) -> ((all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A_1,hAPP_nat_nat(suc,N_1))),hAPP_nat_real(A_1,N_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_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),A_1))))))) # label(fact_5064_summable) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1767 (all A_59 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_59),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_59),A_59)) # label(fact_1900_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1768 (all Xa all Ya (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Ya),Xa)) & hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Xa),Ya)) <-> Xa = Ya)) # label(fact_1704_order__eq__iff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1769 (all Ta all B all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D)) -> (all X_1 (is_int(X_1) -> ((all Xa_1 (is_int(Xa_1) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B)) -> X_1 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa_1))))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Ta)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),D)),Ta)))))))) # label(fact_5091_bset_I6_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1770 (all B_2 all D_3 all A_3 (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),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_4353_coprime__mult__nat) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1771 (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)) = one_one_nat) & (zero_zero_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))) = hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N))))) # label(fact_3829_fact__add__num__eq__if__nat) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1772 (all X_42 all Y_36 (X_42 = Y_36 -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_42),Y_36)))) # label(fact_1700_order__eq__refl) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1773 (all Xa hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),Xa))) = hAPP_nat_nat(suc,Xa)) # label(fact_4750_card__bdd__nat__set__le) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1774 (all Ya all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) -> hBOOL(hAPP_f448129468l_bool(nat_nat_set,hAPP_i1948725293t_bool(ord_at875362053st_int(Xa),Ya))))) # label(fact_5069_SetInterval_Otransfer__nat__int__set__function__closures) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1775 (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_2910_MultInv__prop5) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1776 (all A_198 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_198),one_one_nat) = A_198) # label(fact_320_power__one__right) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1777 (all Na (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Na)) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Na),one_one_int) = 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),Na))))))) # label(fact_4774_int__card__bdd__int__set__l__l) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1778 (all X_6 all N_11 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_11)) -> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_11),one_one_nat))),X_6) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_6),N_11))) # label(fact_2560_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1779 (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_4484_zgcd__zdvd__zgcd__zmult) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1780 (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]. 4.77/4.84 1781 (all P_1 collec1979865426at_nat(P_1) = P_1) # label(fact_145_Collect__def) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1782 (all X_40 all Y_34 (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_40),Y_34)) -> (Y_34 != X_40 -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_34),X_40))))) # label(fact_1715_linorder__cases) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1783 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),hAPP_int_int(bit1,K))))) # label(fact_758_rel__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1784 (all Xa all Ya (Xa = Ya | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Xa),Ya)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Xa),Ya)))) # label(fact_2014_order__le__less) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1785 (all A_1 all P_5 setS(A_1,P_5) = image_275383677t_bool(multInvPair(A_1,P_5),hAPP_i1948725293t_bool(sRStar,P_5))) # label(fact_4605_SetS__def) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1786 (all A_1 all D_1 all Na hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Na),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),D_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_1)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),D_1))),hAPP_n1699378549t_bool(ord_lessThan_nat,Na)))) # label(fact_4952_arith__series__nat) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1787 (all N all A_3 all B_2 (one_one_int = hAPP_int_int(legacy_zgcd(A_3),B_2) -> 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]. 4.77/4.84 1788 (all C_42 all D_15 all A_109 all B_79 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_109),B_79)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_42),D_15)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_109)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_42)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_109),C_42)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_79),D_15)))))))) # label(fact_1343_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1789 (all A_1 all A all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma)) -> (hBOOL(hAPP_int_bool(is_RRset(A),Ma)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),A)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(rRset2norRR(A,Ma),A_1)),hAPP_i1948725293t_bool(norRRset,Ma))))))) # label(fact_4591_RRset2norRR__correct2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1790 (all N_23 all A_76 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_76)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_76),N_23)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_76),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_76),N_23)))))) # label(fact_1608_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1791 (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_3739_real__of__nat__div2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1792 (all A_133 zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_133),zero_zero_real)) # label(fact_1149_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1793 (all A_82 all C_28 all B_55 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_82),C_28)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_55),C_28))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_28)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_82),B_55))))) # label(fact_1548_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1794 (all X (hBOOL(hAPP_int_bool(even_odd_even_int,X)) -> 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)))) = 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_3283_even__plus__one__div__two) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1795 (all M all 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)) = 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))))) # label(fact_3145_div__Suc__eq__div__add3) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1796 (all Ma all Na (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ma),hAPP_int_int(number_number_of_int,Na))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,Ma)),hAPP_int_real(number267125858f_real,Na))))) # label(fact_4441_number__of__less__real__of__int__iff2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1797 (all X_3 all Y_3 hAPP_real_real(cos,hAPP_real_real(arccos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_3),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_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,Y_3),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_3),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_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,Y_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3680_cos__arccos__lemma1) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1798 (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_2651_dvd_Oless__irrefl) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1799 (all X_52 all N_25 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_52),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_25)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_52),N_25)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_52),N_25))) # label(fact_1513_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1800 (all Lx_6 all Rx_6 all Ry_4 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_6),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,Rx_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_6),Ry_4))) # label(fact_1042_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1801 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa)) -> hBOOL(hAPP_real_bool(isCont_real_real(ln),Xa)))) # label(fact_5156_isCont__ln) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1802 (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,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)),X)))) # label(fact_3902_real__root__mult__exp) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1803 (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_4416_le__floor) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1804 (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(K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N))))) # label(fact_4510_zgcd__zmult__eq__self) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1805 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),one_one_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,Xa)),one_one_real)))) # label(fact_3535_real__sqrt__lt__1__iff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1806 (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_4571_norm__frac_Opinduct) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1807 (all A_54 all M_11 all N_21 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_54),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_11),N_21)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_54),M_11)),N_21)) # label(fact_2093_power__mult) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1808 (all K_2 all M hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),K_2),M))) # label(fact_2545_zcong__refl) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1809 (all N_24 all A_77 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_77)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_77),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_77),N_24)))))) # label(fact_1600_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1810 (all Xa -(hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd)) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven)))) # label(fact_3345_even__odd__conj) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1811 (all N all D_3 (is_int(D_3) -> (D_3 != zero_zero_int -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_3),N)) -> hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N),D_3)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,N)),hAPP_int_real(real_int,D_3)))))) # label(fact_4449_real__of__int__div) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1812 (all X_32 all Y_26 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_32),Y_26)) -> Y_26 != X_32)) # label(fact_1823_less__imp__neq) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1813 (all M_20 all N_38 all A_175 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_175)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_175),M_20)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_175),N_38))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_20),N_38))))) # label(fact_625_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1814 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ya),Xa)) <-> -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)))) # label(fact_2047_linorder__not__less) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1815 (all C_74 all A_185 all B_128 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_185),B_128)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_74),A_185)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_74),B_128))))) # label(fact_530_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1816 (all K_2 all L_2 (zero_zero_int != hAPP_int_int(div_mod_int(K_2),hAPP_int_int(uminus_uminus_int,L_2)) -> hAPP_int_int(div_mod_int(K_2),L_2) != zero_zero_int)) # label(fact_3483_zmod__zminus2__not__zero) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1817 (all N_18 one_one_complex = 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_18))) # label(fact_2191_power__m1__even) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1818 (all C_77 all A_188 all B_131 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_77),A_188)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_77),B_131))) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_188),B_131)))) # label(fact_516_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1819 (all M all V_1 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_1)) = 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_1))) # label(fact_3206_Suc__mod__eq__add3__mod__number__of) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1820 (all C_37 all B_65 all A_94 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_65),A_94)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_37),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_94),C_37)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_65),C_37)))))) # label(fact_1436_mult__strict__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1821 (all K_8 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),hAPP_int_complex(number528085621omplex,K_8)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(succ,K_8))) # label(fact_872_number__of__succ) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1822 (all X_10 all Y_7 all A_29 all B_22 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_10),Y_7)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_29),B_22)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_10),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Y_7),B_22))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_10),A_29)),B_22))) # label(fact_2392_mult__diff__mult) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1823 (all L (is_int(L) -> (min = hAPP_int_int(bit1,L) <-> L = min))) # label(fact_2161_rel__simps_I43_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1824 (all Ma all Na (Na = Ma <-> hAPP_nat_rat(semiri151668891at_rat,Na) = hAPP_nat_rat(semiri151668891at_rat,Ma))) # label(fact_322_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1825 (all Xa all Ya (Xa != Ya & hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Ya)) <-> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya)))) # label(fact_2030_order__less__le) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1826 (all A_1 all B_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),A_1)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),hAPP_i1948725293t_bool(d22set,A_1)))))) # label(fact_2946_d22set__mem) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1827 (all Xa all A (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),A)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))))) # label(fact_4611_Nat__Transfer_Otransfer__nat__int__set__function__closures_I6_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1828 (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_2976_Suc__mono) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1829 (all Xa all Ya (hAPP_nat_int(semiri1621563631at_int,Ya) = hAPP_nat_int(semiri1621563631at_int,Xa) <-> Ya = Xa)) # label(fact_326_Nat__Transfer_Otransfer__int__nat__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1830 (all A_6 all B_7 all Y all R_5 (is_int(Y) & is_int(B_7) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_7),Y)),R_5) = A_6 -> ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_7)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_5),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_7),R_5))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_7)) -> 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_7))) -> (B_7 != zero_zero_int -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_6),B_7) = Y))))) # label(fact_2851_divmod__int__rel__div__eq) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1831 (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),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M) -> N = M))) # label(fact_2156_mult__left__cancel) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1832 (all K all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) <-> 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)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),Na))))) # label(fact_3058_Suc__mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1833 (all A_123 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_123),one_on1684967323de_int) = A_123) # label(fact_1228_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1834 (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_2906_Suc__n__div__2__gt__zero) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1835 (all A_165 all N_33 all B_113 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_165),N_33)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_113),N_33))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_113)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_165),B_113))))) # label(fact_752_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1836 (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_293_add__less__mono1) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1837 (all C_19 all B_46 all A_69 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_46),A_69)) -> (B_46 = C_19 -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_19),A_69))))) # label(fact_1741_xt1_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1838 (all X all Y hAPP_complex_real(re,hAPP_real_complex(hAPP_r265291036omplex(complex,X),Y)) = X) # label(fact_4113_Re) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1839 (all Z_11 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_11) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_11),Z_11)) # label(fact_1895_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1840 (all A_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_int_bool(hAPP_i1948725293t_bool(zcong(A_1),zero_zero_int),P_5)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A_1)) -> (all X_1 (hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_1),setS(A_1,P_5))) -> hAPP_f957591787ol_nat(finite_card_int,X_1) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) # label(fact_3324_SetS__elems__card) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1841 (all A_150 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_150),zero_z891286103de_int) = A_150) # label(fact_916_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1842 (all Q_1 all B_2 all R_2 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_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),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_1),C_1))),R_2))))))) # label(fact_3126_zmult2__lemma__aux3) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1843 (all Xa all F ((all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(F,N_1)))) -> ((all N_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))),Xa))) -> hBOOL(summable_real(F))))) # label(fact_4945_pos__summable) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1844 (all A_1 all C all B_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_1),C)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_1),C))) <-> zero_zero_real = C | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_1),B_1)))) # label(fact_2358_dvd__mult__cancel__right) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1845 (all M hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,M)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,M))) # label(fact_3105_int__Suc) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1846 (all A_64 all B_41 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_64),B_41)) -> -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_41),A_64)))) # label(fact_1776_order__less__asym_H) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1847 (all Ta all D_1 all D (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D_1),D)) -> (all X_1 all K_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D_1),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_1),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,K_1),D))),Ta))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D_1),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_1),Ta))))))) # label(fact_2238_inf__period_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1848 (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]. 4.77/4.84 1849 (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]. 4.77/4.84 1850 (all A_133 zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_133),zero_zero_rat)) # label(fact_1145_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1851 (all X_60 all Y_50 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_60),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_50),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_50)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_60),Y_50))))) # label(fact_893_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1852 (all N (N != zero_zero_nat -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)))) # label(fact_298_gr0I) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1853 (all R_2 all A_3 hAPP_real_real(abs_abs_real,R_2) = hAPP_complex_real(norm_norm_complex,rcis(R_2,A_3))) # label(fact_4258_complex__mod__rcis) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1854 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),Xa)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),one_one_real)) -> hBOOL(hAPP_real_bool(isCont_real_real(arcsin),Xa))))) # label(fact_5170_isCont__arcsin) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1855 (all C_74 all A_185 all B_128 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_185),B_128)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_74),A_185)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_74),B_128))))) # label(fact_532_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1856 (all M_15 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,M_15),M_15) = 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_15)) # label(fact_1474_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1857 (all W_15 ((hAPP_int_nat(number_number_of_nat,W_15) = 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_15)) = one_one_nat) & (zero_zero_nat != hAPP_int_nat(number_number_of_nat,W_15) -> 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_15))))) # label(fact_46_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1858 (all B_1 all A_1 (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,B_1),A_1)) -> -hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A_1),B_1)))) # label(fact_1770_xt1_I9_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1859 (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_3241_negDivAlg__correct) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1860 (all A_14 all B_14 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(plus_plus_int,A_14),B_14))),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(abs_abs_int,A_14)),hAPP_int_int(abs_abs_int,B_14))))) # label(fact_2728_abs__triangle__ineq) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1861 (all Z_2 all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,Z_2))),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_2),hAPP_nat_int(semiri1621563631at_int,Ma))))) # label(fact_2659_dvd__int__iff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1862 (all P all Q all R hAPP_int_int(P,hAPP_P1175774780nt_int(Q,R)) = hAPP_P1175774780nt_int(hAPP_f521271395nt_int(hAPP_f11151005nt_int(cOMBB_47643171nt_int,P),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]. 4.77/4.84 1863 (all P_1 all Na all K (is_int(Na) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Na),K)),hAPP_int_int(div_mod_int(Na),K))) <-> (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,J_1),zero_zero_int)) & Na = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),I_2)),J_1) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),J_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,I_2),J_1))))))))) # label(fact_3151_split__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1864 (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]. 4.77/4.84 1865 (all C_47 all A_114 all B_84 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_114),B_84)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_47)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_114),C_47)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_84),C_47)))))) # label(fact_1321_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1866 (all A_160 all B_108 all C_62 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_160),B_108)),C_62) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_160),C_62)),B_108)) # label(fact_819_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1867 (all A_1 all E_2 all C all B_1 all D_1 (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_1),A_1)),E_2)),D_1) = C <-> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_1),E_2)),C) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_1),E_2)),D_1))) # label(fact_2387_eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1868 (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_2128_zmult__int) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1869 (all A_64 all B_41 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_64),B_41)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_41),A_64)))) # label(fact_1782_order__less__asym_H) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1870 (all Ya all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ya)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(nat,Xa)),hAPP_int_nat(nat,Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Xa),Ya)))))) # label(fact_2781_transfer__nat__int__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1871 (all Xa (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd)) | hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven)))) # label(fact_3343_even__odd__disj) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1872 (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)))) -> 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))) = negDivAlg(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_3227_negDivAlg__eqn__1__number__of) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1873 (all N (hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))) = 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_3822_lemma__odd__mod__4__div__2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1874 (all A_88 all M_12 all N_26 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_88),M_12)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_88),N_26)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_88),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_12),N_26))) # label(fact_1493_power__add) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1875 (all M_6 all N_10 ((N_10 != zero_zero_nat -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M_6),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),one_one_nat))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_6),N_10)) & (zero_zero_nat = N_10 -> one_one_int = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_6),N_10)))) # label(fact_2613_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1876 (all Xa (zero_zero_real = Xa <-> hAPP_real_real(exp_real,Xa) = one_one_real)) # label(fact_3403_exp__eq__one__iff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1877 (all Xa all Ya (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ya),Xa)) | Xa = Ya <-> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)))) # label(fact_1843_not__less__iff__gr__or__eq) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1878 (all N (zero_zero_nat != N -> (exists M_1 hAPP_nat_nat(suc,M_1) = N))) # label(fact_3982_not0__implies__Suc) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1879 (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_3414_ln__div) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1880 (all Ya all Xa (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Xa),Ya)),zEven)))) # label(fact_3337_even__times__either) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1881 (all P_3 all M ((zero_zero_nat != M -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),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)))) & (zero_zero_nat = M -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),M) = one_one_nat))) # label(fact_2553_power__eq__if) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1882 (all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2)) -> (one_one_nat != B_2 -> (exists X_1 exists Y_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),X_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)),one_one_nat))))) # label(fact_4036_coprime__bezout__strong) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1883 (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_2786_abs__power3__distrib) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1884 (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_4356_coprime__rmult__nat) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1885 (all V_19 all V_18 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_18)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_19)) -> hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_18),V_19)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_18)),hAPP_int_complex(number528085621omplex,V_19))))) # label(fact_865_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 4.77/4.84 1886 (all P all Q all R hAPP_int_fun_int_int(P,hAPP_nat_int(Q,R)) = hAPP_nat_fun_int_int(hAPP_f1935805932nt_int(hAPP_f125788821nt_int(cOMBB_859312247nt_nat,P),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]. 4.77/4.84 1887 (all A_25 all B_19 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,A_25)),B_19)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_25),B_19)))) # label(fact_2670_abs__le__D1) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1888 (all 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)) = hAPP_int_rat(number_number_of_rat,K_2)) # label(fact_4641_Frct__code__post_I4_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1889 (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_3525_real__sqrt__ge__zero) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1890 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y)) -> (exists X_1 Y = hAPP_real_real(exp_real,X_1)))) # label(fact_3504_exp__total) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1891 (all X hAPP_real_real(inverse_inverse_real,ratreal(X)) = ratreal(hAPP_rat_rat(inverse_inverse_rat,X))) # label(fact_4705_real__inverse__code) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1892 (all Lx_1 all Ly_1 all Rx_1 all Ry_1 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_1),Ly_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_1),Ry_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_1),Ly_1)),Ry_1))) # label(fact_1077_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1893 (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_67_zadd__int) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1894 (all Xa 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,Xa))))) # label(fact_4834_summable__exp) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1895 (all X_59 all Y_49 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_59),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_49),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_49)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_59),Y_49))))) # label(fact_991_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1896 (all A_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,A_1)),zero_zero_real)) <-> zero_zero_real = A_1)) # label(fact_2709_abs__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1897 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,Xa)),one_one_real)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),hAPP_int_int(bit1,pls))))) # label(fact_122_less__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1898 (all N (hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),N) = 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) = one_one_int)) # label(fact_2601_neg__one__power) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1899 (all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd)) -> -(all K_1 (is_int(K_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) != Xa))))) # label(fact_2954_zOddE) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1900 (all A_121 all B_90 all N_28 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_121),B_90)),N_28) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_121),N_28)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_90),N_28))) # label(fact_1245_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1901 (all Xa (hAPP_nat_nat(div_mod_nat(Xa),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = zero_zero_nat <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa)))) # label(fact_3806_even__nat__equiv__def) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1902 (all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Na))))) # label(fact_245_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1903 (all X_1 hAPP_real_real(sin,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)))) # label(fact_4819_lemma__sin__ext) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1904 (all Ta all B all D (is_int(Ta) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D)) -> (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)) -> (all X_1 (is_int(X_1) -> ((all Xa_1 (is_int(Xa_1) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa_1) != X_1)))))) -> (Ta = X_1 -> Ta = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),D))))))))) # label(fact_5092_bset_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1905 (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_3221_mod__induct) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1906 (all Na all Ma hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(suc,Na)),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,Na))),Ma)) # label(fact_5125_min__Suc1) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1907 (all K all N_1 all F_2 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))) = 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))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),K)))) # label(fact_4896_sumr__offset4) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1908 (all Xa all Ya (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Ya)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Ya)) & -hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ya),Xa)))) # label(fact_2022_less__le__not__le) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1909 (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_4280_fact__ge__one__int) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1910 (all V_14 all W_6 hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_14),W_6)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_14)),hAPP_int_complex(number528085621omplex,W_6))) # label(fact_1097_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1911 (all B_120 all A_174 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_174),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_120),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_174),B_120)),zero_zero_rat))))) # label(fact_641_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1912 (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(bit0,K2))))) # label(fact_559_less__eq__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1913 (all R_2 all A_3 hAPP_complex_real(re,rcis(R_2,A_3)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_2),hAPP_real_real(cos,A_3))) # label(fact_4261_Re__rcis) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1914 (all Ma all Na (hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),Na) = zero_zero_nat <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)))) # label(fact_2507_diff__is__0__eq) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1915 (all Diff all F all Na 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),Na)) & (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),Na)) -> 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))))) & F = hAPP_n546711566l_real(Diff,zero_zero_nat) -> (exists T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_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),Na))),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,Na),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,Na)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),Na))) = hAPP_real_real(F,H) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),T)))))) # label(fact_4933_Maclaurin__objl) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1916 (all R_2 all S_1 ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(number_number_of_rat,R_2)),hAPP_int_rat(number_number_of_rat,S_1))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(number267125858f_real,R_2)),hAPP_int_real(number267125858f_real,S_1))) # label(fact_4667_Ratreal__number__of__quotient2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1917 (all P all Q all R hAPP_f2002277285nt_int(P,hAPP_i1584592887nt_int(Q,R)) = hAPP_i998473677nt_int(hAPP_f142232355nt_int(hAPP_f1377453843nt_int(cOMBB_320287386nt_int,P),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]. 4.77/4.85 1918 (all N hAPP_nat_nat(fact_fact_nat,N) != zero_zero_nat) # label(fact_3831_fact__nonzero__nat) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1919 (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_3713_real__of__nat__Suc__gt__zero) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1920 (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_3101_Suc__pred) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1921 (all Xa all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,Xa)),hAPP_int_int(number_number_of_int,Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)))) # label(fact_509_le__number__of) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1922 (all A_162 all B_110 all C_64 (hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_162),C_64) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_162),B_110) -> B_110 = C_64)) # label(fact_792_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1923 (all X all K_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),K_2) = hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),K_2)))) # label(fact_3903_real__root__power) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1924 (all A_1 all B_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_1),B_1)),zero_zero_int)))) # label(fact_2380_less__iff__diff__less__0) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1925 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa)) -> (zero_zero_real = hAPP_real_real(ln,Xa) <-> one_one_real = Xa))) # label(fact_3363_ln__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1926 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K)),pls)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),pls)))) # label(fact_110_rel__simps_I12_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1927 (all Xa all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Xa),Na)),zEven)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven))))) # label(fact_3367_power__preserves__even) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1928 (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_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),N) = X))) # label(fact_3928_real__root__pow__pos2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1929 (all V_7 all W_3 hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_7),W_3)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(number267125858f_real,V_7)),hAPP_int_real(number267125858f_real,W_3))) # label(fact_2230_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1930 (all Ya all Xa (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Ya),Xa)) -> (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Xa),Ya)) <-> Xa = Ya))) # label(fact_1830_linorder__antisym__conv3) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1931 (all Xa (one_one_real = Xa <-> one_one_real = hAPP_real_real(sqrt,Xa))) # label(fact_3516_real__sqrt__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1932 (all V_25 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_25)),one_one_rat) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_25),hAPP_int_int(bit1,pls)))) # label(fact_27_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1933 (all I_1 all P_1 all K (hBOOL(hAPP_int_bool(P_1,K)) -> ((all I_2 (is_int(I_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),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)) -> (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_4004_int__induct) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1934 (all N N = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),one_one_nat)) # label(fact_2108_nat__mult__1__right) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1935 (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_4219_divmod__nat__rel) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1936 (all A_131 all E_3 all B_93 all C_49 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_131),B_93)),E_3)),C_49) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_131),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_93),E_3)),C_49))) # label(fact_1162_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1937 (all N_23 all A_76 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_76)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_76),N_23)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_76),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_76),N_23)))))) # label(fact_1610_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1938 (all Ma all Na hAPP_nat_nat(nat_case_nat(zero_zero_nat,cOMBI_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),Na)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),hAPP_nat_nat(suc,Na))) # label(fact_4879_diff__Suc) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1939 (all A_24 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(abs_abs_rat,A_24)),hAPP_rat_rat(abs_abs_rat,A_24)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_24),A_24)) # label(fact_2672_abs__mult__self) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1940 (all R_2 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_2))),one_one_real)),R_2))) # label(fact_4457_real__of__int__ceiling__diff__one__le) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1941 (all Ma all Na ((exists K_1 Na = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),K_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)))) # label(fact_3048_less__iff__Suc__add) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1942 (all K 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)) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),K)))) # label(fact_5080_setsum__shift__lb__Suc0__0) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1943 (all A_1 all E_2 all C all B_1 all D_1 (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_1),A_1)),E_2)),D_1))) <-> 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_1),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_1),E_2)),D_1))))) # label(fact_2427_less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1944 (all A_1 all B_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),A_1)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_1)))))) # label(fact_4726_one__le__Fract__iff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1945 (all Lx_3 all Ly_3 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),Rx_3)),Ly_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_3)),Rx_3)) # label(fact_1062_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1946 (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_4287_fact__reduce__int) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1947 (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_3629_minus__sin__cos__eq) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1948 (all X all Y (Y != X | 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]. 4.77/4.85 1949 (all C_24 all D_9 all A_78 all B_51 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_78),B_51)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_24),D_9)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_51)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_24)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_78),C_24)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_51),D_9)))))))) # label(fact_1572_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1950 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(fequal_real,X),Y)) | Y != X)) # label(help_fequal_2_1_fequal_000tc__RealDef__Oreal_T) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1951 (all U all Ma all Na 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)),Na))) <-> 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)),Na)))))) # label(fact_2538_nat__le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1952 (all C_16 all A_66 all B_43 (B_43 = A_66 -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_43),C_16)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_66),C_16))))) # label(fact_1762_ord__eq__less__trans) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1953 (all X_32 all Y_26 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_32),Y_26)) -> Y_26 != X_32)) # label(fact_1824_less__imp__neq) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1954 (all A_199 all N_54 all N_53 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_54),N_53)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_199)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_199),one_on1684967323de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_199),N_53)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_199),N_54))))))) # label(fact_260_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1955 (all A_3 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))))) = 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))))) # label(fact_2804_abs__power2__distrib) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1956 (all V_15 all W_7 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_15)),hAPP_int_real(number267125858f_real,W_7)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_15),W_7))) # label(fact_1094_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1957 (all C_27 all D_12 all A_81 all B_54 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_81),B_54)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_27),D_12)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_81)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_27)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_81),C_27)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_54),D_12)))))))) # label(fact_1556_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1958 (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_1 all Y_1 (hBOOL(hAPP_P1555980039t_bool(accp_P490777396at_nat(nat_gcd_rel),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,X_1),Y_1))) -> ((zero_zero_nat != Y_1 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Y_1),hAPP_nat_nat(div_mod_nat(X_1),Y_1)))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,X_1),Y_1))))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,A0),A1))))) # label(fact_4664_nat__gcd_Opinduct) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1959 (all A_55 all B_37 (is_int(B_37) & is_int(A_55) -> (A_55 != B_37 -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_55),B_37)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_55),B_37)))))) # label(fact_1998_order__neq__le__trans) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1960 (all C_25 all D_10 all A_79 all B_52 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_79),B_52)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_25),D_10)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_79)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_25)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_79),C_25)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_52),D_10)))))))) # label(fact_1566_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1961 (all A (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A)) -> A = image_nat_int(semiri1621563631at_int,image_int_nat(nat,A)))) # label(fact_4614_transfer__nat__int__set__return__embed) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1962 (all A_137 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_137),zero_zero_real) = zero_zero_real) # label(fact_1117_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1963 (all Xa 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),Xa),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,Xa))),hAPP_real_real(cos,Xa))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,Xa))),hAPP_real_real(cos,Xa)))))) # label(fact_4777_DERIV__cos__cos__mult) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1964 (all A_1 all B_1 all K all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ma)) -> (one_one_int = hAPP_int_int(legacy_zgcd(K),Ma) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),B_1),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),A_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),B_1)),Ma)))))) # label(fact_4524_zcong__cancel2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1965 (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_4508_zgcd__zmult__eq__self2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1966 (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_3946_Suc__lessE) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1967 (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]. 4.77/4.85 1968 (all B_118 all C_68 all A_172 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_172)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_118),C_68)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_118),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_172),C_68)))))) # label(fact_655_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1969 (all N_55 ((N_55 = zero_zero_nat -> one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N_55)) & (N_55 != zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N_55) = zero_zero_nat))) # label(fact_257_power__0__left) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1970 (all A_194 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_194),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_194),one_one_int)))) # label(fact_420_less__add__one) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1971 (all F ((all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(F,N_1)))) -> (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_4858_suminf__0__le) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1972 (all B_86 all A_116 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_116),zero_z891286103de_int)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_86)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_116),B_86)),zero_z891286103de_int))))) # label(fact_1311_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1973 (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_1255_mult__Bit0) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1974 (all K hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),K))))) # label(fact_4733_finite__Collect__less__nat) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1975 (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_3360_ln__gt__zero__imp__gt__one) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1976 (all Diff all F all Na 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),Na)) -> (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),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,H),T)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),Na)) -> 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(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),Na))),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,Na),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,Na)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),Na))) = hAPP_real_real(F,H) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H),T))))))))) # label(fact_4935_Maclaurin__minus) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1977 (all I (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),I)) -> hAPP_list_int_nat(size_size_list_int,quickc666637781d_zero(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))) # label(fact_4288_length__around__zero) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1978 (all A_157 all C_59 all D_18 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_157),C_59)),D_18) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_157),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_59),D_18))) # label(fact_835_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1979 (all X hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,one_one_int),zero_zero_int) = bezw(X,zero_zero_nat)) # label(fact_4154_bezw__0) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1980 (all A_74 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,pls))),A_74) = A_74) # label(fact_1617_mult__numeral__1) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1981 (all B_1 all A_1 all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),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_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),A_1)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),hAPP_i1948725293t_bool(wset(A_1),P_5)))))))) # label(fact_2934_wset__mem) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1982 (all Z_9 all X_24 all Y_20 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_24),Y_20)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_20),Z_9)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_24),Z_9))))) # label(fact_1926_order__le__less__trans) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1983 (all A_55 all B_37 (B_37 != A_55 -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_55),B_37)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_55),B_37))))) # label(fact_1995_order__neq__le__trans) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1984 (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_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),P_5))) = hAPP_i1948725293t_bool(sRStar,P_5)) # label(fact_3190_SRStar__def) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1985 (all C_2 hBOOL(hAPP_f448129468l_bool(nat_nat_set,image_nat_int(semiri1621563631at_int,C_2)))) # label(fact_4612_Nat__Transfer_Otransfer__int__nat__set__function__closures_I5_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1986 (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_3124_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1987 (all Xa 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,Xa)),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,Xa)))) # label(fact_4869_cos__paired) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1988 (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_2989_zmod__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1989 (all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) -> (-hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,Na))))) # label(fact_3809_even__num__iff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1990 (all P all Q all R hAPP_i1216375074nt_int(hAPP_f1375227553nt_int(hAPP_f1311564564nt_int(cOMBS_1640193733nt_int,P),Q),R) = hAPP_P1145851913nt_int(hAPP_i2104874254nt_int(P,R),hAPP_i1524277240nt_int(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]. 4.77/4.85 1991 (all C_73 all A_184 all B_127 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_184),B_127)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_184),C_73)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_127),C_73))))) # label(fact_535_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1992 (all P all Q all R hAPP_bool_bool(P,hAPP_nat_bool(Q,R)) = hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f894608603t_bool(cOMBB_bool_bool_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1993 (all A_1 all B_1 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),B_1)),zOdd)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),zOdd)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),zEven))))) # label(fact_3351_even__plus__odd__prop1) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1994 (all Ma all Z_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),Z_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_int_nat(nat,Z_2))))) # label(fact_2779_zless__nat__eq__int__zless) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1995 (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]. 4.77/4.85 1996 (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_2635_dvd_Ole__less__trans) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1997 (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_3765_tan__monotone) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1998 (all C_41 all D_14 all A_108 all B_78 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_108),B_78)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_41),D_14)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_78)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_41)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_108),C_41)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_78),D_14)))))))) # label(fact_1352_mult__mono) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 1999 (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]. 4.77/4.85 2000 (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_2103_mult__le__mono1) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2001 (all A_50 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_50),zero_zero_real))) # label(fact_2210_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2002 (all V_1 ((hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1))) -> zero_zero_int = hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,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_1) = hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_1))))) # label(fact_3264_int__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2003 (all A_16 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,A_16)),zero_zero_real))) # label(fact_2721_abs__not__less__zero) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2004 (all Na all Va hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(suc,Na)),hAPP_int_nat(number_number_of_nat,Va)) = 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,Va)))),zero_zero_nat),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Na),hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va))))))) # label(fact_5118_min__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2005 (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_2525_add__diff__assoc2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2006 (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_4152_DeMoivre) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2007 (all N hAPP_int_int(uminus_uminus_int,N) = archim1246769320r_real(hAPP_real_real(uminus_uminus_real,hAPP_int_real(real_int,N)))) # label(fact_4427_floor__minus__real__of__int) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2008 (all G all F all Xa all D_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_1)) -> ((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),Xa))),D_1)) -> 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),Xa))),D_1)) -> 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),Xa))),D_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,Xa)),hAPP_real_real(F,Z))))))))) # label(fact_5178_lemma__isCont__inj) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2009 (all A_157 all C_59 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_157),C_59)),D_18) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_157),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_59),D_18))) # label(fact_837_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2010 (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_2894_div__less__dividend) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2011 (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(ln,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(ln,X)),hAPP_real_real(ln,Y))))) # label(fact_3371_ln__mult) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2012 (all M_9 all A_31 all N_15 all B_23 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_31),N_15)),B_23)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_9),N_15)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_31),M_9)),B_23))))) # label(fact_2364_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2013 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,Xa)),one_one_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),zero_zero_real)))) # label(fact_3408_exp__less__one__iff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2014 (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_2102_le__cube) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2015 (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_4142_expi__def) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2016 (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]. 4.77/4.85 2017 (all C_30 all A_84 all B_57 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_30),A_84)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_30),B_57))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_30)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_84),B_57))))) # label(fact_1538_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2018 (all Xa all Ya 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,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa),Z_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ya),Z_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya))))) # label(fact_2122_real__mult__less__iff1) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2019 (all Xa all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa)) -> hBOOL(hAPP_real_bool(isCont_real_real(hAPP_n546711566l_real(root,Na)),Xa))))) # label(fact_5167_isCont__root__pos) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2020 (all F all Na 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)),Na))) # label(fact_4899_sumr__one__lb__realpow__zero) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2021 (all A_56 all B_38 (B_38 != A_56 -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_38),A_56)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_38),A_56))))) # label(fact_1985_xt1_I12_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2022 (all M all N (M = 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_1260_less__or__eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2023 (all A_151 A_151 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_151),zero_zero_nat)) # label(fact_912_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2024 (all A_75 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_75),hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,pls))) = A_75) # label(fact_1613_mult__numeral__1__right) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2025 (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_3189_mod__mult__self4) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2026 (exists K_4 ((all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(cnj,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_1)),K_4)))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),K_4)))) # label(fact_4086_cnj_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2027 (all C_41 all D_14 all A_108 all B_78 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_108),B_78)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_41),D_14)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_78)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_41)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_108),C_41)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_78),D_14)))))))) # label(fact_1354_mult__mono) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2028 (all X all Y hAPP_complex_real(im,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_complex_real(im,X)),hAPP_complex_real(im,Y))) # label(fact_4100_Im_Odiff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2029 (all Xa 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),Xa),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,Xa))),hAPP_real_real(sin,Xa))))) # label(fact_4790_DERIV__sin__sin__mult2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2030 (all Q_5 all R_4 all A_3 all B_2 all Q_1 all R_2 (is_int(Q_5) & is_int(Q_1) & 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_1),R_2))) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_5),R_4))) -> (B_2 != zero_zero_int -> Q_1 = Q_5))))) # label(fact_3255_unique__quotient) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2031 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,Xa)),hAPP_real_real(sqrt,Ya))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)))) # label(fact_3509_real__sqrt__le__iff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2032 (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]. 4.77/4.85 2033 (all Va all Wa (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,Wa)),hAPP_int_rat(number_number_of_rat,Va))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,Va)),hAPP_int_rat(number_number_of_rat,Wa))))) # label(fact_689_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2034 (all Z_2 all Xa all Ya (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Ya)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Ya),Z_2)) -> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Z_2))))) # label(fact_1927_order__le__less__trans) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2035 (all A_3 cis(A_3) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(cos,A_3)),hAPP_real_real(sin,A_3))) # label(fact_4151_cis__def) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2036 (all Xa all Ya (hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Xa),Ya))) <-> -hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa)) & -hBOOL(hAPP_nat_bool(even_odd_even_nat,Ya)) | hBOOL(hAPP_nat_bool(even_odd_even_nat,Ya)) & hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)))) # label(fact_3794_even__difference__nat) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2037 (all Y_43 all X_49 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_49)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_43)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_43),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_49),Y_43)),X_49)))))) # label(fact_1594_mult__right__le__one__le) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2038 (all A_140 all B_98 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_140),B_98) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_98),A_140)) # label(fact_1033_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2039 (all X_20 all Y_16 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_20),Y_16)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_20),Y_16)))) # label(fact_1973_order__less__imp__le) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2040 (all A_163 all B_111 all C_65 (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_163),C_65) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_163),B_111) -> C_65 = B_111)) # label(fact_787_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2041 (all N_55 ((N_55 != zero_zero_nat -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),N_55) = zero_zero_int) & (N_55 = zero_zero_nat -> one_one_int = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),N_55)))) # label(fact_258_power__0__left) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2042 (all B_2 hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int) = posDivAlg(zero_zero_int,B_2)) # label(fact_3293_posDivAlg__0) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2043 (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_4039_pred__Bit0) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2044 (all Z_1 hAPP_complex_real(im,hAPP_int_complex(ring_11397209091omplex,Z_1)) = zero_zero_real) # label(fact_4581_complex__Im__of__int) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2045 (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_3078_pos__mod__conj) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2046 (all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N))) # label(fact_1253_less__eq__nat_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2047 (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_3601_cos__add) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2048 (all Diff all Na all Xa (zero_zero_real = Xa -> (Na != zero_zero_nat -> 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,Xa))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na))))) # label(fact_4907_Maclaurin__zero) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2049 (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_3410_exp__ge__add__one__self) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2050 (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_4738_Nat__Transfer_Otransfer__nat__int__set__functions_I5_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2051 (all V_2 all V_1 ((-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1))) -> (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_int_nat(nat,hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,V_1)),hAPP_int_int(number_number_of_int,V_2))) = hAPP_nat_nat(div_mod_nat(hAPP_int_nat(number_number_of_nat,V_1)),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_int_nat(number_number_of_nat,V_1) = hAPP_nat_nat(div_mod_nat(hAPP_int_nat(number_number_of_nat,V_1)),hAPP_int_nat(number_number_of_nat,V_2)))) & (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1))) -> hAPP_nat_nat(div_mod_nat(hAPP_int_nat(number_number_of_nat,V_1)),hAPP_int_nat(number_number_of_nat,V_2)) = zero_zero_nat))) # label(fact_3275_mod__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2052 (all A_56 all B_38 (A_56 != B_38 -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_38),A_56)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_38),A_56))))) # label(fact_1990_xt1_I12_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2053 (all B_1 all A_1 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_1),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_1),hAPP_i1948725293t_bool(wset(A_1),P_5))) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(inv(P_5),B_1)),hAPP_i1948725293t_bool(wset(A_1),P_5)))))))) # label(fact_2923_wset__mem__inv__mem) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2054 (all Xa all K (Xa = 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) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven)))) # label(fact_3374_zEvenI) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2055 (all K_2 all L_2 hAPP_int_int(bit0,L_2) != hAPP_int_int(bit1,K_2)) # label(fact_133_rel__simps_I50_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2056 (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_3008_Suc__lessI) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2057 (all A_128 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,one_on1684967323de_int),A_128) = A_128) # label(fact_1193_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2058 (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]. 4.77/4.85 2059 (all A_128 A_128 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,one_one_rat),A_128)) # label(fact_1192_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2060 (all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,N),zero_zero_nat) = N) # label(fact_5193_max__0R) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2061 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(cos,X)),one_one_real))) # label(fact_3603_cos__le__one) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2062 (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(uminus_uminus_int,A_3)),hAPP_int_int(uminus_uminus_int,B_2))) # label(fact_4677_minus__rat__cancel) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2063 (all A_14 all B_14 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_14),B_14))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(abs_abs_rat,A_14)),hAPP_rat_rat(abs_abs_rat,B_14))))) # label(fact_2726_abs__triangle__ineq) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2064 (all M hAPP_int_int(legacy_zgcd(M),one_one_int) = one_one_int) # label(fact_4487_zgcd__1) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2065 (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)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,one_one_int),hAPP_int_int(number_number_of_int,K_2))) # label(fact_4666_Fract__1__number__of) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2066 (all Sb (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Sb)) -> 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,Sb)),zero_zero_real,sequentially)))) # label(fact_5041_LIMSEQ__neg__powr) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2067 (all B_1 all A_1 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),hAPP_i1948725293t_bool(d22set,A_1))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_1)))) # label(fact_2944_d22set__g__1) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2068 (all F all G (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G)) <-> (all X_1 (is_int(X_1) -> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_int_bool(F,X_1)),hAPP_int_bool(G,X_1))))))) # label(fact_1713_le__fun__def) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2069 (all Xa all Ya (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,Xa)),hAPP_nat_int(semiri1621563631at_int,Ya))))) # label(fact_2529_zdvd__int) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2070 (all A_3 all R_2 all B_2 all M all C_1 all D_3 all 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_2),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_2),D_3))),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_2),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))))) # label(fact_2589_xzgcda__linear__aux1) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2071 (all S ((all M_1 exists N_1 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,N_1),S)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)))) <-> -hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S)))) # label(fact_4607_infinite__nat__iff__unbounded) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2072 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),hAPP_int_int(bit0,K))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K)))) # label(fact_2184_rel__simps_I25_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2073 (all C_70 all D_21 all A_178 all B_122 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_178),B_122)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_70),D_21)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_178),C_70)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_122),D_21)))))) # label(fact_602_add__le__less__mono) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2074 (all A_129 all M_17 all B_91 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_129),M_17)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_91),M_17)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_129),B_91)),M_17)) # label(fact_1181_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2075 (all Xa all Ya (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Ya)) -> Xa != Ya)) # label(fact_1798_order__less__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2076 (all Lx_5 all Rx_5 all Ry_3 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_5),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_5),Ry_3)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_5),Rx_5)),Ry_3)) # label(fact_1045_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2077 (all Na all Xa exists T (hAPP_real_real(cos,Xa) = 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,Xa))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na))),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,Na))),pi)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,Na)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa),Na))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,Xa))))) # label(fact_4914_Maclaurin__cos__expansion) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2078 (all A_124 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_124),one_on1684967323de_int) = A_124) # label(fact_1221_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.85 2079 (all A_1 (hBOOL(tendsto_nat_real(A_1,zero_zero_real,sequentially)) -> (hBOOL(monoseq_real(A_1)) -> 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_1))),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_1)),sequentially))))) # label(fact_5051_summable__Leibniz_I4_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2080 (all A_127 (is_int(A_127) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),A_127) = A_127)) # label(fact_1205_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2081 (all X_7 all Y_6 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_7),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_6),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_7)),Y_6)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_7),Y_6)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_2458_power2__diff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2082 (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_3544_real__sqrt__abs) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2083 (all X hAPP_real_real(uminus_uminus_real,hAPP_real_real(arctan,X)) = hAPP_real_real(arctan,hAPP_real_real(uminus_uminus_real,X))) # label(fact_3523_arctan__minus) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2084 (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]. 4.77/4.86 2085 (all Z_2 all Wa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat,Wa)),hAPP_int_nat(nat,Z_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_2))))) # label(fact_2784_nat__less__eq__zless) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2086 (all M_14 all A_90 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_90),one_one_int)),M_14) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M_14),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_90),M_14))) # label(fact_1483_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2087 (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_3423_abs__ln__one__plus__x__minus__x__bound__nonpos) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2088 (all M all I all N 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)) = 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)) # label(fact_5123_min__diff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2089 (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))) # label(fact_3132_tsub__eq) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2090 (all B_2 all A_3 (is_int(B_2) & is_int(A_3) -> (zero_zero_int != A_3 | zero_zero_int != B_2 -> (exists C_3 exists D_2 (is_int(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(C_3)))))) # label(fact_4538_make__zrelprime) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2091 (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]. 4.77/4.86 2092 (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_4104_Im__sgn) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2093 (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_2576_zcong__zmult) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2094 (all A_165 all N_33 all B_113 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_165),N_33)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_113),N_33))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_113)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_165),B_113))))) # label(fact_749_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2095 (all A_130 all B_92 all C_48 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_130),B_92)),C_48) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_130),C_48)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_92),C_48))) # label(fact_1179_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2096 (all C_21 all A_71 all B_48 (B_48 = A_71 -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_21),B_48)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_21),A_71))))) # label(fact_1671_xt1_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2097 (all A_1 all B_1 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_1),B_1)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A_1)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B_1)))))) # label(fact_1373_mult__less__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2098 (all A_1 (zero_zero_real = A_1 <-> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_1),A_1) = zero_zero_real)) # label(fact_129_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2099 (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_4319_gcd__semilattice__nat_Ole__infI2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2100 (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_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = 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))))))) # label(fact_4228_div__neg__neg) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2101 (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_4328_gcd__semilattice__nat_Ole__infE) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2102 (all Xa all Ya (is_int(Ya) & is_int(Xa) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ya),Xa)) <-> Xa = Ya))) # label(fact_1707_order__eq__iff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2103 (all M_24 all N_48 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(semiri984289939at_nat,M_24)),hAPP_nat_nat(semiri984289939at_nat,N_48)) = hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_24),N_48))) # label(fact_374_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2104 (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_3002_zdvd__zmod__imp__zdvd) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2105 (all Lx_2 all Ly_2 all Rx_2 all Ry_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_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_2),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_2),Ly_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_2),Ry_2))) # label(fact_1072_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2106 (all X_26 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_26),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,X_26),X_26)) # label(fact_1908_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2107 (all Xa hAPP_real_real(ln,Xa) = 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)),Xa))) # label(fact_4883_ln__def) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2108 (all M_21 -hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_21)),zero_z891286103de_int))) # label(fact_453_of__nat__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2109 (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]. 4.77/4.86 2110 (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_3520_arctan__monotone_H) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2111 (all X all N_4 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_4)) -> (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_4),X)))))))) # label(fact_3932_real__root__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2112 (all A_163 all B_111 all C_65 (is_int(C_65) & is_int(B_111) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_163),C_65) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_163),B_111) -> B_111 = C_65))) # label(fact_789_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2113 (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_3619_cos__periodic__pi) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2114 (all B_2 all A_3 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,B_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),C_1))) # label(fact_4305_gcd__left__commute__nat) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2115 (all X0 all F all R_3 ((all X_1 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),hAPP_r1134773055l_bool(ord_gr788844697n_real(hAPP_real_real(uminus_uminus_real,R_3)),R_3))) -> 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_1)))))) -> (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X0),hAPP_r1134773055l_bool(ord_gr788844697n_real(hAPP_real_real(uminus_uminus_real,R_3)),R_3))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_3)) -> 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_4841_DERIV__power__series_H) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2116 (all W_4 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_int_rat(number_number_of_rat,W_4)),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_int_rat(number_number_of_rat,W_4)),hAPP_int_rat(number_number_of_rat,W_4))) # label(fact_1916_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2117 (all Z_8 all Y_19 all X_23 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_19),X_23)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_8),Y_19)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_8),X_23))))) # label(fact_1939_xt1_I7_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2118 (all Z_17 all X_44 all Y_38 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_44),Y_38)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_38),Z_17)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_44),Z_17))))) # label(fact_1646_order__trans) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2119 (all A_3 all C_1 hAPP_int_rat(hAPP_int_fun_int_rat(fract,zero_zero_int),C_1) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,zero_zero_int),A_3)) # label(fact_4673_eq__rat_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2120 (all Z_11 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_11) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Z_11),Z_11)) # label(fact_1896_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2121 (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(A_2) & is_int(B_4) -> (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_3321_negDivAlg_Opinduct) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2122 (all C all D_1 all A_1 all B_1 (is_int(D_1) & is_int(B_1) & is_int(A_1) & is_int(C) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),C)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),D_1)) != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),D_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),C)) <-> D_1 != C & B_1 != A_1))) # label(fact_1174_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2123 (all D_1 (one_one_nat = D_1 <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_1),zero_zero_nat)))) # label(fact_4023_coprime__0) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2124 (all W all V_1 ((hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,V_1)),hAPP_int_nat(number_number_of_nat,W)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,W))) & (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,V_1)),hAPP_int_nat(number_number_of_nat,W)) = hAPP_int_nat(nat,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,V_1)),hAPP_int_nat(number_number_of_nat,W)))))) # label(fact_3273_power__nat__number__of__number__of) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2125 (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_4130_complex__Re__mult) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2126 (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)) -> hAPP_real_real(cos,hAPP_real_real(arccos,Y)) = Y))) # label(fact_3674_cos__arccos) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2127 (all Xa 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,Xa))))) # label(fact_4830_summable__sin) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2128 (all A_1 (hBOOL(tendsto_nat_real(A_1,zero_zero_real,sequentially)) -> (hBOOL(monoseq_real(A_1)) -> 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_1))),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_1)),sequentially))))) # label(fact_5031_summable__Leibniz_I5_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2129 (all M_22 all N_45 all A_193 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_193)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_193),M_22)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_193),N_45))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_22),N_45))))) # label(fact_434_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2130 (all N_27 all M_16 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),M_16)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N_27)) -> 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_16),N_27)))))) # label(fact_1454_less__1__mult) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2131 (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_3015_add__Suc__shift) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2132 (all P all Q all R hAPP_complex_real(P,hAPP_nat_complex(Q,R)) = hAPP_nat_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Complex__Ocomplex_000tc__RealDef__Oreal_000tc__Nat__) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2133 (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_2889_aux______1) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2134 (all V_21 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21)) -> hAPP_int_complex(number528085621omplex,hAPP_int_int(succ,V_21)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_21)),one_one_complex))) # label(fact_740_semiring__number__of__add__1) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2135 (all A_32 all C_8 all B_24 all D_7 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_32),B_24)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C_8),D_7)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_32),C_8)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_24),D_7))) # label(fact_2349_add__diff__add) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2136 (all C_21 all A_71 all B_48 (A_71 = B_48 -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_21),B_48)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_21),A_71))))) # label(fact_1676_xt1_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2137 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(semiri151668891at_rat,Ma)),hAPP_nat_rat(semiri151668891at_rat,Na))))) # label(fact_510_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2138 (all V_21 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21)) -> hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_21)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_21)),one_one_nat))) # label(fact_742_semiring__number__of__add__1) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2139 (all P all Q all R hAPP_r931665397l_real(hAPP_f1607917947l_real(hAPP_f1076489843l_real(cOMBB_415306361l_real,P),Q),R) = hAPP_b1482221219l_real(P,hAPP_real_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__prod_Itc__RealDef__Oreal_) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2140 (all Na ((exists M_1 Na = 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))) <-> -hBOOL(hAPP_nat_bool(even_odd_even_nat,Na)))) # label(fact_3817_odd__Suc__mult__two__ex) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2141 (all A_121 all B_90 all N_28 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_121),N_28)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_90),N_28)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_121),B_90)),N_28)) # label(fact_1241_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2142 (all A_127 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),A_127) = A_127) # label(fact_1203_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2143 (all X_56 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_56),X_56))) # label(fact_1015_order__refl) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2144 (all V_16 all W_8 all Z_20 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_16),W_8))),Z_20) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_16)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,W_8)),Z_20))) # label(fact_1090_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2145 (all B_86 all A_116 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_116),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_86)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_116),B_86)),zero_zero_real))))) # label(fact_1313_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2146 (all Lx_1 all Ly_1 all Rx_1 all Ry_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_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),Ly_1)),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),Ly_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_1),Ry_1))) # label(fact_1078_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2147 (all C_70 all D_21 all A_178 all B_122 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_178),B_122)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_70),D_21)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_178),C_70)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_122),D_21)))))) # label(fact_605_add__le__less__mono) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2148 (all Xa (is_int(Xa) -> ((exists 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) = Xa & is_int(Y_1))) <-> -hBOOL(hAPP_int_bool(even_odd_even_int,Xa))))) # label(fact_3280_odd__equiv__def) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2149 (all A_147 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A_147) = A_147) # label(fact_939_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2150 (all A_105 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_105),A_105)),zero_zero_int))) # label(fact_1366_not__square__less__zero) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2151 (all A_3 hAPP_real_real(log(A_3),one_one_real) = zero_zero_real) # label(fact_3435_log__one) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2152 (all M zero_zero_nat = hAPP_nat_nat(div_mod_nat(M),hAPP_nat_nat(suc,zero_zero_nat))) # label(fact_3176_mod__1) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2153 (all A_1 all B_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_1)),one_one_rat)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B_1))))) # label(fact_4727_Fract__le__one__iff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2154 (all P_1 all P1 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> ((all X_1 all K_1 (is_int(X_1) & is_int(K_1) -> (hBOOL(hAPP_int_bool(P1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),D_1)))) <-> hBOOL(hAPP_int_bool(P1,X_1))))) -> ((exists Z all X_1 (is_int(X_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Z)) -> (hBOOL(hAPP_int_bool(P1,X_1)) <-> hBOOL(hAPP_int_bool(P_1,X_1)))))) -> ((exists X1 hBOOL(hAPP_int_bool(P1,X1))) -> (exists X1 (hBOOL(hAPP_int_bool(P_1,X1)) & is_int(X1)))))))) # label(fact_4005_minusinfinity) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2155 (all P all Q all R hAPP_r474017924t_real(hAPP_f1993765077t_real(hAPP_f895854793t_real(cOMBB_1948233853l_real,P),Q),R) = hAPP_f1888375463t_real(P,hAPP_r474017924t_real(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]. 4.77/4.86 2156 (all C_45 all A_112 all B_82 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_112),B_82)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_45)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_45),A_112)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_45),B_82)))))) # label(fact_1336_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2157 (all Z_7 all X_22 all Y_18 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_22),Y_18)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_18),Z_7)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_22),Z_7))))) # label(fact_1943_order__less__le__trans) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2158 (all C_39 all A_96 all B_67 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_96),B_67)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_39)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_39),A_96)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_39),B_67)))))) # label(fact_1426_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2159 (all A_128 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,one_on1645066479umeral),A_128) = A_128) # label(fact_1195_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2160 (all A_28 all B_20 all V_5 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_28),hAPP_int_real(number267125858f_real,V_5))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_20),hAPP_int_real(number267125858f_real,V_5))) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_28),B_20)),hAPP_int_real(number267125858f_real,V_5))) # label(fact_2399_left__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2161 (all X_55 all Y_47 all Z_19 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_55),Y_47)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_55),Z_19)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_55),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_47),Z_19))) # label(fact_1168_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2162 (all N hAPP_nat_real(real_nat,N) = hAPP_int_real(real_int,hAPP_nat_int(semiri1621563631at_int,N))) # label(fact_4394_real__of__int__of__nat__eq) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2163 (all V_1 hAPP_int_nat(number_number_of_nat,V_1) = hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,V_1))) # label(fact_2745_nat__number__of__def) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2164 (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]. 4.77/4.86 2165 (all P_1 all I_1 (is_int(I_1) -> (hBOOL(hAPP_nat_bool(P_1,hAPP_int_nat(nat,I_1))) <-> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),zero_zero_int)) -> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))) & (all N_1 (I_1 = hAPP_nat_int(semiri1621563631at_int,N_1) -> hBOOL(hAPP_nat_bool(P_1,N_1))))))) # label(fact_2788_split__nat) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2166 (all V_13 all V_12 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_12)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_13)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_12)),hAPP_int_nat(number_number_of_nat,V_13)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_12),V_13))))) # label(fact_1281_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2167 (all P all Q all R hAPP_i1153444985nt_int(hAPP_f55007289nt_int(hAPP_f747475781nt_int(cOMBB_222539774nt_int,P),Q),R) = hAPP_f1375227553nt_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_038) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2168 (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 = zero_zero_nat | hAPP_nat_nat(suc,zero_zero_nat) = X)) # label(fact_3038_nat__lt__two__imp__zero__or__one) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2169 (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_3294_posDivAlg__correct) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2170 (all A_1 all C all D_1 all B_1 (is_int(D_1) & is_int(B_1) -> (B_1 != zero_zero_int -> (D_1 != zero_zero_int -> (hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_1) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,C),D_1) <-> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B_1) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),D_1)))))) # label(fact_4671_eq__rat_I1_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2171 (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_int_int(div_mod_int(A_3),B_2) = hAPP_P1175774780nt_int(product_snd_int_int,negDivAlg(A_3,B_2))))) # label(fact_4233_mod__neg__pos) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2172 (all P all Q all R hAPP_P731461727l_real(P,hAPP_n332733730l_real(Q,R)) = hAPP_nat_real(hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2173 (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(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,Y),A_3)) = hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)),A_3)))) # label(fact_4430_powr__mult) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2174 (all I all K_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K_2)) -> (K_2 != hAPP_nat_nat(suc,I) -> -(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_3947_lessE) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2175 (all K_2 all N all M hAPP_nat_nat(div_mod_nat(M),N) = 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)) # label(fact_3182_mod__mult__self3) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2176 (all M all N all K_2 (hAPP_int_int(legacy_zgcd(N),K_2) = one_one_int -> (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_4526_zrelprime__zdvd__zmult__aux) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2177 (all P_1 all Xa all Ya (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Xa),Ya)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Ya),Xa)) -> hBOOL(P_1)))) # label(fact_1786_order__less__imp__triv) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2178 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y)) -> (exists 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),hAPP_real_real(tan,X_1))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)))))) # label(fact_3777_lemma__tan__total) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2179 (all Ma all Na (is_int(Ma) & is_int(Na) -> (Na != zero_zero_int | Ma != zero_zero_int <-> 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),Na)))))) # label(fact_4997_gcd__pos__int) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2180 (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_2974_Nat__Transfer_Otransfer__nat__int__functions_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2181 (all A_146 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),A_146) = A_146) # label(fact_948_add__0__left) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2182 (all Xa all A all Ma (hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,A),rsetR(Ma))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (one_one_int = hAPP_int_int(legacy_zgcd(Xa),Ma) -> 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),Xa),A)),rsetR(Ma))))))) # label(fact_4788_RsetR__zmult__mono) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2183 (all X_14 all P_6 all Q_7 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_14),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_6),Q_7)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_14),P_6)),Q_7)) # label(fact_2091_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2184 (all X hAPP_complex_complex(uminus473333897omplex,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)))) # label(fact_4134_complex__minus__def) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2185 (all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Na))))) # label(fact_329_zero__less__int__conv) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2186 (all X all Y hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_complex_real(re,X)),hAPP_complex_real(re,Y))) # label(fact_4121_Re_Oadd) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2187 (all P all Q all R hAPP_f151095827nt_int(P,hAPP_i1216375074nt_int(Q,R)) = hAPP_i1836591008nt_int(hAPP_f400200275nt_int(hAPP_f1731308757nt_int(cOMBB_735549356nt_int,P),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]. 4.77/4.86 2188 (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]. 4.77/4.86 2189 (all X_62 one_on1645066479umeral = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_62),zero_zero_nat)) # label(fact_479_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2190 (exists K_4 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),K_4)) & (all X_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(im,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_1)),K_4)))))) # label(fact_4107_Im_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2191 (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_3185_Divides_Otransfer__int__nat__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2192 (all A_1 all C all B_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_1),B_1)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_1),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_1),C))))) # label(fact_983_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2193 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),hAPP_nat_nat(suc,Ma))) <-> Na = Ma))) # label(fact_3052_le__less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2194 (all Ma all Na (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),Na) = hAPP_nat_nat(suc,zero_zero_nat) <-> Ma = hAPP_nat_nat(suc,zero_zero_nat) & Na = hAPP_nat_nat(suc,zero_zero_nat))) # label(fact_3046_mult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2195 (all Xa all Na (-hBOOL(hAPP_nat_bool(even_odd_even_nat,Na)) -> (Xa != zero_zero_real -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root,Na),Xa),hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,Na)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),hAPP_nat_nat(suc,zero_zero_nat)))))))))) # label(fact_3935_DERIV__odd__real__root) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2196 (all Na (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Na)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,Na)),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),Na)))))) # label(fact_4776_card__bdd__int__set__l__l) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2197 (all B_117 all C_67 all A_171 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_171)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_117),C_67)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_117),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_171),C_67)))))) # label(fact_659_add__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2198 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) -> hAPP_real_real(ln,Xa) = 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,Xa),one_one_real))),suc)))))) # label(fact_4825_ln__series) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2199 (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)) -> Y = X))) # label(fact_2638_dvd_Oantisym) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2200 (all X_33 all Y_27 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_33),Y_27)) -> -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_27),X_33)))) # label(fact_1811_order__less__not__sym) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2201 (all A_3 all B_2 (is_int(A_3) & is_int(B_2) -> (zero_zero_int != hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) -> (exists A_4 exists B_3 (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)) & 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 & hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_4),B_3) = one_one_int & is_int(B_3) & is_int(A_4)))))) # label(fact_5117_gcd__coprime__exists__int) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2202 (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_3233_anything__times__even) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2203 (all V_4 all W_2 all C_6 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_4),W_2))),C_6) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_4)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_int_rat(number_number_of_rat,W_2)),C_6))) # label(fact_2437_add__number__of__diff1) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2204 (all A_14 all B_14 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,A_14),B_14))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(abs_abs_real,A_14)),hAPP_real_real(abs_abs_real,B_14))))) # label(fact_2727_abs__triangle__ineq) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2205 (all M_6 all N_10 ((N_10 != zero_zero_nat -> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,M_6),N_10) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,M_6),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,M_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),one_one_nat)))) & (N_10 = zero_zero_nat -> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,M_6),N_10) = one_one_rat))) # label(fact_2607_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2206 (all P_1 all Na all K (hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(div_mod_nat(Na),K))) <-> (K = zero_zero_nat -> hBOOL(hAPP_nat_bool(P_1,Na))) & (zero_zero_nat != K -> (all I_2 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),K)) -> (Na = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),I_2)),J_1) -> hBOOL(hAPP_nat_bool(P_1,J_1)))))))) # label(fact_3197_split__mod) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2207 (all X_34 all Y_28 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_34),Y_28)) -> -hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_28),X_34)))) # label(fact_1807_order__less__imp__not__less) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2208 (all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),pls)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,Xa)),zero_zero_real)))) # label(fact_612_le__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2209 (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(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) = 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)))) # label(fact_2971_neg__zmod__mult__2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2210 (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_3508_real__sqrt__le__mono) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2211 (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_1271_add__le__mono1) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2212 (all R_2 all S_1 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,R_2),S_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,R_2),hAPP_real_real(uminus_uminus_real,S_1))) # label(fact_3432_real__diff__def) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2213 (all Xa all A (hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,Xa),A)) <-> hBOOL(hAPP_f448129468l_bool(A,Xa)))) # label(fact_142_mem__def) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2214 (all A_1 all B_1 ((all D_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),B_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),A_1)) <-> D_2 = one_one_nat)) <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_1),B_1) = one_one_nat)) # label(fact_4349_coprime__nat) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2215 (all A_46 all B_35 all C_14 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_46),B_35)),C_14)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,B_35),C_14)))) # label(fact_2257_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2216 (all B_89 all A_119 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_119)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_89)) -> 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_119),B_89)))))) # label(fact_1297_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2217 (all P_1 all Na ((exists X_1 (hBOOL(hAPP_nat_bool(P_1,X_1)) & hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na))))) <-> (exists M_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),Na)) & hBOOL(hAPP_nat_bool(P_1,M_1)))))) # label(fact_4900_ex__nat__less__eq) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2218 (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_2600_zpower__zdvd__prop1) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2219 (all N_24 all A_77 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_77)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_77),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_77),N_24)))))) # label(fact_1601_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2220 (all P_1 all K (hBOOL(hAPP_nat_bool(P_1,K)) -> ((all N_1 (hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_1))) -> hBOOL(hAPP_nat_bool(P_1,N_1)))) -> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))))) # label(fact_4001_zero__induct) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2221 (all N_24 all A_77 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_77)) -> 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_77),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_77),N_24)))))) # label(fact_1604_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2222 (all R_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,R_2),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,archim1246769320r_real(R_2))),one_one_real)))) # label(fact_4450_real__of__int__floor__add__one__ge) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2223 (all A_151 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_151),zero_zero_rat) = A_151) # label(fact_908_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2224 (all A_1 all Ma all Xa (is_int(Xa) -> (Xa != zero_zero_int -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),Ma)) -> big_co1548731110nt_int(cOMBI_int,image_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,times_times_int),Xa),hAPP_i1948725293t_bool(bnorRset(A_1),Ma))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(bnorRset(A_1),Ma))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Xa),hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(bnorRset(A_1),Ma)))))))) # label(fact_4811_Bnor__prod__power) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2225 (all A_138 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,zero_z126310315umeral),A_138) = zero_z126310315umeral) # label(fact_1109_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2226 (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)) -> zero_zero_int = A_3))))) # label(fact_2598_zcong__zless__0) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2227 (all A_59 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_59),A_59) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_59),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_1899_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2228 (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_4719_diff__rat) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2229 (all A_1 all Na (A_1 = zero_zero_real & zero_zero_nat != Na <-> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_1),Na) = zero_zero_real)) # label(fact_449_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2230 (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_2818_nat__aux__def) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2231 (all C_40 all A_97 all B_68 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_97),B_68)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_40)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_97),C_40)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_68),C_40)))))) # label(fact_1421_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2232 (all Xa all Ya all Xaa (Ya = zero_zero_real & Xaa = Xa <-> of_real_complex(Xaa) = hAPP_real_complex(hAPP_r265291036omplex(complex,Xa),Ya))) # label(fact_3983_complex__eq__cancel__iff2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2233 (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_2141_four__x__squared) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2234 -(all X_1 all Y_1 (is_int(X_1) & is_int(Y_1) -> 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)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))) != 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_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_2072__096_B_Bthesis_O_A_I_B_Bx_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_061_A_I4_) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2235 (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_3958_complex__minus) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2236 (all X all K_2 all N all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (N != zero_zero_nat -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N) -> (exists I_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),I_2) = X))))) # label(fact_4203_prime__power__exp) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2237 (all C all A_1 all B_1 (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_1),B_1)) -> (B_1 = C -> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_1),C))))) # label(fact_1665_ord__le__eq__trans) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2238 (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_2083_nat__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2239 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Xa)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),one_one_real)) -> hBOOL(monoseq_real(hAPP_r474017924t_real(power_power_real,Xa)))))) # label(fact_4805_monoseq__realpow) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2240 (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_4565_fact_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2241 (all Z_7 all X_22 all Y_18 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_22),Y_18)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_18),Z_7)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_22),Z_7))))) # label(fact_1944_order__less__le__trans) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2242 (all B_117 all C_67 all A_171 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_171)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_117),C_67)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_117),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_171),C_67)))))) # label(fact_663_add__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2243 (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_3630_cos__monotone__0__pi) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2244 (all W_16 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),hAPP_int_complex(number528085621omplex,W_16)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,pls)),W_16))) # label(fact_24_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2245 (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_4180_prime__g__one) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2246 (all B_2 all N all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_real_real(log(B_2),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(log(B_2),X)))) # label(fact_3744_log__nat__power) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2247 (all A_35 A_35 = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_35),zero_zero_real)) # label(fact_2331_diff__0__right) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2248 (all A_1 all B_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),zero_zero_int)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_1)),zero_zero_rat))))) # label(fact_4716_Fract__less__zero__iff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2249 (all A (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A)) -> hAPP_f957591787ol_nat(finite_card_int,A) = hAPP_f22106695ol_nat(finite_card_nat,image_int_nat(nat,A)))) # label(fact_4616_Nat__Transfer_Otransfer__int__nat__set__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2250 (all P all Q all R hAPP_f713949180nt_int(P,hAPP_i1153444985nt_int(Q,R)) = hAPP_i1836591008nt_int(hAPP_f617310012nt_int(hAPP_f197974549nt_int(cOMBB_638531587nt_int,P),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]. 4.77/4.86 2251 (all Ya all Xa (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ya),Xa)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) <-> Xa = Ya))) # label(fact_1690_order__antisym__conv) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2252 (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(inverse_divide_real,X),Y)) = 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)))) # label(fact_3906_real__root__divide) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2253 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa)) -> hBOOL(hAPP_real_bool(deriv_real(ln,Xa),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),Xa))))) # label(fact_3870_DERIV__ln__divide) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2254 (all A_148 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,zero_zero_real),A_148) = A_148) # label(fact_935_add__0) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2255 (all A_130 all B_92 all C_48 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_130),C_48)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_92),C_48)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_130),B_92)),C_48)) # label(fact_1178_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2256 (all C_22 all A_72 all B_49 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_72),B_49)) -> (C_22 = B_49 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_72),C_22))))) # label(fact_1669_ord__le__eq__trans) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2257 (all M all N (is_int(N) -> (N != zero_zero_int -> 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_4722_Fract__add__one) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2258 (all M_24 all N_48 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(semiri132038758t_real,M_24)),hAPP_nat_real(semiri132038758t_real,N_48)) = hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_24),N_48))) # label(fact_373_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2259 (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_2908_MultInv__zcong__prop1) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2260 (all A_176 all N_40 all N_39 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_40),N_39)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_176)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_176),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_176),N_39)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_176),N_40))))))) # label(fact_617_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2261 (all V_1 all W hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,V_1))),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_1)),hAPP_int_int(number_number_of_int,W)))) # label(fact_3123_zmod__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2262 (all X hAPP_real_real(uminus_uminus_real,ratreal(X)) = ratreal(hAPP_rat_rat(uminus_uminus_rat,X))) # label(fact_4702_real__uminus__code) # label(axiom) # label(non_clause). [assumption]. 4.77/4.86 2263 (all A_1 all Wa (zero_zero_nat != hAPP_int_nat(number_number_of_nat,Wa) & zero_z891286103de_int = A_1 <-> zero_z891286103de_int = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_1),hAPP_int_nat(number_number_of_nat,Wa)))) # label(fact_181_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2264 (all Lx_1 all Ly_1 all Rx_1 all Ry_1 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_1),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_1),Ly_1)),Ry_1)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_1),Ly_1)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_1),Ry_1))) # label(fact_1076_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2265 (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]. 4.77/4.87 2266 (all C_38 all A_95 all B_66 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_95),B_66)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_38)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_38),A_95)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_38),B_66)))))) # label(fact_1433_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2267 (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_4827_summable__rabs__cancel) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2268 (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))) -> N = archim1246769320r_real(X))))) # label(fact_4452_floor__eq) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2269 (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_4501_zgcd__self) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2270 (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_2465_one__not__neg__one__mod__m) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2271 (all X all A_3 all 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)) = hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),B_2))) # label(fact_4411_powr__add) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2272 (all A_1 all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Xa)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),natfloor(Xa))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,A_1)),Xa))))) # label(fact_3729_le__natfloor__eq) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2273 (all C_37 all B_65 all A_94 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_65),A_94)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_37),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_94),C_37)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_65),C_37)))))) # label(fact_1437_mult__strict__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2274 (all V_24 all W_14 all Z_21 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_24)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,W_14)),Z_21)) = 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_24),W_14))),Z_21)) # label(fact_187_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2275 (all L all U hAPP_n1699378549t_bool(ord_at238088361st_nat(L),U) = hAPP_n1699378549t_bool(ord_at4362885an_nat(L),hAPP_nat_nat(suc,U))) # label(fact_5071_atLeastLessThanSuc__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2276 (all Z_2 all Xa all Ya (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Ya)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ya),Z_2)) -> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Z_2))))) # label(fact_1644_order__trans) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2277 (all A_44 all B_33 all K_6 (hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_33),K_6) = A_44 -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,B_33),A_44)))) # label(fact_2266_dvdI) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2278 (all C_31 all A_85 all B_58 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_31),A_85)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_31),B_58))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_31)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_85),B_58))))) # label(fact_1529_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2279 (all A_135 all B_95 (zero_z891286103de_int = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_135),B_95) -> A_135 = zero_z891286103de_int | B_95 = zero_z891286103de_int)) # label(fact_1132_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2280 (all W all Z1 all 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)) = 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))) # label(fact_1263_zadd__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2281 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N)) -> N = zero_zero_nat | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)))) # label(fact_2952_divides__le) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2282 (all A_3 all B_2 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)) = 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)))))) # label(fact_2455_zspecial__product) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2283 (all A_20 all N_9 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(abs_abs_real,A_20)),N_9) = hAPP_real_real(abs_abs_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_20),N_9))) # label(fact_2688_power__abs) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2284 (all Na all A_1 all B_1 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_1),B_1))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1)) -> -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_1,B_1),Na))),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A_1,B_1),Na))))))))) # label(fact_4246_not__P__Bolzano__bisect) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2285 (all X all Y (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(fequal_rat,X),Y)) | Y = X)) # label(help_fequal_1_1_fequal_000tc__Rat__Orat_T) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2286 (all P_3 hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int) = normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),zero_zero_int))) # label(fact_4659_normalize__denom__zero) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2287 (all X all N (-hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) -> hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N)) = X)) # label(fact_3895_odd__real__root__power__cancel) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2288 (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,X),Y)) | X = Y)) # label(fact_2643_dvd_Ole__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2289 (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_2627_dvd_Oord__less__eq__trans) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2290 (all B_115 all A_169 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_169)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_115)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_169),B_115)))))) # label(fact_672_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2291 (all A image_int_nat(nat,image_nat_int(semiri1621563631at_int,A)) = A) # label(fact_4602_transfer__int__nat__set__return__embed) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2292 (all X all Y hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_complex_real(re,X)),hAPP_complex_real(re,Y))) # label(fact_4125_Re_Odiff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2293 (all Na (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Na)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat,Na)),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_eq_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),Na)))))) # label(fact_4775_card__bdd__int__set__le) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2294 (all X_50 all Y_44 -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_50),X_50)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_44),Y_44))),zero_zero_rat))) # label(fact_1587_not__sum__squares__lt__zero) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2295 (all A_134 zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,zero_zero_real),A_134)) # label(fact_1142_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2296 (all Sb all Ta all A_1 all B_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B_1)) -> 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_1),Sb)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,B_1),Ta))),pair_less)))) # label(fact_4540_pair__lessI1) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2297 (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]. 4.77/4.87 2298 (all N_16 all X_12 all Y_9 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_12),Y_9)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_12),N_16)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_9),N_16))))) # label(fact_2316_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2299 (all Ma all Na ((hBOOL(hAPP_nat_bool(even_odd_even_nat,Na)) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,Ma))) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),Na))))) # label(fact_3786_even__add) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2300 (all Wa all Xa (Xa = hAPP_int_complex(number528085621omplex,Wa) <-> hAPP_int_complex(number528085621omplex,Wa) = Xa)) # label(fact_90_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2301 (all Xa all Ya all F ((all X_1 hBOOL(hAPP_real_bool(deriv_real(F,X_1),zero_zero_real))) -> hAPP_real_real(F,Ya) = hAPP_real_real(F,Xa))) # label(fact_3936_DERIV__isconst__all) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2302 (all Na (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_n522471361de_int(semiri1424489471de_int,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)))) # label(fact_241_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2303 (all Xa all A_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),hAPP_int_real(real_int,A_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(Xa)),A_1)))) # label(fact_4421_ceiling__le__eq) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2304 (all X_40 all Y_34 (is_int(Y_34) & is_int(X_40) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_40),Y_34)) -> (X_40 != Y_34 -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_34),X_40)))))) # label(fact_1719_linorder__cases) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2305 (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_3107_diff__Suc__diff__eq2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2306 (all C_35 all D_13 all A_92 all B_63 all R_6 (zero_zero_complex != R_6 -> (B_63 = A_92 & D_13 != C_35 -> hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_92),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,R_6),C_35)) != hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_63),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,R_6),D_13))))) # label(fact_1446_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2307 (all Xa all Ya (is_int(Ya) & is_int(Xa) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)) <-> Xa = Ya)))) # label(fact_1972_linorder__antisym__conv2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2308 (all A_1 all B_1 all C (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_1),B_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_1),C) <-> C = B_1)) # label(fact_811_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2309 (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_2483_diff__add__inverse) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2310 (all Na all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma)) -> (Na = 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),Na)),Ma))))) # label(fact_2551_dvd__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2311 (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_5122_min__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2312 (all V_16 all W_8 all Z_20 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_16)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,W_8)),Z_20)) = 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_16),W_8))),Z_20)) # label(fact_1088_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2313 (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_4324_gcd__semilattice__nat_Oinf__greatest) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2314 (all Ya all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ya),Xa)) -> Xa = comple124823625p_real(hAPP_r1134773055l_bool(ord_gr788844697n_real(Ya),Xa)))) # label(fact_4631_Sup__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2315 (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_3704_real__of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2316 (all Z_15 all X_37 all Y_31 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_37),Y_31)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_31),Z_15)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_37),Z_15))))) # label(fact_1740_order__less__trans) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2317 (all X_8 all N_12 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_12)) | X_8 = one_one_nat -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_8),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_8),N_12))))) # label(fact_2448_dvd__power) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2318 (all Z_2 all Ya all Xa (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ya),Xa)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Z_2),Ya)) -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Z_2),Xa))))) # label(fact_1921_xt1_I8_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2319 (all A_1 all E_2 all C all B_1 all D_1 (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_1),A_1)),E_2)),D_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,A_1),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_1),E_2)),D_1))))) # label(fact_2421_le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2320 (all X_2 ((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_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_3),N_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,hAPP_nat_real(X_2,M_1)),hAPP_nat_real(X_2,N_1)))),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,J_1))))))))) <-> hBOOL(cauchy_real(X_2)))) # label(fact_4374_Cauchy__iff2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2321 (all Xa (Xa = zero_zero_real <-> zero_zero_real = hAPP_real_real(sqrt,Xa))) # label(fact_3513_real__sqrt__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2322 (all A_133 zero_zero_complex = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_133),zero_zero_complex)) # label(fact_1147_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2323 (all X_16 all Y_12 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_12),X_16)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_16),Y_12)))) # label(fact_2038_linorder__le__less__linear) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2324 (all Xa (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven)) -> hAPP_int_int(div_mod_int(Xa),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_int)) # label(fact_3375_even__div__2__prop1) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2325 (all U hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(ord_at641636577an_int(zero_zero_int),U)) = hAPP_int_nat(nat,U)) # label(fact_4918_card__atLeastZeroLessThan__int) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2326 (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_2841_zdiv__mono2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2327 (all K (is_int(K) -> (min = hAPP_int_int(bit1,K) <-> min = K))) # label(fact_2162_rel__simps_I47_J) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2328 (all Xa all Ya 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,Z_2),Xa)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_2),Ya))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya))))) # label(fact_2123_real__mult__le__cancel__iff2) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2329 (all K hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),K))))) # label(fact_4734_finite__Collect__le__nat) # label(axiom) # label(non_clause). [assumption]. 4.77/4.87 2330 (all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) <-> zero_zero_nat != Na)) # label(fact_304_neq0__conv) # label(axiom) # label(non_clause). [assumption]. 4.91/5.01 2331 (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]. 4.91/5.01 2332 (all C_47 all A_114 all B_84 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_114),B_84)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_47)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_114),C_47)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_84),C_47)))))) # label(fact_1322_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 4.91/5.01 2333 (all V_7 all W_3 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(number_number_of_int,V_7)),hAPP_int_int(number_number_of_int,W_3)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_7),W_3))) # label(fact_2231_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 4.91/5.01 2334 (all C_24 all D_9 all A_78 all B_51 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_78),B_51)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_24),D_9)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_51)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_24)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_78),C_24)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_51),D_9)))))))) # label(fact_1573_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 4.91/5.01 2335 (all A_138 zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A_138)) # label(fact_1106_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 4.91/5.01 2336 (all B all P_2 all P_1 all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D)) -> ((exists Z all X_1 (is_int(X_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Z)) -> (hBOOL(hAPP_int_bool(P_1,X_1)) <-> hBOOL(hAPP_int_bool(P_2,X_1)))))) -> ((all X_1 (is_int(X_1) -> ((all Xa_1 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D))) -> (all Xb (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa_1) != X_1)))) -> (hBOOL(hAPP_int_bool(P_1,X_1)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),D))))))) -> ((all X_1 all K_1 (is_int(K_1) & is_int(X_1) -> (hBOOL(hAPP_int_bool(P_2,X_1)) <-> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),D))))))) -> ((exists X1 (hBOOL(hAPP_int_bool(P_1,X1)) & is_int(X1))) <-> (exists X_1 (is_int(X_1) & (exists Xa_1 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),B)) & hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xa_1),X_1))) & is_int(Xa_1))) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D))))) | (exists X_1 (is_int(X_1) & hBOOL(hAPP_int_bool(P_2,X_1)) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D))))))))))) # label(fact_5180_cpmi) # label(axiom) # label(non_clause). [assumption]. 4.91/5.01 2337 (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_3912_real__root__ge__zero) # label(axiom) # label(non_clause). [assumption]. 4.91/5.01 2338 (all 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)))) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,N))) # label(fact_3137_add__2__eq__Suc_H) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2339 (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_3609_sin__ge__zero) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2340 (all M_18 all N_30 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(semiri132038758t_real,M_18)),hAPP_nat_real(semiri132038758t_real,N_30)) = hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_18),N_30))) # label(fact_1103_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2341 (all V_20 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_20)) -> hAPP_int_rat(number_number_of_rat,hAPP_int_int(succ,V_20)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,V_20)))) # label(fact_744_semiring__1__add__number__of) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2342 (all C_18 all A_68 all B_45 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_68),B_45)) -> (C_18 = B_45 -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_68),C_18))))) # label(fact_1754_ord__less__eq__trans) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2343 (all M all N all K_2 (hAPP_int_int(legacy_zgcd(N),K_2) = one_one_int -> (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_4516_zrelprime__zdvd__zmult) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2344 (all A_156 all C_58 all D_17 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_58),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_156),D_17)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_156),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_58),D_17))) # label(fact_843_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2345 (all P all Q all R hAPP_real_bool(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,P),Q),R) = hAPP_bool_bool(hAPP_r1938957037l_bool(P,R),hAPP_real_bool(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__HOL__Obool_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2346 (all X all Y (X = Y | -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y)))) # label(help_fequal_1_1_fequal_000tc__Nat__Onat_T) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2347 (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_4408_powr__less__cancel) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2348 (all Xa (hAPP_real_real(cos,Xa) != zero_zero_real -> hBOOL(hAPP_real_bool(isCont_real_real(tan),Xa)))) # label(fact_5157_isCont__tan) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2349 (all C_33 all A_87 all B_60 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_33),A_87)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_33),B_60))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_33)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_87),B_60))))) # label(fact_1517_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2350 (all V_1 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_int_nat(number_number_of_nat,V_1)) = one_one_nat) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_int_nat(number_number_of_nat,V_1)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_1))))) # label(fact_502_nat__1__add__number__of) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2351 (all A_33 all B_25 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_33),B_25)),B_25) = A_33) # label(fact_2344_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2352 (all B_1 all A_1 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),zEven)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),zOdd)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),B_1)),zOdd))))) # label(fact_3352_IntNatAux_Oeven__plus__odd) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2353 (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_4838_rat__inverse__code) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2354 (all X -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X)),zero_zero_real))) # label(fact_3400_not__exp__less__zero) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2355 (all Xa all Ya (hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa)) | hBOOL(hAPP_nat_bool(even_odd_even_nat,Ya)) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Xa),Ya))))) # label(fact_3791_even__product__nat) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2356 (all Xa all P_5 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(sRStar,P_5))) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Xa),zero_zero_int),P_5)))) # label(fact_3196_StandardRes__SRStar__prop1a) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2357 (all Na zero_zero_real = 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),Na))) # label(fact_4911_lemma__STAR__cos2) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2358 (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_1505_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2359 (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_4214_divmod__nat__rel__unique) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2360 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),Xa)) -> hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_r474017924t_real(power_power_real,Xa)),zero_zero_real,sequentially)))) # label(fact_5033_LIMSEQ__inverse__realpow__zero) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2361 (all A_86 all C_32 all B_59 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_86),C_32)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_59),C_32))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_32)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_86),B_59))))) # label(fact_1526_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2362 (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,J_2),I)),K_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))) # label(fact_2526_diff__add__assoc2) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2363 (all A_159 all B_107 all C_61 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_159),B_107)),C_61) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_159),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_107),C_61))) # label(fact_822_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2364 (all A_19 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(abs_abs_rat,A_19)))) # label(fact_2705_abs__ge__zero) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2365 (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_2848_div__eq__minus1) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2366 (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_2569_Euler_Oaux2) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2367 (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_1 (hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1)) & hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_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))))) = X))))) # label(fact_3823_sin__zero__lemma) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2368 (all A_121 all B_90 all N_28 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_121),B_90)),N_28) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_121),N_28)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_90),N_28))) # label(fact_1244_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2369 (all R_2 all A_3 all B_2 hAPP_complex_complex(scaleR1652505878omplex(R_2),hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_2),A_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_2),B_2))) # label(fact_5110_complex__scaleR) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2370 (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_4402_powr__ge__pzero) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2371 (all Lx_2 all Ly_2 all Rx_2 all Ry_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_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_2),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_2),Ly_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_2),Ry_2))) # label(fact_1073_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2372 (all A_1 all E_2 all C all B_1 all D_1 (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_1),A_1)),E_2)),D_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,A_1),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_1),E_2)),D_1))))) # label(fact_2423_le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2373 (all P_1 all Na all A_1 all B_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1)) -> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_1),A_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Na)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A_1,B_1),Na))),hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A_1,B_1),Na))))) # label(fact_4241_Bolzano__bisect__diff) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2374 (all N_46 all A_195 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_195)) -> 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_195),N_46))))) # label(fact_409_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2375 (all Xa all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa)),one_one_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),one_one_real))))) # label(fact_3917_real__root__le__1__iff) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2376 (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_2603_zcong__zprime__prod__zero__contra) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2377 (all P all Q all R hAPP_real_real(P,hAPP_nat_real(Q,R)) = hAPP_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__Nat__Ona) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2378 (all A_123 (is_int(A_123) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_123),one_one_int) = A_123)) # label(fact_1233_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2379 (all X_2 (hBOOL(summable_complex(X_2)) -> hBOOL(summable_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_2))))) # label(fact_4846_Im_Osummable) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2380 (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_3537_real__sqrt__abs2) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2381 (all Z_2 all Xa all Ya (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Ya)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ya),Z_2)) -> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Z_2))))) # label(fact_1941_order__less__le__trans) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2382 (all A_91 all B_61 all V_10 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_91),hAPP_int_complex(number528085621omplex,V_10))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_61),hAPP_int_complex(number528085621omplex,V_10))) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_91),B_61)),hAPP_int_complex(number528085621omplex,V_10))) # label(fact_1465_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2383 (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_2958_dvd__pos__nat) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2384 (all X all Y (is_int(X) & is_int(Y) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Y)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),X)) | Y = X)) # label(fact_57_zless__linear) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2385 (all Ya all Xa all A_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Ya)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(log(A_1),Xa)),hAPP_real_real(log(A_1),Ya)))))))) # label(fact_3445_log__less__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2386 (all A_1 all B_1 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_1))),hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,plus_plus_int),hAPP_int_fun_int_int(times_times_int,B_1)))))),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_1))),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_1))))),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_1)))),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_1))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),zero_zero_int)))))) = divmod_int_rel(A_1,B_1)) # label(fact_4836_divmod__int__rel__def) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2387 (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]. 4.91/5.02 2388 (all V_19 all V_18 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_18)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_19)) -> hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_18),V_19)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_18)),hAPP_int_int(number_number_of_int,V_19))))) # label(fact_868_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2389 (all A_163 all B_111 all C_65 (hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_163),C_65) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_163),B_111) -> C_65 = B_111)) # label(fact_786_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2390 (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]. 4.91/5.02 2391 (all Xa all A (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),A)) -> hBOOL(hAPP_int_bool(nat_is_nat,Xa))))) # label(fact_5131_Nat__Transfer_Otransfer__int__nat__set__function__closures_I6_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2392 (all Na all Ma (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,Na)),hAPP_nat_real(real_nat,Ma))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),Ma)))) # label(fact_3703_real__of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2393 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ya),hAPP_real_real(uminus_uminus_real,Xa))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa),Ya)),zero_zero_real)))) # label(fact_3438_real__add__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2394 (all M_22 all N_45 all A_193 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_193)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_193),M_22)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_193),N_45))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_22),N_45))))) # label(fact_437_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2395 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),one_one_real)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(Xa)),one_one_nat)))) # label(fact_3316_natceiling__le__eq__one) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2396 (all Xa all Ya (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Ya)) -> -hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Ya),Xa)))) # label(fact_1812_order__less__not__sym) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2397 (all W_4 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_i769753017umeral(number1443263063umeral,W_4)),hAPP_i769753017umeral(number1443263063umeral,W_4)) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_i769753017umeral(number1443263063umeral,W_4)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_1918_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2398 (all Lx_5 all Rx_5 all Ry_3 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_5),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,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_5),Rx_5)),Ry_3)) # label(fact_1048_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2399 (all Na all G all Xa all Ma (hBOOL(hAPP_real_bool(deriv_real(G,Xa),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)),Na),Xa),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,Na)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(G,Xa)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)))),Ma))))) # label(fact_4773_DERIV__fun__pow) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2400 (all B_77 all A_107 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_77),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_107),zero_zero_real)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_77)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_107)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_107),B_77))))) # label(fact_1356_split__mult__pos__le) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2401 (all Xa all Ya (Ya = zero_zero_rat & zero_zero_rat = Xa <-> 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,Xa),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,Ya),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_rat)))) # label(fact_994_sum__power2__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2402 (all A_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),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_1),A_1)),zero_zero_int)))) # label(fact_597_double__add__le__zero__iff__single__add__le__zero) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2403 (all Na all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),Ma)) -> big_co1705425894at_nat(cOMBI_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)),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,Na)))) # label(fact_5082_fact__div__fact) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2404 (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_4403_powr__gt__zero) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2405 (all Va all K all A_1 all B_1 (A_1 != B_1 -> ((all X_1 hBOOL(hAPP_real_bool(deriv_real(Va,X_1),K))) -> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(Va,A_1)),hAPP_real_real(Va,B_1))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(Va,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_1),B_1)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3875_DERIV__const__average) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2406 (all A_3 all B_2 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),one_one_int))) # label(fact_2575_zcong__1) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2407 (all A_1 all B_1 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_1),B_1))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1)) -> (all N_1 -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_1,B_1),N_1))),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A_1,B_1),N_1)))))))))) # label(fact_4247_not__P__Bolzano__bisect_H) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2408 (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]. 4.91/5.02 2409 (all Z_12 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_12) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_12),Z_12)) # label(fact_1892_mult__2) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2410 (all V_1 ((-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1))) -> hAPP_nat_real(real_nat,hAPP_int_nat(number_number_of_nat,V_1)) = hAPP_int_real(number267125858f_real,V_1)) & (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1))) -> zero_zero_real = hAPP_nat_real(real_nat,hAPP_int_nat(number_number_of_nat,V_1))))) # label(fact_3740_real__of__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2411 (all P all Q all R hAPP_real_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,P),Q),R) = hAPP_real_real(hAPP_r1250527377l_real(P,R),hAPP_real_real(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__RealDef_) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2412 (all Na (is_int(Na) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Na)) -> 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),Na))))) = Na))) # label(fact_4770_int__card__bdd__int__set__l__le) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2413 (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_3717_real__of__nat__div4) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2414 (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))) -> 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_2182_Legendre__1mod4) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2415 (all Y_2 all X_2 ((exists A_2 ((all N_1 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(X_2,N_1))),A_2))) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_2)))) -> (hBOOL(vanishes(Y_2)) -> 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_2)),Y_2)))))) # label(fact_5126_vanishes__mult__bounded) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2416 (all Na 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,Na),Ma)),Ma)) <-> Na = one_one_nat))) # label(fact_2552_dvd__mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2417 (all A_1 all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),archim1246769320r_real(Xa))) <-> 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_1)),one_one_real)),Xa)))) # label(fact_4467_less__floor__eq) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2418 (all B_2 all Q_5 all R_4 all Q_1 all R_2 (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_5)),R_4)),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_1)),R_2))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_4)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_4),B_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_5),Q_1))))))) # label(fact_2064_unique__quotient__lemma) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2419 (all N_31 all A_153 all B_104 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_153),B_104)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_153)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_153),N_31)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_104),N_31))))))) # label(fact_877_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2420 (all P_1 ((exists X_1 (is_int(X_1) & hBOOL(hAPP_int_bool(P_1,X_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)))) <-> (exists X_1 hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X_1)))))) # label(fact_735_transfer__int__nat__quantifiers_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2421 (all A_1 all B_1 all Na (zero_zero_nat != Na -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_1),B_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_1),Na)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_1),Na)))))) # label(fact_2957_pow__divides__eq__nat) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2422 (all Ya all Xa (is_int(Xa) & is_int(Ya) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ya),Xa)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)) <-> Xa = Ya)))) # label(fact_1691_order__antisym__conv) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2423 (all N one_one_real = 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))) # label(fact_3762_cos__2npi) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2424 (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]. 4.91/5.02 2425 (all B_1 all A_1 all F all G all Xa all D (hBOOL(hAPP_real_bool(deriv_real(F,hAPP_real_real(G,Xa)),D)) -> (D != zero_zero_real -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),Xa)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),B_1)) -> ((all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),Y_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),B_1)) -> Y_1 = hAPP_real_real(F,hAPP_real_real(G,Y_1)))) -> (hBOOL(hAPP_real_bool(isCont_real_real(G),Xa)) -> hBOOL(hAPP_real_bool(deriv_real(G,Xa),hAPP_real_real(inverse_inverse_real,D)))))))))) # label(fact_5172_DERIV__inverse__function) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2426 (all N all M hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,N),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)))) # label(fact_4559_times__code__int__def) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2427 (all X all 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)))) = hAPP_int_int(int_lcm(X),Y)) # label(fact_4623_Nitpick_Oint__lcm__def) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2428 (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_1277_real__add__left__mono) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2429 (all A_129 all M_17 all B_91 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_129),M_17)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_91),M_17)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_129),B_91)),M_17)) # label(fact_1184_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2430 (all B_118 all C_68 all A_172 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_172)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_118),C_68)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_118),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_172),C_68)))))) # label(fact_658_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2431 (all Z_18 all Y_40 all X_46 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_40),X_46)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_18),Y_40)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_18),X_46))))) # label(fact_1635_xt1_I6_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2432 (all Z_2 all Xa all Ya (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ya),Z_2)) -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Z_2))))) # label(fact_1942_order__less__le__trans) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2433 (all C_12 all D_8 all A_43 all B_32 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_43),B_32)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,C_12),D_8)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_43),C_12)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_32),D_8)))))) # label(fact_2275_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2434 (all V_25 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_25)),one_one_complex) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_25),hAPP_int_int(bit1,pls)))) # label(fact_28_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2435 (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_4663_Nitpick_Oint__gcd__def) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2436 (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_2830_int__div__less__self) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2437 (all Z_1 hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_complex_real(re,Z_1)),hAPP_complex_real(im,Z_1)) = Z_1) # label(fact_4119_complex__surj) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2438 (all A_1 all B_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_1),B_1))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_1)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_1),zero_zero_real)))) # label(fact_1287_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2439 (all K all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,K)),hAPP_int_int(number_number_of_int,L))))) # label(fact_58_less__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2440 (all N_37 all A_168 all B_114 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_168),B_114)) -> (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,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_168),N_37)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_114),N_37)))))) # label(fact_698_power__mono) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2441 (all 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)))))) = hAPP_complex_real(norm_norm_complex,Z_1)) # label(fact_4137_cmod__def) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2442 (all B_73 all A_102 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_102)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_73),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_73),A_102)),zero_zero_rat))))) # label(fact_1388_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2443 (all Z_2 all Xa all Ya (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Ya)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ya),Z_2)) -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Z_2))))) # label(fact_1645_order__trans) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2444 (all X_40 all Y_34 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_40),Y_34)) -> (X_40 != Y_34 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_34),X_40))))) # label(fact_1718_linorder__cases) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2445 (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_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = 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))))))) # label(fact_4227_div__pos__neg) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2446 (all Wa all Ya all Xa all Z_2 (Wa = Xa | Z_2 = Ya <-> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Wa),Z_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa),Ya)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Wa),Ya)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa),Z_2)))) # label(fact_1189_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2447 (all A_1 all B_1 all C (zero_zero_real != C -> (A_1 = B_1 <-> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_1),C) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_1),C)))) # label(fact_2098_real__mult__right__cancel) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2448 (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_2533_dvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2449 (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_3244_odd__pow) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2450 (all K all I_1 all J image_nat_nat(hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),K),hAPP_n1699378549t_bool(ord_at4362885an_nat(I_1),J)) = hAPP_n1699378549t_bool(ord_at4362885an_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K))) # label(fact_4905_image__add__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2451 (all Xa all Ya all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) -> (Xa = Ya <-> hAPP_real_real(hAPP_n546711566l_real(root,Na),Ya) = hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa)))) # label(fact_3891_real__root__eq__iff) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2452 (all Ta all A all D (is_int(Ta) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D)) -> (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)) -> (all X_1 (is_int(X_1) -> ((all Xa_1 (is_int(Xa_1) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D))) -> (all Xb (is_int(Xb) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A)) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa_1) != X_1)))))) -> (Ta = X_1 -> Ta = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),D))))))))) # label(fact_5093_aset_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2453 (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_2890_div__le__mono2) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2454 (all C_19 all B_46 all A_69 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_46),A_69)) -> (C_19 = B_46 -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_19),A_69))))) # label(fact_1744_xt1_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2455 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N)))) # label(fact_2542_zdiff__int) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2456 (all B_73 all A_102 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_102)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_73),zero_z126310315umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_73),A_102)),zero_z126310315umeral))))) # label(fact_1390_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2457 (all Na all Xa 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,Xa))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na))),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,Na))),pi)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,Na)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa),Na))) = hAPP_real_real(sin,Xa) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,Xa))))) # label(fact_4913_Maclaurin__sin__expansion2) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2458 (all M_23 all N_47 hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_23),N_47)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(semiri984289939at_nat,M_23)),N_47)) # label(fact_385_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2459 (all A_45 all B_34 all C_13 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_45),B_34)),C_13)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_45),C_13)))) # label(fact_2261_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2460 (all A_1 hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(d22set,A_1)))) # label(fact_3942_d22set__fin) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2461 (all A_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_1),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_1),A_1)),zero_zero_rat)))) # label(fact_486_double__add__less__zero__iff__single__add__less__zero) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2462 (all Ma all K all Na (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Na),K) <-> K = zero_zero_nat | Na = Ma)) # label(fact_2088_mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2463 (all X_62 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_62),zero_zero_nat) = one_one_rat) # label(fact_476_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2464 (all W hAPP_complex_complex(cnj,hAPP_int_complex(number528085621omplex,W)) = hAPP_int_complex(number528085621omplex,W)) # label(fact_4077_complex__cnj__number__of) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2465 (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]. 4.91/5.02 2466 (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_3960_complex__mult) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2467 (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_4959_gcd__dvd2__int) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2468 (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_3494_power3__minus) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2469 (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_3511_real__sqrt__less__mono) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2470 (all Xa all Ya (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Xa),Ya)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Ya),Xa)) | Xa = Ya)) # label(fact_1840_not__less__iff__gr__or__eq) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2471 (all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> X = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y))) # label(fact_4320_gcd__proj1__if__dvd__nat) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2472 (all P all Q all R hAPP_i1524277240nt_int(hAPP_f1735705551nt_int(hAPP_f713949180nt_int(cOMBS_1891347093nt_int,P),Q),R) = hAPP_P1975530577nt_int(hAPP_i1216375074nt_int(P,R),hAPP_i1524277240nt_int(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]. 4.91/5.02 2473 (all Ta all D_1 all D (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D_1),D)) -> (all X_1 all K_1 (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Ta))) <-> -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,K_1),D))),Ta))))))) # label(fact_2236_inf__period_I4_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2474 (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_1262_zmult__1__right) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2475 (all A_75 (is_int(A_75) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_75),hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,pls))) = A_75)) # label(fact_1614_mult__numeral__1__right) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2476 (all A_138 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A_138) = zero_zero_nat) # label(fact_1111_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2477 (all C all B_1 all A_1 (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,B_1),A_1)) -> (C = B_1 -> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,C),A_1))))) # label(fact_1742_xt1_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2478 (all A_1 all B_1 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,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A_1)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B_1))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_1),A_1))))) # label(fact_1406_mult__less__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2479 (all N hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(real_nat,N)))) # label(fact_3687_real__of__nat__ge__zero) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2480 (all B_1 all A_1 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),zOdd)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),zOdd)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),B_1)),zEven))))) # label(fact_3353_IntNatAux_Oodd__plus__odd) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2481 (all R_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R_2)) -> (exists N_1 (is_int(N_1) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_2),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_1),one_one_int)))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_1)),R_2)))))) # label(fact_4476_reals__Archimedean__6b__int) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2482 (all Ra hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(times_times_real,Ra)),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))))),Ra,sequentially))) # label(fact_5047_LIMSEQ__inverse__real__of__nat__add__minus__mult) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2483 (all C_26 all D_11 all A_80 all B_53 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_80),B_53)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_26),D_11)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_80)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_26)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_80),C_26)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_53),D_11)))))))) # label(fact_1560_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2484 (all X all Y ((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)) & (Y = zero_zero_nat -> bezw(X,Y) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,one_one_int),zero_zero_int)))) # label(fact_4229_bezw_Osimps) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2485 (all Na (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(real_nat,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)))) # label(fact_3724_real__of__nat__gt__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2486 (all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M)) -> 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)))))) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M),one_one_int)))) # label(fact_3162_inv__is__inv__aux) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2487 (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_4991_gcd__commute__int) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2488 (all P all Q all R hAPP_int_int(hAPP_int_fun_int_int(P,R),Q) = hAPP_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2489 (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_4463_real__of__int__div2) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2490 (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))) & (hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M),N)) = N -> zero_zero_nat = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M)),N)))) # label(fact_3177_mod__Suc) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2491 (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,B_2),one_one_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)))) # label(fact_2857_neg__zdiv__mult__2) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2492 (all Z_2 all Ya all Xa (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Ya),Xa)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Z_2),Ya)) -> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Z_2),Xa))))) # label(fact_1934_xt1_I7_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2493 (all X X = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),X)) # label(fact_4310_gcd__idem__nat) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2494 (all B_87 all A_117 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_117)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_87),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_87),A_117)),zero_zero_nat))))) # label(fact_1308_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2495 (all X (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(sqrt,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3546_real__sqrt__pow2) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2496 (all B_69 all A_98 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_98),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_69),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_98),B_69)))))) # label(fact_1416_mult__neg__neg) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2497 (all A_89 all M_13 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_89),one_one_complex)),M_13) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_89),M_13)),M_13)) # label(fact_1486_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2498 (all A_83 all C_29 all B_56 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_83),C_29)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_56),C_29))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_29)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_83),B_56))))) # label(fact_1543_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2499 (all N hAPP_int_nat(nat,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N))) = zero_zero_nat) # label(fact_3487_nat__zminus__int) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2500 (all A_134 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,zero_z891286103de_int),A_134) = zero_z891286103de_int) # label(fact_1139_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2501 (all A_82 all C_28 all B_55 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_82),C_28)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_55),C_28))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_28)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_82),B_55))))) # label(fact_1549_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2502 (all A_1 all B_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_1),B_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,B_1),A_1)))) # label(fact_4010_coprime__commute) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2503 (all X hAPP_int_nat(nat,archim856651990g_real(X)) = natceiling(X)) # label(fact_4363_natceiling__def) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2504 (all X_39 all Y_33 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_39),Y_33)) -> -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_33),X_39)))) # label(fact_1724_order__less__asym) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2505 (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]. 4.91/5.02 2506 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,Xa)),hAPP_int_real(real_int,Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)))) # label(fact_4387_real__of__int__less__iff) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2507 (all N all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> hAPP_int_nat(nat,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(nat,X)),N))) # label(fact_2800_Nat__Transfer_Otransfer__nat__int__functions_I4_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2508 (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_4232_mod__int__def) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2509 (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_2909_MultInv__def) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2510 (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_282_nat__add__left__commute) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2511 (all A_139 all B_97 all 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_139),B_97)),C_51) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_139),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_97),C_51))) # label(fact_1050_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2512 (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_2486_diff__cancel2) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2513 (all Ma all K all F ((all M_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,M_1)),hAPP_nat_nat(F,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(F,Ma)),K)),hAPP_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),K)))))) # label(fact_2153_mono__nat__linear__lb) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2514 (all Na all Ma all F ((all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_1)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_1)))) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,hAPP_nat_nat(suc,N_1))),hAPP_nat_nat(F,N_1))))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),Ma)) <-> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,Ma)),hAPP_nat_nat(F,Na))) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,Na)),hAPP_nat_nat(F,Ma)))))) # label(fact_3214_dvd_Olift__Suc__mono__less__iff) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2515 (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_3370_ln__ge__zero__imp__ge__one) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2516 (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(bit1,L_2)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),L_2))) # label(fact_200_add__Bit0__Bit1) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2517 (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]. 4.91/5.02 2518 (all X hAPP_real_real(log(hAPP_real_real(exp_real,one_one_real)),X) = hAPP_real_real(ln,X)) # label(fact_3446_log__ln) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2519 (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)) & (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> M = hAPP_nat_nat(div_mod_nat(M),N)))) # label(fact_3181_mod__if) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2520 (all C all A_1 all B_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B_1))) <-> 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_1),A_1)) | 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_1),B_1)))) # label(fact_1371_mult__less__cancel__left__disj) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2521 (all N_36 all A_167 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_167)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_167),N_36))))) # label(fact_704_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2522 (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_3612_cos__monotone__0__pi_H) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2523 (all A_54 all M_11 all N_21 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_54),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_11),N_21)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_54),M_11)),N_21)) # label(fact_2096_power__mult) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2524 (all K_8 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(number_number_of_int,K_8)) = hAPP_int_int(number_number_of_int,hAPP_int_int(succ,K_8))) # label(fact_874_number__of__succ) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2525 (all A_1 all B_1 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_1),B_1),Ma)) <-> hAPP_int_int(div_mod_int(B_1),Ma) = hAPP_int_int(div_mod_int(A_1),Ma)))) # label(fact_3083_zcong__zmod__eq) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2526 (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 -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),M) = one_one_int)) # label(fact_5006_invertible__coprime__int) # label(axiom) # label(non_clause). [assumption]. 4.91/5.02 2527 (all Xa quotient_of(Xa) = 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,Xa)),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))))) # label(fact_5022_quotient__of__def) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2528 (all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> -hBOOL(hAPP_nat_bool(even_odd_even_nat,P_3)) | hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) = P_3)) # label(fact_4195_prime__odd) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2529 (all X_55 all Y_47 all Z_19 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_55),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Y_47),Z_19)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_55),Y_47)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_55),Z_19))) # label(fact_1164_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2530 (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_2120_real__mult__less__mono2) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2531 (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]. 4.91/5.03 2532 (all C_38 all A_95 all B_66 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_95),B_66)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_38)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_38),A_95)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_38),B_66)))))) # label(fact_1430_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2533 (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(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)))) & (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(hAPP_int_fun_int_int(div_div_int,K_2),L_2)))) # label(fact_3564_z3div__def) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2534 (all X_55 all Y_47 all Z_19 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_55),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Y_47),Z_19)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_55),Y_47)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_55),Z_19))) # label(fact_1167_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2535 (all M_26 all N_50 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_26),N_50)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_26)),hAPP_n522471361de_int(semiri1424489471de_int,N_50))))) # label(fact_361_less__imp__of__nat__less) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2536 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),min)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K)),min)))) # label(fact_2179_rel__simps_I28_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2537 (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_2808_decr__lemma) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2538 (all Xa all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Xa)) | Na = 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,Xa),Na))))) # label(fact_302_nat__zero__less__power__iff) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2539 (all A_1 (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_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) <-> zero_zero_rat != A_1)) # label(fact_208_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2540 (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_3239_Parity_Oeven__plus__even) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2541 (all Wa all Xa (hAPP_int_nat(number_number_of_nat,Wa) = Xa <-> Xa = hAPP_int_nat(number_number_of_nat,Wa))) # label(fact_94_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2542 (all Na (zero_zero_nat = Na <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Na)),zero_zero_int)))) # label(fact_771_int__le__0__conv) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2543 (all X_3 all N_5 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_5)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_3)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N_5),X_3)))))) # label(fact_3923_real__root__pos__pos) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2544 (all Wa all Xa (hAPP_int_real(number267125858f_real,Wa) = Xa <-> hAPP_int_real(number267125858f_real,Wa) = Xa)) # label(fact_92_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2545 (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_1 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_1))))) # label(fact_4206_bezout__prime) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2546 (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_5135_Nat__Transfer_Otransfer__int__nat__function__closures_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2547 (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_3741_real__of__nat__div3) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2548 (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_2860_Euler__part2) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2549 (all X_10 all Y_7 all A_29 all B_22 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_10),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Y_7),B_22))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_10),A_29)),B_22)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_10),Y_7)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_29),B_22))) # label(fact_2389_mult__diff__mult) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2550 (all Lx_4 all Ly_4 all Rx_4 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_4)),Rx_4) = 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_4),Rx_4))) # label(fact_1061_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2551 (all C all A_1 all B_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),A_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),zero_zero_int)))) # label(fact_1372_mult__less__cancel__left__disj) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2552 (all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),pls)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,Xa)),zero_zero_rat)))) # label(fact_611_le__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2553 (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]. 4.91/5.03 2554 (all X all Y 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))))) = zero_zero_real) # label(fact_3632_sin__cos__add) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2555 (all L all U hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_at875362053st_int(L),U)))) # label(fact_5066_finite__atLeastAtMost__int) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2556 (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]. 4.91/5.03 2557 (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,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_4766_card__less__Suc2) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2558 (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_3532_real__sqrt__ge__one) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2559 (all P_1 all I_1 all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),K)) -> (hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K),one_one_int))) -> ((all I_2 (is_int(I_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),K)) -> (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_2926_int__less__induct) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2560 (all Z_1 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),Z_1) = Z_1) # label(fact_2111_real__mult__1) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2561 (all B_88 all A_118 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_118)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_88),zero_z126310315umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_118),B_88)),zero_z126310315umeral))))) # label(fact_1300_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2562 (all V_1 hAPP_int_real(number267125858f_real,V_1) = hAPP_int_real(real_int,hAPP_int_int(number_number_of_int,V_1))) # label(fact_4377_real__number__of) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2563 (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_2825_div__neg__pos__less0) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2564 (all A_163 all B_111 all C_65 (hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_163),C_65) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_163),B_111) -> B_111 = C_65)) # label(fact_785_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2565 (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_3090_zdiv__zmod__equality2) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2566 (all P all Q all R hAPP_i16292313t_bool(hAPP_f601898971t_bool(hAPP_f857296063t_bool(cOMBB_214503962ol_int,P),Q),R) = hAPP_r2115560837t_bool(P,hAPP_int_rat(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]. 4.91/5.03 2567 (all L all F ((all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_1)),hAPP_nat_real(F,hAPP_nat_nat(suc,N_1))))) -> ((all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_1)),L))) -> ((all E (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),E)) -> (exists N_1 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_1)),E)))))) -> hBOOL(tendsto_nat_real(F,L,sequentially)))))) # label(fact_5153_increasing__LIMSEQ) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2568 (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,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))) # label(fact_4357_gcd__diff2__nat) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2569 (all A_3 all B_2 all 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)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(legacy_zgcd(A_3),B_2)),N)) # label(fact_4491_zgcd__power__distrib) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2570 (all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Ya)))) # label(fact_616_le__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2571 (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_3217_gr0__implies__Suc) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2572 (exists X_1 (is_int(X_1) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(s1),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(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) & (all Y_1 (is_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))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1)) -> X_1 = Y_1))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,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(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)))) # label(fact_2214__096EX_B_As_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_A1_A_G_A_091s) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2573 (all Q_1 all R_2 hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,Q_1),R_2) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Q_1),hAPP_rat_rat(inverse_inverse_rat,R_2))) # label(fact_4636_divide__rat__def) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2574 (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,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_1),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_1))),hAPP_real_real(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_4802_DERIV__sin__circle__all) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2575 (all B_70 all A_99 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_99),zero_z126310315umeral)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_70)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_99),B_70)),zero_z126310315umeral))))) # label(fact_1411_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2576 (all C_30 all A_84 all B_57 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_30),A_84)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_30),B_57))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_30)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_84),B_57))))) # label(fact_1535_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2577 (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_2780_dvd__imp__le__int) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2578 (all A_1 all B_1 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_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),A_1))))) # label(fact_1408_mult__less__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2579 (all L all U hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_at641636577an_int(L),U)))) # label(fact_4915_finite__atLeastLessThan__int) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2580 (all Xa all Ya all Z_2 (Xa = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Ya),Z_2) <-> Ya = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa),Z_2))) # label(fact_2935_eq__diff__eq_H) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2581 (all X all Y hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),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)))) # label(fact_4133_complex__add__def) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2582 (all Ma all Na (zero_zero_nat != Na | Ma != zero_zero_nat <-> 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),Na))))) # label(fact_4343_gcd__pos__nat) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2583 (all A_161 all B_109 all C_63 all 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_161),C_63)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_109),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_161),B_109)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_63),D_19))) # label(fact_800_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2584 (all X hAPP_complex_real(re,hAPP_complex_complex(uminus473333897omplex,X)) = hAPP_real_real(uminus_uminus_real,hAPP_complex_real(re,X))) # label(fact_4123_Re_Ominus) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2585 (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_1261_zmult__1) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2586 (all K all Ma all Na (Na = Ma <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),Na) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),Ma))) # label(fact_284_nat__add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 4.91/5.03 2587 (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_3967_cos__arg__i__mult__