0.03/0.11 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.03/0.12 % Command : tptp2X_and_run_prover9 %d %s 0.12/0.33 % Computer : n020.cluster.edu 0.12/0.33 % Model : x86_64 x86_64 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.12/0.33 % Memory : 8042.1875MB 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64 0.12/0.33 % CPULimit : 960 0.12/0.33 % WCLimit : 120 0.12/0.33 % DateTime : Tue Aug 9 04:37:05 EDT 2022 0.12/0.33 % CPUTime : 8.57/8.66 ============================== Prover9 =============================== 8.57/8.66 Prover9 (32) version 2009-11A, November 2009. 8.57/8.66 Process 25537 was started by sandbox2 on n020.cluster.edu, 8.57/8.66 Tue Aug 9 04:37:12 2022 8.57/8.66 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 960 -f /tmp/Prover9_25383_n020.cluster.edu". 8.57/8.66 ============================== end of head =========================== 8.57/8.66 8.57/8.66 ============================== INPUT ================================= 8.57/8.66 8.57/8.66 % Reading from file /tmp/Prover9_25383_n020.cluster.edu 8.57/8.66 8.57/8.66 set(prolog_style_variables). 8.57/8.66 set(auto2). 8.57/8.66 % set(auto2) -> set(auto). 8.57/8.66 % set(auto) -> set(auto_inference). 8.57/8.66 % set(auto) -> set(auto_setup). 8.57/8.66 % set(auto_setup) -> set(predicate_elim). 8.57/8.66 % set(auto_setup) -> assign(eq_defs, unfold). 8.57/8.66 % set(auto) -> set(auto_limits). 8.57/8.66 % set(auto_limits) -> assign(max_weight, "100.000"). 8.57/8.66 % set(auto_limits) -> assign(sos_limit, 20000). 8.57/8.66 % set(auto) -> set(auto_denials). 8.57/8.66 % set(auto) -> set(auto_process). 8.57/8.66 % set(auto2) -> assign(new_constants, 1). 8.57/8.66 % set(auto2) -> assign(fold_denial_max, 3). 8.57/8.66 % set(auto2) -> assign(max_weight, "200.000"). 8.57/8.66 % set(auto2) -> assign(max_hours, 1). 8.57/8.66 % assign(max_hours, 1) -> assign(max_seconds, 3600). 8.57/8.66 % set(auto2) -> assign(max_seconds, 0). 8.57/8.66 % set(auto2) -> assign(max_minutes, 5). 8.57/8.66 % assign(max_minutes, 5) -> assign(max_seconds, 300). 8.57/8.66 % set(auto2) -> set(sort_initial_sos). 8.57/8.66 % set(auto2) -> assign(sos_limit, -1). 8.57/8.66 % set(auto2) -> assign(lrs_ticks, 3000). 8.57/8.66 % set(auto2) -> assign(max_megs, 400). 8.57/8.66 % set(auto2) -> assign(stats, some). 8.57/8.66 % set(auto2) -> clear(echo_input). 8.57/8.66 % set(auto2) -> set(quiet). 8.57/8.66 % set(auto2) -> clear(print_initial_clauses). 8.57/8.66 % set(auto2) -> clear(print_given). 8.57/8.66 assign(lrs_ticks,-1). 8.57/8.66 assign(sos_limit,10000). 8.57/8.66 assign(order,kbo). 8.57/8.66 set(lex_order_vars). 8.57/8.66 clear(print_given). 8.57/8.66 8.57/8.66 % formulas(sos). % not echoed (5492 formulas) 8.57/8.66 8.57/8.66 ============================== end of input ========================== 8.57/8.66 8.57/8.66 % From the command line: assign(max_seconds, 960). 8.57/8.66 8.57/8.66 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 8.57/8.66 8.57/8.66 % Formulas that are not ordinary clauses: 8.57/8.66 1 (all K_2 all L_2 hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),L_2)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit0,K_2)),hAPP_int_int(bit0,L_2))) # label(fact_155_add__Bit0__Bit0) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 2 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),A_3)) -> zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)))) # label(fact_2835_div__neg__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 3 (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]. 8.57/8.66 4 (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]. 8.57/8.66 5 (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]. 8.57/8.66 6 (all N hAPP_nat_nat(suc,hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_3137_add__2__eq__Suc_H) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 7 (all N all M 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)) # label(fact_2483_diff__add__inverse) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 8 (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]. 8.57/8.66 9 (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]. 8.57/8.66 10 (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]. 8.57/8.66 11 (all Ma all Va (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 <-> hAPP_int_int(number_number_of_int,Va) = hAPP_nat_int(semiri1621563631at_int,Ma))) # label(fact_2797_int__eq__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 12 (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]. 8.57/8.66 13 (all A_152 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_152),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_897_zero__le__power2) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 14 (all N hAPP_real_real(hAPP_n546711566l_real(root,N),zero_zero_real) = zero_zero_real) # label(fact_3889_real__root__zero) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 15 (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]. 8.57/8.66 16 (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]. 8.57/8.66 17 (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_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_30),M_7)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_30),N_13))))) # label(fact_2374_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 18 (all N_51 one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,one_one_nat),N_51)) # label(fact_353_power__one) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 19 (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]. 8.57/8.66 20 (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]. 8.57/8.66 21 (all Xa (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,Xa)),one_one_rat)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),hAPP_int_int(bit1,pls))))) # label(fact_121_less__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 22 (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]. 8.57/8.66 23 (all Z_1 hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_complex_real(re,Z_1)),hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,Z_1))) = hAPP_complex_complex(cnj,Z_1)) # label(fact_4131_cnj__def) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 24 (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_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))) & (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))))) # label(fact_3227_negDivAlg__eqn__1__number__of) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 25 (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(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A_1,B_1),hAPP_nat_nat(suc,N_1)))),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A_1,B_1),N_1))))))) # label(fact_4245_Bolzano__bisect__Suc__le__snd) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 26 (all Xa all Ya (hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xa),Ya))) <-> -hBOOL(hAPP_int_bool(even_odd_even_int,Ya)) & -hBOOL(hAPP_int_bool(even_odd_even_int,Xa)) | hBOOL(hAPP_int_bool(even_odd_even_int,Ya)) & hBOOL(hAPP_int_bool(even_odd_even_int,Xa)))) # label(fact_3243_even__difference) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 27 (all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Xa),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = Xa))) # label(fact_3378_even__div__2__prop2) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 28 (all N hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N)))) # label(fact_733_zero__zle__int) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 29 (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]. 8.57/8.66 30 (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]. 8.57/8.66 31 (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]. 8.57/8.66 32 (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]. 8.57/8.66 33 (all Xa comple124823625p_real(hAPP_r1134773055l_bool(ord_lessThan_real,Xa)) = Xa) # label(fact_4956_Sup__lessThan) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 34 (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]. 8.57/8.66 35 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa)) -> (hAPP_real_real(ln,Xa) = zero_zero_real <-> Xa = one_one_real))) # label(fact_3363_ln__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 36 (all A_1 all B_1 ((all D_2 (one_one_nat = 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)))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_1),B_1)))) # label(fact_4027_coprime) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 37 (all Xa all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,Xa)),hAPP_nat_int(semiri1621563631at_int,Ya))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)))) # label(fact_2530_Nat__Transfer_Otransfer__int__nat__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 38 (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]. 8.57/8.66 39 (all N all V_1 ((-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1))) -> 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)) = 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)) & (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1))) -> 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)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N)))) # label(fact_3276_Suc__nat__number__of__add) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 40 (all P all Q all R hAPP_int_bool(P,hAPP_int_int(Q,R)) = hAPP_int_bool(hAPP_f429935748t_bool(hAPP_f1512942609t_bool(cOMBB_int_bool_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__HOL__Obool_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 41 (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]. 8.57/8.66 42 (all B_1_1 all B_2_1 all B_3_1 is_bool(tendsto_nat_real(B_1_1,B_2_1,B_3_1))) # label(gsy_c_Limits_Otendsto_000tc__Nat__Onat_000tc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 43 (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]. 8.57/8.66 44 (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 (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))) = hAPP_real_real(F,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_4939_Maclaurin__all__le) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 45 (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,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) <-> 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)) # label(fact_2387_eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 46 (all X all Y hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(uminus_uminus_int,X)),Y) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)) # label(fact_4995_gcd__neg1__int) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 47 (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]. 8.57/8.66 48 (all Xa all Ya (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Xa),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)))) # label(fact_2024_less__le__not__le) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 49 (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]. 8.57/8.66 50 (all P all Q all R hAPP_nat_int(hAPP_int_fun_nat_int(P,R),Q) = hAPP_int_int(hAPP_nat_fun_int_int(hAPP_f2105620693nt_int(cOMBC_int_nat_int,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Nat__Onat_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 51 (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]. 8.57/8.66 52 (all N (is_int(N) -> archim856651990g_real(hAPP_int_real(real_int,N)) = N)) # label(fact_4397_ceiling__real__of__int) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 53 (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]. 8.57/8.66 54 (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,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)) = 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)) # label(fact_1160_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 55 (all K_2 hAPP_int_int(pred,hAPP_int_int(bit0,K_2)) = hAPP_int_int(bit1,hAPP_int_int(pred,K_2))) # label(fact_4039_pred__Bit0) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 56 (all A_3 all B_2 all P_3 all 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,A_3),B_2)) -> hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,P_3),Q_1))) # label(fact_4691_normalize__eq) # label(axiom) # label(non_clause). [assumption]. 8.57/8.66 57 (all M (hAPP_nat_nat(div_mod_nat(M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_nat | hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))) = hAPP_nat_nat(div_mod_nat(M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) | hAPP_nat_nat(div_mod_nat(M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) | hAPP_nat_nat(div_mod_nat(M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = one_one_nat)) # label(fact_3207_mod__exhaust__less__4) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 58 (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]. 8.57/8.67 59 (all Na (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Na)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_int_real(real_int,Na))))) # label(fact_4439_real__of__int__ge__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 60 (all B_40 all A_58 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_40),A_58)) -> (B_40 != A_58 -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_40),A_58))))) # label(fact_1951_xt1_I11_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 61 (all I all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat,I)),N) = hAPP_nat_nat(nat_aux(I),N)) # label(fact_2818_nat__aux__def) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 62 (all X_62 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_62),zero_zero_nat) = one_one_nat) # label(fact_481_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 63 (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(ord_less_eq_nat,A_3),B_2)) | zero_zero_nat = B_2)) # label(fact_2511_divides__ge) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 64 (all M_21 -hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(semiri1619134803umeral,M_21)),zero_z126310315umeral))) # label(fact_454_of__nat__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 65 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_complex_real(norm_norm_complex,X))),hAPP_complex_real(norm_norm_complex,X)))) # label(fact_3425_complex__mod__minus__le__complex__mod) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 66 (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)) -> 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_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]. 8.57/8.67 67 (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)) | Ya = Xa)) # label(fact_1840_not__less__iff__gr__or__eq) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 68 (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]. 8.57/8.67 69 (all A_121 all B_90 all N_28 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_121),B_90)),N_28) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_121),N_28)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_90),N_28))) # label(fact_1242_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 70 (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]. 8.57/8.67 71 (all X all Y hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_fst_int_int,bezw(X,Y))),hAPP_nat_int(semiri1621563631at_int,X))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(X,Y))),hAPP_nat_int(semiri1621563631at_int,Y))) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y))) # label(fact_4296_bezw__aux) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 72 (all Xa all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (hAPP_int_int(legacy_zgcd(Xa),Ma) = one_one_int -> hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(noXRRset(Ma),Xa)) = hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(norRRset,Ma))))) # label(fact_4586_card__nor__eq__noX) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 73 (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_eq_int,hAPP_int_int(bit0,K)),hAPP_int_int(bit1,L))))) # label(fact_727_rel__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 74 (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]. 8.57/8.67 75 (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]. 8.57/8.67 76 (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]. 8.57/8.67 77 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_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)))))) & hAPP_real_real(tan,X_1) = Y & 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]. 8.57/8.67 78 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) & Na != Ma <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)))) # label(fact_1256_nat__less__le) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 79 (all A_137 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_137),zero_zero_nat) = zero_zero_nat) # label(fact_1118_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 80 (all C_52 all A_141 all B_99 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_141),B_99)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_141),C_52)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_99),C_52))))) # label(fact_976_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 81 (all P (is_bool(P) -> P = fFalse | fTrue = P)) # label(help_If_3_1_If_000tc__Int__Oint_T) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 82 (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]. 8.57/8.67 83 (all A_144 all C_55 all B_102 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_144),C_55)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_102),C_55))) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_144),B_102)))) # label(fact_958_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 84 (all N all X (X = hAPP_int_real(real_int,archim1246769320r_real(X)) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,archim1246769320r_real(X)),N) = archim1246769320r_real(hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N)))) # label(fact_4424_floor__power) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 85 (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]. 8.57/8.67 86 (all B all A (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A)) -> (hBOOL(hAPP_f448129468l_bool(nat_nat_set,B)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,image_int_nat(nat,A)),image_int_nat(nat,B))) <-> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A),B)))))) # label(fact_4608_transfer__int__nat__set__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 87 (all A_89 all M_13 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_89),M_13)),M_13) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_89),one_on1645066479umeral)),M_13)) # label(fact_1487_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 88 (all X_10 all Y_7 all A_29 all B_22 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_10),Y_7)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_29),B_22)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_10),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_7),B_22))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_10),A_29)),B_22))) # label(fact_2391_mult__diff__mult) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 89 (all L_3 all K_5 (hAPP_int_int(abs_abs_int,K_5) = hAPP_int_int(abs_abs_int,L_3) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,L_3),K_5)))) # label(fact_2690_dvd__if__abs__eq) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 90 (all A_123 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_123),one_one_rat) = A_123) # label(fact_1227_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 91 (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]. 8.57/8.67 92 (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(suc,Ma)),hAPP_nat_nat(suc,Na))))) # label(fact_3012_Suc__less__eq) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 93 (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_512_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 94 (all C all A_1 all B_1 (B_1 = A_1 -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C),B_1)) -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C),A_1))))) # label(fact_1673_xt1_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 95 (all Xa all Ya (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))))) <-> Ya = zero_zero_real & Xa = zero_zero_real)) # label(fact_239_realpow__two__sum__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 96 (all Ma all Na all A_1 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_1)) -> (Ma = Na <-> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_1),Ma) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_1),Na)))) # label(fact_423_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 97 (all A all B (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_nat_int(semiri1621563631at_int,A)),image_nat_int(semiri1621563631at_int,B))))) # label(fact_4604_transfer__nat__int__set__relations_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 98 (all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N) = hAPP_nat_nat(suc,N)) # label(fact_3064_Suc__eq__plus1__left) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 99 (all M all N all K_2 (one_one_int = hAPP_int_int(legacy_zgcd(N),K_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M)))))) # label(fact_4526_zrelprime__zdvd__zmult__aux) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 100 (all X_40 all Y_34 (-hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_40),Y_34)) -> (Y_34 != X_40 -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_34),X_40))))) # label(fact_1716_linorder__cases) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 101 (all X_42 all Y_36 (X_42 = Y_36 -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_42),Y_36)))) # label(fact_1694_order__eq__refl) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 102 (all X_57 (is_int(X_57) -> X_57 = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_57),one_one_nat))) # label(fact_1003_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 103 (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]. 8.57/8.67 104 (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]. 8.57/8.67 105 (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]. 8.57/8.67 106 (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]. 8.57/8.67 107 (all Y_42 all X_48 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_48)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_42)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_42),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_42),X_48)),X_48)))))) # label(fact_1597_mult__left__le__one__le) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 108 (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]. 8.57/8.67 109 (all B_86 all A_116 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_116),zero_z126310315umeral)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_86)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_116),B_86)),zero_z126310315umeral))))) # label(fact_1312_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 110 (all X hAPP_complex_complex(sgn_sgn_complex,X) = hAPP_complex_complex(scaleR1652505878omplex(hAPP_real_real(inverse_inverse_real,hAPP_complex_real(norm_norm_complex,X))),X)) # label(fact_5113_complex__sgn__def) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 111 (all A_74 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(bit1,pls))),A_74) = A_74) # label(fact_1616_mult__numeral__1) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 112 (all A_1 (zero_zero_real != A_1 <-> 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]. 8.57/8.67 113 (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]. 8.57/8.67 114 (all X all Y (Y != X | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(fequal_rat,X),Y)))) # label(help_fequal_2_1_fequal_000tc__Rat__Orat_T) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 115 (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]. 8.57/8.67 116 (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]. 8.57/8.67 117 (all A_159 all B_107 all C_61 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_159),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_107),C_61)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_159),B_107)),C_61)) # label(fact_820_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 118 (all A_1 all Na (A_1 = zero_z126310315umeral & Na != zero_zero_nat <-> zero_z126310315umeral = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_1),Na))) # label(fact_448_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 119 (all Xa hBOOL(hAPP_real_bool(deriv_real(cos,Xa),hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,Xa))))) # label(fact_3868_DERIV__cos) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 120 (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]. 8.57/8.67 121 (all Z_1 all Z_4 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_4)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_4),Z_1)) -> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_1),Z_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,Z_1)),hAPP_int_nat(nat,Z_4))))) # label(fact_2798_nat__diff__distrib) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 122 (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]. 8.57/8.67 123 (all A_154 all N_32 all B_105 (hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_154),N_32) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_105),N_32) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_154)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_105)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32)) -> B_105 = A_154))))) # label(fact_858_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 124 (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]. 8.57/8.67 125 (all C_26 all D_11 all A_80 all B_53 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_80),B_53)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_26),D_11)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_80)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_26)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_80),C_26)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_53),D_11)))))))) # label(fact_1557_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 126 (all V_1 ((-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1))) -> hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_1)) = hAPP_int_int(number_number_of_int,V_1)) & (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1))) -> hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_1)) = zero_zero_int))) # label(fact_3264_int__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 127 (all A_160 all B_108 all C_62 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_160),C_62)),B_108) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_160),B_108)),C_62)) # label(fact_815_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 128 (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]. 8.57/8.67 129 (all C_1 all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_1),A_3)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_1),B_2))))) # label(fact_2498_divides__mul__l) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 130 (all P all Q all R hAPP_i1524277240nt_int(P,hAPP_int_int(Q,R)) = hAPP_i1524277240nt_int(hAPP_f883232462nt_int(hAPP_f2002277285nt_int(cOMBB_1846624359nt_int,P),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]. 8.57/8.67 131 (all X_63 all Y_52 (is_int(Y_52) & is_int(X_63) -> (X_63 != Y_52 -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_63),Y_52)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_52),X_63)))))) # label(fact_313_linorder__neqE__linordered__idom) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 132 (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]. 8.57/8.67 133 (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]. 8.57/8.67 134 (all Ma all Na (one_one_nat = Ma & Na = one_one_nat <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),Na) = one_one_nat)) # label(fact_2107_nat__mult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 135 (all L_2 hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),L_2)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),hAPP_int_int(bit0,L_2))) # label(fact_2445_diff__bin__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 136 (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]. 8.57/8.67 137 (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]. 8.57/8.67 138 (all A_36 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_36),A_36) = zero_zero_real) # label(fact_2327_diff__self) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 139 (all C_9 all A_38 all B_27 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_38),B_27)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_38),C_9)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_38),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_27),C_9)))))) # label(fact_2302_dvd__add) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 140 (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]. 8.57/8.67 141 (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]. 8.57/8.67 142 (all B_1_1 is_bool(summable_real(B_1_1))) # label(gsy_c_Series_Osummable_000tc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 143 (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]. 8.57/8.67 144 (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]. 8.57/8.67 145 (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(plus_plus_int,Xa),Ya)),zEven))))) # label(fact_3339_EvenOdd_Oeven__plus__even) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 146 (all A_157 all C_59 all 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)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_157),C_59)),D_18)) # label(fact_835_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 147 (all M all V_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V_1)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),hAPP_int_nat(number_number_of_nat,V_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(pred,V_1))))) # label(fact_4050_Suc__diff__number__of) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 148 (all V_16 all W_8 all Z_20 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_16),W_8))),Z_20) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_16)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,W_8)),Z_20))) # label(fact_1089_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 149 (all M all N hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)),hAPP_nat_nat(div_mod_nat(M),N)) = divmod_nat(M,N)) # label(fact_4210_divmod__nat__div__mod) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 150 (all X all K_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),K_2)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),K_2))) # label(fact_3903_real__root__power) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 151 (all B_2 all B_5 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_5),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_5)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))))))) # label(fact_2836_zdiv__mono2__neg) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 152 (all A_36 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_36),A_36) = zero_zero_int) # label(fact_2328_diff__self) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 153 (all P_1 P_1 = collect_nat(P_1)) # label(fact_149_Collect__def) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 154 (all Xa all Ya (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Xa),Xa)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ya),Ya)))) <-> zero_zero_rat != Xa | Ya != zero_zero_rat)) # label(fact_1590_sum__squares__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 155 (all A_3 all B_2 hAPP_rat_rat(uminus_uminus_rat,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)),B_2)) # label(fact_4669_minus__rat) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 156 (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(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_4),X_4)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(abs_abs_rat,Y_4)),X_4))) # label(fact_2751_abs__mult__pos) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 157 (all Na (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)))) # label(fact_329_zero__less__int__conv) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 158 (all A_1 all B_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,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_1),B_1)),zero_zero_real)))) # label(fact_2379_less__iff__diff__less__0) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 159 (all N_51 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,one_one_real),N_51) = one_one_real) # label(fact_352_power__one) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 160 (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]. 8.57/8.67 161 (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]. 8.57/8.67 162 (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) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_1),B_1)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C),D_1))))) # label(fact_2333_diff__eq__diff__less__eq) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 163 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),hAPP_int_int(div_mod_int(A_3),B_2))))) # label(fact_3030_neg__mod__bound) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 164 (all W_11 hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,W_11)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),hAPP_int_complex(number528085621omplex,W_11))),hAPP_int_complex(number528085621omplex,W_11))) # label(fact_221_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 165 (all P_1 all A_1 all B_1 ((all D_2 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_1),D_2) = A_1 -> hBOOL(hAPP_nat_bool(P_1,D_2)))) & (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B_1)) -> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))) <-> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A_1),B_1))))) # label(fact_2532_nat__diff__split) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 166 (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]. 8.57/8.67 167 (all A_3 all B_2 all C_1 all 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)) = 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))) # label(fact_4689_mult__rat) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 168 (all Xa all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (hAPP_int_int(legacy_zgcd(Xa),Ma) = one_one_int -> hBOOL(hAPP_int_bool(is_RRset(hAPP_i1948725293t_bool(noXRRset(Ma),Xa)),Ma))))) # label(fact_4587_noX__is__RRset) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 169 (all C 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(plus_plus_rat,C),A_1)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),B_1))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_1),B_1)))) # label(fact_540_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 170 (all M hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,M)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,M))) # label(fact_3105_int__Suc) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 171 (all A_34 all B_26 A_34 = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_34),B_26)),B_26)) # label(fact_2341_add__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 172 (all U all Ma all Na all J all I_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J)),U)),Ma)),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_2541_nat__le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 173 (all P_1 all L ((exists 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))) & hBOOL(hAPP_complex_bool(P_1,X_1)))) <-> (exists X_1 hBOOL(hAPP_complex_bool(P_1,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,L),X_1)))))) # label(fact_2418_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 174 (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]. 8.57/8.67 175 (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_eq_int,B_1),A_1)))) # label(fact_2943_d22set__le) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 176 (all X hAPP_int_nat(nat,archim1246769320r_real(X)) = natfloor(X)) # label(fact_4020_natfloor__def) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 177 (all A_161 all B_109 all C_63 all D_19 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_161),B_109)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_63),D_19)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_161),C_63)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_109),D_19))) # label(fact_799_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 178 (all X all A_3 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,archim856651990g_real(X)),A_3) = archim856651990g_real(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_int_real(real_int,A_3)))) # label(fact_4428_ceiling__subtract) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 179 (all P all Q all R hAPP_r1250527377l_real(hAPP_r337325687l_real(hAPP_f1644353557l_real(cOMBC_1508019322l_real,P),Q),R) = hAPP_r1250527377l_real(hAPP_r337325687l_real(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__fun_Itc_) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 180 (all X_2 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (hBOOL(hAPP_f448129468l_bool(resSet(Ma),X_2)) -> hBOOL(hAPP_f448129468l_bool(finite_finite_int,X_2))))) # label(fact_4052_ResSet__finite) # label(axiom) # label(non_clause). [assumption]. 8.57/8.67 181 (all V_21 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21)) -> hAPP_int_rat(number_number_of_rat,hAPP_int_int(succ,V_21)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_21)),one_one_rat))) # label(fact_739_semiring__number__of__add__1) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 182 (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]. 8.57/8.68 183 (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]. 8.57/8.68 184 (all N all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,P_3),N)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),N)))) # label(fact_4178_prime__coprime__strong) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 185 (all Z_1 hAPP_complex_real(re,hAPP_complex_complex(sgn_sgn_complex,Z_1)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_complex_real(re,Z_1)),hAPP_complex_real(norm_norm_complex,Z_1))) # label(fact_4128_Re__sgn) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 186 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N))) # label(fact_3017_add__Suc__right) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 187 (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)))) -> (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]. 8.57/8.68 188 (all F all F_1 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,X_1),B_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),X_1)) -> hBOOL(hAPP_real_bool(deriv_real(F,X_1),hAPP_real_real(F_1,X_1))))) -> (exists Z (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z),B_1)) & hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_1),A_1)),hAPP_real_real(F_1,Z)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B_1)),hAPP_real_real(F,A_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),Z))))))) # label(fact_3888_MVT2) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 189 (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]. 8.57/8.68 190 (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(semiri984289939at_nat,Ma)),hAPP_nat_nat(semiri984289939at_nat,Na))))) # label(fact_514_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 191 (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]. 8.57/8.68 192 (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]. 8.57/8.68 193 (all C (zero_zero_rat = C <-> hBOOL(vanishes(cOMBK_rat_nat(C))))) # label(fact_5140_vanishes__const) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 194 (all A_1 all B_1 (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A_1),B_1)) -> -hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,B_1),A_1)))) # label(fact_1777_order__less__asym_H) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 195 (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,A_1),X_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),B_1)) -> hBOOL(hAPP_real_bool(deriv_real(F,X_1),zero_zero_real)))) -> hAPP_real_real(F,A_1) = hAPP_real_real(F,B_1))))) # label(fact_5176_DERIV__isconst__end) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 196 (all N hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)))) # label(fact_3747_two__realpow__gt) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 197 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Y)),one_one_real)) -> hAPP_real_real(sin,hAPP_real_real(arccos,Y)) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3681_sin__arccos__abs) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 198 (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]. 8.57/8.68 199 (all A_3 all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_3)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)))))) # label(fact_4431_ge__one__powr__ge__zero) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 200 (all K_2 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),min) = hAPP_int_int(succ,K_2)) # label(fact_2413_diff__bin__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 201 (all P all Q all R hAPP_P1333854989l_bool(P,hAPP_r1195171167l_real(Q,R)) = hAPP_real_bool(hAPP_f330985925l_bool(hAPP_f940061907l_bool(cOMBB_1795223523l_real,P),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]. 8.57/8.68 202 (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)) & (N_10 = zero_zero_nat -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_6),N_10) = one_one_int))) # label(fact_2613_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 203 (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]. 8.57/8.68 204 (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]. 8.57/8.68 205 (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(bit1,L_2)) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),L_2))) # label(fact_2408_diff__bin__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 206 (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]. 8.57/8.68 207 (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]. 8.57/8.68 208 (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]. 8.57/8.68 209 (all Y_15 all X_19 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_15),X_19)) -> -hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_19),Y_15)))) # label(fact_1981_leD) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 210 (all X all Y hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(cnj,X)),hAPP_complex_complex(cnj,Y)) = hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),Y))) # label(fact_4073_cnj_Odiff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 211 (all X all Y hAPP_nat_nat(nat_gcd(X),Y) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)) # label(fact_4657_gcd__eq__nitpick__gcd) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 212 (all M all N hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M),N))),N) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M)),N)) # label(fact_3170_mod__Suc__eq__Suc__mod) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 213 (all Z_10 all Y_21 all X_25 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_21),X_25)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_10),Y_21)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_10),X_25))))) # label(fact_1923_xt1_I8_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 214 (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]. 8.57/8.68 215 (all X hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_complex_real(re,X)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) = hAPP_complex_complex(invers1449016382omplex,X)) # label(fact_4109_complex__inverse__def) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 216 (all B_1_1 is_int(quickcheck_int_of(B_1_1))) # label(gsy_c_Quickcheck__Narrowing_Oint__of) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 217 (all Z1 all Z2 all W hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z1),W)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z2),W)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z1),Z2)),W)) # label(fact_2112_real__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 218 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)))) # label(fact_3051_Suc__le__lessD) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 219 (all A_125 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_125),one_on1684967323de_int) = A_125) # label(fact_1214_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 220 (all A_149 A_149 = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_149),zero_z126310315umeral)) # label(fact_928_add__0__right) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 221 (all X hAPP_real_real(uminus_uminus_real,hAPP_real_real(sqrt,X)) = hAPP_real_real(sqrt,hAPP_real_real(uminus_uminus_real,X))) # label(fact_3519_real__sqrt__minus) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 222 (all P all Q all R hAPP_nat_real(hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,P),Q),R) = hAPP_P731461727l_real(P,hAPP_n332733730l_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 223 (all P all Q all R hAPP_nat_bool(hAPP_f1154658180t_bool(hAPP_f1361476881t_bool(cOMBB_int_bool_nat,P),Q),R) = hAPP_int_bool(P,hAPP_nat_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__HOL__Obool_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 224 (all Z_14 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Z_14),Z_14) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Z_14),hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_1880_mult__2__right) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 225 (all A_101 all B_72 (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_101),B_72))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_101)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_72))))) # label(fact_1394_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 226 (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]. 8.57/8.68 227 (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]. 8.57/8.68 228 (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]. 8.57/8.68 229 (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]. 8.57/8.68 230 (all Va all 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)) = 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)))) # label(fact_5189_max__Suc__number__of) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 231 (all Lx_3 all Ly_3 all Rx_3 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_3),Rx_3)),Ly_3) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_3),Ly_3)),Rx_3)) # label(fact_1064_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 232 (all Ma all K all Na (Na = Ma <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Na),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),K))) # label(fact_285_nat__add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 233 (all A_12 all B_12 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(abs_abs_int,A_12)),hAPP_int_int(abs_abs_int,B_12))),hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_12),B_12))))) # label(fact_2734_abs__triangle__ineq2) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 234 (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]. 8.57/8.68 235 (all X_26 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_26),X_26) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_26),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_1906_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 236 (all R_2 all A_3 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,A_3),R_2) = 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))) # label(fact_3392_real__divide__square__eq) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 237 (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]. 8.57/8.68 238 (all A_3 all B_2 (is_int(B_2) -> (zero_zero_int != B_2 -> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),hAPP_int_int(div_mod_int(A_3),B_2))))))) # label(fact_3265_divmod__int__rel__div__mod) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 239 (all N_17 all M_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_17),M_10)) -> hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_10),N_17)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_nat_complex(semiri2020571505omplex,M_10)),hAPP_nat_complex(semiri2020571505omplex,N_17)))) # label(fact_2242_of__nat__diff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 240 (all C_16 all A_66 all B_43 (B_43 = A_66 -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_43),C_16)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_66),C_16))))) # label(fact_1765_ord__eq__less__trans) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 241 (all K_2 all L_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),L_2)) -> (exists N_1 L_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),N_1)))) # label(fact_3333_le__Suc__ex) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 242 (all Ma all Na (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(semiri1424489471de_int,Ma)),hAPP_n522471361de_int(semiri1424489471de_int,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)))) # label(fact_513_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 243 (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]. 8.57/8.68 244 (all M all N hAPP_nat_nat(div_mod_nat(M),N) = hAPP_P1654798560at_nat(product_snd_nat_nat,divmod_nat(M,N))) # label(fact_4239_mod__nat__def) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 245 (all G all Xa (hBOOL(hAPP_real_bool(sums_real(G),Xa)) -> hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_r195310020l_real(hAPP_f2126667875l_real(cOMBC_746811033l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),zero_zero_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,G),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,div_div_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,minus_minus_nat),one_one_nat))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),Xa)))) # label(fact_4867_sums__if_H) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 246 (all A_46 all B_35 all C_14 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_46),B_35)),C_14)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_35),C_14)))) # label(fact_2252_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 247 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(uminus_uminus_real,X)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),X))))))) # label(fact_3455_ln__one__minus__pos__lower__bound) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 248 (all A_75 A_75 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_75),hAPP_int_complex(number528085621omplex,hAPP_int_int(bit1,pls)))) # label(fact_1612_mult__numeral__1__right) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 249 (all X_43 all Y_37 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_43),Y_37)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_37),X_43)) -> X_43 = Y_37))) # label(fact_1654_order__antisym) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 250 (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,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) <-> 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))) # label(fact_2385_eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 251 (all M all N xzgcda(M,N,M,N,one_one_int,zero_zero_int,zero_zero_int,one_one_int) = xzgcd(M,N)) # label(fact_4200_xzgcd__def) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 252 (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]. 8.57/8.68 253 (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]. 8.57/8.68 254 (all Z_1 hAPP_real_real(sqrt,hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_1),hAPP_complex_complex(cnj,Z_1)))) = hAPP_complex_real(norm_norm_complex,Z_1)) # label(fact_4132_complex__mod__sqrt__Re__mult__cnj) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 255 (all A_101 all B_72 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_101),B_72))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_101)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_72))))) # label(fact_1395_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 256 (all N (-hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)))) # label(fact_3793_odd__pos) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 257 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),hAPP_int_int(bit1,K))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K)))) # label(fact_758_rel__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 258 (all N N = hAPP_int_nat(nat,hAPP_nat_int(semiri1621563631at_int,N))) # label(fact_2703_nat__int) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 259 (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]. 8.57/8.68 260 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y)))))) # label(fact_2826_Divides_Otransfer__nat__int__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 261 (all A_162 all B_110 all C_64 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_162),B_110) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_162),C_64) -> C_64 = B_110)) # label(fact_794_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 262 (all Lx_1 all Ly_1 all Rx_1 all Ry_1 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_1),Ly_1)),Ry_1)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_1),Ly_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_1),Ry_1))) # label(fact_1079_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 263 (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]. 8.57/8.68 264 (all X all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hAPP_real_real(hAPP_n546711566l_real(root,M),hAPP_real_real(hAPP_n546711566l_real(root,N),X)) = hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(hAPP_n546711566l_real(root,M),X))))) # label(fact_3892_real__root__commute) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 265 (all B_1 all C all A_1 all D_1 (one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_1),D_1) -> (one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B_1),C) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_1),D_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_1),C) <-> B_1 = A_1 & C = D_1)))) # label(fact_4354_coprime__crossproduct__nat) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 266 (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]. 8.57/8.68 267 (all P all Q all R hAPP_i1948725293t_bool(P,hAPP_int_int(Q,R)) = hAPP_i1948725293t_bool(hAPP_f1791153283t_bool(hAPP_f1399575567t_bool(cOMBB_118231410ol_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 268 (all A_1 hBOOL(hAPP_real_bool(isCont_real_real(abs_abs_real),A_1))) # label(fact_5158_isCont__abs) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 269 (all X (-hBOOL(hAPP_nat_bool(even_odd_even_nat,X)) -> X = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))))))) # label(fact_3810_odd__nat__div__two__times__two__plus__one) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 270 (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]. 8.57/8.68 271 (all N_8 hAPP_real_real(abs_abs_real,hAPP_nat_real(semiri132038758t_real,N_8)) = hAPP_nat_real(semiri132038758t_real,N_8)) # label(fact_2700_abs__of__nat) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 272 (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]. 8.57/8.68 273 (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_1721_order__less__asym) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 274 (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]. 8.57/8.68 275 (all A_158 all B_106 all C_60 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_158),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_106),C_60)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_158),B_106)),C_60)) # label(fact_828_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 276 (all X all Y X = hAPP_real_real(hAPP_r1250527377l_real(hAPP_b646336293l_real(if_real,fTrue),X),Y)) # label(help_If_1_1_If_000tc__RealDef__Oreal_T) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 277 (all Xa all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,Xa)),hAPP_int_rat(number_number_of_rat,Ya))))) # label(fact_60_less__number__of) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 278 (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)),zOdd)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd))))) # label(fact_2945_power__preserves__odd) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 279 (all A_1 all B_1 (B_1 != A_1 -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_1)) -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A_1),B_1))))) # label(fact_1994_order__neq__le__trans) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 280 (all B_124 all C_71 all A_180 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_180)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_124),C_71)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_124),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_180),C_71)))))) # label(fact_580_add__increasing) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 281 (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]. 8.57/8.68 282 (all Xa (zero_zero_rat = Xa <-> Xa = zero_zero_rat)) # label(fact_763_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 283 (all A_198 (is_int(A_198) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_198),one_one_nat) = A_198)) # label(fact_321_power__one__right) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 284 (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]. 8.57/8.68 285 (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]. 8.57/8.68 286 (all A_55 all B_37 (is_int(A_55) & is_int(B_37) -> (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]. 8.57/8.68 287 (all Ya all Xa (-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_1833_linorder__antisym__conv3) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 288 (all N_16 all X_12 all Y_9 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_12),Y_9)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_12),N_16)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_9),N_16))))) # label(fact_2314_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 289 (all A_194 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_194),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_194),one_one_nat)))) # label(fact_419_less__add__one) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 290 (all Ya all Xa (is_int(Ya) & is_int(Xa) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ya),Xa)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)) <-> Ya = Xa)))) # label(fact_1834_linorder__antisym__conv3) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 291 (all N_23 all A_76 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_76)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_76),N_23)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_76),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_76),N_23)))))) # label(fact_1609_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 292 (all R_2 all S_1 all T_1 all R_4 all S_3 all M all T_2 all 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_3),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_2),N)) -> (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(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)) = hAPP_int_int(div_mod_int(R_4),R_2)))) # label(fact_3111_xzgcda__linear__aux2) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 293 (all F all A (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A)) -> ((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(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(F,X_1)))))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),big_co1548731110nt_int(F,A)))))) # label(fact_4815_transfer__nat__int__sum__prod__closure_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 294 (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]. 8.57/8.68 295 (all Xa ((exists Y_1 Xa = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),Y_1)) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa)))) # label(fact_3799_even__nat__equiv__def2) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 296 (all X_55 all Y_47 all Z_19 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_55),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y_47),Z_19)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_55),Y_47)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_55),Z_19))) # label(fact_1169_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 297 (all K 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),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Na),K))))) # label(fact_2496_dvd__reduce) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 298 (all 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))) & (hAPP_int_nat(number_number_of_nat,W_15) = zero_zero_nat -> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),hAPP_int_nat(number_number_of_nat,W_15)) = one_on1684967323de_int))) # label(fact_42_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 8.57/8.68 299 (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]. 8.57/8.68 300 (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]. 8.57/8.69 301 (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]. 8.57/8.69 302 (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]. 8.57/8.69 303 (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]. 8.57/8.69 304 (all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) -> X != Y)) # label(fact_2630_dvd_Oless__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 305 (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]. 8.57/8.69 306 (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]. 8.57/8.69 307 (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]. 8.57/8.69 308 (all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),hAPP_int_int(bit1,pls))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,Xa)),one_one_real)))) # label(fact_122_less__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 309 (all M all R_2 all S_1 all T_1 all N (is_int(R_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N)) -> (hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,R_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,S_1),T_1)) = xzgcd(M,N) -> R_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_1),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_1),N)))))) # label(fact_4198_xzgcd__linear) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 310 (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]. 8.57/8.69 311 (all A_1 all P_5 hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(wset(A_1),P_5)))) # label(fact_3941_wset__fin) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 312 (all M all N all R_4 all S_3 all S_1 all T_2 all T_1 all Rn all Sn_1 all Tn_1 all R_2 (is_int(Rn) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),R_2)) -> (xzgcda(M,N,R_4,R_2,S_3,S_1,T_2,T_1) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,Rn),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Sn_1),Tn_1)) -> (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_3),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,T_2),N)) -> (R_2 = 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)) -> Rn = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Sn_1),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Tn_1),N)))))))) # label(fact_4201_xzgcda__linear) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 313 (all A_1 all A all Ma (hBOOL(hAPP_int_bool(is_RRset(A),Ma)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),A)) -> hAPP_int_int(legacy_zgcd(A_1),Ma) = one_one_int))) # label(fact_4588_RRset__gcd) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 314 (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,S_2),N) = B_2 & A_3 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,R_1),N)))))) # label(fact_4479_coprime__pow) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 315 (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]. 8.57/8.69 316 (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_eq_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_eq_nat,Xa),Ya))))) # label(fact_629_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 317 (all I quickcheck_nat_of(I) = hAPP_int_nat(nat,quickcheck_int_of(I))) # label(fact_4268_nat__of__def) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 318 (all X hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_40_quartic__square__square) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 319 (all A_135 all B_95 (is_int(B_95) & is_int(A_135) -> (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_135),B_95) = zero_zero_int -> zero_zero_int = B_95 | A_135 = zero_zero_int))) # label(fact_1137_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 320 (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]. 8.57/8.69 321 (all W_5 hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,W_5)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),one_one_real)),hAPP_int_real(number267125858f_real,W_5))) # label(fact_1868_double__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 322 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),zero_zero_int)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_3)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_2),one_one_int)),A_3))) # label(fact_2857_neg__zdiv__mult__2) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 323 (all M_6 all N_10 ((N_10 = zero_zero_nat -> one_on1645066479umeral = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,M_6),N_10)) & (N_10 != zero_zero_nat -> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,M_6),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)))))) # label(fact_2610_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 324 (all Z_16 all Y_32 all X_38 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_32),X_38)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Z_16),Y_32)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Z_16),X_38))))) # label(fact_1730_xt1_I10_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 325 (all Z1 all Z2 all Z3 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z2),Z3)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z1),Z2)),Z3)) # label(fact_100_zadd__assoc) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 326 (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_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),Na)) & 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_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))))) -> (exists T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) & hAPP_real_real(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_real,zero_zero_real),T)))))))) # label(fact_4936_Maclaurin2) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 327 (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]. 8.57/8.69 328 (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]. 8.57/8.69 329 (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]. 8.57/8.69 330 (all Lx_4 all Ly_4 all Rx_4 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_4),Ly_4)),Rx_4) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_4),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ly_4),Rx_4))) # label(fact_1059_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 331 (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) -> C_64 = B_110)) # label(fact_792_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 332 (all A_154 all N_32 all B_105 (is_int(A_154) & is_int(B_105) -> (hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_105),N_32) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_154),N_32) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_154)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_105)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32)) -> B_105 = A_154)))))) # label(fact_863_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 333 (all C_75 all D_22 all A_186 all B_129 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_186),B_129)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_75),D_22)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_186),C_75)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_129),D_22)))))) # label(fact_525_add__mono) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 334 (all Xa (is_int(Xa) -> ((exists A_2 exists B_4 (is_int(B_4) & Xa = hAPP_P1175774780nt_int(twoSqu1907779896sum2sq,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_2),B_4)) & is_int(A_2))) <-> hBOOL(hAPP_int_bool(twoSqu1770640450sum2sq,Xa))))) # label(fact_2816_is__sum2sq__def) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 335 (all P_5 hAPP_P1175774780nt_int(produc1298267108nt_int(div_div_int),quotient_of(P_5)) = archim791455193or_rat(P_5)) # label(fact_4832_rat__floor__code) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 336 (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_nat_real(real_nat,Na)),one_one_real)),hAPP_nat_real(real_nat,Ma))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),Ma)))) # label(fact_3725_nat__less__real__le) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 337 (all X (zero_zero_real != hAPP_real_real(cos,X) -> (zero_zero_real != hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X)) -> hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(tan,X))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(tan,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3684_tan__double) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 338 (all Xa all Ya (Xa = 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)))) # label(fact_2650_dvd_Oeq__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 339 (all B_2 all A_3 (is_int(B_2) & is_int(A_3) -> (zero_zero_int != A_3 | B_2 != zero_zero_int -> (exists C_3 exists D_2 (hAPP_int_int(legacy_zgcd(C_3),D_2) = one_one_int & B_2 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(legacy_zgcd(A_3),B_2)),D_2) & A_3 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(legacy_zgcd(A_3),B_2)),C_3) & is_int(D_2) & is_int(C_3)))))) # label(fact_4538_make__zrelprime) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 340 (all A_53 all N_20 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_53),N_20)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_53),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_20))) # label(fact_2134_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 341 (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]. 8.57/8.69 342 (all Xa all F all G (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G)) -> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_int_bool(F,Xa)),hAPP_int_bool(G,Xa))))) # label(fact_1692_le__funD) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 343 (all B_85 all A_115 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_115),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_85),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_115),B_85)))))) # label(fact_1318_mult__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 344 (all B_1_1 all B_2_1 all B_3_1 is_bool(tendsto_real_real(B_1_1,B_2_1,B_3_1))) # label(gsy_c_Limits_Otendsto_000tc__RealDef__Oreal_000tc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 345 (all Lx_4 all Ly_4 all 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)) = 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)) # label(fact_1061_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 346 (all C_25 all D_10 all A_79 all B_52 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_79),B_52)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_25),D_10)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_79)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_25)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_79),C_25)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_52),D_10)))))))) # label(fact_1563_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 347 (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]. 8.57/8.69 348 (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]. 8.57/8.69 349 (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_eq_real,one_one_real),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(hAPP_n546711566l_real(root,Na),Ya)))))) # label(fact_3916_real__root__ge__1__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 350 (all Na (zero_zero_nat = Na <-> 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]. 8.57/8.69 351 (all A_1 all Wa (hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_1),hAPP_int_nat(number_number_of_nat,Wa)) = zero_z126310315umeral <-> A_1 = zero_z126310315umeral & zero_zero_nat != hAPP_int_nat(number_number_of_nat,Wa))) # label(fact_183_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 352 (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]. 8.57/8.69 353 (all A_159 all B_107 all C_61 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_159),B_107)),C_61) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_159),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_107),C_61))) # label(fact_821_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 354 (all X hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_complex_real(re,X)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_complex_real(re,hAPP_complex_complex(invers1449016382omplex,X))) # label(fact_4138_complex__Re__inverse) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 355 (all Xa all Ya all P_1 ((-hBOOL(P_1) -> hAPP_int_nat(nat,Ya) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,P_1),Xa),Ya))) & (hBOOL(P_1) -> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,P_1),Xa),Ya)) = hAPP_int_nat(nat,Xa)))) # label(fact_2653_nat__if__cong) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 356 (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]. 8.57/8.69 357 (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]. 8.57/8.69 358 (all C_22 all A_72 all B_49 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_72),B_49)) -> (C_22 = B_49 -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_72),C_22))))) # label(fact_1667_ord__le__eq__trans) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 359 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_2),one_one_int))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2)))) # label(fact_2620_norR__mem__unique__aux) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 360 (all Va all V (is_int(Va) & is_int(V) -> (hAPP_int_nat(number_number_of_nat,Va) = hAPP_int_nat(number_number_of_nat,V) <-> (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,Va))) -> (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V))) -> hAPP_int_int(number_number_of_int,Va) = zero_zero_int) & (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V))) -> V = Va)) & (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,Va))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,V)),zero_zero_int)))))) # label(fact_3262_eq__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 361 (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]. 8.57/8.69 362 (all C_10 all A_41 all B_30 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_41),B_30)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_41),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_30),C_10))))) # label(fact_2285_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 363 (all A_121 all B_90 all 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)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_121),B_90)),N_28)) # label(fact_1245_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 364 (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]. 8.57/8.69 365 (all X all Y Y = hAPP_complex_real(im,hAPP_real_complex(hAPP_r265291036omplex(complex,X),Y))) # label(fact_4091_Im) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 366 (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]. 8.57/8.69 367 (all X all Y hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X)),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3557_real__sqrt__ge__abs1) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 368 (all A_3 one_one_real = hAPP_real_real(hAPP_r1250527377l_real(powr,one_one_real),A_3)) # label(fact_4381_powr__one__eq__one) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 369 (all N all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (X != zero_zero_real -> hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_nat_real(real_nat,N)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N)) & (zero_zero_real = X -> zero_zero_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N))))) # label(fact_4474_powr__realpow2) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 370 (all B_86 all A_116 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_116),zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_86)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_116),B_86)),zero_zero_nat))))) # label(fact_1314_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 371 (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]. 8.57/8.69 372 (all X hAPP_real_real(sgn_sgn_real,X) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_real_real(abs_abs_real,X))) # label(fact_3608_real__sgn__eq) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 373 (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,A_1),K)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),K)),Ma)))))) # label(fact_4525_zcong__cancel) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 374 (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]. 8.57/8.69 375 (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]. 8.57/8.69 376 (all C_47 all A_114 all B_84 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_114),B_84)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_47)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_114),C_47)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_84),C_47)))))) # label(fact_1319_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 377 (all P all Q all R hAPP_r474017924t_real(hAPP_f1501741374t_real(hAPP_f1135648319t_real(cOMBC_1329014627t_real,P),Q),R) = hAPP_f1676298106t_real(hAPP_r50993370t_real(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Nat__Onat_Mtc__Nat__O) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 378 (all C_74 all A_185 all B_128 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_185),B_128)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_74),A_185)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_74),B_128))))) # label(fact_531_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 379 (all Ma all D_1 ((exists Q_4 Ma = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,D_1),Q_4)) <-> zero_zero_nat = hAPP_nat_nat(div_mod_nat(Ma),D_1))) # label(fact_3179_mod__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 380 (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]. 8.57/8.69 381 (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]. 8.57/8.69 382 (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]. 8.57/8.69 383 (all Ma all K all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,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))) <-> (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,Ma),Na))))) # label(fact_2131_mult__le__cancel2) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 384 (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_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_192)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_192),N_44)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_192),N_43)))))) # label(fact_444_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 385 (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]. 8.57/8.69 386 (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]. 8.57/8.69 387 (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]. 8.57/8.69 388 (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]. 8.57/8.69 389 (all P hAPP_real_real(cOMBI_real,P) = P) # label(help_COMBI_1_1_COMBI_000tc__RealDef__Oreal_U) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 390 (all Y all X all Z_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Z_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_1))))) # label(fact_4159_termination__basic__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 391 (all A_1 all B_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_1),B_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),B_1),Ma)))) # label(fact_2594_zcong__def) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 392 (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]. 8.57/8.69 393 (all A_44 all B_33 all K_6 (A_44 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_33),K_6) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,B_33),A_44)))) # label(fact_2267_dvdI) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 394 (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]. 8.57/8.69 395 (all Xa all Ya (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Xa),Ya)) -> (Ya = Xa <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Xa),Ya))))) # label(fact_1999_linorder__antisym__conv1) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 396 (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]. 8.57/8.69 397 (all Lx_2 all Ly_2 all Rx_2 all Ry_2 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ly_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_2),Ry_2))) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_2),Ly_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_2),Ry_2))) # label(fact_1071_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 398 (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]. 8.57/8.69 399 (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]. 8.57/8.69 400 (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]. 8.57/8.69 401 (all B_132 all A_189 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_189),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_132),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_189),B_132)),zero_zero_rat))))) # label(fact_495_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 402 (all X (is_int(X) -> X = hAPP_int_int(succ,hAPP_int_int(pred,X)))) # label(fact_4038_succ__pred) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 403 (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]. 8.57/8.69 404 (all C all D_1 all A_1 all B_1 (is_int(C) & is_int(D_1) & is_int(B_1) & is_int(A_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)) != 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)) <-> B_1 != A_1 & D_1 != C))) # label(fact_1174_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 405 (all B_2 zero_zero_int = hAPP_int_int(div_mod_int(zero_zero_int),B_2)) # label(fact_2987_zmod__zero) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 406 (all X_57 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_57),one_one_nat) = X_57) # label(fact_1002_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 407 (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,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_1),B_1)),E_2)),C)),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))))) # label(fact_2425_le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 408 (all P all Q all R hAPP_real_real(hAPP_n546711566l_real(P,R),Q) = hAPP_nat_real(hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__RealDef__Orea) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 409 (all N (N != zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3157_Suc__Suc__mult__two__diff__two) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 410 (all X_51 all Y_45 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_51),X_51)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_45),Y_45))))) # label(fact_1583_sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 411 (all X one_one_real = 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)))))) # label(fact_3621_sin__cos__squared__add) # label(axiom) # label(non_clause). [assumption]. 8.57/8.69 412 (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]. 8.57/8.70 413 (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]. 8.57/8.70 414 (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]. 8.57/8.70 415 (all X all Y hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),Y)) = hAPP_complex_real(dist_dist_complex(X),Y)) # label(fact_4545_dist__complex__def) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 416 (all Y all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (one_one_real != A_3 -> hAPP_real_real(log(A_3),hAPP_real_real(hAPP_r1250527377l_real(powr,A_3),Y)) = Y))) # label(fact_4461_log__powr__cancel) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 417 (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]. 8.57/8.70 418 (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]. 8.57/8.70 419 (all Wa all Ya all Xa all Z_2 (Wa = Xa | Z_2 = Ya <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Wa),Z_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Xa),Ya)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Wa),Ya)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Xa),Z_2)))) # label(fact_1190_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 420 (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]. 8.57/8.70 421 (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]. 8.57/8.70 422 (all N_18 one_one_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_18))) # label(fact_2192_power__m1__even) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 423 (all B_112 all A_164 all C_66 (is_int(B_112) & is_int(C_66) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_112),A_164) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_66),A_164) -> C_66 = B_112))) # label(fact_783_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 424 (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]. 8.57/8.70 425 (all A_3 all B_2 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,expi(A_3)),expi(B_2)) = expi(hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_3),B_2))) # label(fact_3996_expi__add) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 426 (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]. 8.57/8.70 427 (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]. 8.57/8.70 428 (all A_147 (is_int(A_147) -> A_147 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),A_147))) # label(fact_944_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 429 (all A_3 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(abs_abs_int,A_3)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(abs_abs_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_2804_abs__power2__distrib) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 430 (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]. 8.57/8.70 431 (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]. 8.57/8.70 432 (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]. 8.57/8.70 433 (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]. 8.57/8.70 434 (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]. 8.57/8.70 435 (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]. 8.57/8.70 436 (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]. 8.57/8.70 437 (all X all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),hAPP_int_int(multInv(P_3),X))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),hAPP_int_int(multInv(P_3),X))),hAPP_int_int(multInv(P_3),hAPP_int_int(multInv(P_3),X)))),P_3)))))) # label(fact_2919_Int2_Oaux____1) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 438 (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,zero_zero_int),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,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]. 8.57/8.70 439 (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]. 8.57/8.70 440 (all A_151 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_151),zero_zero_real) = A_151) # label(fact_911_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 441 (all K_2 all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N))) # label(fact_2100_add__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 442 (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]. 8.57/8.70 443 (all N all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N))))) # label(fact_737_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 444 (all C_25 all D_10 all A_79 all B_52 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_79),B_52)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_25),D_10)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_79)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_25)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_79),C_25)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_52),D_10)))))))) # label(fact_1567_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 445 (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]. 8.57/8.70 446 (all X all Y hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X),Y)) = hAPP_int_int(nat_tsub(hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y))) # label(fact_3133_Nat__Transfer_Otransfer__int__nat__functions_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 447 (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]. 8.57/8.70 448 (all C all A_1 all B_1 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_1),B_1)) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),A_1)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),B_1))))) # label(fact_979_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 449 (all Ma all No 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))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,Ma)),hAPP_nat_real(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),No)))))) # label(fact_3292_lemma__f__mono__add) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 450 (all A_1 all C all B_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,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_1),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_1),C))))) # label(fact_985_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 451 (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]. 8.57/8.70 452 (all Wa all Z_2 (is_int(Z_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Wa),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)))))) # label(fact_738_zle__iff__zadd) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 453 (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]. 8.57/8.70 454 (all M_25 all N_49 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_25)),hAPP_n522471361de_int(semiri1424489471de_int,N_49))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_25),N_49)))) # label(fact_366_of__nat__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 455 (all A_1 all P_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> hBOOL(hAPP_f448129468l_bool(finite_finite_int,comple219730294t_bool(setS(A_1,P_5)))))) # label(fact_3938_Union__SetS__finite) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 456 (all A_3 all B_2 all C_1 hAPP_nat_nat(div_mod_nat(A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),C_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),B_2)),C_1))),hAPP_nat_nat(div_mod_nat(A_3),B_2))) # label(fact_3191_mod__mult2__eq) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 457 (all B_77 all A_107 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_77)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_107)) | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_77),zero_zero_rat)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_107),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_107),B_77))))) # label(fact_1355_split__mult__pos__le) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 458 (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]. 8.57/8.70 459 (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]. 8.57/8.70 460 (all Xa ((exists 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))))) = Xa & -hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1)))) | (exists N_1 (-hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1)) & hAPP_real_real(uminus_uminus_real,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)))))) = Xa)) <-> hAPP_real_real(cos,Xa) = zero_zero_real)) # label(fact_3784_cos__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 461 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Xa)),one_one_real)) -> hAPP_real_real(arctan,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_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))))))) # label(fact_4816_arctan__series) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 462 (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_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),Na)))))) # label(fact_4739_bdd__int__set__l__le__finite) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 463 (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,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,Z_1),X)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z_1))))) # label(fact_2624_dvd_Oless__trans) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 464 (all F all K_3 all Na ((all P_4 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,P_4),Na)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,P_4)),K_3)))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,Na)),K_3))))) # label(fact_4941_real__setsum__nat__ivl__bounded) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 465 (all A_3 all B_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(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),A_3)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) # label(fact_1022_zadd__power3) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 466 (all N (is_int(N) -> N = archim1246769320r_real(hAPP_int_real(real_int,N)))) # label(fact_4383_floor__real__of__int) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 467 (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]. 8.57/8.70 468 (all A_45 all B_34 all C_13 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_45),B_34)),C_13)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_45),C_13)))) # label(fact_2259_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 469 (all Xa (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(sqrt,Xa)),zero_zero_real)))) # label(fact_3529_real__sqrt__lt__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 470 (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]. 8.57/8.70 471 (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]. 8.57/8.70 472 (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,one_one_int),B_1)))))) # label(fact_2930_wset__g__1) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 473 (all Xa all Ya (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Ya),Xa)) | Ya = Xa <-> -hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Xa),Ya)))) # label(fact_1841_not__less__iff__gr__or__eq) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 474 (all Q all P (hBOOL(P) | hBOOL(hAPP_bool_bool(fimplies(P),Q)))) # label(help_fimplies_1_1_U) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 475 (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]. 8.57/8.70 476 (all X all Y all Z_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),Z_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)),Z_1)) # label(fact_4302_gcd__semilattice__nat_Oinf__assoc) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 477 (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]. 8.57/8.70 478 (all Xa all Ya (is_int(Ya) & is_int(Xa) -> (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 = Ya & zero_zero_int = Xa))) # label(fact_4_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 479 (all A_154 all N_32 all B_105 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_154),N_32) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_105),N_32) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_154)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_105)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32)) -> A_154 = B_105))))) # label(fact_862_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 480 (all A_39 all B_28 hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_39),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_39),B_28)))) # label(fact_2297_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 481 (all K all Ma all Na (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),Na) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),Ma) <-> Na = Ma)) # label(fact_3025_Suc__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 482 (all Ma all K all Na (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,K),Na) = one_one_nat -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K),Ma))))) # label(fact_4367_coprime__dvd__mult__iff__nat) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 483 (all A all B (image_nat_int(semiri1621563631at_int,A) = image_nat_int(semiri1621563631at_int,B) <-> A = B)) # label(fact_4600_transfer__nat__int__set__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 484 -(all S1 (is_int(S1) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,S1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))))) # label(fact_2203__096_B_Bthesis_O_A_I_B_Bs1_O_A_091s1_A_094_A2_A_061_A_N1_093_A_Imod_A4) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 485 (all A_1 all E_2 all C all B_1 all D_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_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_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_2428_less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 486 (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]. 8.57/8.70 487 (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]. 8.57/8.70 488 (all N all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(suc,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_nat_nat(suc,N))) = X)) # label(fact_3922_real__root__pos) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 489 (all I (is_int(I) -> (zero_zero_int != I -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),I)) -> hAPP_int_int(sgn_sgn_int,I) = one_one_int) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),I)) -> hAPP_int_int(uminus_uminus_int,one_one_int) = hAPP_int_int(sgn_sgn_int,I))) & (I = zero_zero_int -> zero_zero_int = hAPP_int_int(sgn_sgn_int,I)))) # label(fact_3670_zsgn__def) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 490 (all X_54 all Y_46 all Q_9 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_54),Q_9)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_46),Q_9)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_54),Y_46)),Q_9)) # label(fact_1246_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 491 (all C_53 all A_142 all B_100 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_142),B_100)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_53),A_142)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_53),B_100))))) # label(fact_969_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 492 (all X (is_int(X) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),zero_zero_int)) -> (exists N_1 hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_1))) = X)))) # label(fact_3503_negD) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 493 (all A_47 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,zero_z891286103de_int),A_47)) -> zero_z891286103de_int = A_47)) # label(fact_2248_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 494 (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]. 8.57/8.70 495 (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]. 8.57/8.70 496 (all V_21 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21)) -> hAPP_int_int(number_number_of_int,hAPP_int_int(succ,V_21)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_21)),one_one_int))) # label(fact_743_semiring__number__of__add__1) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 497 (all M_19 hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_n1108039445umeral(semiri1619134803umeral,M_19)))) # label(fact_721_zero__le__imp__of__nat) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 498 (all Na hAPP_nat_nat(suc,Na) = hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),Na)))) # label(fact_4749_card__Collect__le__nat) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 499 (all A_165 all N_33 all B_113 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_165),N_33)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_113),N_33))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_113)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_165),B_113))))) # label(fact_750_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 500 (all A_40 all B_29 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_40),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_29),A_40)))) # label(fact_2289_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 501 (all Ya (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_int_rat(number_number_of_rat,Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Ya)))) # label(fact_614_le__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 502 (all X hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,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))))))) # label(fact_3629_minus__sin__cos__eq) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 503 (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]. 8.57/8.70 504 (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]. 8.57/8.70 505 (all B_115 all A_169 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_169)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_115)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_169),B_115)))))) # label(fact_673_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 506 (all A_146 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A_146) = A_146) # label(fact_946_add__0__left) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 507 (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]. 8.57/8.70 508 (all B_42 all A_65 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_42),A_65)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_65),B_42)))) # label(fact_1774_xt1_I9_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 509 (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]. 8.57/8.70 510 (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]. 8.57/8.70 511 (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]. 8.57/8.70 512 (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]. 8.57/8.70 513 (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]. 8.57/8.70 514 (all A_187 all C_76 all B_130 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_187),C_76)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_130),C_76))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_187),B_130)))) # label(fact_523_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 515 (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]. 8.57/8.70 516 (all A_50 hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_50),zero_z126310315umeral))) # label(fact_2209_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 517 (all M_18 all N_30 hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_18),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))) # label(fact_1103_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 518 (all X_63 all Y_52 (Y_52 != X_63 -> (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_63),Y_52)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_52),X_63))))) # label(fact_311_linorder__neqE__linordered__idom) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 519 (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]. 8.57/8.70 520 (all Wa all Ma (is_int(Wa) -> (Ma = hAPP_int_nat(nat,Wa) <-> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa)) -> Ma = zero_zero_nat) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa)) -> hAPP_nat_int(semiri1621563631at_int,Ma) = Wa)))) # label(fact_2789_nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 521 (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]. 8.57/8.70 522 (all X all A_3 hAPP_real_real(exp_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),hAPP_real_real(ln,X))) = hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)) # label(fact_4436_powr__def) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 523 (all Xa all Ya all B_1 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),B_1)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_1),Xa)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_1),Ya))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya))))) # label(fact_630_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 524 (all A_33 all B_25 A_33 = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_33),B_25)),B_25)) # label(fact_2344_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 525 (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_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ya),zEven))) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat,Ya)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat,Xa)))))) # label(fact_3380_neg__one__power__parity) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 526 (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]. 8.57/8.70 527 (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)) & F = hAPP_n546711566l_real(Diff,zero_zero_nat) & (all M_1 all T (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),Na)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))))) -> (exists T (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_real,T),H)))))) # label(fact_4933_Maclaurin__objl) # label(axiom) # label(non_clause). [assumption]. 8.57/8.70 528 (all I_1 all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),I_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,I_1),K)))))) # label(fact_2843_pos__imp__zdiv__pos__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 529 (all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_3)) -> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_2975_inv__inv__aux) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 530 (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]. 8.57/8.71 531 (all A_194 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_194),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_194),one_one_rat)))) # label(fact_415_less__add__one) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 532 (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]. 8.57/8.71 533 (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_2563_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 534 (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]. 8.57/8.71 535 (all Code_numeral_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,code_c271388182l_size(Code_numeral_1)),hAPP_nat_nat(suc,zero_zero_nat)) = code_c271388182l_size(code_S1047413653umeral(Code_numeral_1))) # label(fact_4061_code__numeral_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 536 (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]. 8.57/8.71 537 (all A_1 all C all D_1 all B_1 (is_int(B_1) & is_int(D_1) -> (B_1 != zero_zero_int -> (D_1 != zero_zero_int -> (hAPP_int_rat(hAPP_int_fun_int_rat(fract,C),D_1) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_1) <-> 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,C),B_1)))))) # label(fact_4671_eq__rat_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 538 (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]. 8.57/8.71 539 (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]. 8.57/8.71 540 (all B_2 all D_3 all A_3 (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),A_3) -> (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),B_2) -> one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2))))) # label(fact_5000_coprime__mult__int) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 541 (all M_25 all N_49 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_25)),hAPP_nat_nat(semiri984289939at_nat,N_49))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_25),N_49)))) # label(fact_368_of__nat__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 542 (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]. 8.57/8.71 543 (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]. 8.57/8.71 544 (all A_8 all B_9 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,A_8),B_9))),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(abs_abs_int,A_8)),hAPP_int_int(abs_abs_int,B_9))))) # label(fact_2762_abs__triangle__ineq4) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 545 (all Xa all Ya all Z_2 (Ya = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa),Z_2) <-> hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Ya),Z_2) = Xa)) # label(fact_2935_eq__diff__eq_H) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 546 (all Xa hBOOL(hAPP_real_bool(isCont_real_real(cos),Xa))) # label(fact_5160_isCont__cos) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 547 (all Xa all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)) <-> 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)))) # label(fact_2027_less__le__not__le) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 548 (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]. 8.57/8.71 549 (all X all Y hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(cnj,X)),hAPP_complex_complex(cnj,Y)) = hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),Y))) # label(fact_4066_cnj_Oadd) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 550 (all B_1_1 all B_2_1 is_bool(hAPP_P62333565t_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__prod_Itc__) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 551 (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]. 8.57/8.71 552 (all B_1_1 all B_2_1 is_bool(hAPP_P1555980039t_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 553 (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]. 8.57/8.71 554 (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]. 8.57/8.71 555 (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_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_192)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_192),N_44)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_192),N_43)))))) # label(fact_441_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 556 (all Xa all Ra (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ra)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,Ra)),Xa)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Xa)),Ra)))) # label(fact_3431_abs__le__interval__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 557 (all A_187 all C_76 all B_130 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_187),C_76)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_130),C_76))) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_187),B_130)))) # label(fact_521_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 558 (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]. 8.57/8.71 559 (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 ((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) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1))))))) # label(fact_1008_realpow__pos__nth__unique) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 560 (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]. 8.57/8.71 561 (all Ma all K all Na (zero_zero_nat = K | Na = Ma <-> 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))) # label(fact_2088_mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 562 (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]. 8.57/8.71 563 (all Z_1 all W hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(uminus_uminus_int,Z_1)),W) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),W))) # label(fact_3463_zmult__zminus) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 564 (all P_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,P_5),zOdd)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_2938_Euler_Oaux__2) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 565 (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_2127_int__mult) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 566 (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]. 8.57/8.71 567 (all M zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),M)) # label(fact_2477_diff__self__eq__0) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 568 (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]. 8.57/8.71 569 (all Y_2 all X_2 ((exists A_2 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),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(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]. 8.57/8.71 570 (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]. 8.57/8.71 571 (all Z_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_2),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z_2)),Z_2)),zero_zero_int)))) # label(fact_127_odd__less__0) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 572 (all A_1 all B_1 ((all D_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),A_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),B_1)) <-> hAPP_nat_nat(suc,zero_zero_nat) = D_2)) <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_1),B_1) = one_one_nat)) # label(fact_4371_coprime__Suc__0__nat) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 573 (all A_3 all B_2 hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(uminus_uminus_int,A_3)),hAPP_int_int(uminus_uminus_int,B_2)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)) # label(fact_3469_zdiv__zminus__zminus) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 574 (all X all Y hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_complex_real(norm_norm_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,X),Y))) # label(fact_3949_complex__norm) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 575 (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]. 8.57/8.71 576 (all X_35 all Y_29 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_35),Y_29)) -> Y_29 != X_35)) # label(fact_1797_order__less__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 577 (all A_64 all B_41 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_64),B_41)) -> -hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_41),A_64)))) # label(fact_1779_order__less__asym_H) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 578 (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]. 8.57/8.71 579 (all M all N hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),N)),N) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N)) # label(fact_4971_gcd__add1__int) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 580 (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]. 8.57/8.71 581 (all C_23 all B_50 all A_73 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_50),A_73)) -> (B_50 = C_23 -> 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]. 8.57/8.71 582 (all F (hBOOL(summable_real(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(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_f352196356l_real(suminf_real,F)))))) # label(fact_4859_suminf__ge__zero) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 583 (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]. 8.57/8.71 584 (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_2234_inf__period_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 585 (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]. 8.57/8.71 586 (all Xa (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(Xa)),one_one_nat)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),one_one_real)))) # label(fact_3316_natceiling__le__eq__one) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 587 (all Xa hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),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_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)))))))) = hAPP_real_real(exp_real,Xa)) # label(fact_4823_exp__first__two__terms) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 588 (all Z_4 all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),zero_zero_int)) -> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),Z_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat,hAPP_int_int(uminus_uminus_int,Z_1))),hAPP_int_nat(nat,hAPP_int_int(uminus_uminus_int,Z_4))))) # label(fact_3497_nat__mult__distrib__neg) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 589 (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]. 8.57/8.71 590 (all P all Q all R hAPP_nat_bool(hAPP_n1699378549t_bool(P,R),Q) = hAPP_nat_bool(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 591 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),M)) -> M = N | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)),M)) | zero_zero_nat = M)) # label(fact_2570_divides__cases) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 592 (all M all N all K_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K_2))) # label(fact_2494_diff__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 593 (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]. 8.57/8.71 594 (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]. 8.57/8.71 595 (all Na all Xa 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))) & 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]. 8.57/8.71 596 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arcsin,Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3583_arcsin__lt__bounded) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 597 (all P all Q all R hAPP_f429384717t_bool(P,hAPP_i627165055t_real(Q,R)) = hAPP_i1948725293t_bool(hAPP_f351634443t_bool(hAPP_f1576199059t_bool(cOMBB_95956018ol_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__RealDef__Oreal_J_000tc__fun_) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 598 (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]. 8.57/8.71 599 (all P all Q all R hAPP_real_bool(hAPP_f1402056515l_bool(hAPP_f824790735l_bool(cOMBB_real_bool_real,P),Q),R) = hAPP_real_bool(P,hAPP_real_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__HOL__Obool_000tc__RealDef__Ore) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 600 (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(hAPP_int_int(div_mod_int(A_1),Ma)),hAPP_int_int(div_mod_int(B_1),Ma)),Ma)))) # label(fact_3001_zcong__zmod) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 601 (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]. 8.57/8.71 602 (all X_5 all Y_5 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_5),Y_5)) -> (Y_5 != X_5 -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_5),Y_5))))) # label(fact_2562_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 603 (all P all Q (-hBOOL(Q) | hBOOL(fdisj(P,Q)))) # label(help_fdisj_2_1_U) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 604 (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,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Wa),one_one_int)),Z_2)))) # label(fact_776_add1__zle__eq) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 605 (all Na all K all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Ma)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),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(minus_minus_nat,Ma),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),K))))))) # label(fact_2487_le__diff__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 606 (all Wa all Ya all Xa all Z_2 (hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Wa),Ya)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa),Z_2)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Wa),Z_2)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa),Ya)) <-> Z_2 = Ya | Xa = Wa)) # label(fact_1188_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 607 (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]. 8.57/8.71 608 (all Xa all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,Xa)),hAPP_int_rat(number_number_of_rat,Ya))))) # label(fact_507_le__number__of) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 609 (all M zero_zero_real = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,pi),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M))))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3772_cos__pi__eq__zero) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 610 (all M all K_2 all N (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,K_2),N) = one_one_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),N))) # label(fact_4352_gcd__mult__cancel__nat) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 611 (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,Na)),hAPP_nat_nat(F,Ma))) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,Ma)),hAPP_nat_nat(F,Na)))))) # label(fact_3214_dvd_Olift__Suc__mono__less__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 612 (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]. 8.57/8.71 613 (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_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_176)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_176),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_176),N_39)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_176),N_40))))))) # label(fact_620_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 614 (all A_1 all B_1 (is_int(A_1) & is_int(B_1) -> (B_1 = A_1 <-> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_1),B_1) = zero_zero_int))) # label(fact_2324_eq__iff__diff__eq__0) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 615 (all X all Y hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),X)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),Y)))))) # label(fact_3539_real__sqrt__mult__self__sum__ge__zero) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 616 (all P_1 collect_real(P_1) = P_1) # label(fact_146_Collect__def) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 617 (all Z_1 all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_1))))) # label(fact_4157_termination__basic__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 618 (all Xa all Ya (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)) -> (Xa = Ya <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya))))) # label(fact_2001_linorder__antisym__conv1) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 619 (all Xa all Ya (is_int(Xa) & is_int(Ya) -> (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,Xa),Xa)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ya),Ya)) <-> zero_zero_int = Ya & Xa = zero_zero_int))) # label(fact_1444_sum__squares__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 620 (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]. 8.57/8.71 621 (all M all 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)) = 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))) # label(fact_3143_Suc__div__eq__add3__div__number__of) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 622 (all M hAPP_nat_nat(suc,M) != zero_zero_nat) # label(fact_2997_Suc__neq__Zero) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 623 (all B_1 all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa)) -> hBOOL(hAPP_real_bool(deriv_real(log(B_1),Xa),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(ln,B_1)),Xa)))))) # label(fact_4791_DERIV__log) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 624 (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]. 8.57/8.71 625 (all N all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_nat_real(real_nat,N))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,natfloor(X)),N))))) # label(fact_3738_less__natfloor) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 626 (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)) -> (Xa = Ya <-> hAPP_real_real(ln,Ya) = hAPP_real_real(ln,Xa))))) # label(fact_3347_ln__inj__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 627 (all A_3 all 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))) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int))) -> 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,B_2),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2))) -> negDivAlg(A_3,B_2) = hAPP_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_3279_negDivAlg_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 628 (all K_2 hAPP_int_int(uminus_uminus_int,hAPP_int_int(bit1,K_2)) = hAPP_int_int(bit1,hAPP_int_int(pred,hAPP_int_int(uminus_uminus_int,K_2)))) # label(fact_4045_minus__Bit1) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 629 (all X_61 all Y_51 (hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_51),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,X_61),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_61)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_51)) -> Y_51 = X_61)))) # label(fact_884_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 630 (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]. 8.57/8.71 631 (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]. 8.57/8.71 632 (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]. 8.57/8.71 633 (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,A_1),zero_zero_int)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_1)),zero_zero_rat))))) # label(fact_4724_Fract__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 634 (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]. 8.57/8.71 635 (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,D_1),B_1)),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D_1),A_1)),Ma))))) # label(fact_2579_zcong__zmult__prop2) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 636 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),pls)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K)),pls)))) # label(fact_757_rel__simps_I29_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 637 (all A_74 A_74 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit1,pls))),A_74)) # label(fact_1615_mult__numeral__1) # label(axiom) # label(non_clause). [assumption]. 8.57/8.71 638 (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]. 8.57/8.71 639 (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]. 8.57/8.71 640 (all Q_1 all R_2 hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),hAPP_int_int(uminus_uminus_int,R_2)) = hAPP_P1975530577nt_int(negateSnd,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_2))) # label(fact_4188_negateSnd__eq) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 641 (all K_2 hAPP_int_int(uminus_uminus_int,hAPP_int_int(bit0,K_2)) = hAPP_int_int(bit0,hAPP_int_int(uminus_uminus_int,K_2))) # label(fact_3461_minus__Bit0) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 642 (all A_3 all B_2 ((-(hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int))) -> hAPP_P1975530577nt_int(adjust(B_2),posDivAlg(A_3,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))) = posDivAlg(A_3,B_2)) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2)) -> posDivAlg(A_3,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),A_3)))) # label(fact_3288_posDivAlg_Osimps) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 643 (all P all Q all R hAPP_r481142414t_bool(hAPP_f1625314454t_bool(hAPP_f340152631t_bool(cOMBB_1453147152l_real,P),Q),R) = hAPP_f2119767738t_bool(P,hAPP_r632163725t_bool(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]. 8.57/8.72 644 (all D_3 all A_3 all B_2 (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)) -> 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]. 8.57/8.72 645 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,Xa)),one_one_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),one_one_real)))) # label(fact_3535_real__sqrt__lt__1__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 646 (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]. 8.57/8.72 647 (all N all K_2 (is_int(K_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2)) -> hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N)),K_2) = K_2))) # label(fact_4508_zgcd__zmult__eq__self2) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 648 (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]. 8.57/8.72 649 (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]. 8.57/8.72 650 (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),sin)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),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_4794_DERIV__sin__realpow2a) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 651 (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]. 8.57/8.72 652 (all V_1 hAPP_complex_real(im,hAPP_int_complex(number528085621omplex,V_1)) = zero_zero_real) # label(fact_4098_complex__Im__number__of) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 653 (all B_40 all A_58 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_40),A_58)) -> (B_40 != A_58 -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_40),A_58))))) # label(fact_1947_xt1_I11_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 654 (all X all Q_1 all N all R_2 (X = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Q_1),N)),R_2) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),R_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,R_2),N)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),X)))))) # label(fact_2547_divides__div__not) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 655 (all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),A_3)) -> -(-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),A_3))))) # label(fact_2625_dvd_Oless__asym_H) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 656 (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]. 8.57/8.72 657 (all P all Q all R hAPP_f1325804733t_bool(P,hAPP_P2053772734t_bool(Q,R)) = hAPP_P1400155668t_bool(hAPP_f223623489t_bool(hAPP_f1815301987t_bool(cOMBB_650041899nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__f) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 658 (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)) -> Ya = Xa))) # label(fact_1651_order__antisym) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 659 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arctan,Y))) & Y = hAPP_real_real(tan,hAPP_real_real(arctan,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_3769_arctan) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 660 (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]. 8.57/8.72 661 (all X all A_3 all B_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_3),B_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_2)))))) # label(fact_4406_powr__mono) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 662 (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_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_176)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_176),one_on1684967323de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_176),N_39)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_176),N_40))))))) # label(fact_618_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 663 (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]. 8.57/8.72 664 (all P all Q all R hAPP_nat_rat(hAPP_f2062659861at_rat(hAPP_f1082758869at_rat(cOMBB_rat_rat_nat,P),Q),R) = hAPP_rat_rat(P,hAPP_nat_rat(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Rat__Orat_000tc__Rat__Orat_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 665 (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]. 8.57/8.72 666 (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]. 8.57/8.72 667 (all Y hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arctan,Y)))) # label(fact_3581_arctan__lbound) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 668 (all Y_2 all X_2 (hBOOL(vanishes(X_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,plus_plus_rat),X_2)),Y_2)))))) # label(fact_5142_vanishes__add) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 669 (all Xa all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) -> (hBOOL(hAPP_nat_bool(even_odd_even_nat,Na)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,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_real_real(uminus_uminus_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_3886_DERIV__even__real__root) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 670 (all Ma all Na (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,Ma)),hAPP_int_real(number267125858f_real,Na))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ma),hAPP_int_int(number_number_of_int,Na))))) # label(fact_4444_number__of__le__real__of__int__iff2) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 671 (all X all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_nat_real(real_nat,A_3))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X)),A_3)))) # label(fact_3721_natceiling__le) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 672 (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))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arccos,Y)),pi))))) # label(fact_3677_arccos__bounded) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 673 (all X all A_3 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)) = hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_real_real(uminus_uminus_real,A_3))) # label(fact_4435_powr__minus__divide) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 674 (all X hAPP_real_real(abs_abs_real,hAPP_int_real(real_int,X)) = hAPP_int_real(real_int,hAPP_int_int(abs_abs_int,X))) # label(fact_4399_real__of__int__abs) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 675 (all Nat hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(size_size_nat,Nat)),hAPP_nat_nat(suc,zero_zero_nat)) = hAPP_nat_nat(size_size_nat,hAPP_nat_nat(suc,Nat))) # label(fact_4056_nat_Osize_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 676 (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]. 8.57/8.72 677 (all Ya all Xa (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Ya),Xa)) -> (Ya = Xa <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Xa),Ya))))) # label(fact_1688_order__antisym__conv) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 678 (all C_17 all A_67 all B_44 (A_67 = B_44 -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_17),B_44)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_17),A_67))))) # label(fact_1758_xt1_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 679 (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]. 8.57/8.72 680 (all Xa all Ya all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) -> (hAPP_real_real(hAPP_n546711566l_real(root,Na),Ya) = hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa) <-> Ya = Xa))) # label(fact_3891_real__root__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 681 (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]. 8.57/8.72 682 (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,re),X_2),hAPP_complex_real(re,A_1),sequentially)))) # label(fact_5055_Re_OLIMSEQ) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 683 (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]. 8.57/8.72 684 (all P (is_bool(P) -> P = fFalse | P = fTrue)) # label(help_If_3_1_If_000tc__RealDef__Oreal_T) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 685 (all A_1 all 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)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),archim856651990g_real(Xa))))) # label(fact_4471_le__ceiling__eq) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 686 (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]. 8.57/8.72 687 (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),B_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),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)))))) # label(fact_2842_nonneg1__imp__zdiv__pos__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 688 (all K_2 all M all N hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,K_2)),hAPP_int_int(legacy_zgcd(M),N)) = hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N))) # label(fact_4505_zgcd__zmult__distrib2__abs) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 689 (all A_148 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,zero_z891286103de_int),A_148) = A_148) # label(fact_933_add__0) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 690 (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_2,X_1)) <-> hBOOL(hAPP_int_bool(P_1,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]. 8.57/8.72 691 (all P_3 all A_3 hAPP_int_int(div_mod_int(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),P_3) = hAPP_int_int(inv(P_3),A_3)) # label(fact_3161_WilsonRuss_Oinv__def) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 692 (all X hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,one_one_nat),X) = one_one_nat) # label(fact_4337_gcd__lcm__complete__lattice__nat_Oinf__bot__left) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 693 (all P all Q all R hAPP_bool_bool(P,hAPP_int_bool(Q,R)) = hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f627970963t_bool(cOMBB_bool_bool_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 694 (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]. 8.57/8.72 695 (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]. 8.57/8.72 696 (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]. 8.57/8.72 697 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),Xa)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),natfloor(Xa))))) # label(fact_3315_le__natfloor__eq__one) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 698 (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]. 8.57/8.72 699 (all N_34 hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_n522471361de_int(semiri1424489471de_int,N_34)))) # label(fact_714_of__nat__0__le__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 700 (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]. 8.57/8.72 701 (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]. 8.57/8.72 702 (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]. 8.57/8.72 703 (all K_2 all L_2 hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),L_2)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,K_2)),hAPP_int_int(bit0,L_2))) # label(fact_199_add__Bit1__Bit0) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 704 (all X_47 all Y_41 (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_47),Y_41)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_41),X_47)))) # label(fact_1622_linorder__le__cases) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 705 (all B_76 all A_106 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_106),zero_zero_nat)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_76)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_106)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_76),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_106),B_76)),zero_zero_nat)))) # label(fact_1362_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 706 (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]. 8.57/8.72 707 (all Ya all Xa (-hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Ya),Xa)) -> (-hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Xa),Ya)) <-> Xa = Ya))) # label(fact_1831_linorder__antisym__conv3) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 708 (all K_2 all L_2 hAPP_int_int(bit1,K_2) != hAPP_int_int(bit0,L_2)) # label(fact_133_rel__simps_I50_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 709 (all A_1 all Ma hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(bnorRset(A_1),Ma)))) # label(fact_4293_Bnor__fin) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 710 (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]. 8.57/8.72 711 (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]. 8.57/8.72 712 (all C_30 all A_84 all B_57 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_30),A_84)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_30),B_57))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_30)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_84),B_57))))) # label(fact_1536_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 713 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),Y)) -> (exists X_1 (Y = hAPP_real_real(exp_real,X_1) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y),one_one_real))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_1)))))) # label(fact_3422_lemma__exp__total) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 714 (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]. 8.57/8.72 715 (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]. 8.57/8.72 716 (all C_47 all A_114 all B_84 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_114),B_84)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_47)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_114),C_47)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_84),C_47)))))) # label(fact_1324_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 717 (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]. 8.57/8.72 718 (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]. 8.57/8.72 719 (all M hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),M)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = M) # label(fact_2898_add__self__div__2) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 720 (all P all Q all R hAPP_f465821005nt_int(P,hAPP_i1778081398nt_int(Q,R)) = hAPP_i1778081398nt_int(hAPP_f2094497995nt_int(hAPP_f534361363nt_int(cOMBB_1657320178nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__prod) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 721 (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(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat)))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,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(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat)))),hAPP_real_real(sin,Xa)))) # label(fact_4862_sin__paired) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 722 (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]. 8.57/8.72 723 (all B_124 all C_71 all A_180 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_180)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_124),C_71)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_124),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_180),C_71)))))) # label(fact_577_add__increasing) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 724 (all N -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat))) # label(fact_301_less__nat__zero__code) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 725 (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]. 8.57/8.72 726 (all X hAPP_real_real(arctan,hAPP_real_real(uminus_uminus_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(arctan,X))) # label(fact_3523_arctan__minus) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 727 (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_le1912455174t_bool,Ya),Z_2)) -> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Z_2))))) # label(fact_1735_order__less__trans) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 728 (all M all N all K_2 hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N),K_2))),N) = hAPP_int_int(legacy_zgcd(M),N)) # label(fact_4499_zgcd__zadd__zmult) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 729 (all P_1 all A0 all A1 (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(lazy_small_lazy_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1))) -> ((all D_2 all I_2 (is_int(I_2) & is_int(D_2) -> (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(lazy_small_lazy_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,D_2),I_2))) -> ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,D_2),I_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,D_2),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int)))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,D_2),I_2)))))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1))))) # label(fact_4573_small__lazy_H_Opinduct) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 730 (all X_1 hAPP_nat_real(diffs_real(cos_coeff),X_1) = hAPP_real_real(uminus_uminus_real,hAPP_nat_real(sin_coeff,X_1))) # label(fact_4877_cos__fdiffs) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 731 (all A_124 (is_int(A_124) -> A_124 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_124),one_one_int))) # label(fact_1226_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 732 (all X_62 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_62),zero_zero_nat) = one_one_real) # label(fact_480_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 733 (all Ma (Ma = hAPP_nat_nat(suc,zero_zero_nat) <-> 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]. 8.57/8.72 734 (all C 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(plus_plus_int,C),A_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),B_1))))) # label(fact_982_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 735 (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]. 8.57/8.72 736 (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]. 8.57/8.72 737 (all N exists P_4 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),P_4)) & hBOOL(hAPP_nat_bool(prime,P_4)))) # label(fact_4221_euclid) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 738 (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]. 8.57/8.72 739 (all Z_1 (is_int(Z_1) -> Z_1 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),Z_1))) # label(fact_158_zadd__0) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 740 (all N all A_3 all B_2 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N)),A_3)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N)),B_2)))))) # label(fact_4193_prime__divprod__pow) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 741 (all A_139 all B_97 all C_51 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_139),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_97),C_51)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_139),B_97)),C_51)) # label(fact_1053_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 742 (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]. 8.57/8.72 743 (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]. 8.57/8.72 744 (all Xa (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xa),one_one_int)),zEven)))) # label(fact_3368_odd__minus__one__even) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 745 (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]. 8.57/8.72 746 (all V_1 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_1),pls)) -> zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V_1)),one_one_nat)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_1),pls)) -> hAPP_int_nat(number_number_of_nat,hAPP_int_int(pred,V_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V_1)),one_one_nat)))) # label(fact_4037_nat__number__of__diff__1) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 747 (all X_14 all P_6 all Q_7 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_14),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_6),Q_7)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_14),P_6)),Q_7)) # label(fact_2089_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 748 (all B_2 all D_3 all A_3 (one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),A_3) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),B_2) = one_one_nat -> one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2))))) # label(fact_4353_coprime__mult__nat) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 749 (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]. 8.57/8.72 750 (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]. 8.57/8.72 751 (all I all K_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),J_2)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),K_2)))) # label(fact_2524_diff__add__assoc) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 752 (all M all K_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2)),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)))) # label(fact_1274_add__leD1) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 753 (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]. 8.57/8.72 754 (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]. 8.57/8.72 755 (all Xa all Ya (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,Xa),Xa)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ya),Ya))),zero_zero_rat)) <-> zero_zero_rat = Xa & zero_zero_rat = Ya)) # label(fact_1584_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 756 (all Xa all Na all Ya (Ya = Xa <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Ya),hAPP_nat_nat(suc,Na)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Xa),hAPP_nat_nat(suc,Na)))) # label(fact_3028_exp__mono__eq) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 757 (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,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_2383_eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 758 (all P all Q all R hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,P),Q),R) = hAPP_bool_bool(hAPP_i68813070l_bool(P,R),hAPP_int_bool(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__HOL__Obool_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 759 (all B_1 all A_1 all C (is_int(B_1) & is_int(C) -> (B_1 = C <-> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_1),A_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),A_1)))) # label(fact_807_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 760 (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]. 8.57/8.72 761 (all B_120 all A_174 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_174),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_120),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_174),B_120)),zero_zero_real))))) # label(fact_643_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 762 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X))))))) # label(fact_3381_ln__one__plus__pos__lower__bound) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 763 (all Na all Ma (hAPP_nat_real(real_nat,Na) = hAPP_nat_real(real_nat,Ma) <-> Ma = Na)) # label(fact_3690_real__of__nat__inject) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 764 (all X all Y X = hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,fTrue),X),Y)) # label(help_If_1_1_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_T) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 765 (all V_4 all W_2 all 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)) = 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)) # label(fact_2437_add__number__of__diff1) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 766 (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]. 8.57/8.72 767 (all A_3 all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> natceiling(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_nat_real(real_nat,A_3))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natceiling(X)),A_3))) # label(fact_3746_natceiling__add) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 768 (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]. 8.57/8.72 769 (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]. 8.57/8.72 770 (all X all Y hAPP_int_int(div_mod_int(hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(div_mod_nat(X),Y))) # label(fact_3185_Divides_Otransfer__int__nat__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.72 771 (all A_3 all B_2 (B_2 != zero_zero_nat -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),B_2)))) # label(fact_4344_gcd__le2__nat) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 772 (all R_2 all X hAPP_complex_real(re,hAPP_complex_complex(scaleR1652505878omplex(R_2),X)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_2),hAPP_complex_real(re,X))) # label(fact_5112_complex__Re__scaleR) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 773 (all B_5 all Q_5 all R_4 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_5)),R_4))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_4),B_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_5)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Q_5)))))) # label(fact_2062_q__pos__lemma) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 774 (all Xa (Xa = zero_zero_nat <-> Xa = zero_zero_nat)) # label(fact_768_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 775 (all Ma all Na (Ma = zero_zero_nat & Na = zero_zero_nat <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),Na) = zero_zero_nat)) # label(fact_315_add__is__0) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 776 (all D_3 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(abs_abs_int,D_3)),D_3))) # label(fact_2948_zdvd__self__abs2) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 777 (all X all B_2 all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (A_3 != one_one_real -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_2)) -> (one_one_real != B_2 -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_real_real(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]. 8.57/8.73 778 (all V_4 all W_2 all 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) = 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))) # label(fact_2439_add__number__of__diff1) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 779 (all C all A_1 all 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))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1)))) # label(fact_542_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 780 (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_1806_order__less__imp__not__less) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 781 (all A_3 all B_2 all C_1 all Q_1 all R_2 (is_int(C_1) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(B_2,C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_2))) -> (C_1 != zero_zero_int -> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2),C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Q_1)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),R_2)),C_1))),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),R_2)),C_1)))))))) # label(fact_3284_zmult1__lemma) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 782 (all A_1 all B_1 (is_int(B_1) & is_int(A_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)) -> (B_1 = zero_zero_int | A_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)) & (-(A_1 = zero_zero_int | zero_zero_int = B_1) -> 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))))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),zero_zero_int)) -> norm_frac(hAPP_int_int(uminus_uminus_int,A_1),hAPP_int_int(uminus_uminus_int,B_1)) = norm_frac(A_1,B_1))))) # label(fact_4803_norm__frac_Opsimps) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 783 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hAPP_P1175774780nt_int(product_snd_int_int,negDivAlg(A_3,B_2)) = hAPP_int_int(div_mod_int(A_3),B_2)))) # label(fact_4233_mod__neg__pos) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 784 (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]. 8.57/8.73 785 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),hAPP_int_int(bit0,K))))) # label(fact_729_rel__simps_I21_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 786 (all K_2 all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N))) # label(fact_2495_diff__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 787 (all A_61 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_61),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_61),A_61)),A_61)) # label(fact_1870_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 788 (all M_15 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,M_15),M_15) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,one_on1645066479umeral),one_on1645066479umeral)),M_15)) # label(fact_1473_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 789 (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,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),hAPP_real_real(hAPP_n546711566l_real(root,N_4),X)))))))) # label(fact_3931_real__root__increasing) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 790 (all A_1 all B_1 all Na (Na != zero_zero_nat -> (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))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_1),B_1))))) # label(fact_2957_pow__divides__eq__nat) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 791 (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,N_1))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),one_one_nat))),R_2)))))) # label(fact_3778_reals__Archimedean6) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 792 (all A_1 all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),archim856651990g_real(Xa))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,A_1)),Xa)))) # label(fact_4420_less__ceiling__eq) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 793 (all A_1 all P_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int))),one_one_int),P_5)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)),P_5)))) # label(fact_2232_inv__not__p__minus__1__aux) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 794 (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]. 8.57/8.73 795 (all K_2 all L_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),L_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),L_2)))))) # label(fact_1272_add__le__mono) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 796 (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]. 8.57/8.73 797 (all B_2 all D_3 all A_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),A_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),B_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2)))))) # label(fact_4014_coprime__mul) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 798 (all P all Q all R hAPP_f766218899t_real(P,hAPP_i371647666l_real(Q,R)) = hAPP_i1671359920t_real(hAPP_f74147283t_real(hAPP_f1169087445t_real(cOMBB_2129825580al_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__031) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 799 (all X_1 ((-hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)) -> zero_zero_real = hAPP_nat_real(cos_coeff,X_1)) & (hBOOL(hAPP_nat_bool(even_odd_even_nat,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))) = hAPP_nat_real(cos_coeff,X_1)))) # label(fact_3828_cos__coeff__def) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 800 (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_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),tan)),Ya)))) = hAPP_real_real(arctan,Ya)) # label(fact_4881_arctan__def) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 801 (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]. 8.57/8.73 802 (all X all M hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),hAPP_int_int(div_mod_int(X),M)),M))) # label(fact_3000_mod__mod__is__mod) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 803 (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]. 8.57/8.73 804 (all N hAPP_nat_nat(suc,N) != N) # label(fact_2985_n__not__Suc__n) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 805 (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]. 8.57/8.73 806 (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]. 8.57/8.73 807 (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,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)) -> 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,Ta),X_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ta),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),D)))))))))) # label(fact_5090_bset_I8_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 808 (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]. 8.57/8.73 809 (all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) -> X != Y)) # label(fact_2629_dvd_Oless__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 810 (all A hAPP_f22106695ol_nat(finite_card_nat,A) = hAPP_f957591787ol_nat(finite_card_int,image_nat_int(semiri1621563631at_int,A))) # label(fact_4599_Nat__Transfer_Otransfer__nat__int__set__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 811 (all Xa all Ya (hBOOL(hAPP_f526364711l_bool(hAPP_P1062890741l_bool(member2143287562nt_int,hAPP_P230595783nt_int(hAPP_P941956607nt_int(produc883642259nt_int,Xa),Ya)),ratrel)) <-> zero_zero_int != hAPP_P1175774780nt_int(product_snd_int_int,Ya) & hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_fst_int_int,Ya)),hAPP_P1175774780nt_int(product_snd_int_int,Xa)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_fst_int_int,Xa)),hAPP_P1175774780nt_int(product_snd_int_int,Ya)) & zero_zero_int != hAPP_P1175774780nt_int(product_snd_int_int,Xa))) # label(fact_4665_ratrel__iff) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 812 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) <-> (exists K_1 Na = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),K_1)))) # label(fact_1267_le__iff__add) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 813 (all P all Q all R hAPP_int_rat(P,hAPP_int_int(Q,R)) = hAPP_int_rat(hAPP_f1900950721nt_rat(hAPP_f1270196437nt_rat(cOMBB_int_rat_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__Rat__Orat_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 814 (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_eq_int,hAPP_int_int(number_number_of_int,K)),hAPP_int_int(number_number_of_int,L))))) # label(fact_564_less__eq__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 815 (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]. 8.57/8.73 816 (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]. 8.57/8.73 817 (all Z_14 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_14),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_14),Z_14)) # label(fact_1882_mult__2__right) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 818 (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]. 8.57/8.73 819 (all N (zero_zero_nat != N -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),N) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),N)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3156_lemma__Suc__Suc__4n__diff__2) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 820 (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]. 8.57/8.73 821 (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))) -> natfloor(X) = N))) # label(fact_3745_natfloor__eq) # label(axiom) # label(non_clause). [assumption]. 8.57/8.73 822 (all W all Z1 all Z2 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z1),Z2)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z2))) # label(fact_2354_zdiff__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 823 (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)) -> (one_one_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,U_2),V_8) -> 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]. 8.65/8.73 824 (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]. 8.65/8.73 825 (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_1025_real__less__def) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 826 (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),hAPP_nat_real(semiri132038758t_real,Na))))) # label(fact_243_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 827 (all V_22 all 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)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_22),W_12))) # label(fact_198_number__of__add) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 828 (all A_149 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_149),zero_zero_complex) = A_149) # label(fact_924_add__0__right) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 829 (all P all Q all R hAPP_f289318786t_bool(hAPP_r653725362t_bool(P,R),Q) = hAPP_r373117745t_bool(hAPP_f1115375056t_bool(hAPP_f271976045t_bool(cOMBC_1801450329t_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__O_021) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 830 (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]. 8.65/8.73 831 (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]. 8.65/8.73 832 (all A_1 all C 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,A_1),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_1),C))))) # label(fact_547_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 833 (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]. 8.65/8.73 834 (all W_16 hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,pls)),W_16)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_int_real(number267125858f_real,W_16))) # label(fact_25_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 835 (all A_88 all M_12 all N_26 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_88),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_12),N_26)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_88),M_12)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_88),N_26))) # label(fact_1491_power__add) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 836 (all Xa (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,Xa)),zero_zero_rat)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),pls)))) # label(fact_104_less__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 837 (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]. 8.65/8.73 838 (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]. 8.65/8.73 839 (all Xa all Ya (hBOOL(hAPP_int_bool(even_odd_even_int,Xa)) & hBOOL(hAPP_int_bool(even_odd_even_int,Ya)) | -hBOOL(hAPP_int_bool(even_odd_even_int,Ya)) & -hBOOL(hAPP_int_bool(even_odd_even_int,Xa)) <-> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xa),Ya))))) # label(fact_3240_even__sum) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 840 (all X all Y hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),hAPP_nat_nat(div_mod_nat(X),Y)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)) # label(fact_4334_gcd__red__nat) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 841 (all C_23 all B_50 all A_73 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_50),A_73)) -> (C_23 = B_50 -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_23),A_73))))) # label(fact_1661_xt1_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 842 (all L all U image_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,plus_plus_int),L),hAPP_i1948725293t_bool(ord_at641636577an_int(zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,U),L))) = hAPP_i1948725293t_bool(ord_at641636577an_int(L),U)) # label(fact_4921_image__add__int__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 843 (all Ma all Z_2 (is_int(Z_2) -> (Z_2 = hAPP_nat_int(semiri1621563631at_int,Ma) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2)) & Ma = hAPP_int_nat(nat,Z_2)))) # label(fact_2776_int__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 844 (all X all D_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),D_3)) -> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,X)),hAPP_nat_real(real_nat,D_3)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),D_3))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,hAPP_nat_nat(div_mod_nat(X),D_3))),hAPP_nat_real(real_nat,D_3))))) # label(fact_3749_real__of__nat__div__aux) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 845 (all A_56 all B_38 (is_int(B_38) & is_int(A_56) -> (A_56 != B_38 -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_38),A_56)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_38),A_56)))))) # label(fact_1991_xt1_I12_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 846 (all Ya all Xa (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Xa)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Ya)) -> (zero_zero_nat = Xa & Ya = zero_zero_nat <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Xa),Ya) = zero_zero_nat)))) # label(fact_586_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 847 (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_3707_power__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 848 (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]. 8.65/8.73 849 (all Na (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)))) # label(fact_245_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 850 (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) & is_int(Tn) & hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,K_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Sn),Tn)) = xzgcda(M,N,R_4,R_2,S_3,S_1,T_2,T_1)))))) # label(fact_4534_xzgcd__correct__aux1) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 851 (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]. 8.65/8.73 852 (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]. 8.65/8.73 853 (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,G),F)) & hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G)))) # label(fact_2077_less__fun__def) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 854 (all K all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),Na)) | K = zero_zero_nat <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_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_2514_nat__mult__dvd__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 855 (all P all Q all R hAPP_nat_real(hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,P),Q),R) = hAPP_nat_real(P,hAPP_nat_nat(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 856 (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]. 8.65/8.73 857 (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]. 8.65/8.73 858 (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]. 8.65/8.73 859 (all Xa (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,Xa)),one_one_rat)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),hAPP_int_int(bit1,pls))))) # label(fact_635_le__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 860 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(uminus_uminus_real,Y))) # label(fact_3433_minus__real__def) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 861 (all B_88 all A_118 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_118)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_88),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_118),B_88)),zero_zero_nat))))) # label(fact_1302_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 862 (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]. 8.65/8.73 863 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(real_nat,M)),hAPP_nat_real(real_nat,N)))) # label(fact_3716_real__of__nat__diff) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 864 (all N all M hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(div_mod_nat(M),N))) # label(fact_3194_mult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 865 (all B_2 all A_3 all C_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),C_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B_2),C_1))) # label(fact_4305_gcd__left__commute__nat) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 866 (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]. 8.65/8.73 867 (all A_126 A_126 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,one_on1684967323de_int),A_126)) # label(fact_1207_mult__1) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 868 (all X all N hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,N)),pi))) = hAPP_real_real(tan,X)) # label(fact_3735_tan__periodic__n) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 869 (all A_40 all B_29 hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_40),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_29),A_40)))) # label(fact_2292_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 870 (all F all L all C (hBOOL(tendsto_real_real(F,L,at_real(C))) -> (zero_zero_real != L -> (exists R_1 ((all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),X_1))),R_1)) & X_1 != C -> hAPP_real_real(F,X_1) != zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1))))))) # label(fact_5087_LIM__fun__not__zero) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 871 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)))) # label(fact_1258_less__imp__le__nat) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 872 (all C_16 all A_66 all B_43 (A_66 = B_43 -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_43),C_16)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_66),C_16))))) # label(fact_1767_ord__eq__less__trans) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 873 (all B_1 all A_1 (B_1 = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_1),A_1) <-> zero_zero_real = A_1)) # label(fact_905_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 874 (all P all Q all R hAPP_nat_nat(hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,P),Q),R) = hAPP_nat_nat(P,hAPP_nat_nat(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 875 (all Va all Wa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,Va)),hAPP_int_int(number_number_of_int,Wa))) <-> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,Wa)),hAPP_int_int(number_number_of_int,Va))))) # label(fact_693_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 876 (all P all Q all R hAPP_r794344869l_real(hAPP_f517738447l_real(hAPP_f1879982819l_real(cOMBB_731595089l_real,P),Q),R) = hAPP_f1607917947l_real(P,hAPP_r1134773055l_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J_000tc__fun_043) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 877 (all P all Q all R hAPP_rat_rat(hAPP_nat_fun_rat_rat(P,R),hAPP_nat_rat(Q,R)) = hAPP_nat_rat(hAPP_f2062659861at_rat(hAPP_f1131305186at_rat(cOMBS_nat_rat_rat,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__Rat__Orat_000tc__Rat__Orat_U) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 878 (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]. 8.65/8.73 879 (all N_17 all M_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_17),M_10)) -> hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(semiri132038758t_real,M_10)),hAPP_nat_real(semiri132038758t_real,N_17)) = hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_10),N_17)))) # label(fact_2243_of__nat__diff) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 880 (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]. 8.65/8.73 881 (all X of_real_complex(X) = hAPP_complex_complex(cnj,of_real_complex(X))) # label(fact_4063_complex__cnj__complex__of__real) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 882 (all C_17 all A_67 all B_44 (A_67 = B_44 -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_17),B_44)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_17),A_67))))) # label(fact_1755_xt1_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 883 (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]. 8.65/8.73 884 (all A_158 all B_106 all C_60 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_158),B_106)),C_60) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_158),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_106),C_60))) # label(fact_826_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 885 (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]. 8.65/8.73 886 (all K all Ma all 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))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)))) # label(fact_3062_Suc__mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 887 (all M all K_2 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2) != M -> (hAPP_nat_nat(suc,zero_zero_nat) != M -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M))))) # label(fact_3099_one__less__m) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 888 (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]. 8.65/8.73 889 (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]. 8.65/8.73 890 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)),N) = M)) # label(fact_2892_div__mult__self__is__m) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 891 (all Lx all Ly all Rx all Ry hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx),Rx)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ly),Ry)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx),Ly)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx),Ry))) # label(fact_1083_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.73 892 (all A_1 all B_1 all Ma (is_int(B_1) -> ((exists K_1 (is_int(K_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,Ma),K_1)) = B_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),B_1),Ma))))) # label(fact_2591_zcong__iff__lin) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 893 (all A_1 all B_1 all C (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,A_1),B_1) <-> B_1 = C)) # label(fact_812_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 894 (all X all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(inverse_inverse_real,X)) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)))) # label(fact_3909_real__root__inverse) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 895 (all N all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_2),N))))) # label(fact_4009_coprime__exp__imp) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 896 (all A_1 (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_1),A_1))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_1)))) # label(fact_483_zero__less__double__add__iff__zero__less__single__add) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 897 (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]. 8.65/8.74 898 (all X_31 all Y_25 (X_31 != Y_25 -> (-hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_31),Y_25)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_25),X_31))))) # label(fact_1826_linorder__neqE) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 899 (all A_47 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,zero_zero_rat),A_47)) -> zero_zero_rat = A_47)) # label(fact_2247_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 900 (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]. 8.65/8.74 901 (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,A_1),B_1)) <-> 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)))))) # label(fact_1579_mult__le__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 902 (all A_3 all B_2 hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(uminus_uminus_int,A_3)),hAPP_int_int(uminus_uminus_int,B_2)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)) # label(fact_4677_minus__rat__cancel) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 903 (all P all Q all R hAPP_r481142414t_bool(hAPP_f1341793266t_bool(hAPP_f571820015t_bool(cOMBB_1159313332l_real,P),Q),R) = hAPP_f41603166t_bool(P,hAPP_r373117745t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Nat__Onat_Mtc__HOL_) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 904 (all A_53 all N_20 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_53),N_20)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_53),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_20))) # label(fact_2135_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 905 (all Ya all Xa (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_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))))),Ya) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Xa),Ya)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3377_even__prod__div__2) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 906 (all A_3 all B_2 hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(uminus_uminus_real,B_2)),A_3) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)),ii)) # label(fact_3977_Complex__mult__i) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 907 (all L all U hAPP_n1699378549t_bool(ord_gr660468384an_nat(L),U) = hAPP_n1699378549t_bool(ord_at4362885an_nat(hAPP_nat_nat(suc,L)),U)) # label(fact_4904_atLeastSucLessThan__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 908 (all A_3 all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,A_3)),X)) -> natfloor(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_nat_real(real_nat,A_3))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,natfloor(X)),A_3))) # label(fact_3732_natfloor__subtract) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 909 (all X_43 all Y_37 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_43),Y_37)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_37),X_43)) -> X_43 = Y_37))) # label(fact_1650_order__antisym) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 910 (all P all Q all R hAPP_f883232462nt_int(hAPP_i998473677nt_int(P,R),Q) = hAPP_i1584592887nt_int(hAPP_f1574055885nt_int(hAPP_f960630521nt_int(cOMBC_1805044090nt_int,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_019) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 911 (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]. 8.65/8.74 912 (all A_121 all B_90 all 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)) = 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)) # label(fact_1244_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 913 (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]. 8.65/8.74 914 (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,hAPP_int_nat(number_number_of_nat,V_1)),one_one_nat) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,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,hAPP_int_nat(number_number_of_nat,V_1)),one_one_nat) = one_one_nat))) # label(fact_1006_nat__number__of__add__1) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 915 (all C_40 all A_97 all B_68 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_97),B_68)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_40)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_97),C_40)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_68),C_40)))))) # label(fact_1418_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 916 (all B_126 all A_182 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_182),zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_126),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_182),B_126)),zero_zero_nat))))) # label(fact_568_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 917 (all A_26 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_26),hAPP_rat_rat(abs_abs_rat,A_26)))) # label(fact_2666_abs__ge__self) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 918 (all Xa all Ya (zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Xa),Ya) <-> Ya = zero_zero_nat & Xa = zero_zero_nat)) # label(fact_4341_gcd__lcm__complete__lattice__nat_Oinf__eq__top__iff) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 919 (all X_60 all Y_50 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_60),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_50),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_50)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_60),Y_50))))) # label(fact_891_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 920 (all A_160 all B_108 all C_62 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_160),B_108)),C_62) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_160),C_62)),B_108)) # label(fact_816_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 921 (all X_32 all Y_26 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_32),Y_26)) -> X_32 != Y_26)) # label(fact_1818_less__imp__neq) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 922 (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]. 8.65/8.74 923 (all A_3 all B_2 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),B_2))) # label(fact_2455_zspecial__product) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 924 (all A_7 all B_8 all C_4 all D_5 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_int_int(hAPP_int_fun_int_int(plus_plus_int,A_7),B_8)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_4),D_5)))),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,A_7),C_4))),hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_8),D_5)))))) # label(fact_2765_abs__diff__triangle__ineq) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 925 (all B_1_1 all B_2_1 is_bool(hAPP_C24501650l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__Code____Numeral__Ocode____numeral_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 926 (all Xa all Ya (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)) <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Xa),Ya) = Xa)) # label(fact_4314_gcd__semilattice__nat_Ole__iff__inf) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 927 (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,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,A_1),B_1))))) # label(fact_1373_mult__less__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 928 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(sin,X))),one_one_real))) # label(fact_3613_abs__sin__le__one) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 929 (all V_1 ((-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1))) -> hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_1)) = hAPP_nat_nat(suc,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_nat_nat(suc,hAPP_int_nat(number_number_of_nat,V_1)) = one_one_nat))) # label(fact_3268_Suc__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 930 (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_nat,Ya),Xa)) <-> Ya != Xa)) # label(fact_1848_linorder__neq__iff) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 931 (all A_121 all B_90 all 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) = 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))) # label(fact_1241_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 932 (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]. 8.65/8.74 933 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_real_real(sqrt,X) != zero_zero_real)) # label(fact_3530_real__sqrt__not__eq__zero) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 934 (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]. 8.65/8.74 935 (all N all A_3 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,cis(A_3)),cis(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),A_3))) = cis(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N))),A_3))) # label(fact_4153_cis__real__of__nat__Suc__mult) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 936 (all K_2 (is_int(K_2) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),pls) = K_2)) # label(fact_2351_diff__bin__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 937 (all X all N hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(real_int,X)),N) = hAPP_int_real(real_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N))) # label(fact_4392_real__of__int__power) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 938 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(div_mod_nat(M),N)),N)))) # label(fact_3178_mod__less__divisor) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 939 (all X_53 all P_7 all Q_8 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_53),P_7)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_53),Q_8)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_53),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_8))) # label(fact_1501_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 940 (all C_40 all A_97 all B_68 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_97),B_68)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_40)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_97),C_40)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_68),C_40)))))) # label(fact_1419_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 941 (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]. 8.65/8.74 942 (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]. 8.65/8.74 943 (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]. 8.65/8.74 944 (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]. 8.65/8.74 945 (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]. 8.65/8.74 946 (all M hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),hAPP_nat_nat(suc,zero_zero_nat)) = hAPP_nat_nat(suc,zero_zero_nat)) # label(fact_4342_gcd__Suc__0) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 947 (all P all Q P = hAPP_f957591787ol_nat(cOMBK_1643584600t_bool(P),Q)) # label(help_COMBK_1_1_COMBK_000tc__Nat__Onat_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 948 (all X_63 all Y_52 (X_63 != Y_52 -> (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_63),Y_52)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_52),X_63))))) # label(fact_312_linorder__neqE__linordered__idom) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 949 (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]. 8.65/8.74 950 (all A_123 A_123 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_123),one_one_complex)) # label(fact_1229_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 951 (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]. 8.65/8.74 952 (all X all N (-hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) -> X = hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N)))) # label(fact_3895_odd__real__root__power__cancel) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 953 (all C_31 all A_85 all B_58 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_31),A_85)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_31),B_58))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_31)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_85),B_58))))) # label(fact_1530_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 954 (all A_132 all B_94 all C_50 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_132),C_50)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_94),C_50)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_132),B_94)),C_50)) # label(fact_1152_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 955 (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]. 8.65/8.74 956 (all B_1_1 all B_2_1 is_bool(hAPP_P603027463t_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 957 (all Xa all F all G (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G)) -> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_int_bool(F,Xa)),hAPP_int_bool(G,Xa))))) # label(fact_1627_le__funE) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 958 (all A_44 all B_33 all K_6 (hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_33),K_6) = A_44 -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,B_33),A_44)))) # label(fact_2268_dvdI) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 959 (all X_36 all Y_30 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_36),Y_30)) -> X_36 != Y_30)) # label(fact_1793_order__less__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 960 (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]. 8.65/8.74 961 (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]. 8.65/8.74 962 (all N all M all K_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N) -> M = N))) # label(fact_2156_mult__left__cancel) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 963 (all C_56 all A_145 all B_103 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_56),A_145)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_56),B_103))) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_145),B_103)))) # label(fact_954_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 964 (all X all Y (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,X),Y)) | X != Y)) # label(help_fequal_2_1_fequal_000tc__Int__Oint_T) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 965 (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]. 8.65/8.74 966 (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]. 8.65/8.74 967 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,M),N))) # label(fact_5195_max__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 968 (all W_15 ((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))) & (zero_zero_nat = hAPP_int_nat(number_number_of_nat,W_15) -> one_one_rat = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),hAPP_int_nat(number_number_of_nat,W_15))))) # label(fact_41_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 969 (all P all Q all R hAPP_i1584592887nt_int(hAPP_i1778081398nt_int(P,R),hAPP_int_int(Q,R)) = hAPP_i1584592887nt_int(hAPP_f1574055885nt_int(hAPP_f434654626nt_int(cOMBS_2011279539nt_int,P),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]. 8.65/8.74 970 (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]. 8.65/8.74 971 (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]. 8.65/8.74 972 (all N all M zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M))) # label(fact_2505_diff__add__0) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 973 (all X all A_3 all B_2 (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) = one_one_int -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X),B_2)) -> hAPP_int_int(abs_abs_int,X) = one_one_int)))) # label(fact_5015_coprime__common__divisor__int) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 974 (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)) -> big_co1548731110nt_int(cOMBI_int,comple219730294t_bool(setS(A_1,P_5))) = hAPP_int_int(zfact,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)))))) # label(fact_4814_Union__SetS__setprod__prop2) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 975 (all Xa all K (Xa = 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) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd)))) # label(fact_2951_zOddI) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 976 (all X_52 all N_25 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_52),N_25)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_52),N_25)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,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))) # label(fact_1509_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 977 (all A_162 all B_110 all C_64 (is_int(B_110) & is_int(C_64) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_162),B_110) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_162),C_64) -> C_64 = B_110))) # label(fact_795_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 978 (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]. 8.65/8.74 979 (all Y_15 all X_19 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_15),X_19)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_19),Y_15)))) # label(fact_1984_leD) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 980 (all X_9 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_9),X_9)),one_one_complex) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_9),one_one_complex)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_9),one_one_complex))) # label(fact_2434_real__squared__diff__one__factored) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 981 (all Xa all Ya (Ya = Xa -> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Ya)))) # label(fact_1695_order__eq__refl) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 982 (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]. 8.65/8.74 983 (all I all M hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),I))))) # label(fact_3049_less__add__Suc2) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 984 (all A all K (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_n1699378549t_bool(ord_at4362885an_nat(K),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),hAPP_f22106695ol_nat(finite_card_nat,A))))) -> hAPP_n1699378549t_bool(ord_at4362885an_nat(K),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),hAPP_f22106695ol_nat(finite_card_nat,A))) = A)) # label(fact_4908_subset__card__intvl__is__intvl) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 985 (all Na (Na = zero_zero_nat <-> 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]. 8.65/8.74 986 (all M_14 all A_90 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,M_14),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_90),M_14)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_90),one_on1645066479umeral)),M_14)) # label(fact_1480_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 987 (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,R_2),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_4)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q_5))))))) # label(fact_2066_unique__quotient__lemma__neg) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 988 (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]. 8.65/8.74 989 (all X_52 all N_25 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_52),N_25)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_52),N_25)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,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))) # label(fact_1514_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 990 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Xa)),one_one_real)) -> hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)))),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,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)))))))) # label(fact_4826_summable__arctan__series) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 991 (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_real,Xa),Ya)) -> (exists X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),X_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Ya)) & hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),field_1210416355s_real))))))) # label(fact_4627_Rats__dense__in__nn__real) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 992 (all A_125 A_125 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_125),one_one_complex)) # label(fact_1215_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 993 (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),B_109)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_63),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))) # label(fact_798_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 994 (all Z_6 all X_13 all Y_10 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_13),Y_10)) -> (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_13),Z_6)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_13),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Y_10),Z_6)))))) # label(fact_2217_dvd__diff) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 995 (all M hAPP_nat_int(semiri1621563631at_int,M) = hAPP_int_int(abs_abs_int,hAPP_nat_int(semiri1621563631at_int,M))) # label(fact_2702_abs__int__eq) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 996 (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]. 8.65/8.74 997 (all Y_2 all X_2 (hBOOL(vanishes(X_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,minus_minus_rat),X_2)),Y_2)))))) # label(fact_5141_vanishes__diff) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 998 (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]. 8.65/8.74 999 (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]. 8.65/8.74 1000 (all Na all Xa (zero_zero_real != Xa -> (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))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,Xa))) & 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)))))))) # label(fact_4924_Maclaurin__exp__lt) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 1001 (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]. 8.65/8.74 1002 (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]. 8.65/8.74 1003 (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]. 8.65/8.74 1004 (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]. 8.65/8.74 1005 (all U all Ma all Na all J all I_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,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))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J)),U)),Ma)),Na))))) # label(fact_2549_nat__less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 1006 (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_lessThan_nat,K_1)))))) # label(fact_4954_finite__nat__bounded) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 1007 (all X1_2 all X2 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (hAPP_int_int(standardRes(Ma),X1_2) = hAPP_int_int(standardRes(Ma),X2) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X1_2),X2),Ma))))) # label(fact_3122_StandardRes__prop2) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 1008 (all C_10 all A_41 all B_30 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_41),B_30)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_41),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_30),C_10))))) # label(fact_2283_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 8.65/8.74 1009 (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]. 8.65/8.75 1010 (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]. 8.65/8.75 1011 (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]. 8.65/8.75 1012 (all Z_2 all X_2 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Z_2),X_2)) -> ((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)))) -> comple124823625p_real(X_2) = Z_2))) # label(fact_4642_Sup__eq__maximum) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1013 (all K_3 all K all F ((all N_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,K),hAPP_nat_real(F,N_1))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_1)),K_3)))) -> hBOOL(bseq_real(F)))) # label(fact_4947_BseqI2) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1014 (all N all M -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,N)),hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,M))))) # label(fact_3477_not__int__zless__negative) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1015 (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]. 8.65/8.75 1016 (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]. 8.65/8.75 1017 (all Xa all A_1 (Xa = A_1 <-> zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa),hAPP_real_real(uminus_uminus_real,A_1)))) # label(fact_3430_real__add__minus__iff) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1018 (all Xa all Ya (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ya),Xa)))) # label(fact_2042_linorder__not__le) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1019 (all F all L all C (hBOOL(tendsto_real_real(F,L,at_real(C))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,L),zero_zero_real)) -> (exists R_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1)) & (all X_1 (X_1 != C & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),X_1))),R_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,X_1)),zero_zero_real))))))))) # label(fact_5085_LIM__fun__less__zero) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1020 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y)))))) # label(fact_3584_arcsin__lbound) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1021 (all P_1 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)) -> hBOOL(P_1)))) # label(fact_1784_order__less__imp__triv) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1022 (all A_151 (is_int(A_151) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_151),zero_zero_int) = A_151)) # label(fact_913_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1023 (all Z_1 (is_int(Z_1) -> (-hBOOL(hAPP_int_bool(nat_neg,Z_1)) -> Z_1 = hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,Z_1))))) # label(fact_3253_not__neg__nat) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1024 (all C_18 all A_68 all B_45 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_68),B_45)) -> (B_45 = C_18 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_68),C_18))))) # label(fact_1753_ord__less__eq__trans) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1025 (all M all N (one_one_int = hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N)) -> hAPP_int_int(abs_abs_int,M) = one_one_int)) # label(fact_2740_abs__zmult__eq__1) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1026 (all A_1 (is_int(A_1) -> (A_1 = zero_zero_int <-> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),A_1) = zero_zero_int))) # label(fact_130_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1027 (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]. 8.65/8.75 1028 (all D2 all D1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D2)) -> (exists E (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,E),D1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,E),D2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),E))))))) # label(fact_3948_real__lbound__gt__zero) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1029 (all D_1 all A_1 all B_1 (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_1),B_1) = one_one_int & one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_1),A_1) <-> one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),B_1)))) # label(fact_4998_coprime__mul__eq__int) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1030 (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]. 8.65/8.75 1031 (all Xa all Ya (zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,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 = Ya & zero_zero_rat = Xa)) # label(fact_2_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1032 (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)) -> (hAPP_int_int(legacy_zgcd(M),N) = K_2 -> (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]. 8.65/8.75 1033 (all Xa all Ya (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Ya)) -> Xa != Ya)) # label(fact_1819_less__imp__neq) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1034 (all A_24 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_24),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))) # label(fact_2672_abs__mult__self) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1035 (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]. 8.65/8.75 1036 (all V_2 ((-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,zero_zero_nat)),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_nat(nat,hAPP_int_int(div_mod_int(one_one_int),hAPP_int_int(number_number_of_int,V_2)))) & (hBOOL(hAPP_int_bool(nat_neg,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)) = hAPP_nat_nat(suc,zero_zero_nat)))) # label(fact_3224_one__mod__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1037 (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]. 8.65/8.75 1038 (all B_116 all A_170 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_170)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_116)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_170),B_116)))))) # label(fact_669_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1039 (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]. 8.65/8.75 1040 (all X (hBOOL(hAPP_nat_bool(even_odd_even_nat,X)) -> hAPP_nat_nat(div_mod_nat(X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = zero_zero_nat)) # label(fact_3804_even__nat__mod__two__eq__zero) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1041 (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]. 8.65/8.75 1042 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) -> N = hAPP_nat_nat(suc,zero_zero_nat) | zero_zero_nat = N)) # label(fact_3153_less__2__cases) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1043 (all K_2 all N all M hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N)),M)),N) = hAPP_nat_nat(div_mod_nat(M),N)) # label(fact_3182_mod__mult__self3) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1044 (all X (-hBOOL(hAPP_nat_bool(even_odd_even_nat,X)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,X)),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = hAPP_nat_nat(suc,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_3801_odd__nat__plus__one__div__two) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1045 (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]. 8.65/8.75 1046 (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]. 8.65/8.75 1047 (all C_17 all A_67 all B_44 (A_67 = B_44 -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_17),B_44)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_17),A_67))))) # label(fact_1761_xt1_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1048 (all P all Q all R hAPP_P1668131738nt_int(hAPP_f167782601nt_int(hAPP_f400538607nt_int(cOMBB_866192175nt_int,P),Q),R) = hAPP_i61245889nt_int(P,hAPP_P1175774780nt_int(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]. 8.65/8.75 1049 (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]. 8.65/8.75 1050 (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_eq_real,zero_zero_real),hAPP_nat_real(F,M_1))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na))),hAPP_f352196356l_real(suminf_real,F)))))) # label(fact_4929_series__pos__le) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1051 (all P all Q (is_bool(P) -> hAPP_nat_bool(cOMBK_bool_nat(P),Q) = P)) # label(help_COMBK_1_1_COMBK_000tc__HOL__Obool_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1052 (all C_69 all D_20 all A_177 all B_121 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_177),B_121)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_69),D_20)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_177),C_69)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_121),D_20)))))) # label(fact_606_add__less__le__mono) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1053 (all P all Q all R hAPP_r224952523l_real(hAPP_f1663701695l_real(hAPP_f774948957l_real(cOMBB_3698607l_real,P),Q),R) = hAPP_f1908301369l_real(P,hAPP_r1230691443l_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__fun_Itc__prod_Itc__Real) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1054 (all X all Y hAPP_int_int(int_lcm(X),Y) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(nat_lcm(hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,X))),hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,Y))))) # label(fact_4623_Nitpick_Oint__lcm__def) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1055 (all X_58 all Y_48 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_58),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_48),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_997_sum__power2__ge__zero) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1056 (all A_1 all B_1 (B_1 = A_1 <-> zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_1),B_1))) # label(fact_2321_eq__iff__diff__eq__0) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1057 (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]. 8.65/8.75 1058 (all U_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),U_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),U_1)))) # label(fact_3549_lemma__real__divide__sqrt__less) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1059 (all Z_1 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Z_1),hAPP_complex_complex(cnj,Z_1)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_complex_real(im,Z_1)))),ii)) # label(fact_4089_complex__diff__cnj) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1060 (all P all Q all R hAPP_r265291036omplex(P,hAPP_nat_real(Q,R)) = hAPP_n1983812447omplex(hAPP_f23851874omplex(hAPP_f406530075omplex(cOMBB_1714920583ex_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__C) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1061 (all X all Y (is_int(Y) -> hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,fFalse),X),Y) = Y)) # label(help_If_2_1_If_000tc__Int__Oint_T) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1062 (all C_19 all B_46 all A_69 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_46),A_69)) -> (C_19 = B_46 -> 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]. 8.65/8.75 1063 (all Xa all Na (hBOOL(hAPP_int_bool(even_odd_even_int,Xa)) & Na != zero_zero_nat <-> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Xa),Na))))) # label(fact_3251_even__power) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1064 (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]. 8.65/8.75 1065 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N))) # label(fact_2522_le__add__diff__inverse2) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1066 (all X all Y all Z_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),Z_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Z_1))) # label(fact_4304_gcd__semilattice__nat_Oinf__left__commute) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1067 (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]. 8.65/8.75 1068 (all Xa all Ya all Z_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),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),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Ya),Z_2))))) # label(fact_4316_gcd__semilattice__nat_Ole__inf__iff) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1069 (all A_1 all Wa (A_1 = zero_zero_rat & zero_zero_nat != hAPP_int_nat(number_number_of_nat,Wa) <-> zero_zero_rat = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_1),hAPP_int_nat(number_number_of_nat,Wa)))) # label(fact_180_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1070 (all A_3 all B_2 all C_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),C_1))) # label(fact_2832_zdiv__zmult2__eq) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1071 (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]. 8.65/8.75 1072 (all A_53 all N_20 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_53),N_20)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_53),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_20))) # label(fact_2136_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1073 (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]. 8.65/8.75 1074 (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]. 8.65/8.75 1075 (all Q_1 -(all A_2 all B_4 (is_int(A_2) & is_int(B_4) -> (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)) -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_4) != one_one_int))))) # label(fact_5026_Rat__cases) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1076 (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]. 8.65/8.75 1077 (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]. 8.65/8.75 1078 (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(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_5053_summable__Leibniz_H_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1079 (all 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)) = 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)))) # label(fact_3159_one__div__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1080 (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]. 8.65/8.75 1081 (all B_132 all A_189 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_189),zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_132),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_189),B_132)),zero_zero_nat))))) # label(fact_498_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1082 (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]. 8.65/8.75 1083 (all Wa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,Wa)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),zero_zero_int)))) # label(fact_115_bin__less__0__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1084 (all Xa all Ya (is_int(Ya) & is_int(Xa) -> (hAPP_int_complex(number528085621omplex,Ya) = hAPP_int_complex(number528085621omplex,Xa) <-> Xa = Ya))) # label(fact_85_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1085 (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]. 8.65/8.75 1086 (all P all Q all R hAPP_f448129468l_bool(P,hAPP_r481142414t_bool(Q,R)) = hAPP_real_bool(hAPP_f101635158l_bool(hAPP_f1925507189l_bool(cOMBB_1121934354l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obo_006) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1087 (all A_163 all B_111 all C_65 (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_163),B_111) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_163),C_65) -> B_111 = C_65)) # label(fact_784_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1088 (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]. 8.65/8.75 1089 (all X all Y hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)),X))) # label(fact_4312_gcd__semilattice__nat_Oinf__le1) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1090 (all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(suc,zero_zero_nat))))) # label(fact_4181_prime__nd__one) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1091 (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]. 8.65/8.75 1092 (all C_20 all A_70 all B_47 (A_70 = B_47 -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_47),C_20)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_70),C_20))))) # label(fact_1683_ord__eq__le__trans) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1093 (all N_58 hAPP_int_rat(number_number_of_rat,hAPP_nat_int(semiri1621563631at_int,N_58)) = hAPP_nat_rat(semiri151668891at_rat,N_58)) # label(fact_203_number__of__int) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1094 (all L_2 all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),L_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_2),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_2),M)))))) # label(fact_2481_diff__less__mono2) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1095 (all A_3 all B_2 ((zero_zero_int = hAPP_int_int(div_mod_int(A_3),B_2) -> zero_zero_int = hAPP_int_int(div_mod_int(A_3),hAPP_int_int(uminus_uminus_int,B_2))) & (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(div_mod_int(A_3),B_2)),B_2) = hAPP_int_int(div_mod_int(A_3),hAPP_int_int(uminus_uminus_int,B_2))))) # label(fact_3491_zmod__zminus2__eq__if) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1096 (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]. 8.65/8.75 1097 (all Ya all Xa (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ya),Xa)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Ya)) <-> Ya = Xa))) # label(fact_1686_order__antisym__conv) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1098 (all Xa (hAPP_real_real(sin,Xa) = zero_zero_real <-> (exists N_1 (Xa = 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))))) & hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1)))) | (exists N_1 (Xa = hAPP_real_real(uminus_uminus_real,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)))))) & hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1)))))) # label(fact_3783_sin__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1099 (all P_3 hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3)) = frct(P_3)) # label(fact_4711_Frct__def) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1100 (all P all Q all R hAPP_i1836591008nt_int(hAPP_f400200275nt_int(hAPP_f1731308757nt_int(cOMBB_735549356nt_int,P),Q),R) = hAPP_f151095827nt_int(P,hAPP_i1216375074nt_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__p) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1101 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(ln,X)),zero_zero_real))))) # label(fact_3361_ln__less__zero) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1102 (all N all D_3 (is_int(D_3) -> (zero_zero_int != D_3 -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_3),N)) -> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,N)),hAPP_int_real(real_int,D_3)) = hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N),D_3)))))) # label(fact_4449_real__of__int__div) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1103 (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]. 8.65/8.75 1104 (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),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),P_3)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),one_one_int),P_3)))))) # label(fact_2469_zcong__square) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1105 (all X natceiling(X) = hAPP_int_nat(nat,archim856651990g_real(X))) # label(fact_4363_natceiling__def) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1106 (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]. 8.65/8.75 1107 (all L_2 hAPP_int_int(bit0,L_2) != min) # label(fact_2165_rel__simps_I42_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1108 (all N all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,M),N)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),M))))) # label(fact_2401_zdvd__not__zless) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1109 (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]. 8.65/8.75 1110 (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]. 8.65/8.75 1111 (all Ma all D_1 (is_int(Ma) -> ((exists Q_4 (is_int(Q_4) & Ma = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D_1),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]. 8.65/8.75 1112 (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]. 8.65/8.75 1113 (all A_51 all N_19 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_51),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_19))))) # label(fact_2146_zero__le__even__power_H) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1114 (all N N = hAPP_nat_nat(size_size_nat,N)) # label(fact_4058_nat__size) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1115 (all C_18 all A_68 all B_45 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_68),B_45)) -> (B_45 = C_18 -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_68),C_18))))) # label(fact_1752_ord__less__eq__trans) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1116 (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(bit1,K)),hAPP_int_int(bit0,L))))) # label(fact_112_rel__simps_I16_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1117 (all A_1 all B_1 (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))) <-> 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,zero_zero_int),A_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_eq_int,A_1),zero_zero_int)))) # label(fact_1288_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1118 (all B_1_1 is_bool(bseq_real(B_1_1))) # label(gsy_c_SEQ_OBseq_000tc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1119 (all Ya all Xa (is_int(Xa) & is_int(Ya) -> (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)) -> (Xa = Ya <-> hAPP_int_nat(nat,Ya) = hAPP_int_nat(nat,Xa)))))) # label(fact_2742_transfer__nat__int__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1120 (all B_133 all A_190 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_190)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_133)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_190),B_133)))))) # label(fact_494_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1121 (all P_1 all A_1 all B_1 (-(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B_1)) & -hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) | (exists D_2 (-hBOOL(hAPP_nat_bool(P_1,D_2)) & hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_1),D_2) = A_1))) <-> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A_1),B_1))))) # label(fact_2531_nat__diff__split__asm) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1122 (all A_40 all B_29 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_40),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_29),A_40)))) # label(fact_2293_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1123 (all N (-hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_3820_lemma__not__even__div2) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 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]. 8.65/8.75 1125 (all M all K_2 all N (one_one_int = hAPP_int_int(legacy_zgcd(K_2),N) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),N)),hAPP_int_int(legacy_zgcd(M),N))))) # label(fact_4515_zgcd__zmult__zdvd__zgcd) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1126 (all A_13 all B_13 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_13)),hAPP_rat_rat(abs_abs_rat,B_13))),hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B_13),A_13))))) # label(fact_2729_abs__triangle__ineq2__sym) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1127 (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,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_eq_int,zero_zero_int),A_1))))) # label(fact_2844_pos__imp__zdiv__nonneg__iff) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1128 (all M zero_zero_nat != hAPP_nat_nat(suc,M)) # label(fact_2996_Zero__neq__Suc) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1129 (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]. 8.65/8.75 1130 (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]. 8.65/8.75 1131 (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]. 8.65/8.75 1132 (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]. 8.65/8.75 1133 (all A_34 all B_26 (is_int(A_34) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_34),B_26)),B_26) = A_34)) # label(fact_2342_add__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1134 (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_eq_int,hAPP_int_int(bit0,K)),hAPP_int_int(bit0,L))))) # label(fact_560_rel__simps_I31_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1135 (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]. 8.65/8.75 1136 (all A_34 all B_26 A_34 = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_34),B_26)),B_26)) # label(fact_2340_add__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1137 (all Z_10 all Y_21 all X_25 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_21),X_25)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_10),Y_21)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_10),X_25))))) # label(fact_1925_xt1_I8_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1138 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,Xa)),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa),Ya))))) # label(fact_3437_real__0__le__add__iff) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1139 (all Z_13 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_13),Z_13) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_13),hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_1885_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1140 (all X_15 all Y_11 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_15),Y_11)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_15),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_11),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_15)),Y_11))) # label(fact_2052_power2__sum) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1141 (all A_9 all N_7 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,hAPP_rat_rat(abs_abs_rat,A_9)),N_7)))) # label(fact_2757_zero__le__power__abs) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1142 (all A_12 all B_12 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_12)),hAPP_real_real(abs_abs_real,B_12))),hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_12),B_12))))) # label(fact_2733_abs__triangle__ineq2) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1143 (all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2)) -> (exists K_1 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_1) = J_2 & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1)))))) # label(fact_474_less__imp__add__positive) # label(axiom) # label(non_clause). [assumption]. 8.65/8.75 1144 (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]. 8.65/8.76 1145 (all Xa all Ya (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Ya)) -> Ya != Xa)) # label(fact_1798_order__less__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1146 (all C_38 all A_95 all B_66 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_95),B_66)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_38)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_38),A_95)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_38),B_66)))))) # label(fact_1431_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1147 (all B_1_1 all B_2_1 is_int(hAPP_f1594865479ol_int(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__Int__Oint) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1148 (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_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),Na)))))) # label(fact_4760_bdd__int__set__l__l__finite) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1149 (all K_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),I)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),J_2))))) # label(fact_2104_mult__le__mono2) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1150 (all A_35 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_35),zero_zero_complex) = A_35) # label(fact_2330_diff__0__right) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1151 (all V_22 all W_12 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_22)),hAPP_int_complex(number528085621omplex,W_12)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_22),W_12))) # label(fact_196_number__of__add) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1152 (all M all K_2 all N (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,K_2),N) = one_one_nat -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M))))) # label(fact_4368_coprime__dvd__mult__nat) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1153 (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]. 8.65/8.76 1154 (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_real,Ya),Xa)))) # label(fact_1842_not__less__iff__gr__or__eq) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1155 (all X all Y all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),Y)))))) # label(fact_3897_real__root__le__mono) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1156 (all P all Q (-hBOOL(hAPP_bool_bool(fimplies(P),Q)) | hBOOL(Q) | -hBOOL(P))) # label(help_fimplies_3_1_U) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1157 (all A_1 all B_1 (zero_zero_complex = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_1),B_1) <-> B_1 = zero_zero_complex | zero_zero_complex = A_1)) # label(fact_1121_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1158 (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_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,one_on1684967323de_int),A_183)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_183),N_42)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_183),N_41)))))) # label(fact_551_power__increasing) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1159 (all A_147 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,zero_z891286103de_int),A_147) = A_147) # label(fact_940_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1160 (all N all M all R_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),R_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),R_2)) -> (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N) = one_one_int -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N)),R_2)))))) # label(fact_4962_divides__mult__int) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1161 (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,Xa),Ya)) <-> 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)))))) # label(fact_2122_real__mult__less__iff1) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1162 (all X X = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),zero_zero_nat)) # label(fact_4339_gcd__0__nat) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1163 (all A_3 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M)) -> (exists X_1 (is_int(X_1) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),M)) & (all Y_1 (is_int(Y_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),Y_1),M)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),M)) -> Y_1 = X_1))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),X_1),M)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)))))) # label(fact_2618_zcong__zless__unique) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1164 (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,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_3500_neg__even__power) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1165 (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_3238_Parity_Oeven__plus__odd) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1166 (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]. 8.65/8.76 1167 (all X all Y hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X),Y)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y))) # label(fact_2129_Nat__Transfer_Otransfer__int__nat__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1168 (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]. 8.65/8.76 1169 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),K)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),hAPP_int_int(bit0,K))))) # label(fact_114_rel__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1170 (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]. 8.65/8.76 1171 (all Z_4 all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_1),Z_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat,Z_1)),hAPP_int_nat(nat,Z_4)))) # label(fact_2793_nat__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1172 (all N hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,pi),hAPP_nat_real(real_nat,N))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),N)) # label(fact_3751_cos__npi2) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1173 (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_le1568362934t_bool,Z_2),Ya)) -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Z_2),Xa))))) # label(fact_1631_xt1_I6_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1174 (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]. 8.65/8.76 1175 (all Na all Ma (Ma = zero_zero_nat & zero_zero_nat = Na <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Na)),hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,Ma)))))) # label(fact_3485_int__zle__neg) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1176 (all X hAPP_complex_complex(uminus473333897omplex,X) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),X))) # label(fact_3973_complex__i__mult__minus) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1177 (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]. 8.65/8.76 1178 (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,Ya),Xa)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(tan,Ya)),hAPP_real_real(tan,Xa))))))))) # label(fact_3764_tan__monotone_H) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1179 (all B_125 all A_181 all C_72 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_72)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_125),A_181)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_125),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_181),C_72)))))) # label(fact_573_add__increasing2) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1180 (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]. 8.65/8.76 1181 (all Z_12 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_12),Z_12) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_12)) # label(fact_1892_mult__2) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1182 (all Xa hAPP_real_real(cos,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),cos_coeff)),hAPP_r474017924t_real(power_power_real,Xa)))) # label(fact_4820_cos__def) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1183 (all A_36 zero_zero_complex = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_36),A_36)) # label(fact_2326_diff__self) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1184 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> (M != N -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))))) # label(fact_1259_le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1185 (all P all Q all R hAPP_i998473677nt_int(hAPP_f142232355nt_int(hAPP_f1377453843nt_int(cOMBB_320287386nt_int,P),Q),R) = hAPP_f2002277285nt_int(P,hAPP_i1584592887nt_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1186 (all M zero_zero_nat != hAPP_nat_nat(suc,M)) # label(fact_2994_Suc__not__Zero) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1187 (all A_3 all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,X)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),A_3)))))) # label(fact_4376_ln__powr__bound) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1188 (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]. 8.65/8.76 1189 (all N hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,N)),one_one_real) = hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N))) # label(fact_3715_real__of__nat__Suc) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1190 (all C_39 all A_96 all B_67 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_96),B_67)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_39)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_39),A_96)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_39),B_67)))))) # label(fact_1429_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1191 (all Xa all Ya (is_int(Ya) & is_int(Xa) -> (zero_zero_int = Xa & Ya = zero_zero_int <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Xa),Xa)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ya),Ya))),zero_zero_int))))) # label(fact_1586_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1192 (all Y all X (zero_zero_real != hAPP_real_real(cos,X) -> (zero_zero_real != hAPP_real_real(cos,Y) -> hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(tan,X)),hAPP_real_real(tan,Y))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y)))))) # label(fact_3760_lemma__tan__add1) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1193 (all Q_1 ((zero_zero_rat = Q_1 -> zero_zero_rat = hAPP_rat_rat(sgn_sgn_rat,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))))) # label(fact_4635_sgn__rat__def) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1194 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K1)),hAPP_int_int(bit1,K2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)))) # label(fact_74_less__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1195 (all I_1 all J all 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)) <-> 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))))) # label(fact_2509_less__diff__conv) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1196 (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]. 8.65/8.76 1197 (all A_130 all B_92 all C_48 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_130),C_48)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_92),C_48)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_130),B_92)),C_48)) # label(fact_1177_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1198 (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]. 8.65/8.76 1199 (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,A_1),X_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),B_1)) -> (exists Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_1)) & hBOOL(hAPP_real_bool(deriv_real(F,X_1),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]. 8.65/8.76 1200 (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]. 8.65/8.76 1201 (all R all S_4 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,R),S_4) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R),hAPP_real_real(inverse_inverse_real,S_4))) # label(fact_3853_real__divide__def) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1202 (all N archim1246769320r_real(hAPP_nat_real(real_nat,N)) = hAPP_nat_int(semiri1621563631at_int,N)) # label(fact_4019_floor__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1203 (all A_147 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,zero_z126310315umeral),A_147) = A_147) # label(fact_945_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1204 (all Z_1 (is_int(Z_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> -(all M_1 Z_1 != hAPP_nat_int(semiri1621563631at_int,M_1))))) # label(fact_1011_nonneg__eq__int) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1205 (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]. 8.65/8.76 1206 (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]. 8.65/8.76 1207 (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]. 8.65/8.76 1208 (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_3795_even__nat__def) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1209 (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]. 8.65/8.76 1210 (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]. 8.65/8.76 1211 (all B_1 all A_1 (zero_zero_complex = A_1 <-> hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_1),A_1) = B_1)) # label(fact_904_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1212 (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]. 8.65/8.76 1213 (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)) -> hAPP_nat_nat(suc,N) = M))) # label(fact_3023_le__SucE) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1214 (all P_1 all Na ((exists X_1 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na))) & hBOOL(hAPP_nat_bool(P_1,X_1)))) <-> (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]. 8.65/8.76 1215 (all Z_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),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))))) # label(fact_3139_one__less__nat__eq) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1216 (all P_1 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_nat,Ya),Xa)) -> hBOOL(P_1)))) # label(fact_1788_order__less__imp__triv) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1217 (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]. 8.65/8.76 1218 (all R_2 of_real_complex(R_2) = hAPP_real_complex(hAPP_r265291036omplex(complex,R_2),zero_zero_real)) # label(fact_3984_complex__of__real__def) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1219 (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]. 8.65/8.76 1220 (all A_137 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_137),zero_z126310315umeral) = zero_z126310315umeral) # label(fact_1116_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1221 (all X_57 X_57 = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_57),one_one_nat)) # label(fact_1001_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1222 (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]. 8.65/8.76 1223 (all Diff all Na all Xa (zero_zero_real = Xa -> (Na != zero_zero_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_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_n546711566l_real(Diff,zero_zero_nat),zero_zero_real)))) # label(fact_4907_Maclaurin__zero) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1224 (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_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)))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2)) -> posDivAlg(A_3,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),A_3)))) # label(fact_3291_posDivAlg__eqn) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1225 (all A_120 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_120),A_120)))) # label(fact_1284_zero__le__square) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1226 (all Ta all B all D_1 all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),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)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa_1) != X_1)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Ta))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),D)),Ta))))))))) # label(fact_5103_bset_I9_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1227 (all A_156 all C_58 all D_17 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_156),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_58),D_17)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_58),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_156),D_17))) # label(fact_840_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1228 (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]. 8.65/8.76 1229 (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,one_one_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N_4),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),X))))))) # label(fact_3921_real__root__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1230 (all P all Q all R hAPP_f237669397t_bool(P,hAPP_P989584703t_bool(Q,R)) = hAPP_P989584703t_bool(hAPP_f1781059817t_bool(hAPP_f326728547t_bool(cOMBB_2086486755nt_int,P),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]. 8.65/8.76 1231 (all Y all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Y)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,natfloor(X)),Y) = natfloor(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_nat_real(real_nat,Y)))))) # label(fact_3755_natfloor__div__nat) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1232 (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]. 8.65/8.76 1233 (all A_28 all B_20 all V_5 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_28),B_20)),hAPP_int_rat(number_number_of_rat,V_5)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_28),hAPP_int_rat(number_number_of_rat,V_5))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_20),hAPP_int_rat(number_number_of_rat,V_5)))) # label(fact_2397_left__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1234 (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]. 8.65/8.76 1235 (all Y_14 all X_18 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_14),X_18)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_18),Y_14)))) # label(fact_2007_not__leE) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1236 (all N all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_real_real(ln,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_real_real(ln,X)))) # label(fact_3737_ln__realpow) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1237 (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_eq_int),Na)))))) # label(fact_4741_bdd__int__set__le__finite) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1238 (all X_30 all Y_24 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_24),X_30)) | Y_24 = X_30 | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_30),Y_24)))) # label(fact_1835_linorder__less__linear) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1239 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N))) # label(fact_4358_gcd__diff1__nat) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1240 (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))) -> (zero_zero_int != hAPP_P1175774780nt_int(product_snd_int_int,P_3) -> normalize(P_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3))))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_P1175774780nt_int(product_snd_int_int,P_3)),hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,P_3)),hAPP_P1175774780nt_int(product_snd_int_int,P_3)))))) & (zero_zero_int = hAPP_P1175774780nt_int(product_snd_int_int,P_3) -> normalize(P_3) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int))) & (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]. 8.65/8.76 1241 (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),Rx)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ly),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))) # label(fact_1085_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1242 (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]. 8.65/8.76 1243 (all X_31 all Y_25 (X_31 != Y_25 -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_31),Y_25)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_25),X_31))))) # label(fact_1828_linorder__neqE) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1244 (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]. 8.65/8.76 1245 (all P all Q all R hAPP_int_bool(hAPP_f41603166t_bool(hAPP_f389116883t_bool(cOMBB_1593174431ol_int,P),Q),R) = hAPP_f54304608l_bool(P,hAPP_i418383825t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obo) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1246 (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(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)))) = 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)))))))) # label(fact_3379_even__sum__div__2) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1247 (all A_157 all C_59 all 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)) = 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)) # label(fact_837_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1248 (all N_8 hAPP_int_int(abs_abs_int,hAPP_nat_int(semiri1621563631at_int,N_8)) = hAPP_nat_int(semiri1621563631at_int,N_8)) # label(fact_2701_abs__of__nat) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1249 (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]. 8.65/8.76 1250 (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]. 8.65/8.76 1251 (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]. 8.65/8.76 1252 (all A_3 all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N)) -> (one_one_int = hAPP_int_int(legacy_zgcd(A_3),N) -> (exists X_1 (is_int(X_1) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),X_1)),one_one_int),N))))))) # label(fact_4533_zcong__lineq__ex) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1253 (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]. 8.65/8.76 1254 (all X_21 all Y_17 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_21),Y_17)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_21),Y_17)) | X_21 = Y_17)) # label(fact_1961_order__le__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1255 (all A_128 (is_int(A_128) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),A_128) = A_128)) # label(fact_1198_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1256 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),pi)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sin,X)))))) # label(fact_3610_sin__gt__zero__pi) # label(axiom) # label(non_clause). [assumption]. 8.65/8.76 1257 (all C_75 all D_22 all A_186 all B_129 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_186),B_129)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_75),D_22)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_186),C_75)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_129),D_22)))))) # label(fact_528_add__mono) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1258 (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]. 8.65/8.77 1259 (all B_1 adjust(B_1) = produc1518849193nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,hAPP_f617310012nt_int(hAPP_f197974549nt_int(cOMBB_638531587nt_int,cOMBS_1891347093nt_int),hAPP_f55007289nt_int(hAPP_f747475781nt_int(cOMBB_222539774nt_int,hAPP_f1311564564nt_int(cOMBS_1640193733nt_int,hAPP_f251264923nt_int(hAPP_f279307923nt_int(cOMBB_662120866nt_int,if_Pro1731782967nt_int),hAPP_f429935748t_bool(hAPP_f1512942609t_bool(cOMBB_int_bool_int,hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int)),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,minus_minus_int),B_1))))),hAPP_f1574055885nt_int(hAPP_f960630521nt_int(cOMBC_1805044090nt_int,hAPP_f142232355nt_int(hAPP_f1377453843nt_int(cOMBB_320287386nt_int,cOMBB_1846624359nt_int),hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,plus_plus_int),hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_int)))),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,minus_minus_int),B_1))))),hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_4840_adjust__def) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1260 (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]. 8.65/8.77 1261 (all P all Q all R hAPP_i1796122964t_bool(hAPP_f1902329768t_bool(hAPP_f1182713365t_bool(cOMBB_2087181719ol_int,P),Q),R) = hAPP_f561022312t_bool(P,hAPP_i614188481l_bool(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]. 8.65/8.77 1262 (all R_2 all A_3 rcis(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),R_2),hAPP_real_real(uminus_uminus_real,A_3)) = hAPP_complex_complex(invers1449016382omplex,rcis(R_2,A_3))) # label(fact_4266_rcis__inverse) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1263 (all X (-hBOOL(hAPP_int_bool(even_odd_even_int,X)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),one_one_int) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3282_odd__plus__one__div__two) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1264 (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]. 8.65/8.77 1265 (all A_1 all B_1 (is_int(B_1) & is_int(A_1) -> (B_1 = A_1 <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),B_1),zero_zero_int))))) # label(fact_2573_IntPrimes_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1266 (all B_134 all C_78 all A_196 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_196)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_134),C_78)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_134),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_196),C_78)))))) # label(fact_391_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1267 (all L_2 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,pls),hAPP_int_int(bit1,L_2)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),L_2))) # label(fact_2443_diff__bin__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1268 (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_2) <-> 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))))) # label(fact_2050_conj__le__cong) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1269 (all C_37 all B_65 all A_94 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_65),A_94)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_37),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_94),C_37)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_65),C_37)))))) # label(fact_1438_mult__strict__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1270 (all A_33 all B_25 A_33 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_33),B_25)),B_25)) # label(fact_2343_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1271 (all A_201 -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_201),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),zero_zero_rat))) # label(fact_211_power2__less__0) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1272 (all Z_1 all W hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),hAPP_int_int(uminus_uminus_int,W)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_1),W)) # label(fact_3478_diff__int__def__symmetric) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1273 (all A_1 (A_1 = zero_zero_nat <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_1)))) # label(fact_2960_gcd__lcm__complete__lattice__nat_Otop__unique) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1274 (all N all F_3 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,F_3),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,F_3),N)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,F_3),hAPP_int_int(div_mod_int(M),N)))))) # label(fact_3003_zdvd__zmod) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1275 (all C_45 all A_112 all B_82 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_112),B_82)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_45)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_45),A_112)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_45),B_82)))))) # label(fact_1332_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1276 (all B_2 posDivAlg(zero_zero_int,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int)) # label(fact_3293_posDivAlg__0) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1277 (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]. 8.65/8.77 1278 (all P (is_bool(P) -> P = fTrue | P = fFalse)) # label(help_fTrue_1_1_T) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1279 (all Wa all Xa (hAPP_int_complex(number528085621omplex,Wa) = Xa <-> hAPP_int_complex(number528085621omplex,Wa) = Xa)) # label(fact_90_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1280 (all M_6 all N_10 ((zero_zero_nat = N_10 -> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,M_6),N_10) = one_one_rat) & (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)))))) # label(fact_2607_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1281 (all B_75 all A_104 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_104)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_75)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_104),B_75)))))) # label(fact_1379_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1282 (all Na all Ma (hBOOL(hAPP_int_bool(nat_is_nat,Ma)) -> (hBOOL(hAPP_int_bool(nat_is_nat,Na)) -> hAPP_i1948725293t_bool(ord_at875362053st_int(Ma),Na) = image_nat_int(semiri1621563631at_int,hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_int_nat(nat,Ma)),hAPP_int_nat(nat,Na)))))) # label(fact_5127_SetInterval_Otransfer__int__nat__set__functions) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1283 (all A_146 A_146 = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,zero_z126310315umeral),A_146)) # label(fact_952_add__0__left) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1284 (all Q_2 all P_1 ((all A_2 all B_4 (is_int(B_4) & is_int(A_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_4)) -> hBOOL(hAPP_rat_bool(P_1,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_4)))))) -> hBOOL(hAPP_rat_bool(P_1,Q_2)))) # label(fact_4728_Rat__induct__pos) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1285 (all P_1 all Ma all Na ((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Na),Ma))) -> ((Ma = Na -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Na),Ma))) -> ((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),Ma)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Na),Ma))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Na),Ma)))))) # label(fact_297_nat__less__cases) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1286 (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]. 8.65/8.77 1287 (all P_1 collec1979865426at_nat(P_1) = P_1) # label(fact_145_Collect__def) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1288 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = X)) # label(fact_3546_real__sqrt__pow2) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1289 (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)) -> Y_51 = X_61)))) # label(fact_886_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1290 (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]. 8.65/8.77 1291 (all A_17 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_17)) -> hAPP_real_real(abs_abs_real,A_17) = A_17)) # label(fact_2715_abs__of__pos) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1292 (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_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_18)),hAPP_int_nat(number_number_of_nat,V_19)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_18),V_19))))) # label(fact_867_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1293 (all P all Q all R hAPP_r285062775l_bool(hAPP_f834119977l_bool(hAPP_f848516149l_bool(cOMBB_270132509l_real,P),Q),R) = hAPP_f252743847l_bool(P,hAPP_r373117745t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Nat__Onat_Mtc__HOL__035) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1294 (all P (is_bool(P) -> fTrue = P | P = fFalse)) # label(help_fFalse_1_1_T) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1295 (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]. 8.65/8.77 1296 (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]. 8.65/8.77 1297 (all Xa (is_int(Xa) -> (zero_zero_real = hAPP_int_real(real_int,Xa) <-> zero_zero_int = Xa))) # label(fact_4378_real__of__int__zero__cancel) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1298 (all M_18 all N_30 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(semiri2020571505omplex,M_18)),hAPP_nat_complex(semiri2020571505omplex,N_30)) = hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_18),N_30))) # label(fact_1102_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1299 (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]. 8.65/8.77 1300 (all P all Q (hBOOL(P) | hBOOL(Q) | -hBOOL(fdisj(P,Q)))) # label(help_fdisj_3_1_U) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1301 (all B all A (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A)) -> (hBOOL(hAPP_f448129468l_bool(nat_nat_set,B)) -> (A = B <-> image_int_nat(nat,A) = image_int_nat(nat,B))))) # label(fact_4613_transfer__int__nat__set__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1302 (all B_124 all C_71 all A_180 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_180)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_124),C_71)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_124),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_180),C_71)))))) # label(fact_582_add__increasing) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1303 (all A_137 zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_137),zero_zero_int)) # label(fact_1119_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1304 (all P all Q all R hAPP_i1584592887nt_int(hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,P),Q),R) = hAPP_i1584592887nt_int(P,hAPP_int_int(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]. 8.65/8.77 1305 (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]. 8.65/8.77 1306 (all N_57 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,N_57)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_57)))) # label(fact_234_of__nat__less__two__power) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1307 (all Z_9 all X_24 all Y_20 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_24),Y_20)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_20),Z_9)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_24),Z_9))))) # label(fact_1932_order__le__less__trans) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1308 (all N hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),hAPP_nat_nat(fact_fact_nat,N))) # label(fact_3843_fact__plus__one__nat) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1309 (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]. 8.65/8.77 1310 (all L_2 hAPP_int_int(bit1,L_2) != pls) # label(fact_132_rel__simps_I39_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1311 (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_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_192)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_192),N_44)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_192),N_43)))))) # label(fact_442_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1312 (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,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_16),W_8))),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))) # label(fact_1091_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1313 (all N all A_3 all B_2 (hAPP_int_int(legacy_zgcd(A_3),B_2) = one_one_int -> hAPP_int_int(legacy_zgcd(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),N)) = one_one_int)) # label(fact_4503_zgcd__1__power__distrib) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1314 (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]. 8.65/8.77 1315 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K)),pls)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),pls)))) # label(fact_113_rel__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1316 (all A_1 all B_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_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_eq_real,A_1),B_1)))) # label(fact_2376_le__iff__diff__le__0) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1317 (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]. 8.65/8.77 1318 (all N_34 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(semiri132038758t_real,N_34)))) # label(fact_716_of__nat__0__le__iff) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1319 (all A_32 all C_8 all B_24 all D_7 hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_32),C_8)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_24),D_7)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_32),B_24)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C_8),D_7))) # label(fact_2350_add__diff__add) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1320 (all A_3 (hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_one_int -> hAPP_int_int(abs_abs_int,A_3) = one_one_int)) # label(fact_2807_power2__eq1__iff) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1321 (all D_1 all K all P_5 (hBOOL(hAPP_nat_bool(prime,P_5)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_5),K))) <-> (exists I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),K)) & hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_5),I_2) = D_1))))) # label(fact_4184_divides__primepow) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1322 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,X)),X)))) # label(fact_3385_ln__bound) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1323 (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]. 8.65/8.77 1324 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(div_mod_int(B_2),A_3))) = 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_3163_pos__zmod__mult__2) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1325 (all Xa all Ya (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya)) -> Ya != Xa)) # label(fact_1820_less__imp__neq) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1326 (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]. 8.65/8.77 1327 (all B_1_1 is_bool(trivial_limit_nat(B_1_1))) # label(gsy_c_Limits_Otrivial__limit_000tc__Nat__Onat) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1328 (all X_21 all Y_17 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_21),Y_17)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_21),Y_17)) | Y_17 = X_21)) # label(fact_1966_order__le__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1329 (all N all M hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)))) # label(fact_1265_le__add2) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1330 (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]. 8.65/8.77 1331 (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]. 8.65/8.77 1332 (all X all N all Y (hAPP_nat_nat(div_mod_nat(X),N) = hAPP_nat_nat(div_mod_nat(Y),N) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y),X)) -> (exists Q_4 X = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),Q_4)))))) # label(fact_3213_nat__mod__eq__lemma) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1333 (all A_154 all N_32 all B_105 (hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_105),N_32) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_154),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]. 8.65/8.77 1334 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)),N)) = hAPP_nat_nat(div_mod_nat(M),N)) # label(fact_3195_Divides_Omod__div__equality_H) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1335 (all X_36 all Y_30 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_36),Y_30)) -> Y_30 != X_36)) # label(fact_1790_order__less__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1336 (all N all B_2 all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,P_3),N)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,P_3),N)),B_2)))))) # label(fact_2446_zprime__power__zdvd__cancel__left) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1337 (all Xa all Ya (Xa = Ya -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Ya)))) # label(fact_1696_order__eq__refl) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1338 (all Xa all Ya (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Ya)) -> (Xa != Ya -> 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]. 8.65/8.77 1339 (all Z_2 (is_int(Z_2) -> (zero_zero_int = Z_2 <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(abs_abs_int,Z_2)),one_one_int))))) # label(fact_2771_zabs__less__one__iff) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1340 (all P_1 all A0 all A1 (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(upto_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1))) -> ((all I_2 all J_1 (is_int(J_1) & is_int(I_2) -> (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(upto_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,I_2),J_1))) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),J_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I_2),one_one_int)),J_1))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,I_2),J_1)))))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1))))) # label(fact_4572_upto_Opinduct) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1341 (all Ma all Na (is_int(Ma) & is_int(Na) -> (zero_zero_int != Na | zero_zero_int != Ma <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Ma),Na)))))) # label(fact_4997_gcd__pos__int) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1342 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (exists N_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),Y)),X)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_1))),Y)))))))) # label(fact_3779_reals__Archimedean4) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1343 (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]. 8.65/8.77 1344 (all X all Y hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(real_int,X)),hAPP_int_real(real_int,Y))) # label(fact_4390_real__of__int__mult) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1345 (all X_57 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_57),one_one_nat) = X_57) # label(fact_1000_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1346 (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]. 8.65/8.77 1347 (all A_20 all N_9 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_rat_rat(abs_abs_rat,A_20)),N_9) = hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_20),N_9))) # label(fact_2687_power__abs) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1348 (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]. 8.65/8.77 1349 (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]. 8.65/8.77 1350 (all Xa all Na (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,Na))) -> (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,hAPP_int_real(number267125858f_real,Na)),Xa)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,Na)),natfloor(Xa))))))) # label(fact_3320_le__natfloor__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1351 (all Va (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Va),pls)) <-> hAPP_int_nat(number_number_of_nat,Va) = zero_zero_nat)) # label(fact_755_eq__number__of__0) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1352 (all Xa (hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(div_mod_nat(Xa),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) <-> -hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa)))) # label(fact_3805_odd__nat__equiv__def) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1353 (all K all N_1 all F_2 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,F_2),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),K))),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))) = 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)))) # label(fact_4896_sumr__offset4) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1354 (all Na hAPP_n1699378549t_bool(ord_atMost_nat,Na) = hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),Na)) # label(fact_5187_atLeast0AtMost) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1355 (all A_202 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_202),hAPP_int_real(number267125858f_real,pls)) = A_202) # label(fact_178_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1356 (all A_163 all B_111 all C_65 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_163),C_65) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_163),B_111) -> B_111 = C_65)) # label(fact_788_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1357 (all X all A_3 all B_2 hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),B_2) = hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_2)),A_3)) # label(fact_4385_powr__powr__swap) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1358 (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),zero_zero_real))) # label(fact_4793_DERIV__sin__circle__all__zero) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1359 (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]. 8.65/8.77 1360 (all A_3 all X hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),X) = hAPP_real_real(scaleR_scaleR_real(A_3),X)) # label(fact_5109_real__scaleR__def) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1361 (all A_129 all M_17 all B_91 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_129),B_91)),M_17) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_129),M_17)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_91),M_17))) # label(fact_1182_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1362 (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]. 8.65/8.77 1363 (all A_1 all B_1 ((all D_2 (is_int(D_2) -> (D_2 = one_one_int <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_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))))) <-> 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]. 8.65/8.77 1364 (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]. 8.65/8.77 1365 (all Z_1 (hBOOL(hAPP_int_bool(nat_neg,Z_1)) -> hAPP_int_nat(nat,Z_1) = zero_zero_nat)) # label(fact_3250_neg__nat) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1366 (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]. 8.65/8.77 1367 (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]. 8.65/8.77 1368 (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(arccos),Xa))))) # label(fact_5169_isCont__arccos) # label(axiom) # label(non_clause). [assumption]. 8.65/8.77 1369 (all A_149 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_149),zero_zero_nat) = A_149) # label(fact_926_add__0__right) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 1370 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_complex_real(re,X)),hAPP_complex_real(re,Y)) = hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),Y))) # label(fact_4125_Re_Odiff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 1371 (all B_1 all A_1 (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_1),A_1)) -> (A_1 != B_1 -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,B_1),A_1))))) # label(fact_1949_xt1_I11_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 1372 (all A_1 all B_1 all Na (zero_zero_nat != Na -> (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]. 8.94/9.02 1373 (all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat) = hAPP_nat_nat(suc,N)) # label(fact_3063_Suc__eq__plus1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 1374 (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]. 8.94/9.02 1375 (all K_2 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),K_2) = hAPP_int_int(bit0,K_2)) # label(fact_156_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 1376 (all A_1 all P_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,setS(A_1,P_5))))) # label(fact_3945_SetS__finite) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 1377 (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]. 8.94/9.02 1378 (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_2,X_1)) <-> hBOOL(hAPP_int_bool(P_1,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(X_1) & is_int(K_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 X_1 (is_int(X_1) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D))) & hBOOL(hAPP_int_bool(P_2,X_1)))) | (exists X_1 (is_int(X_1) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D))) & (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))))) <-> (exists X1 (is_int(X1) & hBOOL(hAPP_int_bool(P_1,X1)))))))))) # label(fact_5180_cpmi) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 1379 (all P all Q all R hAPP_i157317923t_real(hAPP_f1613846310t_real(hAPP_f1720068861t_real(cOMBC_295337211t_real,P),Q),R) = hAPP_f1888375463t_real(hAPP_i447943416t_real(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Or) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 1380 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(semiri132038758t_real,Ma)),hAPP_nat_real(semiri132038758t_real,Na))))) # label(fact_511_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 1381 (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)) -> (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa),Ya) = zero_zero_real <-> Ya = zero_zero_real & zero_zero_real = Xa)))) # label(fact_585_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 1382 (all A_3 zero_zero_int = hAPP_int_int(div_mod_int(A_3),hAPP_int_int(number_number_of_int,min))) # label(fact_3085_zmod__minus1__right) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 1383 (all X (zero_zero_real = hAPP_real_real(sin,X) -> one_one_real = hAPP_real_real(abs_abs_real,hAPP_real_real(cos,X)))) # label(fact_3598_sin__zero__abs__cos__one) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 1384 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),M))))) # label(fact_2506_nat__dvd__not__less) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 1385 (all B_125 all A_181 all C_72 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_72)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_125),A_181)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_125),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_181),C_72)))))) # label(fact_576_add__increasing2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 1386 (all M all N (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) = M -> N = zero_zero_nat)) # label(fact_314_add__eq__self__zero) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 1387 (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]. 8.94/9.02 1388 (all X all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),X)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),one_one_int),P_3))))))) # label(fact_2862_Euler__part3) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 1389 (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]. 8.94/9.02 1390 (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]. 8.94/9.02 1391 (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_le1912455174t_bool,Z_2),Ya)) -> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Z_2),Xa))))) # label(fact_1728_xt1_I10_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 1392 (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]. 8.94/9.02 1393 (all B_86 all A_116 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_116),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_86)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_116),B_86)),zero_zero_rat))))) # label(fact_1310_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 1394 (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]. 8.94/9.02 1395 (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]. 8.94/9.02 1396 (all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),zero_zero_int)) -> zero_zero_nat = hAPP_int_nat(nat,Z_1))) # label(fact_2770_nat__le__0) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 1397 (all A_1 all C all B_1 (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)) <-> 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))))) # label(fact_1368_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 1398 (all A_101 all B_72 (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_101),B_72))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_101)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_72))))) # label(fact_1399_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 8.94/9.02 1399 (all A_3 all B_2 all Q_1 all R_2 (A_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_1)),R_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,R_2),B_2)) & 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,zero_zero_int),B_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,B_2),R_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)))))) # label(fact_3281_divmod__int__relI) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1400 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat))) = N)) # label(fact_3101_Suc__pred) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1401 (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(bit0,K)),hAPP_int_int(bit1,L))))) # label(fact_762_rel__simps_I15_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1402 (all Y_15 all X_19 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_15),X_19)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_19),Y_15)))) # label(fact_1983_leD) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1403 (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),B_109)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_63),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))) # label(fact_796_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1404 (all V_20 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_20)) -> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_int_real(number267125858f_real,V_20)) = hAPP_int_real(number267125858f_real,hAPP_int_int(succ,V_20)))) # label(fact_746_semiring__1__add__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1405 (all X (is_int(X) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) -> X = one_one_int | X = zero_zero_int)))) # label(fact_1004_int__pos__lt__two__imp__zero__or__one) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1406 (all C all A_1 all B_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_1),B_1)) | zero_zero_rat = C <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_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_2360_dvd__mult__cancel__left) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1407 (all A_130 all B_92 all 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) = 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))) # label(fact_1175_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1408 (all X_64 all Y_53 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_64),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_53),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_int))) # label(fact_219_not__sum__power2__lt__zero) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1409 (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 X1 (hBOOL(hAPP_int_bool(P_1,X1)) & is_int(X1))) <-> (exists X_1 (is_int(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)))))))) # label(fact_5100_periodic__finite__ex) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1410 (all Xa (hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(uminus_uminus_int,Xa))) <-> hBOOL(hAPP_int_bool(even_odd_even_int,Xa)))) # label(fact_3474_even__neg) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1411 (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]. 8.94/9.03 1412 (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]. 8.94/9.03 1413 (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]. 8.94/9.03 1414 (all A_83 all C_29 all B_56 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_83),C_29)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_56),C_29))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_29)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_83),B_56))))) # label(fact_1540_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1415 (all P all Q all R hAPP_P2110489235nt_int(hAPP_f1855025978nt_int(hAPP_f1038672605nt_int(cOMBB_1506897658nt_int,P),Q),R) = hAPP_int_fun_int_int(P,hAPP_P1175774780nt_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_012) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1416 (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]. 8.94/9.03 1417 (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]. 8.94/9.03 1418 (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]. 8.94/9.03 1419 (all A_126 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),A_126) = A_126) # label(fact_1211_mult__1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1420 (all Xa all Ya (is_int(Ya) & is_int(Xa) -> (hAPP_int_rat(number_number_of_rat,Xa) = hAPP_int_rat(number_number_of_rat,Ya) <-> Xa = Ya))) # label(fact_84_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1421 (all A_1 all B_1 all C (B_1 = C <-> hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_1),B_1) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_1),C))) # label(fact_810_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1422 (all A_3 all B_2 all C_1 all D_3 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)),hAPP_real_complex(hAPP_r265291036omplex(complex,C_1),D_3)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_3),C_1)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_2),D_3))) # label(fact_3959_complex__diff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1423 (all C_44 all B_81 all A_111 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_81),A_111)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_44),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_111),C_44)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_81),C_44)))))) # label(fact_1337_mult__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1424 (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]. 8.94/9.03 1425 (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]. 8.94/9.03 1426 (all Lx_1 all Ly_1 all Rx_1 all Ry_1 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_1),Ly_1)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_1),Ry_1)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_1),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_1),Ly_1)),Ry_1))) # label(fact_1075_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1427 (all C all A_1 all 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,A_1),B_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,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B_1))))) # label(fact_1371_mult__less__cancel__left__disj) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1428 (all Z_1 all X all D_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Z_1))),one_one_int)),D_3)))))) # label(fact_2809_incr__lemma) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1429 (all A_3 all C_1 all D_3 all B_2 (is_int(B_2) & is_int(D_3) -> (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]. 8.94/9.03 1430 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) = hAPP_nat_nat(fact_fact_nat,N))) # label(fact_3846_fact__reduce__nat) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1431 (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_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A_183)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_183),N_42)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_183),N_41)))))) # label(fact_555_power__increasing) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1432 (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]. 8.94/9.03 1433 (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]. 8.94/9.03 1434 (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]. 8.94/9.03 1435 (all C_19 all B_46 all A_69 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_46),A_69)) -> (C_19 = B_46 -> 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]. 8.94/9.03 1436 (all Xa (is_int(Xa) -> (hBOOL(hAPP_int_bool(even_odd_even_int,Xa)) <-> (exists Y_1 (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Y_1) = Xa & is_int(Y_1)))))) # label(fact_3266_even__equiv__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1437 (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]. 8.94/9.03 1438 (all C_42 all D_15 all A_109 all B_79 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_109),B_79)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_42),D_15)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_109)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_42)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_109),C_42)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_79),D_15)))))))) # label(fact_1347_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1439 (all B_87 all A_117 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_117)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_87),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_87),A_117)),zero_zero_rat))))) # label(fact_1304_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1440 (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,A_3),Y_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),B_2))))))))) # label(fact_3287_lemma__interval__lt) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1441 (all X hAPP_real_real(uminus_uminus_real,hAPP_real_real(cos,X)) = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),pi))) # label(fact_3619_cos__periodic__pi) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1442 (all X hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,pi),X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X))) # label(fact_3616_sin__periodic__pi2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1443 (all A_88 all M_12 all N_26 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_88),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_12),N_26)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_88),M_12)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_88),N_26))) # label(fact_1495_power__add) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1444 (all Na (-hBOOL(hAPP_nat_bool(even_odd_even_nat,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))))) # label(fact_3817_odd__Suc__mult__two__ex) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1445 (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]. 8.94/9.03 1446 (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)) -> (one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)),R_2)))))) # label(fact_4366_divides__mult__nat) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1447 (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]. 8.94/9.03 1448 (all A_105 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_105),A_105)),zero_zero_real))) # label(fact_1365_not__square__less__zero) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1449 (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,X_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))))))) -> ((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(P1,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]. 8.94/9.03 1450 (all Na all Diff all F all H (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H)) & (all M_1 all T (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),Na)) & 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_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))))) & hAPP_n546711566l_real(Diff,zero_zero_nat) = F -> (exists 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)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),T)))))) # label(fact_4934_Maclaurin2__objl) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1451 (all Xa all Ya (Ya = Xa | 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)))) # label(fact_2018_order__le__less) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1452 (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]. 8.94/9.03 1453 (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]. 8.94/9.03 1454 (all A_59 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_59),A_59) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_59),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_1898_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1455 (all X all Y all R_2 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_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(plus_plus_real,X),R_2)),Y)) # label(fact_3988_Complex__add__complex__of__real) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1456 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K1)),hAPP_int_int(bit1,K2))))) # label(fact_761_less__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1457 (all Wa all Xa (hAPP_i1732201573de_int(number1226105091de_int,Wa) = Xa <-> Xa = hAPP_i1732201573de_int(number1226105091de_int,Wa))) # label(fact_89_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1458 (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]. 8.94/9.03 1459 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa),Ya)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ya),hAPP_real_real(uminus_uminus_real,Xa))))) # label(fact_3440_real__add__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1460 (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]. 8.94/9.03 1461 (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]. 8.94/9.03 1462 (all C_21 all A_71 all B_48 (A_71 = B_48 -> (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]. 8.94/9.03 1463 (all N hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),N) = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),pi))) # label(fact_3750_cos__npi) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1464 (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]. 8.94/9.03 1465 (all A_1 all B_1 all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),Xa)) -> (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(powr,Xa),A_1)),hAPP_real_real(hAPP_r1250527377l_real(powr,Xa),B_1)))))) # label(fact_4432_powr__le__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1466 (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)))))) & Y = hAPP_real_real(sin,hAPP_real_real(arcsin,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)))))),hAPP_real_real(arcsin,Y)))))) # label(fact_3589_arcsin) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1467 (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_eq_real,hAPP_real_real(log(A_1),Xa)),hAPP_real_real(log(A_1),Ya))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya))))))) # label(fact_3447_log__le__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1468 (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]. 8.94/9.03 1469 (all A_3 -(-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),zero_zero_nat)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_3)))) # label(fact_2963_gcd__lcm__complete__lattice__nat_Onot__top__less) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1470 (all A_46 all B_35 all C_14 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_46),B_35)),C_14)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_35),C_14)))) # label(fact_2253_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1471 (all Xa (Xa = one_one_rat <-> one_one_rat = Xa)) # label(fact_850_one__reorient) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1472 (all A (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A)) -> (hBOOL(hAPP_f448129468l_bool(finite_finite_int,A)) <-> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_int_nat(nat,A)))))) # label(fact_4615_transfer__int__nat__set__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1473 (all Ma all Na (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,Ma)),hAPP_int_real(number267125858f_real,Na))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ma),hAPP_int_int(number_number_of_int,Na))))) # label(fact_4441_number__of__less__real__of__int__iff2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1474 (all Z_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(nat,Z_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_2)))) # label(fact_2782_zero__less__nat__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1475 (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]. 8.94/9.03 1476 (all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) <-> hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_nat_int(semiri1621563631at_int,A))))) # label(fact_4598_transfer__nat__int__set__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1477 (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)))) # label(fact_1974_order__less__imp__le) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1478 (all A_3 zero_zero_complex = rcis(zero_zero_real,A_3)) # label(fact_4260_rcis__zero__mod) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1479 (all P all Q all R hAPP_f727283836t_bool(P,hAPP_i1823918693l_bool(Q,R)) = hAPP_i1529485324t_bool(hAPP_f1545556668t_bool(hAPP_f85237525t_bool(cOMBB_389152643ol_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__HOL__Obool_Mtc__HOL) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1480 (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_leq)))) # label(fact_4542_pair__leqI1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1481 (all X hAPP_real_real(cos,X) = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi)))) # label(fact_3651_cos__periodic) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1482 (all B_2 all M all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),A_3)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M))))))) # label(fact_2588_zcong__not) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1483 (all C_44 all B_81 all A_111 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_81),A_111)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_44),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_111),C_44)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_81),C_44)))))) # label(fact_1339_mult__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1484 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(inverse_inverse_real,X))) # label(fact_3863_real__inv__sqrt__pow2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1485 (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)) -> 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,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]. 8.94/9.03 1486 (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]. 8.94/9.03 1487 (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]. 8.94/9.03 1488 (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]. 8.94/9.03 1489 (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]. 8.94/9.03 1490 (all Ma all Na (hAPP_nat_complex(semiri2020571505omplex,Na) = hAPP_nat_complex(semiri2020571505omplex,Ma) <-> Na = Ma)) # label(fact_323_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1491 (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]. 8.94/9.03 1492 (all C_54 all D_16 all A_143 all B_101 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_143),B_101)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_54),D_16)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_143),C_54)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_101),D_16)))))) # label(fact_965_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1493 (all A_3 (is_int(A_3) -> (zero_zero_int != A_3 -> one_one_int = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),A_3)))) # label(fact_2822_zdiv__self) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1494 (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,B_1),A_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_1406_mult__less__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1495 (all M all 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)) = 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))) # label(fact_3206_Suc__mod__eq__add3__mod__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1496 (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_2634_dvd_Oless__imp__neq) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1497 (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]. 8.94/9.03 1498 (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]. 8.94/9.03 1499 (all M all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),M))))) # label(fact_4160_prime__dvd__power__two) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1500 (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]. 8.94/9.03 1501 (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]. 8.94/9.03 1502 (all P all Q all R hAPP_f448129468l_bool(P,hAPP_i1948725293t_bool(Q,R)) = hAPP_int_bool(hAPP_f2119767738t_bool(hAPP_f423804115t_bool(cOMBB_1418110531ol_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obo) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1503 (all R_2 all X hAPP_complex_real(im,hAPP_complex_complex(scaleR1652505878omplex(R_2),X)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_2),hAPP_complex_real(im,X))) # label(fact_5111_complex__Im__scaleR) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1504 (all Na all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),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)) = 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)))) # label(fact_5082_fact__div__fact) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1505 (all X all A_3 all B_2 hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_3),B_2)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_2))) # label(fact_4412_powr__divide2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.03 1506 (all Xa all Ya (Xa = zero_zero_real & Ya = zero_zero_real <-> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa),Xa)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ya),Ya)) = zero_zero_real)) # label(fact_2126_real__two__squares__add__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1507 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K)),min)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),min)))) # label(fact_2183_rel__simps_I11_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1508 (all A_3 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,one_one_nat),A_3))) # label(fact_4016_coprime__1_H) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1509 (all M zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),zero_zero_nat)) # label(fact_2085_mult__0__right) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1510 (all K_2 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(suc,zero_zero_nat)),K_2))) # label(fact_2979_dvd__1__left) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1511 (all K_2 all A_3 all B_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),K_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),K_2)),M)))) # label(fact_2578_zcong__scalar) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1512 (all A_6 all B_7 all Q_6 all Y (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),Q_6)),Y) = 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_int,B_7),Y)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),zero_zero_int))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_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))) -> (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]. 8.94/9.04 1513 (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]. 8.94/9.04 1514 (all C_26 all D_11 all A_80 all B_53 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_80),B_53)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_26),D_11)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_80)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_26)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_80),C_26)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_53),D_11)))))))) # label(fact_1559_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1515 (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(bit1,K1)),hAPP_int_int(bit1,K2))))) # label(fact_556_less__eq__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1516 (all B_2 all A_3 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_2),A_3))),hAPP_complex_real(norm_norm_complex,B_2))),hAPP_complex_real(norm_norm_complex,A_3)))) # label(fact_3427_complex__mod__triangle__ineq2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1517 (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]. 8.94/9.04 1518 (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]. 8.94/9.04 1519 (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(J_1) & is_int(I_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),J_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_1),zero_zero_int)) & hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),I_2)),J_1) = Na -> 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]. 8.94/9.04 1520 (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]. 8.94/9.04 1521 (all P all Q all R hAPP_f617735571l_real(hAPP_r712953929l_real(P,R),Q) = hAPP_r2019696015l_real(hAPP_f1591894897l_real(hAPP_f644105375l_real(cOMBC_467724138l_real,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1522 (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(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,P_3),Y_1)),one_one_nat))))) # label(fact_4206_bezout__prime) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1523 (all A_36 zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_36),A_36)) # label(fact_2325_diff__self) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1524 (all P all Q all R hAPP_int_int(hAPP_P2110489235nt_int(P,R),Q) = hAPP_P1175774780nt_int(hAPP_i61245889nt_int(hAPP_f578362681nt_int(cOMBC_1417754532nt_int,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1525 (all M all N all R_4 all R_2 all S_3 all S_1 all T_2 all T_1 all K_2 (is_int(K_2) -> ((exists Sn exists Tn xzgcda(M,N,R_4,R_2,S_3,S_1,T_2,T_1) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,K_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Sn),Tn))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),R_2)) -> K_2 = hAPP_int_int(legacy_zgcd(R_4),R_2))))) # label(fact_4531_xzgcd__correct__aux2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1526 (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]. 8.94/9.04 1527 (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]. 8.94/9.04 1528 (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]. 8.94/9.04 1529 (all P_5 hBOOL(hAPP_f448129468l_bool(resSet(P_5),hAPP_i1948725293t_bool(sRStar,P_5)))) # label(fact_4054_ResSet__SRStar__prop) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1530 (all A_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_1)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_1),A_1))))) # label(fact_484_zero__less__double__add__iff__zero__less__single__add) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1531 (all A_1 hBOOL(hAPP_complex_bool(isCont_complex_real(im),A_1))) # label(fact_5164_Im_OisCont) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1532 (all K all Ma all 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))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)))) # label(fact_3058_Suc__mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1533 (all M_20 all N_38 all A_175 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_175)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_175),M_20)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_175),N_38))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_20),N_38))))) # label(fact_626_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1534 (all Lx_6 all Rx_6 all Ry_4 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_6),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_6),Ry_4)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_6),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_6),Ry_4))) # label(fact_1041_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1535 (all K_2 (is_int(K_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2)) -> -(all N_1 K_2 != hAPP_nat_int(semiri1621563631at_int,N_1))))) # label(fact_1012_nonneg__int__cases) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1536 (all A_3 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),one_one_nat))) # label(fact_4017_coprime__1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1537 (all A_127 A_127 = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,one_on1645066479umeral),A_127)) # label(fact_1202_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1538 (all A_74 A_74 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,pls))),A_74)) # label(fact_1617_mult__numeral__1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1539 (all Ma 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)),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),hAPP_nat_int(semiri1621563631at_int,Ma)))))) # label(fact_2805_nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1540 (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]. 8.94/9.04 1541 (all X all Y hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),X)) # label(fact_4308_gcd__semilattice__nat_Oinf__commute) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1542 (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]. 8.94/9.04 1543 (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_int -> 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]. 8.94/9.04 1544 (all A_159 all B_107 all C_61 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_159),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_107),C_61)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_159),B_107)),C_61)) # label(fact_822_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1545 (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_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)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1))))))) # label(fact_3127_zmult2__lemma__aux4) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1546 (all Va all Wa (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_i1732201573de_int(number1226105091de_int,Va)),hAPP_i1732201573de_int(number1226105091de_int,Wa))) <-> -hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_i1732201573de_int(number1226105091de_int,Wa)),hAPP_i1732201573de_int(number1226105091de_int,Va))))) # label(fact_690_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1547 (all X all Y all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),Y),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),X)),hAPP_int_int(multInv(P_3),Y)),P_3))))) # label(fact_2905_MultInv__prop1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1548 (all X all A_3 all B_2 hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),B_2)) = hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),B_2)) # label(fact_4386_powr__powr) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1549 (all Xa all Ya (zero_zero_real = Ya & zero_zero_real = Xa <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,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)))))),zero_zero_real)))) # label(fact_995_sum__power2__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1550 (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]. 8.94/9.04 1551 (all X all Y all C_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_1),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_1),hAPP_real_real(abs_abs_real,Y)))) -> X = zero_zero_real))) # label(fact_2925_rabs__ratiotest__lemma) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1552 (all A_1 all Wa (is_int(A_1) -> (hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_1),hAPP_int_nat(number_number_of_nat,Wa)) = zero_zero_int <-> hAPP_int_nat(number_number_of_nat,Wa) != zero_zero_nat & A_1 = zero_zero_int))) # label(fact_186_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1553 (all A_3 all B_2 all C_1 all 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))) = 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))) # label(fact_3960_complex__mult) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1554 (all P all Q all R hAPP_f1902329768t_bool(P,hAPP_r285062775l_bool(Q,R)) = hAPP_r1562208522t_bool(hAPP_f2084827628t_bool(hAPP_f1121609875t_bool(cOMBB_2125215742l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Nat__Onat_Mtc__fun_) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1555 (all A_191 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_191),zero_zero_nat) = one_one_rat) # label(fact_458_power__0) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1556 (all X all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),hAPP_int_int(number_number_of_int,min)),P_3)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),one_one_int),P_3))))) # label(fact_2464_zcong__neg__1__impl__ne__1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1557 (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_4707_rat__number__collapse_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1558 (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]. 8.94/9.04 1559 (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_int,A_1),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)))))) # label(fact_1375_mult__less__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1560 (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(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_5059_summable__Leibniz_H_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1561 (all Z_11 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_11) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_11),Z_11)) # label(fact_1897_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1562 (all M all N all Z_1 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N))),Z_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,N)),Z_1))) # label(fact_66_zadd__int__left) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1563 (all Na 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)) = zero_zero_real) # label(fact_4911_lemma__STAR__cos2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1564 (all B_2 all Y all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y)) -> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),hAPP_real_real(log(B_2),X)) = hAPP_real_real(log(B_2),hAPP_real_real(hAPP_r1250527377l_real(powr,X),Y))))) # label(fact_4472_log__powr) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1565 (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]. 8.94/9.04 1566 (all A_105 -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_105),A_105)),zero_zero_rat))) # label(fact_1364_not__square__less__zero) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1567 (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]. 8.94/9.04 1568 (all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N)) -> hAPP_int_int(fact_fact_int,N) != zero_zero_int)) # label(fact_4277_fact__nonzero__int) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1569 (all Ma all Na ((hBOOL(hAPP_nat_bool(even_odd_even_nat,Ma)) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,Na))) <-> 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]. 8.94/9.04 1570 (all K all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K)),hAPP_int_int(bit0,L))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L)))) # label(fact_78_rel__simps_I14_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1571 (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]. 8.94/9.04 1572 (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) <-> 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,A_1),B_1)),E_2)),C))) # label(fact_2382_eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1573 (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]. 8.94/9.04 1574 (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]. 8.94/9.04 1575 (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]. 8.94/9.04 1576 (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]. 8.94/9.04 1577 (all A_137 zero_zero_complex = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_137),zero_zero_complex)) # label(fact_1115_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1578 (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]. 8.94/9.04 1579 (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_le1568362934t_bool,Z_2),Ya)) -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Z_2),Xa))))) # label(fact_1935_xt1_I7_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1580 (all C_35 all D_13 all A_92 all B_63 all R_6 (zero_zero_complex != R_6 -> (C_35 != D_13 & A_92 = B_63 -> 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]. 8.94/9.04 1581 (all A_23 all B_18 hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_23),B_18)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(abs_abs_real,A_23)),hAPP_real_real(abs_abs_real,B_18))) # label(fact_2676_abs__mult) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1582 (all X X = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),X)) # label(fact_4338_gcd__0__left__nat) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1583 (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]. 8.94/9.04 1584 (all V_1 all W hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_1),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))) # label(fact_1276_times__numeral__code_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1585 (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]. 8.94/9.04 1586 (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]. 8.94/9.04 1587 (all Lx_5 all Rx_5 all Ry_3 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_5),Rx_5)),Ry_3) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_5),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_5),Ry_3))) # label(fact_1046_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1588 (all A_202 A_202 = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_202),hAPP_int_complex(number528085621omplex,pls))) # label(fact_177_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1589 (all K_2 all I all J_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,K_2)),I)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,K_2)),J_2)))))) # label(fact_1876_zmult__zless__mono2__lemma) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1590 (all A_1 all B_1 all C all D_1 (hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C),D_1) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_1),B_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),D_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B_1))))) # label(fact_2338_diff__eq__diff__less) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1591 (all Xa all Ya (is_int(Ya) & is_int(Xa) -> (zero_zero_int != Ya | zero_zero_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_int_int(hAPP_int_fun_int_int(times_times_int,Xa),Xa)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ya),Ya))))))) # label(fact_1592_sum__squares__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1592 (all K all L (is_int(K) & is_int(L) -> (hAPP_int_int(bit1,K) = hAPP_int_int(bit1,L) <-> L = K))) # label(fact_95_rel__simps_I51_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1593 (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]. 8.94/9.04 1594 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(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)) = hAPP_real_real(log(hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X))) # label(fact_3424_log__base__10__eq1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1595 (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]. 8.94/9.04 1596 (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]. 8.94/9.04 1597 (all A_3 all B_2 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),B_2)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),A_3))))) # label(fact_4169_prime__divprod) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1598 (all A_134 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,zero_zero_int),A_134) = zero_zero_int) # label(fact_1144_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1599 (all A_134 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A_134) = zero_zero_nat) # label(fact_1143_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1600 (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]. 8.94/9.04 1601 (all W_10 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),hAPP_int_complex(number528085621omplex,W_10))),hAPP_int_complex(number528085621omplex,W_10)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(bit1,W_10))) # label(fact_225_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1602 (all A_159 all B_107 all C_61 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_159),B_107)),C_61) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_159),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_107),C_61))) # label(fact_823_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1603 (all X all Y hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,X))),hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,Y)))) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)) # label(fact_5020_gcd__int__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1604 (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,X_1),Ta))) <-> -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))))))) # label(fact_2235_inf__period_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1605 (all A_56 all B_38 (A_56 != B_38 -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_38),A_56)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_38),A_56))))) # label(fact_1989_xt1_I12_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1606 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_complex_real(im,X))),hAPP_complex_real(norm_norm_complex,X)))) # label(fact_4102_abs__Im__le__cmod) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1607 (all P all Q (hBOOL(P) | -hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)))) # label(help_fconj_2_1_U) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1608 (all D_3 all A_3 all B_2 (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)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),A_3) = one_one_nat)) # label(fact_4355_coprime__lmult__nat) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1609 (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]. 8.94/9.04 1610 (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]. 8.94/9.04 1611 (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]. 8.94/9.04 1612 (all V_7 all W_3 hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_7),W_3)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_int_rat(number_number_of_rat,V_7)),hAPP_int_rat(number_number_of_rat,W_3))) # label(fact_2228_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1613 (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]. 8.94/9.04 1614 (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]. 8.94/9.04 1615 (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]. 8.94/9.04 1616 (all P all Q all R hAPP_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,P),Q),R) = hAPP_int_int(hAPP_int_fun_int_int(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 8.94/9.04 1617 (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_1638_xt1_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1618 (all Xa all Ya (is_int(Xa) & is_int(Ya) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,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)) <-> zero_zero_int = Xa & zero_zero_int = Ya))) # label(fact_996_sum__power2__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1619 (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]. 8.94/9.05 1620 (all Xa all P_5 (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(sr,P_5))) -> Xa = hAPP_int_int(standardRes(P_5),Xa)))) # label(fact_2980_StandardRes__SR__prop) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1621 (all A_151 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_151),zero_zero_nat) = A_151) # label(fact_912_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1622 (all Ma all Na all Qr (hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(Ma,Na),Qr)) <-> (zero_zero_nat = Na -> hAPP_P1654798560at_nat(product_fst_nat_nat,Qr) = zero_zero_nat) & (zero_zero_nat != Na -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_P1654798560at_nat(product_snd_nat_nat,Qr))) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_P1654798560at_nat(product_snd_nat_nat,Qr)),Na))) & (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_P1654798560at_nat(product_snd_nat_nat,Qr)),zero_zero_nat)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),hAPP_P1654798560at_nat(product_snd_nat_nat,Qr))))) & Ma = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_P1654798560at_nat(product_fst_nat_nat,Qr)),Na)),hAPP_P1654798560at_nat(product_snd_nat_nat,Qr)))) # label(fact_4240_divmod__nat__rel__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1623 (all B_133 all A_190 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_190)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_133)) -> 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_190),B_133)))))) # label(fact_493_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1624 (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_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))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),A_1))))) # label(fact_1577_mult__le__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1625 (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]. 8.94/9.05 1626 (all A_39 all B_28 hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_39),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_39),B_28)))) # label(fact_2298_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1627 (all C_27 all D_12 all A_81 all B_54 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_81),B_54)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_27),D_12)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_81)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_27)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_81),C_27)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_54),D_12)))))))) # label(fact_1552_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1628 (all C all Xa all Ta all A_1 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_1),D_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xa),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),D_1))),Ta))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xa),Ta)))))) # label(fact_2407_zdvd__period) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1629 (all X_43 all Y_37 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_43),Y_37)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_37),X_43)) -> Y_37 = X_43))) # label(fact_1653_order__antisym) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1630 (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]. 8.94/9.05 1631 (all Q_5 all R_4 all A_3 all B_2 all Q_1 all R_2 (is_int(R_4) & is_int(B_2) & is_int(R_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]. 8.94/9.05 1632 (all Xa comple124823625p_real(hAPP_r1134773055l_bool(ord_atMost_real,Xa)) = Xa) # label(fact_5190_Sup__atMost) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1633 (all B_2 all D_3 all A_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),A_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_3),B_2))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),B_2))))) # label(fact_2497_divides__add__revr) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1634 (all C_26 all D_11 all A_80 all B_53 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_80),B_53)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_26),D_11)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_80)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_26)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_80),C_26)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_53),D_11)))))))) # label(fact_1558_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1635 (all Y all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y)) -> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(ln,X)),hAPP_real_real(ln,Y)) = hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y))))) # label(fact_3371_ln__mult) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1636 (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]. 8.94/9.05 1637 (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]. 8.94/9.05 1638 (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]. 8.94/9.05 1639 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(nat,X)),hAPP_int_nat(nat,Y)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y)))) # label(fact_2897_nat__div__distrib) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1640 (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]. 8.94/9.05 1641 (all P all Q all R hAPP_f351634443t_bool(hAPP_r1392521531t_bool(P,R),Q) = hAPP_r632163725t_bool(hAPP_f956464319t_bool(hAPP_f422375781t_bool(cOMBC_243356514t_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Int__Oint_Mtc__fun_It) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1642 (all P_1 all A0 all A1 (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(posDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A0),A1))) -> ((all A_2 all B_4 (is_int(A_2) & is_int(B_4) -> (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(posDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_2),B_4))) -> ((-(hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_4),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),B_4))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_4)))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A_2),B_4)))))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,A0),A1))))) # label(fact_3322_posDivAlg_Opinduct) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1643 (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) -> one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,P_3),Q_1))) # label(fact_5017_normalize__coprime) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1644 (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]. 8.94/9.05 1645 (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]. 8.94/9.05 1646 (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]. 8.94/9.05 1647 (all C_24 all D_9 all A_78 all B_51 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_78),B_51)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_24),D_9)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_51)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_24)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_78),C_24)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_51),D_9)))))))) # label(fact_1570_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1648 (all A_39 all B_28 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_39),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_39),B_28)))) # label(fact_2295_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1649 (all Xa all A (hBOOL(hAPP_P880235623l_bool(A,Xa)) <-> hBOOL(hAPP_f1469637905l_bool(hAPP_P1681069257l_bool(member308914164t_bool,Xa),A)))) # label(fact_138_mem__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1650 (all Z_1 (is_int(Z_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> Z_1 = hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,Z_1))))) # label(fact_2775_nat__0__le) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1651 (all N all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N)),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N)),one_one_real))) -> archim856651990g_real(X) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N),one_one_int)))) # label(fact_4473_ceiling__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1652 (all A_1 all B_1 (is_int(B_1) & is_int(A_1) -> (A_1 = B_1 <-> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_1),B_1) = zero_zero_int))) # label(fact_2320_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1653 (all X all Y all Z_1 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Y)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Z_1)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_1))) # label(fact_1508_zpower__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1654 (all V_25 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))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_25)),one_one_rat)) # label(fact_27_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1655 (all A_1 all C 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,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)) <-> 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))))) # label(fact_1369_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1656 (all P_1 all Na ((all X_1 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),Na))) -> hBOOL(hAPP_nat_bool(P_1,X_1)))) <-> (all M_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),Na)) -> hBOOL(hAPP_nat_bool(P_1,M_1)))))) # label(fact_5068_all__nat__less) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1657 (all A_28 all B_20 all V_5 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_28),hAPP_int_complex(number528085621omplex,V_5))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_20),hAPP_int_complex(number528085621omplex,V_5))) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_28),B_20)),hAPP_int_complex(number528085621omplex,V_5))) # label(fact_2398_left__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1658 (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]. 8.94/9.05 1659 (all K all L (is_int(K) & is_int(L) -> (L = K <-> hAPP_int_int(bit0,L) = hAPP_int_int(bit0,K)))) # label(fact_96_rel__simps_I48_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1660 (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_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_4086_cnj_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1661 (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_at238088361st_nat(zero_zero_nat),Na))))) <-> (exists M_1 (hBOOL(hAPP_nat_bool(P_1,M_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),Na)))))) # label(fact_5067_ex__nat__less) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1662 (all Xa (hBOOL(hAPP_int_bool(even_odd_even_int,Xa)) <-> 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_3270_even__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1663 (all N_56 all A_200 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_200)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_56)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_200),N_56)))))) # label(fact_249_one__less__power) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1664 (all Xa all Ya (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ya),Xa)) <-> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)))) # label(fact_2043_linorder__not__le) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1665 (all X all Y hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Y)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X),Y))) # label(fact_2571_diff__square) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1666 (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_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_1)),zero_zero_rat)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),zero_zero_int))))) # label(fact_4716_Fract__less__zero__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1667 (all N (zero_zero_nat != N -> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)),one_one_nat)) = 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))) # label(fact_3160_Suc__mult__two__diff__one) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1668 (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]. 8.94/9.05 1669 (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]. 8.94/9.05 1670 (all A_138 zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,zero_zero_int),A_138)) # label(fact_1112_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1671 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)))))) # label(fact_3095_n__less__m__mult__n) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1672 (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]. 8.94/9.05 1673 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),zero_zero_int))))) # label(fact_2837_div__nonpos__pos__le0) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1674 (all B_31 all A_42 all C_11 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_42),C_11)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_42),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_31),C_11))))) # label(fact_2279_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1675 (all N_55 ((zero_zero_nat = N_55 -> one_one_rat = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),N_55)) & (N_55 != zero_zero_nat -> zero_zero_rat = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),N_55)))) # label(fact_252_power__0__left) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1676 (all X_1 ((hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)) -> zero_zero_real = hAPP_nat_real(sin_coeff,X_1)) & (-hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1)) -> hAPP_nat_real(sin_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,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X_1),hAPP_nat_nat(suc,zero_zero_nat))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,X_1)))))) # label(fact_3826_sin__coeff__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1677 (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]. 8.94/9.05 1678 (all Xa all Ya (is_int(Ya) & is_int(Xa) -> (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_1707_order__eq__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1679 (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]. 8.94/9.05 1680 (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_eq_int,hAPP_int_int(bit1,K)),hAPP_int_int(bit1,L))))) # label(fact_557_rel__simps_I34_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1681 (all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,W)),zero_zero_int)) -> hAPP_int_int(div_mod_int(one_one_int),hAPP_int_int(number_number_of_int,W)) = hAPP_P1175774780nt_int(product_snd_int_int,hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,one_one_int),hAPP_int_int(uminus_uminus_int,hAPP_int_int(number_number_of_int,W))))))) # label(fact_4231_mod__pos__neg__1__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1682 (all V_1 all 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)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_1),W))) # label(fact_59_plus__numeral__code_I9_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1683 (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]. 8.94/9.05 1684 (all M all N all K_2 (one_one_int = hAPP_int_int(legacy_zgcd(N),K_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M))))) # label(fact_4516_zrelprime__zdvd__zmult) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1685 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = N)) # label(fact_3116_Suc__diff__1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1686 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> zero_zero_nat != N)) # label(fact_299_gr__implies__not0) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1687 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Xa),Ya)),zero_zero_real)))) # label(fact_2502_real__le__eq__diff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1688 (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]. 8.94/9.05 1689 (all R_2 all Q_1 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,R_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Q_1)) = A_3 -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),A_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Q_1)))))) # label(fact_2058_self__quotient__aux1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1690 (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]. 8.94/9.05 1691 (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]. 8.94/9.05 1692 (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]. 8.94/9.05 1693 (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]. 8.94/9.05 1694 (all M all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(div_mod_nat(M),N)),M))) # label(fact_3172_mod__less__eq__dividend) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1695 (all A_1 (zero_zero_real = hAPP_real_real(abs_abs_real,A_1) <-> zero_zero_real = A_1)) # label(fact_2661_abs__eq__0) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1696 (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]. 8.94/9.05 1697 (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]. 8.94/9.05 1698 (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,re),X_2)),hAPP_complex_real(re,A_1))))) # label(fact_4871_Re_Osums) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1699 (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]. 8.94/9.05 1700 (all X all Y hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),hAPP_complex_complex(uminus473333897omplex,Y)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),Y)) # label(fact_3458_complex__diff__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1701 (all Ma all Wa (is_int(Wa) -> (hAPP_int_nat(nat,Wa) = Ma <-> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa)) -> hAPP_nat_int(semiri1621563631at_int,Ma) = Wa) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa)) -> Ma = zero_zero_nat)))) # label(fact_2790_nat__eq__iff2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1702 (all B_1 all D_1 all A_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_1),B_1)) & (all E (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,E),B_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,E),A_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,E),D_1)))) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_1),A_1)) <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_1),B_1) = D_1)) # label(fact_4317_gcd__unique__nat) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1703 (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]. 8.94/9.05 1704 (all P all Q all R hAPP_f892584630t_bool(hAPP_P1400155668t_bool(P,R),hAPP_P1668131738nt_int(Q,R)) = hAPP_P989584703t_bool(hAPP_f1187443546t_bool(hAPP_f11709033t_bool(cOMBS_1676841879t_bool,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc_) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1705 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Y)),one_one_real)) -> Y = hAPP_real_real(cos,hAPP_real_real(arccos,Y)))) # label(fact_3672_cos__arccos__abs) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1706 (all A_134 zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A_134)) # label(fact_1138_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1707 (all B_1_1 all B_2_1 is_bool(hAPP_f1469637905l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__prod_Itc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_Mtc__f) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1708 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),zero_zero_int)) -> hAPP_int_int(div_mod_int(A_3),B_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)))) # label(fact_3110_mod__pos__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1709 (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]. 8.94/9.05 1710 (all Na hAPP_nat_real(diffs_real(sin_coeff),Na) = hAPP_nat_real(cos_coeff,Na)) # label(fact_4873_sin__fdiffs2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1711 (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]. 8.94/9.05 1712 (all X hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_real_real(sin,X))),hAPP_real_real(cos,X)) = hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X))) # label(fact_3607_sin__double) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1713 (all X_1 hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,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]. 8.94/9.05 1714 (all M_21 -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,M_21)),zero_zero_rat))) # label(fact_452_of__nat__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1715 (all Y (Y != zero_z126310315umeral -> -(all Code_numeral Y != code_S1047413653umeral(Code_numeral)))) # label(fact_4090_code__numeral_Oexhaust) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1716 (all Ya all Xa (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),Xa)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),Ya)) -> (Ya = zero_z891286103de_int & zero_z891286103de_int = Xa <-> zero_z891286103de_int = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,Xa),Ya))))) # label(fact_584_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1717 (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]. 8.94/9.05 1718 (all A_128 A_128 = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,one_on1645066479umeral),A_128)) # label(fact_1195_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1719 (all Xa hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_int_real(real_int,Xa)),ring_1_Ints_real))) # label(fact_4730_Ints__real__of__int) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1720 (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]. 8.94/9.05 1721 (all B_1 all A_1 (zero_zero_nat = A_1 <-> B_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_1),A_1))) # label(fact_906_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1722 (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]. 8.94/9.05 1723 (all V_11 all B_62 all C_34 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_11)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_62),C_34)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_11)),B_62)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_11)),C_34))) # label(fact_1459_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1724 (all K_2 all L_2 hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),L_2)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(bit0,K_2)),hAPP_int_int(bit0,L_2))) # label(fact_2352_diff__bin__simps_I7_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1725 (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]. 8.94/9.05 1726 (all M_15 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat)),M_15) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_15),M_15)) # label(fact_1475_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1727 (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]. 8.94/9.05 1728 (all A_75 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_75),hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit1,pls))) = A_75) # label(fact_1611_mult__numeral__1__right) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1729 (all A_131 all E_3 all B_93 all C_49 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_131),E_3)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_93),E_3)),C_49)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_131),B_93)),E_3)),C_49)) # label(fact_1163_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1730 (all N_57 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,N_57)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_57)))) # label(fact_235_of__nat__less__two__power) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1731 (all P_5 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),P_5)) & (all M_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M_1),P_5)) -> one_one_nat = M_1 | M_1 = P_5)) <-> hBOOL(hAPP_nat_bool(prime,P_5)))) # label(fact_4183_prime__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1732 (all M_14 all A_90 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,M_14),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_90),M_14)) = 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)) # label(fact_1478_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1733 (all P all Q all R hAPP_f513299955t_bool(P,hAPP_P921862901l_bool(Q,R)) = hAPP_P877703507t_bool(hAPP_f1965248563t_bool(hAPP_f1273270095t_bool(cOMBB_595786433nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_000tc__fun_Itc_040) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1734 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),A_3)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),A_3))) # label(fact_2856_pos__zdiv__mult__2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1735 (all C all A_1 all B_1 (A_1 = B_1 -> (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,C),B_1)) -> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,C),A_1))))) # label(fact_1756_xt1_I1_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1736 (all B_2 all Bq all Br all A_3 all C_1 all Aq all Ar (hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(A_3,C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Aq),Ar))) -> (hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(B_2,C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Bq),Br))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_1)) -> hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_3),B_2),C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Aq),Bq)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ar),Br)),C_1))),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ar),Br)),C_1)))))))) # label(fact_4213_divmod__nat__rel__add1__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1737 (all X_42 all Y_36 (Y_36 = X_42 -> 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]. 8.94/9.05 1738 (all Xa (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Xa),field_1210416355s_real)) -> -(all M_1 all N_1 (zero_zero_nat != N_1 -> (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) -> one_one_nat != hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M_1),N_1)))))) # label(fact_4622_Rats__abs__nat__div__natE) # label(axiom) # label(non_clause). [assumption]. 8.94/9.05 1739 (all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W))) -> hAPP_P1175774780nt_int(product_snd_int_int,posDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W))) = hAPP_int_int(div_mod_int(one_one_int),hAPP_int_int(number_number_of_int,W)))) # label(fact_4235_mod__pos__pos__1__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1740 (all M all K_2 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2) != M -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),K_2))))) # label(fact_3098_one__less__k) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1741 (all A_51 all N_19 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_51),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_19))))) # label(fact_2147_zero__le__even__power_H) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1742 (all A_1 all B_1 (zero_zero_real = B_1 & one_one_real = A_1 <-> one_one_complex = hAPP_real_complex(hAPP_r265291036omplex(complex,A_1),B_1))) # label(fact_3955_Complex__eq__1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1743 (all N -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N)),zero_zero_real))) # label(fact_3701_not__real__of__nat__less__zero) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1744 (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]. 8.94/9.06 1745 (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]. 8.94/9.06 1746 (all X one_one_real = hAPP_real_real(hAPP_r1250527377l_real(powr,X),zero_zero_real)) # label(fact_4407_powr__zero__eq__one) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1747 (all Na (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,Na)),zero_zero_real)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Na),zero_zero_int)))) # label(fact_4437_real__of__int__lt__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1748 (all J_2 all K_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),J_2)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),K_2)),M)) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),J_2) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),K_2)))) # label(fact_2462_neg__one__power__eq__mod__m) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1749 (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]. 8.94/9.06 1750 (all X hAPP_real_real(cos,hAPP_real_real(arctan,X)) != zero_zero_real) # label(fact_3606_cos__arctan__not__zero) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1751 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(div_mod_int(A_3),B_2)),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),hAPP_int_int(div_mod_int(A_3),B_2))))) # label(fact_3081_neg__mod__conj) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1752 (all C_9 all A_38 all B_27 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_38),B_27)) -> (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_38),C_9)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_38),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_27),C_9)))))) # label(fact_2304_dvd__add) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1753 (all X ratreal(hAPP_rat_rat(uminus_uminus_rat,X)) = hAPP_real_real(uminus_uminus_real,ratreal(X))) # label(fact_4702_real__uminus__code) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1754 (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]. 8.94/9.06 1755 (all Ya all Xa (is_int(Ya) & is_int(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)) -> (Xa = zero_zero_int & Ya = zero_zero_int <-> zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xa),Ya)))))) # label(fact_587_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1756 (all K all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(pred,K)),L)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L)))) # label(fact_4041_le__iff__pred__less) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1757 (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_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_199)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_199),one_one_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_199),N_53)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_199),N_54))))))) # label(fact_264_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1758 (all F all L (hBOOL(tendsto_nat_real(F,L,sequentially)) -> hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F),hAPP_real_real(abs_abs_real,L),sequentially)))) # label(fact_5038_LIMSEQ__imp__rabs) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1759 (all V_14 all W_6 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_14)),hAPP_int_real(number267125858f_real,W_6)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_14),W_6))) # label(fact_1098_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1760 (all P all Q all R hAPP_n215258509l_bool(P,hAPP_nat_nat(Q,R)) = hAPP_n215258509l_bool(hAPP_f66927821l_bool(hAPP_f2037783381l_bool(cOMBB_1146692694ol_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HO) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1761 (all A_3 all P_3 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3)) -> hAPP_int_int(legendre(A_3),P_3) = zero_zero_int) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),A_3)) -> hAPP_int_int(legendre(A_3),P_3) = hAPP_int_int(number_number_of_int,min)) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),A_3)) -> one_one_int = hAPP_int_int(legendre(A_3),P_3))))) # label(fact_2466_Legendre__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1762 (all A_91 all B_61 all V_10 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_91),hAPP_i769753017umeral(number1443263063umeral,V_10))),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_61),hAPP_i769753017umeral(number1443263063umeral,V_10))) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_91),B_61)),hAPP_i769753017umeral(number1443263063umeral,V_10))) # label(fact_1469_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1763 (all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),N) = N) # label(fact_5194_max__0L) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1764 (all P all Q all R hAPP_b589554111l_bool(P,hAPP_nat_bool(Q,R)) = hAPP_n1006566506l_bool(hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_005) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1765 (all C_1 all A_3 all B_2 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),A_3)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) -> (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]. 8.94/9.06 1766 (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]. 8.94/9.06 1767 (all Xa (zero_zero_real = Xa <-> zero_zero_real = Xa)) # label(fact_767_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1768 (all Y all X all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Y),zero_zero_int),P_3)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)),zero_zero_int),P_3)))))) # label(fact_2602_zcong__zmult__prop3) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1769 (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]. 8.94/9.06 1770 (all X hAPP_int_int(abs_abs_int,X) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,zero_zero_int),X)) # label(fact_4968_gcd__0__left__int) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1771 (all P all Q all R hAPP_f1509287679l_real(P,hAPP_r474017924t_real(Q,R)) = hAPP_r1409264866l_real(hAPP_f1523861863l_real(hAPP_f1067681227l_real(cOMBB_1934313333l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__fun__022) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1772 (all P all Q all R hAPP_i1948725293t_bool(hAPP_f720654810t_bool(hAPP_f908327869t_bool(cOMBB_1290694363ol_int,P),Q),R) = hAPP_f429935748t_bool(P,hAPP_int_fun_int_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1773 (all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),pls)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,Xa)),zero_zero_int)))) # label(fact_613_le__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1774 (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]. 8.94/9.06 1775 (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]. 8.94/9.06 1776 (all X_9 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_9),X_9)),one_one_real) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_9),one_one_real)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_9),one_one_real))) # label(fact_2435_real__squared__diff__one__factored) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1777 (all X_32 all Y_26 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_32),Y_26)) -> Y_26 != X_32)) # label(fact_1821_less__imp__neq) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1778 (all A_55 all B_37 (B_37 != A_55 -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_55),B_37)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_55),B_37))))) # label(fact_1996_order__neq__le__trans) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1779 (all M all N all K_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),K_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N)),hAPP_nat_nat(suc,K_2))) # label(fact_3027_Suc__diff__diff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1780 (all X_61 all Y_51 (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)))) = 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)))) -> (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)) -> Y_51 = X_61)))) # label(fact_887_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1781 (all A_52 all K_7 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,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_rat)) -> zero_zero_rat = A_52)) # label(fact_2142_even__power__le__0__imp__0) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1782 (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_2019_order__le__less) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1783 (all Xa all 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,Xa),hAPP_i1948725293t_bool(sr,P_5))))) # label(fact_3187_SRStar__SR__prop) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1784 (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]. 8.94/9.06 1785 (all 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_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]. 8.94/9.06 1786 (all B_1 all P_5 all A_1 (is_int(A_1) & is_int(B_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_1)) -> (-hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),hAPP_i1948725293t_bool(wset(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_1),one_one_int)),P_5))) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),hAPP_i1948725293t_bool(wset(A_1),P_5))) -> B_1 = A_1 | B_1 = hAPP_int_int(inv(P_5),A_1)))))) # label(fact_2932_wset__mem__imp__or) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1787 (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,zero_zero_int),B_2)) -> divmod_int(A_3,B_2) = hAPP_P1975530577nt_int(negateSnd,posDivAlg(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> divmod_int(A_3,B_2) = negDivAlg(A_3,B_2))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_2)) -> (A_3 != zero_zero_int -> hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2))) = divmod_int(A_3,B_2)) & (A_3 = zero_zero_int -> divmod_int(A_3,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_2)) -> divmod_int(A_3,B_2) = posDivAlg(A_3,B_2))))) # label(fact_4161_divmod__int__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1788 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)))) # label(fact_3005_Suc__lessD) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1789 (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]. 8.94/9.06 1790 (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]. 8.94/9.06 1791 (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]. 8.94/9.06 1792 (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]. 8.94/9.06 1793 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),zero_zero_real)) -> (exists R_1 exists A_2 (hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(sin,A_2)) = Y & X = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(cos,A_2)))))) # label(fact_3849_polar__ex2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1794 (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]. 8.94/9.06 1795 (all X all Y (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)))))) -> Y = zero_zero_real)) # label(fact_3553_real__sqrt__sum__squares__eq__cancel) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1796 (all P all Q all R hAPP_nat_complex(hAPP_f219720582omplex(hAPP_f1052840010omplex(cOMBS_696648398omplex,P),Q),R) = hAPP_real_complex(hAPP_n1983812447omplex(P,R),hAPP_nat_real(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__Complex__Ocom) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1797 (all X hAPP_real_real(abs_abs_real,hAPP_real_real(exp_real,X)) = hAPP_real_real(exp_real,X)) # label(fact_3393_abs__exp__cancel) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1798 (all A_148 (is_int(A_148) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),A_148) = A_148)) # label(fact_937_add__0) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1799 (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)),zero_zero_real)))) # label(fact_5166_isCont__root__zero) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1800 (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]. 8.94/9.06 1801 (all A_21 all B_16 hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_16),A_21)) = hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_21),B_16))) # label(fact_2685_abs__minus__commute) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1802 (all X hAPP_real_real(uminus_uminus_real,hAPP_complex_real(re,X)) = hAPP_complex_real(re,hAPP_complex_complex(uminus473333897omplex,X))) # label(fact_4123_Re_Ominus) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1803 (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]. 8.94/9.06 1804 (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]. 8.94/9.06 1805 (all N one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),N)) # label(fact_4350_coprime__plus__one__nat) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1806 (all V_4 all W_2 all C_6 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_4),W_2))),C_6) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_4)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_int_complex(number528085621omplex,W_2)),C_6))) # label(fact_2438_add__number__of__diff1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1807 (all N all A_3 hAPP_complex_real(re,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,cis(A_3)),N)) = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),A_3))) # label(fact_4144_cos__n__Re__cis__pow__n) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1808 (all K_2 all 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)) = 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))) # label(fact_4709_Fract__number__of__quotient) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1809 (all Lx_5 all Rx_5 all Ry_3 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_5),Rx_5)),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))) # label(fact_1045_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1810 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_2)) -> hAPP_int_int(div_mod_int(A_3),B_2) = hAPP_P1175774780nt_int(product_snd_int_int,posDivAlg(A_3,B_2))))) # label(fact_4234_mod__pos__pos) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1811 (all Na (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),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_int,zero_zero_int))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_eq_int),Na)))) = hAPP_int_nat(nat,Na))) # label(fact_4743_card__bdd__int__set__l__le) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1812 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> (N != hAPP_nat_nat(suc,M) -> 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]. 8.94/9.06 1813 (all K_2 hAPP_int_int(succ,K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),one_one_int)) # label(fact_857_succ__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1814 (all M_19 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(semiri984289939at_nat,M_19)))) # label(fact_723_zero__le__imp__of__nat) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1815 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K1)),hAPP_int_int(bit0,K2))))) # label(fact_77_less__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1816 (all P (-hBOOL(P) | -hBOOL(hAPP_bool_bool(fNot,P)))) # label(help_fNot_1_1_U) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1817 (all A_3 zero_zero_int = hAPP_int_int(div_mod_int(A_3),A_3)) # label(fact_2986_zmod__self) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1818 (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]. 8.94/9.06 1819 (all Z_1 Z_1 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),Z_1)) # label(fact_2111_real__mult__1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1820 (all B_89 all A_119 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_119)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_89)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_119),B_89)))))) # label(fact_1296_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1821 (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))) -> zero_zero_nat = 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))))) # label(fact_3274_div__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1822 (all Y_14 all X_18 (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_14),X_18)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_18),Y_14)))) # label(fact_2004_not__leE) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1823 (all M_5 all D_4 (hAPP_nat_nat(div_mod_nat(M_5),D_4) = zero_zero_nat -> (exists Q_4 M_5 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,D_4),Q_4)))) # label(fact_3220_mod__eq__0D) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1824 (all X -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(abs_abs_real,X)),one_one_real)),X))) # label(fact_2846_abs__add__one__not__less__self) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1825 (all L all K (is_int(K) & is_int(L) -> (K = zero_zero_int -> 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 -> divmod_int(K,L) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),K)) & (zero_zero_int != L -> divmod_int(K,L) = hAPP_P1975530577nt_int(produc713050258nt_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(sgn_sgn_int,L))),hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,fdisj(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),L)),hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K)),hAPP_bool_bool(hAPP_b589554111l_bool(fconj,hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,L),zero_zero_int)),hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),zero_zero_int)))),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)))))))) # label(fact_4853_divmod__int__pdivmod) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1826 (all X_35 all Y_29 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_35),Y_29)) -> X_35 != Y_29)) # label(fact_1801_order__less__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1827 (all Xa all Ya (is_int(Xa) & is_int(Ya) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)) -> (Ya = Xa <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)))))) # label(fact_2003_linorder__antisym__conv1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1828 (all A_61 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_61),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_61),A_61)),A_61)) # label(fact_1875_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1829 (all K_2 hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_nat_int(semiri1621563631at_int,K_2)),one_one_int) = hAPP_nat_rat(semiri151668891at_rat,K_2)) # label(fact_4686_of__nat__rat) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1830 (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]. 8.94/9.06 1831 (all Xa all Ya (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)) | Ya = Xa)) # label(fact_1026_less__eq__real__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1832 (all Y_39 all X_45 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_39),X_45)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_45),Y_39)) -> Y_39 = X_45))) # label(fact_1640_xt1_I5_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1833 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,pi)),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(cos,Y)),hAPP_real_real(cos,X))))))) # label(fact_3631_cos__monotone__minus__pi__0_H) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1834 (all A_88 all M_12 all N_26 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_88),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_12),N_26)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_88),M_12)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_88),N_26))) # label(fact_1496_power__add) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1835 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(arccos,Y))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arccos,Y)),pi))))) # label(fact_3678_arccos__lt__bounded) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1836 (all P_5 (is_int(P_5) -> (hBOOL(hAPP_int_bool(zprime,P_5)) <-> (all M_1 (is_int(M_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M_1),P_5)) -> P_5 = M_1 | one_one_int = M_1))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),P_5))))) # label(fact_2470_zprime__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1837 (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]. 8.94/9.06 1838 (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]. 8.94/9.06 1839 (all A_156 all C_58 all D_17 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_156),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_58),D_17)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_58),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_156),D_17))) # label(fact_842_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1840 (all Xa (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,Xa)),zero_zero_rat)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),pls)))) # label(fact_611_le__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1841 (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_2030_order__less__le) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1842 (all X_21 all Y_17 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_21),Y_17)) -> Y_17 = X_21 | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_21),Y_17)))) # label(fact_1965_order__le__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1843 (all B_2 negDivAlg(hAPP_int_int(number_number_of_int,min),B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_2),one_one_int))) # label(fact_3277_negDivAlg__minus1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1844 (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)) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(rRset2norRR(A,Ma),A_1)),hAPP_i1948725293t_bool(norRRset,Ma))))))) # label(fact_4590_RRset2norRR__correct) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1845 (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,Xa),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 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 N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A_1,N_1)),Xa)))))) # label(fact_5042_monoseq__le) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1846 (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]. 8.94/9.06 1847 (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]. 8.94/9.06 1848 (all A_202 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_202),hAPP_int_rat(number_number_of_rat,pls)) = A_202) # label(fact_176_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1849 (all X all Y zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(cos,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(sin,Y))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(sin,Y))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3632_sin__cos__add) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1850 (all K_2 hAPP_int_real(number267125858f_real,K_2) = ratreal(hAPP_int_rat(number_number_of_rat,K_2))) # label(fact_4696_Ratreal__number__collapse_I3_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1851 (all X_2 (hBOOL(summable_complex(X_2)) -> hAPP_f352196356l_real(suminf_real,hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_2)) = hAPP_complex_real(im,hAPP_f213450978omplex(suminf_complex,X_2)))) # label(fact_4848_Im_Osuminf) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1852 (all M all N (zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),M) = zero_zero_nat -> N = M))) # label(fact_2476_diffs0__imp__equal) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1853 (all A_132 all B_94 all C_50 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_132),B_94)),C_50) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_132),C_50)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_94),C_50))) # label(fact_1157_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1854 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_3),B_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_2),A_3)) -> hAPP_int_int(abs_abs_int,B_2) = hAPP_int_int(abs_abs_int,A_3)))) # label(fact_2704_zdvd__antisym__abs) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1855 (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]. 8.94/9.06 1856 (all Xa (Xa = one_one_complex <-> one_one_complex = Xa)) # label(fact_852_one__reorient) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1857 (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]. 8.94/9.06 1858 (all X all N hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(real_int,X)),N) = hAPP_int_real(real_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N))) # label(fact_4393_power__real__of__int) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1859 (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]. 8.94/9.06 1860 (all A_45 all B_34 all C_13 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_45),B_34)),C_13)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_45),C_13)))) # label(fact_2263_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1861 (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]. 8.94/9.06 1862 (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]. 8.94/9.06 1863 (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]. 8.94/9.06 1864 (all P all Q all R hAPP_r1250527377l_real(P,hAPP_nat_real(Q,R)) = hAPP_n546711566l_real(hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R_008) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1865 (all A all Xa (hBOOL(hAPP_int_bool(nat_is_nat,Xa)) -> (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_int_nat(nat,Xa)),image_int_nat(nat,A))) <-> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),A)))))) # label(fact_5147_transfer__int__nat__set__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1866 (all B_1 all A_1 all Ma (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),hAPP_i1948725293t_bool(bnorRset(A_1),Ma))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),A_1)))) # label(fact_4291_Bnor__mem__zle) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1867 (all Lx_2 all Ly_2 all Rx_2 all Ry_2 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_2),Ly_2)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_2),Ry_2)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_2),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Ly_2),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_2),Ry_2)))) # label(fact_1070_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.06 1868 (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(log(hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(exp_real,one_one_real))),hAPP_real_real(ln,X)))) # label(fact_3454_log__base__10__eq2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1869 (all K_2 all A_3 all B_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),A_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),B_2)),M)))) # label(fact_2577_zcong__scalar2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1870 (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]. 8.94/9.07 1871 (all V_2 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,hAPP_int_nat(number_number_of_nat,V_1)),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_nat(number_number_of_nat,V_2)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_1)),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_1),V_2))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls)) -> hAPP_int_nat(number_number_of_nat,V_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_1)),hAPP_int_nat(number_number_of_nat,V_2)))))) # label(fact_80_add__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1872 (all R_2 all X all Y hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_2)),hAPP_real_complex(hAPP_r265291036omplex(complex,X),Y)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_2),X)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_2),Y))) # label(fact_3986_complex__of__real__mult__Complex) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1873 (all N_56 all A_200 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_200)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_56)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_200),N_56)))))) # label(fact_251_one__less__power) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1874 (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]. 8.94/9.07 1875 (all Xa hBOOL(hAPP_real_bool(isCont_real_real(sin),Xa))) # label(fact_5159_isCont__sin) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1876 (all N hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))))) # label(fact_3858_inv__real__of__nat__fact__ge__zero) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1877 (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]. 8.94/9.07 1878 (all S ((exists K_1 hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,S),hAPP_n1699378549t_bool(ord_lessThan_nat,K_1)))) <-> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S)))) # label(fact_4950_finite__nat__iff__bounded) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1879 (all X_42 all Y_36 (Y_36 = X_42 -> 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]. 8.94/9.07 1880 (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]. 8.94/9.07 1881 (all A_37 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,one_one_real),A_37))) # label(fact_2312_one__dvd) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1882 (all W_4 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,W_4)),hAPP_int_nat(number_number_of_nat,W_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,W_4)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_1914_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1883 (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]. 8.94/9.07 1884 (all A_1 all C all B_1 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_1),C)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_1),C))) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_1),B_1)))) # label(fact_546_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1885 (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]. 8.94/9.07 1886 (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_int,Xa),Ya)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat,Xa)),hAPP_int_nat(nat,Ya))))))) # label(fact_2785_transfer__nat__int__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1887 (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,Xa),Ya)))) # label(fact_1975_order__less__imp__le) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1888 (all Xa all Ya (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_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(power_power_nat,Xa),Ya))))) # label(fact_3807_even__power__nat) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1889 (all A_3 all B_2 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))))) = 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))))) # label(fact_1023_zadd__power2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1890 (all Na (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_int_real(real_int,Na))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Na)))) # label(fact_4438_real__of__int__gt__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1891 (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]. 8.94/9.07 1892 (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]. 8.94/9.07 1893 (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),Rx_3)),Ly_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)) # label(fact_1063_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1894 (all A_50 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_50),zero_zero_rat))) # label(fact_2206_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1895 (all Ma all Na ((-(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) | Na = zero_zero_nat) -> hAPP_P1586233937at_nat(produc1391996073at_nat(hAPP_f837737749at_nat(hAPP_f999082675at_nat(cOMBB_348438404at_nat,product_Pair_nat_nat),suc)),divmod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),Na),Na)) = divmod_nat(Ma,Na)) & (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) | zero_zero_nat = Na -> divmod_nat(Ma,Na) = hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,zero_zero_nat),Ma)))) # label(fact_4850_divmod__nat__if) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1896 (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]. 8.94/9.07 1897 (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))) <-> Na = Ma))) # label(fact_3011_not__less__less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1898 (all Z_1 (is_int(Z_1) -> -(all M_1 all N_1 Z_1 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,M_1)),hAPP_nat_int(semiri1621563631at_int,N_1))))) # label(fact_3326_int__diff__cases) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1899 (all X hAPP_real_real(abs_abs_real,hAPP_nat_real(real_nat,X)) = hAPP_nat_real(real_nat,X)) # label(fact_3692_abs__real__of__nat__cancel) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1900 (all N all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,M),N)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(fact_fact_int,M)),hAPP_int_int(fact_fact_int,N)))))) # label(fact_4275_fact__less__mono__int) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1901 (all Xa all X_2 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Xa),X_2)) -> ((exists Z 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)))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),comple124823625p_real(X_2)))))) # label(fact_4632_Sup__upper__EX) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1902 (all Lx_6 all Rx_6 all 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)) = 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))) # label(fact_1038_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1903 (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]. 8.94/9.07 1904 (all P all Q all R hAPP_P1982799375l_real(hAPP_r931665397l_real(P,R),hAPP_r1195171167l_real(Q,R)) = hAPP_r938550413l_real(hAPP_f1293550189l_real(hAPP_f183342685l_real(cOMBS_794971624l_real,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__prod_Itc__RealDef__Oreal_Mtc___037) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1905 (all Na all Ma (Na = zero_zero_nat & zero_zero_nat = Ma <-> hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,Na)) = hAPP_nat_int(semiri1621563631at_int,Ma))) # label(fact_3476_negative__eq__positive) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1906 (all Xa all Ya (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Xa),Ya)) & Xa != Ya <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Xa),Ya)))) # label(fact_2028_order__less__le) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1907 (all N all M ((M = zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) = zero_zero_nat) & (zero_zero_nat != M -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N))))) # label(fact_2548_mult__eq__if) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1908 (all B_75 all A_104 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_104)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_75)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_104),B_75)))))) # label(fact_1378_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1909 (all P_1 all Na all K (is_int(K) & is_int(Na) -> ((K = zero_zero_int -> hBOOL(hAPP_int_bool(P_1,Na))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K)) -> (all I_2 all J_1 (is_int(J_1) & is_int(I_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,J_1),K)) & 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_eq_int,zero_zero_int),J_1)) -> hBOOL(hAPP_int_bool(P_1,J_1)))))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),zero_zero_int)) -> (all I_2 all J_1 (is_int(I_2) & is_int(J_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),J_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_1),zero_zero_int)) & hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),I_2)),J_1) = Na -> hBOOL(hAPP_int_bool(P_1,J_1)))))) <-> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(div_mod_int(Na),K)))))) # label(fact_3125_split__zmod) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1910 (all P all Q all R hAPP_i418383825t_bool(hAPP_f1222178131t_bool(hAPP_f743790483t_bool(cOMBB_1647910634ol_int,P),Q),R) = hAPP_f737665877t_bool(P,hAPP_i157317923t_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__fun_) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1911 (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]. 8.94/9.07 1912 (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]. 8.94/9.07 1913 (all X_8 all N_12 (X_8 = one_one_complex | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_12)) -> 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]. 8.94/9.07 1914 (all B_112 all A_164 all C_66 (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_66),A_164) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_112),A_164) -> C_66 = B_112)) # label(fact_778_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1915 (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]. 8.94/9.07 1916 (all A_1 all D_1 all Na 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))) = 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))))) # label(fact_4952_arith__series__nat) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1917 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(cos,X)))))) # label(fact_3662_cos__ge__zero) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1918 (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)) <-> hAPP_int_int(legacy_zgcd(Na),Ma) = Ma)))) # label(fact_4511_zdvd__iff__zgcd) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1919 (all D all Xa (zero_zero_real != Xa -> ((hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),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)))) = D) -> ((hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),zero_zero_real)) -> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,Xa)))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = D) -> hBOOL(hAPP_real_bool(deriv_real(sqrt,Xa),D)))))) # label(fact_3874_DERIV__real__sqrt__generic) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1920 (all B_134 all C_78 all A_196 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_196)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_134),C_78)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_134),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_196),C_78)))))) # label(fact_393_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1921 (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]. 8.94/9.07 1922 (all X_59 all Y_49 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_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(power_power_nat,Y_49),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_49)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_59),Y_49))))) # label(fact_992_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1923 (all A_3 all B_2 hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = hAPP_P1175774780nt_int(product_fst_int_int,divmod_int(A_3,B_2))) # label(fact_4223_div__int__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1924 (all N zero_zero_real = hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,pi),hAPP_nat_real(real_nat,N)))) # label(fact_3728_sin__npi2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1925 (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]. 8.94/9.07 1926 (all K_2 ((zero_z891286103de_int = K_2 -> zero_zero_int = quickcheck_int_of(K_2)) & (K_2 != zero_z891286103de_int -> (zero_z891286103de_int = hAPP_Q1762011733de_int(div_mo231679042de_int(K_2),hAPP_i1732201573de_int(number1226105091de_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),quickcheck_int_of(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,K_2),hAPP_i1732201573de_int(number1226105091de_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = quickcheck_int_of(K_2)) & (zero_z891286103de_int != hAPP_Q1762011733de_int(div_mo231679042de_int(K_2),hAPP_i1732201573de_int(number1226105091de_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) -> quickcheck_int_of(K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),quickcheck_int_of(hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,K_2),hAPP_i1732201573de_int(number1226105091de_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),one_one_int))))) # label(fact_4249_Quickcheck__Narrowing_Oint__of__code) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1927 (all A_1 all B_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B_1)) -> -hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),hAPP_i1948725293t_bool(d22set,A_1))))) # label(fact_2942_d22set__le__swap) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1928 (all M all N 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,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(plus_plus_nat,M),N)),K_2)) # label(fact_2099_add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1929 (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]. 8.94/9.07 1930 (all E_1 all B_2 all D_3 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_3),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,E_1),B_2)) -> (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) = one_one_int -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),E_1) = one_one_int)))) # label(fact_4983_coprime__divisors__nat) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1931 (all A_3 all B_2 all C_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),C_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),B_2)),C_1)) # label(fact_2883_div__mult2__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1932 (all P all Q all R hAPP_nat_nat(hAPP_f1914919701at_nat(hAPP_f1408247010at_nat(cOMBS_nat_nat_nat,P),Q),R) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(P,R),hAPP_nat_nat(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1933 (all X all Y hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X),hAPP_complex_complex(invers1449016382omplex,Y)) = hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,X),Y)) # label(fact_3966_complex__divide__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1934 (all K_2 all L_2 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2)) -> hAPP_int_int(div_mod_int(K_2),L_2) = hAPP_int_int(z3mod(K_2),L_2)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2)) -> hAPP_int_int(z3mod(K_2),L_2) = hAPP_int_int(div_mod_int(K_2),hAPP_int_int(uminus_uminus_int,L_2))))) # label(fact_3563_z3mod__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.07 1935 (all P all Q all R hAPP_bool_bool(hAPP_r1938957037l_bool(P,R),hAPP_real_bool(Q,R)) = hAPP_real_bool(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__HOL__Obool_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1936 (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]. 8.94/9.08 1937 (all P all Q all R hAPP_real_bool(P,hAPP_int_real(Q,R)) = hAPP_int_bool(hAPP_f429384717t_bool(hAPP_f1173902867t_bool(cOMBB_real_bool_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__HOL__Obool_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1938 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K1)),hAPP_int_int(bit1,K2))))) # label(fact_726_less__eq__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1939 (all K_2 min != hAPP_int_int(bit0,K_2)) # label(fact_2166_rel__simps_I45_J) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1940 (all A_52 all K_7 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,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_real)) -> zero_zero_real = A_52)) # label(fact_2143_even__power__le__0__imp__0) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1941 (all Na all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Na),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))))) # label(fact_4445_int__less__real__le) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1942 (all X hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X)) = hAPP_complex_real(im,hAPP_complex_complex(cnj,X))) # label(fact_4092_complex__Im__cnj) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1943 (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]. 8.94/9.08 1944 (all 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)))))) <-> hBOOL(vanishes(X_2)))) # label(fact_5150_vanishes__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1945 (all Z_14 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_14),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_14),Z_14)) # label(fact_1883_mult__2__right) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1946 (all P all Q all R hAPP_f2144054103l_bool(P,hAPP_i1948725293t_bool(Q,R)) = hAPP_i1823918693l_bool(hAPP_f1344929775l_bool(hAPP_f1830214065l_bool(cOMBB_793166024ol_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__fun_Itc__020) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1947 (all A_1 (is_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)))) = zero_zero_int <-> zero_zero_int = A_1))) # label(fact_22_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1948 (all A_1 all B_1 all C all D_1 all Xa (is_int(A_1) & is_int(C) & is_int(B_1) -> (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),B_1)),C) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,D_1),hAPP_int_nat(nat,Xa)) -> (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)) -> (one_one_int = hAPP_int_int(legacy_zgcd(A_1),B_1) -> (one_one_int = hAPP_int_int(legacy_zgcd(B_1),C) -> (one_one_int = hAPP_int_int(legacy_zgcd(C),A_1) -> (exists K_1 exists L_1 exists M_1 (is_int(K_1) & is_int(M_1) & C = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_1),hAPP_int_nat(nat,Xa)) & B_1 = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,L_1),hAPP_int_nat(nat,Xa)) & hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K_1),hAPP_int_nat(nat,Xa)) = A_1 & is_int(L_1))))))))))) # label(fact_4535_int__triple__relprime__odd__power__divisors) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1949 (all Xa all A (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),A)) <-> hBOOL(hAPP_int_bool(A,Xa)))) # label(fact_144_mem__def) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1950 (all B_2 all A_3 all X (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),X)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),A_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),X)) & -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_4329_gcd__semilattice__nat_Oless__infI1) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1951 (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]. 8.94/9.08 1952 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_2899_div__2__gt__zero) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1953 (all A_1 all C all B_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_1),B_1)) | zero_zero_rat = C <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_1),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_1),C))))) # label(fact_2356_dvd__mult__cancel__right) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1954 (all C_40 all A_97 all B_68 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_97),B_68)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_40)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_97),C_40)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_68),C_40)))))) # label(fact_1423_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1955 (all M_5 all D_4 (is_int(M_5) -> (hAPP_int_int(div_mod_int(M_5),D_4) = zero_zero_int -> (exists Q_4 (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D_4),Q_4) = M_5 & is_int(Q_4)))))) # label(fact_3166_zmod__eq__0D) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1956 (all Y all X all U_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),U_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),U_1)))))))) # label(fact_3505_lemma__sqrt__hcomplex__capprox) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1957 (all A_3 all N all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),N))) -> (zero_zero_nat != N -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_3),B_2))))) # label(fact_2956_pow__divides__pow__int) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1958 (all R_2 exists X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_P1175774780nt_int(product_snd_int_int,X_1))) & (all Y_1 (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,Y_1)),hAPP_P1175774780nt_int(product_snd_int_int,Y_1)) = one_one_int & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_P1175774780nt_int(product_snd_int_int,Y_1))) & R_2 = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_P1175774780nt_int(product_fst_int_int,Y_1)),hAPP_P1175774780nt_int(product_snd_int_int,Y_1)) -> Y_1 = X_1)) & hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,X_1)),hAPP_P1175774780nt_int(product_snd_int_int,X_1)) = one_one_int & hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_P1175774780nt_int(product_fst_int_int,X_1)),hAPP_P1175774780nt_int(product_snd_int_int,X_1)) = R_2)) # label(fact_5023_quotient__of__unique) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1959 (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)) -> (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) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I_1),U)),Ma) <-> Ma = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),I_1)),U)),Na)))) # label(fact_2536_nat__eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1960 (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]. 8.94/9.08 1961 (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]. 8.94/9.08 1962 (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]. 8.94/9.08 1963 (all P_1 all I_1 (is_int(I_1) -> (hBOOL(hAPP_nat_bool(P_1,hAPP_int_nat(nat,I_1))) <-> (all N_1 (hAPP_nat_int(semiri1621563631at_int,N_1) = I_1 -> hBOOL(hAPP_nat_bool(P_1,N_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)))))) # label(fact_2788_split__nat) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1964 (all X hAPP_complex_real(re,X) = hAPP_complex_real(re,hAPP_complex_complex(cnj,X))) # label(fact_4114_complex__Re__cnj) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1965 (all M_20 all N_38 all A_175 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_175)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_175),M_20)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_175),N_38))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_20),N_38))))) # label(fact_628_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1966 (all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> X = Y | -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)))) # label(fact_2643_dvd_Ole__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1967 (all Xa all Ya (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) <-> Xa != Ya & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)))) # label(fact_2033_order__less__le) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1968 (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(arccos,Xa),hAPP_real_real(inverse_inverse_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))))))) # label(fact_3865_DERIV__arccos) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1969 (all A_35 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_35),zero_zero_real) = A_35) # label(fact_2331_diff__0__right) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1970 (all X hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(sin,hAPP_real_real(uminus_uminus_real,X))),hAPP_real_real(sin,X))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(cos,hAPP_real_real(uminus_uminus_real,X))),hAPP_real_real(cos,X))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_real) # label(fact_3633_sin__cos__minus) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1971 (all X hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),natceiling(X)))) # label(fact_3309_zero__le__natceiling) # label(axiom) # label(non_clause). [assumption]. 8.94/9.08 1972 (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]. 9.02/9.08 1973 (all A_45 all B_34 all C_13 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_45),B_34)),C_13)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_45),C_13)))) # label(fact_2258_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 1974 (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)) & 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))))) & zero_zero_real = hAPP_real_real(cos,Y_1) -> 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]. 9.02/9.08 1975 (all X_17 all Y_13 (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_17),Y_13)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_13),X_17)))) # label(fact_2009_leI) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 1976 (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]. 9.02/9.08 1977 (all X all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),hAPP_int_int(multInv(P_3),X))),hAPP_int_int(multInv(P_3),hAPP_int_int(multInv(P_3),X)))),X),P_3)))))) # label(fact_2918_Int2_Oaux____2) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 1978 (all N_24 all A_77 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_77)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_77),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_77),N_24)))))) # label(fact_1602_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 1979 (all N hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N))))) # label(fact_3798_real__of__nat__fact__gt__zero) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 1980 (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]. 9.02/9.08 1981 (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]. 9.02/9.08 1982 (all P all Q all R hAPP_r1134773055l_bool(hAPP_f1884005689l_bool(hAPP_f172451459l_bool(cOMBB_421479067l_real,P),Q),R) = hAPP_f330985925l_bool(P,hAPP_r2019696015l_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__prod_Itc__RealDef__Orea) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 1983 (all P all Q all R hAPP_nat_fun_int_int(hAPP_f1935805932nt_int(hAPP_f125788821nt_int(cOMBB_859312247nt_nat,P),Q),R) = hAPP_int_fun_int_int(P,hAPP_nat_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_001) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 1984 (all X_64 all Y_53 -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_64),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_53),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_rat))) # label(fact_217_not__sum__power2__lt__zero) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 1985 (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]. 9.02/9.08 1986 (all Ma all Na all A_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_1)) -> (Na = Ma <-> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_1),Na) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_1),Ma)))) # label(fact_424_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 1987 (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]. 9.02/9.08 1988 -(all S_2 all W_1 (is_int(S_2) & is_int(W_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(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(abs_abs_int,W_1))),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,y),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_2),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n)))) = W_1))) # label(fact_2615__096_B_Bthesis_O_A_I_B_Bs_Aw_O_Aw_A_061_Ay_A_N_As_A_K_A_I1_A_L_Aint_An) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 1989 (all X hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_3623_sin__squared__eq) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 1990 (all X all Y hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y))) # label(fact_4975_abs__gcd__int) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 1991 (all Xa all Ya (is_int(Xa) & is_int(Ya) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)) | Ya = Xa <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya))))) # label(fact_2020_order__le__less) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 1992 (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]. 9.02/9.08 1993 (all B_2 all Q_1 all R_2 all B_5 all Q_5 all 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) = 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) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_5)),R_4))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_4),B_5)) -> (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,zero_zero_int),B_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_5),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q_5))))))))) # label(fact_2065_zdiv__mono2__lemma) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 1994 (all B_116 all A_170 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_170)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_116)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_170),B_116)))))) # label(fact_668_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 1995 (all A_147 A_147 = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,zero_zero_real),A_147)) # label(fact_942_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 1996 (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)) -> 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,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]. 9.02/9.08 1997 (all B_76 all A_106 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_76),zero_z891286103de_int)) & hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_106)) | hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_106),zero_z891286103de_int)) & hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_76)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_106),B_76)),zero_z891286103de_int)))) # label(fact_1359_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 1998 (all Xa all Ya all B_1 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),B_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_1),Xa)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_1),Ya)))))) # label(fact_429_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 1999 (all N hAPP_C498520661umeral(hAPP_C1594335432umeral(minus_1690775515umeral,code_S1047413653umeral(N)),one_on1645066479umeral) = N) # label(fact_4084_Suc__code__numeral__minus__one) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2000 (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]. 9.02/9.08 2001 (all A_122 all N_29 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_122),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_122),N_29)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_122),N_29)),A_122)) # label(fact_1235_power__commutes) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2002 (all X hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = 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))) # label(fact_3628_cos__double) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2003 (all A_139 all B_97 all C_51 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_139),B_97)),C_51) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_139),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_97),C_51))) # label(fact_1054_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2004 (all A_59 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_59),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,A_59),A_59)) # label(fact_1901_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2005 (all X_41 all Y_35 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_35),X_41)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_41),Y_35)))) # label(fact_1711_linorder__linear) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2006 (all A_57 all B_39 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_57),B_39)) -> (A_57 != B_39 -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_57),B_39))))) # label(fact_1958_order__le__neq__trans) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2007 (all Z_1 all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),W)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z_1)))) # label(fact_505_zle__linear) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2008 (all A_1 all B_1 (is_int(B_1) & is_int(A_1) -> (zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),B_1) <-> zero_zero_int = A_1 | zero_zero_int = B_1))) # label(fact_1123_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2009 (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]. 9.02/9.08 2010 (all M all K_2 all N (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K_2),N) = 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_4964_coprime__dvd__mult__int) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2011 (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_eq_real,zero_zero_real),hAPP_real_real(ln,Xa))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),Xa))))) # label(fact_3369_ln__ge__zero__iff) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2012 (all C_31 all A_85 all B_58 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_31),A_85)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_31),B_58))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_31)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_85),B_58))))) # label(fact_1531_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2013 (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]. 9.02/9.08 2014 (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]. 9.02/9.08 2015 (all R_2 all A_3 hAPP_complex_real(im,rcis(R_2,A_3)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_2),hAPP_real_real(sin,A_3))) # label(fact_4262_Im__rcis) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2016 (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]. 9.02/9.08 2017 (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]. 9.02/9.08 2018 (all N_27 all M_16 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),M_16)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),N_27)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,M_16),N_27)))))) # label(fact_1451_less__1__mult) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2019 (all A_1 all B_1 all C all Xa (is_int(A_1) -> (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),B_1) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,C),hAPP_int_nat(nat,Xa)) -> (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)) -> (one_one_int = hAPP_int_int(legacy_zgcd(A_1),B_1) -> (exists K_1 (A_1 = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K_1),hAPP_int_nat(nat,Xa)) & is_int(K_1))))))))) # label(fact_4528_int__relprime__odd__power__divisors) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2020 (all B_5 all A_5 all A_3 all B_2 (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) != zero_zero_int -> (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)) -> (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)) -> one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_5),B_5))))) # label(fact_5004_gcd__coprime__int) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2021 (all X hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) # label(fact_3624_cos__squared__eq) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2022 (all Ma all Na (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(semiri1424489471de_int,Ma)),hAPP_n522471361de_int(semiri1424489471de_int,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)))) # label(fact_356_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2023 (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]. 9.02/9.08 2024 (all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Y)))) # label(fact_4158_termination__basic__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2025 (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]. 9.02/9.08 2026 (all Lx_2 all Ly_2 all Rx_2 all Ry_2 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_2),Ly_2)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_2),Ry_2)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_2),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ly_2),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_2),Ry_2)))) # label(fact_1068_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2027 (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]. 9.02/9.08 2028 (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]. 9.02/9.08 2029 (all Xa all Ya (Xa = Ya | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ya),Xa)) <-> -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]. 9.02/9.08 2030 (all D_3 all A_3 all B_2 (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)) -> one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),A_3))) # label(fact_5001_coprime__lmult__int) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2031 (all P all Q all R hAPP_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,P),Q),R) = hAPP_real_real(P,hAPP_nat_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__Nat__Ona) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2032 (all B_96 all A_136 (zero_zero_nat != A_136 -> (zero_zero_nat != B_96 -> zero_zero_nat != hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_136),B_96)))) # label(fact_1129_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2033 (all Wa all Ya all Xa all Z_2 (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)) = 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)) <-> Z_2 = Ya | Xa = Wa)) # label(fact_1189_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2034 (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]. 9.02/9.08 2035 (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]. 9.02/9.08 2036 (all M_6 all N_10 ((zero_zero_nat = N_10 -> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,M_6),N_10) = one_one_complex) & (zero_zero_nat != N_10 -> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,M_6),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)))))) # label(fact_2609_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2037 (all A_126 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,one_one_rat),A_126) = A_126) # label(fact_1206_mult__1) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2038 (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]. 9.02/9.08 2039 (all A_55 all B_37 (B_37 != A_55 -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_55),B_37)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_55),B_37))))) # label(fact_1997_order__neq__le__trans) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2040 (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]. 9.02/9.08 2041 (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_real,Xa),one_one_real)) -> hBOOL(tendsto_nat_real(hAPP_r474017924t_real(power_power_real,Xa),zero_zero_real,sequentially))))) # label(fact_5046_LIMSEQ__realpow__zero) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2042 (all A_127 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,one_one_rat),A_127) = A_127) # label(fact_1199_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2043 (all Xa all Ya (-hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Xa),Ya)) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Ya),Xa)))) # label(fact_2046_linorder__not__less) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2044 (all Ma all Na all A_1 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_1)) -> (Na = Ma <-> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_1),Na) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_1),Ma)))) # label(fact_422_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2045 (all A_22 all B_17 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(abs_abs_real,A_22)),hAPP_real_real(abs_abs_real,B_17)) = hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(abs_abs_real,A_22)),hAPP_real_real(abs_abs_real,B_17)))) # label(fact_2679_abs__add__abs) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2046 (all A_34 all B_26 hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_34),B_26)),B_26) = A_34) # label(fact_2339_add__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2047 (all B_71 all A_100 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_71),A_100))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_100)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_71))))) # label(fact_1405_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2048 (all B_1_1 is_int(archim856651990g_real(B_1_1))) # label(gsy_c_Archimedean__Field_Oceiling_000tc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2049 (all A_150 A_150 = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_150),zero_z126310315umeral)) # label(fact_921_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2050 (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]. 9.02/9.08 2051 (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]. 9.02/9.08 2052 (all X_29 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_29),X_29))) # label(fact_1856_order__less__irrefl) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2053 (all S ((all X_1 (is_int(X_1) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),S)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))))) <-> hBOOL(hAPP_f448129468l_bool(nat_nat_set,S)))) # label(fact_4610_nat__set__def) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2054 (all M all N hAPP_real_real(hAPP_r1250527377l_real(plus_plus_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(plus_plus_nat,M),N))) # label(fact_3705_real__of__nat__add) # label(axiom) # label(non_clause). [assumption]. 9.02/9.08 2055 (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]. 9.02/9.08 2056 (all A_1 all Na all B_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_1),B_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_1),hAPP_nat_nat(suc,Na))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_1),hAPP_nat_nat(suc,Na)))))) # label(fact_4029_coprime__exp2) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2057 (all N hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(real_nat,N))),pi)) = zero_zero_real) # label(fact_3761_sin__2npi) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2058 (all L all U hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,U),L)),one_one_int)) = hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(ord_at875362053st_int(L),U))) # label(fact_5081_card__atLeastAtMost__int) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2059 (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]. 9.02/9.09 2060 (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]. 9.02/9.09 2061 (all N all M hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N))),hAPP_nat_int(semiri1621563631at_int,M)))) # label(fact_3456_negative__zle) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2062 (all B_1 all A_1 all C (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_1),A_1) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),A_1) <-> B_1 = C)) # label(fact_802_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2063 (all V_15 all W_7 hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_15),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))) # label(fact_1094_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2064 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),one_one_real)),hAPP_nat_real(real_nat,natfloor(X))))) # label(fact_3733_real__natfloor__gt__diff__one) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2065 (all Ma all Na (-(-hBOOL(hAPP_nat_bool(even_odd_even_nat,Ma)) <-> -hBOOL(hAPP_nat_bool(even_odd_even_nat,Na))) <-> -hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),Na))))) # label(fact_3785_odd__add) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2066 (all N all A_3 all B_2 (one_one_int = hAPP_int_int(legacy_zgcd(A_3),B_2) -> one_one_int = hAPP_int_int(legacy_zgcd(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N)),B_2))) # label(fact_4504_zgcd__1__power__left__distrib) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2067 (all Xa all Ya (Ya != Xa <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ya),Xa)))) # label(fact_1847_linorder__neq__iff) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2068 (all Xa all Ya (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Xa),Ya)) -> (-hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Xa),Ya)) <-> Xa = Ya))) # label(fact_1969_linorder__antisym__conv2) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2069 (all P all Q all R hAPP_i345030060t_bool(hAPP_f791698359t_bool(hAPP_f847817363t_bool(cOMBB_562416518ol_int,P),Q),R) = hAPP_f428220345t_bool(P,hAPP_i345030060t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__HOL__026) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2070 (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]. 9.02/9.09 2071 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arcsin,Y))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arcsin,Y)),pi)) & hAPP_real_real(sin,hAPP_real_real(arcsin,Y)) = Y))) # label(fact_3665_arcsin__pi) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2072 (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]. 9.02/9.09 2073 (all Ma all Na (Ma = hAPP_nat_nat(suc,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,Ma),hAPP_nat_nat(suc,Na))))) # label(fact_3020_le__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2074 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_real_real(exp_real,hAPP_real_real(ln,X)) = X)) # label(fact_3411_exp__ln) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2075 (all J_2 all I -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),I)),I))) # label(fact_278_not__add__less2) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2076 (all X_55 all Y_47 all 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)) = 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))) # label(fact_1164_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2077 (all M all K_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),K_2)) -> one_one_nat = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M))),K_2))) # label(fact_3199_Suc__times__mod__eq) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2078 (all W hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,W)),hAPP_int_nat(number_number_of_nat,W)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,W))) # label(fact_4747_nat__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2079 (all U hAPP_int_nat(nat,U) = hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(ord_at641636577an_int(zero_zero_int),U))) # label(fact_4918_card__atLeastZeroLessThan__int) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2080 (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,zero_zero_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_1))))) # label(fact_4723_zero__le__Fract__iff) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2081 (all F ((all I_2 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,hAPP_nat_nat(suc,I_2))),hAPP_nat_nat(F,I_2)))) -> (exists I_2 all K_1 hAPP_nat_nat(F,I_2) = hAPP_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),K_1))))) # label(fact_4570_weak__decr__stable) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2082 (all Ma all Na (Na = Ma <-> hAPP_nat_int(semiri1621563631at_int,Na) = hAPP_nat_int(semiri1621563631at_int,Ma))) # label(fact_325_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2083 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,Na))))) # label(fact_309_zless__int) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2084 (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]. 9.02/9.09 2085 (all M_6 all N_10 ((zero_zero_nat != 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))) = 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) = one_one_nat))) # label(fact_2612_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2086 (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]. 9.02/9.09 2087 (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(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,Xa)),hAPP_real_real(sin,Xa))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,Xa)),hAPP_real_real(sin,Xa)))))) # label(fact_4746_DERIV__sin__sin__mult) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2088 (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,zero_zero_real),X_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),pi)) & hAPP_real_real(cos,X_1) = Y & (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)) & hAPP_real_real(cos,Y_1) = Y -> X_1 = Y_1))))))) # label(fact_3667_cos__total) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2089 (all A_101 all B_72 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_101),B_72))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_101)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_72))))) # label(fact_1396_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2090 (all Na all P_1 (-hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(P_1,Na)) -> (exists K_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),Na)) & hBOOL(hAPP_nat_bool(P_1,K_1)) & (all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),K_1)) -> -hBOOL(hAPP_nat_bool(P_1,I_2))))))))) # label(fact_2154_ex__least__nat__le) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2091 (all C_56 all A_145 all B_103 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_56),A_145)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_56),B_103))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_145),B_103)))) # label(fact_955_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2092 (all B_1_1 is_bool(cauchy_real(B_1_1))) # label(gsy_c_SEQ_OCauchy_000tc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2093 (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]. 9.02/9.09 2094 (all C_54 all D_16 all A_143 all B_101 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_143),B_101)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_54),D_16)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_143),C_54)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_101),D_16)))))) # label(fact_966_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2095 (all A_138 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A_138) = zero_zero_rat) # label(fact_1106_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2096 (all X_16 all Y_12 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_12),X_16)) | hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_16),Y_12)))) # label(fact_2036_linorder__le__less__linear) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2097 (all B_2 all Bq all Br all A_3 all C_1 all Aq all Ar (is_int(C_1) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Aq),Ar))) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(B_2,C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Bq),Br))) -> (C_1 != zero_zero_int -> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2),C_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Aq),Bq)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ar),Br)),C_1))),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Ar),Br)),C_1))))))))) # label(fact_3269_zadd1__lemma) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2098 (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]. 9.02/9.09 2099 (all A_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma)) -> (one_one_int = hAPP_int_int(legacy_zgcd(A_1),Ma) -> (exists X_1 ((all Y_1 (is_int(Y_1) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_1),hAPP_i1948725293t_bool(norRRset,Ma))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),Y_1),Ma)) -> Y_1 = X_1))) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(norRRset,Ma))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),X_1),Ma)) & is_int(X_1)))))) # label(fact_4585_norR__mem__unique) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2100 (all N (zero_zero_nat != N -> one_one_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))) # label(fact_4365_coprime__minus__one__nat) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2101 (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]. 9.02/9.09 2102 (all Z_1 hAPP_complex_real(im,of_real_complex(Z_1)) = zero_zero_real) # label(fact_4093_Im__complex__of__real) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2103 (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]. 9.02/9.09 2104 (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]. 9.02/9.09 2105 (all C_1 all A_3 all B_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),C_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_2),C_1)),M)))) # label(fact_2583_zcong__shift) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2106 (all A_1 all B_1 all C (B_1 = C <-> 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]. 9.02/9.09 2107 (all P all Q all R hAPP_P1175774780nt_int(hAPP_f521271395nt_int(hAPP_f11151005nt_int(cOMBB_47643171nt_int,P),Q),R) = hAPP_int_int(P,hAPP_P1175774780nt_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__Int__Oint_000tc__prod_Itc__Int__Oin) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2108 (all A_37 hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,one_one_complex),A_37))) # label(fact_2310_one__dvd) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2109 t = one_one_int -> (exists X_1 exists Y_1 (is_int(Y_1) & 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))))) = 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) & is_int(X_1))) # label(fact_2075__096t_A_061_A1_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2110 (all A_1 all Ra (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Ra)) -> (hBOOL(summable_real(A_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,hAPP_real_real(abs_abs_real,hAPP_f352196356l_real(suminf_real,hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,A_1),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),N_1))))),Ra))))))) # label(fact_4852_suminf__exist__split) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2111 (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]. 9.02/9.09 2112 (all X_29 -hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_29),X_29))) # label(fact_1853_order__less__irrefl) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2113 (all Z_16 all Y_32 all X_38 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_32),X_38)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_16),Y_32)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_16),X_38))))) # label(fact_1732_xt1_I10_J) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2114 (all Xa all Na all Ya (hAPP_int_int(div_mod_int(Xa),Na) = hAPP_int_int(div_mod_int(Ya),Na) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Na),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xa),Ya))))) # label(fact_3043_zmod__eq__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2115 (all A_1 all B_1 (hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_1),B_1) = zero_zero_real <-> A_1 = B_1)) # label(fact_2323_eq__iff__diff__eq__0) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2116 (all P all Q all R hAPP_f747162069nt_int(P,hAPP_int_fun_int_int(Q,R)) = hAPP_i744251628nt_int(hAPP_f836602773nt_int(hAPP_f888249301nt_int(cOMBB_1875755114nt_int,P),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]. 9.02/9.09 2117 (all R1 all A_3 all R2 all B_2 rcis(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R1),R2),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),B_2)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,rcis(R1,A_3)),rcis(R2,B_2))) # label(fact_4263_rcis__mult) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2118 (all I (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),I)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_nat(nat,I))),one_one_nat) = hAPP_list_int_nat(size_size_list_int,quickc666637781d_zero(I)))) # label(fact_4288_length__around__zero) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2119 (all B_87 all A_117 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_117)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_87),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_87),A_117)),zero_zero_real))))) # label(fact_1307_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2120 (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,Xa),zero_zero_real)) | 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),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]. 9.02/9.09 2121 (all Xa all Ya (is_int(Ya) & is_int(Xa) -> (hAPP_int_real(number267125858f_real,Ya) = hAPP_int_real(number267125858f_real,Xa) <-> Xa = Ya))) # label(fact_86_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2122 (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]. 9.02/9.09 2123 (all C_9 all A_38 all B_27 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_38),B_27)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_38),C_9)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_38),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_27),C_9)))))) # label(fact_2305_dvd__add) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2124 (all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> phi(P_3) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)))) # label(fact_2937_phi__prime) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2125 (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]. 9.02/9.09 2126 (all C_1 all A_3 all B_2 all Q_1 all R_2 (hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(A_3,B_2),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),B_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_1)) -> hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(A_3,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),C_1)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,Q_1),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),hAPP_nat_nat(div_mod_nat(Q_1),C_1))),R_2)))))))) # label(fact_4211_divmod__nat__rel__mult2__eq) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2127 (all X all Y hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(cnj,X)),hAPP_complex_complex(cnj,Y)) = hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X),Y))) # label(fact_4078_complex__cnj__mult) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2128 (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]. 9.02/9.09 2129 (all Lx_4 all Ly_4 all Rx_4 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_4),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Ly_4),Rx_4)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_4),Ly_4)),Rx_4)) # label(fact_1058_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2130 (all P all Q all R hAPP_r2019696015l_real(hAPP_f1591894897l_real(hAPP_f2114902373l_real(cOMBB_237479429l_real,P),Q),R) = hAPP_r2019696015l_real(P,hAPP_real_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__p) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2131 (all K (is_int(K) -> (K = min <-> min = hAPP_int_int(bit1,K)))) # label(fact_2162_rel__simps_I47_J) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2132 (all A_134 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,zero_z126310315umeral),A_134) = zero_z126310315umeral) # label(fact_1141_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2133 (all A_1 all Wa (hAPP_int_nat(number_number_of_nat,Wa) != zero_zero_nat & A_1 = zero_zero_real <-> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_1),hAPP_int_nat(number_number_of_nat,Wa)) = zero_zero_real)) # label(fact_184_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2134 (all S (-hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S)) <-> (all M_1 exists N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) & hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,N_1),S)))))) # label(fact_4606_infinite__nat__iff__unbounded__le) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2135 (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)) -> X_1 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa_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]. 9.02/9.09 2136 (all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))) -> A_3 != hAPP_int_int(inv(P_3),A_3))))) # label(fact_2904_inv__distinct) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2137 (all X archim791455193or_rat(X) = archim1246769320r_real(ratreal(X))) # label(fact_4679_real__floor__code) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2138 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),hAPP_nat_nat(suc,Ma))) <-> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)))) # label(fact_3014_not__less__eq) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2139 (all W_11 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),hAPP_int_int(number_number_of_int,W_11))),hAPP_int_int(number_number_of_int,W_11)) = hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W_11))) # label(fact_223_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2140 (all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven)) -> -(all K_1 (is_int(K_1) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),K_1) != Xa))))) # label(fact_3382_zEvenE) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2141 (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]. 9.02/9.09 2142 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(suc,Na))) <-> Na = Ma | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)))) # label(fact_3013_less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2143 (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]. 9.02/9.09 2144 (all A_3 all N all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),A_3))))) # label(fact_2412_zprime__zdvd__power) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2145 (all A_3 all P_3 (is_int(A_3) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),A_3)),one_one_int),P_3)) -> one_one_int = A_3 | hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int) = A_3)))))) # label(fact_2468_zcong__square__zless) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2146 (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(B_2),zero_zero_int),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(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]. 9.02/9.09 2147 (all K_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),K_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),K_2))))) # label(fact_1028_le__trans) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2148 (all I all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),N)) -> I = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),I)))) # label(fact_2490_diff__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2149 (all Lx_5 all Rx_5 all 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) = 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))) # label(fact_1049_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2150 (all Y_43 all X_49 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_49)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_43)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_43),one_one_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_49),Y_43)),X_49)))))) # label(fact_1595_mult__right__le__one__le) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2151 (all N_27 all M_16 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),M_16)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),N_27)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,M_16),N_27)))))) # label(fact_1453_less__1__mult) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2152 (all A_139 all B_97 all C_51 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_139),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_97),C_51)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_139),B_97)),C_51)) # label(fact_1052_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2153 (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]. 9.02/9.09 2154 (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]. 9.02/9.09 2155 (all A_19 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(abs_abs_int,A_19)))) # label(fact_2707_abs__ge__zero) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2156 (all I_1 all J (is_int(I_1) & is_int(J) -> (zero_zero_int = J & I_1 = zero_zero_int <-> hAPP_int_int(legacy_zgcd(I_1),J) = zero_zero_int))) # label(fact_4489_zgcd0) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2157 (all N (one_one_nat != N -> (exists P_4 (hBOOL(hAPP_nat_bool(prime,P_4)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),N)))))) # label(fact_4204_prime__factor) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2158 (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]. 9.02/9.09 2159 (all U_1 all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,U_1),U_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),X)))) # label(fact_3428_real__minus__mult__self__le) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2160 (all A_1 all B_1 all Ma (hAPP_int_int(legacy_zgcd(B_1),Ma) = one_one_int -> (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_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),hAPP_i1948725293t_bool(bnorRset(A_1),Ma))))))) # label(fact_4527_Bnor__mem__if) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2161 (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]. 9.02/9.09 2162 (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]. 9.02/9.09 2163 (all A_135 all B_95 (zero_z126310315umeral = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_135),B_95) -> A_135 = zero_z126310315umeral | zero_z126310315umeral = B_95)) # label(fact_1134_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2164 (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]. 9.02/9.09 2165 (all A_3 all N all B_2 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)),N) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),N))) # label(fact_4986_gcd__exp__int) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2166 (all C all A_1 all B_1 (B_1 = A_1 -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_1),C)) -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),C))))) # label(fact_1680_ord__eq__le__trans) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2167 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_complex_real(im,X)),hAPP_complex_real(im,Y)) = hAPP_complex_real(im,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),Y))) # label(fact_4100_Im_Odiff) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2168 (all Na all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Na)),Ma)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),Ma)))) # label(fact_3057_less__eq__Suc__le) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2169 (all M all N all K_2 all L_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_2),L_2)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),L_2) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))))) # label(fact_295_less__add__eq__less) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2170 (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) -> B_111 = C_65)) # label(fact_787_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2171 (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]. 9.02/9.09 2172 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hAPP_P1175774780nt_int(product_fst_int_int,negDivAlg(A_3,B_2)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)))) # label(fact_4224_div__neg__pos) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2173 (all A_11 all B_11 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_int_int(abs_abs_int,A_11)),hAPP_int_int(abs_abs_int,B_11)))),hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_11),B_11))))) # label(fact_2737_abs__triangle__ineq3) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2174 (all A_1 all B_1 all 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,A_1),B_1)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,Xa),A_1)),hAPP_real_real(hAPP_r1250527377l_real(powr,Xa),B_1)))))) # label(fact_4410_powr__less__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2175 (all Y_39 all X_45 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_39),X_45)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_45),Y_39)) -> Y_39 = X_45))) # label(fact_1636_xt1_I5_J) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2176 (all P all Q all R hAPP_f1576199059t_bool(P,hAPP_r847664445t_bool(Q,R)) = hAPP_r1392521531t_bool(hAPP_f332806115t_bool(hAPP_f1177545951t_bool(cOMBB_1459832681l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__fun_Itc__Int__Oint_Mtc__RealDef__Oreal_J_Mt) # label(axiom) # label(non_clause). [assumption]. 9.02/9.09 2177 (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]. 9.02/9.09 2178 (all P all Q all R hAPP_int_int(P,hAPP_nat_int(Q,R)) = hAPP_nat_int(hAPP_f1139079189at_int(hAPP_f1431025877at_int(cOMBB_int_int_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__Int__Oint_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 9.02/9.10 2179 (all V_23 all W_13 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_23)),hAPP_int_complex(number528085621omplex,W_13)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_23),W_13))) # label(fact_192_add__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 9.02/9.10 2180 (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]. 9.02/9.10 2181 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa)) <-> Xa = hAPP_real_real(exp_real,hAPP_real_real(ln,Xa)))) # label(fact_3412_exp__ln__iff) # label(axiom) # label(non_clause). [assumption]. 9.02/9.10 2182 (all Xa (one_one_real = Xa <-> Xa = one_one_real)) # label(fact_854_one__reorient) # label(axiom) # label(non_clause). [assumption]. 9.02/9.10 2183 (all A_1 all B_1 all C (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_1),C) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_1),B_1) <-> C = B_1)) # label(fact_811_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 9.02/9.10 2184 (all A_45 all B_34 all C_13 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_45),B_34)),C_13)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_45),C_13)))) # label(fact_2260_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 9.02/9.10 2185 (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,hAPP_int_int(standardRes(P_5),Xa)),hAPP_i1948725293t_bool(sRStar,P_5))))))) # label(fact_3209_StandardRes__SRStar__prop4) # label(axiom) # label(non_clause). [assumption]. 9.02/9.10 2186 (all Ma all Na (is_int(Na) & is_int(Ma) -> (Na = zero_zero_int & Ma = zero_zero_int <-> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Ma),Na) = zero_zero_int))) # label(fact_4996_gcd__zero__int) # label(axiom) # label(non_clause). [assumption]. 9.02/9.10 2187 (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]. 9.02/9.10 2188 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N))))) # label(fact_3054_le__imp__less__Suc) # label(axiom) # label(non_clause). [assumption]. 9.02/9.10 2189 (all Ma all K all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),Na)) | zero_zero_nat = K <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,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))))) # label(fact_2515_nat__mult__dvd__cancel__disj_H) # label(axiom) # label(non_clause). [assumption]. 9.02/9.10 2190 (all B_119 all A_173 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_173),zero_z891286103de_int)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_119),zero_z891286103de_int)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_173),B_119)),zero_z891286103de_int))))) # label(fact_648_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption]. 9.02/9.10 2191 (all A_135 all B_95 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_135),B_95) = zero_zero_nat -> zero_zero_nat = B_95 | zero_zero_nat = A_135)) # label(fact_1136_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 9.02/9.10 2192 (all C_69 all D_20 all A_177 all B_121 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_177),B_121)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_69),D_20)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_177),C_69)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_121),D_20)))))) # label(fact_608_add__less__le__mono) # label(axiom) # label(non_clause). [assumption]. 9.02/9.10 2193 (all X hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X)) = hAPP_complex_real(im,hAPP_complex_complex(uminus473333897omplex,X))) # label(fact_4099_Im_Ominus) # label(axiom) # label(non_clause). [assumption]. 9.02/9.10 2194 (all C_17 all A_67 all B_44 (A_67 = B_44 -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_17),B_44)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_17),A_67))))) # label(fact_1760_xt1_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.02/9.10 2195 (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]. 9.02/9.10 2196 (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(plus_plus_nat,hAPP_int_nat(nat,X)),hAPP_int_nat(nat,Y)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y))))) # label(fact_2795_Nat__Transfer_Otransfer__nat__int__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.02/9.10 2197 (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]. 9.02/9.10 2198 (all A_3 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)),A_3) = natfloor(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_nat_real(real_nat,A_3))))) # label(fact_3742_natfloor__add) # label(axiom) # label(non_clause). [assumption]. 9.02/9.10 2199 (all A_1 all A all Ma ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma)) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),A)) & hBOOL(hAPP_int_bool(is_RRset(A),Ma)) -> hAPP_int_int(rRset2norRR(A,Ma),A_1) = hilbert_Eps_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,zcong(A_1)),Ma))),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),hAPP_i1948725293t_bool(norRRset,Ma))))) & (-(hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma)) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),A)) & hBOOL(hAPP_int_bool(is_RRset(A),Ma))) -> zero_zero_int = hAPP_int_int(rRset2norRR(A,Ma),A_1)))) # label(fact_4891_RRset2norRR__def) # label(axiom) # label(non_clause). [assumption]. 9.02/9.10 2200 (all X_64 all Y_53 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_64),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_53),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_real))) # label(fact_218_not__sum__power2__lt__zero) # label(axiom) # label(non_clause). [assumption]. 9.02/9.10 2201 (all X_2 (hBOOL(cauchy_real(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))))))))))) # label(fact_4374_Cauchy__iff2) # label(axiom) # label(non_clause). [assumption]. 9.02/9.10 2202 (all N all I ((zero_z126310315umeral = I -> hAPP_nat_nat(code_nat_of_aux(I),N) = N) & (zero_z126310315umeral != I -> hAPP_nat_nat(code_nat_of_aux(hAPP_C498520661umeral(hAPP_C1594335432umeral(minus_1690775515umeral,I),one_on1645066479umeral)),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(code_nat_of_aux(I),N)))) # label(fact_3384_nat__of__aux__code) # label(axiom) # label(non_clause). [assumption]. 9.02/9.10 2203 (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]. 9.02/9.10 2204 (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]. 9.02/9.10 2205 (all V_9 all U_3 all Y_23 all X_28 all A_63 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_28),A_63)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_23),A_63)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),U_3)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),V_9)) -> (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,U_3),V_9) = one_one_real -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,U_3),X_28)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,V_9),Y_23))),A_63)))))))) # label(fact_1858_convex__bound__le) # label(axiom) # label(non_clause). [assumption]. 9.02/9.10 2206 (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]. 9.27/9.35 2207 (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]. 9.27/9.35 2208 (all A 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,Z),X_1)) -> (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),A)) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_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(plus_plus_int,X_1),D))))))) -> ((all X_1 all K_1 (is_int(X_1) & is_int(K_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)))) <-> hBOOL(hAPP_int_bool(P_2,X_1))))) -> ((exists X1 (is_int(X1) & hBOOL(hAPP_int_bool(P_1,X1)))) <-> (exists X_1 (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D))) & hBOOL(hAPP_int_bool(P_2,X_1)) & is_int(X_1))) | (exists X_1 (is_int(X_1) & (exists Xa_1 (is_int(Xa_1) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),A)) & hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xa_1),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_5181_cppi) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2209 (all B_1_1 all B_2_1 is_bool(hAPP_f54304608l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2210 (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]. 9.27/9.35 2211 (all Xa 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,Ya),Xa)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)))) # label(fact_2026_less__le__not__le) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2212 (all P_3 hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_2859_Euler_Oaux____2) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2213 (all V_6 all B_21 all C_7 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_6)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_21),C_7)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_6)),B_21)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_6)),C_7))) # label(fact_2395_right__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2214 (all Na hAPP_real_real(uminus_uminus_real,hAPP_nat_real(sin_coeff,Na)) = hAPP_nat_real(diffs_real(cos_coeff),Na)) # label(fact_4878_cos__fdiffs2) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2215 (all Z_1 all X all Y all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),Y),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Z_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y),Z_1)),M)))) # label(fact_2585_zcong__zpower) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2216 (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]. 9.27/9.35 2217 (all Z_4 all Z_1 all W_9 all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_9),W)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_4),Z_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W_9),Z_4)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),Z_1)))))) # label(fact_732_zadd__zless__mono) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2218 (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]. 9.27/9.35 2219 (all X all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)) = hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(uminus_uminus_real,X)))) # label(fact_3907_real__root__minus) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2220 (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]. 9.27/9.35 2221 (all N all X (is_int(N) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,N)),one_one_real)),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_int_real(real_int,N))) -> archim856651990g_real(X) = N)))) # label(fact_4469_ceiling__eq3) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2222 (all X (zero_zero_real != X -> hAPP_real_real(arctan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),X)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sgn_sgn_real,X)),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_real_real(arctan,X)))) # label(fact_3587_arctan__inverse) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2223 (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,Xa)),hAPP_real_real(F,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa),H_1)))))))))))) # label(fact_3882_DERIV__pos__inc__right) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2224 (all A_3 all Q_1 all R_2 (is_int(A_3) & is_int(Q_1) -> (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))) -> (A_3 != zero_zero_int -> one_one_int = Q_1)))) # label(fact_3259_self__quotient) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2225 (all Z_17 all X_44 all Y_38 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_44),Y_38)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_38),Z_17)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_44),Z_17))))) # label(fact_1647_order__trans) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2226 (all P_1 all I_1 hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),P_1)),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),I_1)))))) # label(fact_4765_finite__M__bounded__by__nat) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2227 (all P all Q all R hAPP_f237669397t_bool(hAPP_P877703507t_bool(P,R),hAPP_P989584703t_bool(Q,R)) = hAPP_P989584703t_bool(hAPP_f1781059817t_bool(hAPP_f125837377t_bool(cOMBS_1610675060t_bool,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc__033) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2228 (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]. 9.27/9.35 2229 (all C_41 all D_14 all A_108 all B_78 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_108),B_78)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_41),D_14)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_78)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_41)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_108),C_41)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_78),D_14)))))))) # label(fact_1349_mult__mono) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2230 (all A_3 cis(A_3) = rcis(one_one_real,A_3)) # label(fact_4259_cis__rcis__eq) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2231 (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]. 9.27/9.35 2232 (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]. 9.27/9.35 2233 (all P all Q all R hAPP_f1512942609t_bool(P,hAPP_i1948725293t_bool(Q,R)) = hAPP_i872162435t_bool(hAPP_f1251097615t_bool(hAPP_f1475822739t_bool(cOMBB_2027861294ol_int,P),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]. 9.27/9.35 2234 (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]. 9.27/9.35 2235 (all P_5 (hBOOL(hAPP_int_bool(zprime,P_5)) -> hAPP_i1948725293t_bool(d22set,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_i1948725293t_bool(wset(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_5))) # label(fact_2936_d22set__eq__wset) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2236 (all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Ya)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_int_rat(number_number_of_rat,Ya))))) # label(fact_107_less__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2237 (all N all K_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),N)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N)))))) # label(fact_4989_gcd__greatest__int) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2238 (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]. 9.27/9.35 2239 (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]. 9.27/9.35 2240 (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]. 9.27/9.35 2241 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X))),X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3383_abs__ln__one__plus__x__minus__x__bound) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2242 (all Ma all Na all A_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_1)) -> (Na = Ma <-> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_1),Ma) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_1),Na)))) # label(fact_421_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2243 (all C_40 all A_97 all B_68 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_97),B_68)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_40)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_97),C_40)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_68),C_40)))))) # label(fact_1420_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2244 (all Xa all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Xa)) | zero_zero_nat = Na <-> 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_306_zero__less__power__nat__eq) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2245 (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) -> (D_1 = C <-> B_1 = A_1))) # label(fact_2225_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2246 (all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),pls)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,Xa)),zero_zero_int)))) # label(fact_106_less__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2247 (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]. 9.27/9.35 2248 (all P all Q all R hAPP_r2019696015l_real(hAPP_f320596107l_real(hAPP_f850851693l_real(cOMBS_193881978l_real,P),Q),R) = hAPP_f617735571l_real(hAPP_r712953929l_real(P,R),hAPP_r1250527377l_real(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2249 (all A_130 all B_92 all C_48 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_130),B_92)),C_48) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_130),C_48)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_92),C_48))) # label(fact_1180_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2250 (all K_2 hAPP_int_real(number267125858f_real,K_2) = ratreal(hAPP_int_rat(number_number_of_rat,K_2))) # label(fact_4695_number__of__real__code) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2251 (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]. 9.27/9.35 2252 (all X all Y Y = hAPP_real_real(hAPP_r1250527377l_real(hAPP_b646336293l_real(if_real,fFalse),X),Y)) # label(help_If_2_1_If_000tc__RealDef__Oreal_T) # label(axiom) # label(non_clause). [assumption]. 9.27/9.35 2253 (all N (-hBOOL(hAPP_nat_bool(prime,N)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N)) -> (exists M_1 exists K_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),K_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N)) & N = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),K_1) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_1),N)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M_1)))))) # label(fact_4207_not__prime__ex__mk) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2254 (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]. 9.27/9.36 2255 (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]. 9.27/9.36 2256 (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]. 9.27/9.36 2257 (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]. 9.27/9.36 2258 (all A_3 all B_2 all C_1 all D_3 hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),D_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C_1),D_3))) # label(fact_4678_divide__rat) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2259 (all N (zero_zero_nat != N -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),N)))) # label(fact_4030_coprime__minus1) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2260 (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]. 9.27/9.36 2261 (all A_1 all E_2 all C all B_1 all 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,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_1),B_1)),E_2)),C)),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_2430_less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2262 (all A_19 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(abs_abs_real,A_19)))) # label(fact_2706_abs__ge__zero) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2263 (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]. 9.27/9.36 2264 (all B_2 all Q_1 all R_2 all B_5 all Q_5 all 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) = 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) -> (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_int,R_2),B_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,zero_zero_int),B_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_5),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_5),Q_1))))))))) # label(fact_2148_zdiv__mono2__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2265 (all Z_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Z_1),Z_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_1)) # label(fact_2137_nat__mult__2) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2266 (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]. 9.27/9.36 2267 (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]. 9.27/9.36 2268 (all X hAPP_real_real(sin,hAPP_real_real(uminus_uminus_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X))) # label(fact_3592_sin__minus) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2269 (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]. 9.27/9.36 2270 (all M_26 all N_50 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_26),N_50)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_26)),hAPP_nat_int(semiri1621563631at_int,N_50))))) # label(fact_364_less__imp__of__nat__less) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2271 (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]. 9.27/9.36 2272 (all X all Y 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)))) # label(fact_3601_cos__add) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2273 (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]. 9.27/9.36 2274 (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]. 9.27/9.36 2275 (all Z_1 hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_1),hAPP_complex_complex(cnj,Z_1))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(norm_norm_complex,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_4062_complex__mod__mult__cnj) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2276 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(suc,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)))) # label(fact_3056_less__Suc__eq__le) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2277 (all A_82 all C_28 all B_55 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_82),C_28)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_55),C_28))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_28)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_82),B_55))))) # label(fact_1545_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2278 (all A_1 (is_int(A_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(abs_abs_int,A_1))) <-> A_1 != zero_zero_int))) # label(fact_2719_zero__less__abs__iff) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2279 (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]. 9.27/9.36 2280 (all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),zero_zero_nat)) <-> zero_zero_nat = Na)) # label(fact_1252_le__0__eq) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2281 (all X all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),X)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_int_int(number_number_of_int,min)),P_3))))))) # label(fact_2820_Euler__part1) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2282 (all P_1 all Xa all 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,Xa),Ya)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) -> hBOOL(P_1)))) # label(fact_2628_dvd_Oless__imp__triv) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2283 (all A A = image_int_nat(nat,image_nat_int(semiri1621563631at_int,A))) # label(fact_4602_transfer__int__nat__set__return__embed) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2284 (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]. 9.27/9.36 2285 (all Na all Va (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_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_eq_nat,Na),hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va)))))))) # label(fact_4785_le__Suc__number__of) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2286 (all P all Q all R hAPP_b1482221219l_real(P,hAPP_real_bool(Q,R)) = hAPP_r931665397l_real(hAPP_f1607917947l_real(hAPP_f1076489843l_real(cOMBB_415306361l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__prod_Itc__RealDef__Oreal_) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2287 (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]. 9.27/9.36 2288 (all Code_numeral_5 all Code_numeral_4 (code_S1047413653umeral(Code_numeral_5) = code_S1047413653umeral(Code_numeral_4) <-> Code_numeral_5 = Code_numeral_4)) # label(fact_4070_code__numeral_Oinject) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2289 (all M all N all K_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)),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))) # label(fact_2083_nat__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2290 (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]. 9.27/9.36 2291 (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]. 9.27/9.36 2292 (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]. 9.27/9.36 2293 (all A_1 all B_1 all C all D_1 (is_int(B_1) & is_int(D_1) & is_int(C) & is_int(A_1) -> (hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C),D_1) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_1),B_1) -> (D_1 = C <-> A_1 = B_1)))) # label(fact_2227_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2294 (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]. 9.27/9.36 2295 (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]. 9.27/9.36 2296 (all Q all P (-hBOOL(P) | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)) | -hBOOL(Q))) # label(help_fconj_1_1_U) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2297 (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_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_17)) = zero_zero_int) & (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]. 9.27/9.36 2298 (all M_6 all N_10 ((N_10 = zero_zero_nat -> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,M_6),N_10) = one_on1684967323de_int) & (N_10 != zero_zero_nat -> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,M_6),N_10) = 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)))))) # label(fact_2608_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2299 (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]. 9.27/9.36 2300 (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]. 9.27/9.36 2301 (all Xa all Ya (Xa = Ya <-> hAPP_real_real(exp_real,Xa) = hAPP_real_real(exp_real,Ya))) # label(fact_3388_exp__inj__iff) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2302 (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]. 9.27/9.36 2303 (all Z_17 all X_44 all Y_38 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_44),Y_38)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_38),Z_17)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_44),Z_17))))) # label(fact_1648_order__trans) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2304 (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]. 9.27/9.36 2305 (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]. 9.27/9.36 2306 (all C_73 all A_184 all B_127 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_184),B_127)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_184),C_73)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_127),C_73))))) # label(fact_538_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2307 (all Xa all Na all Diff all F (F = hAPP_n546711566l_real(Diff,zero_zero_nat) -> ((all M_1 all T (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),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))) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))))) -> (exists T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,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_4932_Maclaurin__bi__le) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2308 (all M hAPP_int_int(legacy_zgcd(one_one_int),M) = one_one_int) # label(fact_4488_zgcd__1__left) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2309 (all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,P_3)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_2812_Int2_Oaux__1) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2310 (all A_162 all B_110 all C_64 (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_162),B_110) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_162),C_64) -> B_110 = C_64)) # label(fact_793_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2311 (all A_3 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))))) # label(fact_1910_cube__square) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2312 (all X_61 all Y_51 (is_int(X_61) & is_int(Y_51) -> (hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_51),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_61),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_61)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_51)) -> Y_51 = X_61))))) # label(fact_889_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2313 (all X all Y hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(abs_abs_int,X)),Y)) # label(fact_4979_gcd__abs1__int) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2314 (all L all U hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_gr660468384an_nat(L),U)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,U),hAPP_nat_nat(suc,L))) # label(fact_4626_card__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2315 (all Xa (is_int(Xa) -> (one_one_int = Xa <-> one_one_int = Xa))) # label(fact_856_one__reorient) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2316 (all C_73 all A_184 all B_127 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_184),B_127)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_184),C_73)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_127),C_73))))) # label(fact_537_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2317 (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))) # label(fact_3639_sin__arcsin) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2318 (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]. 9.27/9.36 2319 (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,B_1),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_1)))) # label(fact_1291_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2320 (all Xa hBOOL(hAPP_real_bool(deriv_real(arctan,Xa),hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3873_DERIV__arctan) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2321 (all X all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> X = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),N)))) # label(fact_3930_real__root__pow__pos) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2322 (all Na all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),Ma)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,Na)),hAPP_nat_real(real_nat,Ma))))) # label(fact_3703_real__of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2323 (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]. 9.27/9.36 2324 (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]. 9.27/9.36 2325 (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]. 9.27/9.36 2326 (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))) & (K_2 != zero_zero_nat -> 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_2885_nat__mult__div__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2327 (all Z_2 (hAPP_complex_complex(cnj,Z_2) = zero_zero_complex <-> zero_zero_complex = Z_2)) # label(fact_4076_complex__cnj__zero__iff) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2328 (all A_1 all E_2 all C all B_1 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_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))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_1),B_1)),E_2)),C)),D_1)))) # label(fact_2432_less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2329 (all A_50 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_50),zero_zero_nat))) # label(fact_2204_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2330 (all X_1 hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),cos_coeff)),hAPP_r474017924t_real(power_power_real,X_1))) = hAPP_real_real(cos,X_1)) # label(fact_4821_lemma__cos__ext) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2331 (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]. 9.27/9.36 2332 (all Y all X (hBOOL(hAPP_int_bool(nat_is_nat,X)) -> (hBOOL(hAPP_int_bool(nat_is_nat,Y)) -> hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(div_mod_int(X),Y)))))) # label(fact_5133_Divides_Otransfer__int__nat__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2333 (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]. 9.27/9.36 2334 (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]. 9.27/9.36 2335 (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]. 9.27/9.36 2336 (all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,W)),zero_zero_int)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,one_one_int),hAPP_int_int(number_number_of_int,W)) = hAPP_P1175774780nt_int(product_fst_int_int,hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,one_one_int),hAPP_int_int(uminus_uminus_int,hAPP_int_int(number_number_of_int,W))))))) # label(fact_4222_div__pos__neg__1__number__of) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2337 (all Y all X all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,Y),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3))))))) # label(fact_4405_powr__less__mono2__neg) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2338 (all Z_1 all W hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),W)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(uminus_uminus_int,Z_1)),hAPP_int_int(uminus_uminus_int,W))) # label(fact_3464_zminus__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2339 (all A_1 all C all D_1 all B_1 (is_int(B_1) & is_int(D_1) -> (zero_zero_int != B_1 -> (D_1 != 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,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]. 9.27/9.36 2340 (all X_62 one_on1684967323de_int = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_62),zero_zero_nat)) # label(fact_477_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2341 (all P_1 all L ((exists X_1 (is_int(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))) & hBOOL(hAPP_int_bool(P_1,X_1)))) <-> (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))))) # label(fact_2415_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2342 (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]. 9.27/9.36 2343 (all Q_2 all P_1 ((all A_2 all B_4 (is_int(A_2) & is_int(B_4) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_4)) -> (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_2),B_4) -> hBOOL(hAPP_rat_bool(P_1,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_4))))))) -> hBOOL(hAPP_rat_bool(P_1,Q_2)))) # label(fact_5025_Rat__induct) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2344 (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]. 9.27/9.36 2345 (all A_3 all B_2 all C_1 all D_3 all N (hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,D_3),N) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)),C_1) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N)) -> (one_one_int = hAPP_int_int(legacy_zgcd(A_3),B_2) -> (hAPP_int_int(legacy_zgcd(B_2),C_1) = one_one_int -> (one_one_int = hAPP_int_int(legacy_zgcd(C_1),A_3) -> (exists K_1 exists L_1 exists M_1 (is_int(K_1) & hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_1),N) = hAPP_int_int(abs_abs_int,C_1) & hAPP_int_int(abs_abs_int,B_2) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,L_1),N) & hAPP_int_int(abs_abs_int,A_3) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K_1),N) & is_int(M_1) & is_int(L_1))))))))) # label(fact_4537_int__triple__relprime__power__divisors) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2346 (all B_119 all A_173 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_173),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_119),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_173),B_119)),zero_zero_int))))) # label(fact_651_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2347 (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]. 9.27/9.36 2348 (all A_61 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_61),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_61),A_61)),A_61)) # label(fact_1874_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2349 (all Nat hAPP_nat_nat(nat_size,hAPP_nat_nat(suc,Nat)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(nat_size,Nat)),hAPP_nat_nat(suc,zero_zero_nat))) # label(fact_4055_nat_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2350 (all X_39 all Y_33 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_39),Y_33)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_33),X_39)))) # label(fact_1726_order__less__asym) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2351 (all P all Q all R hAPP_f352196356l_real(P,hAPP_r474017924t_real(Q,R)) = hAPP_real_real(hAPP_f1122023986l_real(hAPP_f1786887945l_real(cOMBB_1896127834l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__Real) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2352 (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]. 9.27/9.36 2353 (all A_23 all B_18 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,A_23)),hAPP_int_int(abs_abs_int,B_18)) = hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_23),B_18))) # label(fact_2677_abs__mult) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2354 (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]. 9.27/9.36 2355 (all B_1_1 is_bool(monoseq_real(B_1_1))) # label(gsy_c_SEQ_Omonoseq_000tc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2356 (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(bit1,K)),pls)))) # label(fact_110_rel__simps_I12_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2357 (all L all U hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,U),L)) = hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(ord_at641636577an_int(L),U))) # label(fact_4919_card__atLeastLessThan__int) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2358 (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]. 9.27/9.36 2359 (all Xa all Ya (Xa = 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)))) # label(fact_1704_order__eq__iff) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2360 (all B_1 all A_1 all C (B_1 = C <-> hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_1),A_1) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C),A_1))) # label(fact_804_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2361 (all P all Q all R hAPP_real_real(hAPP_r1250527377l_real(P,R),hAPP_real_real(Q,R)) = hAPP_real_real(hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__RealDef_) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2362 (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]. 9.27/9.36 2363 (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(times_times_nat,X),Y) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),K_2) -> (exists I_2 exists J_1 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),I_2) = X & hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),J_1) = Y))))) # label(fact_4480_prime__power__mult) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2364 (all P all Q all R hAPP_r50993370t_real(hAPP_f1163894827t_real(hAPP_f1998755145t_real(cOMBB_421464275l_real,P),Q),R) = hAPP_f1566856189t_real(P,hAPP_r474017924t_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Nat__Onat_Mtc__RealDef__Oreal_J_000tc__fun__029) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2365 (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]. 9.27/9.36 2366 (all N hAPP_nat_int(semiri1621563631at_int,N) = archim856651990g_real(hAPP_nat_real(real_nat,N))) # label(fact_4362_ceiling__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 9.27/9.36 2367 (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]. 9.27/9.36 2368 (all K all F all Na (is_int(K) -> ((all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),Na)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),one_one_nat))),hAPP_nat_int(F,I_2)))),one_one_int)))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K)) -> (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))))))))) # label(fact_2814_nat0__intermed__int__val) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2369 (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]. 9.27/9.37 2370 (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]. 9.27/9.37 2371 (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]. 9.27/9.37 2372 (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]. 9.27/9.37 2373 (all A_3 all X (is_int(X) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),A_3)) -> (X != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),one_one_int) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),one_one_int)))))))) # label(fact_2596_Euler_Oaux1) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2374 (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]. 9.27/9.37 2375 (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]. 9.27/9.37 2376 (all Xa all P_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Xa),zero_zero_int),P_5)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_5),Xa)))) # label(fact_2592_zcong__eq__zdvd__prop) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2377 (all X hAPP_real_real(tan,X) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,X)),hAPP_real_real(cos,X))) # label(fact_3723_tan__def) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2378 (all Na all S (hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,S)) -> ((all X_1 (hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_1),S)) -> hBOOL(hAPP_f448129468l_bool(finite_finite_int,X_1)))) -> ((all X_1 (hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_1),S)) -> hAPP_f957591787ol_nat(finite_card_int,X_1) = Na)) -> hAPP_f521865025ol_nat(hAPP_f93439879ol_nat(big_co1758102091ol_nat,finite_card_int),S) = hAPP_f521865025ol_nat(hAPP_f93439879ol_nat(big_co1758102091ol_nat,cOMBK_1643584600t_bool(Na)),S))))) # label(fact_5027_card__setsum__aux) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2379 (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]. 9.27/9.37 2380 (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]. 9.27/9.37 2381 (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]. 9.27/9.37 2382 (all X all Y hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),Y))) # label(fact_2887_Divides_Otransfer__int__nat__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2383 (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]. 9.27/9.37 2384 (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]. 9.27/9.37 2385 (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(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Xa) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,Ya),Z_2))) # label(fact_3415_eq__divide__2__times__iff) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2386 (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]. 9.27/9.37 2387 (all N_23 all A_76 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_76)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_76),N_23)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_76),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_76),N_23)))))) # label(fact_1605_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2388 (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]. 9.27/9.37 2389 (all A_1 all Ma all Xa (is_int(Xa) -> (zero_zero_int != Xa -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,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)))) = 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))))))) # label(fact_4811_Bnor__prod__power) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2390 (all F all G ((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_real_real(abs_abs_real,hAPP_nat_real(F,N_1))),hAPP_nat_real(G,N_1))))) -> (hBOOL(summable_real(G)) -> hBOOL(summable_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F)))))) # label(fact_4828_summable__rabs__comparison__test) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2391 (all B_133 all A_190 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_190)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_133)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_190),B_133)))))) # label(fact_491_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2392 (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]. 9.27/9.37 2393 (all P all Q all R hAPP_f1293550189l_real(hAPP_r1043989325l_real(P,R),hAPP_r2019696015l_real(Q,R)) = hAPP_r1230691443l_real(hAPP_f324002831l_real(hAPP_f677014621l_real(cOMBS_107495110l_real,P),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]. 9.27/9.37 2394 (all F 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_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)) = zero_zero_real) # label(fact_4899_sumr__one__lb__realpow__zero) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2395 (all Xa all Ya (Xa = 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)))) # label(fact_1705_order__eq__iff) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2396 (all A_1 all B_1 (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_1)) -> (A_1 != B_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]. 9.27/9.37 2397 (all P all Q all R hAPP_i1216375074nt_int(hAPP_P621635040nt_int(hAPP_f1582586579nt_int(cOMBC_707974837nt_int,P),Q),R) = hAPP_P1145851913nt_int(hAPP_i2104874254nt_int(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2398 (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]. 9.27/9.37 2399 (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]. 9.27/9.37 2400 (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(bseq_real(hAPP_r474017924t_real(power_power_real,Xa)))))) # label(fact_4735_Bseq__realpow) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2401 (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]. 9.27/9.37 2402 (all W pls = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,pls),W)) # label(fact_1254_mult__Pls) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2403 (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]. 9.27/9.37 2404 (all A_124 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_124),one_one_complex) = A_124) # label(fact_1222_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2405 (all P all Q all R hAPP_i345030060t_bool(hAPP_f725770969t_bool(hAPP_f983569339t_bool(cOMBB_550562460ol_int,P),Q),R) = hAPP_f1791153283t_bool(P,hAPP_int_fun_int_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc___018) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2406 (all A_3 zero_zero_real = hAPP_real_real(log(A_3),one_one_real)) # label(fact_3435_log__one) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2407 (all K_2 zero_zero_rat = hAPP_int_rat(hAPP_int_fun_int_rat(fract,zero_zero_int),K_2)) # label(fact_4675_rat__number__collapse_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2408 (all X all Y all M (one_one_int = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)),M) -> 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]. 9.27/9.37 2409 (all M all D_3 all R_2 (R_2 = hAPP_nat_nat(div_mod_nat(M),D_3) -> (exists Q_4 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,R_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Q_4),D_3)) = M))) # label(fact_3218_mod__eqD) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2410 (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]. 9.27/9.37 2411 (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]. 9.27/9.37 2412 (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(cos,Y)),hAPP_real_real(sin,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,Y)),hAPP_real_real(cos,X)))) # label(fact_3591_sin__diff2) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2413 (all P all Q all R hAPP_int_bool(hAPP_f1448486952t_bool(hAPP_f696844121t_bool(cOMBB_nat_bool_int,P),Q),R) = hAPP_nat_bool(P,hAPP_int_nat(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__HOL__Obool_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2414 (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]. 9.27/9.37 2415 (all Xa all Na all 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)))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)))) # label(fact_3073_exp__mono__le) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2416 (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]. 9.27/9.37 2417 (all V_7 all W_3 hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_7),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))) # label(fact_2231_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2418 (all B_5 all A_5 all A_3 all B_2 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2) != zero_zero_nat -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)) = A_3 -> (B_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)) -> one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_5),B_5))))) # label(fact_4364_gcd__coprime__nat) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2419 (all A_134 zero_zero_complex = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,zero_zero_complex),A_134)) # label(fact_1140_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2420 (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]. 9.27/9.37 2421 (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]. 9.27/9.37 2422 (all K_2 hAPP_int_int(bit1,K_2) = hAPP_int_int(succ,hAPP_int_int(bit0,K_2))) # label(fact_774_succ__Bit0) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2423 (all N hAPP_complex_real(im,hAPP_nat_complex(semiri2020571505omplex,N)) = zero_zero_real) # label(fact_4101_complex__Im__of__nat) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2424 (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]. 9.27/9.37 2425 (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,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,Na),Ya))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Ya))))) # label(fact_3913_real__root__gt__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2426 (all P all Q all R hAPP_f743790483t_bool(P,hAPP_r279757445t_bool(Q,R)) = hAPP_r2052264067t_bool(hAPP_f77155443t_bool(hAPP_f403082607t_bool(cOMBB_984071017l_real,P),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]. 9.27/9.37 2427 (all Xa all Ya (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Ya),Xa)) | hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Xa),Ya)) <-> Ya != Xa)) # label(fact_1846_linorder__neq__iff) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2428 (all Ma all Na (Ma = Na <-> hAPP_nat_int(semiri1621563631at_int,Na) = hAPP_nat_int(semiri1621563631at_int,Ma))) # label(fact_103_int__int__eq) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2429 (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]. 9.27/9.37 2430 (all A_3 all B_2 all C_1 all N (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,C_1),N) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N)) -> (one_one_int = hAPP_int_int(legacy_zgcd(A_3),B_2) -> (exists K_1 (is_int(K_1) & hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K_1),N) = hAPP_int_int(abs_abs_int,A_3))))))) # label(fact_4530_int__relprime__power__divisors) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2431 (all B_120 all A_174 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_174),zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_120),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_174),B_120)),zero_zero_nat))))) # label(fact_644_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2432 (all A_74 (is_int(A_74) -> 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))) # label(fact_1618_mult__numeral__1) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2433 (all B_87 all A_117 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_117)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_87),zero_z126310315umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_87),A_117)),zero_z126310315umeral))))) # label(fact_1306_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2434 (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]. 9.27/9.37 2435 (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]. 9.27/9.37 2436 (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]. 9.27/9.37 2437 (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]. 9.27/9.37 2438 (all A_25 all B_19 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(abs_abs_rat,A_25)),B_19)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_25),B_19)))) # label(fact_2669_abs__le__D1) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2439 (all P all Q all R hAPP_f1222178131t_bool(hAPP_r2052264067t_bool(P,R),Q) = hAPP_r373117745t_bool(hAPP_f1459420095t_bool(hAPP_f693283245t_bool(cOMBC_1113900202t_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Int__Oint_Mtc__fun_It_028) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2440 (all Z_2 all Z_3 ((hBOOL(hAPP_int_bool(nat_neg,Z_3)) -> hAPP_int_nat(nat,Z_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,Z_2)),hAPP_int_nat(nat,Z_3))) & (-hBOOL(hAPP_int_bool(nat_neg,Z_3)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,hAPP_int_bool(nat_neg,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_2),Z_3))),zero_zero_nat),hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z_2),Z_3))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,Z_2)),hAPP_int_nat(nat,Z_3))))) # label(fact_4745_diff__nat__eq__if) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2441 (all M all K_2 all N hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),N)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N)) # label(fact_4969_gcd__add__mult__int) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2442 (all B_2 all A_3 all A_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),A_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_5),B_2)))))) # label(fact_2834_zdiv__mono1) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2443 (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]. 9.27/9.37 2444 (all P_1 all Xa all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ya),Xa)) -> hBOOL(P_1)))) # label(fact_1789_order__less__imp__triv) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2445 (all A_150 (is_int(A_150) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_150),zero_zero_int) = A_150)) # label(fact_920_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2446 (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]. 9.27/9.37 2447 (all L_3 all K_5 (hAPP_rat_rat(abs_abs_rat,K_5) = hAPP_rat_rat(abs_abs_rat,L_3) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,L_3),K_5)))) # label(fact_2691_dvd__if__abs__eq) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2448 (all Xa all 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)) <-> Xa = Ya)) # label(fact_2652_divides__antisym) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2449 (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]. 9.27/9.37 2450 (all A_1 all B_1 all Q_2 all Ra (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_1,B_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),Ra))) -> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(hAPP_int_int(uminus_uminus_int,A_1),B_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,Ra),zero_zero_int)),hAPP_int_int(uminus_uminus_int,Q_2)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(uminus_uminus_int,Q_2)),one_one_int))),hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,Ra),zero_zero_int)),zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_1),Ra))))))) # label(fact_3502_zminus1__lemma) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2451 (all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(zfact,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int))),hAPP_int_int(number_number_of_int,min)),P_3)))) # label(fact_2819_Wilson__Russ) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2452 (all X all M all Y 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,hAPP_int_int(div_mod_int(X),M)),Y)),M)) # label(fact_2990_zdiff__zmod__left) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2453 (all B_2 all A_3 (is_int(A_3) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2)) -> A_3 = hAPP_int_int(div_mod_int(A_3),B_2))))) # label(fact_3079_mod__pos__pos__trivial) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2454 (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,produc1935615926l_real),bolzano_bisect(P_1,A_1,B_1)),L_1,sequentially)) & 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))))) & (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]. 9.27/9.37 2455 (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]. 9.27/9.37 2456 (all B_1_1 all B_2_1 is_bool(hAPP_f115998247l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__prod_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__p) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2457 (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]. 9.27/9.37 2458 (all K_2 all L_2 hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),L_2)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(bit1,K_2)),hAPP_int_int(bit0,L_2))) # label(fact_2409_diff__bin__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2459 (all Z_7 all X_22 all Y_18 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_22),Y_18)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_18),Z_7)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_22),Z_7))))) # label(fact_1945_order__less__le__trans) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2460 (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]. 9.27/9.37 2461 (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]. 9.27/9.37 2462 (all Ma all 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)) = hAPP_i1948725293t_bool(noXRRset(Ma),Xa)) # label(fact_4772_noXRRset__def) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2463 (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]. 9.27/9.37 2464 (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]. 9.27/9.37 2465 (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))) -> (zero_zero_int != B_2 -> (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]. 9.27/9.37 2466 (all Z_10 all Y_21 all X_25 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_21),X_25)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z_10),Y_21)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z_10),X_25))))) # label(fact_1919_xt1_I8_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2467 (all Lx_3 all Ly_3 all Rx_3 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_3),Rx_3)),Ly_3) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_3),Ly_3)),Rx_3)) # label(fact_1067_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2468 (all B_1_1 all B_2_1 all B_3_1 is_bool(tendsto_nat_complex(B_1_1,B_2_1,B_3_1))) # label(gsy_c_Limits_Otendsto_000tc__Nat__Onat_000tc__Complex__Ocomplex) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2469 (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]. 9.27/9.37 2470 (all A_140 all B_98 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_98),A_140) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_140),B_98)) # label(fact_1033_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2471 (all A_1 all B_1 (B_1 = A_1 <-> quotient_of(A_1) = quotient_of(B_1))) # label(fact_4649_quotient__of__inject__eq) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2472 (all A_3 all B_2 (is_int(A_3) -> A_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))),hAPP_int_int(div_mod_int(A_3),B_2)))) # label(fact_3087_zmod__zdiv__equality) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2473 (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)) <-> Ya = Xa))) # label(fact_1687_order__antisym__conv) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2474 (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]. 9.27/9.37 2475 (all Na all K all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Ma)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Na)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),K) <-> Ma = Na)))) # label(fact_2489_eq__diff__iff) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2476 (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]. 9.27/9.37 2477 (all A_21 all B_16 hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_21),B_16)) = hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B_16),A_21))) # label(fact_2684_abs__minus__commute) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2478 (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]. 9.27/9.37 2479 (all L all U hAPP_i1948725293t_bool(ord_at875362053st_int(L),U) = hAPP_i1948725293t_bool(ord_at641636577an_int(L),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,U),one_one_int))) # label(fact_5079_atLeastLessThanPlusOne__atLeastAtMost__int) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2480 (all K_2 all I all J_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),I)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),J_2)))))) # label(fact_1504_zmult__zless__mono2) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2481 (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]. 9.27/9.37 2482 (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]. 9.27/9.37 2483 (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]. 9.27/9.37 2484 (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]. 9.27/9.37 2485 (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]. 9.27/9.37 2486 (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]. 9.27/9.37 2487 (all Wa all Ya all Xa all Z_2 (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Wa),Ya)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Xa),Z_2)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Wa),Z_2)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Xa),Ya)) <-> Ya = Z_2 | Wa = Xa)) # label(fact_1187_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2488 (all Xa all Ya (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa),Ya) = zero_zero_real <-> Ya = hAPP_real_real(uminus_uminus_real,Xa))) # label(fact_3429_real__add__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2489 (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,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)) = 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)) # label(fact_2189_left__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2490 (all A_44 all B_33 all K_6 (A_44 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_33),K_6) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,B_33),A_44)))) # label(fact_2266_dvdI) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2491 (all B_40 all A_58 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_40),A_58)) -> (A_58 != B_40 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_40),A_58))))) # label(fact_1952_xt1_I11_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2492 (all C_56 all A_145 all B_103 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_56),A_145)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_56),B_103))) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_145),B_103)))) # label(fact_953_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 9.27/9.37 2493 (all X_54 all Y_46 all Q_9 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_54),Q_9)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_46),Q_9)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_54),Y_46)),Q_9)) # label(fact_1249_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2494 (all B_2 all A_3 all C_1 hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),C_1)) = 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))) # label(fact_4992_gcd__int_Oleft__commute) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2495 (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]. 9.27/9.38 2496 (all A_13 all B_13 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(abs_abs_int,A_13)),hAPP_int_int(abs_abs_int,B_13))),hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_13),A_13))))) # label(fact_2731_abs__triangle__ineq2__sym) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2497 (all B_132 all A_189 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_189),zero_z126310315umeral)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_132),zero_z126310315umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_189),B_132)),zero_z126310315umeral))))) # label(fact_500_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2498 (all A_3 all P_3 (is_int(A_3) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),P_3)) -> hAPP_int_int(inv(P_3),hAPP_int_int(inv(P_3),A_3)) = A_3)))))) # label(fact_2866_inv__inv) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2499 (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]. 9.27/9.38 2500 (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]. 9.27/9.38 2501 (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]. 9.27/9.38 2502 (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]. 9.27/9.38 2503 (all K_2 all L_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_2),L_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),L_2)))))) # label(fact_294_add__less__mono) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2504 (all A_1 (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)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_1),zero_zero_rat)))) # label(fact_97_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2505 (all Xa all Na (is_int(Xa) -> (zero_zero_int != Xa | Na = zero_zero_nat <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(abs_abs_int,Xa)),Na)))))) # label(fact_2787_zero__less__zpower__abs__iff) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2506 (all A_1 all C all B_1 (is_int(C) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_1),B_1)) | C = zero_zero_int <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_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)))))) # label(fact_2355_dvd__mult__cancel__right) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2507 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(fact,M)),hAPP_nat_nat(fact,N))))) # label(fact_4558_fact__mono) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2508 (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]. 9.27/9.38 2509 (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]. 9.27/9.38 2510 (all Q_5 all R_4 all A_3 all B_2 all Q_1 all R_2 (is_int(Q_5) & is_int(B_2) & is_int(Q_1) -> (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 -> Q_5 = Q_1))))) # label(fact_3255_unique__quotient) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2511 (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]. 9.27/9.38 2512 (all X_47 all Y_41 (-hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_47),Y_41)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_41),X_47)))) # label(fact_1623_linorder__le__cases) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2513 (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]. 9.27/9.38 2514 (all N one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(suc,N)),N)) # label(fact_4346_coprime__Suc__nat) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2515 (all B_112 all A_164 all C_66 (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_66),A_164) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_112),A_164) -> B_112 = C_66)) # label(fact_781_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2516 (all C all A_1 all B_1 (B_1 = A_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]. 9.27/9.38 2517 (all Wa all Z_2 (is_int(Wa) & is_int(Z_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_2)) | Z_2 = Wa <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_2),one_one_int)))))) # label(fact_119_zless__add1__eq) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2518 (all P all Q all R hAPP_b589554111l_bool(P,hAPP_P603027463t_bool(Q,R)) = hAPP_P921862901l_bool(hAPP_f365317245l_bool(hAPP_f1961001761l_bool(cOMBB_1550063917nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_016) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2519 (all C_16 all A_66 all B_43 (A_66 = B_43 -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_43),C_16)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_66),C_16))))) # label(fact_1768_ord__eq__less__trans) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2520 (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]. 9.27/9.38 2521 (exists X_1 (is_int(X_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(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_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]. 9.27/9.38 2522 (all Real1 all Real2 complex_size(hAPP_real_complex(hAPP_r265291036omplex(complex,Real1),Real2)) = zero_zero_nat) # label(fact_3963_complex_Osize_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2523 (all N_51 one_on1645066479umeral = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,one_on1645066479umeral),N_51)) # label(fact_351_power__one) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2524 (all Ya all Xa (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ya),Xa)) -> (Ya = Xa <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya))))) # label(fact_1690_order__antisym__conv) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2525 (all Va all 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))) <-> (-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))))) # label(fact_4786_le__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2526 (all Y (Y != zero_zero_real -> 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__zero) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2527 (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)) -> A_52 = zero_zero_int))) # label(fact_2144_even__power__le__0__imp__0) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2528 (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]. 9.27/9.38 2529 (all K_2 all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N)))) # label(fact_4327_dvd__gcd__D2__nat) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2530 (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]. 9.27/9.38 2531 (all Xa all Ya (Ya = Xa <-> hAPP_complex_real(im,Ya) = hAPP_complex_real(im,Xa) & hAPP_complex_real(re,Xa) = hAPP_complex_real(re,Ya))) # label(fact_4116_complex__eq__iff) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2532 (all A_1 all B_1 (A_1 != B_1 -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_1),A_1)) -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,B_1),A_1))))) # label(fact_1987_xt1_I12_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2533 (all R_2 all P_3 all Q_1 (quotient_of(R_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_1) -> one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,P_3),Q_1))) # label(fact_5016_quotient__of__coprime) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2534 (all I all M hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),M))))) # label(fact_3050_less__add__Suc1) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2535 (all C_36 all B_64 all A_93 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_64),A_93)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_36),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_36),A_93)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_36),B_64)))))) # label(fact_1439_mult__strict__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2536 (all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,min)),B_2) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_2),one_one_int))) # label(fact_3152_zmod__minus1) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2537 (all N all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y),N))))) # label(fact_2501_divides__exp) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2538 (all X hAPP_int_int(abs_abs_int,X) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),zero_zero_int)) # label(fact_4967_gcd__0__int) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2539 (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_eq_int,hAPP_int_int(bit1,K)),hAPP_int_int(bit0,L))))) # label(fact_760_rel__simps_I33_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2540 (all A_1 all B_1 all C all D_1 (hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C),D_1) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_1),B_1) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),D_1)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_1),B_1))))) # label(fact_2336_diff__eq__diff__less) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2541 (all X_2 (hBOOL(summable_complex(X_2)) -> hBOOL(summable_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_2))))) # label(fact_4844_cnj_Osummable) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2542 (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]. 9.27/9.38 2543 (all X one_one_real = 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)))))) # label(fact_3622_sin__cos__squared__add2) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2544 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sqrt,X))))) # label(fact_3531_real__sqrt__gt__zero) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2545 (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]. 9.27/9.38 2546 (all A_56 all B_38 (A_56 != B_38 -> (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]. 9.27/9.38 2547 (all Na all N_3 ((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_nat,X_1),Na)))) -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_3)))) # label(fact_4619_bounded__nat__set__is__finite) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2548 (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]. 9.27/9.38 2549 (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]. 9.27/9.38 2550 (all A_149 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_149),zero_z891286103de_int) = A_149) # label(fact_923_add__0__right) # label(axiom) # label(non_clause). [assumption]. 9.27/9.38 2551 (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]. 9.27/9.39 2552 (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,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))) <-> 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)))))) # label(fact_2538_nat__le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2553 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_3),B_2)) -> zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)))) # label(fact_2839_div__pos__pos__trivial) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2554 (all Na all Ma (is_int(Ma) -> (zero_zero_int != Ma -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),Na)),Ma)) <-> one_one_int = hAPP_int_int(abs_abs_int,Na))))) # label(fact_2796_zdvd__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2555 (all Xa all Na (one_one_nat = Xa | Na = zero_zero_nat <-> one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Xa),Na))) # label(fact_475_exp__eq__1) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2556 (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]. 9.27/9.39 2557 (all A_130 all B_92 all 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) = 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))) # label(fact_1178_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2558 (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]. 9.27/9.39 2559 (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]. 9.27/9.39 2560 (all X all D_3 (is_int(D_3) -> (D_3 != zero_zero_int -> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),D_3))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,hAPP_int_int(div_mod_int(X),D_3))),hAPP_int_real(real_int,D_3))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,X)),hAPP_int_real(real_int,D_3))))) # label(fact_4465_real__of__int__div__aux) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2561 (all P_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_5)) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int) = hAPP_nat_int(semiri1621563631at_int,hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(sRStar,P_5))))) # label(fact_3302_SRStar__card) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2562 (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(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) = 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)))) & (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(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)))) # label(fact_2970_zmod__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2563 (all K all F (zero_zero_nat = hAPP_nat_nat(F,zero_zero_nat) -> hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),K)) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_nat_nat(suc,zero_zero_nat)),K)))) # label(fact_5080_setsum__shift__lb__Suc0__0) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2564 (all Na (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_nat_rat(semiri151668891at_rat,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)))) # label(fact_240_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2565 (all Xa all Ya all B_1 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),B_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_1),Xa)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_1),Ya)))))) # label(fact_428_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2566 (all B_10 all A_10 ((hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_10)) | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_10),zero_zero_rat))) & (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))) -> 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]. 9.27/9.39 2567 (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]. 9.27/9.39 2568 (all C_35 all D_13 all A_92 all B_63 all R_6 (is_int(C_35) & is_int(D_13) & is_int(R_6) -> (R_6 != zero_zero_int -> (B_63 = A_92 & C_35 != D_13 -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_92),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_6),C_35)) != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_63),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_6),D_13)))))) # label(fact_1449_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2569 (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)) -> zero_zero_int != hAPP_int_int(div_mod_int(K_2),L_2))) # label(fact_3483_zmod__zminus2__not__zero) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2570 (all Ya all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ya),Xa)) -> comple124823625p_real(hAPP_r1134773055l_bool(ord_gr788844697n_real(Ya),Xa)) = Xa)) # label(fact_4631_Sup__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2571 (all I all J_2 all K_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),J_2)),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))) # label(fact_2484_diff__diff__left) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2572 (all A_83 all C_29 all B_56 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_83),C_29)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_56),C_29))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_29)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_83),B_56))))) # label(fact_1544_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2573 (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]. 9.27/9.39 2574 (all A_40 all B_29 hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_40),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_29),A_40)))) # label(fact_2291_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2575 (all Xa all Ya (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) <-> Xa = Ya))) # label(fact_1971_linorder__antisym__conv2) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2576 (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]. 9.27/9.39 2577 (all D_1 int_ge_less_than2(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)))) # label(fact_4860_int__ge__less__than2__def) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2578 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N))) # label(fact_2508_diff__is__0__eq_H) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2579 (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]. 9.27/9.39 2580 (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]. 9.27/9.39 2581 (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]. 9.27/9.39 2582 (all P all Q all R hAPP_real_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,P),Q),R) = hAPP_nat_real(hAPP_r474017924t_real(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__Nat__Onat_000tc__RealDef__Orea) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2583 (all Xa all Ya (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)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ya),hAPP_real_real(uminus_uminus_real,Xa))))) # label(fact_3438_real__add__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2584 (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]. 9.27/9.39 2585 (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]. 9.27/9.39 2586 (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]. 9.27/9.39 2587 (all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,pls)),Ya)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(number_number_of_int,Ya))))) # label(fact_126_less__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2588 (all K_2 (is_int(K_2) -> hAPP_int_int(number_number_of_int,K_2) = K_2)) # label(fact_279_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2589 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(suc,N))))) # label(fact_3022_le__SucI) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2590 (all B_31 all A_42 all C_11 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_42),C_11)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_42),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_31),C_11))))) # label(fact_2280_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2591 (all V_11 all B_62 all C_34 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_11)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_62),C_34)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_11)),B_62)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_11)),C_34))) # label(fact_1460_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2592 (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]. 9.27/9.39 2593 (all B_2 all A_3 (is_int(B_2) & is_int(A_3) -> (A_3 != zero_zero_int | zero_zero_int != B_2 -> one_one_int = hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(legacy_zgcd(A_3),B_2))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),hAPP_int_int(legacy_zgcd(A_3),B_2)))))) # label(fact_4512_div__zgcd__relprime) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2594 (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,B_1),A_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)))))) # label(fact_1407_mult__less__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2595 (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]. 9.27/9.39 2596 (all P all Q all R hAPP_real_bool(hAPP_r1134773055l_bool(P,R),Q) = hAPP_real_bool(hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__HOL__Obo) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2597 (all M all K_2 all N (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K_2),N) = one_one_int -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),N))) # label(fact_4999_gcd__mult__cancel__int) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2598 (all A_3 all B_2 (is_int(B_2) -> (zero_zero_int != B_2 -> (hAPP_int_int(div_mod_int(A_3),B_2) != zero_zero_int -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))),one_one_int) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(uminus_uminus_int,A_3)),B_2)) & (hAPP_int_int(div_mod_int(A_3),B_2) = zero_zero_int -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(uminus_uminus_int,A_3)),B_2) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)))))) # label(fact_3499_zdiv__zminus1__eq__if) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2599 (all I all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),N))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),I)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))))) # label(fact_2554_power__dvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2600 (all P all Q all R hAPP_nat_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,P),Q),R) = hAPP_real_real(hAPP_n546711566l_real(P,R),hAPP_nat_real(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__RealDef__Orea) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2601 (all Va all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,Va)),hAPP_nat_nat(suc,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))))) # label(fact_4784_less__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2602 (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]. 9.27/9.39 2603 (all N all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N)),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N)),one_one_real))) -> archim856651990g_real(X) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N),one_one_int)))) # label(fact_4375_ceiling__eq2) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2604 (all X_32 all Y_26 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_32),Y_26)) -> X_32 != Y_26)) # label(fact_1823_less__imp__neq) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2605 (all A_1 all B_1 all Wa (B_1 = zero_zero_real & A_1 = hAPP_int_real(number267125858f_real,Wa) <-> hAPP_real_complex(hAPP_r265291036omplex(complex,A_1),B_1) = hAPP_int_complex(number528085621omplex,Wa))) # label(fact_3954_Complex__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2606 (all K (is_int(K) -> (pls = hAPP_int_int(bit0,K) <-> K = pls))) # label(fact_135_rel__simps_I44_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2607 (all Ma all Na all K (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K)) -> (Ma = Na <-> 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_2194_nat__mult__eq__cancel1) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2608 (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]. 9.27/9.39 2609 (all B_76 all A_106 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_76),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_106)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_76)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_106),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_106),B_76)),zero_zero_real)))) # label(fact_1361_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2610 (all P all Q all R hAPP_r1250527377l_real(hAPP_n682674106l_real(P,R),hAPP_nat_real(Q,R)) = hAPP_n546711566l_real(hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__fun_Itc__Real) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2611 -(hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_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_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(twoSqu1770640450sum2sq,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)))))) # label(fact_2061_smaller_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2612 (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]. 9.27/9.39 2613 (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]. 9.27/9.39 2614 (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,A_1),X_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),B_1)) -> (exists Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),zero_zero_real)) & hBOOL(hAPP_real_bool(deriv_real(F,X_1),Y_1)))))) -> 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]. 9.27/9.39 2615 (all Ta all A all D_1 all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),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),A)) -> X_1 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa_1))))))) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Ta))) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),D)),Ta))))))))) # label(fact_5101_aset_I10_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2616 (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]. 9.27/9.39 2617 (all X hAPP_int_int(fact_fact_int,hAPP_nat_int(semiri1621563631at_int,X)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(fact_fact_nat,X))) # label(fact_4279_transfer__int__nat__factorial) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2618 (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]. 9.27/9.39 2619 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(div_mod_int(X),Y)))))) # label(fact_3034_Divides_Otransfer__nat__int__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2620 (all X all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),hAPP_int_int(multInv(P_3),X))),one_one_int),P_3)))))) # label(fact_2921_MultInv__prop2) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2621 (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]. 9.27/9.39 2622 (all A all Ma (hBOOL(hAPP_int_bool(is_RRset(A),Ma)) <-> hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,A),rsetR(Ma))) & hAPP_f957591787ol_nat(finite_card_int,A) = phi(Ma))) # label(fact_4593_is__RRset__def) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2623 (all M hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),hAPP_nat_nat(suc,zero_zero_nat)) = M) # label(fact_3068_div__1) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2624 (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 ((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)))))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))))))) # label(fact_3880_DERIV__pos__inc__left) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2625 (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_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))) = 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))))) # label(fact_3273_power__nat__number__of__number__of) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2626 (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]. 9.27/9.39 2627 (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]. 9.27/9.39 2628 (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]. 9.27/9.39 2629 (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]. 9.27/9.39 2630 (all C all D_1 all A_1 all B_1 (B_1 != A_1 & D_1 != C <-> hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_1),C)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_1),D_1)) != hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_1),D_1)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_1),C)))) # label(fact_1171_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2631 (all 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)))))) -> hBOOL(vanishes(X_2)))) # label(fact_5151_vanishesI) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2632 (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]. 9.27/9.39 2633 (all A_124 A_124 = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_124),one_on1645066479umeral)) # label(fact_1223_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2634 (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]. 9.27/9.39 2635 (all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,min)),B_2) = hAPP_int_int(number_number_of_int,min))) # label(fact_2848_div__eq__minus1) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2636 (all W_4 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_i1732201573de_int(number1226105091de_int,W_4)),hAPP_i1732201573de_int(number1226105091de_int,W_4)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_i1732201573de_int(number1226105091de_int,W_4)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_1917_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2637 (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_eq_int,B_1),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,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))) = posDivAlg(A_1,B_1)))) # label(fact_3301_posDivAlg_Opsimps) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2638 (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]. 9.27/9.39 2639 (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]. 9.27/9.39 2640 (all M ((zero_zero_nat = M -> one_one_nat = hAPP_nat_nat(fact_fact_nat,M)) & (M != zero_zero_nat -> hAPP_nat_nat(fact_fact_nat,M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))))) # label(fact_3845_fact__num__eq__if__nat) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2641 (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]. 9.27/9.39 2642 (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]. 9.27/9.39 2643 (all M_18 all N_30 hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_18),N_30)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(semiri984289939at_nat,M_18)),hAPP_nat_nat(semiri984289939at_nat,N_30))) # label(fact_1104_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2644 (all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (one_one_real != A_3 -> one_one_real = hAPP_real_real(log(A_3),A_3)))) # label(fact_3444_log__eq__one) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2645 (all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> hAPP_nat_nat(suc,hAPP_int_nat(nat,Z_1)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),Z_1)))) # label(fact_3142_Suc__nat__eq__nat__zadd1) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2646 (all M_24 all N_48 hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_24),N_48)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_nat_complex(semiri2020571505omplex,M_24)),hAPP_nat_complex(semiri2020571505omplex,N_48))) # label(fact_372_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2647 (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]. 9.27/9.39 2648 (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]. 9.27/9.39 2649 (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,C),A_1) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B_1)))) # label(fact_2097_real__mult__left__cancel) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2650 (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]. 9.27/9.39 2651 (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]. 9.27/9.39 2652 (all L_3 all K_5 (hAPP_real_real(abs_abs_real,K_5) = hAPP_real_real(abs_abs_real,L_3) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,L_3),K_5)))) # label(fact_2692_dvd__if__abs__eq) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2653 (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(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),B_2)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),A_3)) & hBOOL(hAPP_nat_bool(prime,P_3))))) # label(fact_4177_coprime__prime) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2654 (all C_19 all B_46 all A_69 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_46),A_69)) -> (C_19 = B_46 -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_19),A_69))))) # label(fact_1745_xt1_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2655 (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),Xa)) & hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Ya)))) # label(fact_2023_less__le__not__le) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2656 (all N ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N),zero_zero_int)) -> one_one_int = hAPP_int_int(zfact,N)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N),zero_zero_int)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N),hAPP_int_int(zfact,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,N),one_one_int))) = hAPP_int_int(zfact,N)))) # label(fact_2863_zfact_Osimps) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2657 (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))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_1),B_1)) | C = zero_zero_real)) # label(fact_2358_dvd__mult__cancel__right) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2658 (all A_1 all B_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_1),B_1)),zero_zero_rat)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_1),B_1)))) # label(fact_2378_less__iff__diff__less__0) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2659 (all P all Q all R hAPP_r1195171167l_real(hAPP_r2019696015l_real(P,R),hAPP_real_real(Q,R)) = hAPP_r1195171167l_real(hAPP_f617735571l_real(hAPP_f1730463541l_real(cOMBS_511527878l_real,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__prod_Itc) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2660 (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]. 9.27/9.39 2661 (all M all N all K_2 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N)))) # label(fact_4299_gcd__dvd__prod__nat) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2662 (all A_1 (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 <-> zero_zero_rat = A_1)) # label(fact_19_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2663 (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]. 9.27/9.39 2664 (all Xa -hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Xa))) # label(fact_1851_order__less__irrefl) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2665 (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,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))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_1),B_1)),E_2)),C)),D_1)))) # label(fact_2424_le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2666 (all N all M (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,M))) -> N = M))) # label(fact_3010_less__antisym) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2667 (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(B_1),A_1),Ma)))) # label(fact_2546_zcong__sym) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2668 (all B_2 all A_3 (is_int(A_3) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),A_3)) -> hAPP_int_int(div_mod_int(A_3),B_2) = A_3)))) # label(fact_3082_mod__neg__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2669 (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)))) -> (exists E (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),E)) & (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,Y_1),hAPP_real_real(F,Xa)))),E)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_1),hAPP_real_real(F,Xa))))) -> Y_1 = hAPP_real_real(F,hAPP_real_real(G,Y_1)))))))))) # label(fact_5177_isCont__inv__fun__inv) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2670 (all B_1 all C all A_1 all D_1 (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_1),D_1) = one_one_int -> (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B_1),C) = one_one_int -> (hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,A_1)),hAPP_int_int(abs_abs_int,C)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,B_1)),hAPP_int_int(abs_abs_int,D_1)) <-> hAPP_int_int(abs_abs_int,B_1) = hAPP_int_int(abs_abs_int,A_1) & hAPP_int_int(abs_abs_int,C) = hAPP_int_int(abs_abs_int,D_1))))) # label(fact_5007_coprime__crossproduct__int) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2671 (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_eq_real,hAPP_int_real(real_int,N_1)),R_2)) & 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)))) & is_int(N_1))))) # label(fact_4476_reals__Archimedean__6b__int) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2672 (all B_133 all A_190 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_190)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_133)) -> 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_190),B_133)))))) # label(fact_489_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2673 (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(cos,Y)),hAPP_real_real(cos,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,Y)),hAPP_real_real(sin,X))) = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y))) # label(fact_3599_cos__diff2) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2674 (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,X_1),B_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),X_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,X_1),B_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),X_1)) -> hAPP_real_real(F,X_1) = hAPP_real_real(F,A_1))))))) # label(fact_5174_DERIV__isconst1) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2675 (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]. 9.27/9.39 2676 (all Z_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Z_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Z_1),Z_1)) # label(fact_2138_nat__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2677 (all A_86 all C_32 all B_59 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_86),C_32)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_59),C_32))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_32)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_86),B_59))))) # label(fact_1522_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2678 (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]. 9.27/9.39 2679 (all X_1 hAPP_f352196356l_real(suminf_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),sin_coeff)),hAPP_r474017924t_real(power_power_real,X_1))) = hAPP_real_real(sin,X_1)) # label(fact_4819_lemma__sin__ext) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2680 (all A_3 all B_2 (is_int(B_2) -> (zero_zero_int != B_2 -> (hAPP_int_int(div_mod_int(A_3),B_2) = zero_zero_int -> hAPP_int_int(hAPP_int_fun_int_int(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))) & (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)))))) # label(fact_3498_zdiv__zminus2__eq__if) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2681 (all Ma (one_one_nat = hAPP_nat_nat(div_mod_nat(Ma),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(div_mod_nat(Ma),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3208_mod2__gr__0) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2682 (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]. 9.27/9.39 2683 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (exists N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_1)))),X))))) # label(fact_3864_reals__Archimedean) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2684 (all Z_2 (is_int(Z_2) -> (hAPP_int_int(abs_abs_int,Z_2) = one_one_int <-> Z_2 = hAPP_int_int(number_number_of_int,min) | one_one_int = Z_2))) # label(fact_2748_abs__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2685 (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]. 9.27/9.39 2686 (all Wa all Z_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),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_2773_nat__mono__iff) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2687 (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(zcong(big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(wset(A_1),P_5))),one_one_int),P_5)))))) # label(fact_4813_wset__zcong__prod__1) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2688 (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]. 9.27/9.39 2689 (all A_39 all B_28 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_39),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_39),B_28)))) # label(fact_2299_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2690 (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]. 9.27/9.39 2691 (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]. 9.27/9.39 2692 (all X_40 all Y_34 (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_40),Y_34)) -> (X_40 != Y_34 -> 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]. 9.27/9.39 2693 (all X_2 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_int_int(standardRes(Ma),X_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))),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,ord_less_int),Ma))))))) # label(fact_4787_SR__pos) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2694 (all X all Y hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)),X))) # label(fact_4960_gcd__dvd1__int) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2695 (all X all Y (is_int(X) -> X = hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,fTrue),X),Y))) # label(help_If_1_1_If_000tc__Int__Oint_T) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2696 (all B_10 all A_10 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_10)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_10),zero_zero_int))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_10),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_10))) -> hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_10),B_10)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,A_10)),hAPP_int_int(abs_abs_int,B_10)))) # label(fact_2756_abs__eq__mult) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2697 (all A_1 all B_1 (hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_1),B_1) = zero_zero_rat <-> zero_zero_rat = B_1 | zero_zero_rat = A_1)) # label(fact_1120_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2698 (all A_127 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,one_one_complex),A_127) = A_127) # label(fact_1201_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2699 (all B_2 all A_3 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M)) -> (hAPP_int_int(legacy_zgcd(A_3),M) = one_one_int -> (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]. 9.27/9.39 2700 (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]. 9.27/9.39 2701 (all Xa (hAPP_real_real(cos,Xa) != zero_zero_real -> hBOOL(hAPP_real_bool(deriv_real(tan,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_3872_DERIV__tan) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2702 (all X all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(inverse_inverse_real,X)) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X))))) # label(fact_3934_real__root__inverse__lemma) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2703 (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]. 9.27/9.39 2704 (all Ma all Na all A_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_1)) -> (Ma = Na <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_1),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_1),Na)))) # label(fact_425_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2705 (all A_1 all B_1 all C (hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_1),B_1) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_1),C) <-> B_1 = C)) # label(fact_809_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2706 (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]. 9.27/9.39 2707 (all R_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_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_4451_real__of__int__floor__add__one__gt) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2708 (all K_8 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,K_8)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(succ,K_8))) # label(fact_871_number__of__succ) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2709 (all X all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M)) -> (one_one_int = hAPP_int_int(legacy_zgcd(X),M) -> 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]. 9.27/9.39 2710 (all A_155 all C_57 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_155),C_57) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_57),A_155)) # label(fact_846_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2711 (all V_24 all W_14 all Z_21 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_24),W_14))),Z_21) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_24)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,W_14)),Z_21))) # label(fact_190_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2712 (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]. 9.27/9.39 2713 (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]. 9.27/9.39 2714 (all A_140 all B_98 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_98),A_140) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_140),B_98)) # label(fact_1035_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2715 (all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,W))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(number_number_of_int,W))) -> posDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W)) = hAPP_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,one_one_int),hAPP_int_int(number_number_of_int,W))) -> posDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W)) = hAPP_P1975530577nt_int(adjust(hAPP_int_int(number_number_of_int,W)),posDivAlg(one_one_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(number_number_of_int,W))))))) # label(fact_3289_posDivAlg__eqn__1__number__of) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2716 (all B_125 all A_181 all C_72 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_72)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_125),A_181)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_125),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_181),C_72)))))) # label(fact_572_add__increasing2) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2717 (all Xa all A (hBOOL(hAPP_P62333565t_bool(A,Xa)) <-> hBOOL(hAPP_f526364711l_bool(hAPP_P1062890741l_bool(member2143287562nt_int,Xa),A)))) # label(fact_139_mem__def) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2718 (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_nat_rat(semiri151668891at_rat,hAPP_int_nat(number_number_of_nat,V_17)) = zero_zero_rat) & (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_rat(number_number_of_rat,V_17) = hAPP_nat_rat(semiri151668891at_rat,hAPP_int_nat(number_number_of_nat,V_17))))) # label(fact_899_of__nat__number__of__lemma) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2719 (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,Xa),Ya)) <-> 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)))))) # label(fact_2123_real__mult__le__cancel__iff2) # label(axiom) # label(non_clause). [assumption]. 9.27/9.39 2720 (all Xa all Ya (zero_zero_rat != Ya | zero_zero_rat != Xa <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,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))))))))) # label(fact_214_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 9.27/9.40 2721 (all W_15 ((hAPP_int_nat(number_number_of_nat,W_15) != zero_zero_nat -> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),hAPP_int_nat(number_number_of_nat,W_15)) = zero_zero_complex) & (hAPP_int_nat(number_number_of_nat,W_15) = zero_zero_nat -> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),hAPP_int_nat(number_number_of_nat,W_15)) = one_one_complex))) # label(fact_43_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 9.27/9.40 2722 (all Y all X (hBOOL(hAPP_int_bool(nat_is_nat,X)) -> (hBOOL(hAPP_int_bool(nat_is_nat,Y)) -> hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y)))))) # label(fact_5134_Divides_Otransfer__int__nat__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.27/9.40 2723 (all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),N))) # label(fact_1031_le__refl) # label(axiom) # label(non_clause). [assumption]. 9.27/9.40 2724 (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]. 9.27/9.40 2725 (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]. 9.27/9.40 2726 (all Xa all A (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Xa),A)) <-> hBOOL(hAPP_nat_bool(A,Xa)))) # label(fact_141_mem__def) # label(axiom) # label(non_clause). [assumption]. 9.27/9.40 2727 (all V_21 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21)) -> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_21)),one_one_real) = hAPP_int_real(number267125858f_real,hAPP_int_int(succ,V_21)))) # label(fact_741_semiring__number__of__add__1) # label(axiom) # label(non_clause). [assumption]. 9.27/9.40 2728 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(tan,X)))))) # label(fact_3763_tan__gt__zero) # label(axiom) # label(non_clause). [assumption]. 9.27/9.40 2729 (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]. 9.27/9.40 2730 (all I all J_2 hAPP_int_int(legacy_zgcd(hAPP_int_int(uminus_uminus_int,I)),J_2) = hAPP_int_int(legacy_zgcd(I),J_2)) # label(fact_4497_zgcd__zminus) # label(axiom) # label(non_clause). [assumption]. 9.27/9.40 2731 (all Ma all Z_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,Z_2)))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,Ma)),Z_2)))) # label(fact_2658_int__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 9.27/9.40 2732 (all Va all V ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls)) -> 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)) = hAPP_int_nat(number_number_of_nat,Va)) & (-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))))) # label(fact_4795_diff__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2733 (all X_35 all Y_29 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_35),Y_29)) -> X_35 != Y_29)) # label(fact_1802_order__less__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2734 (all R_2 exists N_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_1)),R_2)) & 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)))) & is_int(N_1))) # label(fact_4478_real__lb__ub__int) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2735 (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]. 9.34/9.40 2736 (all C all A_1 all B_1 (B_1 = A_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]. 9.34/9.40 2737 (all A_1 (hBOOL(tendsto_nat_real(A_1,zero_zero_real,sequentially)) -> (hBOOL(monoseq_real(A_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_5048_summable__Leibniz_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2738 (all A_1 (is_int(A_1) -> (A_1 = zero_zero_int <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,A_1)),zero_zero_int))))) # label(fact_2710_abs__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2739 (all X_15 all Y_11 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_15),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_11),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_15)),Y_11)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_15),Y_11)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_2056_power2__sum) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2740 (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]. 9.34/9.40 2741 (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]. 9.34/9.40 2742 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> M = hAPP_nat_nat(div_mod_nat(M),N))) # label(fact_3171_mod__less) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2743 (all V_6 all B_21 all C_7 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_6)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B_21),C_7)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_6)),B_21)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_6)),C_7))) # label(fact_2393_right__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2744 (all Lx_2 all Ly_2 all Rx_2 all Ry_2 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_2),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Ly_2),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_2),Ry_2))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_2),Ly_2)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_2),Ry_2))) # label(fact_1069_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2745 (all Na all P_1 (hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) -> ((all N_1 (hBOOL(hAPP_nat_bool(P_1,N_1)) -> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_1))))) -> hBOOL(hAPP_nat_bool(P_1,Na))))) # label(fact_3999_nat__induct) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2746 (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]. 9.34/9.40 2747 (all C all B_1 all A_1 (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,B_1),A_1)) -> (C = B_1 -> 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]. 9.34/9.40 2748 (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]. 9.34/9.40 2749 (all A_3 all B_2 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,cis(A_3)),cis(B_2)) = cis(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),B_2))) # label(fact_4147_cis__mult) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2750 (all V_20 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_20)) -> hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_20)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_int_nat(number_number_of_nat,V_20)))) # label(fact_747_semiring__1__add__number__of) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2751 (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]. 9.34/9.40 2752 (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]. 9.34/9.40 2753 (all A_44 all B_33 all K_6 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_33),K_6) = A_44 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_33),A_44)))) # label(fact_2264_dvdI) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2754 (all X X = hAPP_real_real(ln,hAPP_real_real(exp_real,X))) # label(fact_3394_ln__exp) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2755 (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]. 9.34/9.40 2756 (all A_37 hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,one_on1645066479umeral),A_37))) # label(fact_2311_one__dvd) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2757 (all Xa all Ya (hAPP_real_real(sqrt,Ya) = hAPP_real_real(sqrt,Xa) <-> Ya = Xa)) # label(fact_3510_real__sqrt__eq__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2758 (all K all F all Na (is_int(K) -> ((all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),Na)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),one_one_nat))),hAPP_nat_int(F,I_2)))),one_one_int)))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),hAPP_nat_int(F,Na))) -> (exists I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),Na)) & K = hAPP_nat_int(F,I_2)))))))) # label(fact_2815_int__val__lemma) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2759 (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),hAPP_real_real(ln,Xa))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),Xa))))) # label(fact_3364_ln__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2760 (all Xa (is_int(Xa) -> (Xa = zero_zero_int <-> zero_zero_int = Xa))) # label(fact_769_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2761 (all Z_1 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) != zero_zero_int) # label(fact_202_odd__nonzero) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2762 (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,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))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_1),B_1)),E_2)),C)),D_1)))) # label(fact_2426_le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2763 (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]. 9.34/9.40 2764 (all A_135 all B_95 (hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_135),B_95) = zero_z891286103de_int -> B_95 = zero_z891286103de_int | A_135 = zero_z891286103de_int)) # label(fact_1132_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2765 (all K hAPP_n1699378549t_bool(ord_lessThan_nat,hAPP_nat_nat(suc,K)) = hAPP_n1699378549t_bool(ord_atMost_nat,K)) # label(fact_5185_lessThan__Suc__atMost) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2766 (all 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_nat,X_1),M_1)))) <-> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_3)))) # label(fact_4618_finite__nat__set__iff__bounded) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2767 (all Xa all A_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,A_1)),one_one_real))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim1246769320r_real(Xa)),A_1)))) # label(fact_4468_floor__le__eq) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2768 (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]. 9.34/9.40 2769 (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 ((all H_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),H_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H_1),D_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa),H_1))),hAPP_real_real(F,Xa)))))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))))))) # label(fact_3881_DERIV__neg__dec__right) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2770 (all Na all Ma (is_int(Ma) & is_int(Na) -> (hAPP_i1732201573de_int(quickcheck_of_int,Ma) = hAPP_i1732201573de_int(quickcheck_of_int,Na) <-> Ma = Na))) # label(fact_4557_Quickcheck__Narrowing_Oof__int__inject) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2771 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_complex_real(im,X)),hAPP_complex_real(im,Y)) = hAPP_complex_real(im,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),Y))) # label(fact_4097_Im_Oadd) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2772 (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]. 9.34/9.40 2773 (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]. 9.34/9.40 2774 (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]. 9.34/9.40 2775 (all X hAPP_complex_complex(cnj,hAPP_complex_complex(uminus473333897omplex,X)) = hAPP_complex_complex(uminus473333897omplex,hAPP_complex_complex(cnj,X))) # label(fact_4067_cnj_Ominus) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2776 (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)) -> (C_1 = B_2 -> (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]. 9.34/9.40 2777 (all P all Q (hBOOL(hAPP_bool_bool(fimplies(P),Q)) | -hBOOL(Q))) # label(help_fimplies_2_1_U) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2778 (all Z_2 hilbert_Eps_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_f1402056515l_bool(hAPP_f824790735l_bool(cOMBB_real_bool_real,hAPP_r1134773055l_bool(fequal_real,hAPP_complex_real(re,hAPP_complex_complex(sgn_sgn_complex,Z_2)))),cos))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_f1402056515l_bool(hAPP_f824790735l_bool(cOMBB_real_bool_real,hAPP_r1134773055l_bool(fequal_real,hAPP_complex_real(im,hAPP_complex_complex(sgn_sgn_complex,Z_2)))),sin))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,pi)))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_eq_real),pi))))) = arg(Z_2)) # label(fact_4807_arg__def) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2779 (all W_5 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),one_one_int)),hAPP_int_int(number_number_of_int,W_5)) = hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W_5))) # label(fact_1869_double__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2780 (all V_4 all W_2 all C_6 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_4)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(number_number_of_int,W_2)),C_6)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_4),W_2))),C_6)) # label(fact_2440_add__number__of__diff1) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2781 (all N (is_int(N) -> N = quickcheck_int_of(hAPP_i1732201573de_int(quickcheck_of_int,N)))) # label(fact_4554_int__of__of__int) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2782 (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]. 9.34/9.40 2783 (all Ya all Xa (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ya),Xa)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Ya)) -> Xa = Ya))) # label(fact_1637_xt1_I5_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2784 (all B_2 all A_3 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_2),A_3)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_2),A_3)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),A_3)) # label(fact_3420_real__average__minus__second) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2785 (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)) -> (X = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),N) -> hAPP_real_real(hAPP_n546711566l_real(root,N),X) = Y)))) # label(fact_3929_real__root__pos__unique) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2786 (all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),hAPP_int_int(bit1,pls))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,Xa)),one_one_int)))) # label(fact_637_le__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2787 (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]. 9.34/9.40 2788 (all W_11 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),hAPP_int_rat(number_number_of_rat,W_11))),hAPP_int_rat(number_number_of_rat,W_11)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,W_11))) # label(fact_220_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2789 (all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Ya)))) # label(fact_109_less__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2790 (all B_1_1 all B_2_1 is_bool(hAPP_Q2096512830t_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__Quickcheck____Narrowing__Ocode____int_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2791 (all A_150 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_150),zero_zero_real) = A_150) # label(fact_918_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2792 (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]. 9.34/9.40 2793 (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]. 9.34/9.40 2794 (all A_140 all B_98 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_98),A_140) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_140),B_98)) # label(fact_1034_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2795 (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]. 9.34/9.40 2796 (all A_7 all B_8 all C_4 all D_5 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(hAPP_rat_fun_rat_rat(plus_plus_rat,A_7),B_8)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_4),D_5)))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_7),C_4))),hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B_8),D_5)))))) # label(fact_2763_abs__diff__triangle__ineq) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2797 (all Ma (one_one_nat = Ma <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),one_one_nat)))) # label(fact_2500_nat__dvd__1__iff__1) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2798 (all R1 all A_3 all R2 all B_2 rcis(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,R1),R2),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_3),B_2)) = hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,rcis(R1,A_3)),rcis(R2,B_2))) # label(fact_4264_rcis__divide) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2799 (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(isCont_real_real(hAPP_n546711566l_real(root,Na)),Xa))))) # label(fact_5168_isCont__root__neg) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2800 (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_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_nat,Ma),Na))))) # label(fact_2157_nat__mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2801 (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]. 9.34/9.40 2802 (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]. 9.34/9.40 2803 (all A_3 all B_2 all C_1 (is_int(A_3) -> (hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),B_2) = C_1 -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_1),B_2) = A_3))) # label(fact_2574_Int2_Oaux1) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2804 (all V_1 all 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)) = 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)))) # label(fact_3488_minus__numeral__code_I6_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2805 (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]. 9.34/9.40 2806 (all N_55 ((zero_zero_nat = N_55 -> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),N_55) = one_on1645066479umeral) & (N_55 != zero_zero_nat -> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),N_55) = zero_z126310315umeral))) # label(fact_255_power__0__left) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2807 (all I_1 all K (is_int(K) -> (hAPP_int_int(hAPP_int_fun_int_int(div_div_int,I_1),K) = zero_zero_int <-> K = zero_zero_int | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),K)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),I_1)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),I_1))))) # label(fact_2845_zdiv__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2808 (all A_191 one_one_complex = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_191),zero_zero_nat)) # label(fact_460_power__0) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2809 (all X all N hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(cnj,X)),N) = hAPP_complex_complex(cnj,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X),N))) # label(fact_4065_complex__cnj__power) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2810 (all N zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),N)) # label(fact_5121_min__0L) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2811 (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]. 9.34/9.40 2812 (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]. 9.34/9.40 2813 (all L_2 hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),L_2)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,min),hAPP_int_int(bit1,L_2))) # label(fact_2444_diff__bin__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2814 (all X hAPP_complex_complex(cnj,hAPP_complex_complex(invers1449016382omplex,X)) = hAPP_complex_complex(invers1449016382omplex,hAPP_complex_complex(cnj,X))) # label(fact_4074_complex__cnj__inverse) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2815 (all K all Ma all Na (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Ma),Na))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),Na)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),Ma)))) # label(fact_4990_gcd__greatest__iff__int) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2816 (all P all Q all R hAPP_P621635040nt_int(P,hAPP_i1524277240nt_int(Q,R)) = hAPP_i1153444985nt_int(hAPP_f1148171384nt_int(hAPP_f334321603nt_int(cOMBB_1484610687nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc_) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2817 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(div_mod_int(A_3),B_2))))) # label(fact_3077_pos__mod__sign) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2818 (all Ma all Na (Na = zero_zero_nat | Ma = zero_zero_nat <-> zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),Na))) # label(fact_2086_mult__is__0) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2819 (all A_203 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,pls)),A_203) = A_203) # label(fact_174_add__numeral__0) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2820 (all Xa all Ya (Xa = 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)))) # label(fact_2015_order__le__less) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2821 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,ratreal(Xa)),ratreal(Ya))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Xa),Ya)))) # label(fact_4692_real__less__code) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2822 (all Y all X all N (-hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) -> (X = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),N) -> Y = hAPP_real_real(hAPP_n546711566l_real(root,N),X)))) # label(fact_3893_odd__real__root__unique) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2823 (all B_1_1 all B_2_1 is_bool(hAPP_nat_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__Nat__Onat_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2824 (all I_1 all J hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_nat_nat(suc,I_1)),hAPP_nat_nat(suc,J)) = image_nat_nat(suc,hAPP_n1699378549t_bool(ord_at238088361st_nat(I_1),J))) # label(fact_5075_image__Suc__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2825 (all Y all X (hBOOL(hAPP_int_bool(nat_is_nat,X)) -> (hBOOL(hAPP_int_bool(nat_is_nat,Y)) -> hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)))))) # label(fact_5136_Nat__Transfer_Otransfer__int__nat__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2826 (all Na all Q_3 all P_1 ((all X_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(Q_3,X_1)),hAPP_nat_nat(P_1,X_1)))) -> hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,hAPP_f1914919701at_nat(hAPP_f1408247010at_nat(cOMBS_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,minus_minus_nat),P_1)),Q_3)),hAPP_n1699378549t_bool(ord_lessThan_nat,Na)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,P_1),hAPP_n1699378549t_bool(ord_lessThan_nat,Na))),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,Q_3),hAPP_n1699378549t_bool(ord_lessThan_nat,Na))))) # label(fact_4955_sum__diff__distrib) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2827 (all P_1 all A0 (hBOOL(hAPP_int_bool(accp_int(quickc1265749348ro_rel),A0)) -> ((all I_2 (is_int(I_2) -> (hBOOL(hAPP_int_bool(accp_int(quickc1265749348ro_rel),I_2)) -> ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),zero_zero_int)) -> (zero_zero_int != I_2 -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,I_2),one_one_int))))) -> hBOOL(hAPP_int_bool(P_1,I_2)))))) -> hBOOL(hAPP_int_bool(P_1,A0))))) # label(fact_4269_around__zero_Opinduct) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2828 (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]. 9.34/9.40 2829 (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]. 9.34/9.40 2830 (all P all Q all R hAPP_f291087797t_bool(P,hAPP_P1668131738nt_int(Q,R)) = hAPP_P2053772734t_bool(hAPP_f1198736473t_bool(hAPP_f2042071795t_bool(cOMBB_773384995nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__I) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2831 (all X all Y hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(abs_abs_int,X)),hAPP_int_int(abs_abs_int,Y))) # label(fact_4977_gcd__abs__int) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2832 (all A_1 all Na (hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_1),Na) = zero_z891286103de_int <-> zero_z891286103de_int = A_1 & Na != zero_zero_nat)) # label(fact_446_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2833 (all C_10 all A_41 all B_30 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_41),B_30)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_41),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_30),C_10))))) # label(fact_2287_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2834 (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]. 9.34/9.40 2835 (all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) -> (B_2 != A_3 -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),A_3)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))))) # label(fact_2642_dvd_Ole__neq__trans) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2836 (all Xa (hAPP_int_int(abs_abs_int,Xa) = one_one_int <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Xa),one_one_int)))) # label(fact_2749_zdvd1__eq) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2837 (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]. 9.34/9.40 2838 (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]. 9.34/9.40 2839 (all P all Q all R hAPP_int_fun_int_rat(P,hAPP_P1175774780nt_int(Q,R)) = hAPP_P507092031nt_rat(hAPP_f1547007718nt_rat(hAPP_f382317021nt_rat(cOMBB_170828966nt_int,P),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]. 9.34/9.40 2840 (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]. 9.34/9.40 2841 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real)) -> zero_zero_nat = natceiling(X))) # label(fact_3314_natceiling__neg) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2842 (all N all M hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,N),M) = hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,quickcheck_int_of(N)),quickcheck_int_of(M)))) # label(fact_4560_plus__code__int__def) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2843 (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]. 9.34/9.40 2844 (all Z_1 zero_zero_real = hAPP_complex_real(im,hAPP_int_complex(ring_11397209091omplex,Z_1))) # label(fact_4581_complex__Im__of__int) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2845 (all B_134 all C_78 all A_196 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_196)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_134),C_78)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_134),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_196),C_78)))))) # label(fact_396_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2846 (all Y_14 all X_18 (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_14),X_18)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_18),Y_14)))) # label(fact_2008_not__leE) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2847 (all X all Y hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y))) # label(fact_5003_transfer__int__nat__gcd_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2848 (all Z_12 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_12),Z_12) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_12)) # label(fact_1891_mult__2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2849 (all B_123 all A_179 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_179)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_123)) -> 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_179),B_123)))))) # label(fact_593_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2850 (all X_8 all N_12 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_12)) | one_on1645066479umeral = X_8 -> 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]. 9.34/9.40 2851 (all N_35 all A_166 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,one_on1684967323de_int),A_166)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,one_on1684967323de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_166),N_35))))) # label(fact_708_one__le__power) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2852 (all P_1 all Na all K ((zero_zero_nat = K -> hBOOL(hAPP_nat_bool(P_1,Na))) & (K != zero_zero_nat -> (all I_2 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),K)) -> (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) = Na -> hBOOL(hAPP_nat_bool(P_1,J_1)))))) <-> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(div_mod_nat(Na),K))))) # label(fact_3197_split__mod) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2853 (all A_89 all M_13 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_89),M_13)),M_13) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_89),one_one_int)),M_13)) # label(fact_1490_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2854 (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]. 9.34/9.40 2855 (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]. 9.34/9.40 2856 (all X all Y ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,ratreal(X)),ratreal(Y))) # label(fact_4704_real__times__code) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2857 (all M all N M = 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)) # label(fact_2482_diff__add__inverse2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2858 (all X_62 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_62),zero_zero_nat) = one_one_complex) # label(fact_478_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2859 (all A_151 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_151),zero_z891286103de_int) = A_151) # label(fact_909_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 9.34/9.40 2860 (all B_2 all A_3 all K_2 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))),hAPP_int_int(div_mod_int(A_3),B_2))),K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),K_2)) # label(fact_3089_zdiv__zmod__equality) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2861 (all B_124 all C_71 all A_180 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_180)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_124),C_71)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_124),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_180),C_71)))))) # label(fact_579_add__increasing) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2862 (all X ((zero_zero_real != X -> (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_real_real(sgn_sgn_real,X) = hAPP_real_real(uminus_uminus_real,one_one_real)) & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_real_real(sgn_sgn_real,X) = one_one_real)) & (X = zero_zero_real -> hAPP_real_real(sgn_sgn_real,X) = zero_zero_real))) # label(fact_3635_real__sgn__def) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2863 (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]. 9.34/9.41 2864 (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]. 9.34/9.41 2865 (all N hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N),one_one_int)),N) = one_one_int) # label(fact_4980_coprime__plus__one__int) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2866 (all A_1 all B_1 (hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(negDivAlg_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,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),B_1))) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),zero_zero_int)) -> negDivAlg(A_1,B_1) = 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_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_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),B_1)))) -> negDivAlg(A_1,B_1) = hAPP_P1975530577nt_int(adjust(B_1),negDivAlg(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_3298_negDivAlg_Opsimps) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2867 (all Ma all A_1 all B_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B_1)) -> -hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),hAPP_i1948725293t_bool(bnorRset(A_1),Ma))))) # label(fact_4292_Bnor__mem__zle__swap) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2868 (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]. 9.34/9.41 2869 (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]. 9.34/9.41 2870 (all X all Y (is_int(Y) & is_int(X) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Y)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),X)) | X = Y)) # label(fact_57_zless__linear) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2871 (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_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_int,A_1),B_1))))) # label(fact_4721_Fract__less__one__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2872 (all X all A_3 hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)) = hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_real_real(uminus_uminus_real,A_3))) # label(fact_4413_powr__minus) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2873 (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]. 9.34/9.41 2874 (all Na (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Na)) -> hAPP_int_int(fact_fact_int,Na) = big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),Na)))) # label(fact_5083_fact__altdef__int) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2875 (all B_2 all D_3 all A_3 all C_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),C_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),D_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,C_1),D_3)))))) # label(fact_4325_gcd__semilattice__nat_Oinf__mono) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2876 (all A_138 zero_z126310315umeral = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,zero_z126310315umeral),A_138)) # label(fact_1109_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2877 (all Y all U_1 all X1_1 all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X1_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X1_1),Y)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),U_1))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),hAPP_real_real(inverse_inverse_real,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,U_1),hAPP_real_real(inverse_inverse_real,X1_1)))))))) # label(fact_3855_real__mult__inverse__cancel2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2878 (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]. 9.34/9.41 2879 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,natfloor(X))),X)))) # label(fact_3718_real__natfloor__le) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2880 (all Xa all Ya (Ya = one_one_real & zero_zero_real = Xa <-> ii = hAPP_real_complex(hAPP_r265291036omplex(complex,Xa),Ya))) # label(fact_3976_Complex__eq__i) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2881 (all Y all X all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_3)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,Y),A_3))))))) # label(fact_4429_powr__mono2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2882 (all Ma all K all Na (is_int(K) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Na)) -> ((exists S_2 exists T (is_int(T) & hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,K),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,S_2),T)) = xzgcd(Ma,Na) & is_int(S_2))) <-> K = hAPP_int_int(legacy_zgcd(Ma),Na))))) # label(fact_4529_xzgcd__correct) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2883 (all B_1_1 all B_2_1 is_bool(hAPP_f448129468l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_J_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2884 (all X ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> hAPP_int_int(fact_fact_int,X) = zero_zero_int) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> hAPP_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]. 9.34/9.41 2885 (all N_52 all A_197 (A_197 != zero_zero_rat -> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_197),N_52) != zero_zero_rat)) # label(fact_344_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2886 (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]. 9.34/9.41 2887 (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]. 9.34/9.41 2888 (all Z_11 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_11),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)) # label(fact_1895_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2889 (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]. 9.34/9.41 2890 (all B_2 all A_3 all X (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),X)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),X)))) # label(fact_4318_gcd__semilattice__nat_Ole__infI1) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2891 (all Z_2 all Xa all X_2 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Xa),X_2)) -> ((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,Xa),comple124823625p_real(X_2)))))) # label(fact_4633_SupInf_OSup__upper) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2892 (all V_8 all U_2 all Y_22 all X_27 all A_60 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_27),A_60)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_22),A_60)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),U_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),V_8)) -> (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,U_2),V_8) = one_one_real -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,U_2),X_27)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,V_8),Y_22))),A_60)))))))) # label(fact_1878_convex__bound__lt) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2893 (all A_163 all B_111 all C_65 (is_int(B_111) & is_int(C_65) -> (hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_163),B_111) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_163),C_65) -> B_111 = C_65))) # label(fact_789_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2894 (all P all Q all R hAPP_f1735705551nt_int(hAPP_i1836591008nt_int(P,R),hAPP_i1584592887nt_int(Q,R)) = hAPP_i1584592887nt_int(hAPP_f465821005nt_int(hAPP_f1081447804nt_int(cOMBS_1470252563nt_int,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__prod_Itc__I) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2895 (all Na (hBOOL(hAPP_nat_bool(even_odd_even_nat,Na)) <-> (exists M_1 Na = 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)))) # label(fact_3815_even__mult__two__ex) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2896 (all X_53 all P_7 all Q_8 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_53),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_8)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_53),P_7)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_53),Q_8))) # label(fact_1502_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2897 (all A_3 all B_2 all K_2 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,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) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),K_2)) # label(fact_3090_zdiv__zmod__equality2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2898 (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,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_real,Y_1),zero_zero_real)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,B_1)),hAPP_real_real(F,A_1)))))) # label(fact_4002_DERIV__neg__imp__decreasing) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2899 (all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),hAPP_nat_nat(fact_fact_nat,N)))) # label(fact_3840_fact__ge__one__nat) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2900 (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_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(div_mod_int(Q_1),C_1))),R_2)),zero_zero_int)))))) # label(fact_3129_zmult2__lemma__aux2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2901 (all X_26 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_26),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))))) # label(fact_1908_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2902 (all X_54 all Y_46 all Q_9 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_54),Y_46)),Q_9) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_54),Q_9)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_46),Q_9))) # label(fact_1250_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2903 (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)))))) -> X = hAPP_real_real(arcsin,hAPP_real_real(sin,X))))) # label(fact_3664_arcsin__sin) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2904 (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(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)))) & (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)))) # label(fact_3830_fact__add__num__eq__if2__nat) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2905 (all A_158 all B_106 all C_60 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_158),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_106),C_60)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_158),B_106)),C_60)) # label(fact_830_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2906 (all N all M hAPP_Q1762011733de_int(div_mo231679042de_int(N),M) = hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(div_mod_int(quickcheck_int_of(N)),quickcheck_int_of(M)))) # label(fact_4552_mod__code__int__def) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2907 (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]. 9.34/9.41 2908 (all Xa all Ya (Ya = zero_zero_real & Xa = 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))))) = zero_zero_real)) # label(fact_3_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2909 (all N hAPP_complex_real(re,hAPP_nat_complex(semiri2020571505omplex,N)) = hAPP_nat_real(semiri132038758t_real,N)) # label(fact_4126_complex__Re__of__nat) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2910 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)))))) # label(fact_5012_transfer__nat__int__gcd__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2911 (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]. 9.34/9.41 2912 (all Xa (is_int(Xa) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd)) -> -(all K_1 (is_int(K_1) -> Xa != 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)))))) # label(fact_2954_zOddE) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2913 (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]. 9.34/9.41 2914 (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]. 9.34/9.41 2915 (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]. 9.34/9.41 2916 (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]. 9.34/9.41 2917 (all C all A_1 all B_1 (is_int(C) -> (zero_zero_int = C | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_1),B_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_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_2359_dvd__mult__cancel__left) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2918 (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]. 9.34/9.41 2919 (all A_39 all B_28 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_39),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_39),B_28)))) # label(fact_2294_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2920 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),N)))) # label(fact_3782_lemma__STAR__cos) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2921 (all Ya all Xa (is_int(Ya) & is_int(Xa) -> (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]. 9.34/9.41 2922 (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_2031_order__less__le) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2923 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M)) # label(fact_281_nat__add__commute) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2924 (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]. 9.34/9.41 2925 (all X hAPP_real_real(tan,hAPP_real_real(uminus_uminus_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(tan,X))) # label(fact_3688_tan__minus) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2926 (all X all Y all M hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),hAPP_int_int(div_mod_int(Y),M))),M) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y)),M)) # label(fact_2991_zdiff__zmod__right) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2927 (all P all Q all R hAPP_f448129468l_bool(hAPP_i2112223885l_bool(P,R),Q) = hAPP_int_bool(hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2928 (all N all M hAPP_C2132160860umeral(hAPP_C1614127039umeral(produc2136830103umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,N),M)),hAPP_C498520661umeral(div_mo1740067990umeral(N),M)) = code_d418564891umeral(N,M)) # label(fact_4197_div__mod__code__numeral__def) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2929 (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]. 9.34/9.41 2930 (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]. 9.34/9.41 2931 (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]. 9.34/9.41 2932 (all C_17 all A_67 all B_44 (A_67 = B_44 -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_17),B_44)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_17),A_67))))) # label(fact_1759_xt1_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2933 (all 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))) = hAPP_real_real(sin,Xa)) # label(fact_4818_sin__def) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2934 (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]. 9.34/9.41 2935 (all Xa all A_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim1246769320r_real(Xa)),A_1)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),hAPP_int_real(real_int,A_1))))) # label(fact_4415_floor__less__eq) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2936 (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]. 9.34/9.41 2937 (all Ma all 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) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,Ma),hAPP_nat_nat(suc,Na))) # label(fact_5191_max__Suc2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2938 (all A_3 all B_2 all N hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(legacy_zgcd(A_3),B_2)),N) = hAPP_int_int(legacy_zgcd(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),N))) # label(fact_4491_zgcd__power__distrib) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2939 (all X_6 all N_11 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_11)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_11),one_one_nat))),X_6) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_6),N_11))) # label(fact_2557_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2940 (all N all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N) = hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_nat_real(real_nat,N)))) # label(fact_4458_powr__realpow) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2941 (all P_1 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_real,Ya),Xa)) -> hBOOL(P_1)))) # label(fact_1787_order__less__imp__triv) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2942 (all A_18 (is_int(A_18) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_18)) -> A_18 = hAPP_int_int(abs_abs_int,A_18)))) # label(fact_2713_abs__of__nonneg) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2943 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(arctan,Y))) & 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_3582_arctan__bounded) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2944 (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]. 9.34/9.41 2945 (all B_96 all A_136 (is_int(B_96) & is_int(A_136) -> (zero_zero_int != A_136 -> (zero_zero_int != B_96 -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_136),B_96) != zero_zero_int)))) # label(fact_1130_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2946 (all A_1 all B_1 (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,zero_zero_rat),B_1)) & 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,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_1),B_1)),zero_zero_rat)))) # label(fact_1289_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2947 (all P all Q all R hAPP_bool_bool(hAPP_n1006566506l_bool(P,R),hAPP_nat_bool(Q,R)) = hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__HOL__Obool_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2948 (all N_52 all A_197 (A_197 != zero_zero_real -> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_197),N_52) != zero_zero_real)) # label(fact_346_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2949 (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),cos_coeff)),hAPP_r474017924t_real(power_power_real,Xa))))) # label(fact_4831_summable__cos) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2950 (all K all Na all Ma (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)))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),Na)))) # label(fact_2406_zdvd__reduce) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2951 (all A_21 all B_16 hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_16),A_21)) = hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_21),B_16))) # label(fact_2686_abs__minus__commute) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2952 (all B_1 all P_5 all A_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_1)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),hAPP_i1948725293t_bool(wset(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_1),one_one_int)),P_5))) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),hAPP_i1948725293t_bool(wset(A_1),P_5)))))) # label(fact_2929_wset__subset) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2953 (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]. 9.34/9.41 2954 (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]. 9.34/9.41 2955 (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(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),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]. 9.34/9.41 2956 (all Xa (one_on1684967323de_int = Xa <-> Xa = one_on1684967323de_int)) # label(fact_851_one__reorient) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2957 (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]. 9.34/9.41 2958 (all P all Q all R hAPP_i447943416t_real(hAPP_f1295083291t_real(hAPP_f212164053t_real(cOMBB_929094628al_int,P),Q),R) = hAPP_f1690286043t_real(P,hAPP_i371647666l_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__032) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2959 (all K_2 hAPP_int_int(pred,K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,min),K_2)) # label(fact_4047_add__Min) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2960 (all Q_1 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(prime,Q_1)) -> (Q_1 != P_3 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,P_3),Q_1)))))) # label(fact_4165_distinct__prime__coprime) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2961 (all A_54 all M_11 all N_21 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_54),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_11),N_21)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_54),M_11)),N_21)) # label(fact_2095_power__mult) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2962 (all Wa (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,Wa))) <-> hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,Wa)))))) # label(fact_3246_neg__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2963 (all P_1 ((exists X_1 (is_int(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)))) <-> (exists X1 hBOOL(hAPP_nat_bool(P_1,X1))))) # label(fact_2744_ex__nat) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2964 (all A_3 all B_2 hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),hAPP_real_real(uminus_uminus_real,B_2)) = hAPP_complex_complex(cnj,hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2))) # label(fact_4082_complex__cnj) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2965 (all V_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,V_1))) -> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V_1)),one_one_nat)) = hAPP_int_nat(number_number_of_nat,V_1))) # label(fact_3141_expand__Suc) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2966 (all B_112 all A_164 all C_66 (hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_66),A_164) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_112),A_164) -> B_112 = C_66)) # label(fact_779_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2967 (all A_146 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,zero_z891286103de_int),A_146) = A_146) # label(fact_947_add__0__left) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2968 (all A_133 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_133),zero_zero_int) = zero_zero_int) # label(fact_1151_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2969 (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]. 9.34/9.41 2970 (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,Xa),Z_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ya),Z_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya))))) # label(fact_2124_real__mult__le__cancel__iff1) # label(axiom) # label(non_clause). [assumption]. 9.34/9.41 2971 (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]. 9.34/9.41 2972 (all A_1 all C all D_1 all B_1 (is_int(D_1) & is_int(B_1) -> (B_1 != zero_zero_int -> (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]. 9.34/9.42 2973 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,Xa)),hAPP_real_real(exp_real,Ya))))) # label(fact_3389_exp__le__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 2974 (all Na all Xa all Diff all F (F = hAPP_n546711566l_real(Diff,zero_zero_nat) & Xa != zero_zero_real & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) & (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_real,zero_zero_real),hAPP_real_real(abs_abs_real,T))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_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_4930_Maclaurin__all__lt__objl) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 2975 (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]. 9.34/9.42 2976 (all Na hAPP_n1699378549t_bool(ord_atMost_nat,Na) = image_int_nat(nat,hAPP_i1948725293t_bool(ord_at875362053st_int(zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Na)))) # label(fact_5182_SetInterval_Otransfer__nat__int__set__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 2977 (all A_3 all B_2 all P_3 all Q_1 hAPP_P1175774780nt_int(twoSqu1907779896sum2sq,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),P_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),Q_1))),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Q_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),P_3)))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(twoSqu1907779896sum2sq,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_3),B_2))),hAPP_P1175774780nt_int(twoSqu1907779896sum2sq,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_1)))) # label(fact_2750_mult__sum2sq) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 2978 (all Y all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X)),one_one_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,Y)),one_one_real)) -> hAPP_real_real(arctan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(arctan,X)),hAPP_real_real(arctan,Y))))) # label(fact_3506_arctan__add) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 2979 (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]. 9.34/9.42 2980 (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]. 9.34/9.42 2981 (all N_51 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,one_one_complex),N_51) = one_one_complex) # label(fact_350_power__one) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 2982 (all P all Q all R hAPP_b646336293l_real(P,hAPP_nat_bool(Q,R)) = hAPP_n682674106l_real(hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,P),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]. 9.34/9.42 2983 (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)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(ln,Xa)),hAPP_real_real(ln,Ya))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)))))) # label(fact_3346_ln__less__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 2984 (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]. 9.34/9.42 2985 (all A_1 (is_int(A_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,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_210_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 2986 (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]. 9.34/9.42 2987 (all S ((exists K_1 hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,S),hAPP_n1699378549t_bool(ord_atMost_nat,K_1)))) <-> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S)))) # label(fact_5184_finite__nat__iff__bounded__le) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 2988 (all Diff all F all Na all H (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))))) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H),zero_zero_real)) -> (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_4937_Maclaurin__minus__objl) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 2989 (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]. 9.34/9.42 2990 (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]. 9.34/9.42 2991 (all C_30 all A_84 all B_57 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_30),A_84)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_30),B_57))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_30)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_84),B_57))))) # label(fact_1537_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 2992 (all N hAPP_complex_complex(cnj,hAPP_nat_complex(semiri2020571505omplex,N)) = hAPP_nat_complex(semiri2020571505omplex,N)) # label(fact_4071_complex__cnj__of__nat) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 2993 (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]. 9.34/9.42 2994 (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]. 9.34/9.42 2995 (all M all N all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (zero_zero_nat != N -> (N = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_3),M) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)))))) # label(fact_4182_prime__factor__lt) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 2996 (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]. 9.34/9.42 2997 (all M_24 all N_48 hAPP_n522471361de_int(semiri1424489471de_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_24),N_48)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_24)),hAPP_n522471361de_int(semiri1424489471de_int,N_48))) # label(fact_371_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 2998 (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]. 9.34/9.42 2999 (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]. 9.34/9.42 3000 (all K all I_1 all J (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J)) -> (J = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),I_1) <-> K = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),I_1)))) # label(fact_2523_le__imp__diff__is__add) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3001 (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_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(hAPP_int_fun_int_int(times_times_int,A_1),hAPP_int_int(multInv(P_5),Xa)))),hAPP_i1948725293t_bool(sRStar,P_5)))))))) # label(fact_3164_SRStar__mult__prop2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3002 (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]. 9.34/9.42 3003 (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]. 9.34/9.42 3004 (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]. 9.34/9.42 3005 (all A_3 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)))) = 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)))))) # label(fact_3494_power3__minus) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3006 (all M all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)),M))) # label(fact_2882_div__le__dividend) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3007 (all Xa all Ya (Xa = Ya <-> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ya),Xa)) & hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Ya)))) # label(fact_1702_order__eq__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3008 (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]. 9.34/9.42 3009 (all P_1 collec50511176nt_int(P_1) = P_1) # label(fact_148_Collect__def) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3010 (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]. 9.34/9.42 3011 (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]. 9.34/9.42 3012 (all Na all Xa 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(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))))) # label(fact_4925_Maclaurin__exp__le) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3013 (all X_33 all Y_27 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_33),Y_27)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_27),X_33)))) # label(fact_1816_order__less__not__sym) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3014 (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]. 9.34/9.42 3015 (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]. 9.34/9.42 3016 (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]. 9.34/9.42 3017 (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(times_times_int,Xa),Ya)),zOdd))))) # label(fact_2940_EvenOdd_Oodd__times__odd) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3018 (all X all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),hAPP_int_int(multInv(P_3),X))),X),P_3)))))) # label(fact_2911_MultInv__prop4) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3019 (all B_120 all A_174 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_174),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_120),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_174),B_120)),zero_zero_int))))) # label(fact_645_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3020 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(cos,X)))))) # label(fact_3663_cos__gt__zero__pi) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3021 (all W_10 hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit1,W_10)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,W_10))),hAPP_int_rat(number_number_of_rat,W_10))) # label(fact_224_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3022 (all A_46 all B_35 all C_14 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_46),B_35)),C_14)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,B_35),C_14)))) # label(fact_2254_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3023 (all K all Ma all Na (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) <-> zero_zero_nat = K | Ma = Na)) # label(fact_2087_mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3024 (all Va all V (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Va),V)) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Va),V)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,Va)),hAPP_int_nat(number_number_of_nat,V))))) # label(fact_310_less__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3025 (all X all Y hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),hAPP_int_int(abs_abs_int,Y)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)) # label(fact_4978_gcd__abs2__int) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3026 (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]. 9.34/9.42 3027 (all A_139 all B_97 all C_51 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_139),B_97)),C_51) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_139),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_97),C_51))) # label(fact_1051_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3028 (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]. 9.34/9.42 3029 (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_nat_complex(semiri2020571505omplex,hAPP_int_nat(number_number_of_nat,V_17)) = hAPP_int_complex(number528085621omplex,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_complex(semiri2020571505omplex,hAPP_int_nat(number_number_of_nat,V_17)) = zero_zero_complex))) # label(fact_900_of__nat__number__of__lemma) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3030 (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]. 9.34/9.42 3031 (all P all Q all R hAPP_real_bool(hAPP_i688218208l_bool(P,R),Q) = hAPP_int_bool(hAPP_r481142414t_bool(hAPP_f1845081853t_bool(cOMBC_int_real_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__RealDef__Oreal_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3032 (all Xa all Ya (hBOOL(hAPP_P1555980039t_bool(accp_P490777396at_nat(nat_gcd_rel),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Xa),Ya))) -> (zero_zero_nat = Ya -> hAPP_nat_nat(nat_gcd(Xa),Ya) = Xa) & (zero_zero_nat != Ya -> hAPP_nat_nat(nat_gcd(Ya),hAPP_nat_nat(div_mod_nat(Xa),Ya)) = hAPP_nat_nat(nat_gcd(Xa),Ya)))) # label(fact_4662_nat__gcd_Opsimps) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3033 (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]. 9.34/9.42 3034 (all X_31 all Y_25 (is_int(Y_25) & is_int(X_31) -> (X_31 != Y_25 -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_31),Y_25)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_25),X_31)))))) # label(fact_1829_linorder__neqE) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3035 (all Z_10 all Y_21 all X_25 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_21),X_25)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_10),Y_21)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_10),X_25))))) # label(fact_1924_xt1_I8_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3036 (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]. 9.34/9.42 3037 (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]. 9.34/9.42 3038 (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)) -> natceiling(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,natceiling(X)),hAPP_int_nat(number_number_of_nat,N))))) # label(fact_3297_natceiling__add__number__of) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3039 (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]. 9.34/9.42 3040 (all P (is_bool(P) -> P = fFalse | fTrue = P)) # label(help_If_3_1_If_000tc__Nat__Onat_T) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3041 (all B_77 all 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,zero_zero_int),A_107)) | 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]. 9.34/9.42 3042 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_int_nat(nat,X)),hAPP_int_nat(nat,Y)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y))))) # label(fact_5019_transfer__nat__int__gcd_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3043 (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)) -> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,S_2),T) != R_2)))))) # label(fact_4647_obtain__pos__sum) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3044 (all Na all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Na)),hAPP_nat_nat(suc,Ma))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),Ma)))) # label(fact_3021_Suc__le__mono) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3045 (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]. 9.34/9.42 3046 (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]. 9.34/9.42 3047 (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_2240_inf__period_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3048 (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]. 9.34/9.42 3049 (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]. 9.34/9.42 3050 (all W_15 ((zero_zero_nat != hAPP_int_nat(number_number_of_nat,W_15) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,W_15)) = zero_zero_nat) & (hAPP_int_nat(number_number_of_nat,W_15) = zero_zero_nat -> one_one_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]. 9.34/9.42 3051 (all A_120 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_120),A_120)))) # label(fact_1283_zero__le__square) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3052 (all A_3 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),P_3)) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),hAPP_int_nat(nat,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))))) # label(fact_2810_Euler_Oaux__1) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3053 (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]. 9.34/9.42 3054 (all N hAPP_nat_nat(suc,N) != N) # label(fact_2984_Suc__n__not__n) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3055 (all Xa hBOOL(hAPP_real_bool(isCont_real_real(sqrt),Xa))) # label(fact_5161_isCont__real__sqrt) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3056 (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]. 9.34/9.42 3057 (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]. 9.34/9.42 3058 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N)) # label(fact_4331_gcd__add2__nat) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3059 (all A_1 all B_1 (one_one_int = hAPP_int_int(legacy_zgcd(A_1),B_1) <-> -(exists P_4 (hBOOL(hAPP_int_bool(zprime,P_4)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_4),B_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_4),A_1)) & is_int(P_4))))) # label(fact_4521_zgcd1__iff__no__common__primedivisor) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3060 (all V_2 all K_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_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_1)),K_2)) & (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_1),V_2))),K_2))) & (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,V_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),K_2)))) # label(fact_3263_nat__number__of__add__left) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3061 (all N hAPP_nat_nat(fact_fact_nat,N) != zero_zero_nat) # label(fact_3831_fact__nonzero__nat) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3062 (all P all X (-hBOOL(hAPP_nat_bool(P,X)) | hBOOL(hAPP_f54304608l_bool(fEx_nat,P)))) # label(help_fEx_1_1_fEx_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3063 (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]. 9.34/9.42 3064 (all X all N all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),X))))) # label(fact_4174_prime__divexp) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3065 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(div_mod_int(A_3),B_2)),B_2)))) # label(fact_3031_pos__mod__bound) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3066 (all Xa all Na all Diff all F (F = hAPP_n546711566l_real(Diff,zero_zero_nat) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) -> (zero_zero_real != Xa -> ((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_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_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,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))) = hAPP_real_real(F,Xa) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(abs_abs_real,T)))))))))) # label(fact_4931_Maclaurin__all__lt) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3067 (all V_7 all W_3 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_int_complex(number528085621omplex,V_7)),hAPP_int_complex(number528085621omplex,W_3)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_7),W_3))) # label(fact_2229_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3068 (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]. 9.34/9.42 3069 (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]. 9.34/9.42 3070 (all X all Y hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X)),Y))) # label(fact_2150_real__sum__squared__expand) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3071 (all B_1_1 all B_2_1 (is_int(B_2_1) -> is_bool(hAPP_int_bool(B_1_1,B_2_1)))) # label(gsy_c_hAPP_000tc__Int__Oint_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3072 (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]. 9.34/9.42 3073 (all A_39 all B_28 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_39),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_39),B_28)))) # label(fact_2296_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3074 (all X all Y (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X),Y)) -> hAPP_int_int(abs_abs_int,X) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y))) # label(fact_4973_gcd__proj1__if__dvd__int) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3075 (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]. 9.34/9.42 3076 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),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,zero_zero_nat),Ma)))) # label(fact_303_add__gr__0) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3077 (all C_77 all A_188 all B_131 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_77),A_188)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_77),B_131))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_188),B_131)))) # label(fact_518_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3078 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N))),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) = divmod_nat(M,N)))) # label(fact_4205_divmod__nat__step) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3079 (all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> hAPP_int_nat(nat,hAPP_int_int(fact_fact_int,X)) = hAPP_nat_nat(fact_fact_nat,hAPP_int_nat(nat,X)))) # label(fact_4284_transfer__nat__int__factorial) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3080 -(all T (is_int(T) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,s),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),one_one_int) != hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)),T))) # label(fact_2151__096_B_Bthesis_O_A_I_B_Bt_O_As_____A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A) # label(axiom) # label(non_clause). [assumption]. 9.34/9.42 3081 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y)) -> (exists R_1 exists A_2 (Y = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(sin,A_2)) & hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(cos,A_2)) = X)))) # label(fact_3850_polar__ex1) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3082 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y)) -> zero_zero_real = hAPP_real_real(cos,arg(hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),Y))))) # label(fact_3964_cos__arg__i__mult__zero__pos) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3083 (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]. 9.34/9.43 3084 (all C_33 all A_87 all B_60 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_33),A_87)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_33),B_60))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_33)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_87),B_60))))) # label(fact_1515_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3085 (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]. 9.34/9.43 3086 (all N hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),pi)) = zero_zero_real) # label(fact_3695_tan__npi) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3087 (all Xa all Ya (is_int(Xa) & is_int(Ya) -> (Xa != Ya & 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))))) # label(fact_2034_order__less__le) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3088 (all M_9 all A_31 all N_15 all B_23 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_31),N_15)),B_23)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_9),N_15)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_31),M_9)),B_23))))) # label(fact_2366_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3089 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M)) -> N != M)) # label(fact_289_less__not__refl2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3090 (all D_3 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_3),hAPP_int_int(abs_abs_int,D_3)))) # label(fact_2947_zdvd__self__abs1) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3091 (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]. 9.34/9.43 3092 (all X all Y hAPP_complex_complex(invers1449016382omplex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,of_real_complex(X)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),of_real_complex(Y)))) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,Y),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) # label(fact_3981_complex__inverse__complex__split) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3093 (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]. 9.34/9.43 3094 (all A_37 hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,one_on1684967323de_int),A_37))) # label(fact_2309_one__dvd) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3095 (all P_1 ((all X_1 exists D_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2)) & (all A_2 all B_4 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),X_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_4),A_2)),D_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),B_4)) -> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4))))))) & (all A_2 all B_4 all C_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),B_4)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_4),C_3)) & hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4))) & hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,B_4),C_3))) -> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),C_3))))) -> (all A_2 all B_4 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),B_4)) -> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4))))))) # label(fact_4290_lemma__BOLZANO2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3096 (all L_2 all K_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2)) -> 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)) = posDivAlg(K_2,L_2)))) # label(fact_3296_posDivAlg__div__mod) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3097 (all A_1 all B_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_1),B_1)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_1),zero_zero_real)) & 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,zero_zero_real),B_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),zero_zero_real)))) # label(fact_1290_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3098 (all X 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))))) = 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))))) # label(fact_2141_four__x__squared) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3099 (all A_128 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,one_one_rat),A_128) = A_128) # label(fact_1192_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3100 (all P_1 all Ma all Na ((exists Q_4 (hBOOL(hAPP_nat_bool(P_1,Q_4)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Na),hAPP_nat_nat(suc,Q_4)))) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Na),Q_4)),Ma)))) | hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) & Na = zero_zero_nat <-> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,Ma),Na))))) # label(fact_3147_split__div_H) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3101 (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]. 9.34/9.43 3102 (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) -> (A_1 = B_1 <-> D_1 = C))) # label(fact_2226_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3103 (all Xa all Ya (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(semiri1621563631at_int,Xa)),hAPP_nat_int(semiri1621563631at_int,Ya))))) # label(fact_307_Nat__Transfer_Otransfer__int__nat__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3104 (all Z_16 all Y_32 all X_38 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_32),X_38)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_16),Y_32)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_16),X_38))))) # label(fact_1733_xt1_I10_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3105 (all B_76 all A_106 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_76)) & hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_106),zero_z126310315umeral)) | hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_76),zero_z126310315umeral)) & hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_106)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_106),B_76)),zero_z126310315umeral)))) # label(fact_1360_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3106 (all Ma all K all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,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))) <-> 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)))) # label(fact_2116_mult__less__cancel2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3107 (all A_156 all C_58 all 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)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_156),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_58),D_17))) # label(fact_839_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3108 (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]. 9.34/9.43 3109 (all C all D_1 all A_1 all B_1 (B_1 != A_1 & 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),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_1),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),D_1)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_1),C)))) # label(fact_1170_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3110 (all A_50 hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_50),zero_z891286103de_int))) # label(fact_2207_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3111 (all M all N hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,M)))),N) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),M)),N)) # label(fact_3205_Suc__mod__eq__add3__mod) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3112 (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]. 9.34/9.43 3113 (all P_1 all Na all K (hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,Na),K))) <-> (K = zero_zero_nat -> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))) & (K != zero_zero_nat -> (all I_2 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),K)) -> (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) = Na -> hBOOL(hAPP_nat_bool(P_1,I_2)))))))) # label(fact_2895_split__div) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3114 (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,hAPP_int_int(abs_abs_int,Ma)),K)))) # label(fact_2693_abs__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3115 (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]. 9.34/9.43 3116 (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),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_1),D_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)) <-> C != D_1 & B_1 != A_1)) # label(fact_1173_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3117 (all Ma all Na the_Pr588456374at_nat(divmod_nat_rel(Ma,Na)) = divmod_nat(Ma,Na)) # label(fact_4888_divmod__nat__def) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3118 (all Xa all Na (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,Na))) -> (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),hAPP_int_real(number267125858f_real,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(Xa)),hAPP_int_nat(number_number_of_nat,Na))))))) # label(fact_3300_natceiling__le__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3119 (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]. 9.34/9.43 3120 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> 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_3183_le__mod__geq) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3121 (all C_42 all D_15 all A_109 all B_79 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_109),B_79)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_42),D_15)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_109)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_42)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_109),C_42)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_79),D_15)))))))) # label(fact_1348_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3122 (all A_122 all N_29 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_122),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_122),N_29)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_122),N_29)),A_122)) # label(fact_1234_power__commutes) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3123 (all A_147 A_147 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A_147)) # label(fact_943_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3124 (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_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_192)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_192),N_44)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_192),N_43)))))) # label(fact_443_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3125 (all F all G (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G)) <-> (all X_1 hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,X_1)),hAPP_nat_bool(G,X_1)))))) # label(fact_1714_le__fun__def) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3126 (all A_155 all C_57 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_155),C_57) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_57),A_155)) # label(fact_844_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3127 (all Ta all D_1 all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),D)) -> (all X_1 all K_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Ta))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,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))),Ta))))))) # label(fact_2237_inf__period_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3128 (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]. 9.34/9.43 3129 (all N_31 all A_153 all B_104 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_153),B_104)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_153)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_153),N_31)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_104),N_31))))))) # label(fact_879_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3130 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,Xa)),hAPP_real_real(exp_real,Ya))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)))) # label(fact_3391_exp__less__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3131 (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]. 9.34/9.43 3132 (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]. 9.34/9.43 3133 (all Ma all Na (one_one_nat = Na & Ma = one_one_nat <-> one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),Na))) # label(fact_2109_nat__1__eq__mult__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3134 (all C_22 all A_72 all B_49 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_72),B_49)) -> (B_49 = C_22 -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_72),C_22))))) # label(fact_1668_ord__le__eq__trans) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3135 (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,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) <-> 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))) # label(fact_2381_eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3136 (all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,N)))) # label(fact_2977_lessI) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3137 (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]. 9.34/9.43 3138 (all X all Y hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Y)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y))) # label(fact_389_Nat__Transfer_Otransfer__int__nat__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3139 (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]. 9.34/9.43 3140 (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]. 9.34/9.43 3141 (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]. 9.34/9.43 3142 (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]. 9.34/9.43 3143 (all P all Q all R hAPP_i1948725293t_bool(hAPP_f1791153283t_bool(hAPP_f735247205t_bool(cOMBC_983868720t_bool,P),Q),R) = hAPP_f429935748t_bool(hAPP_i872162435t_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_013) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3144 (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]. 9.34/9.43 3145 (all B_96 all A_136 (zero_zero_rat != A_136 -> (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]. 9.34/9.43 3146 (all Z_2 all Wa (is_int(Wa) & is_int(Z_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_2),Wa)) & Wa != Z_2 <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_2),Wa))))) # label(fact_562_zless__le) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3147 (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]. 9.34/9.43 3148 (all K_2 all I all J_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),J_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,I),K_2)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,J_2),K_2))))) # label(fact_79_zadd__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3149 (all Xa (-hBOOL(hAPP_int_bool(nat_neg,Xa)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)))) # label(fact_3247_not__neg__eq__ge__0) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3150 (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]. 9.34/9.43 3151 (all X_17 all Y_13 (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_17),Y_13)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_13),X_17)))) # label(fact_2013_leI) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3152 (all K_2 all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(div_mod_nat(M),N)) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N))) # label(fact_3173_mod__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3153 (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]. 9.34/9.43 3154 (all I all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,I))),N)))) # label(fact_3100_diff__Suc__less) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3155 (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]. 9.34/9.43 3156 (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]. 9.34/9.43 3157 (all K all L hAPP_int_int(legacy_zgcd(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(legacy_zgcd(L),hAPP_int_int(div_mod_int(hAPP_int_int(abs_abs_int,K)),hAPP_int_int(abs_abs_int,L)))))) # label(fact_4519_zgcd__code) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3158 (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]. 9.34/9.43 3159 (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]. 9.34/9.43 3160 (all I_1 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,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,K),I_1)))))) # label(fact_3334_zero__induct__lemma) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3161 (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]. 9.34/9.43 3162 (all A_148 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),A_148) = A_148) # label(fact_934_add__0) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3163 (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]. 9.34/9.43 3164 (all C all B_1 all A_1 (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B_1),A_1)) -> (B_1 = C -> 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]. 9.34/9.43 3165 (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]. 9.34/9.43 3166 (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_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_12),V_13)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_12)),hAPP_int_real(number267125858f_real,V_13))))) # label(fact_1280_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3167 (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]. 9.34/9.43 3168 (all A_1 (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_1),A_1) = zero_zero_rat <-> zero_zero_rat = A_1)) # label(fact_128_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3169 (all A_32 all C_8 all B_24 all 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)) = 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))) # label(fact_2347_add__diff__add) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3170 (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]. 9.34/9.43 3171 (all P all Q all R hAPP_r579619957l_real(hAPP_f823702159l_real(hAPP_f1040184293l_real(cOMBB_557258983l_real,P),Q),R) = hAPP_f1591894897l_real(P,hAPP_r1250527377l_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__036) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3172 (all P all Q all R hAPP_complex_complex(P,hAPP_nat_complex(Q,R)) = hAPP_nat_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Complex__Ocomplex_000tc__Complex__Ocomplex_000tc__Na) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3173 (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]. 9.34/9.43 3174 (all Z_12 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_12),Z_12) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Z_12)) # label(fact_1890_mult__2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3175 (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]. 9.34/9.43 3176 (all A_1 all C all B_1 (C = zero_zero_complex | hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_1),B_1)) <-> 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]. 9.34/9.43 3177 (all A_133 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_133),zero_z891286103de_int) = zero_z891286103de_int) # label(fact_1146_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3178 (all A_24 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_24),A_24) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(abs_abs_real,A_24)),hAPP_real_real(abs_abs_real,A_24))) # label(fact_2673_abs__mult__self) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3179 (all Xa (Xa = hAPP_int_real(real_int,archim1246769320r_real(Xa)) <-> (exists N_1 (Xa = hAPP_int_real(real_int,N_1) & is_int(N_1))))) # label(fact_4382_real__of__int__floor__cancel) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3180 (all Z_1 all 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))) = 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)))) # label(fact_4756_zpower__number__of__even) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3181 (all A_17 (is_int(A_17) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_17)) -> hAPP_int_int(abs_abs_int,A_17) = A_17))) # label(fact_2716_abs__of__pos) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3182 (all K_2 pls != hAPP_int_int(bit1,K_2)) # label(fact_131_rel__simps_I46_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3183 (all D_1 (D_1 = one_one_nat <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,zero_zero_nat),D_1)))) # label(fact_4024_coprime__0_H) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3184 (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]. 9.34/9.43 3185 (all K all Ma all Na (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_nat,zero_zero_nat),K)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na))))) # label(fact_2132_mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3186 (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]. 9.34/9.43 3187 (all A_160 all B_108 all C_62 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_160),C_62)),B_108) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_160),B_108)),C_62)) # label(fact_818_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3188 (all Z_1 (is_int(Z_1) -> Z_1 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),zero_zero_int))) # label(fact_157_zadd__0__right) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3189 (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]. 9.34/9.43 3190 (all C_69 all D_20 all A_177 all B_121 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_177),B_121)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_69),D_20)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_177),C_69)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_121),D_20)))))) # label(fact_610_add__less__le__mono) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3191 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) -> hAPP_int_nat(nat,hAPP_int_int(div_mod_int(X),Y)) = hAPP_nat_nat(div_mod_nat(hAPP_int_nat(nat,X)),hAPP_int_nat(nat,Y))))) # label(fact_3202_Divides_Otransfer__nat__int__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3192 (all A_3 all B_2 hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(uminus_uminus_real,A_3)),hAPP_real_real(uminus_uminus_real,B_2)) = hAPP_complex_complex(uminus473333897omplex,hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2))) # label(fact_3958_complex__minus) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3193 (all X_40 all Y_34 (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_40),Y_34)) -> (X_40 != Y_34 -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_34),X_40))))) # label(fact_1717_linorder__cases) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3194 (all M_21 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_21)),zero_zero_int))) # label(fact_457_of__nat__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3195 (all N_55 ((N_55 != zero_zero_nat -> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),N_55) = zero_zero_real) & (N_55 = zero_zero_nat -> 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]. 9.34/9.43 3196 (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]. 9.34/9.43 3197 (all P all Q all R hAPP_real_real(P,hAPP_int_real(Q,R)) = hAPP_int_real(hAPP_f269654879t_real(hAPP_f766218899t_real(cOMBB_real_real_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000tc__Int__Oin) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3198 (all N_51 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,one_one_int),N_51) = one_one_int) # label(fact_354_power__one) # label(axiom) # label(non_clause). [assumption]. 9.34/9.43 3199 (all J_2 all A_3 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),zero_zero_int),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),A_3)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),J_2))),P_3)))))))) # label(fact_2867_MultInvPair__distinct) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3200 (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_eq_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,Na),Ya))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Ya))))) # label(fact_3910_real__root__ge__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3201 (all N all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (exists R_1 (hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_1),hAPP_nat_nat(suc,N)) = A_3 & 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]. 9.34/9.44 3202 (all A_35 (is_int(A_35) -> A_35 = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_35),zero_zero_int))) # label(fact_2332_diff__0__right) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3203 (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]. 9.34/9.44 3204 (all M all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),hAPP_nat_nat(suc,M)))) # label(fact_3059_diff__less__Suc) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3205 (all Ya all Xa (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Xa)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Ya)) -> (Xa = zero_zero_rat & zero_zero_rat = Ya <-> zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Xa),Ya))))) # label(fact_583_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3206 (all P all Q all R hAPP_b589554111l_bool(P,hAPP_int_bool(Q,R)) = hAPP_i68813070l_bool(hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3207 (all Xa all Ya (is_int(Ya) & is_int(Xa) -> (Xa != Ya <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ya),Xa)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya))))) # label(fact_1849_linorder__neq__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3208 (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]. 9.34/9.44 3209 (all N_55 ((zero_zero_nat = N_55 -> one_on1684967323de_int = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),N_55)) & (N_55 != zero_zero_nat -> zero_z891286103de_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]. 9.34/9.44 3210 (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(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A_1,B_1),N_1))))))) # label(fact_4242_Bolzano__bisect__le) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3211 (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]. 9.34/9.44 3212 (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(minus_minus_rat,A_1),B_1)),zero_zero_rat)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_1),B_1)))) # label(fact_2375_le__iff__diff__le__0) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3213 (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]. 9.34/9.44 3214 (all X all Y (is_int(Y) & is_int(X) -> 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]. 9.34/9.44 3215 (all Na all K all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Ma)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Na)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),K))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)))))) # label(fact_2513_less__diff__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3216 (all D_1 all A_1 all B_1 (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)) = one_one_nat <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_1),A_1) = one_one_nat & one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_1),B_1))) # label(fact_4351_coprime__mul__eq__nat) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3217 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)))) # label(fact_2519_le__add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3218 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),zero_zero_real)) -> hAPP_int_real(number267125858f_real,min) = hAPP_real_real(sgn_sgn_real,X))) # label(fact_3634_real__sgn__neg) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3219 (all B_88 all A_118 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_118)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_88),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_118),B_88)),zero_zero_int))))) # label(fact_1303_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3220 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(fact_fact_nat,M)),hAPP_nat_nat(fact_fact_nat,N))))) # label(fact_3832_fact__mono__nat) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3221 (all Z_2 all Ya all Xa (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ya),Xa)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Z_2),Ya)) -> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Z_2),Xa))))) # label(fact_1920_xt1_I8_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3222 (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]. 9.34/9.44 3223 (all A_1 all B_1 (one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_1),B_1) <-> (all D_2 (one_one_nat = 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)))))) # label(fact_4349_coprime__nat) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3224 (all C_41 all D_14 all A_108 all B_78 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_108),B_78)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_41),D_14)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_78)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_41)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_108),C_41)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_78),D_14)))))))) # label(fact_1350_mult__mono) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3225 (all A_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_3)) -> zero_zero_nat = A_3)) # label(fact_2959_gcd__lcm__complete__lattice__nat_Otop__le) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3226 (all A_163 all B_111 all C_65 (hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_163),B_111) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_163),C_65) -> B_111 = C_65)) # label(fact_786_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3227 (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]. 9.34/9.44 3228 (all Q_1 ((hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Q_1),zero_zero_rat)) -> hAPP_rat_rat(uminus_uminus_rat,Q_1) = hAPP_rat_rat(abs_abs_rat,Q_1)) & (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Q_1),zero_zero_rat)) -> Q_1 = hAPP_rat_rat(abs_abs_rat,Q_1)))) # label(fact_4645_abs__rat__def) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3229 (all M_22 all N_45 all A_193 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_193)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_193),M_22)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_193),N_45))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_22),N_45))))) # label(fact_435_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3230 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> natceiling(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),one_one_real)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natceiling(X)),one_one_nat))) # label(fact_3319_natceiling__add__one) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3231 (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]. 9.34/9.44 3232 (all Xa all Ya (Ya != zero_zero_real | Xa != zero_zero_real <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,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_215_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3233 (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]. 9.34/9.44 3234 (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]. 9.34/9.44 3235 (all Z_1 exists A_2 exists R_1 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_1)),expi(A_2)) = Z_1) # label(fact_4051_complex__expi__Ex) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3236 (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]. 9.34/9.44 3237 (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]. 9.34/9.44 3238 (all A_88 all M_12 all N_26 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_88),M_12)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_88),N_26)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_88),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_12),N_26))) # label(fact_1494_power__add) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3239 (all P all X (hBOOL(hAPP_f448129468l_bool(fEx_int,P)) | -hBOOL(hAPP_int_bool(P,X)))) # label(help_fEx_1_1_fEx_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3240 (all X all Y (Y != zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),hAPP_nat_nat(div_mod_nat(X),Y)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y))) # label(fact_4360_gcd__non__0__nat) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3241 (all K_2 all M 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,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),N))) # label(fact_2485_Nat_Odiff__cancel) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3242 (all A_1 all Na (hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_1),Na) = zero_zero_nat <-> Na != zero_zero_nat & zero_zero_nat = A_1)) # label(fact_450_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3243 (all A_3 all B_2 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,B_2),A_3)) # label(fact_4309_gcd__commute__nat) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3244 (all Ta all A all D_1 all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),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),A)) -> X_1 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa_1))))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Ta))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),D)),Ta))))))))) # label(fact_5104_aset_I9_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3245 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_complex_real(re,X)),hAPP_complex_real(re,Y)) = hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),Y))) # label(fact_4121_Re_Oadd) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3246 (all Z_1 (is_int(Z_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> Z_1 = hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,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)) = zero_zero_int))) # label(fact_2777_int__nat__eq) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3247 (all C_12 all D_8 all A_43 all B_32 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_43),B_32)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_12),D_8)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_43),C_12)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_32),D_8)))))) # label(fact_2270_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3248 (all P all Q all R hAPP_rat_bool(hAPP_i16292313t_bool(P,R),Q) = hAPP_int_bool(hAPP_r1900510169t_bool(hAPP_f1480506641t_bool(cOMBC_int_rat_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Rat__Orat_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3249 (all C_1 all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_3),B_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_1),A_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A_3),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,B_2),C_1)))))) # label(fact_2512_diff__less__mono) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3250 (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]. 9.34/9.44 3251 (all Xa all 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)) <-> Xa != Ya & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)))) # label(fact_2648_dvd_Oless__le) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3252 (all X hAPP_real_real(cos,X) = hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X))) # label(fact_3625_cos__sin__eq) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3253 (all K all Xa hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,cos),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,plus_plus_real),K)),Xa),hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa),K)))))) # label(fact_4736_DERIV__cos__add) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3254 (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]. 9.34/9.44 3255 (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]. 9.34/9.44 3256 (all K all 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))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L)))) # label(fact_58_less__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3257 (all N_58 hAPP_nat_complex(semiri2020571505omplex,N_58) = hAPP_int_complex(number528085621omplex,hAPP_nat_int(semiri1621563631at_int,N_58))) # label(fact_204_number__of__int) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3258 (all A_27 hAPP_int_int(abs_abs_int,hAPP_int_int(abs_abs_int,A_27)) = hAPP_int_int(abs_abs_int,A_27)) # label(fact_2656_abs__idempotent) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3259 (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]. 9.34/9.44 3260 (all A_129 all M_17 all B_91 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_129),M_17)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_91),M_17)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_129),B_91)),M_17)) # label(fact_1183_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3261 (all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = X)) # label(fact_4320_gcd__proj1__if__dvd__nat) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3262 (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_eq_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_leq))))) # label(fact_4541_pair__leqI2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3263 (all X_36 all Y_30 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_36),Y_30)) -> X_36 != Y_30)) # label(fact_1795_order__less__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3264 (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,Ma),Na)),Ma)) <-> one_one_nat = Na))) # label(fact_2551_dvd__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3265 (all A_3 all B_2 (is_int(B_2) & is_int(A_3) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int)) -> (B_2 = zero_zero_int | zero_zero_int = A_3 -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int) = norm_frac(A_3,B_2)) & (-(B_2 = zero_zero_int | A_3 = zero_zero_int) -> norm_frac(A_3,B_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(int_gcd,A_3),B_2))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(int_gcd,A_3),B_2))))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int)) -> norm_frac(A_3,B_2) = norm_frac(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2))))) # label(fact_4806_norm__frac_Osimps) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3266 (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,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))) <-> 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))))) # label(fact_2421_le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3267 (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]. 9.34/9.44 3268 (all K_2 all I all J_2 (one_one_int = hAPP_int_int(legacy_zgcd(I),J_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),J_2))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),K_2))))) # label(fact_4517_zrelprime__dvd__mult) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3269 (all Xa all Ya (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)) <-> Ya = Xa))) # label(fact_1970_linorder__antisym__conv2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3270 (all I all J_2 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(legacy_zgcd(I),J_2)))) # label(fact_4500_zgcd__pos) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3271 (all N all M all A_3 all B_2 (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2) = one_one_int -> one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),M)))) # label(fact_4961_coprime__exp2__int) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3272 (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]. 9.34/9.44 3273 (all A_1 all Wa (zero_zero_complex = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_1),hAPP_int_nat(number_number_of_nat,Wa)) <-> zero_zero_complex = A_1 & hAPP_int_nat(number_number_of_nat,Wa) != zero_zero_nat)) # label(fact_182_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3274 (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]. 9.34/9.44 3275 (all Lx_1 all Ly_1 all Rx_1 all 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)) = 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))) # label(fact_1077_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3276 (all A_86 all C_32 all B_59 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_86),C_32)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_59),C_32))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_32)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_86),B_59))))) # label(fact_1524_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3277 (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]. 9.34/9.44 3278 (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]. 9.34/9.44 3279 (all A_91 all B_61 all 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)) = 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)))) # label(fact_1467_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3280 (all C (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,C)),one_one_real)) -> hBOOL(tendsto_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(abs_abs_real,C)),zero_zero_real,sequentially)))) # label(fact_5035_LIMSEQ__rabs__realpow__zero) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3281 (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]. 9.34/9.44 3282 (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]. 9.34/9.44 3283 (all Z_1 (is_int(Z_1) -> Z_1 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),Z_1))) # label(fact_1261_zmult__1) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3284 (all Xa all Ya (zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Xa),Xa)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ya),Ya)) <-> zero_zero_rat = Ya & zero_zero_rat = Xa)) # label(fact_1442_sum__squares__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3285 (all K all P_1 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> ((all X_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(plus_plus_int,X_1),D_1)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K)) -> (all X_1 (hBOOL(hAPP_int_bool(P_1,X_1)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D_1)))))))))) # label(fact_2060_incr__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3286 (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]. 9.34/9.44 3287 (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)) <-> Ya != Xa)) # label(fact_1845_linorder__neq__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3288 (all P all Q all R hAPP_r474017924t_real(P,hAPP_real_real(Q,R)) = hAPP_r474017924t_real(hAPP_f1407095736t_real(hAPP_f1796904975t_real(cOMBB_599996122l_real,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__Nat__Onat_Mtc__RealDe) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3289 (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_nat_bool(P_1,hAPP_int_nat(nat,X_1)))))) <-> (all X1 hBOOL(hAPP_nat_bool(P_1,X1))))) # label(fact_2743_all__nat) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3290 (all A_1 (is_int(A_1) -> (A_1 = zero_zero_int <-> zero_zero_int = hAPP_int_int(abs_abs_int,A_1)))) # label(fact_2662_abs__eq__0) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3291 (all P_1 collect_int(P_1) = P_1) # label(fact_150_Collect__def) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3292 (all A_26 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_26),hAPP_real_real(abs_abs_real,A_26)))) # label(fact_2667_abs__ge__self) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3293 (all X exists N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_nat_real(real_nat,N_1)))) # label(fact_3937_reals__Archimedean2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3294 (all X_35 all Y_29 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_35),Y_29)) -> X_35 != Y_29)) # label(fact_1803_order__less__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3295 (all V_7 all 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)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_7),W_3))) # label(fact_2230_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3296 (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]. 9.34/9.44 3297 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(fact_fact_nat,M)),hAPP_nat_nat(fact_fact_nat,N)))))) # label(fact_3837_fact__less__mono__nat) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3298 (all X all Y all M (hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X),Y)),M) = one_one_nat -> one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),M))) # label(fact_4370_invertible__coprime__nat) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3299 (all C_19 all B_46 all A_69 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_46),A_69)) -> (B_46 = C_19 -> 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]. 9.34/9.44 3300 (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]. 9.34/9.44 3301 (all A_161 all B_109 all C_63 all D_19 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_161),B_109)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_63),D_19)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_161),C_63)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_109),D_19))) # label(fact_797_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3302 (all Z_2 all P_1 ((all N_1 hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,N_1)))) -> ((all N_1 hBOOL(hAPP_int_bool(P_1,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_1)))))) -> hBOOL(hAPP_int_bool(P_1,Z_2))))) # label(fact_4549_int__of__nat__induct) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3303 (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(plus_plus_nat,K),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),Na))))) # label(fact_1268_nat__add__left__cancel__le) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3304 (all B_125 all A_181 all C_72 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_72)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_125),A_181)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_125),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_181),C_72)))))) # label(fact_574_add__increasing2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3305 (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]. 9.34/9.44 3306 (all Xa all Ya (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya)) -> Xa != Ya)) # label(fact_1792_order__less__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3307 (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]. 9.34/9.44 3308 (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]. 9.34/9.44 3309 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X))),X)))) # label(fact_3373_ln__add__one__self__le__self2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.44 3310 (all B all K all Diff all H all 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))))) -> (hAPP_nat_nat(suc,K) = 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_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,minus_minus_real),hAPP_n546711566l_real(Diff,M_1))),hAPP_f1950183573l_real(hAPP_f2069516469l_real(cOMBS_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,plus_plus_real),hAPP_f2119584768l_real(hAPP_f59594445l_real(cOMBC_2010481033l_real,hAPP_f1523861863l_real(hAPP_f1067681227l_real(cOMBB_1934313333l_real,big_co604158596t_real),hAPP_f1993765077t_real(hAPP_f895854793t_real(cOMBB_1948233853l_real,hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f1904307534l_real(hAPP_f224989205l_real(cOMBB_1419615765al_nat,Diff),hAPP_nat_fun_nat_nat(plus_plus_nat,M_1))),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))))),power_power_real))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),M_1))))),hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,hAPP_r1250527377l_real(times_times_real,B)),hAPP_r1250527377l_real(hAPP_f203520653l_real(cOMBC_real_real_real,hAPP_f556082547l_real(hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,power_power_real),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),M_1)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),M_1))))))),T),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,inverse_divide_real),hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f1904307534l_real(hAPP_f224989205l_real(cOMBB_1419615765al_nat,Diff),hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M_1)))),zero_zero_real))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat)))),hAPP_r474017924t_real(power_power_real,T))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),hAPP_nat_nat(suc,M_1))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,T),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),hAPP_nat_nat(suc,M_1)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),hAPP_nat_nat(suc,M_1))))))))))))))) # label(fact_4923_Maclaurin__lemma2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3311 (all J 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(J),zero_zero_int),P_5)) -> hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(multInvPair(A_1,P_5),J)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3323_MultInvPair__card__two) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3312 (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]. 9.34/9.45 3313 (all N N = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N)),one_one_nat)) # label(fact_3066_diff__Suc__1) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3314 (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(X_1) & is_int(K_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_2,X_1)) <-> hBOOL(hAPP_int_bool(P_1,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]. 9.34/9.45 3315 (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]. 9.34/9.45 3316 (all A_24 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_24),A_24) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,A_24)),hAPP_int_int(abs_abs_int,A_24))) # label(fact_2674_abs__mult__self) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3317 (all K_2 hAPP_int_int(uminus_uminus_int,K_2) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,min),K_2)) # label(fact_3481_mult__Min) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3318 (all X (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))) -> X = hAPP_nat_nat(suc,zero_zero_nat) | X = zero_zero_nat)) # label(fact_3038_nat__lt__two__imp__zero__or__one) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3319 (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]. 9.34/9.45 3320 (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]. 9.34/9.45 3321 (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]. 9.34/9.45 3322 (all Na all Xa (one_one_real != Xa -> hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_r474017924t_real(power_power_real,Xa)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa),Na)),one_one_real)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Xa),one_one_real)))) # label(fact_4903_sumr__geometric) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3323 (all X all Y hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,fFalse),X),Y) = Y) # label(help_If_2_1_If_000tc__Nat__Onat_T) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3324 (all A_3 all B_2 all C_1 (is_int(C_1) -> (zero_zero_int != C_1 -> hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),A_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),B_2)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)))) # label(fact_4672_mult__rat__cancel) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3325 (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]. 9.34/9.45 3326 (all A_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,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_1),A_1))))) # label(fact_599_zero__le__double__add__iff__zero__le__single__add) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3327 (all A_1 all Wa (hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_1),hAPP_int_nat(number_number_of_nat,Wa)) = zero_z891286103de_int <-> zero_z891286103de_int = A_1 & zero_zero_nat != hAPP_int_nat(number_number_of_nat,Wa))) # label(fact_181_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3328 (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]. 9.34/9.45 3329 (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]. 9.34/9.45 3330 (all A_18 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_18)) -> A_18 = hAPP_real_real(abs_abs_real,A_18))) # label(fact_2712_abs__of__nonneg) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3331 (all X_26 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_26),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_26),X_26)) # label(fact_1907_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3332 (all A_46 all B_35 all C_14 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_46),B_35)),C_14)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,B_35),C_14)))) # label(fact_2255_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3333 (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]. 9.34/9.45 3334 (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]. 9.34/9.45 3335 (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]. 9.34/9.45 3336 (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]. 9.34/9.45 3337 (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)) <-> Ya = Xa))) # label(fact_1832_linorder__antisym__conv3) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3338 (all M_25 all N_49 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_25)),hAPP_nat_real(semiri132038758t_real,N_49))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_25),N_49)))) # label(fact_367_of__nat__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3339 (all C_69 all D_20 all A_177 all B_121 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_177),B_121)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_69),D_20)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_177),C_69)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_121),D_20)))))) # label(fact_607_add__less__le__mono) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3340 (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]. 9.34/9.45 3341 (all B_1 all A_1 all C (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,B_1),A_1) <-> C = B_1)) # label(fact_806_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3342 (all A_3 all B_2 all C_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B_2),C_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),C_1)) # label(fact_4303_gcd__assoc__nat) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3343 (all Xa 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,X_1),B_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),X_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)))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),Xa)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),B_1)) -> hAPP_real_real(F,A_1) = hAPP_real_real(F,Xa))))))) # label(fact_5175_DERIV__isconst2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3344 (all C_25 all D_10 all A_79 all B_52 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_79),B_52)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_25),D_10)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_79)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_25)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_79),C_25)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_52),D_10)))))))) # label(fact_1564_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3345 (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]. 9.34/9.45 3346 (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]. 9.34/9.45 3347 (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_eq_nat,hAPP_nat_nat(suc,Ma)),Na)))) # label(fact_3055_Suc__le__eq) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3348 (all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(suc,N)))) # label(fact_2978_zero__less__Suc) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3349 (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]. 9.34/9.45 3350 (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]. 9.34/9.45 3351 (all L all U hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,U),L) = hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_at4362885an_nat(L),U))) # label(fact_4902_card__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3352 (all N_37 all A_168 all B_114 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_168),B_114)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_168)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_168),N_37)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_114),N_37)))))) # label(fact_700_power__mono) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3353 (all M hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),M)))) # label(fact_2101_le__square) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3354 (all C_18 all A_68 all B_45 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_68),B_45)) -> (B_45 = C_18 -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_68),C_18))))) # label(fact_1748_ord__less__eq__trans) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3355 (all M all I all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,M),N)),I) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),I)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),I))) # label(fact_5123_min__diff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3356 (all X_14 all P_6 all Q_7 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_14),P_6)),Q_7) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_14),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_6),Q_7))) # label(fact_2090_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3357 (all P all Q all R hAPP_r847664445t_bool(hAPP_f1920544743t_bool(hAPP_f866672739t_bool(cOMBB_1247074793l_real,P),Q),R) = hAPP_f1173902867t_bool(P,hAPP_r1134773055l_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__RealDef__Oreal_Mtc__HOL__Obool_J_000tc__fun) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3358 (all D_3 all I (is_int(I) -> (I != zero_zero_int -> (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]. 9.34/9.45 3359 (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]. 9.34/9.45 3360 (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]. 9.34/9.45 3361 (all F all Na all Ra hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,minus_minus_real),F)),Ra)),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),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,Na)),Ra))) # label(fact_4898_sumr__diff__mult__const) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3362 (all A_3 all B_2 all M hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),B_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,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)))) # label(fact_3112_zcong__zmod__aux) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3363 (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]. 9.34/9.45 3364 (all A_82 all C_28 all B_55 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_82),C_28)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_55),C_28))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_28)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_82),B_55))))) # label(fact_1546_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3365 (all A_126 A_126 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),A_126)) # label(fact_1210_mult__1) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3366 (all X_42 all Y_36 (X_42 = Y_36 -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_42),Y_36)))) # label(fact_1697_order__eq__refl) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3367 (all Z_1 all X all Y all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Y)),one_one_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Y),Z_1))),one_one_int),P_3)))) # label(fact_2405_zcong__zpower__zmult) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3368 (all N all M all R_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,M)),R_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_2),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N)),one_one_real))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M),N))))) # label(fact_4462_lemma__floor) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3369 (all A_132 all B_94 all C_50 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_132),B_94)),C_50) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_132),C_50)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_94),C_50))) # label(fact_1153_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3370 (all A_140 all B_98 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_140),B_98) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_98),A_140)) # label(fact_1036_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3371 (all F all A_1 all B_1 hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,A_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B_1)),hAPP_real_real(F,A_1))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_1),A_1))),A_1)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B_1)),hAPP_real_real(F,A_1))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_1),A_1))),B_1))) # label(fact_3404_lemma__MVT) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3372 (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]. 9.34/9.45 3373 (all Xa all Ya (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ya),zEven)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_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))))))))) # label(fact_3376_even__div__2__l) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3374 (all Code_numeral_1 hAPP_C2129356693al_nat(size_s945831648umeral,code_S1047413653umeral(Code_numeral_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_C2129356693al_nat(size_s945831648umeral,Code_numeral_1)),hAPP_nat_nat(suc,zero_zero_nat))) # label(fact_4088_code__numeral_Osize_I4_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3375 (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,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_real,zero_zero_real),X_1)))))) # label(fact_3777_lemma__tan__total) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3376 (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]. 9.34/9.45 3377 (all P_5 all Na (hBOOL(hAPP_nat_bool(prime,P_5)) & one_one_nat = Na <-> hBOOL(hAPP_nat_bool(prime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_5),Na))))) # label(fact_4175_prime__exp) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3378 (all A_1 (zero_zero_rat = hAPP_rat_rat(abs_abs_rat,A_1) <-> A_1 = zero_zero_rat)) # label(fact_2660_abs__eq__0) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3379 (all Z1 all Z2 all 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) = 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))) # label(fact_1264_zadd__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3380 (all M M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),zero_zero_nat)) # label(fact_2478_minus__nat_Odiff__0) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3381 (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]. 9.34/9.45 3382 (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]. 9.34/9.45 3383 (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]. 9.34/9.45 3384 (all X_52 all N_25 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_52),N_25)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_52),N_25)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,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))) # label(fact_1512_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3385 (all Z_1 (is_int(Z_1) -> Z_1 = hAPP_int_int(uminus_uminus_int,hAPP_int_int(uminus_uminus_int,Z_1)))) # label(fact_3459_zminus__zminus) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3386 (all 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) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Z_11),Z_11)) # label(fact_1893_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3387 (all A_1 all B_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (hAPP_int_int(div_mod_int(B_1),Ma) = hAPP_int_int(div_mod_int(A_1),Ma) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),B_1),Ma))))) # label(fact_3083_zcong__zmod__eq) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3388 (all B_126 all A_182 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_182),zero_z126310315umeral)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_126),zero_z126310315umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_182),B_126)),zero_z126310315umeral))))) # label(fact_570_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3389 (all A_158 all B_106 all 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)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_158),B_106)),C_60)) # label(fact_829_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3390 (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]. 9.34/9.45 3391 (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]. 9.34/9.45 3392 (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]. 9.34/9.45 3393 (all X all A_3 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,archim856651990g_real(X)),A_3) = archim856651990g_real(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_int_real(real_int,A_3)))) # label(fact_4426_ceiling__add) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3394 (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]. 9.34/9.45 3395 (all A_3 all B_2 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),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]. 9.34/9.45 3396 (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,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))))))) -> (X_1 != Ta -> Ta != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),D))))))))) # label(fact_5098_aset_I4_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3397 (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]. 9.34/9.45 3398 (all F all 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))))) <-> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G)))) # label(fact_1713_le__fun__def) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3399 (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]. 9.34/9.45 3400 (all Na (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),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_eq_int),Na)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat,Na)),one_one_nat))) # label(fact_4775_card__bdd__int__set__le) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3401 (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]. 9.34/9.45 3402 (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]. 9.34/9.45 3403 (all A_133 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_133),zero_zero_rat) = zero_zero_rat) # label(fact_1145_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3404 (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)) & (zero_zero_nat != N_55 -> zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N_55)))) # label(fact_257_power__0__left) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3405 (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]. 9.34/9.45 3406 (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]. 9.34/9.45 3407 (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]. 9.34/9.45 3408 (all C_42 all D_15 all A_109 all B_79 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_109),B_79)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_42),D_15)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_109)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_42)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_109),C_42)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_79),D_15)))))))) # label(fact_1345_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3409 (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]. 9.34/9.45 3410 (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]. 9.34/9.45 3411 (all P_3 all M ((M != zero_zero_nat -> 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]. 9.34/9.45 3412 (all 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,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_1),A_1)),zero_zero_real)))) # label(fact_596_double__add__le__zero__iff__single__add__le__zero) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3413 (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]. 9.34/9.45 3414 (all K (hBOOL(hAPP_rat_bool(iszero_rat,hAPP_int_rat(number_number_of_rat,K))) <-> hBOOL(hAPP_int_bool(iszero_int,hAPP_int_int(number_number_of_int,K))))) # label(fact_4629_iszero__rat) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3415 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(fequal_real,X),Y)) | X != Y)) # label(help_fequal_2_1_fequal_000tc__RealDef__Oreal_T) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3416 (all A_1 all B_1 (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_1),B_1)) -> (A_1 != B_1 -> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A_1),B_1))))) # label(fact_1955_order__le__neq__trans) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3417 (all M hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(fact_fact_int,M)))) # label(fact_4276_fact__ge__zero__int) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3418 (all Q all P (hBOOL(fdisj(P,Q)) | -hBOOL(P))) # label(help_fdisj_1_1_U) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3419 (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_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_1)))) -> negDivAlg(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)),negDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(number_number_of_int,V_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,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_1)))) -> negDivAlg(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,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_1)))))) # label(fact_3285_negDivAlg__eqn__number__of) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3420 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)) -> (exists X_1 (hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),field_1210416355s_real)) & 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)))))) # label(fact_4630_Rats__dense__in__real) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3421 (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]. 9.34/9.45 3422 (all M_9 all A_31 all N_15 all B_23 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_31),N_15)),B_23)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_9),N_15)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_31),M_9)),B_23))))) # label(fact_2363_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3423 (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]. 9.34/9.45 3424 (all A_122 all N_29 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_122),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_122),N_29)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_122),N_29)),A_122)) # label(fact_1237_power__commutes) # label(axiom) # label(non_clause). [assumption]. 9.34/9.45 3425 (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]. 9.34/9.46 3426 (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]. 9.34/9.46 3427 (all M hAPP_nat_nat(suc,M) != zero_zero_nat) # label(fact_2992_Zero__not__Suc) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3428 (all A_155 all C_57 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_57),A_155) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_155),C_57)) # label(fact_848_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3429 (all Va (zero_zero_nat = 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)))))) # label(fact_4049_neg__number__of__pred__iff__0) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3430 (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]. 9.34/9.46 3431 (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]. 9.34/9.46 3432 (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]. 9.34/9.46 3433 (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]. 9.34/9.46 3434 (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]. 9.34/9.46 3435 (all A_162 all B_110 all C_64 (hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_162),C_64) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_162),B_110) -> B_110 = C_64)) # label(fact_791_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3436 (all K all L (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K)),hAPP_int_int(bit1,L))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L)))) # label(fact_75_rel__simps_I17_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3437 (all P all Q all R hAPP_r1962736578t_bool(hAPP_f1063001928t_bool(hAPP_f2050994651t_bool(cOMBB_2138578194l_real,P),Q),R) = hAPP_f1545556668t_bool(P,hAPP_r1150855451l_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__fun_) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3438 (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]. 9.34/9.46 3439 (all A_3 all B_2 hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B_2),A_3) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)) # label(fact_4991_gcd__commute__int) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3440 (all C_54 all D_16 all A_143 all B_101 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_143),B_101)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_54),D_16)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_143),C_54)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_101),D_16)))))) # label(fact_963_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3441 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)),B_2)))) # label(fact_5010_gcd__le2__int) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3442 (all A_159 all B_107 all 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) = 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))) # label(fact_825_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3443 (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]. 9.34/9.46 3444 (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)) -> X_45 = Y_39))) # label(fact_1639_xt1_I5_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3445 (all X_21 all Y_17 (is_int(Y_17) & is_int(X_21) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_21),Y_17)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_21),Y_17)) | Y_17 = X_21))) # label(fact_1967_order__le__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3446 (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]. 9.34/9.46 3447 (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]. 9.34/9.46 3448 (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]. 9.34/9.46 3449 (all B_2 all A_3 all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N)) -> (one_one_int = hAPP_int_int(legacy_zgcd(A_3),N) -> (exists X_1 (is_int(X_1) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) & (all Y_1 (is_int(Y_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),N)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Y_1)),B_2),N)) -> X_1 = Y_1))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),X_1)),B_2),N)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),N))))))) # label(fact_4532_zcong__lineq__unique) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3450 (all N hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N)))) # label(fact_1911_two__realpow__ge__one) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3451 (all C_43 all B_80 all A_110 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_80),A_110)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_43),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_43),A_110)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_43),B_80)))))) # label(fact_1341_mult__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3452 (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)) -> one_one_int = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat,Xa))))) # label(fact_3332_EvenOdd_Oneg__one__even__power) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3453 (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,B_1),E_2)),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,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_1),B_1)),E_2)),C) = D_1))) # label(fact_2384_eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3454 (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]. 9.34/9.46 3455 (all V_8 all U_2 all Y_22 all X_27 all A_60 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_27),A_60)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_22),A_60)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),U_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),V_8)) -> (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,U_2),V_8) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,U_2),X_27)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_8),Y_22))),A_60)))))))) # label(fact_1879_convex__bound__lt) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3456 (all M all N all K_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(div_mod_nat(M),N)),K_2) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),K_2))) # label(fact_3174_mod__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3457 (all V_20 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_20)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(number_number_of_int,V_20)) = hAPP_int_int(number_number_of_int,hAPP_int_int(succ,V_20)))) # label(fact_748_semiring__1__add__number__of) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3458 (all Na all Ma (is_int(Na) & is_int(Ma) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (Ma = one_one_int & Na = one_one_int <-> one_one_int = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),Na))))) # label(fact_1620_pos__zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3459 (all Z_16 all Y_32 all X_38 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_32),X_38)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z_16),Y_32)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z_16),X_38))))) # label(fact_1727_xt1_I10_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3460 (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]. 9.34/9.46 3461 (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]. 9.34/9.46 3462 (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]. 9.34/9.46 3463 (all A_1 all E_2 all C all B_1 all D_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_1),B_1)),E_2)),C)),D_1)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_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_2431_less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3464 (all 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)))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_59),A_59)) # label(fact_1902_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3465 (all A_3 all B_2 hAPP_rat_rat(abs_abs_rat,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(abs_abs_int,A_3)),hAPP_int_int(abs_abs_int,B_2))) # label(fact_4690_abs__rat) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3466 (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]. 9.34/9.46 3467 (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_99_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3468 (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]. 9.34/9.46 3469 (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]. 9.34/9.46 3470 (all P all Q all R hAPP_r1250527377l_real(hAPP_n682674106l_real(P,R),Q) = hAPP_n546711566l_real(hAPP_r195310020l_real(hAPP_f2126667875l_real(cOMBC_746811033l_real,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__RealDef__Oreal_000tc__fun_Itc__Real) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3471 (all B_10 all A_10 ((hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_10)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_10),zero_zero_real))) & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_10)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_10),zero_zero_real))) -> hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_10),B_10)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(abs_abs_real,A_10)),hAPP_real_real(abs_abs_real,B_10)))) # label(fact_2755_abs__eq__mult) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3472 (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]. 9.34/9.46 3473 (all K all P_1 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> ((all X_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),D_1)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K)) -> (all 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),D_1)))))))))) # label(fact_2616_decr__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3474 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(suc,N))) # label(fact_3067_diff__Suc__eq__diff__pred) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3475 (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]. 9.34/9.46 3476 (all Xa ((exists N_1 (hAPP_int_real(real_int,N_1) = Xa & is_int(N_1))) <-> hAPP_int_real(real_int,archim856651990g_real(Xa)) = Xa)) # label(fact_4396_real__of__int__ceiling__cancel) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3477 (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]. 9.34/9.46 3478 (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]. 9.34/9.46 3479 (all B_96 all A_136 (A_136 != zero_zero_complex -> (zero_zero_complex != B_96 -> zero_zero_complex != hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_136),B_96)))) # label(fact_1126_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3480 (all A_3 all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,A_3)),X)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,natceiling(X)),A_3) = natceiling(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_nat_real(real_nat,A_3))))) # label(fact_3734_natceiling__subtract) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3481 (all Ta all B all D_1 all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),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(dvd_dvd_int,D_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Ta))) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),D)),Ta))))))))) # label(fact_5102_bset_I10_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3482 (all P all Q all R hAPP_i1948725293t_bool(hAPP_i345030060t_bool(hAPP_f791698359t_bool(cOMBC_241247961t_bool,P),Q),R) = hAPP_i1948725293t_bool(hAPP_i345030060t_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__fun_Itc__Int__Oint) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3483 (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]. 9.34/9.46 3484 (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)))),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))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_int_nat(number_number_of_nat,Va)),hAPP_nat_nat(suc,Na))) # label(fact_5119_min__Suc__number__of) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3485 (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]. 9.34/9.46 3486 (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),sin)),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(cos,Xa)),hAPP_real_real(sin,Xa))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,Xa)),hAPP_real_real(sin,Xa)))))) # label(fact_4796_DERIV__sin__realpow2) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3487 (all Ma all Na ((exists K_1 hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),K_1)) = Na) <-> 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]. 9.34/9.46 3488 (all B_85 all A_115 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_115),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_85),zero_zero_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_115),B_85)))))) # label(fact_1316_mult__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3489 (all Ma phi(Ma) = hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(norRRset,Ma))) # label(fact_4584_phi__def) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3490 (all K all L (quickcheck_int_of(L) = quickcheck_int_of(K) <-> K = L)) # label(fact_4254_Quickcheck__Narrowing_Oint__of__inject) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3491 (all K all L (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),L)) <-> (exists N_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),N_1) = L))) # label(fact_2933_le__Suc__ex__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3492 (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]. 9.34/9.46 3493 (all A_1 all B_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_1)) & 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),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_1),B_1))))) # label(fact_1286_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3494 (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]. 9.34/9.46 3495 (all M_18 all N_30 hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_18),N_30)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,M_18)),hAPP_nat_int(semiri1621563631at_int,N_30))) # label(fact_1105_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3496 (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]. 9.34/9.46 3497 (all Xa all Ya (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_1703_order__eq__iff) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3498 (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]. 9.34/9.46 3499 (all A_144 all C_55 all B_102 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_144),C_55)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_102),C_55))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_144),B_102)))) # label(fact_962_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3500 (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 ((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))))))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))))))) # label(fact_3879_DERIV__neg__dec__left) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3501 (all X_61 all Y_51 (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)))) = 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)))) -> (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]. 9.34/9.46 3502 (all C_70 all D_21 all A_178 all B_122 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_178),B_122)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_70),D_21)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_178),C_70)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_122),D_21)))))) # label(fact_601_add__le__less__mono) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3503 (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)))))) -> (X_1 = Ta -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),D) = Ta)))))))) # label(fact_5093_aset_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3504 (all U hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_atMost_nat,U)) = hAPP_nat_nat(suc,U)) # label(fact_5186_card__atMost) # label(axiom) # label(non_clause). [assumption]. 9.34/9.46 3505 (all X all Y ((Y = zero_zero_nat -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = X) & (zero_zero_nat != Y -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),hAPP_nat_nat(div_mod_nat(X),Y))))) # label(fact_4359_gcd__nat_Osimps) # label(axiom) # label(non_clause). [assumption]. 9.41/9.46 3506 (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(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(minus_minus_real,X),Y))) # label(fact_3600_cos__diff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.46 3507 (all Xa all Ya (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_2032_order__less__le) # label(axiom) # label(non_clause). [assumption]. 9.41/9.46 3508 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),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,zero_zero_nat),Ma)))) # label(fact_2118_nat__0__less__mult__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.46 3509 (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]. 9.41/9.46 3510 (all P_5 all A_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_1)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),hAPP_i1948725293t_bool(wset(A_1),P_5))))) # label(fact_2928_wset__mem__mem) # label(axiom) # label(non_clause). [assumption]. 9.41/9.46 3511 (all A_1 all B_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_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))))) # label(fact_2824_neg__imp__zdiv__neg__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.46 3512 (all Xa all Ya (hBOOL(hAPP_int_bool(even_odd_even_int,Xa)) | hBOOL(hAPP_int_bool(even_odd_even_int,Ya)) <-> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Xa),Ya))))) # label(fact_3235_Parity_Oeven__product) # label(axiom) # label(non_clause). [assumption]. 9.41/9.46 3513 (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]. 9.41/9.46 3514 (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]. 9.41/9.46 3515 (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(A_2) & is_int(B_4) -> (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]. 9.41/9.46 3516 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> M = N))) # label(fact_1027_le__antisym) # label(axiom) # label(non_clause). [assumption]. 9.41/9.46 3517 (all N (hAPP_nat_nat(div_mod_nat(N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = one_one_nat -> 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_3821_lemma__even__mod__4__div__2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.46 3518 (all Ma all Na all P_1 ((all M_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,M_1),zero_zero_nat))) -> ((all M_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_1),hAPP_nat_nat(div_mod_nat(M_1),N_1))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,M_1),N_1))))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Ma),Na))))) # label(fact_4550_gcd__nat__induct) # label(axiom) # label(non_clause). [assumption]. 9.41/9.46 3519 (all Xa all Ya (is_int(Ya) & is_int(Xa) -> (hAPP_int_real(real_int,Xa) = hAPP_int_real(real_int,Ya) <-> Ya = Xa))) # label(fact_4384_real__of__int__inject) # label(axiom) # label(non_clause). [assumption]. 9.41/9.46 3520 (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]. 9.41/9.46 3521 (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]. 9.41/9.46 3522 (all X all Y all Xa_2 all Ya_1 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Ya_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(sqrt,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Ya_1),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_3555_real__sqrt__sum__squares__mult__squared__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.46 3523 (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]. 9.41/9.46 3524 (all X_53 all P_7 all Q_8 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_53),P_7)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_53),Q_8)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_53),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_8))) # label(fact_1497_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.46 3525 (all C_33 all A_87 all B_60 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_33),A_87)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_33),B_60))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_33)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_87),B_60))))) # label(fact_1520_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 9.41/9.46 3526 (all Xa all Ya (zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa),Xa)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ya),Ya)) <-> zero_zero_real = Ya & zero_zero_real = Xa)) # label(fact_1443_sum__squares__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.46 3527 (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]. 9.41/9.46 3528 (all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) -> (hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_int_nat(nat,Xa))) <-> hBOOL(hAPP_int_bool(even_odd_even_int,Xa))))) # label(fact_3812_pos__int__even__equiv__nat__even) # label(axiom) # label(non_clause). [assumption]. 9.41/9.46 3529 (all X_15 all Y_11 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_15),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_11),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_15)),Y_11)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_15),Y_11)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_2055_power2__sum) # label(axiom) # label(non_clause). [assumption]. 9.41/9.46 3530 (all Ma all Na (hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),Na) <-> hAPP_nat_nat(suc,zero_zero_nat) = Na & hAPP_nat_nat(suc,zero_zero_nat) = Ma)) # label(fact_3046_mult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.46 3531 (all A_1 all B_1 all C (is_int(C) & is_int(B_1) -> (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,A_1),B_1) <-> C = B_1))) # label(fact_813_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 9.41/9.46 3532 (all X_31 all Y_25 (X_31 != Y_25 -> (-hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_31),Y_25)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_25),X_31))))) # label(fact_1825_linorder__neqE) # label(axiom) # label(non_clause). [assumption]. 9.41/9.46 3533 (all A_1 all B_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,B_1),A_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_1),B_1)))) # label(fact_4010_coprime__commute) # label(axiom) # label(non_clause). [assumption]. 9.41/9.46 3534 (all A_23 all B_18 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(abs_abs_rat,A_23)),hAPP_rat_rat(abs_abs_rat,B_18)) = hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_23),B_18))) # label(fact_2675_abs__mult) # label(axiom) # label(non_clause). [assumption]. 9.41/9.46 3535 (all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(fact_fact_nat,N)))) # label(fact_3836_fact__gt__zero__nat) # label(axiom) # label(non_clause). [assumption]. 9.41/9.46 3536 (all V_2 all K_2 all V_1 ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_2)),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_1),V_2))),K_2)) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_2)),K_2)) = zero_zero_nat))) # label(fact_2139_nat__number__of__mult__left) # label(axiom) # label(non_clause). [assumption]. 9.41/9.46 3537 (all N all J_2 all K_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_2),K_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_2),N)),K_2)))) # label(fact_2480_less__imp__diff__less) # label(axiom) # label(non_clause). [assumption]. 9.41/9.46 3538 (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(ord_less_eq_nat,hAPP_int_nat(nat,Xa)),hAPP_int_nat(nat,Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)))))) # label(fact_2774_transfer__nat__int__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.46 3539 (all P all Q all R hAPP_b589554111l_bool(P,hAPP_real_bool(Q,R)) = hAPP_r1938957037l_bool(hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,P),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]. 9.41/9.46 3540 (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]. 9.41/9.46 3541 (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]. 9.41/9.46 3542 (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]. 9.41/9.46 3543 (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]. 9.41/9.46 3544 (all M_6 all N_10 ((zero_zero_nat = N_10 -> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,M_6),N_10) = one_one_real) & (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)))))) # label(fact_2611_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3545 (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]. 9.41/9.47 3546 (all K_2 all M all N hAPP_int_int(legacy_zgcd(hAPP_int_int(legacy_zgcd(K_2),M)),N) = hAPP_int_int(legacy_zgcd(K_2),hAPP_int_int(legacy_zgcd(M),N))) # label(fact_4496_zgcd__assoc) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3547 (all Na (hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(div_mod_nat(Na),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,Na)))) # label(fact_3816_even__even__mod__4__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3548 (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]. 9.41/9.47 3549 (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]. 9.41/9.47 3550 (all A_3 ((zero_zero_real != A_3 -> (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> hAPP_real_real(uminus_uminus_real,one_one_real) = hAPP_real_real(sgn_sgn_real,A_3)) & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> hAPP_real_real(sgn_sgn_real,A_3) = one_one_real)) & (A_3 = zero_zero_real -> zero_zero_real = hAPP_real_real(sgn_sgn_real,A_3)))) # label(fact_3636_sgn__real__def) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3551 (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]. 9.41/9.47 3552 (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]. 9.41/9.47 3553 (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_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_199)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_199),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_199),N_53)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_199),N_54))))))) # label(fact_262_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3554 (all Ya all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ya),Xa)) -> Xa = comple124823625p_real(hAPP_r1134773055l_bool(ord_at1589558736t_real(Ya),Xa)))) # label(fact_5063_Sup__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3555 (all P all Q all R hAPP_nat_fun_nat_nat(P,hAPP_nat_nat(Q,R)) = hAPP_nat_fun_nat_nat(hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3556 (all P_1 all L ((exists X_1 hBOOL(hAPP_real_bool(P_1,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,L),X_1)))) <-> (exists X_1 (hBOOL(hAPP_real_bool(P_1,X_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,L),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),zero_zero_real))))))) # label(fact_2419_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3557 (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 | one_one_nat = Q_1)) # label(fact_4171_prime__product) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3558 (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]. 9.41/9.47 3559 (all B_1_1 all B_2_1 is_bool(hAPP_rat_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__Rat__Orat_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3560 (all X_41 all Y_35 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_41),Y_35)) | hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_35),X_41)))) # label(fact_1709_linorder__linear) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3561 (all A_191 hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_191),zero_zero_nat) = one_on1645066479umeral) # label(fact_461_power__0) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3562 (all F all K all A_1 all B_1 (B_1 != A_1 -> ((all X_1 hBOOL(hAPP_real_bool(deriv_real(F,X_1),K))) -> K = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B_1)),hAPP_real_real(F,A_1))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_1),A_1))))) # label(fact_3883_DERIV__const__ratio__const2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3563 (all K_2 hAPP_int_int(pred,K_2) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K_2),one_one_int)) # label(fact_4044_pred__def) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3564 (all Na hAPP_nat_nat(fact_fact_nat,Na) = big_co1705425894at_nat(cOMBI_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(one_one_nat),Na))) # label(fact_5072_fact__altdef__nat) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3565 (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]. 9.41/9.47 3566 (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]. 9.41/9.47 3567 (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]. 9.41/9.47 3568 (all B_73 all A_102 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_102)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_73),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_73),A_102)),zero_zero_real))))) # label(fact_1391_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3569 (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]. 9.41/9.47 3570 (all C_12 all D_8 all A_43 all B_32 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_43),B_32)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,C_12),D_8)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_43),C_12)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_32),D_8)))))) # label(fact_2272_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3571 (all M hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(M),zero_zero_int),M))) # label(fact_2572_zcong__id) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3572 (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]. 9.41/9.47 3573 (all M_15 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M_15),M_15) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),one_one_int)),M_15)) # label(fact_1476_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3574 (all A_27 hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(abs_abs_rat,A_27)) = hAPP_rat_rat(abs_abs_rat,A_27)) # label(fact_2654_abs__idempotent) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3575 (all P_3 (is_int(P_3) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (P_3 != hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) -> (P_3 != hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),P_3))))))) # label(fact_2202_prime__g__5) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3576 (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]. 9.41/9.47 3577 (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]. 9.41/9.47 3578 (all X hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),one_one_nat) = one_one_nat) # label(fact_4335_gcd__lcm__complete__lattice__nat_Oinf__bot__right) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3579 (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]. 9.41/9.47 3580 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),N)) # label(fact_4332_gcd__add1__nat) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3581 (all Xa (zero_zero_complex = Xa <-> Xa = zero_zero_complex)) # label(fact_765_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3582 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)))))) # label(fact_3093_one__less__mult) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3583 (all A_148 A_148 = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,zero_zero_real),A_148)) # label(fact_935_add__0) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3584 (all M 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)))) = hAPP_nat_nat(div_mod_nat(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]. 9.41/9.47 3585 (all A_47 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,zero_z126310315umeral),A_47)) -> zero_z126310315umeral = A_47)) # label(fact_2250_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3586 (all M all K_2 all N (hAPP_int_int(legacy_zgcd(K_2),N) = one_one_int -> hAPP_int_int(legacy_zgcd(M),N) = hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),N))) # label(fact_4485_zgcd__zmult__cancel) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3587 (all K_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(div_mod_int(M),K_2)),M)))) # label(fact_3033_zmod__le__nonneg__dividend) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3588 (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]. 9.41/9.47 3589 (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),A_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,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]. 9.41/9.47 3590 (all B_77 all A_107 (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,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),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]. 9.41/9.47 3591 (all A_1 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,hAPP_nat_real(real_nat,A_1)),Xa)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),natfloor(Xa)))))) # label(fact_3729_le__natfloor__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3592 (all A_1 all B_1 (B_1 = A_1 <-> hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_1),B_1) = zero_zero_complex)) # label(fact_2322_eq__iff__diff__eq__0) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3593 (all P_3 all Q_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Q_1)) -> (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,P_3),Q_1) = one_one_int -> 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]. 9.41/9.47 3594 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Ma)),Na)) <-> hBOOL(hAPP_nat_bool(nat_case_bool(fFalse,hAPP_n1699378549t_bool(ord_less_eq_nat,Ma)),Na)))) # label(fact_4876_less__eq__nat_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3595 (all X_30 all Y_24 (is_int(X_30) & is_int(Y_24) -> X_30 = Y_24 | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_24),X_30)) | 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]. 9.41/9.47 3596 (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]. 9.41/9.47 3597 (all Xa (-hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa)) <-> hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(suc,Xa))))) # label(fact_3789_even__Suc) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3598 (all A_1 all Na (zero_zero_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_1),Na) <-> Na != zero_zero_nat & A_1 = zero_zero_real)) # label(fact_449_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3599 (all K_2 all L_2 pdivmod(K_2,L_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(abs_abs_int,K_2)),hAPP_int_int(abs_abs_int,L_2))),hAPP_int_int(div_mod_int(hAPP_int_int(abs_abs_int,K_2)),hAPP_int_int(abs_abs_int,L_2)))) # label(fact_3222_pdivmod__def) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3600 (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]. 9.41/9.47 3601 (all A_129 all M_17 all B_91 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_129),M_17)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_91),M_17)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_129),B_91)),M_17)) # label(fact_1186_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3602 (all J_2 all K_2 all A_3 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),P_3)) -> (hBOOL(hAPP_int_bool(zprime,P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),zero_zero_int),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),zero_zero_int),P_3)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),zero_zero_int),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),J_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(multInv(P_3),K_2))),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(J_2),K_2),P_3))))))))) # label(fact_2917_MultInv__zcong__prop3) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3603 (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(hAPP_int_fun_int_int(minus_minus_int,K),L)),zero_zero_int)))) # label(fact_2411_less__bin__lemma) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3604 (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]. 9.41/9.47 3605 (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]. 9.41/9.47 3606 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),M)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N))) # label(fact_4357_gcd__diff2__nat) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3607 (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]. 9.41/9.47 3608 (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,Ma),Na)) <-> 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)))))) # label(fact_2152_nat__mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3609 (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]. 9.41/9.47 3610 (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)) -> hAPP_nat_int(semiri1621563631at_int,hAPP_f521865025ol_nat(finite1876863882t_bool,setS(A_1,P_5))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3330_SetS__card) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3611 (all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),N) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,N)))) # label(fact_3115_Suc3__eq__add__3) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3612 (all Xa (hBOOL(hAPP_nat_bool(even_odd_even_nat,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)))) # label(fact_3813_nat__even__iff__2__dvd) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3613 (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]. 9.41/9.47 3614 (all N_18 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)) = one_one_complex) # label(fact_2191_power__m1__even) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3615 (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]. 9.41/9.47 3616 (all C_35 all D_13 all A_92 all B_63 all R_6 (zero_zero_nat != R_6 -> (D_13 != C_35 & A_92 = B_63 -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_92),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,R_6),C_35)) != hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_63),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,R_6),D_13))))) # label(fact_1448_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3617 (all Ma all Ta all K (is_int(K) -> (K != zero_zero_int -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),Ma)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),Ta))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),Ta)))))) # label(fact_2403_zdvd__mono) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3618 (all X_54 all Y_46 all Q_9 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_54),Y_46)),Q_9) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_54),Q_9)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_46),Q_9))) # label(fact_1248_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3619 (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]. 9.41/9.47 3620 (all X all Y (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)))))) -> zero_zero_real = X)) # label(fact_3554_real__sqrt__sum__squares__eq__cancel2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3621 (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]. 9.41/9.47 3622 (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),Xa)) -> Xa = Ya))) # label(fact_1652_order__antisym) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3623 (all P all Q all R hAPP_int_rat(hAPP_P507092031nt_rat(P,R),hAPP_P1175774780nt_int(Q,R)) = hAPP_P1853363688nt_rat(hAPP_f1065357839nt_rat(hAPP_f1120137083nt_rat(cOMBS_549334880nt_rat,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin_002) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3624 (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]. 9.41/9.47 3625 (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]. 9.41/9.47 3626 (all Y all X (X = hAPP_real_real(exp_real,Y) -> hAPP_real_real(ln,X) = Y)) # label(fact_3396_ln__unique) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3627 (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(div_mod_nat(hAPP_int_nat(nat,X)),hAPP_int_nat(nat,Y)) = hAPP_int_nat(nat,hAPP_int_int(div_mod_int(X),Y))))) # label(fact_3201_nat__mod__distrib) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3628 (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(hAPP_i1948725293t_bool(zcong(Xa),zero_zero_int),P_5)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(standardRes(P_5),Xa)),hAPP_i1948725293t_bool(sRStar,P_5)))))) # label(fact_3210_StandardRes__SRStar__prop1) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3629 (all X one_one_real = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(sin,X)))) # label(fact_3597_sin__cos__squared__add3) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3630 (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]. 9.41/9.47 3631 (all N hAPP_int_nat(number_number_of_nat,N) = natfloor(hAPP_int_real(number267125858f_real,N))) # label(fact_3306_natfloor__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3632 (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 (hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_1),N) = A_3 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1))))))) # label(fact_1009_realpow__pos__nth) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3633 (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]. 9.41/9.47 3634 (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]. 9.41/9.47 3635 (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),M)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),N))))) # label(fact_2595_zprime__zdvd__zmult__better) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3636 (all Xa all Ya (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Xa),Ya)) -> (Xa = Ya <-> -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Xa),Ya))))) # label(fact_1968_linorder__antisym__conv2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3637 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N))) # label(fact_3061_mult__Suc__right) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3638 (all Wa all Xa (Xa = hAPP_int_nat(number_number_of_nat,Wa) <-> Xa = hAPP_int_nat(number_number_of_nat,Wa))) # label(fact_94_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3639 (all Va (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Va)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,Va))))) # label(fact_328_less__0__number__of) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3640 (all Y all X all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y)) -> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),Y)) = hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)))))) # label(fact_3925_real__root__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3641 (all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,N),zero_zero_nat) = zero_zero_nat) # label(fact_5120_min__0R) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3642 (all Na all Xa all Diff all F (hAPP_n546711566l_real(Diff,zero_zero_nat) = F & (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]. 9.41/9.47 3643 (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]. 9.41/9.47 3644 (all A_1 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,hAPP_r1250527377l_real(inverse_divide_real,A_1)),hAPP_r474017924t_real(power_power_real,Xa)),zero_zero_real,sequentially)))) # label(fact_5034_LIMSEQ__divide__realpow__zero) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3645 (all Real1_2 all Real2_2 all Real1_1 all Real2_1 (hAPP_real_complex(hAPP_r265291036omplex(complex,Real1_1),Real2_1) = hAPP_real_complex(hAPP_r265291036omplex(complex,Real1_2),Real2_2) <-> Real2_2 = Real2_1 & Real1_1 = Real1_2)) # label(fact_3953_complex_Oinject) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3646 (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]. 9.41/9.47 3647 (all A_3 all B_2 all C_1 hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)),C_1) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(div_mod_int(B_2),C_1))),C_1)) # label(fact_2988_zmod__zmult1__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3648 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N))) -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> M = N))) # label(fact_3006_less__SucE) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3649 (all C (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,C)),one_one_real)) -> hBOOL(tendsto_nat_real(hAPP_r474017924t_real(power_power_real,C),zero_zero_real,sequentially)))) # label(fact_5036_LIMSEQ__rabs__realpow__zero2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3650 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N))))) # label(fact_3148_le__div__geq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3651 (all K_2 all L_2 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,K_2)),hAPP_int_int(bit1,L_2)) = hAPP_int_int(bit0,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),hAPP_int_int(succ,L_2)))) # label(fact_875_add__Bit1__Bit1) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3652 (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]. 9.41/9.47 3653 (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)) -> xzgcda(M,N,R_4,R_2,S_3,S_1,T_2,T_1) = 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)))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_2),zero_zero_int)) -> xzgcda(M,N,R_4,R_2,S_3,S_1,T_2,T_1) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,R_4),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,S_3),T_2))))) # label(fact_4199_xzgcda_Osimps) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3654 (all A_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),one_one_nat)) -> one_one_nat = A_3)) # label(fact_2968_gcd__lcm__complete__lattice__nat_Ole__bot) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3655 (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)) -> X = hAPP_real_real(arccos,hAPP_real_real(cos,X))))) # label(fact_3671_arccos__cos) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3656 (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]. 9.41/9.47 3657 (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]. 9.41/9.47 3658 (all Va all Wa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,Va)),hAPP_int_real(number267125858f_real,Wa))) <-> -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,Wa)),hAPP_int_real(number267125858f_real,Va))))) # label(fact_691_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3659 (all Ma all Na (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))) <-> Ma != zero_zero_nat | zero_zero_nat != Na)) # label(fact_4343_gcd__pos__nat) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3660 (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]. 9.41/9.47 3661 (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]. 9.41/9.47 3662 (all X all N hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,X)),N)) # label(fact_390_Nat__Transfer_Otransfer__int__nat__functions_I4_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3663 (all A_152 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_152),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_898_zero__le__power2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3664 (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]. 9.41/9.47 3665 (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]. 9.41/9.47 3666 (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]. 9.41/9.47 3667 (all A_151 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_151),zero_zero_complex) = A_151) # label(fact_910_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3668 (all A_1 big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(d22set,A_1)) = hAPP_int_int(zfact,A_1)) # label(fact_4809_d22set__prod__zfact) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3669 (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]. 9.41/9.47 3670 (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]. 9.41/9.47 3671 (all M all N (M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) -> N = one_one_nat | zero_zero_nat = M)) # label(fact_2125_mult__eq__self__implies__10) # label(axiom) # label(non_clause). [assumption]. 9.41/9.47 3672 (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)) -> (one_one_int = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,U_3),V_9) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,U_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]. 9.41/9.47 3673 (all A_3 all B_2 all X (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),B_2)) & 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)) & -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_4330_gcd__semilattice__nat_Oless__infI2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3674 (all A_135 all B_95 (hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_135),B_95) = zero_zero_rat -> B_95 = zero_zero_rat | zero_zero_rat = A_135)) # label(fact_1131_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3675 (all V_11 all B_62 all C_34 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_11)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_62),C_34)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_11)),B_62)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_11)),C_34))) # label(fact_1458_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3676 (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]. 9.41/9.48 3677 (all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) -> (hBOOL(hAPP_nat_bool(even_odd_even_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)))))) # label(fact_3809_even__num__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3678 (all N all M hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(minus_534354567de_int,N),M) = hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,quickcheck_int_of(N)),quickcheck_int_of(M)))) # label(fact_4561_minus__code__int__def) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3679 (all Ma all Xa (hAPP_int_int(standardRes(Ma),Xa) = zero_zero_int <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Xa),zero_zero_int),Ma)))) # label(fact_3076_StandardRes__eq__zcong) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3680 -(all M1 (is_int(M1) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,v),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,w),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) != hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))),M1))) # label(fact_2880__096_B_Bthesis_O_A_I_B_Bm1_O_Av_A_094_A2_A_L_Aw_A_094_A2_A_061_A_I1_A_) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3681 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),zero_zero_real)) -> zero_zero_real = hAPP_real_real(cos,arg(hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),Y))))) # label(fact_3965_cos__arg__i__mult__zero__neg) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3682 (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]. 9.41/9.48 3683 (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]. 9.41/9.48 3684 (all Ma all X_2 ((all Y1 all Y2 (is_int(Y1) & is_int(Y2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Y1),Y2),Ma)) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y2),X_2)) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y1),X_2)) -> Y1 = Y2))) <-> hBOOL(hAPP_f448129468l_bool(resSet(Ma),X_2)))) # label(fact_4053_ResSet__def) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3685 (all P all Q all R hAPP_P1668131738nt_int(hAPP_f167782601nt_int(hAPP_f1769710159nt_int(cOMBC_2037038898nt_int,P),Q),R) = hAPP_f521271395nt_int(hAPP_P752050971nt_int(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc_) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3686 (all A_198 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_198),one_one_nat) = A_198) # label(fact_318_power__one__right) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3687 (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]. 9.41/9.48 3688 (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]. 9.41/9.48 3689 (all R_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_2),zero_zero_real)) -> (exists 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)) & is_int(N_1))))) # label(fact_4477_reals__Archimedean__6c__int) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3690 (all K_2 all M all N hAPP_int_int(legacy_zgcd(M),hAPP_int_int(legacy_zgcd(K_2),N)) = hAPP_int_int(legacy_zgcd(K_2),hAPP_int_int(legacy_zgcd(M),N))) # label(fact_4495_zgcd__left__commute) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3691 (all B_75 all A_104 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_104)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_75)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_104),B_75)))))) # label(fact_1377_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3692 (all P all Q all R hAPP_P1975530577nt_int(P,hAPP_i1524277240nt_int(Q,R)) = hAPP_i1524277240nt_int(hAPP_f1735705551nt_int(hAPP_f151095827nt_int(cOMBB_1389251822nt_int,P),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]. 9.41/9.48 3693 (all Lx_1 all Ly_1 all Rx_1 all Ry_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_1),Ly_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_1),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))) # label(fact_1078_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3694 (all M all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),M))))) # label(fact_291_trans__less__add1) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3695 (all X_41 all Y_35 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_35),X_41)) | hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_41),Y_35)))) # label(fact_1708_linorder__linear) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3696 (all C all D_1 all A_1 all B_1 (C != D_1 & A_1 != B_1 <-> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_1),C)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_1),D_1)) != hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_1),D_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_1),C)))) # label(fact_1172_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3697 (all M all N all K_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),K_2)) # label(fact_283_nat__add__assoc) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3698 (all P all Q all R hAPP_int_bool(hAPP_i1948725293t_bool(P,R),Q) = hAPP_int_bool(hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3699 (all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N),one_one_int)),hAPP_int_int(fact_fact_int,N)) = hAPP_int_int(fact_fact_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N),one_one_int)))) # label(fact_4286_fact__plus__one__int) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3700 (all X_20 all Y_16 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_20),Y_16)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_20),Y_16)))) # label(fact_1977_order__less__imp__le) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3701 (all I_1 all J hAPP_n1699378549t_bool(ord_at4362885an_nat(hAPP_nat_nat(suc,I_1)),hAPP_nat_nat(suc,J)) = image_nat_nat(suc,hAPP_n1699378549t_bool(ord_at4362885an_nat(I_1),J))) # label(fact_4906_image__Suc__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3702 (all K_2 hAPP_int_int(bit1,K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),K_2)),K_2)) # label(fact_201_Bit1__def) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3703 (all P all Q all R hAPP_i371647666l_real(hAPP_f1352333353l_real(hAPP_f318350629l_real(cOMBB_1946221070al_int,P),Q),R) = hAPP_r1250527377l_real(P,hAPP_int_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__R) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3704 (all Y all U_1 all X1_1 all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X1_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X1_1),Y)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),U_1))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,X)),Y)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,X1_1)),U_1))))))) # label(fact_3856_real__mult__inverse__cancel) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3705 (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]. 9.41/9.48 3706 (all Xa all A_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim856651990g_real(Xa)),A_1)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,A_1)),one_one_real))))) # label(fact_4470_ceiling__less__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3707 (all N_46 all A_195 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_195)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_195),N_46))))) # label(fact_413_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3708 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (hAPP_real_real(sqrt,X) = zero_zero_real -> X = zero_zero_real))) # label(fact_3524_real__sqrt__eq__zero__cancel) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3709 (all K all F all Na all Ma (is_int(K) -> ((all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),I_2)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),Na)) -> 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 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),I_2)) & K = hAPP_nat_int(F,I_2) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),Na)))))))))) # label(fact_2817_nat__intermed__int__val) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3710 (all Ta all D_1 all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),D)) -> (all X_1 all K_1 (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,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))),Ta))) <-> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Ta))))))) # label(fact_2233_inf__period_I4_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3711 (all B all A (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A)) -> (hBOOL(hAPP_f448129468l_bool(nat_nat_set,B)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B)) <-> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,image_int_nat(nat,A)),image_int_nat(nat,B))))))) # label(fact_4617_transfer__int__nat__set__relations_I5_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3712 (all Lx_2 all Ly_2 all Rx_2 all Ry_2 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_2),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,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)))) # label(fact_1073_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3713 (all A_1 all B_1 all C all D_1 (hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C),D_1) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_1),B_1) -> (D_1 = C <-> A_1 = B_1))) # label(fact_2224_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3714 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,Ma)),hAPP_nat_nat(semiri984289939at_nat,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)))) # label(fact_358_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3715 (all 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)))) = 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)) # label(fact_1873_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3716 (all X hAPP_real_real(tan,X) = hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi)))) # label(fact_3757_tan__periodic) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3717 (all B_70 all A_99 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_99),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_70)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_99),B_70)),zero_zero_rat))))) # label(fact_1409_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3718 (all P_3 all Q_1 divmod_int(P_3,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))) # label(fact_4196_divmod__int__mod__div) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3719 (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]. 9.41/9.48 3720 (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]. 9.41/9.48 3721 (all A_47 (is_int(A_47) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,zero_zero_int),A_47)) -> A_47 = zero_zero_int))) # label(fact_2246_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3722 (all A_124 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_124),one_one_rat) = A_124) # label(fact_1220_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3723 (all P_1 all L ((exists X_1 (hBOOL(hAPP_C24501650l_bool(P_1,X_1)) & hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,L),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_1),zero_z126310315umeral))))) <-> (exists X_1 hBOOL(hAPP_C24501650l_bool(P_1,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,L),X_1)))))) # label(fact_2420_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3724 (all C_30 all A_84 all B_57 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_30),A_84)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_30),B_57))) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_30)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_84),B_57))))) # label(fact_1533_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3725 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,ratreal(X)),ratreal(Y)) = ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,X),Y))) # label(fact_4703_real__divide__code) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3726 (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)) -> (F = hAPP_n546711566l_real(Diff,zero_zero_nat) -> ((all M_1 all T (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),H)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),T)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),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)) & 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)))))))) # label(fact_4938_Maclaurin) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3727 (all R_2 all A_3 hAPP_real_real(abs_abs_real,R_2) = hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_2)),hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(cos,A_3)),hAPP_real_real(sin,A_3))))) # label(fact_3994_cmod__complex__polar) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3728 (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]. 9.41/9.48 3729 (all X all A_3 archim1246769320r_real(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_int_real(real_int,A_3))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,archim1246769320r_real(X)),A_3)) # label(fact_4425_floor__subtract) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3730 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),hAPP_real_real(cos,X)))) # label(fact_3614_cos__ge__minus__one) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3731 (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]. 9.41/9.48 3732 (all B_96 all A_136 (A_136 != zero_z891286103de_int -> (B_96 != zero_z891286103de_int -> zero_z891286103de_int != hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_136),B_96)))) # label(fact_1125_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3733 (all M_20 all N_38 all A_175 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_175)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_175),M_20)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_175),N_38))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_20),N_38))))) # label(fact_623_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3734 (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_3236_Parity_Oodd__plus__odd) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3735 (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_real,hAPP_real_real(sqrt,Xa)),hAPP_real_real(sqrt,Ya))))) # label(fact_3512_real__sqrt__less__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3736 (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]. 9.41/9.48 3737 (all X all N (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,N))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natfloor(X)),hAPP_int_nat(number_number_of_nat,N)) = natfloor(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_int_real(number267125858f_real,N)))))) # label(fact_3299_natfloor__add__number__of) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3738 (all A_1 all C all B_1 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_1),B_1)) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_1),C)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_1),C))))) # label(fact_984_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3739 (all P all Q all R hAPP_r1150855451l_bool(hAPP_f381769257l_bool(hAPP_f512004221l_bool(cOMBB_188663909l_real,P),Q),R) = hAPP_f1344929775l_bool(P,hAPP_r632163725t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__Int__Oint_Mtc__HOL__034) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3740 (all Xa (one_on1645066479umeral = Xa <-> Xa = one_on1645066479umeral)) # label(fact_853_one__reorient) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3741 (all N all X (is_int(N) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N)),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N)),one_one_real))) -> archim1246769320r_real(X) = N)))) # label(fact_4452_floor__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3742 (all E_1 all B_2 all D_3 all A_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),A_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,E_1),B_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),B_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),E_1)))))) # label(fact_4011_coprime__divisors) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3743 (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]. 9.41/9.48 3744 (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]. 9.41/9.48 3745 (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_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_12)),hAPP_int_rat(number_number_of_rat,V_13)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_12),V_13))))) # label(fact_1278_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3746 (all P_5 hAPP_P1975530577nt_int(produc1518849193nt_int(hAPP_f1574055885nt_int(hAPP_f2054677283nt_int(cOMBB_782625832nt_int,product_Pair_int_int),abs_abs_int)),quotient_of(P_5)) = quotient_of(hAPP_rat_rat(abs_abs_rat,P_5))) # label(fact_4835_rat__abs__code) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3747 (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]. 9.41/9.48 3748 (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]. 9.41/9.48 3749 (all Xa all Ya (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xa),Ya)),zEven)) <-> (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ya),zEven))))) # label(fact_3340_even__diff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3750 (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]. 9.41/9.48 3751 (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)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),zero_zero_real))))) # label(fact_3911_real__root__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3752 (all X_10 all Y_7 all A_29 all B_22 hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_10),Y_7)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_29),B_22)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_10),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Y_7),B_22))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_10),A_29)),B_22))) # label(fact_2390_mult__diff__mult) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3753 (all B_1_1 is_bool(vanishes(B_1_1))) # label(gsy_c_RealDef_Ovanishes) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3754 (all Z_11 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Z_11),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)) # label(fact_1896_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3755 -(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]. 9.41/9.48 3756 (all Z_1 hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(cnj,Z_1)) = hAPP_complex_real(norm_norm_complex,Z_1)) # label(fact_4072_complex__mod__cnj) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3757 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),Y)))) # label(fact_3418_real__gt__half__sum) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3758 (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]. 9.41/9.48 3759 (all F all Ya all G all Xa (hBOOL(hAPP_real_bool(sums_real(G),Xa)) -> (hBOOL(hAPP_real_bool(sums_real(F),Ya)) -> hBOOL(hAPP_real_bool(sums_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f335634176l_real(cOMBS_337378985l_real,hAPP_f1129044203l_real(hAPP_f264196187l_real(cOMBB_1480366270al_nat,if_real),even_odd_even_nat)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,F),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,div_div_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,G),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,div_div_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,minus_minus_nat),one_one_nat))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa),Ya)))))) # label(fact_4868_sums__if) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3760 (all Wa all Xa (is_int(Xa) -> (hAPP_int_int(number_number_of_int,Wa) = Xa <-> Xa = hAPP_int_int(number_number_of_int,Wa)))) # label(fact_93_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3761 (all Xa (zero_zero_nat = hAPP_nat_nat(div_mod_nat(Xa),hAPP_nat_nat(suc,hAPP_nat_nat(suc,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]. 9.41/9.48 3762 (all Y all X all U_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,X)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,U_1),hAPP_real_real(sqrt,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),U_1))))) # label(fact_3561_real__sqrt__sum__squares__less) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3763 (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]. 9.41/9.48 3764 (all X_61 all Y_51 (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)))) = 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)))) -> (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]. 9.41/9.48 3765 (all X_62 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_62),zero_zero_nat) = one_one_int) # label(fact_482_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3766 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)),zero_zero_int)) -> hAPP_int_int(number_number_of_int,min) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)))) # label(fact_2853_div__pos__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3767 (all X_17 all Y_13 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_17),Y_13)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_13),X_17)))) # label(fact_2012_leI) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3768 (all Ta all A 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),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,Ta),X_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ta),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),D))))))))) # label(fact_5088_aset_I8_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3769 (all X_52 all N_25 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,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_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_52),N_25)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_52),N_25))) # label(fact_1511_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3770 (all Z_1 all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_1),N)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),N))))) # label(fact_2441_zdvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3771 (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]. 9.41/9.48 3772 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3417_real__less__half__sum) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3773 (all A_3 all B_2 hAPP_int_int(div_mod_int(A_3),hAPP_int_int(uminus_uminus_int,B_2)) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,A_3)),B_2))) # label(fact_3473_zmod__zminus2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.48 3774 (all L all K (is_int(K) & is_int(L) -> (K != zero_zero_int -> (L != zero_zero_int -> hAPP_P1975530577nt_int(produc713050258nt_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(sgn_sgn_int,L))),hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,hAPP_int_int(sgn_sgn_int,K)),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 -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),K) = divmod_int(K,L))) & (K = zero_zero_int -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int) = divmod_int(K,L)))) # label(fact_4855_divmod__int__code) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3775 (all F all L all C (hBOOL(tendsto_real_real(F,L,at_real(C))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),L)) -> (exists R_1 ((all X_1 (C != X_1 & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),X_1))),R_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(F,X_1))))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_1))))))) # label(fact_5086_LIM__fun__gt__zero) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3776 (all N all D_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),D_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),N)) -> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(real_nat,D_3)) = hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),D_3))))) # label(fact_3736_real__of__nat__div) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3777 (all Y Y = hAPP_real_real(tan,hAPP_real_real(arctan,Y))) # label(fact_3694_tan__arctan) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3778 (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]. 9.41/9.49 3779 (all N_24 all A_77 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_77)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_77),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_77),N_24)))))) # label(fact_1599_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3780 (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]. 9.41/9.49 3781 (all B_1_1 all B_2_1 is_bool(hAPP_P1333854989l_bool(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_000tc__HOL__Obo) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3782 (all Xa hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,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))))) # label(fact_4822_lemma__sin__minus) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3783 (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)))))) -> (Ta != X_1 -> 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]. 9.41/9.49 3784 (all Xa hBOOL(hAPP_real_bool(isCont_real_real(arctan),Xa))) # label(fact_5162_isCont__arctan) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3785 (all C_39 all A_96 all B_67 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_96),B_67)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_39)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_39),A_96)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_39),B_67)))))) # label(fact_1427_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3786 (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]. 9.41/9.49 3787 (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_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_4),C_3)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),B_4)) & hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,B_4),C_3))) -> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),C_3))))) -> ((all X_1 exists D_2 ((all A_2 all B_4 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_4),A_2)),D_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),B_4)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),X_1)) -> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4))))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2)))) -> (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,A_1),B_1))))))) # label(fact_4295_lemma__BOLZANO) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3788 (all A_146 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,zero_zero_real),A_146) = A_146) # label(fact_949_add__0__left) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3789 (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]. 9.41/9.49 3790 (all X all Y (Y = X | -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(fequal_real,X),Y)))) # label(help_fequal_1_1_fequal_000tc__RealDef__Oreal_T) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3791 (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]. 9.41/9.49 3792 (all A_151 hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_151),zero_z126310315umeral) = A_151) # label(fact_914_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3793 (all A_47 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_47)) -> zero_zero_nat = A_47)) # label(fact_2245_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3794 (all A (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A)) -> image_nat_int(semiri1621563631at_int,image_int_nat(nat,A)) = A)) # label(fact_4614_transfer__nat__int__set__return__embed) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3795 (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]. 9.41/9.49 3796 (all A_3 all B_2 all C_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1)) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),C_1))),hAPP_int_int(div_mod_int(A_3),B_2)) = hAPP_int_int(div_mod_int(A_3),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1)))) # label(fact_3131_zmod__zmult2__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3797 (all Wa (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,Wa))) <-> hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,Wa)))))) # label(fact_3249_neg__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3798 (all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),hAPP_int_int(bit1,pls))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,Xa)),one_one_real)))) # label(fact_636_le__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3799 (all B_1 all A_1 (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,B_1),A_1)) -> -hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A_1),B_1)))) # label(fact_1771_xt1_I9_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3800 (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]. 9.41/9.49 3801 (all Z_1 all W hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_1),W) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,W),Z_1)) # label(fact_2158_real__mult__commute) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3802 (all P P = hAPP_nat_nat(cOMBI_nat,P)) # label(help_COMBI_1_1_COMBI_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3803 (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(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,min))),hAPP_real_real(cos,Xa))),hAPP_real_real(sin,Xa))))) # label(fact_4797_DERIV__cos__realpow2a) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3804 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(cos,X)))))) # label(fact_3658_cos__gt__zero) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3805 (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)))) <-> 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)) # label(fact_3149_neq__one__mod__two) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3806 (all N one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,N),one_one_int)),N)) # label(fact_4981_coprime__minus__one__int) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3807 (all N all D_3 all A_3 (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),A_3) = one_one_int -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N)) = one_one_int)) # label(fact_4984_coprime__exp__int) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3808 (all B_70 all A_99 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_99),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_70)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_99),B_70)),zero_zero_real))))) # label(fact_1412_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3809 (all N_8 hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(semiri151668891at_rat,N_8)) = hAPP_nat_rat(semiri151668891at_rat,N_8)) # label(fact_2699_abs__of__nat) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3810 (all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_int_real(number267125858f_real,Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,pls)),Ya)))) # label(fact_125_less__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3811 (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]. 9.41/9.49 3812 (all P (is_bool(P) -> fTrue = P | fFalse = P)) # label(help_If_3_1_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_T) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3813 (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]. 9.41/9.49 3814 (all Y all X (hAPP_real_real(cos,X) != zero_zero_real -> (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]. 9.41/9.49 3815 (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))) -> 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)))) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W))) -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,W))) = one_one_int))) # label(fact_4789_zpower__number__of__odd) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3816 (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_1628_le__funE) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3817 (all P all Q all R hAPP_i1778081398nt_int(hAPP_f2004554147nt_int(hAPP_f1228585167nt_int(cOMBB_1102189010nt_int,P),Q),R) = hAPP_f1574055885nt_int(P,hAPP_int_fun_int_int(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__fun_Itc___024) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3818 (all A_1 all B_1 (B_1 = zero_zero_real | zero_zero_real = A_1 <-> zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_1),B_1))) # label(fact_1122_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3819 (all B_1_1 all B_2_1 is_int(hAPP_f659380387ol_int(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__Int__Oint) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3820 (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))) -> zero_zero_real = hAPP_nat_real(semiri132038758t_real,hAPP_int_nat(number_number_of_nat,V_17))))) # label(fact_901_of__nat__number__of__lemma) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3821 (all A_148 A_148 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A_148)) # label(fact_936_add__0) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3822 (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]. 9.41/9.49 3823 (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]. 9.41/9.49 3824 (all N all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M),N)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),hAPP_int_int(fact_fact_int,N)))))) # label(fact_4283_dvd__fact__int) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3825 (all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,W))) -> hAPP_P1175774780nt_int(product_fst_int_int,posDivAlg(one_one_int,hAPP_int_int(number_number_of_int,W))) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,one_one_int),hAPP_int_int(number_number_of_int,W)))) # label(fact_4226_div__pos__pos__1__number__of) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3826 (all A_134 zero_z891286103de_int = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,zero_z891286103de_int),A_134)) # label(fact_1139_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3827 (all Lx all Ly all Rx all Ry hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx),Rx)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ly),Ry)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx),Ly)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx),Ry))) # label(fact_1080_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3828 (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]. 9.41/9.49 3829 (all X all Y hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Y),hAPP_int_int(div_mod_int(X),Y)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)) # label(fact_4985_gcd__red__int) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3830 (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]. 9.41/9.49 3831 (all Z_2 all Ya all Xa (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ya),Xa)) -> (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Z_2),Ya)) -> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Z_2),Xa))))) # label(fact_1630_xt1_I6_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3832 (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]. 9.41/9.49 3833 (all A_26 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_26),hAPP_int_int(abs_abs_int,A_26)))) # label(fact_2668_abs__ge__self) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3834 (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]. 9.41/9.49 3835 (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]. 9.41/9.49 3836 (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)) | Ya = Xa)) # label(fact_2649_dvd_Ole__less) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3837 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X)) -> (exists N_1 (is_int(N_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(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Y),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_1),X))))),X)))))) # label(fact_2813_best__division__abs) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3838 (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)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_4),hAPP_nat_nat(suc,hAPP_nat_nat(fact,N)))))) # label(fact_4569_euclid__bound) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3839 (all Wa all Xa (hAPP_int_rat(number_number_of_rat,Wa) = Xa <-> hAPP_int_rat(number_number_of_rat,Wa) = Xa)) # label(fact_88_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3840 (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]. 9.41/9.49 3841 (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(inverse_divide_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(inverse_divide_real,X),Y)),A_3)))) # label(fact_4434_powr__divide) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3842 (all N all D_3 all A_3 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_3),A_3) = 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(power_power_nat,A_3),N)))) # label(fact_4361_coprime__exp__nat) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3843 (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]. 9.41/9.49 3844 (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,hAPP_real_real(ln,Xa)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),one_one_real))))) # label(fact_3362_ln__less__zero__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3845 (all Xa all Ya (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ya),Xa)))) # label(fact_2049_linorder__not__less) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3846 (all B_120 all A_174 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_174),zero_z126310315umeral)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_120),zero_z126310315umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_174),B_120)),zero_z126310315umeral))))) # label(fact_646_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3847 (all P_1 collec1347809874nt_int(P_1) = P_1) # label(fact_147_Collect__def) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3848 (all X hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X)) = hAPP_real_real(sqrt,hAPP_real_real(inverse_inverse_real,X))) # label(fact_3851_real__sqrt__inverse) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3849 (all X_5 all Y_5 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_5),Y_5)) -> (Y_5 != X_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]. 9.41/9.49 3850 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(fact_fact_nat,N)),hAPP_nat_nat(fact_fact_nat,M))))) # label(fact_3839_fact__dvd) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3851 (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]. 9.41/9.49 3852 (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]. 9.41/9.49 3853 (all A_1 all B_1 all P_5 (hBOOL(hAPP_nat_bool(prime,P_5)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_1),B_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_5),A_1)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_5),B_1))))) # label(fact_4168_prime__divprod__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3854 (all X_53 all P_7 all Q_8 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_53),P_7)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_53),Q_8)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_53),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_8))) # label(fact_1500_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3855 (all Na (hAPP_nat_int(semiri1621563631at_int,Na) = zero_zero_int <-> zero_zero_nat = Na)) # label(fact_71_int__eq__0__conv) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3856 (all A_139 all B_97 all C_51 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_139),B_97)),C_51) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_139),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_97),C_51))) # label(fact_1055_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3857 (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(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(log(A_3),X)),hAPP_real_real(log(A_3),Y)) = hAPP_real_real(log(A_3),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),Y))))))) # label(fact_3452_log__divide) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3858 (all K_2 all L_2 (is_int(L_2) -> (L_2 = zero_zero_int -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),hAPP_int_int(abs_abs_int,K_2)) = pdivmod(K_2,L_2)) & (L_2 != zero_zero_int -> posDivAlg(hAPP_int_int(abs_abs_int,K_2),hAPP_int_int(abs_abs_int,L_2)) = pdivmod(K_2,L_2)))) # label(fact_3295_pdivmod__posDivAlg) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3859 (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]. 9.41/9.49 3860 (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]. 9.41/9.49 3861 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> M = 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))) # label(fact_2891_div__mult__self1__is__m) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3862 (all Z_8 all Y_19 all X_23 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_19),X_23)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Z_8),Y_19)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z_8),X_23))))) # label(fact_1933_xt1_I7_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3863 (all W_15 ((zero_zero_nat = hAPP_int_nat(number_number_of_nat,W_15) -> one_one_real = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),hAPP_int_nat(number_number_of_nat,W_15))) & (hAPP_int_nat(number_number_of_nat,W_15) != zero_zero_nat -> 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]. 9.41/9.49 3864 (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]. 9.41/9.49 3865 (all I all J_2 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(legacy_zgcd(I),J_2)),J_2))) # label(fact_4482_zgcd__zdvd2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3866 (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]. 9.41/9.49 3867 (all B_118 all C_68 all A_172 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_172)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_118),C_68)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_118),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_172),C_68)))))) # label(fact_656_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3868 (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,M),hAPP_nat_nat(suc,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,M),K_2)),hAPP_nat_nat(suc,J_2)))) # label(fact_3108_diff__Suc__diff__eq1) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3869 (all K_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,I),K_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,J_2),K_2)))))) # label(fact_2115_mult__less__mono1) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3870 (all B_2 all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_3)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,archim1246769320r_real(A_3)),archim1246769320r_real(B_2))),archim1246769320r_real(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),B_2))))))) # label(fact_4033_le__mult__floor) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3871 (all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real)) -> Y = hAPP_real_real(cos,hAPP_real_real(arccos,Y)) & 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_3679_arccos) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3872 (all X_36 all Y_30 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_36),Y_30)) -> Y_30 != X_36)) # label(fact_1796_order__less__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3873 (all X_53 all P_7 all 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)) = 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))) # label(fact_1498_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3874 (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]. 9.41/9.49 3875 (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(plus_plus_nat,K),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),Na))))) # label(fact_286_nat__add__left__cancel__less) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3876 (all P all Q all R hAPP_f1805168059t_bool(hAPP_i1529485324t_bool(P,R),Q) = hAPP_i1948725293t_bool(hAPP_f1048215610t_bool(hAPP_f472159229t_bool(cOMBC_1683390479t_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__HOL__Obool__015) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3877 (all C_52 all A_141 all B_99 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_141),B_99)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_141),C_52)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_99),C_52))))) # label(fact_974_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3878 (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]. 9.41/9.49 3879 (all Y_39 all X_45 (is_int(X_45) & is_int(Y_39) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_39),X_45)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_45),Y_39)) -> X_45 = Y_39)))) # label(fact_1642_xt1_I5_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3880 (all A_144 all C_55 all B_102 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_144),C_55)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_102),C_55))) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_144),B_102)))) # label(fact_959_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3881 (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]. 9.41/9.49 3882 (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]. 9.41/9.49 3883 (all K_2 ((K_2 != zero_z126310315umeral -> (hAPP_C498520661umeral(div_mo1740067990umeral(K_2),hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) != zero_z126310315umeral -> code_int_of(K_2) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),code_int_of(hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,K_2),hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),one_one_int)) & (hAPP_C498520661umeral(div_mo1740067990umeral(K_2),hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_z126310315umeral -> code_int_of(K_2) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),code_int_of(hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,K_2),hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) & (K_2 = zero_z126310315umeral -> zero_zero_int = code_int_of(K_2)))) # label(fact_3325_Code__Numeral_Oint__of__code) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3884 (all A_121 all B_90 all 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) = 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))) # label(fact_1243_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3885 (all A_40 all B_29 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_40),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_29),A_40)))) # label(fact_2290_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3886 (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))) -> (zero_zero_int != B_2 -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = Q_1)))) # label(fact_3260_divmod__int__rel__div) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3887 (all Xa all Ya (-hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Xa),Ya)) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Ya),Xa)))) # label(fact_2041_linorder__not__le) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3888 (all A_138 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,zero_z891286103de_int),A_138) = zero_z891286103de_int) # label(fact_1107_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3889 (all K_2 all 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)) = frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,K_2)),hAPP_int_int(number_number_of_int,L_2)))) # label(fact_4640_Frct__code__post_I6_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3890 (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]. 9.41/9.49 3891 (all A_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_1),A_1)),zero_zero_rat)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_1),zero_zero_rat)))) # label(fact_595_double__add__le__zero__iff__single__add__le__zero) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3892 (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]. 9.41/9.49 3893 (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]. 9.41/9.49 3894 (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]. 9.41/9.49 3895 (all Q_1 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(prime,Q_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),Q_1)) -> Q_1 = P_3)))) # label(fact_4162_primes__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3896 (all R_2 all P_3 all Q_1 (quotient_of(R_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_1) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Q_1)))) # label(fact_4651_quotient__of__denom__pos) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3897 (all Q_1 ((all A_2 all B_4 (is_int(A_2) & is_int(B_4) -> (hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_2),B_4) = Q_1 -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_4)) -> (zero_zero_int != A_2 -> 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]. 9.41/9.49 3898 (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,hAPP_nat_real(X_2,N_1)),A_1)))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),A_1))))) # label(fact_5044_LIMSEQ__le__const2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.49 3899 (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]. 9.41/9.50 3900 (all Z_5 (hBOOL(hAPP_int_bool(nat_neg,Z_5)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_5),zero_zero_int)))) # label(fact_3245_neg__def) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3901 (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]. 9.41/9.50 3902 (all N all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y),hAPP_nat_nat(suc,N)))))) # label(fact_3072_divides__rexp) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3903 (all A_3 all B_2 all 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) = 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))) # label(fact_4993_gcd__assoc__int) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3904 (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]. 9.41/9.50 3905 (all Xa all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_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_int,Xa),Ya)))) # label(fact_62_less__number__of) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3906 (all 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)))) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_59),A_59)) # label(fact_1903_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3907 (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]. 9.41/9.50 3908 (all A_137 zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_137),zero_zero_rat)) # label(fact_1113_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3909 (all A_122 all N_29 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)) = 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)) # label(fact_1238_power__commutes) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3910 (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]. 9.41/9.50 3911 (all N_18 one_one_int = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_18))) # label(fact_2193_power__m1__even) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3912 (all X all Y hAPP_complex_real(im,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X),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)))) # label(fact_4129_complex__Im__mult) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3913 (all A_1 all B_1 (B_1 != A_1 -> (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_1),B_1)) -> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A_1),B_1))))) # label(fact_1993_order__neq__le__trans) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3914 (all Xa all Ya all Xaa (zero_zero_real = Ya & Xa = Xaa <-> 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]. 9.41/9.50 3915 (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]. 9.41/9.50 3916 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sin,X)),one_one_real))) # label(fact_3602_sin__le__one) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3917 (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]. 9.41/9.50 3918 (all V_9 all U_3 all Y_23 all X_28 all A_63 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_28),A_63)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_23),A_63)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),U_3)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),V_9)) -> (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,U_3),V_9) = one_one_rat -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,U_3),X_28)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,V_9),Y_23))),A_63)))))))) # label(fact_1857_convex__bound__le) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3919 (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]. 9.41/9.50 3920 (all B_119 all A_173 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_173),zero_z126310315umeral)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_119),zero_z126310315umeral)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_173),B_119)),zero_z126310315umeral))))) # label(fact_652_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3921 (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]. 9.41/9.50 3922 (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]. 9.41/9.50 3923 (all F all G (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,F),G)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G)) & -hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,G),F)))) # label(fact_2076_less__fun__def) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3924 (all A_1 all Wa (A_1 = zero_zero_nat & zero_zero_nat != hAPP_int_nat(number_number_of_nat,Wa) <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_1),hAPP_int_nat(number_number_of_nat,Wa)) = zero_zero_nat)) # label(fact_185_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3925 (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_2017_order__le__less) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3926 (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]. 9.41/9.50 3927 (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]. 9.41/9.50 3928 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_real_real(sgn_sgn_real,X) = one_one_real)) # label(fact_3627_real__sgn__pos) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3929 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Y),X)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(abs_abs_int,X)),hAPP_int_int(abs_abs_int,Y)) = hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y)))) # label(fact_2829_abs__div) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3930 (all C all A_1 all B_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_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))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B_1)))) # label(fact_980_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3931 (all F all Xa all F_1 all Na all Ma ((all R_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),R_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,R_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),Na))) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(F,R_1),Xa),hAPP_real_real(hAPP_n546711566l_real(F_1,R_1),Xa))))) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_f2119584768l_real(hAPP_f59594445l_real(cOMBC_2010481033l_real,hAPP_f1523861863l_real(hAPP_f1067681227l_real(cOMBB_1934313333l_real,big_co604158596t_real),hAPP_f42270917t_real(cOMBC_nat_real_real,F))),hAPP_n1699378549t_bool(ord_at4362885an_nat(Ma),Na)),Xa),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,F_1),Xa)),hAPP_n1699378549t_bool(ord_at4362885an_nat(Ma),Na)))))) # label(fact_4942_DERIV__sumr) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3932 (all Y all X (zero_zero_real != hAPP_real_real(cos,X) -> (hAPP_real_real(cos,Y) != zero_zero_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(inverse_divide_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y)))))) # label(fact_3753_add__tan__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3933 (all K_2 zero_zero_rat = frct(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,K_2),zero_zero_int))) # label(fact_4637_Frct__code__post_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3934 (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]. 9.41/9.50 3935 (all C_15 all A_49 all B_36 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_49),B_36)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_36),C_15)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_49),C_15))))) # label(fact_2220_dvd__trans) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3936 (all A_1 (A_1 != one_one_nat <-> -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)))) # label(fact_2966_gcd__lcm__complete__lattice__nat_Obot__less) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3937 (all K_2 all M all N (is_int(K_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N))) -> (zero_zero_int != K_2 -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),N)))))) # label(fact_2402_zdvd__mult__cancel) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3938 (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]. 9.41/9.50 3939 (all Xa all 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)) <-> 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))))) # label(fact_3086_zdvd__iff__zmod__eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3940 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),hAPP_int_int(bit0,K))))) # label(fact_2184_rel__simps_I25_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3941 (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]. 9.41/9.50 3942 (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]. 9.41/9.50 3943 (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]. 9.41/9.50 3944 (all A_1 (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)))))) <-> zero_zero_real != A_1)) # label(fact_209_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3945 (all X_7 all Y_6 hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_7),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_6),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_7)),Y_6)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_7),Y_6)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_2456_power2__diff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3946 (all X_39 all Y_33 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_39),Y_33)) -> -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_33),X_39)))) # label(fact_1720_order__less__asym) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3947 (all N hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)) != zero_zero_real) # label(fact_3792_real__of__nat__fact__not__zero) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3948 (all Z_2 all Wa (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z_2),Wa)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Z_2),Wa)) & Wa != Z_2)) # label(fact_4646_less__rat__def) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3949 (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]. 9.41/9.50 3950 (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,Xa),Ya)) | Ya = Xa)) # label(fact_1962_order__le__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3951 (all C_41 all D_14 all A_108 all B_78 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_108),B_78)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_41),D_14)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_78)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_41)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_108),C_41)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_78),D_14)))))))) # label(fact_1351_mult__mono) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3952 (all A_161 all B_109 all C_63 all D_19 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_161),B_109)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_63),D_19)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_161),C_63)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_109),D_19))) # label(fact_801_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3953 (all X all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),P_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(standardRes(P_3),X))))) # label(fact_3119_StandardRes__lbound) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3954 (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]. 9.41/9.50 3955 (all X_16 all Y_12 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_16),Y_12)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_12),X_16)))) # label(fact_2039_linorder__le__less__linear) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3956 (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]. 9.41/9.50 3957 (all Xa all Ya (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,Xa),Xa)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ya),Ya))),zero_zero_real)) <-> Xa = zero_zero_real & zero_zero_real = Ya)) # label(fact_1585_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3958 (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]. 9.41/9.50 3959 (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,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,B_1),A_1)),E_2)),D_1) = C <-> 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))) # label(fact_2386_eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3960 (all N_58 hAPP_nat_real(semiri132038758t_real,N_58) = hAPP_int_real(number267125858f_real,hAPP_nat_int(semiri1621563631at_int,N_58))) # label(fact_205_number__of__int) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3961 (all I_1 (is_int(I_1) -> (I_1 != zero_zero_int -> hBOOL(hAPP_f448129468l_bool(finite_finite_int,collect_int(hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,dvd_dvd_int),I_1))))))) # label(fact_4762_finite__divisors__int) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3962 (all M_23 all N_47 hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_23),N_47)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(semiri132038758t_real,M_23)),N_47)) # label(fact_384_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3963 (all X all Y all Z_1 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Z_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y),Z_1))) # label(fact_101_zadd__left__commute) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3964 (all A_1 all B_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_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(div_div_int,A_1),B_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),zero_zero_int))))) # label(fact_2838_neg__imp__zdiv__nonneg__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3965 (all R_2 all A_3 all 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)) = hAPP_complex_complex(scaleR1652505878omplex(R_2),hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2))) # label(fact_5110_complex__scaleR) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3966 (all N hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N)))),zero_zero_int))) # label(fact_3495_negative__zless__0) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3967 (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(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_int,A_1),B_1)))) # label(fact_987_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3968 (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]. 9.41/9.50 3969 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hAPP_real_real(ln,X) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),one_one_real) -> X = one_one_real))) # label(fact_3357_ln__eq__minus__one) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3970 -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]. 9.41/9.50 3971 (all Xa hBOOL(hAPP_real_bool(deriv_real(sin,Xa),hAPP_real_real(cos,Xa)))) # label(fact_3867_DERIV__sin) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3972 (all N all I (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),I)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),N))))) # label(fact_3109_nat__one__le__power) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3973 (all A_1 all B_1 (B_1 = zero_zero_real & zero_zero_real = A_1 <-> hAPP_real_complex(hAPP_r265291036omplex(complex,A_1),B_1) = zero_zero_complex)) # label(fact_3951_Complex__eq__0) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3974 (all M all K_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M)),K_2))))) # label(fact_2518_le__add__diff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3975 (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]. 9.41/9.50 3976 (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]. 9.41/9.50 3977 (exists 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_4141_Re_Obounded) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3978 (all A_123 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_123),one_one_real) = A_123) # label(fact_1231_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3979 (all I all J_2 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),I))) # label(fact_277_not__add__less1) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3980 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,Xa)),one_one_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),one_one_real)))) # label(fact_3533_real__sqrt__le__1__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3981 (all Z1 all Z2 all Z3 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z2),Z3)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z1),Z2)),Z3)) # label(fact_2159_real__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3982 (all C_44 all B_81 all A_111 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_81),A_111)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_44),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_111),C_44)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_81),C_44)))))) # label(fact_1338_mult__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3983 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> (M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) -> hAPP_nat_nat(suc,zero_zero_nat) = N))) # label(fact_3096_mn__eq__m__one) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3984 (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]. 9.41/9.50 3985 (all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_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))) = hAPP_int_int(fact_fact_int,N))) # label(fact_4287_fact__reduce__int) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3986 (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]. 9.41/9.50 3987 (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]. 9.41/9.50 3988 (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]. 9.41/9.50 3989 (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]. 9.41/9.50 3990 (all Na all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),Ma)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,Na)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,Ma)),one_one_real))))) # label(fact_3726_nat__le__real__less) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3991 (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]. 9.41/9.50 3992 (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]. 9.41/9.50 3993 (all A_187 all C_76 all B_130 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_187),C_76)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_130),C_76))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_187),B_130)))) # label(fact_524_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3994 (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]. 9.41/9.50 3995 (all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y)) | X != Y)) # label(help_fequal_2_1_fequal_000tc__Nat__Onat_T) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3996 (all Ma all Na (Ma != Na <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),Ma)))) # label(fact_280_nat__neq__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3997 (all Code_numeral_3 zero_z126310315umeral != code_S1047413653umeral(Code_numeral_3)) # label(fact_4080_code__numeral_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 3998 (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]. 9.41/9.50 3999 (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]. 9.41/9.50 4000 (all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) -> -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))))) # label(fact_2633_dvd_Oless__not__sym) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 4001 (all X_14 all P_6 all 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) = 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))) # label(fact_2091_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 4002 (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]. 9.41/9.50 4003 (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)) -> A_3 = B_2))))))) # label(fact_2599_zcong__zless__imp__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 4004 (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)) -> (zero_zero_real != D -> (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)) -> hAPP_real_real(F,hAPP_real_real(G,Y_1)) = 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]. 9.41/9.50 4005 (all A_55 all B_37 (A_55 != B_37 -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_55),B_37)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_55),B_37))))) # label(fact_1992_order__neq__le__trans) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 4006 (all N_52 all A_197 (is_int(A_197) -> (A_197 != zero_zero_int -> zero_zero_int != hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_197),N_52)))) # label(fact_347_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 4007 (all I (is_int(I) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),zero_zero_int)) -> I = hAPP_int_int(abs_abs_int,I)) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),zero_zero_int)) -> hAPP_int_int(uminus_uminus_int,I) = hAPP_int_int(abs_abs_int,I)))) # label(fact_3489_zabs__def) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 4008 (all Ya all Xa (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Ya),Xa)) -> (Xa = Ya <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Xa),Ya))))) # label(fact_1685_order__antisym__conv) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 4009 (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,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_131),B_93)),E_3)),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))) # label(fact_1161_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 4010 (all Wa all Xa (hAPP_i769753017umeral(number1443263063umeral,Wa) = Xa <-> hAPP_i769753017umeral(number1443263063umeral,Wa) = Xa)) # label(fact_91_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 4011 (all D_3 all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),A_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_3),B_2))))) # label(fact_4026_coprime__divprod) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 4012 (all B_96 all A_136 (zero_z126310315umeral != A_136 -> (B_96 != zero_z126310315umeral -> zero_z126310315umeral != hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_136),B_96)))) # label(fact_1127_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 4013 (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]. 9.41/9.50 4014 (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]. 9.41/9.50 4015 (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]. 9.41/9.50 4016 (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]. 9.41/9.50 4017 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)),A_3)))) # label(fact_5009_gcd__le1__int) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 4018 (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]. 9.41/9.50 4019 (all M all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),M))))) # label(fact_4192_prime__dvd__square) # label(axiom) # label(non_clause). [assumption]. 9.41/9.50 4020 (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]. 9.41/9.50 4021 (all P_1 all A_1 all B_1 all Na hAPP_P1860904029l_real(produc595218619l_real(hAPP_f2072293513l_real(hAPP_f1763467369l_real(cOMBS_163651580l_real,hAPP_f1663701695l_real(hAPP_f774948957l_real(cOMBB_3698607l_real,cOMBS_902912708l_real),hAPP_f324002831l_real(hAPP_f677014621l_real(cOMBS_107495110l_real,hAPP_f1161935955l_real(hAPP_f328572565l_real(cOMBB_520243667l_real,cOMBS_794971624l_real),hAPP_f517738447l_real(hAPP_f1879982819l_real(cOMBB_731595089l_real,hAPP_f1076489843l_real(cOMBB_415306361l_real,if_Pro313124157l_real)),hAPP_f1884005689l_bool(hAPP_f172451459l_bool(cOMBB_421479067l_real,hAPP_f940061907l_bool(cOMBB_1795223523l_real,P_1)),hAPP_f320596107l_real(hAPP_f850851693l_real(cOMBS_193881978l_real,hAPP_f1119546819l_real(hAPP_f555557763l_real(cOMBB_422423973l_real,cOMBB_2107848501l_real),produc865579683l_real)),hAPP_r337325687l_real(hAPP_f1644353557l_real(cOMBC_1508019322l_real,hAPP_f1644353557l_real(hAPP_f1874145443l_real(cOMBB_1768400739l_real,cOMBC_real_real_real),hAPP_f1732848529l_real(hAPP_f1310479593l_real(cOMBB_2092576745l_real,hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real)),plus_plus_real))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))),hAPP_f1591894897l_real(hAPP_f644105375l_real(cOMBC_467724138l_real,hAPP_f1108539711l_real(hAPP_f1109256153l_real(cOMBB_44084907l_real,cOMBS_511527878l_real),hAPP_f823702159l_real(hAPP_f1040184293l_real(cOMBB_557258983l_real,hAPP_f2114902373l_real(cOMBB_237479429l_real,produc865579683l_real)),hAPP_r337325687l_real(hAPP_f1644353557l_real(cOMBC_1508019322l_real,hAPP_f1644353557l_real(hAPP_f1874145443l_real(cOMBB_1768400739l_real,cOMBC_real_real_real),hAPP_f1732848529l_real(hAPP_f1310479593l_real(cOMBB_2092576745l_real,hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real)),plus_plus_real))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),cOMBI_real)))),hAPP_f320596107l_real(hAPP_f850851693l_real(cOMBS_193881978l_real,hAPP_f1119546819l_real(hAPP_f555557763l_real(cOMBB_422423973l_real,cOMBB_2107848501l_real),produc865579683l_real)),hAPP_r337325687l_real(hAPP_f1644353557l_real(cOMBC_1508019322l_real,hAPP_f1644353557l_real(hAPP_f1874145443l_real(cOMBB_1768400739l_real,cOMBC_real_real_real),hAPP_f1732848529l_real(hAPP_f1310479593l_real(cOMBB_2092576745l_real,hAPP_f1960414057l_real(cOMBB_882763271l_real,inverse_divide_real)),plus_plus_real))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_n332733730l_real(bolzano_bisect(P_1,A_1,B_1),Na)) = hAPP_n332733730l_real(bolzano_bisect(P_1,A_1,B_1),hAPP_nat_nat(suc,Na))) # label(fact_4851_Bolzano__bisect_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4022 (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]. 9.41/9.51 4023 (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(times_times_nat,Y),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)) # label(fact_2113_zpower__zpower) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4024 (all B_2 all A_3 (A_3 != zero_zero_nat -> (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)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))))) # label(fact_4373_bezout__nat) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4025 (all Na (zero_zero_real = hAPP_nat_real(real_nat,Na) <-> zero_zero_nat = Na)) # label(fact_3700_real__of__nat__zero__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4026 (all P all Q all R hAPP_int_bool(hAPP_P252494982t_bool(P,R),hAPP_P1175774780nt_int(Q,R)) = hAPP_P603027463t_bool(hAPP_f892584630t_bool(hAPP_f1325804733t_bool(cOMBS_480885903t_bool,P),Q),R)) # label(help_COMBS_1_1_COMBS_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin_004) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4027 (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]. 9.41/9.51 4028 (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_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_176)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_176),one_one_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_176),N_39)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_176),N_40))))))) # label(fact_621_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4029 (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]. 9.41/9.51 4030 (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,hAPP_real_real(log(A_1),Xa)),hAPP_real_real(log(A_1),Ya))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya))))))) # label(fact_3445_log__less__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4031 (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(hAPP_int_fun_int_int(div_div_int,K_2),L_2)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),L_2)) -> hAPP_int_int(z3div(K_2),L_2) = hAPP_int_int(uminus_uminus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,K_2),hAPP_int_int(uminus_uminus_int,L_2)))))) # label(fact_3564_z3div__def) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4032 (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]. 9.41/9.51 4033 (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]. 9.41/9.51 4034 (all Ma all Na (zero_zero_nat = Na & hAPP_nat_nat(suc,zero_zero_nat) = Ma | Ma = zero_zero_nat & Na = hAPP_nat_nat(suc,zero_zero_nat) <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),Na) = hAPP_nat_nat(suc,zero_zero_nat))) # label(fact_3044_one__is__add) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4035 (all A_47 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,zero_zero_real),A_47)) -> A_47 = zero_zero_real)) # label(fact_2251_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4036 (all C_69 all D_20 all A_177 all B_121 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_177),B_121)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_69),D_20)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_177),C_69)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_121),D_20)))))) # label(fact_609_add__less__le__mono) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4037 (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,one_one_real),hAPP_real_real(exp_real,Xa))))) # label(fact_3406_one__le__exp__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4038 (all Z_2 all Wa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Wa)) | 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,Wa),Z_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(nat,Wa)),hAPP_int_nat(nat,Z_2)))))) # label(fact_2791_nat__le__eq__zle) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4039 (all A_17 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_17)) -> A_17 = hAPP_rat_rat(abs_abs_rat,A_17))) # label(fact_2714_abs__of__pos) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4040 (all Y (zero_zero_nat != Y -> -(all Nat_1 hAPP_nat_nat(suc,Nat_1) != Y))) # label(fact_4000_nat_Oexhaust) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4041 (all N all M ((zero_zero_nat = M -> N = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) & (zero_zero_nat != M -> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)))) # label(fact_3118_add__eq__if) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4042 (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]. 9.41/9.51 4043 (all M_14 all A_90 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)) = 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)) # label(fact_1483_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4044 (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]. 9.41/9.51 4045 (all P_3 all 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)))))) = hAPP_int_int(multInv(P_3),X)) # label(fact_2909_MultInv__def) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4046 (all X_6 all N_11 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_11)) -> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_6),N_11) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_11),one_one_nat))),X_6))) # label(fact_2555_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4047 (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,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa),Xa)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ya),Ya)))) <-> zero_zero_real != Ya | Xa != zero_zero_real)) # label(fact_1591_sum__squares__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4048 (all A_3 all B_2 all C_1 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),C_1))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),hAPP_int_int(div_mod_int(B_2),C_1))),C_1)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2)),C_1)) # label(fact_3088_zdiv__zmult1__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4049 (all K_2 all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N))) # label(fact_4301_gcd__mult__distrib__nat) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4050 (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]. 9.41/9.51 4051 (all B_73 all A_102 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_102)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_73),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_73),A_102)),zero_zero_nat))))) # label(fact_1392_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4052 (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]. 9.41/9.51 4053 (all X all Y hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,hAPP_complex_complex(cnj,X)),hAPP_complex_complex(cnj,Y)) = hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,X),Y))) # label(fact_4064_complex__cnj__divide) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4054 (all K_2 all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),I)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),J_2)))))) # label(fact_2114_mult__less__mono2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4055 (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]. 9.41/9.51 4056 (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)))))) <-> A_1 != zero_zero_rat)) # label(fact_208_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4057 (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]. 9.41/9.51 4058 (all Xa all Ya (Ya = Xa <-> hAPP_complex_complex(cnj,Xa) = hAPP_complex_complex(cnj,Ya))) # label(fact_4069_complex__cnj__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4059 (all N_14 all M_8 all X_11 all Y_8 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_11),Y_8)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_14),M_8)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_11),N_14)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_8),M_8)))))) # label(fact_2368_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4060 (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]. 9.41/9.51 4061 (all C_24 all D_9 all A_78 all B_51 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_78),B_51)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_24),D_9)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_51)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_24)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_78),C_24)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_51),D_9)))))))) # label(fact_1571_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4062 (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]. 9.41/9.51 4063 (all X (hBOOL(hAPP_nat_bool(even_odd_even_nat,X)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,X)),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))))) # label(fact_3802_even__nat__plus__one__div__two) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4064 (all X_9 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_9),one_one_int)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_9),one_one_int)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_9),X_9)),one_one_int)) # label(fact_2436_real__squared__diff__one__factored) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4065 (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)) & -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ya),Xa)))) # label(fact_2025_less__le__not__le) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4066 (all N (hAPP_nat_nat(div_mod_nat(N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls))) -> -hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) # label(fact_3822_lemma__odd__mod__4__div__2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4067 (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]. 9.41/9.51 4068 (all B_74 all A_103 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_103)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_74),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_103),B_74)),zero_zero_int))))) # label(fact_1387_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4069 (all A_44 all B_33 all K_6 (A_44 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_33),K_6) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_33),A_44)))) # label(fact_2265_dvdI) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4070 (all K all Ma all Na (Ma = Na <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),Na))) # label(fact_284_nat__add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4071 (all Ma all X_1 all Xa_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcongm(Ma),X_1),Xa_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),Xa_1),Ma)))) # label(fact_5152_zcongm__def) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4072 (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]. 9.41/9.51 4073 (all A_162 all B_110 all C_64 (hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_162),C_64) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_162),B_110) -> C_64 = B_110)) # label(fact_790_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4074 (all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) <-> Na != zero_zero_nat)) # label(fact_304_neq0__conv) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4075 (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]. 9.41/9.51 4076 (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_eq_real,hAPP_nat_real(real_nat,N_1)),R_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_2),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_1)))))))) # label(fact_3780_reals__Archimedean6a) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4077 (all L all U hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(L),U)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,U)),L)) # label(fact_5070_card__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4078 (all A_202 (is_int(A_202) -> A_202 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_202),hAPP_int_int(number_number_of_int,pls)))) # label(fact_179_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4079 (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]. 9.41/9.51 4080 (all A_157 all C_59 all D_18 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_157),C_59)),D_18) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_157),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_59),D_18))) # label(fact_836_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4081 (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]. 9.41/9.51 4082 (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]. 9.41/9.51 4083 (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]. 9.41/9.51 4084 (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_eq_real,Xa),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa)),hAPP_real_real(hAPP_n546711566l_real(root,Na),Ya)))))) # label(fact_3896_real__root__le__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4085 (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]. 9.41/9.51 4086 (all A_3 all B_2 hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_nat_int(semiri1621563631at_int,A_3)),hAPP_nat_int(semiri1621563631at_int,B_2)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),B_2))) # label(fact_2886_zdiv__int) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4087 (all A_53 all N_20 hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_53),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_20)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_53),N_20)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_2133_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4088 (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(inverse_inverse_real,Xa))))) # label(fact_3869_DERIV__ln) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4089 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(div_mod_nat(M),N)),N)))) # label(fact_3188_mod__le__divisor) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4090 (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]. 9.41/9.51 4091 (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]. 9.41/9.51 4092 (all Lx_4 all Ly_4 all Rx_4 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_4),Ly_4)),Rx_4) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_4),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ly_4),Rx_4))) # label(fact_1060_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4093 (all N all M quickc495462417de_int(N,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))) # label(fact_4267_div__mod__code__int__def) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4094 (all P all Q all R hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,P),Q),R) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4095 (all X all Y hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(legacy_zgcd(X),Y)))) # label(fact_4502_zgcd__geq__zero) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4096 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K)),pls)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),pls)))) # label(fact_728_rel__simps_I27_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4097 (all B_89 all A_119 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_119)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_89)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_119),B_89)))))) # label(fact_1294_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4098 (all A_1 all C all B_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_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))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_1),B_1)))) # label(fact_545_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4099 (all B_15 all D_6 all A_15 all C_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(abs_abs_int,A_15)),C_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(abs_abs_int,B_15)),D_6)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,A_15)),hAPP_int_int(abs_abs_int,B_15))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_5),D_6)))))) # label(fact_2725_abs__mult__less) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4100 (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]. 9.41/9.51 4101 (all B_88 all A_118 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_118)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_88),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_118),B_88)),zero_zero_rat))))) # label(fact_1298_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4102 (all X_36 all Y_30 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_36),Y_30)) -> X_36 != Y_30)) # label(fact_1794_order__less__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4103 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,M))) -> hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N)),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N))))) # label(fact_3808_fact__diff__Suc) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4104 (all R_2 all X hAPP_complex_complex(cnj,hAPP_complex_complex(scaleR1652505878omplex(R_2),X)) = hAPP_complex_complex(scaleR1652505878omplex(R_2),hAPP_complex_complex(cnj,X))) # label(fact_5108_cnj_OscaleR) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4105 (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]. 9.41/9.51 4106 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,X)),hAPP_int_real(real_int,Y)) = hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y))) # label(fact_4395_real__of__int__add) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4107 (all Wa all Z_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Wa),Z_2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_2),one_one_int))))) # label(fact_777_zle__add1__eq__le) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4108 (all A_89 all M_13 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_89),one_one_real)),M_13) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_89),M_13)),M_13)) # label(fact_1488_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4109 (all N all K_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),N)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(legacy_zgcd(M),N)))))) # label(fact_4492_zgcd__greatest) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4110 (all Xa all Ma (hAPP_nat_nat(suc,zero_zero_nat) = Xa | Ma = zero_zero_nat <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Xa),Ma) = hAPP_nat_nat(suc,zero_zero_nat))) # label(fact_3070_nat__power__eq__Suc__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4111 (all X all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (A_3 != one_one_real -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_real_real(log(A_3),hAPP_real_real(inverse_inverse_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(log(A_3),X)))))) # label(fact_3862_log__inverse) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4112 (all Lx_3 all Ly_3 all Rx_3 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_3),Ly_3)),Rx_3) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_3),Rx_3)),Ly_3)) # label(fact_1065_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4113 (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_real,zero_zero_real),K_4)))) # label(fact_4085_cnj_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4114 (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]. 9.41/9.51 4115 (all A_138 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,zero_zero_real),A_138) = zero_zero_real) # label(fact_1110_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4116 (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]. 9.41/9.51 4117 (all Xa all Ya (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya)) | Xa = Ya <-> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Ya)))) # label(fact_2016_order__le__less) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4118 (all P all Q all R hAPP_n1699378549t_bool(P,hAPP_nat_nat(Q,R)) = hAPP_n1699378549t_bool(hAPP_f618557131t_bool(hAPP_f1505651103t_bool(cOMBB_800536526ol_nat,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4119 (all A_3 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2))) -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2)) = negDivAlg(A_3,B_2)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),B_2))) -> negDivAlg(A_3,B_2) = hAPP_P1975530577nt_int(adjust(B_2),negDivAlg(A_3,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),B_2)))))) # label(fact_3286_negDivAlg__eqn) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4120 (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]. 9.41/9.51 4121 (all Na all Ma (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,Na)),hAPP_nat_real(real_nat,Ma))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),Ma)))) # label(fact_3702_real__of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4122 (all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = N)) # label(fact_3117_Suc__pred_H) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4123 (all N N = natfloor(hAPP_nat_real(real_nat,N))) # label(fact_3693_natfloor__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4124 (all P all Q all R hAPP_r1195171167l_real(hAPP_f1736103445l_real(hAPP_f1908301369l_real(cOMBS_902912708l_real,P),Q),R) = hAPP_P1860904029l_real(hAPP_r938550413l_real(P,R),hAPP_r1195171167l_real(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__RealDef__Oreal_000tc__prod_Itc__RealDef__Oreal_Mtc__) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4125 (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]. 9.41/9.51 4126 (all Na all Ma (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,Na)),hAPP_int_real(real_int,Ma))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,Na)),Ma)))) # label(fact_4443_number__of__le__real__of__int__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4127 (all Xa all Ya (-hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Xa),Ya)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Xa),Ya)) <-> Xa = Ya))) # label(fact_2000_linorder__antisym__conv1) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4128 (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]. 9.41/9.51 4129 (all P all Q all R hAPP_int_real(P,hAPP_int_int(Q,R)) = hAPP_int_real(hAPP_f563293398t_real(hAPP_f134192053t_real(cOMBB_int_real_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__Int__Oint_000tc__RealDef__Oreal_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4130 (all P all Q all R hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(hAPP_f465821005nt_int(cOMBC_1964283556nt_int,P),Q),R) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__Int__Oint_000tc__prod_Itc__Int__Oin) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4131 (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]. 9.41/9.51 4132 (all P_3 all R_2 all S_1 all Q_1 (is_int(Q_1) & is_int(S_1) -> (Q_1 != zero_zero_int -> (S_1 != zero_zero_int -> (normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,R_2),S_1)) = normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_1)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_2),Q_1) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,P_3),S_1)))))) # label(fact_4661_normalize__crossproduct) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4133 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(tan,X)),zero_zero_real))))) # label(fact_3768_tan__less__zero) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4134 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_real_real(abs_abs_real,X))))) # label(fact_2849_abs__add__one__gt__zero) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4135 (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]. 9.41/9.51 4136 (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]. 9.41/9.51 4137 (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]. 9.41/9.51 4138 (all A_3 all B_2 hAPP_int_int(div_mod_int(hAPP_nat_int(semiri1621563631at_int,A_3)),hAPP_nat_int(semiri1621563631at_int,B_2)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(div_mod_nat(A_3),B_2))) # label(fact_3186_zmod__int) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4139 (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]. 9.41/9.51 4140 (all Va all Wa (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))) <-> -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))))) # label(fact_689_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4141 (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]. 9.41/9.51 4142 (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_rat,one_one_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),A_1))))) # label(fact_4720_one__less__Fract__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.51 4143 (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]. 9.41/9.52 4144 (all Ma all K (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,Ma),K)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(abs_abs_real,Ma)),K)))) # label(fact_2695_abs__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4145 (all B_74 all A_103 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_103)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_74),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_103),B_74)),zero_zero_rat))))) # label(fact_1382_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4146 (all X_60 all Y_50 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_60),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_50),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_50)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_60),Y_50))))) # label(fact_890_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4147 (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]. 9.41/9.52 4148 (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]. 9.41/9.52 4149 (all Z_1 all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),X))))) # label(fact_2635_dvd_Ole__less__trans) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4150 (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]. 9.41/9.52 4151 (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]. 9.41/9.52 4152 (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]. 9.41/9.52 4153 (all Xa all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) -> (zero_zero_real = Xa <-> zero_zero_real = hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa)))) # label(fact_3898_real__root__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4154 (all Na all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,quickcheck_int_of(Na)),quickcheck_int_of(Ma))) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Na),Ma)))) # label(fact_4253_less__code__int__def) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4155 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),hAPP_nat_nat(fact_fact_nat,N)))))) # label(fact_3842_dvd__fact__nat) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4156 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K1)),hAPP_int_int(bit0,K2))))) # label(fact_111_less__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4157 (all C all A_1 all B_1 (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_1)) -> (B_1 = C -> 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]. 9.41/9.52 4158 (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]. 9.41/9.52 4159 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N)))) # label(fact_3053_Suc__leI) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4160 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real)) -> hAPP_real_real(sin,hAPP_real_real(arccos,X)) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3669_sin__arccos) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4161 (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]. 9.41/9.52 4162 (all A_3 all C_1 all D_3 all B_2 (is_int(B_2) & is_int(D_3) -> (zero_zero_int != B_2 -> (D_3 != zero_zero_int -> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C_1),D_3)) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),D_3)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_1),B_2))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),D_3)))))) # label(fact_4718_add__rat) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4163 (all P all Q all R hAPP_r653725362t_bool(hAPP_f1927674023t_bool(hAPP_f1261108921t_bool(cOMBB_172850899l_real,P),Q),R) = hAPP_f729588221t_bool(P,hAPP_r1562208522t_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__Int__Oint_Mtc__fun_Itc__fun_Itc__Nat__Onat_) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4164 (all X all Y all Xa_2 all Ya_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Ya_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) # label(fact_3559_real__sqrt__sum__squares__mult__ge__zero) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4165 (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]. 9.41/9.52 4166 (all V_25 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))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_25)),one_one_int)) # label(fact_30_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4167 (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_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,A_1),B_1))))) # label(fact_1374_mult__less__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4168 (all A_126 A_126 = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,one_on1645066479umeral),A_126)) # label(fact_1209_mult__1) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4169 (all Y_4 all X_4 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_4)) -> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(abs_abs_real,Y_4)),X_4) = hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_4),X_4)))) # label(fact_2752_abs__mult__pos) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4170 (all Na all P_1 (-hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(P_1,Na)) -> (exists K_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_1),Na)) & (all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),K_1)) -> -hBOOL(hAPP_nat_bool(P_1,I_2)))) & hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_1),one_one_nat)))))))) # label(fact_2059_ex__least__nat__less) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4171 (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]. 9.41/9.52 4172 (all Ma all Na (hAPP_nat_nat(suc,zero_zero_nat) = Na & zero_zero_nat = Ma | Ma = hAPP_nat_nat(suc,zero_zero_nat) & Na = 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]. 9.41/9.52 4173 (all X all Y (Y != X -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y),X))))) # label(fact_287_linorder__neqE__nat) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4174 (all A_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),Ma)) -> hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,hAPP_i1948725293t_bool(bnorRset(A_1),Ma)),rsetR(Ma))))) # label(fact_4596_Bnor__in__RsetR) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4175 (all X_7 all Y_6 hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_7),Y_6)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,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_nat_int(hAPP_int_fun_nat_int(power_power_int,X_7),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_6),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),X_7)),Y_6))) # label(fact_2459_power2__diff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4176 (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]. 9.41/9.52 4177 (all A_150 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_150),zero_zero_rat) = A_150) # label(fact_915_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4178 (all Y_15 all X_19 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_15),X_19)) -> -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_19),Y_15)))) # label(fact_1980_leD) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4179 (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]. 9.41/9.52 4180 (all I all J_2 all K_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),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),K_2)),J_2)) # label(fact_2473_diff__commute) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4181 (all A_83 all C_29 all B_56 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_83),C_29)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_56),C_29))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_29)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_83),B_56))))) # label(fact_1541_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4182 (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]. 9.41/9.52 4183 (all X all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)),X) = hAPP_real_real(hAPP_n546711566l_real(root,M),hAPP_real_real(hAPP_n546711566l_real(root,N),X))))) # label(fact_3902_real__root__mult__exp) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4184 (all X all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)) = hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(abs_abs_real,X)))) # label(fact_3908_real__root__abs) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4185 (all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3)) -> (exists P_4 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_4),A_3)) & hBOOL(hAPP_int_bool(zprime,P_4)) & is_int(P_4))))) # label(fact_2617_zprime__factor__exists) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4186 (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]. 9.41/9.52 4187 (all C_22 all A_72 all B_49 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_72),B_49)) -> (B_49 = C_22 -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_72),C_22))))) # label(fact_1670_ord__le__eq__trans) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4188 (all C_22 all A_72 all B_49 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_72),B_49)) -> (B_49 = C_22 -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_72),C_22))))) # label(fact_1664_ord__le__eq__trans) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4189 (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]. 9.41/9.52 4190 (all C_21 all A_71 all B_48 (B_48 = A_71 -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_21),B_48)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_21),A_71))))) # label(fact_1674_xt1_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4191 (all X (zero_zero_real != hAPP_real_real(cos,X) -> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(cos,X))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(tan,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_3848_tan__sec) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4192 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,X)),hAPP_int_real(real_int,Y)) = hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y))) # label(fact_4391_real__of__int__diff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4193 (all Z_12 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Z_12),Z_12) = 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_12)) # label(fact_1889_mult__2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4194 (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]. 9.41/9.52 4195 (all M all N all I (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),I)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,I),N))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))))) # label(fact_300_nat__power__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4196 (all M all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N)) -> hAPP_int_int(legacy_zgcd(M),N) = hAPP_int_int(legacy_zgcd(N),hAPP_int_int(div_mod_int(M),N)))) # label(fact_4514_zgcd__non__0) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4197 (all A_122 all N_29 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_122),N_29)),A_122) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_122),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_122),N_29))) # label(fact_1239_power__commutes) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4198 (all Z_1 all W (is_int(W) & is_int(Z_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),W)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z_1)) -> W = Z_1)))) # label(fact_503_zle__antisym) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4199 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> (exists K_1 N = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_1))))) # label(fact_3169_less__imp__Suc__add) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4200 (all A_132 all B_94 all 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)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_132),B_94)),C_50)) # label(fact_1155_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4201 (all P_5 hAPP_i1948725293t_bool(sr,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)))) # label(fact_2969_SR__def) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4202 (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,cos),G),Xa),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,hAPP_real_real(G,Xa)))),Ma))))) # label(fact_4764_DERIV__fun__cos) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4203 (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,B_4),C_3))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_4),C_3)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),B_4)) & hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),B_4))) -> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_2),C_3))))) -> (-hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_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]. 9.41/9.52 4204 (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]. 9.41/9.52 4205 (all K_2 all L_2 (zero_zero_int != hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,K_2)),L_2) -> hAPP_int_int(div_mod_int(K_2),L_2) != zero_zero_int)) # label(fact_3482_zmod__zminus1__not__zero) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4206 (all K all I_1 all J hAPP_n1699378549t_bool(ord_at4362885an_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K)),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_at4362885an_nat(I_1),J))) # label(fact_4905_image__add__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4207 (all B_40 all A_58 (is_int(A_58) & is_int(B_40) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_40),A_58)) -> (A_58 != B_40 -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_40),A_58)))))) # label(fact_1953_xt1_I11_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4208 (all W hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),W))) # label(fact_2081_real__le__refl) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4209 (all X (one_one_real = hAPP_real_real(cos,X) -> zero_zero_real = hAPP_real_real(sin,X))) # label(fact_3595_cos__one__sin__zero) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4210 (all Y_42 all X_48 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_48)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_42)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_42),one_one_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_42),X_48)),X_48)))))) # label(fact_1596_mult__left__le__one__le) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4211 (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,Xa),zEven)) <-> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Xa),Na)),zEven))))) # label(fact_3367_power__preserves__even) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4212 (all N all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N)),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N)))) -> hAPP_int_nat(nat,archim1246769320r_real(X)) = N))) # label(fact_4031_floor__eq3) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4213 (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]. 9.41/9.52 4214 (all D_1 all Ta (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),Ta)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(uminus_uminus_int,D_1)),Ta)))) # label(fact_3467_uminus__dvd__conv_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4215 (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]. 9.41/9.52 4216 (all Wa all Z_2 (is_int(Z_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_2)) <-> (exists N_1 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Wa),hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_1))) = Z_2)))) # label(fact_3106_zless__iff__Suc__zadd) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4217 (all P all Q all R hAPP_n1865633855at_nat(hAPP_f837737749at_nat(hAPP_f999082675at_nat(cOMBB_348438404at_nat,P),Q),R) = hAPP_n1865633855at_nat(P,hAPP_nat_nat(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__prod_Itc__N) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4218 (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]. 9.41/9.52 4219 (all A_3 all B_2 all 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)) = 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)) # label(fact_3037_zdiv__zadd1__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4220 (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)) -> (B_39 != A_57 -> 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]. 9.41/9.52 4221 (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)))))) & 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_3585_arcsin__bounded) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4222 (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]. 9.41/9.52 4223 (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]. 9.41/9.52 4224 (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,one_one_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),A_1))))) # label(fact_4726_one__le__Fract__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4225 (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]. 9.41/9.52 4226 (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]. 9.41/9.52 4227 (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]. 9.41/9.52 4228 (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]. 9.41/9.52 4229 (all B_40 all A_58 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_40),A_58)) -> (A_58 != B_40 -> 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]. 9.41/9.52 4230 (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]. 9.41/9.52 4231 (all X all Y hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X),Y) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(re,X)),hAPP_complex_real(re,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(im,X)),hAPP_complex_real(im,Y)))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(re,X)),hAPP_complex_real(im,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(im,X)),hAPP_complex_real(re,Y))))) # label(fact_4135_complex__mult__def) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4232 (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]. 9.41/9.52 4233 (all B_1 all A_1 all D_1 (is_int(D_1) -> (D_1 = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_1),B_1) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),D_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),B_1)) & (all E (is_int(E) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,E),B_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,E),A_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,E),D_1))))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),A_1))))) # label(fact_5008_gcd__unique__int) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4234 (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]. 9.41/9.52 4235 (all A_1 all E_2 all C all B_1 all D_1 (is_int(C) -> (C = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_1),A_1)),E_2)),D_1) <-> 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) = 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)))) # label(fact_2388_eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4236 (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]. 9.41/9.52 4237 (all A_1 hBOOL(hAPP_complex_bool(isCont_complex_real(re),A_1))) # label(fact_5165_Re_OisCont) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4238 (all P_1 all I_1 all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),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,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_2868_int__le__induct) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4239 (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]. 9.41/9.52 4240 (all A_1 hBOOL(hAPP_complex_bool(isCont156215680omplex(cnj),A_1))) # label(fact_5163_cnj_OisCont) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4241 (all A_3 all B_2 all C_1 all D_3 hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),C_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_2),D_3)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)),hAPP_real_complex(hAPP_r265291036omplex(complex,C_1),D_3))) # label(fact_3957_complex__add) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4242 (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]. 9.41/9.52 4243 (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]. 9.41/9.52 4244 (all Xa all Na (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))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Xa)) | zero_zero_nat = Na)) # label(fact_302_nat__zero__less__power__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4245 (all B_119 all A_173 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_173),zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_119),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_173),B_119)),zero_zero_nat))))) # label(fact_650_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4246 (all R_2 rcis(R_2,zero_zero_real) = of_real_complex(R_2)) # label(fact_4257_rcis__zero__arg) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4247 (all A_3 all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),archim1246769320r_real(X))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,A_3)),X)))) # label(fact_4417_real__le__floor) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4248 (all V_25 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_25)),one_one_real) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_25),hAPP_int_int(bit1,pls)))) # label(fact_29_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4249 (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]. 9.41/9.52 4250 (all Na ((exists M_1 Na = hAPP_nat_nat(suc,M_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)))) # label(fact_3041_gr0__conv__Suc) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4251 (all A_1 all B_1 (zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_1),B_1) <-> B_1 = A_1)) # label(fact_2317_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4252 (all Xa all Ya (zero_zero_rat = Xa & zero_zero_rat = Ya <-> 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]. 9.41/9.52 4253 (all P_1 all A_1 all B_1 hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_1),B_1) = hAPP_n332733730l_real(bolzano_bisect(P_1,A_1,B_1),zero_zero_nat)) # label(fact_4243_Bolzano__bisect_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4254 (all A_155 all C_57 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_57),A_155) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_155),C_57)) # label(fact_845_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4255 (all A_126 A_126 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,one_one_complex),A_126)) # label(fact_1208_mult__1) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4256 (all A_57 all B_39 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_57),B_39)) -> (B_39 != A_57 -> 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]. 9.41/9.52 4257 (all Xa all P_5 (hAPP_int_int(standardRes(P_5),Xa) != zero_zero_int <-> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Xa),zero_zero_int),P_5)))) # label(fact_3075_StandardRes__prop3) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4258 (all Z_1 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Z_1)))) # label(fact_736_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4259 (all M all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N)),N))) # label(fact_4298_gcd__dvd2__nat) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4260 (all Na all 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))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Na),Ma)))) # label(fact_4446_int__le__real__less) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4261 (all X hAPP_real_real(sin,X) = hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),pi)))) # label(fact_3644_sin__periodic) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4262 (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]. 9.41/9.52 4263 (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]. 9.41/9.52 4264 (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]. 9.41/9.52 4265 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,Xa)),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))))) # label(fact_3439_real__0__less__add__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4266 (all X_15 all Y_11 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_15),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_11),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_15)),Y_11)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_15),Y_11)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_2054_power2__sum) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4267 (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]. 9.41/9.52 4268 (all 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))) & (-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)))) # label(fact_3740_real__of__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4269 (all F all A (hBOOL(hAPP_f448129468l_bool(nat_nat_set,A)) -> ((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(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(F,X_1)))))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_f1594865479ol_int(hAPP_f1926459811ol_int(big_co230513141nt_int,F),A)))))) # label(fact_4944_transfer__nat__int__sum__prod__closure_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.52 4270 (all A_122 all N_29 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_122),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_122),N_29)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_122),N_29)),A_122)) # label(fact_1236_power__commutes) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4271 (all A_135 all B_95 (hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_135),B_95) = zero_zero_real -> zero_zero_real = B_95 | zero_zero_real = A_135)) # label(fact_1135_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4272 (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]. 9.41/9.53 4273 (all A_133 hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_133),zero_z126310315umeral) = zero_z126310315umeral) # label(fact_1148_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4274 (all Ma all Na (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)))) # label(fact_359_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4275 (all A_203 (is_int(A_203) -> A_203 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,pls)),A_203))) # label(fact_175_add__numeral__0) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4276 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)))) # label(fact_1030_nat__le__linear) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4277 (all Xa all Ya all B_1 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),B_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) <-> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_1),Xa)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_1),Ya)))))) # label(fact_631_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4278 (all K_2 hAPP_int_int(pred,hAPP_int_int(bit1,K_2)) = hAPP_int_int(bit0,K_2)) # label(fact_4040_pred__Bit1) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4279 (all Z_4 all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_4)) -> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Z_1),Z_4)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat,Z_1)),hAPP_int_nat(nat,Z_4))))) # label(fact_2794_nat__add__distrib) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4280 (all N_36 all A_167 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_167)) -> 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_167),N_36))))) # label(fact_701_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4281 (all Y exists X_1 ((all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) & hAPP_real_real(tan,Y_1) = Y & 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)) -> X_1 = Y_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_3776_tan__total) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4282 (all A_1 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),hAPP_nat_real(real_nat,A_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(Xa)),A_1))))) # label(fact_3730_natceiling__le__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4283 (all A_57 all B_39 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_57),B_39)) -> (B_39 != A_57 -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_57),B_39))))) # label(fact_1954_order__le__neq__trans) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4284 (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(power_power_nat,hAPP_int_nat(number_number_of_nat,V_1)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N)) & (-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)),N) = hAPP_int_nat(nat,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,V_1)),N))))) # label(fact_3272_power__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4285 (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]. 9.41/9.53 4286 (all Xa all Ya all B_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),B_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) <-> 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)))))) # label(fact_632_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4287 (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]. 9.41/9.53 4288 (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) = zero_zero_nat) & (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_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)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)))) # label(fact_3120_div__if) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4289 (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]. 9.41/9.53 4290 (all C_53 all A_142 all B_100 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_142),B_100)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_53),A_142)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_53),B_100))))) # label(fact_970_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4291 (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]. 9.41/9.53 4292 (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]. 9.41/9.53 4293 (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]. 9.41/9.53 4294 (all A_158 all B_106 all 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)) = 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)) # label(fact_831_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4295 (all W hAPP_int_complex(number528085621omplex,W) = hAPP_complex_complex(cnj,hAPP_int_complex(number528085621omplex,W))) # label(fact_4077_complex__cnj__number__of) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4296 (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) & (zero_zero_nat = N_55 -> 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]. 9.41/9.53 4297 (all X all M hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,hAPP_int_int(div_mod_int(X),M))),M) = hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,X)),M)) # label(fact_3471_zminus__zmod) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4298 (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]. 9.41/9.53 4299 (all P_1 all Na all K (is_int(Na) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K)) -> ((all I_2 all J_1 (is_int(I_2) & is_int(J_1) -> (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,J_1),K)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_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,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Na),K)),hAPP_int_int(div_mod_int(Na),K))))))) # label(fact_3150_split__pos__lemma) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4300 (all A_154 all N_32 all B_105 (hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_154),N_32) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_105),N_32) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_154)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_105)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32)) -> B_105 = A_154))))) # label(fact_861_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4301 (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,C),D_1)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B_1))))) # label(fact_2337_diff__eq__diff__less) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4302 (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]. 9.41/9.53 4303 (all Xa (Xa = one_one_nat <-> one_one_nat = Xa)) # label(fact_855_one__reorient) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4304 (all Z_1 all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),X))))) # label(fact_2626_dvd_Oless__le__trans) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4305 (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]. 9.41/9.53 4306 (all Ma all Na (zero_zero_nat != Na & Ma = zero_zero_nat <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Ma),Na) = zero_zero_nat)) # label(fact_1005_nat__power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4307 (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]. 9.41/9.53 4308 (all N_58 hAPP_nat_int(semiri1621563631at_int,N_58) = hAPP_int_int(number_number_of_int,hAPP_nat_int(semiri1621563631at_int,N_58))) # label(fact_206_number__of__int) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4309 (all Z1 all Z2 all Z3 hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),Z2)),Z3) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),Z3))) # label(fact_1087_zmult__assoc) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4310 (all Xa all Ya (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Ya),Xa)) <-> -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Xa),Ya)))) # label(fact_2045_linorder__not__less) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4311 (all X_6 all N_11 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_11)) -> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_11),one_one_nat))),X_6) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_6),N_11))) # label(fact_2558_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4312 (all N all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(nat,Z_1)),N) = hAPP_int_nat(nat,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Z_1),N)))) # label(fact_2799_nat__power__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4313 (all U all Ma all Na all J all I_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1)) -> (Na = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I_1),J)),U)),Ma) <-> 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) = 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)))) # label(fact_2539_nat__eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4314 (all A_20 all N_9 hAPP_real_real(abs_abs_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_20),N_9)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(abs_abs_real,A_20)),N_9)) # label(fact_2688_power__abs) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4315 (all X_8 all N_12 (one_on1684967323de_int = X_8 | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_12)) -> 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]. 9.41/9.53 4316 (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]. 9.41/9.53 4317 (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]. 9.41/9.53 4318 (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]. 9.41/9.53 4319 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),Na)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),Ma)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,zero_zero_nat)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),Na))))) # label(fact_3102_one__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4320 (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]. 9.41/9.53 4321 (all M all K_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(fact_fact_int,M)),hAPP_int_int(fact_fact_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),K_2)))))) # label(fact_4282_fact__mono__int__aux) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4322 (all X_59 all Y_49 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_59),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_49),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_49)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_59),Y_49))))) # label(fact_993_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4323 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),hAPP_int_int(bit1,K))))) # label(fact_725_rel__simps_I22_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4324 (all Ma all K (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,Ma),hAPP_real_real(abs_abs_real,K))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,Ma),K)))) # label(fact_2698_dvd__abs__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4325 (all Y all X (hBOOL(hAPP_int_bool(even_odd_even_int,X)) -> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y))))) # label(fact_3234_even__times__anything) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4326 (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]. 9.41/9.53 4327 (all A_3 all B_2 (zero_zero_nat != hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2) -> (exists A_4 exists B_3 (one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_4),B_3) & hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)) = B_2 & A_3 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_4),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)))))) # label(fact_4372_gcd__coprime__exists__nat) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4328 (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)),zEven)) -> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven)) | hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ya),zEven)))) # label(fact_3338_EvenOdd_Oeven__product) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4329 (all P all Q all R hAPP_f1356864277nt_int(hAPP_i744251628nt_int(P,R),Q) = hAPP_int_fun_int_int(hAPP_f1760145644nt_int(hAPP_f808647893nt_int(cOMBC_736024425nt_int,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Int__Oint_000tc__fun_Itc__Int__Oint_Mtc__Int__Oint_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4330 (all A_1 (zero_zero_nat != A_1 <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_1),zero_zero_nat)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_1)))) # label(fact_2961_gcd__lcm__complete__lattice__nat_Oless__top) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4331 (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]. 9.41/9.53 4332 (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]. 9.41/9.53 4333 (all A_1 all B_1 (B_1 = A_1 <-> hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_1),B_1) = zero_zero_complex)) # label(fact_2318_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4334 (all A_156 all C_58 all D_17 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_156),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_58),D_17)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_58),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_156),D_17))) # label(fact_841_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4335 (all C_35 all D_13 all A_92 all B_63 all R_6 (R_6 != zero_zero_real -> (D_13 != C_35 & B_63 = A_92 -> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_92),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_6),C_35)) != hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_63),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_6),D_13))))) # label(fact_1447_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4336 (all R_2 hAPP_real_real(abs_abs_real,R_2) = hAPP_real_real(norm_norm_real,R_2)) # label(fact_3426_real__norm__def) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4337 (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(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_4792_DERIV__cos__cos__mult3) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4338 (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]. 9.41/9.53 4339 (all Ma (hAPP_int_int(legacy_zgcd(zero_zero_int),Ma) = one_one_int <-> one_one_int = hAPP_int_int(abs_abs_int,Ma))) # label(fact_4518_zgcd__0__1__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4340 (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(suc,Na)),Ma)))) # label(fact_3019_not__less__eq__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4341 (all C_18 all A_68 all B_45 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_68),B_45)) -> (C_18 = B_45 -> 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]. 9.41/9.53 4342 (all A_1 all C all B_1 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_1),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_1),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,C),zero_zero_rat)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_1),A_1)))) # label(fact_1367_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4343 (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]. 9.41/9.53 4344 (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]. 9.41/9.53 4345 (all A_3 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),hAPP_nat_nat(suc,zero_zero_nat)))) # label(fact_4021_coprime__Suc0) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4346 (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]. 9.41/9.53 4347 (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]. 9.41/9.53 4348 (all Xa archim791455193or_rat(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)))))) # label(fact_4890_floor__rat__def) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4349 (all X hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_883_power2__ge__self) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4350 (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]. 9.41/9.53 4351 (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]. 9.41/9.53 4352 (all A_127 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),A_127) = A_127) # label(fact_1204_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4353 (all B_1 all Q_2 all Ra hAPP_P1975530577nt_int(adjust(B_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),Ra)) = hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ra),B_1))),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Q_2)),one_one_int)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ra),B_1))),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Q_2)),Ra))) # label(fact_4731_adjust__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4354 (all P_1 ((all X_1 hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X_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))))))) # label(fact_734_transfer__int__nat__quantifiers_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4355 (all Xa all Ya (-hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ya),Xa)) & 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_2022_less__le__not__le) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4356 (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]. 9.41/9.53 4357 (all K_2 (is_int(K_2) -> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,pls),K_2) = K_2)) # label(fact_154_add__Pls) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4358 (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]. 9.41/9.53 4359 (all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(fact_fact_int,X))))) # label(fact_4278_transfer__nat__int__factorial__closure) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4360 (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]. 9.41/9.53 4361 (all N_37 all A_168 all B_114 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_168),B_114)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_168)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_168),N_37)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_114),N_37)))))) # label(fact_697_power__mono) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4362 (all Na hAPP_n1699378549t_bool(ord_lessThan_nat,Na) = hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na)) # label(fact_4951_atLeast0LessThan) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4363 (all A_3 all B_2 all Q_1 all R_2 (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))) -> (zero_zero_int != B_2 -> R_2 = hAPP_int_int(div_mod_int(A_3),B_2))))) # label(fact_3261_divmod__int__rel__mod) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4364 (all M all X (is_int(X) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),M)) -> zero_zero_int = X))))) # label(fact_2597_Int2_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4365 (all A_1 (zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_1),A_1) <-> A_1 = zero_zero_real)) # label(fact_930_double__zero__sym) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4366 (all W_16 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(number_number_of_int,W_16)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(bit1,pls)),W_16))) # label(fact_26_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4367 (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]. 9.41/9.53 4368 (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]. 9.41/9.53 4369 (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]. 9.41/9.53 4370 (all A_135 all B_95 (zero_zero_complex = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_135),B_95) -> B_95 = zero_zero_complex | A_135 = zero_zero_complex)) # label(fact_1133_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4371 (all 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)) = hAPP_real_real(sin,X)) # label(fact_3626_sin__cos__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4372 (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]. 9.41/9.53 4373 (all A_3 all B_2 ((hAPP_int_int(div_mod_int(A_3),B_2) != zero_zero_int -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_2),hAPP_int_int(div_mod_int(A_3),B_2)) = hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,A_3)),B_2)) & (zero_zero_int = hAPP_int_int(div_mod_int(A_3),B_2) -> hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,A_3)),B_2) = zero_zero_int))) # label(fact_3490_zmod__zminus1__eq__if) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4374 (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]. 9.41/9.53 4375 (all W_10 hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,W_10)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_int_real(number267125858f_real,W_10))),hAPP_int_real(number267125858f_real,W_10))) # label(fact_226_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4376 (all Xa all Ya (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Xa),Ya)) | Ya = Xa <-> 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]. 9.41/9.53 4377 (all A_3 all B_2 hAPP_P1175774780nt_int(product_snd_int_int,divmod_int(A_3,B_2)) = hAPP_int_int(div_mod_int(A_3),B_2)) # label(fact_4232_mod__int__def) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4378 (all W all Z_1 hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,W),Z_1))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,W))),hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,Z_1)))) # label(fact_2657_nat__abs__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4379 (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]. 9.41/9.53 4380 (all Z_13 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Z_13),Z_13) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Z_13),hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_1884_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4381 (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]. 9.41/9.53 4382 (all U (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),U)) -> hAPP_i1948725293t_bool(ord_at641636577an_int(zero_zero_int),U) = image_nat_int(semiri1621563631at_int,hAPP_n1699378549t_bool(ord_lessThan_nat,hAPP_int_nat(nat,U))))) # label(fact_4953_image__atLeastZeroLessThan__int) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4383 (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_r481142414t_bool(hAPP_f1845081853t_bool(cOMBC_int_real_bool,hAPP_f801032407l_bool(hAPP_f499530369l_bool(cOMBB_1317819104ol_int,ord_less_eq_real),ring_1_of_int_real)),Xa))),hAPP_f429384717t_bool(hAPP_f1173902867t_bool(cOMBB_real_bool_int,hAPP_r1134773055l_bool(ord_less_real,Xa)),hAPP_f563293398t_real(hAPP_f134192053t_real(cOMBB_int_real_int,ring_1_of_int_real),hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,plus_plus_int),one_one_int))))) = archim1246769320r_real(Xa)) # label(fact_4889_floor__real__def) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4384 (all Xa all Ya (Xa = Ya <-> quickcheck_int_of(Xa) = quickcheck_int_of(Ya))) # label(fact_4255_code__int_Oint__of__inject) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4385 (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_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),Na)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,H),T)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,T),zero_zero_real)) -> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),T),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),T))))) -> (exists T (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_real,T),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,H),T))))))))) # label(fact_4935_Maclaurin__minus) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4386 (all B_96 all A_136 (zero_zero_real != A_136 -> (B_96 != zero_zero_real -> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_136),B_96) != zero_zero_real))) # label(fact_1128_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4387 (all A_35 A_35 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_35),zero_zero_rat)) # label(fact_2329_diff__0__right) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4388 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa)) <-> hAPP_real_real(hAPP_r1250527377l_real(powr,Xa),one_one_real) = Xa)) # label(fact_4433_powr__one__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4389 (all Na (hBOOL(hAPP_nat_bool(even_odd_even_nat,Na)) <-> 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)))),Na)))) # label(fact_3814_even__dvd) # label(axiom) # label(non_clause). [assumption]. 9.41/9.53 4390 (all A_8 all B_9 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,A_8),B_9))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(abs_abs_real,A_8)),hAPP_real_real(abs_abs_real,B_9))))) # label(fact_2761_abs__triangle__ineq4) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4391 (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]. 9.41/9.54 4392 (all A_3 hAPP_complex_complex(invers1449016382omplex,cis(A_3)) = cis(hAPP_real_real(uminus_uminus_real,A_3))) # label(fact_4149_cis__inverse) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4393 (all P all Q all R hAPP_r1134773055l_bool(hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,P),Q),R) = hAPP_r1134773055l_bool(P,hAPP_real_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__H_009) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4394 (all A_191 one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_191),zero_zero_nat)) # label(fact_463_power__0) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4395 (all A_3 all Q_1 all R_2 (is_int(R_2) & is_int(A_3) -> (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,A_3),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_2))) -> (zero_zero_int != A_3 -> zero_zero_int = R_2)))) # label(fact_3256_self__remainder) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4396 (all M_19 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,M_19)))) # label(fact_724_zero__le__imp__of__nat) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4397 (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]. 9.41/9.54 4398 (all V_15 all W_7 hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_15),W_7)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_15)),hAPP_int_complex(number528085621omplex,W_7))) # label(fact_1093_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4399 (all Nat_2 zero_zero_nat != hAPP_nat_nat(suc,Nat_2)) # label(fact_2995_nat_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4400 (all X hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),X) = hAPP_int_int(abs_abs_int,X)) # label(fact_4976_gcd__idem__int) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4401 (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_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_18)),hAPP_int_real(number267125858f_real,V_19)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_18),V_19))))) # label(fact_866_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4402 (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]. 9.41/9.54 4403 (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]. 9.41/9.54 4404 (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]. 9.41/9.54 4405 (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_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_18),V_19)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_18)),hAPP_int_rat(number_number_of_rat,V_19))))) # label(fact_864_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4406 (all P all Q all R hAPP_int_int(hAPP_nat_fun_int_int(P,R),Q) = hAPP_nat_int(hAPP_int_fun_nat_int(hAPP_f1673103573at_int(cOMBC_nat_int_int,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Int__Oint_000tc__Int__Oint_U) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4407 (all L all U hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_at4362885an_nat(L),U)))) # label(fact_4895_finite__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4408 (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)) -> image_int_int(rRset2norRR(A,Ma),A) = hAPP_i1948725293t_bool(norRRset,Ma)))) # label(fact_4594_RRset2norRR__eq__norR) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4409 (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]. 9.41/9.54 4410 (all A_129 all M_17 all B_91 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_129),M_17)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_91),M_17)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_129),B_91)),M_17)) # label(fact_1185_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4411 (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]. 9.41/9.54 4412 (all N all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,N),P_3)) -> one_one_int = hAPP_int_int(legacy_zgcd(N),P_3))))) # label(fact_4523_zless__zprime__imp__zrelprime) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4413 (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]. 9.41/9.54 4414 (all C_1 all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) -> (B_2 = C_1 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),C_1))))) # label(fact_2640_dvd_Oord__le__eq__trans) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4415 (all A_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_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(plus_plus_int,A_1),A_1))))) # label(fact_485_zero__less__double__add__iff__zero__less__single__add) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4416 (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)) & Y = hAPP_real_real(sin,X_1) & (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),Y_1)) & 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 -> Y_1 = 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]. 9.41/9.54 4417 -(all R_1 all V_3 (is_int(V_3) & is_int(R_1) -> -(V_3 = 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)))) & 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))))))) # 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]. 9.41/9.54 4418 (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]. 9.41/9.54 4419 (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(hAPP_int_fun_int_int(minus_minus_int,P_3),one_one_int) != hAPP_int_int(inv(P_3),A_3))))) # label(fact_2902_inv__not__p__minus__1) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4420 (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)) -> (X_5 != Y_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]. 9.41/9.54 4421 (all R_2 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_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_4454_real__of__int__floor__ge__diff__one) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4422 (all K_2 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_2)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K_2),M)),K_2) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(suc,zero_zero_nat)))))) # label(fact_3158_int__power__div__base) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4423 (all A_3 all B_2 all P_3 all Q_1 (quotient_of(hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_1) -> hAPP_int_rat(hAPP_int_fun_int_rat(fract,P_3),Q_1) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2))) # label(fact_4668_quotient__of__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4424 (all X_34 all Y_28 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_34),Y_28)) -> -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_28),X_34)))) # label(fact_1804_order__less__imp__not__less) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4425 (all M all X hAPP_int_int(standardRes(M),X) = hAPP_int_int(div_mod_int(X),M)) # label(fact_2981_StandardRes__def) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4426 (all Y all X (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) -> Y = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y))) # label(fact_4321_gcd__proj2__if__dvd__nat) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4427 (all N -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N)))))) # label(fact_3496_not__zle__0__negative) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4428 (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]. 9.41/9.54 4429 (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]. 9.41/9.54 4430 (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]. 9.41/9.54 4431 (all P_1 all Na all K (is_int(Na) & is_int(K) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),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,zero_zero_int),J_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,J_1),K)) & 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(P_1,I_2)))))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),zero_zero_int)) -> (all I_2 all J_1 (is_int(I_2) & is_int(J_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_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(P_1,I_2)))))) & (K = zero_zero_int -> hBOOL(hAPP_int_bool(P_1,zero_zero_int))) <-> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Na),K)))))) # label(fact_2852_split__zdiv) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4432 (all Ya all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ya),Xa)) -> Xa = comple124823625p_real(hAPP_r1134773055l_bool(ord_at1496968948n_real(Ya),Xa)))) # label(fact_4917_Sup__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4433 (all M_22 all N_45 all A_193 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_193)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_193),M_22)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_193),N_45))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_22),N_45))))) # label(fact_433_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4434 (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,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)) = 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)) # label(fact_1162_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4435 (all A_1 (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)))) <-> zero_zero_real = A_1)) # label(fact_21_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4436 (all Y all X exists R_1 exists A_2 (X = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(cos,A_2)) & Y = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(sin,A_2)))) # label(fact_4543_polar__Ex) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4437 (all Z_2 all Ma ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_2),hAPP_nat_int(semiri1621563631at_int,Ma)))) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2)) -> zero_zero_nat = Ma) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(nat,Z_2)),Ma)))) # label(fact_2806_nat__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4438 (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]. 9.41/9.54 4439 (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]. 9.41/9.54 4440 (all B_73 all A_102 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_102)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_73),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_73),A_102)),zero_zero_int))))) # label(fact_1393_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4441 (all R_2 all A_3 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_2),hAPP_real_real(cos,A_3))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_2),hAPP_real_real(sin,A_3))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_3620_sin__cos__squared__add2__mult) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4442 (all Ma all Na (Ma = Na | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)))) # label(fact_1257_le__eq__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4443 (all C_1 all A_3 all B_2 (A_3 = B_2 -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),C_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),C_1))))) # label(fact_2641_dvd_Oord__eq__le__trans) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4444 (all Xa all Na all 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)))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)))) # label(fact_3071_exp__mono__lt) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4445 (all Lx_1 all Ly_1 all Rx_1 all Ry_1 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_1),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_1),Ly_1)),Ry_1)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_1),Ly_1)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_1),Ry_1))) # label(fact_1074_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4446 (all A_155 all C_57 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_57),A_155) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_155),C_57)) # label(fact_847_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4447 (all X_30 all Y_24 (Y_24 = X_30 | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_24),X_30)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_30),Y_24)))) # label(fact_1837_linorder__less__linear) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4448 (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]. 9.41/9.54 4449 (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]. 9.41/9.54 4450 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)) # label(fact_2082_nat__mult__commute) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4451 (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]. 9.41/9.54 4452 (all K all Ma all 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))) <-> 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),K)))) # label(fact_2117_mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4453 (all X_55 all Y_47 all Z_19 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_55),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Y_47),Z_19)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_55),Y_47)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_55),Z_19))) # label(fact_1166_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4454 (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]. 9.41/9.54 4455 (all Xa (Xa = zero_z891286103de_int <-> Xa = zero_z891286103de_int)) # label(fact_764_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4456 (all I all J_2 hAPP_int_int(legacy_zgcd(I),J_2) = hAPP_int_int(legacy_zgcd(J_2),I)) # label(fact_4494_zgcd__commute) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4457 (all C all A_1 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,C),A_1)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,C),B_1))))) # label(fact_2361_dvd__mult__cancel__left) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4458 (all P all Q all R hAPP_i688218208l_bool(hAPP_f801032407l_bool(hAPP_f499530369l_bool(cOMBB_1317819104ol_int,P),Q),R) = hAPP_r1134773055l_bool(P,hAPP_int_real(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__RealDef__Oreal_000tc__fun_Itc__RealDef__Oreal_Mtc__H) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4459 (all A_1 all Na (is_int(A_1) -> (A_1 = zero_zero_int & Na != zero_zero_nat <-> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_1),Na) = zero_zero_int))) # label(fact_451_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4460 (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 (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)) & one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_4),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)) & is_int(B_3) & is_int(A_4)))))) # label(fact_5117_gcd__coprime__exists__int) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4461 (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]. 9.41/9.54 4462 (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 -> one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_3),B_2))) # label(fact_5002_coprime__rmult__int) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4463 (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]. 9.41/9.54 4464 (all C_77 all A_188 all B_131 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_77),A_188)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_77),B_131))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_188),B_131)))) # label(fact_517_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4465 (all B_125 all A_181 all C_72 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_72)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_125),A_181)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_125),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_181),C_72)))))) # label(fact_571_add__increasing2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4466 (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]. 9.41/9.54 4467 (all X_8 all N_12 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_12)) | one_one_rat = X_8 -> 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]. 9.41/9.54 4468 (all R_2 all X all Y hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,R_2),X)),Y) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,of_real_complex(R_2)),hAPP_real_complex(hAPP_r265291036omplex(complex,X),Y))) # label(fact_3989_complex__of__real__add__Complex) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4469 (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,one_one_real),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(hAPP_n546711566l_real(root,Na),Ya)))))) # label(fact_3919_real__root__gt__1__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4470 (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(div_mod_int(A_3),B_2) = 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))))))) # label(fact_4237_mod__neg__neg) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4471 (all A_1 (one_one_nat = A_1 <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_1),one_one_nat)))) # label(fact_2967_gcd__lcm__complete__lattice__nat_Obot__unique) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4472 (all R_2 all 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)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,ratreal(hAPP_int_rat(number_number_of_rat,R_2))),ratreal(hAPP_int_rat(number_number_of_rat,S_1)))) # label(fact_4713_Ratreal__number__of__quotient) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4473 (all P all Q all R hAPP_f1048215610t_bool(hAPP_r1623423338t_bool(P,R),Q) = hAPP_r632163725t_bool(hAPP_f1832844496t_bool(hAPP_f1990636069t_bool(cOMBC_412863505t_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__RealDef__Oreal_000tc__fun_Itc__Int__Oint_Mtc__HOL__O) # label(axiom) # label(non_clause). [assumption]. 9.41/9.54 4474 (all X all Y all N (N != zero_zero_nat -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N)),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))))) # label(fact_2528_divides__exp2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4475 (all K_2 hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_nat_int(semiri1621563631at_int,K_2)),one_one_int) = hAPP_nat_rat(semiri151668891at_rat,K_2)) # label(fact_4687_Fract__of__nat__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4476 (all A_133 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_133),zero_zero_nat) = zero_zero_nat) # label(fact_1150_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4477 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (hAPP_real_real(cos,X) = zero_zero_real -> (exists N_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_3824_cos__zero__lemma) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4478 (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]. 9.41/9.55 4479 (all W hAPP_int_nat(number_number_of_nat,W) = hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,W))) # label(fact_2746_nat__number__of) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4480 (all B_116 all A_170 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_170)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_116)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_170),B_116)))))) # label(fact_666_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4481 (all Z_1 hAPP_complex_complex(cnj,hAPP_int_complex(ring_11397209091omplex,Z_1)) = hAPP_int_complex(ring_11397209091omplex,Z_1)) # label(fact_4579_complex__cnj__of__int) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4482 (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]. 9.41/9.55 4483 (all V_1 (hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1))) -> hAPP_int_nat(number_number_of_nat,V_1) = zero_zero_nat)) # label(fact_3258_neg__imp__number__of__eq__0) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4484 (all Nat_5 all Nat_4 (Nat_5 = Nat_4 <-> hAPP_nat_nat(suc,Nat_5) = hAPP_nat_nat(suc,Nat_4))) # label(fact_2983_nat_Oinject) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4485 (all A_203 A_203 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,pls)),A_203)) # label(fact_172_add__numeral__0) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4486 (all V_2 all V_1 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_1)),hAPP_int_nat(number_number_of_nat,V_2)) = zero_zero_nat) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_1)),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_1),V_2))))) # label(fact_2140_mult__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4487 (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(number_number_of_int,min) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat,Xa))))) # label(fact_2950_EvenOdd_Oneg__one__odd__power) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4488 (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]. 9.41/9.55 4489 (all V_23 all W_13 hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_23),W_13)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_23)),hAPP_int_int(number_number_of_int,W_13))) # label(fact_194_add__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4490 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(nat,X)),hAPP_int_nat(nat,Y)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y))))) # label(fact_2896_Divides_Otransfer__nat__int__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4491 (all Xa (hBOOL(hAPP_int_bool(even_odd_even_int,Xa)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Xa)))) # label(fact_3267_int__even__iff__2__dvd) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4492 (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)) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa_1) != X_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]. 9.41/9.55 4493 (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]. 9.41/9.55 4494 (all I all M all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),N)))) # label(fact_4987_dvd__gcd__D2__int) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4495 (all B_31 all A_42 all C_11 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_42),C_11)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_42),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_31),C_11))))) # label(fact_2281_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4496 (all Z_1 all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z_1))))) # label(fact_2637_dvd_Oorder__trans) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4497 (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]. 9.41/9.55 4498 (all A_3 all B_2 hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_2),A_3)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),B_2)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),A_3)) # label(fact_3419_real__average__minus__first) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4499 (all X_51 all Y_45 hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_51),X_51)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_45),Y_45))))) # label(fact_1581_sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4500 (all W hAPP_int_complex(number528085621omplex,W) != ii) # label(fact_3972_complex__i__not__number__of) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4501 (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))) -> ((Y_1 != zero_zero_nat -> 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]. 9.41/9.55 4502 (all Xa all Ya (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Ya)) <-> Xa != Ya & hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Ya)))) # label(fact_2029_order__less__le) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4503 (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_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_192)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_192),N_44)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_192),N_43)))))) # label(fact_440_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4504 (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]. 9.41/9.55 4505 (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_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),A_183)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_183),N_42)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_183),N_41)))))) # label(fact_554_power__increasing) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4506 (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]. 9.41/9.55 4507 (all D_3 all A_3 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_3),A_3)))) # label(fact_4013_coprime__rmul2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4508 (all B_126 all A_182 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_182),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_126),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_182),B_126)),zero_zero_int))))) # label(fact_569_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4509 (all A_1 (zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_1),A_1) <-> A_1 = zero_zero_real)) # label(fact_129_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4510 (all Xa all 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)) <-> Ya = Xa)) # label(fact_1701_order__eq__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4511 (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]. 9.41/9.55 4512 (all A_6 all B_7 all Y all R_5 (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),Y)),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_int,R_5),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,zero_zero_int),B_7)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_7),R_5)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_5),zero_zero_int))) -> (zero_zero_int != B_7 -> 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]. 9.41/9.55 4513 (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]. 9.41/9.55 4514 (all P_3 hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(sgn_sgn_int,hAPP_P1175774780nt_int(product_fst_int_int,quotient_of(P_3)))),one_one_int) = quotient_of(hAPP_rat_rat(sgn_sgn_rat,P_3))) # label(fact_4648_rat__sgn__code) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4515 (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(powr,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),hAPP_real_real(ln,X))))) # label(fact_4459_ln__powr) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4516 (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_eq_real,B_1),A_1)) <-> 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)))))) # label(fact_1576_mult__le__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4517 (all B_71 all A_100 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_71),A_100))) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_100)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_71))))) # label(fact_1401_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4518 (all C_16 all A_66 all B_43 (B_43 = A_66 -> (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]. 9.41/9.55 4519 (all X all Y hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sqrt,X)),hAPP_real_real(sqrt,Y)) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y))) # label(fact_3518_real__sqrt__mult) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4520 (all N_51 hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,one_on1684967323de_int),N_51) = one_on1684967323de_int) # label(fact_349_power__one) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4521 (all A_1 all Na (zero_zero_nat != Na & zero_zero_rat = A_1 <-> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_1),Na) = zero_zero_rat)) # label(fact_445_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4522 (all Z_3 all Z_2 (is_int(Z_3) & is_int(Z_2) -> (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)) -> (hAPP_int_nat(nat,Z_2) = hAPP_int_nat(nat,Z_3) <-> Z_2 = Z_3))))) # label(fact_2741_eq__nat__nat__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4523 (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]. 9.41/9.55 4524 (all Xa (zero_z126310315umeral = Xa <-> Xa = zero_z126310315umeral)) # label(fact_766_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4525 (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]. 9.41/9.55 4526 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (all Y_1 exists N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),X)))))) # label(fact_3781_reals__Archimedean3) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4527 (all Q_2 all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) -> (Q_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,Ma),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))))) # label(fact_3146_split__div__lemma) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4528 (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]. 9.41/9.55 4529 (all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Y)) -> bezw(X,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))))))) # label(fact_4230_bezw__non__0) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4530 (all M_14 all A_90 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_90),one_one_complex)),M_14) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,M_14),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_90),M_14))) # label(fact_1479_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4531 (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(hAPP_n1699378549t_bool(ord_less_nat,Xa),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,Ya)) & -hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa)))) # label(fact_3794_even__difference__nat) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4532 (all R_2 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),of_real_complex(R_2)) = hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),R_2)) # label(fact_3991_i__complex__of__real) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4533 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_2)) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = hAPP_P1175774780nt_int(product_fst_int_int,posDivAlg(A_3,B_2))))) # label(fact_4225_div__pos__pos) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4534 (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]. 9.41/9.55 4535 (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]. 9.41/9.55 4536 (all A_148 A_148 = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,zero_z126310315umeral),A_148)) # label(fact_938_add__0) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4537 (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]. 9.41/9.55 4538 (all Xa (hAPP_real_real(sqrt,Xa) = zero_zero_real <-> Xa = zero_zero_real)) # label(fact_3513_real__sqrt__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4539 (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]. 9.41/9.55 4540 (all B_1 all A_1 all Ma (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),hAPP_i1948725293t_bool(bnorRset(A_1),Ma))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1)))) # label(fact_4294_Bnor__mem__zg) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4541 (all X all Y Y = hAPP_P1860904029l_real(hAPP_P1982799375l_real(hAPP_b1482221219l_real(if_Pro313124157l_real,fFalse),X),Y)) # label(help_If_2_1_If_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_T) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4542 (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]. 9.41/9.55 4543 (all C_54 all D_16 all A_143 all B_101 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_143),B_101)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_54),D_16)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_143),C_54)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_101),D_16)))))) # label(fact_964_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4544 (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]. 9.41/9.55 4545 (all X all M all Y hAPP_int_int(div_mod_int(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Y)),M) = 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)) # label(fact_2998_zpower__zmod) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4546 (all P (is_bool(P) -> fFalse = P | fTrue = P)) # label(help_If_3_1_If_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_T) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4547 (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]. 9.41/9.55 4548 (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]. 9.41/9.55 4549 (all N all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N)),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N)))) -> N = hAPP_int_nat(nat,archim1246769320r_real(X))))) # label(fact_4034_floor__eq4) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4550 (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(number_number_of_int,V_1)),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(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))))) # label(fact_2854_zdiv__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4551 (all Y all X (-hBOOL(hAPP_int_bool(even_odd_even_int,X)) -> (-hBOOL(hAPP_int_bool(even_odd_even_int,Y)) -> -hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)))))) # label(fact_3232_Parity_Oodd__times__odd) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4552 (all C_27 all D_12 all A_81 all B_54 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_81),B_54)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_27),D_12)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_81)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_27)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_81),C_27)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_54),D_12)))))))) # label(fact_1551_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4553 (all A_7 all B_8 all C_4 all D_5 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,A_7),B_8)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_4),D_5)))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_7),C_4))),hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_8),D_5)))))) # label(fact_2764_abs__diff__triangle__ineq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4554 (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]. 9.41/9.55 4555 (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]. 9.41/9.55 4556 (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]. 9.41/9.55 4557 (all D_1 all A_1 all B_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_1),B_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_1),B_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_1),A_1)))) # label(fact_4015_coprime__mul__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4558 (all Na all Va (hAPP_int_nat(number_number_of_nat,Va) = hAPP_nat_nat(suc,Na) <-> -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)))) -> Na = hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va)))))) # label(fact_4779_Suc__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4559 (all M all N all Qr_1 (hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),Qr_1)) -> divmod_nat(M,N) = Qr_1)) # label(fact_4216_divmod__nat__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4560 (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(A_1),B_1),hAPP_int_int(uminus_uminus_int,Ma))))) # label(fact_3466_zcong__zminus) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4561 (all A_22 all 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)) = 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)))) # label(fact_2678_abs__add__abs) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4562 (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]. 9.41/9.55 4563 (all X all N all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N)))))) # label(fact_4173_prime__divexp__n) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4564 (all A_130 all B_92 all C_48 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_130),C_48)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_92),C_48)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_130),B_92)),C_48)) # label(fact_1176_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4565 (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,Ya),Xa)))) # label(fact_2048_linorder__not__less) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4566 (all A_3 all N all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_2),N))) -> (N != zero_zero_nat -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))))) # label(fact_2527_divides__rev) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4567 (all A_3 one_one_real = hAPP_complex_real(norm_norm_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(cos,A_3)),hAPP_real_real(sin,A_3)))) # label(fact_3961_cmod__unit__one) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4568 (all C_1 all A_3 all B_2 (B_2 = A_3 -> (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]. 9.41/9.55 4569 (all A_3 all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(ln,X)),A_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(powr,A_3),A_3)),X)))))) # label(fact_4475_ln__powr__bound2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4570 (all X hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X)) = hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),pi))) # label(fact_3617_sin__periodic__pi) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4571 (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]. 9.41/9.55 4572 (all M all N ((hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M),N)) != N -> hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M),N)) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M)),N)) & (N = hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M),N)) -> zero_zero_nat = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M)),N)))) # label(fact_3177_mod__Suc) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4573 (all B_1 all A_1 (zero_zero_rat = A_1 <-> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_1),A_1) = B_1)) # label(fact_903_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4574 (all X_51 all Y_45 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_51),X_51)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_45),Y_45))))) # label(fact_1582_sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4575 (all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_int_real(number267125858f_real,Ya))))) # label(fact_108_less__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4576 (all Xa all Ya (is_int(Xa) & is_int(Ya) -> (Ya = Xa | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ya),Xa)) <-> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya))))) # label(fact_1844_not__less__iff__gr__or__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4577 (all C all A_1 all B_1 (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(plus_plus_int,C),A_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),B_1))))) # label(fact_544_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4578 (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]. 9.41/9.55 4579 (all V_20 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_20)) -> hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),hAPP_int_complex(number528085621omplex,V_20)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(succ,V_20)))) # label(fact_745_semiring__1__add__number__of) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4580 (all A_121 all B_90 all N_28 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_121),N_28)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,B_90),N_28)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_121),B_90)),N_28)) # label(fact_1240_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4581 (all Na all 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)))),hAPP_nat_nat(suc,Na)),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,Na),hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va)))))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(suc,Na)),hAPP_int_nat(number_number_of_nat,Va))) # label(fact_5188_max__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4582 (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]. 9.41/9.55 4583 (all B_116 all A_170 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_170)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_116)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_170),B_116)))))) # label(fact_670_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4584 (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]. 9.41/9.55 4585 (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]. 9.41/9.55 4586 (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]. 9.41/9.55 4587 (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]. 9.41/9.55 4588 (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]. 9.41/9.55 4589 (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]. 9.41/9.55 4590 (all C all A_1 all B_1 (B_1 = A_1 -> (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B_1),C)) -> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_1),C))))) # label(fact_1679_ord__eq__le__trans) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4591 (all A_1 all B_1 all C (C != zero_zero_real -> (B_1 = A_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]. 9.41/9.55 4592 (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]. 9.41/9.55 4593 (all X all Y (Y = X | -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(fequal_rat,X),Y)))) # label(help_fequal_1_1_fequal_000tc__Rat__Orat_T) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4594 (all B_117 all C_67 all A_171 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_171)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_117),C_67)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_117),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_171),C_67)))))) # label(fact_660_add__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 9.41/9.55 4595 (all Xa all Ya (Ya = Xa <-> hAPP_nat_int(semiri1621563631at_int,Xa) = hAPP_nat_int(semiri1621563631at_int,Ya))) # label(fact_326_Nat__Transfer_Otransfer__int__nat__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4596 (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]. 9.41/9.56 4597 (all A_3 -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),one_one_nat)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A_3)))) # label(fact_2964_gcd__lcm__complete__lattice__nat_Onot__less__bot) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4598 (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]. 9.41/9.56 4599 (all N all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N),M)))) # label(fact_2590_zdvd__bounds) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4600 (all Ta all A 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),A)) -> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_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(plus_plus_int,X_1),D))))))))) # label(fact_5094_aset_I7_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4601 (all C_39 all A_96 all B_67 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_96),B_67)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_39)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_39),A_96)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_39),B_67)))))) # label(fact_1428_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4602 (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]. 9.41/9.56 4603 (all N_46 all A_195 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_195)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_195),N_46))))) # label(fact_410_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4604 (all N -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat))) # label(fact_274_less__zeroE) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4605 (all B_123 all A_179 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_179)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_123)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_179),B_123)))))) # label(fact_590_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4606 (all Real1 all Real2 hAPP_complex_nat(size_size_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,Real1),Real2)) = zero_zero_nat) # label(fact_3968_complex_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4607 (all A_148 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A_148) = A_148) # label(fact_932_add__0) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4608 (all Lx_6 all Rx_6 all Ry_4 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_6),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_6),Ry_4)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_6),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_6),Ry_4))) # label(fact_1040_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4609 (all C_10 all A_41 all B_30 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_41),B_30)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_41),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_30),C_10))))) # label(fact_2284_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4610 (all W_4 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,W_4)),hAPP_int_complex(number528085621omplex,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))))) # label(fact_1912_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4611 (all A_149 A_149 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_149),zero_zero_rat)) # label(fact_922_add__0__right) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4612 (all Va all K all A_1 all B_1 (B_1 != A_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]. 9.41/9.56 4613 (all P all Q all R hAPP_f54304608l_bool(hAPP_n215258509l_bool(P,R),Q) = hAPP_nat_bool(hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4614 (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]. 9.41/9.56 4615 (all Z_6 all X_13 all Y_10 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_13),Y_10)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_13),Z_6)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_13),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Y_10),Z_6)))))) # label(fact_2215_dvd__diff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4616 (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]. 9.41/9.56 4617 (all C_20 all A_70 all B_47 (A_70 = B_47 -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_47),C_20)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_70),C_20))))) # label(fact_1682_ord__eq__le__trans) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4618 (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]. 9.41/9.56 4619 (all N_55 ((zero_zero_nat = N_55 -> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),N_55) = one_one_complex) & (zero_zero_nat != N_55 -> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),N_55) = zero_zero_complex))) # label(fact_254_power__0__left) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4620 (all Na (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,Na)),zero_zero_real)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Na),zero_zero_int)))) # label(fact_4440_real__of__int__le__zero__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4621 (all C_36 all B_64 all A_93 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_64),A_93)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_36),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_36),A_93)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_36),B_64)))))) # label(fact_1440_mult__strict__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4622 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,Xa)),hAPP_real_real(sqrt,Ya))))) # label(fact_3509_real__sqrt__le__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4623 (all C all A_1 all B_1 (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A_1),B_1)) -> (C = B_1 -> 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]. 9.41/9.56 4624 (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]. 9.41/9.56 4625 (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]. 9.41/9.56 4626 (all M_14 all A_90 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)) = 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)) # label(fact_1477_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4627 (all X_41 all Y_35 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_35),X_41)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_41),Y_35)))) # label(fact_1712_linorder__linear) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4628 (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]. 9.41/9.56 4629 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Xa)),one_one_real)) -> hBOOL(monoseq_real(hAPP_f1888375463t_real(hAPP_f197615720t_real(cOMBS_nat_real_real,hAPP_f451358433l_real(hAPP_f1092742621l_real(cOMBB_1308947378al_nat,times_times_real),hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,one_one_real)),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,hAPP_f1639111240at_nat(hAPP_f1731313045at_nat(cOMBB_963856155at_nat,plus_plus_nat),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_nat))))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,hAPP_r474017924t_real(power_power_real,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))))))) # label(fact_4804_monoseq__arctan__series) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4630 (all M all N hAPP_nat_nat(div_mod_nat(M),hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,N)))) = hAPP_nat_nat(div_mod_nat(M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))),N))) # label(fact_3204_mod__Suc__eq__mod__add3) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4631 (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]. 9.41/9.56 4632 (exists 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)))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),K_4)))) # label(fact_4106_Im_Opos__bounded) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4633 (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]. 9.41/9.56 4634 (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]. 9.41/9.56 4635 (all N hAPP_nat_int(semiri1621563631at_int,N) = hAPP_int_int(ring_1_of_int_int,hAPP_nat_int(semiri1621563631at_int,N))) # label(fact_4578_of__int__int__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4636 (all M all I all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_2),M))))) # label(fact_1269_trans__le__add1) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4637 (all X all K_2 hAPP_real_real(sqrt,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),K_2)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,X)),K_2)) # label(fact_3517_real__sqrt__power) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4638 (all Xa all Ya (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) -> (Xa = Ya <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya))))) # label(fact_2002_linorder__antisym__conv1) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4639 (all Xa all A (hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(semiri1621563631at_int,Xa)),image_nat_int(semiri1621563631at_int,A))) <-> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Xa),A)))) # label(fact_4601_transfer__nat__int__set__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4640 (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]. 9.41/9.56 4641 (all Na 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),zero_zero_real)) -> hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa) = zero_zero_real) & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),zero_zero_real)) -> hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa) = hAPP_real_real(uminus_uminus_real,the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_real,zero_zero_real))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,power_power_real),Na))),hAPP_real_real(uminus_uminus_real,Xa))))))) & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa)) -> hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa) = the_real(hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(ord_less_real,zero_zero_real))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,power_power_real),Na))),Xa)))))) # label(fact_4884_root__def) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4642 (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]. 9.41/9.56 4643 (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]. 9.41/9.56 4644 (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)))) # label(fact_4087_cnj_Obounded) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4645 (all L (is_int(L) -> (pls = hAPP_int_int(bit0,L) <-> pls = L))) # label(fact_136_rel__simps_I38_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4646 (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)) <-> Ya = Xa)))) # label(fact_1972_linorder__antisym__conv2) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4647 (all M_23 all N_47 hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_nat_rat(semiri151668891at_rat,M_23)),N_47) = hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_23),N_47))) # label(fact_387_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4648 (all M all N (M = N -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)))) # label(fact_1029_eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4649 (all X_52 all N_25 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_52),N_25)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_52),N_25)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,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))) # label(fact_1510_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4650 (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]. 9.41/9.56 4651 (all X all Y hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3550_real__sqrt__sum__squares__ge1) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4652 (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]. 9.41/9.56 4653 (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]. 9.41/9.56 4654 (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))) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N) = Q_1)) # label(fact_4217_div__eq) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4655 (exists 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)))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),K_4)))) # label(fact_4140_Re_Ononneg__bounded) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4656 (all N all 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)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M),hAPP_int_int(div_mod_int(M),N))) # label(fact_3091_zmult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4657 (all M all K_2 all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N)) # label(fact_4347_gcd__add__mult__nat) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4658 (all Na hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),Na))) = Na) # label(fact_4759_card__Collect__less__nat) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4659 (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]. 9.41/9.56 4660 (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]. 9.41/9.56 4661 (all X_41 all Y_35 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_35),X_41)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_41),Y_35)))) # label(fact_1710_linorder__linear) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4662 (all A_194 hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_194),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_194),one_on1645066479umeral)))) # label(fact_417_less__add__one) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4663 (all Y_2 all B_1 all X_2 all A_1 (hBOOL(tendsto_nat_real(X_2,A_1,sequentially)) -> (hBOOL(tendsto_nat_real(Y_2,B_1,sequentially)) -> hBOOL(tendsto_nat_complex(hAPP_f219720582omplex(hAPP_f1052840010omplex(cOMBS_696648398omplex,hAPP_f23851874omplex(hAPP_f406530075omplex(cOMBB_1714920583ex_nat,complex),X_2)),Y_2),hAPP_real_complex(hAPP_r265291036omplex(complex,A_1),B_1),sequentially))))) # label(fact_5057_LIMSEQ__Complex) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4664 (all X_62 one_one_rat = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_62),zero_zero_nat)) # label(fact_476_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4665 (all Va all Na ((-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va)))) -> Na = hAPP_int_nat(nat,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)))) <-> hAPP_int_nat(number_number_of_nat,Va) = hAPP_nat_nat(suc,Na))) # label(fact_4778_eq__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4666 (all 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)) & (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_2),zero_zero_real)) -> hAPP_real_real(abs_abs_real,R_2) = R_2))) # label(fact_3442_real__abs__def) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4667 (all B_117 all C_67 all A_171 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_171)) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_117),C_67)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_117),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_171),C_67)))))) # label(fact_664_add__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4668 (all Ma all K (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,Ma),K)) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,Ma),hAPP_rat_rat(abs_abs_rat,K))))) # label(fact_2697_dvd__abs__iff) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4669 (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]. 9.41/9.56 4670 (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]. 9.41/9.56 4671 (all N -hBOOL(hAPP_int_bool(nat_neg,hAPP_nat_int(semiri1621563631at_int,N)))) # label(fact_3242_not__neg__int) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4672 (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]. 9.41/9.56 4673 (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]. 9.41/9.56 4674 (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]. 9.41/9.56 4675 (all Lx_4 all Ly_4 all Rx_4 hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_4),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ly_4),Rx_4)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_4),Ly_4)),Rx_4)) # label(fact_1056_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4676 (all A_160 all B_108 all C_62 hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_160),C_62)),B_108) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_160),B_108)),C_62)) # label(fact_814_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4677 (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]. 9.41/9.56 4678 (all A_54 all M_11 all 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) = 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))) # label(fact_2094_power__mult) # label(axiom) # label(non_clause). [assumption]. 9.41/9.56 4679 (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),Ly)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx),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))) # label(fact_1081_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.56 4680 (all R_2 all A_3 rcis(R_2,A_3) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_2)),cis(A_3))) # label(fact_4265_rcis__def) # label(axiom) # label(non_clause). [assumption]. 9.53/9.56 4681 (all A_3 all N all B_2 hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_3),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_2),N))) # label(fact_4333_gcd__exp__nat) # label(axiom) # label(non_clause). [assumption]. 9.53/9.56 4682 (all A_3 all B_2 hAPP_int_rat(ring_1_of_int_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(sgn_sgn_int,A_3)),hAPP_int_int(sgn_sgn_int,B_2))) = hAPP_rat_rat(sgn_sgn_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2))) # label(fact_4714_sgn__rat) # label(axiom) # label(non_clause). [assumption]. 9.53/9.56 4683 (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(dvd_dvd_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(dvd_dvd_nat,Ma),Na))))) # label(fact_2534_nat__mult__dvd__cancel1) # label(axiom) # label(non_clause). [assumption]. 9.53/9.56 4684 (all N hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N))))) # label(fact_3797_real__of__nat__fact__ge__zero) # label(axiom) # label(non_clause). [assumption]. 9.53/9.56 4685 (all X_26 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_26),X_26) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_26),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_1904_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.56 4686 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X)) -> (zero_zero_real = hAPP_real_real(sin,X) -> (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]. 9.53/9.56 4687 (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]. 9.53/9.56 4688 (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]. 9.53/9.56 4689 (all A_1 (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_1),A_1))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_1)))) # label(fact_598_zero__le__double__add__iff__zero__le__single__add) # label(axiom) # label(non_clause). [assumption]. 9.53/9.56 4690 (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]. 9.53/9.56 4691 (all A_1 all Na (A_1 = zero_zero_complex & Na != zero_zero_nat <-> zero_zero_complex = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_1),Na))) # label(fact_447_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.53/9.56 4692 (all X hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,X)),hAPP_nat_nat(fact_fact_nat,X)) = hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(suc,X))) # label(fact_3838_Fact_Ofact__Suc) # label(axiom) # label(non_clause). [assumption]. 9.53/9.56 4693 (all Y all X all P_3 (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X),P_3)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),Y))))) # label(fact_2463_Euler_Oaux____1) # label(axiom) # label(non_clause). [assumption]. 9.53/9.56 4694 (all X all Y hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),hAPP_int_int(uminus_uminus_int,Y)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)) # label(fact_4994_gcd__neg2__int) # label(axiom) # label(non_clause). [assumption]. 9.53/9.56 4695 (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]. 9.53/9.56 4696 (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]. 9.53/9.56 4697 (all M hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),zero_zero_nat) = M) # label(fact_316_Nat_Oadd__0__right) # label(axiom) # label(non_clause). [assumption]. 9.53/9.56 4698 (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,Ya),Xa)))) # label(fact_2047_linorder__not__less) # label(axiom) # label(non_clause). [assumption]. 9.53/9.56 4699 (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]. 9.53/9.56 4700 (all R_2 hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),R_2) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_2)),ii)) # label(fact_3992_complex__of__real__i) # label(axiom) # label(non_clause). [assumption]. 9.53/9.56 4701 (all X all N (-hBOOL(hAPP_nat_bool(even_odd_even_nat,N)) -> X = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),N))) # label(fact_3894_odd__real__root__pow) # label(axiom) # label(non_clause). [assumption]. 9.53/9.56 4702 (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]. 9.53/9.56 4703 (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]. 9.53/9.56 4704 (all V_24 all W_14 all 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) = 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))) # label(fact_189_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 9.53/9.56 4705 (all Xa all A (hBOOL(hAPP_P1463840893t_bool(A,Xa)) <-> hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,Xa),A)))) # label(fact_143_mem__def) # label(axiom) # label(non_clause). [assumption]. 9.53/9.56 4706 (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]. 9.53/9.56 4707 (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]. 9.53/9.56 4708 (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]. 9.53/9.56 4709 (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]. 9.53/9.56 4710 (all P all Q all R hAPP_b40753821nt_int(P,hAPP_int_bool(Q,R)) = hAPP_i2104874254nt_int(hAPP_f251264923nt_int(hAPP_f279307923nt_int(cOMBB_662120866nt_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__prod_Itc__Int__Oint_Mtc__) # label(axiom) # label(non_clause). [assumption]. 9.53/9.56 4711 (all N_31 all A_153 all B_104 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_153),B_104)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_153)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_153),N_31)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_104),N_31))))))) # label(fact_876_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 9.53/9.56 4712 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),B_2))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))))) # label(fact_1503_pos__zmult__pos) # label(axiom) # label(non_clause). [assumption]. 9.53/9.56 4713 (all Xa (hBOOL(hAPP_int_bool(nat_is_nat,Xa)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)))) # label(fact_5128_is__nat__def) # label(axiom) # label(non_clause). [assumption]. 9.53/9.56 4714 (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]. 9.53/9.56 4715 (all Xa all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,Xa)),hAPP_int_real(number267125858f_real,Ya))))) # label(fact_61_less__number__of) # label(axiom) # label(non_clause). [assumption]. 9.53/9.56 4716 (all Z_18 all Y_40 all X_46 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_40),X_46)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Z_18),Y_40)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Z_18),X_46))))) # label(fact_1632_xt1_I6_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.56 4717 (all P all Q all R hAPP_f11151005nt_int(P,hAPP_P2110489235nt_int(Q,R)) = hAPP_P752050971nt_int(hAPP_f301246617nt_int(hAPP_f1151807883nt_int(cOMBB_563965291nt_int,P),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]. 9.53/9.56 4718 (all A_1 all E_2 all C all B_1 all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_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))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_1),A_1)),E_2)),D_1))))) # label(fact_2429_less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 9.53/9.56 4719 (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]. 9.53/9.56 4720 (all A_1 all B_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_1),B_1)) <-> (exists X_1 exists 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,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)))))) # label(fact_4035_coprime__bezout) # label(axiom) # label(non_clause). [assumption]. 9.53/9.56 4721 (all N_58 hAPP_int_nat(number_number_of_nat,hAPP_nat_int(semiri1621563631at_int,N_58)) = hAPP_nat_nat(semiri984289939at_nat,N_58)) # label(fact_207_number__of__int) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4722 (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)) -> (exists Y_1 (hBOOL(hAPP_real_bool(deriv_real(F,X_1),Y_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1)))))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,A_1)),hAPP_real_real(F,B_1)))))) # label(fact_4003_DERIV__pos__imp__increasing) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4723 (all A_3 ((-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),zero_zero_real)) -> A_3 = hAPP_real_real(abs_abs_real,A_3)) & (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),zero_zero_real)) -> hAPP_real_real(abs_abs_real,A_3) = hAPP_real_real(uminus_uminus_real,A_3)))) # label(fact_3441_abs__real__def) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4724 (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]. 9.53/9.57 4725 (all I all J_2 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(legacy_zgcd(I),J_2)),I))) # label(fact_4483_zgcd__zdvd1) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4726 (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),hilbert_Eps_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,zcong(A_1)),Ma))),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),hAPP_i1948725293t_bool(norRRset,Ma))))),Ma)) & hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hilbert_Eps_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(hAPP_f428220345t_bool(cOMBC_int_int_bool,zcong(A_1)),Ma))),hAPP_f1805168059t_bool(hAPP_f202917053t_bool(cOMBC_94739984l_bool,member_int),hAPP_i1948725293t_bool(norRRset,Ma))))),hAPP_i1948725293t_bool(norRRset,Ma))))))) # label(fact_4892_aux__some) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4727 (all A_125 A_125 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_125),one_one_real)) # label(fact_1217_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4728 (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]. 9.53/9.57 4729 (all Xa all Ya (hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Xa),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,Xa)) & hBOOL(hAPP_nat_bool(even_odd_even_nat,Ya)))) # label(fact_3787_even__sum__nat) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4730 (all C_53 all A_142 all B_100 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_142),B_100)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_53),A_142)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_53),B_100))))) # label(fact_972_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4731 (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]. 9.53/9.57 4732 (all B_31 all A_42 all C_11 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_42),C_11)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_42),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_31),C_11))))) # label(fact_2276_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4733 (all B_76 all A_106 (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,A_106),zero_zero_int)) | 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,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]. 9.53/9.57 4734 (all Ma all Na (zero_zero_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Ma),Na) <-> zero_zero_nat = Ma & Na = zero_zero_nat)) # label(fact_4340_gcd__zero__nat) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4735 (all N all X (X = hAPP_nat_real(real_nat,natfloor(X)) -> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,natfloor(X)),N) = natfloor(hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N)))) # label(fact_3722_natfloor__power) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4736 (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]. 9.53/9.57 4737 (all A_3 all B_2 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),hAPP_real_complex(hAPP_r265291036omplex(complex,A_3),B_2)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(uminus_uminus_real,B_2)),A_3)) # label(fact_3978_i__mult__Complex) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4738 (all B_1_1 all B_2_1 all B_3_1 is_bool(tendsto_complex_real(B_1_1,B_2_1,B_3_1))) # label(gsy_c_Limits_Otendsto_000tc__Complex__Ocomplex_000tc__RealDef__Oreal) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4739 (all A_83 all C_29 all B_56 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_83),C_29)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_56),C_29))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_29)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_83),B_56))))) # label(fact_1542_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4740 (all A_25 all B_19 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,A_25)),B_19)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_25),B_19)))) # label(fact_2671_abs__le__D1) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4741 (all A_155 all C_57 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_155),C_57) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_57),A_155)) # label(fact_849_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4742 (all C all A_1 all 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))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_1),A_1)) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),zero_zero_rat)) | 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)))) # label(fact_1370_mult__less__cancel__left__disj) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4743 (all X all Y (X = Y -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)))) # label(fact_2645_dvd_Oeq__refl) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4744 (all P_1 hBOOL(hAPP_f448129468l_bool(nat_nat_set,collect_int(hAPP_f1805168059t_bool(hAPP_f727283836t_bool(cOMBS_int_bool_bool,hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int))),P_1))))) # label(fact_4737_Nat__Transfer_Otransfer__int__nat__set__function__closures_I4_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4745 (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]. 9.53/9.57 4746 (all A_128 hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),A_128) = A_128) # label(fact_1197_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4747 (all I_1 (hAPP_int_nat(nat,I_1) = zero_zero_nat <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),zero_zero_int)))) # label(fact_2769_nat__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4748 (all P all Q (hBOOL(Q) | -hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)))) # label(help_fconj_3_1_U) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4749 (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]. 9.53/9.57 4750 (all N all P_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),P_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_3),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(fact,N)))))) # label(fact_4551_divides__fact) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4751 (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]. 9.53/9.57 4752 (all L all U hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_gr1297742076an_int(L),U)))) # label(fact_4621_finite__greaterThanLessThan__int) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4753 (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]. 9.53/9.57 4754 (all N all A_3 all B_2 all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),B_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),B_2))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N)),A_3)))))) # label(fact_4185_prime__power__dvd__cancel__right) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4755 (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]. 9.53/9.57 4756 (all C_9 all A_38 all B_27 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_38),B_27)) -> (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_38),C_9)) -> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_38),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_27),C_9)))))) # label(fact_2303_dvd__add) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4757 (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]. 9.53/9.57 4758 (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_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_199)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_199),one_one_rat)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_199),N_53)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_199),N_54))))))) # label(fact_259_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4759 (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,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))) <-> 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))))) # label(fact_2422_le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4760 (all C_53 all A_142 all B_100 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_142),B_100)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_53),A_142)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_53),B_100))))) # label(fact_971_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4761 (all Xa all Ya (is_int(Xa) & is_int(Ya) -> (zero_zero_int != Xa | zero_zero_int != Ya <-> 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)))))))))) # label(fact_216_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4762 (all X all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_int_real(real_int,A_3))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X)),A_3)))) # label(fact_4422_ceiling__le) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4763 (all X_54 all Y_46 all Q_9 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_54),Q_9)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_46),Q_9)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_54),Y_46)),Q_9)) # label(fact_1247_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4764 (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]. 9.53/9.57 4765 (all Z1 all Z2 all W hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z1),W)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z2),W)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Z1),Z2)),W)) # label(fact_2353_zdiff__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4766 (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]. 9.53/9.57 4767 (all U_1 all X (X = hAPP_real_real(exp_real,U_1) -> hAPP_real_real(ln,X) = U_1)) # label(fact_3395_exp__ln__eq) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4768 (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]. 9.53/9.57 4769 (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]. 9.53/9.57 4770 (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]. 9.53/9.57 4771 (all Xa all Ya (hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Ya)) -> Xa != Ya)) # label(fact_1791_order__less__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4772 (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]. 9.53/9.57 4773 (all W_15 ((zero_zero_nat = hAPP_int_nat(number_number_of_nat,W_15) -> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),hAPP_int_nat(number_number_of_nat,W_15)) = one_on1645066479umeral) & (zero_zero_nat != hAPP_int_nat(number_number_of_nat,W_15) -> zero_z126310315umeral = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),hAPP_int_nat(number_number_of_nat,W_15))))) # label(fact_44_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4774 (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]. 9.53/9.57 4775 (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]. 9.53/9.57 4776 (all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> P_3 = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) | -hBOOL(hAPP_nat_bool(even_odd_even_nat,P_3)))) # label(fact_4195_prime__odd) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4777 (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]. 9.53/9.57 4778 (all X all Y hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X),Y)),hAPP_nat_nat(nat_gcd(X),Y)) = hAPP_nat_nat(nat_lcm(X),Y)) # label(fact_4655_Nitpick_Onat__lcm__def) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4779 (all N hAPP_int_nat(number_number_of_nat,N) = natceiling(hAPP_int_real(number267125858f_real,N))) # label(fact_3311_natceiling__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4780 (exists 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_4108_Im_Obounded) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4781 (all X_55 all Y_47 all Z_19 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_55),Y_47)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_55),Z_19)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_55),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,Y_47),Z_19))) # label(fact_1165_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4782 (all Ma all Na all A_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_1)) -> (hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_1),Ma) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_1),Na) <-> Ma = Na))) # label(fact_426_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4783 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) -> (Ma = Na <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),hAPP_nat_nat(suc,Ma)))))) # label(fact_3052_le__less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4784 (all B_89 all A_119 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_119)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_89)) -> 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_119),B_89)))))) # label(fact_1292_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4785 (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]. 9.53/9.57 4786 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K)),min)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),min)))) # label(fact_2170_rel__simps_I13_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4787 (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]. 9.53/9.57 4788 (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]. 9.53/9.57 4789 (all C_35 all D_13 all A_92 all B_63 all R_6 (R_6 != zero_zero_rat -> (A_92 = B_63 & D_13 != C_35 -> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_92),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,R_6),C_35)) != hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_63),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,R_6),D_13))))) # label(fact_1445_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4790 (all C_10 all A_41 all B_30 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_41),B_30)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_41),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_30),C_10))))) # label(fact_2286_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4791 (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]. 9.53/9.57 4792 (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_eq_nat,Xa),Ya)) <-> Xa = Ya)) # label(fact_1706_order__eq__iff) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4793 (all A_1 (zero_zero_rat = A_1 <-> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_1),A_1) = zero_zero_rat)) # label(fact_929_double__zero__sym) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4794 (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]. 9.53/9.57 4795 (all V_2 all V_1 ((hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1))) -> zero_zero_nat = 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))) -> (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_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)) = 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))))))) # label(fact_3275_mod__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4796 (all B_1_1 all B_2_1 (is_bool(B_1_1) & is_bool(B_2_1) -> is_bool(fdisj(B_1_1,B_2_1)))) # label(gsy_c_fdisj) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4797 (all W_15 ((hAPP_int_nat(number_number_of_nat,W_15) != zero_zero_nat -> zero_zero_int = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),hAPP_int_nat(number_number_of_nat,W_15))) & (hAPP_int_nat(number_number_of_nat,W_15) = zero_zero_nat -> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),hAPP_int_nat(number_number_of_nat,W_15)) = one_one_int))) # label(fact_47_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4798 (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]. 9.53/9.57 4799 (all C_15 all A_49 all B_36 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_49),B_36)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_36),C_15)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_49),C_15))))) # label(fact_2219_dvd__trans) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4800 (all X all Y ((Y != zero_zero_nat -> bezw(X,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)))))) & (Y = zero_zero_nat -> hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,one_one_int),zero_zero_int) = bezw(X,Y)))) # label(fact_4229_bezw_Osimps) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4801 (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(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,min))),hAPP_real_real(cos,Xa))),hAPP_real_real(sin,Xa))))) # label(fact_4782_DERIV__cos__cos__mult2) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4802 (all A_159 all B_107 all 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) = 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))) # label(fact_824_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4803 (all B_119 all A_173 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_173),zero_zero_rat)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_119),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_173),B_119)),zero_zero_rat))))) # label(fact_647_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4804 (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)) -> X_43 = Y_37)))) # label(fact_1656_order__antisym) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4805 (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]. 9.53/9.57 4806 (all X all Y hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y))) # label(fact_4306_gcd__semilattice__nat_Oinf__left__idem) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4807 (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]. 9.53/9.57 4808 (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),cos_coeff)),hAPP_r474017924t_real(power_power_real,Xa))),hAPP_real_real(cos,Xa)))) # label(fact_4864_cos__converges) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4809 (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]. 9.53/9.57 4810 (all Y_4 all X_4 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_4)) -> hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_4),X_4)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,Y_4)),X_4))) # label(fact_2753_abs__mult__pos) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4811 (all D_1 int_ge_less_than(D_1) = collec1347809874nt_int(produc93441119t_bool(hAPP_f428220345t_bool(hAPP_f1878066172t_bool(cOMBS_1720984575t_bool,hAPP_f1315737299t_bool(hAPP_f538565589t_bool(cOMBB_1211207212ol_int,cOMBB_bool_bool_int),hAPP_f2144054103l_bool(hAPP_f1734373249l_bool(cOMBB_1652995168ol_int,fconj),hAPP_i1948725293t_bool(ord_less_eq_int,D_1)))),ord_less_int)))) # label(fact_4861_int__ge__less__than__def) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4812 (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]. 9.53/9.57 4813 (all Va (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Va),pls)) <-> hAPP_int_nat(number_number_of_nat,Va) = zero_zero_nat)) # label(fact_756_eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4814 (all X all A_3 all B_2 (hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2) = one_one_nat -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),A_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),B_2)) -> X = one_one_nat)))) # label(fact_4348_coprime__common__divisor__nat) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4815 (all B_2 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),zero_zero_int)) -> hAPP_P1175774780nt_int(product_snd_int_int,hAPP_P1975530577nt_int(negateSnd,negDivAlg(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)))) = hAPP_int_int(div_mod_int(A_3),B_2)))) # label(fact_4236_mod__pos__neg) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4816 (all N all P_3 (hBOOL(hAPP_nat_bool(prime,P_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,P_3),N)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),N)) | N = one_one_nat)) # label(fact_4187_prime__coprime) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4817 (all B_1_1 all B_2_1 is_int(hAPP_nat_int(B_1_1,B_2_1))) # label(gsy_c_hAPP_000tc__Nat__Onat_000tc__Int__Oint) # label(hypothesis) # label(non_clause). [assumption]. 9.53/9.57 4818 (all 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) | 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]. 9.53/9.57 4819 (all P all Q all R hAPP_f627970963t_bool(P,hAPP_i68813070l_bool(Q,R)) = hAPP_i1529485324t_bool(hAPP_f1315737299t_bool(hAPP_f538565589t_bool(cOMBB_1211207212ol_int,P),Q),R)) # label(help_COMBB_1_1_COMBB_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_000tc__fun_Itc) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4820 (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]. 9.53/9.57 4821 (all P_1 all I_1 all J (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J)) -> (hBOOL(hAPP_nat_bool(P_1,J)) -> ((all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J)) -> (hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,I_2))) -> hBOOL(hAPP_nat_bool(P_1,I_2))))) -> hBOOL(hAPP_nat_bool(P_1,I_1)))))) # label(fact_3168_inc__induct) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4822 (all Xa (is_int(Xa) -> (-hBOOL(hAPP_int_bool(even_odd_even_int,Xa)) <-> (exists Y_1 (is_int(Y_1) & Xa = 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)))))) # label(fact_3280_odd__equiv__def) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4823 (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]. 9.53/9.57 4824 (all M_23 all N_47 hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_23),N_47)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,M_23)),N_47)) # label(fact_386_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4825 (all P_1 collect_nat(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))))) # label(fact_4738_Nat__Transfer_Otransfer__nat__int__set__functions_I5_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4826 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N))) # label(fact_3065_Suc__diff__le) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4827 (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]. 9.53/9.57 4828 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K1)),hAPP_int_int(bit0,K2))))) # label(fact_759_less__eq__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4829 (all L (is_int(L) -> (min = hAPP_int_int(bit1,L) <-> min = L))) # label(fact_2161_rel__simps_I43_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4830 (all Z_1 (is_int(Z_1) -> ((all N_1 Z_1 != hAPP_nat_int(semiri1621563631at_int,N_1)) -> -(all N_1 hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_1))) != Z_1)))) # label(fact_4548_int__cases) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4831 (all A_3 all B_2 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(ring_1_of_int_rat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2))),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2))) & hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2)),hAPP_int_rat(ring_1_of_int_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2)),one_one_int)))))) # label(fact_4725_rat__floor__lemma) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4832 (all Y hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),Y)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_real_real(tan,Y))) # label(fact_3766_tan__inverse) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4833 (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]. 9.53/9.57 4834 (all Ma all Na (is_int(Na) & is_int(Ma) -> (Na = hAPP_int_int(number_number_of_int,min) & Ma = hAPP_int_int(number_number_of_int,min) | one_one_int = Na & one_one_int = Ma <-> one_one_int = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),Na)))) # label(fact_2186_zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4835 (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]. 9.53/9.57 4836 (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]. 9.53/9.57 4837 (all A_156 all C_58 all 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)) = 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))) # label(fact_843_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4838 (all M all N hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(div_mod_nat(M),N))) # label(fact_3193_div__mod__equality_H) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4839 (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]. 9.53/9.57 4840 (all Y all X all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_3)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,Y),A_3))))))) # label(fact_4404_powr__less__mono2) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4841 (all A_3 all B_2 (B_2 != A_3 -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),A_3)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2))))) # label(fact_2646_dvd_Oneq__le__trans) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4842 (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),P_5)) -> hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(bnorRset(A_1),P_5)) = hAPP_int_nat(nat,A_1)))) # label(fact_4289_Bnor__prime) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4843 (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,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))),hAPP_f1861673493l_bool(hAPP_f1010710745l_bool(cOMBS_real_bool_bool,hAPP_f60716669l_bool(hAPP_f1485493409l_bool(cOMBB_624415285l_real,fconj),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,ord_less_eq_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),hAPP_r1134773055l_bool(hAPP_f695504873l_bool(cOMBC_real_real_bool,hAPP_f1393328161l_bool(hAPP_f1900646853l_bool(cOMBB_430461877l_real,fequal_real),sin)),Ya)))) = hAPP_real_real(arcsin,Ya)) # label(fact_4882_arcsin__def) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4844 (all A_165 all N_33 all B_113 (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_165),N_33)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_113),N_33))) -> (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_113)) -> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_165),B_113))))) # label(fact_751_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4845 (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,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_131),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,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))) # label(fact_1158_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4846 (all Z_18 all Y_40 all X_46 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_40),X_46)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Z_18),Y_40)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Z_18),X_46))))) # label(fact_1629_xt1_I6_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4847 (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]. 9.53/9.57 4848 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,X)),one_one_real)) -> (exists Z (X = hAPP_real_real(tan,Z) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,pi),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),Z)))))) # label(fact_3773_tan__total__pi4) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4849 (all K_2 zero_zero_rat = hAPP_int_rat(hAPP_int_fun_int_rat(fract,K_2),zero_zero_int)) # label(fact_4674_rat__number__collapse_I4_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4850 (all N hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),N))) # label(fact_3754_fact__lemma) # label(axiom) # label(non_clause). [assumption]. 9.53/9.57 4851 (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]. 9.53/9.57 4852 (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]. 9.53/9.57 4853 (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]. 9.53/9.58 4854 (all Na (hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_n1108039445umeral(semiri1619134803umeral,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)))) # label(fact_242_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4855 (all X_32 all Y_26 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_32),Y_26)) -> X_32 != Y_26)) # label(fact_1824_less__imp__neq) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4856 (all B_75 all A_104 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_104)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_75)) -> 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_104),B_75)))))) # label(fact_1380_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4857 (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]. 9.53/9.58 4858 (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]. 9.53/9.58 4859 (all K_2 (is_int(K_2) -> K_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),pls))) # label(fact_153_add__Pls__right) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4860 (all Na (Na = zero_zero_nat <-> 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]. 9.53/9.58 4861 (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]. 9.53/9.58 4862 (all A_144 all C_55 all B_102 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_144),C_55)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_102),C_55))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_144),B_102)))) # label(fact_961_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4863 (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]. 9.53/9.58 4864 (all A_50 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_50),zero_zero_int))) # label(fact_2205_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4865 (all M_14 all A_90 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_90),one_one_real)),M_14) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,M_14),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_90),M_14))) # label(fact_1481_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4866 (all Ya all Xa (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)) <-> Ya = Xa))) # label(fact_2644_dvd_Oantisym__conv) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4867 (all X hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,X))) = hAPP_int_int(abs_abs_int,X)) # label(fact_2939_int__nat__abs) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4868 (all V_22 all W_12 hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_22),W_12)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_22)),hAPP_int_real(number267125858f_real,W_12))) # label(fact_197_number__of__add) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4869 (all A_3 all B_2 hAPP_int_int(uminus_uminus_int,hAPP_int_int(div_mod_int(A_3),B_2)) = hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,A_3)),hAPP_int_int(uminus_uminus_int,B_2))) # label(fact_3472_zmod__zminus__zminus) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4870 (all Z_1 hAPP_int_real(ring_1_of_int_real,Z_1) = hAPP_complex_real(re,hAPP_int_complex(ring_11397209091omplex,Z_1))) # label(fact_4580_complex__Re__of__int) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4871 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,ratreal(Xa)),ratreal(Ya))) <-> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Xa),Ya)))) # label(fact_4699_real__less__eq__code) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4872 (all X (X != zero_zero_real -> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,X)),X) = one_one_real)) # label(fact_3857_real__mult__inverse__left) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4873 (all C all A_1 all B_1 (zero_zero_real = C | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_1),B_1)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_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))))) # label(fact_2362_dvd__mult__cancel__left) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4874 (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]. 9.53/9.58 4875 (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]. 9.53/9.58 4876 (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]. 9.53/9.58 4877 (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]. 9.53/9.58 4878 (all B_112 all A_164 all C_66 (hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_112),A_164) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_66),A_164) -> C_66 = B_112)) # label(fact_780_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4879 (all B_42 all A_65 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_42),A_65)) -> -hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_65),B_42)))) # label(fact_1772_xt1_I9_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4880 (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(dvd_dvd_int,Xa),Ya)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(nat,Xa)),hAPP_int_nat(nat,Ya))))))) # label(fact_2781_transfer__nat__int__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4881 (all A_137 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_137),zero_z891286103de_int) = zero_z891286103de_int) # label(fact_1114_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4882 (all Ma all Na ((exists J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),Na)) & Ma = hAPP_nat_nat(suc,J_1))) | Ma = zero_zero_nat <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(suc,Na))))) # label(fact_3039_less__Suc__eq__0__disj) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4883 (all M 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)))) = 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)))))) # label(fact_3140_div2__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4884 (all X all Y (hAPP_nat_nat(suc,Y) = hAPP_nat_nat(suc,X) -> Y = X)) # label(fact_2982_Suc__inject) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4885 (all Xa (one_one_real = hAPP_real_real(sqrt,Xa) <-> Xa = one_one_real)) # label(fact_3516_real__sqrt__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4886 (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]. 9.53/9.58 4887 (all P (hBOOL(hAPP_bool_bool(fNot,P)) | hBOOL(P))) # label(help_fNot_2_1_U) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4888 (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)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,Xa)),hAPP_real_real(ln,Ya))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)))))) # label(fact_3358_ln__le__cancel__iff) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4889 (all Xa all 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,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]. 9.53/9.58 4890 (all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),pls)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,Xa)),zero_zero_real)))) # label(fact_105_less__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4891 (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]. 9.53/9.58 4892 (all C_24 all D_9 all A_78 all B_51 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_78),B_51)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_24),D_9)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_51)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_24)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_78),C_24)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_51),D_9)))))))) # label(fact_1574_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4893 (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]. 9.53/9.58 4894 (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]. 9.53/9.58 4895 (all A_3 all B_2 all C_1 hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,A_3),C_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,B_2),C_1))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(div_mod_nat(A_3),C_1)),hAPP_nat_nat(div_mod_nat(B_2),C_1))),C_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_3),B_2)),C_1)) # label(fact_3184_div__add1__eq) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4896 (all Z_1 exists R_1 exists A_2 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_1)),hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(cos,A_2)),hAPP_real_real(sin,A_2))) = Z_1) # label(fact_3998_complex__split__polar) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4897 (all C_27 all D_12 all A_81 all B_54 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_81),B_54)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_27),D_12)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_81)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_27)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_81),C_27)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_54),D_12)))))))) # label(fact_1554_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4898 (all Z_17 all X_44 all Y_38 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_44),Y_38)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_38),Z_17)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_44),Z_17))))) # label(fact_1649_order__trans) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4899 (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]. 9.53/9.58 4900 (all Na (Na = one_one_nat <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,Na),Na)))) # label(fact_4018_coprime__refl) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4901 (all A_61 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_61),A_61)),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))))) # label(fact_1871_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4902 -(all X_1 all Y_1 (is_int(X_1) & is_int(Y_1) -> 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))))) != 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))))) # 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]. 9.53/9.58 4903 (all M all 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) = zero_zero_nat)) # label(fact_2884_div__less) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4904 (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]. 9.53/9.58 4905 (all X_29 -hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_29),X_29))) # label(fact_1850_order__less__irrefl) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4906 (all B_1 all A_1 all C (C = B_1 <-> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_1),A_1) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),A_1))) # label(fact_803_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4907 (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]. 9.53/9.58 4908 (all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natfloor(X)),natfloor(Y))))) # label(fact_3305_natfloor__mono) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4909 (all R_2 all X hAPP_complex_complex(scaleR1652505878omplex(R_2),X) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_2),hAPP_complex_real(re,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_2),hAPP_complex_real(im,X)))) # label(fact_5114_complex__scaleR__def) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4910 (all L all U hAPP_i1948725293t_bool(ord_at641636577an_int(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,L),one_one_int)),U) = hAPP_i1948725293t_bool(ord_gr1297742076an_int(L),U)) # label(fact_4920_atLeastPlusOneLessThan__greaterThanLessThan__int) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4911 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Y),X)) -> hAPP_int_int(abs_abs_int,Y) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y))) # label(fact_4974_gcd__proj2__if__dvd__int) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4912 (all M all N (N = M | 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]. 9.53/9.58 4913 (all X all A_3 all B_2 hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_3),B_2)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_2))) # label(fact_4411_powr__add) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4914 (all Na all Ma (Ma = one_one_nat & one_one_nat = Na <-> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Na),Ma) = one_one_nat)) # label(fact_2106_nat__mult__eq__one) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4915 (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]. 9.53/9.58 4916 (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]. 9.53/9.58 4917 (all I all J_2 all K_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J_2)),K_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K_2)))) # label(fact_296_add__lessD1) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4918 (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_3479_diff__int__def) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4919 (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]. 9.53/9.58 4920 (all Na all Ma (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Na),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,quickcheck_int_of(Na)),quickcheck_int_of(Ma))))) # label(fact_4251_less__eq__code__int__def) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4921 (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_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_nat,Xa),Ya))))) # label(fact_430_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4922 (all Ma all K all Na (hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K),Na) = one_one_int -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),Na))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),Ma))))) # label(fact_4963_coprime__dvd__mult__iff__int) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4923 (all K_2 hAPP_int_rat(ring_1_of_int_rat,K_2) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,K_2),one_one_int)) # label(fact_4681_Fract__of__int__eq) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4924 (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]. 9.53/9.58 4925 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(exp_real,Xa))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa)))) # label(fact_3407_one__less__exp__iff) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4926 (all P_1 all Na ((all X_1 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na))) -> hBOOL(hAPP_nat_bool(P_1,X_1)))) <-> (all 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_4901_all__nat__less__eq) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4927 (all K_2 all I all J_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I),J_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,J_2),K_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I),K_2))))) # label(fact_2079_real__le__trans) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4928 (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]. 9.53/9.58 4929 (all B_31 all A_42 all C_11 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_42),C_11)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_42),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_31),C_11))))) # label(fact_2278_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4930 (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]. 9.53/9.58 4931 (all Z_14 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_14),Z_14) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_14),hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_1881_mult__2__right) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4932 (all Ma all Na (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,Ma)),hAPP_nat_real(semiri132038758t_real,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)))) # label(fact_357_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4933 (all X all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)),X) = hAPP_real_real(hAPP_n546711566l_real(root,M),hAPP_real_real(hAPP_n546711566l_real(root,N),X)))))) # label(fact_3926_real__root__pos__mult__exp) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4934 (all Na hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_nat_real(real_nat,Na)),field_1210416355s_real))) # label(fact_4625_Rats__real__nat) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4935 (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]. 9.53/9.58 4936 (all X_26 hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_26),X_26) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_26),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_1909_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4937 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),hAPP_int_int(bit1,K))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K)))) # label(fact_2169_rel__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4938 (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]. 9.53/9.58 4939 (all X_2 (hBOOL(summable_complex(X_2)) -> hBOOL(summable_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_2))))) # label(fact_4845_Re_Osummable) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4940 (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]. 9.53/9.58 4941 (all A_51 all N_19 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_51),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_19))))) # label(fact_2145_zero__le__even__power_H) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4942 (all Q_1 all R_2 hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Q_1),R_2) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Q_1),hAPP_rat_rat(uminus_uminus_rat,R_2))) # label(fact_4644_diff__rat__def) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4943 (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]. 9.53/9.58 4944 (all C_75 all D_22 all A_186 all B_129 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_186),B_129)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_75),D_22)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_186),C_75)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_129),D_22)))))) # label(fact_527_add__mono) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4945 (all S ((all M_1 exists N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) & hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,N_1),S)))) <-> -hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S)))) # label(fact_4607_infinite__nat__iff__unbounded) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4946 (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]. 9.53/9.58 4947 (all A_198 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_198),one_one_nat) = A_198) # label(fact_319_power__one__right) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4948 (all M_9 all A_31 all N_15 all B_23 (hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_31),N_15)),B_23)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_9),N_15)) -> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_31),M_9)),B_23))))) # label(fact_2365_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4949 (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]. 9.53/9.58 4950 (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]. 9.53/9.58 4951 (all Ma all Na (Ma = Na <-> hAPP_nat_real(semiri132038758t_real,Ma) = hAPP_nat_real(semiri132038758t_real,Na))) # label(fact_324_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4952 (all A_64 all B_41 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_64),B_41)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_41),A_64)))) # label(fact_1781_order__less__asym_H) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4953 (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,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_24),W_14))),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))) # label(fact_187_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4954 (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]. 9.53/9.58 4955 (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(bit1,K)),min)))) # label(fact_2174_rel__simps_I30_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4956 (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]. 9.53/9.58 4957 (all N archim1246769320r_real(hAPP_real_real(uminus_uminus_real,hAPP_nat_real(real_nat,N))) = hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N))) # label(fact_4032_floor__minus__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4958 (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]. 9.53/9.58 4959 (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]. 9.53/9.58 4960 (all A_56 all B_38 (A_56 != B_38 -> (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]. 9.53/9.58 4961 (all N N = natceiling(hAPP_nat_real(real_nat,N))) # label(fact_3689_natceiling__real__of__nat) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4962 (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]. 9.53/9.58 4963 (all Lx_5 all Rx_5 all Ry_3 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_5),Rx_5)),Ry_3) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_5),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_5),Ry_3))) # label(fact_1047_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4964 (all A_128 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,one_one_complex),A_128) = A_128) # label(fact_1194_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4965 (all N all K_2 all M (one_one_int = hAPP_int_int(legacy_zgcd(K_2),M) -> (one_one_int = hAPP_int_int(legacy_zgcd(N),M) -> one_one_int = hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N)),M)))) # label(fact_4486_zgcd__zgcd__zmult) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4966 (all X all Y ((Y != zero_zero_nat -> hAPP_nat_nat(nat_gcd(X),Y) = hAPP_nat_nat(nat_gcd(Y),hAPP_nat_nat(div_mod_nat(X),Y))) & (Y = zero_zero_nat -> hAPP_nat_nat(nat_gcd(X),Y) = X))) # label(fact_4658_nat__gcd_Osimps) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4967 (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)) -> (Ya = zero_z126310315umeral & zero_z126310315umeral = Xa <-> zero_z126310315umeral = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,Xa),Ya))))) # label(fact_588_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4968 (all B_2 all A_3 (is_int(B_2) & is_int(A_3) -> (zero_zero_int != A_3 | zero_zero_int != B_2 -> one_one_int = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_2),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_3),B_2)))))) # label(fact_4965_div__gcd__coprime__int) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4969 (all N_37 all A_168 all B_114 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_168),B_114)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_168)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_168),N_37)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_114),N_37)))))) # label(fact_699_power__mono) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4970 (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]. 9.53/9.58 4971 (all C_38 all A_95 all B_66 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_95),B_66)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_38)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_38),A_95)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_38),B_66)))))) # label(fact_1435_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4972 (all A_1 all 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)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),archim1246769320r_real(Xa))))) # label(fact_4467_less__floor__eq) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4973 (all P all Q all R hAPP_nat_fun_rat_rat(hAPP_f1830834240at_rat(hAPP_f1073644437at_rat(cOMBB_1926947at_nat,P),Q),R) = hAPP_rat_fun_rat_rat(P,hAPP_nat_rat(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__Rat__Orat_000tc__fun_Itc__Rat__Orat_Mtc__Rat__Orat_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4974 (all X_2 (hBOOL(summable_complex(X_2)) -> hAPP_f352196356l_real(suminf_real,hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_2)) = hAPP_complex_real(re,hAPP_f213450978omplex(suminf_complex,X_2)))) # label(fact_4847_Re_Osuminf) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4975 (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,X_1),Ta))) <-> 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))))))) # label(fact_2238_inf__period_I3_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4976 (all X_55 all Y_47 all 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)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_55),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Y_47),Z_19))) # label(fact_1167_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4977 (all B_76 all 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,zero_zero_rat),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,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]. 9.53/9.58 4978 (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]. 9.53/9.58 4979 (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]. 9.53/9.58 4980 (all X (-hBOOL(hAPP_nat_bool(even_odd_even_nat,X)) -> hAPP_nat_nat(div_mod_nat(X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = hAPP_nat_nat(suc,zero_zero_nat))) # label(fact_3803_odd__nat__mod__two__eq__one) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4981 (all W hAPP_int_complex(number528085621omplex,W) = hAPP_int_complex(ring_11397209091omplex,W)) # label(fact_4577_complex__number__of__def) # label(axiom) # label(non_clause). [assumption]. 9.53/9.58 4982 (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]. 9.53/9.58 4983 (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]. 9.53/9.58 4984 (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]. 9.53/9.59 4985 (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]. 9.53/9.59 4986 (all A_1 all B_1 all C all D_1 (hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C),D_1) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_1),B_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C),D_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B_1))))) # label(fact_2335_diff__eq__diff__less__eq) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 4987 (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]. 9.53/9.59 4988 (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]. 9.53/9.59 4989 (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]. 9.53/9.59 4990 (all X all 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))) = hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X),Y))) # label(fact_4130_complex__Re__mult) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 4991 (all Y all X ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X)) -> hAPP_int_int(nat_tsub(X),Y) = zero_zero_int) & (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_3155_tsub__def) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 4992 (all N_56 all A_200 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_200)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_56)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_200),N_56)))))) # label(fact_250_one__less__power) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 4993 (all F (hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,abs_abs_real),F),zero_zero_real,sequentially)) <-> hBOOL(tendsto_nat_real(F,zero_zero_real,sequentially)))) # label(fact_5040_LIMSEQ__rabs__zero) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 4994 (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]. 9.53/9.59 4995 (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]. 9.53/9.59 4996 (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]. 9.53/9.59 4997 (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]. 9.53/9.59 4998 (all Ma (Ma != zero_zero_nat -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,dvd_dvd_nat),Ma)))))) # label(fact_4763_finite__divisors__nat) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 4999 (all C_1 all D_3 all A_3 all B_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C_1),D_3),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_3),C_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_2),D_3)),M))))) # label(fact_2582_zcong__zadd) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5000 (all K all Ma all Na (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),hAPP_int_int(legacy_zgcd(Ma),Na))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),Na)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),Ma)))) # label(fact_4493_zgcd__greatest__iff) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5001 (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]. 9.53/9.59 5002 (all A_203 A_203 = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,pls)),A_203)) # label(fact_173_add__numeral__0) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5003 (all Z_1 hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_1),hAPP_complex_complex(cnj,Z_1)) = of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(im,Z_1)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) # label(fact_4110_complex__mult__cnj) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5004 (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]. 9.53/9.59 5005 (all X hAPP_real_real(cos,X) = hAPP_real_real(cos,hAPP_real_real(uminus_uminus_real,X))) # label(fact_3593_cos__minus) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5006 (all B_1 all A_1 all C (hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),A_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_1),A_1) <-> B_1 = C)) # label(fact_805_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5007 (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]. 9.53/9.59 5008 (all A_3 hAPP_int_rat(hAPP_int_fun_int_rat(fract,zero_zero_int),one_one_int) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),zero_zero_int)) # label(fact_4670_eq__rat_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5009 (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]. 9.53/9.59 5010 (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]. 9.53/9.59 5011 (all M all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),M)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),M))))) # label(fact_2587_zcong__not__zero) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5012 (all W ((hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,W))) -> zero_zero_nat = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,W))) & (-hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,W))) -> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,W)),hAPP_int_nat(number_number_of_nat,W))) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,W))))) # label(fact_4780_nat__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5013 (all C all A_1 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,C),A_1)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),B_1))))) # label(fact_978_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5014 (all X_56 hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_56),X_56))) # label(fact_1018_order__refl) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5015 (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]. 9.53/9.59 5016 (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]. 9.53/9.59 5017 (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]. 9.53/9.59 5018 (all P_1 image_nat_int(semiri1621563631at_int,collect_nat(hAPP_f1154658180t_bool(hAPP_f1361476881t_bool(cOMBB_int_bool_nat,P_1),semiri1621563631at_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))),P_1))) # label(fact_4771_Nat__Transfer_Otransfer__int__nat__set__functions_I5_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5019 (all X_39 all Y_33 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_39),Y_33)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_33),X_39)))) # label(fact_1725_order__less__asym) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5020 (all Z_1 hAPP_complex_real(re,of_real_complex(Z_1)) = Z_1) # label(fact_4112_Re__complex__of__real) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5021 (all X all Y (hAPP_complex_real(re,X) = hAPP_complex_real(re,Y) -> (hAPP_complex_real(im,X) = hAPP_complex_real(im,Y) -> X = Y))) # label(fact_4115_complex__eqI) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5022 (all A_91 all B_61 all V_10 hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_91),hAPP_int_real(number267125858f_real,V_10))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_61),hAPP_int_real(number267125858f_real,V_10))) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_91),B_61)),hAPP_int_real(number267125858f_real,V_10))) # label(fact_1466_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5023 (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]. 9.53/9.59 5024 (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]. 9.53/9.59 5025 (all V_24 all W_14 all Z_21 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_24)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,W_14)),Z_21)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_24),W_14))),Z_21)) # label(fact_188_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5026 (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]. 9.53/9.59 5027 (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(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(minus_minus_real,X),Y))) # label(fact_3590_sin__diff) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5028 (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]. 9.53/9.59 5029 (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]. 9.53/9.59 5030 (all X_2 (hBOOL(summable_complex(X_2)) -> hAPP_f213450978omplex(suminf_complex,hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_2)) = hAPP_complex_complex(cnj,hAPP_f213450978omplex(suminf_complex,X_2)))) # label(fact_4843_cnj_Osuminf) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5031 (all M all N hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),N)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N)) # label(fact_4972_gcd__add2__int) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5032 (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_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))) = hAPP_real_real(ln,Xa)))) # label(fact_4825_ln__series) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5033 (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]. 9.53/9.59 5034 (all A_128 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),A_128) = A_128) # label(fact_1196_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5035 (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_eq_rat,A_1),B_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_1578_mult__le__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5036 (all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(semiri984289939at_nat,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)))) # label(fact_244_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5037 (all I_1 all M_2 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),M_2)) -> hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_f800510211t_bool(hAPP_f561022312t_bool(cOMBS_nat_bool_bool,hAPP_f1146629647l_bool(hAPP_f1080886329l_bool(cOMBB_1015721476ol_nat,fconj),hAPP_f800510211t_bool(hAPP_f1722879237t_bool(cOMBC_226598744l_bool,member_nat),M_2))),hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),hAPP_nat_nat(suc,I_1))))) = hAPP_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))))))) # label(fact_4768_card__less__Suc) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5038 (all Nat_3 zero_zero_nat != hAPP_nat_nat(suc,Nat_3)) # label(fact_2993_nat_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5039 (all X_26 hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_26),X_26) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_26),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_1905_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5040 (all X_6 all N_11 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_11)) -> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_6),N_11) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_11),one_one_nat))),X_6))) # label(fact_2559_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5041 (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]. 9.53/9.59 5042 (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]. 9.53/9.59 5043 (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]. 9.53/9.59 5044 (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]. 9.53/9.59 5045 (all Z_11 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_11),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)) # label(fact_1894_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5046 (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]. 9.53/9.59 5047 (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]. 9.53/9.59 5048 (all M_18 all N_30 hAPP_n522471361de_int(semiri1424489471de_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_18),N_30)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_18)),hAPP_n522471361de_int(semiri1424489471de_int,N_30))) # label(fact_1101_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5049 (all P_1 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(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Xa),Ya))))) -> ((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,Ya),zero_zero_int)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Xa),hAPP_int_int(uminus_uminus_int,Ya)))))) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ya)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(uminus_uminus_int,Xa)),Ya))))) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ya),zero_zero_int)) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(uminus_uminus_int,Xa)),hAPP_int_int(uminus_uminus_int,Ya)))))) -> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Xa),Ya)))))))) # label(fact_5013_gcd__cases__int) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5050 (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]. 9.53/9.59 5051 (all P all Q all R hAPP_f800361423l_real(P,hAPP_r2019696015l_real(Q,R)) = hAPP_r712953929l_real(hAPP_f1119546819l_real(hAPP_f555557763l_real(cOMBB_422423973l_real,P),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]. 9.53/9.59 5052 (all Wa all Ya all Xa all Z_2 (is_int(Wa) & is_int(Z_2) & is_int(Xa) & is_int(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)) = 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)) <-> Wa = Xa | Z_2 = Ya))) # label(fact_1191_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5053 (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]. 9.53/9.59 5054 (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]. 9.53/9.59 5055 (all X 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)))))) # label(fact_3416_real__sum__of__halves) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5056 (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]. 9.53/9.59 5057 (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]. 9.53/9.59 5058 (all Xa all Ya (is_int(Ya) & is_int(Xa) -> (Xa = Ya <-> hAPP_int_int(number_number_of_int,Xa) = hAPP_int_int(number_number_of_int,Ya)))) # label(fact_87_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5059 (all C all A_1 all B_1 (hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),A_1)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),B_1))) <-> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_1),B_1)))) # label(fact_541_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5060 (all C_45 all A_112 all B_82 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_112),B_82)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_45)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_45),A_112)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_45),B_82)))))) # label(fact_1334_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5061 (all K_2 all I all J_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),J_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),I)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K_2),J_2))))) # label(fact_563_zadd__left__mono) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5062 (all A_1 (zero_zero_rat = A_1 <-> 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]. 9.53/9.59 5063 (all N_14 all M_8 all X_11 all Y_8 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_11),Y_8)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_14),M_8)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_11),N_14)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_8),M_8)))))) # label(fact_2370_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5064 (all A_3 (is_int(A_3) -> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),zero_zero_int) = zero_zero_int & hAPP_int_int(div_mod_int(A_3),zero_zero_int) = A_3)) # label(fact_3036_DIVISION__BY__ZERO) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5065 (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]. 9.53/9.59 5066 (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]. 9.53/9.59 5067 (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),zEven)) -> 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_3355_odd__minus__even) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5068 (all X all Y hAPP_real_real(dist_dist_real(X),Y) = hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y))) # label(fact_4544_dist__real__def) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5069 (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]. 9.53/9.59 5070 (all A_3 hAPP_complex_real(im,cis(A_3)) = hAPP_real_real(sin,A_3)) # label(fact_4146_Im__cis) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5071 (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]. 9.53/9.59 5072 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real)) -> hAPP_real_real(cos,hAPP_real_real(arcsin,X)) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) # label(fact_3588_cos__arcsin) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5073 (all Xa all Ya (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya)) -> Ya != Xa)) # label(fact_1799_order__less__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5074 (all X_15 all Y_11 hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_15),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_11),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_15)),Y_11)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_15),Y_11)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) # label(fact_2053_power2__sum) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5075 (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]. 9.53/9.59 5076 (all K all Ma all Na (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) <-> Na = Ma | K = zero_zero_nat)) # label(fact_2188_nat__mult__eq__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5077 (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]. 9.53/9.59 5078 (all C_43 all B_80 all A_110 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_80),A_110)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_43),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_43),A_110)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_43),B_80)))))) # label(fact_1342_mult__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5079 (all C_52 all A_141 all B_99 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_141),B_99)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_141),C_52)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_99),C_52))))) # label(fact_975_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5080 (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,Xa),Ya)) | Xa = Ya)) # label(fact_1963_order__le__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5081 (all Wa all 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)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_2)))) # label(fact_2772_zless__nat__conj) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5082 (all Z_7 all X_22 all Y_18 (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_22),Y_18)) -> (hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_18),Z_7)) -> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_22),Z_7))))) # label(fact_1940_order__less__le__trans) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5083 (all P all Q all R hAPP_P603027463t_bool(hAPP_i119638232t_bool(hAPP_f2018027565t_bool(cOMBC_33652895t_bool,P),Q),R) = hAPP_int_bool(hAPP_P252494982t_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__Int__Oin_003) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5084 (all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,N))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,N)))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,N)))))) # label(fact_1007_number__of1) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5085 (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]. 9.53/9.59 5086 (all Z_1 Z_1 = hAPP_complex_complex(cnj,hAPP_complex_complex(cnj,Z_1))) # label(fact_4068_complex__cnj__cnj) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5087 (all A_91 all B_61 all V_10 hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_91),hAPP_int_int(number_number_of_int,V_10))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_61),hAPP_int_int(number_number_of_int,V_10))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_91),B_61)),hAPP_int_int(number_number_of_int,V_10))) # label(fact_1468_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5088 (all B_2 zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,zero_zero_int),B_2)) # label(fact_2821_zdiv__zero) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5089 (all X all Y (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Y)) -> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Y),hAPP_int_int(div_mod_int(X),Y)))) # label(fact_5005_gcd__non__0__int) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5090 (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]. 9.53/9.59 5091 (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)) & (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> hAPP_nat_nat(div_mod_nat(M),N) = M))) # label(fact_3181_mod__if) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5092 (all A_3 all B_2 all Qr_1 (hBOOL(hAPP_P603027463t_bool(divmod_int_rel(hAPP_int_int(uminus_uminus_int,A_3),hAPP_int_int(uminus_uminus_int,B_2)),Qr_1)) -> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_P1975530577nt_int(negateSnd,Qr_1))))) # label(fact_4189_divmod__int__rel__neg) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5093 (all R_2 all Q_1 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_3)) -> (A_3 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,R_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Q_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_eq_int,Q_1),one_one_int)))))) # label(fact_2057_self__quotient__aux2) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5094 (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_2631_dvd_Oless__imp__not__less) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5095 (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]. 9.53/9.59 5096 (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]. 9.53/9.59 5097 (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(bit1,L_2)) = hAPP_int_int(bit1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(pred,K_2)),L_2))) # label(fact_4048_diff__bin__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5098 (all M all K_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),M)) -> (hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) = K_2 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),K_2))))))) # label(fact_3097_prod__mn__less__k) # label(axiom) # label(non_clause). [assumption]. 9.53/9.59 5099 (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]. 9.53/9.59 5100 (all X all Y (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)))) # label(fact_2632_dvd_Oless__imp__le) # label(axiom) # label(non_clause). [assumption]. 21.46/21.57 5101 (all A_88 all M_12 all N_26 hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_88),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_12),N_26)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_88),M_12)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_88),N_26))) # label(fact_1492_power__add) # label(axiom) # label(non_clause). [assumption]. 21.46/21.57 5102 -(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) != zero_zero_int) # label(conj_0) # label(negated_conjecture) # label(non_clause). [assumption]. 21.46/21.57 21.46/21.57 ============================== end of process non-clausal formulas === 21.46/21.57 21.46/21.57 ============================== PROCESS INITIAL CLAUSES =============== 21.46/21.57 21.46/21.57 ============================== PREDICATE ELIMINATION ================= 21.46/21.57 21.46/21.57 ============================== end predicate elimination ============= 21.46/21.57 21.46/21.57 Auto_denials: (non-Horn, no changes). 21.46/21.57 21.46/21.57 Term ordering decisions: 21.46/21.57 Function symbol KB weights: ord_less_int=1. zero_zero_int=1. member_int=1. one_one_int=1. minus_minus_int=1. times_times_int=1. plus_plus_int=1. zero_zero_real=1. zero_zero_nat=1. pls=1. bit1=1. ord_less_real=1. ord_less_eq_real=1. bit0=1. ord_less_eq_int=1. ord_less_nat=1. number_number_of_nat=1. times_times_real=1. times_times_nat=1. number_number_of_int=1. ord_less_eq_nat=1. suc=1. plus_plus_nat=1. plus_plus_real=1. power_power_real=1. number267125858f_real=1. zero_zero_rat=1. one_one_real=1. dvd_dvd_nat=1. times_times_rat=1. power_power_int=1. one_one_nat=1. semiri1621563631at_int=1. ord_less_rat=1. inverse_divide_real=1. ord_less_eq_rat=1. dvd_dvd_int=1. real_nat=1. nat=1. div_div_int=1. minus_minus_nat=1. plus_plus_rat=1. minus_minus_real=1. abs_abs_real=1. power_power_nat=1. times_times_complex=1. min=1. abs_abs_int=1. ord_le1860547276de_int=1. ord_le258702272de_int=1. cos=1. zero_z126310315umeral=1. zero_z891286103de_int=1. gcd_gcd_nat=1. gcd_gcd_int=1. cOMBS_nat_real_real=1. cOMBB_1308947378al_nat=1. times_123202395de_int=1. pi=1. plus_plus_complex=1. product_Pair_int_int=1. sin=1. fact_fact_nat=1. real_int=1. uminus_uminus_real=1. uminus_uminus_int=1. number_number_of_rat=1. even_odd_even_nat=1. plus_p1446045655de_int=1. power_power_rat=1. div_div_nat=1. sqrt=1. ord_le1304079648umeral=1. root=1. ord_le565307924umeral=1. produc865579683l_real=1. one_one_rat=1. cOMBB_nat_real_nat=1. number528085621omplex=1. fract=1. big_co604158596t_real=1. power_881366806de_int=1. zero_zero_complex=1. nat_neg=1. abs_abs_rat=1. even_odd_even_int=1. times_1655362735umeral=1. ln=1. power_power_complex=1. complex=1. re=1. im=1. minus_minus_rat=1. zprime=1. coprime=1. fconj=1. power_2100829034umeral=1. dvd_dvd_real=1. inverse_inverse_real=1. ord_le1568362934t_bool=1. sequentially=1. powr=1. cnj=1. ord_le951220754t_bool=1. dvd_dvd_rat=1. exp_real=1. cOMBC_nat_nat_nat=1. dvd_dvd_complex=1. n=1. tan=1. one_on1645066479umeral=1. one_one_complex=1. prime=1. zEven=1. cOMBC_nat_real_real=1. one_on1684967323de_int=1. pred=1. minus_minus_complex=1. zOdd=1. ord_le1912455174t_bool=1. ord_le382113706t_bool=1. product_snd_int_int=1. semiri132038758t_real=1. semiri151668891at_rat=1. suminf_real=1. v=1. w=1. cOMBB_1652995168ol_int=1. m=1. cOMBB_882763271l_real=1. cOMBS_int_bool_bool=1. cOMBB_real_real_nat=1. plus_p1627245867umeral=1. norm_norm_complex=1. nat_nat_set=1. finite_finite_nat=1. nat_is_nat=1. member_real=1. product_Pair_nat_nat=1. member_nat=1. arctan=1. product_fst_int_int=1. product_snd_nat_nat=1. finite_card_int=1. ii=1. semiri984289939at_nat=1. succ=1. cOMBS_real_real_real=1. fact_fact_int=1. cOMBC_int_int_bool=1. dvd_dv1760642554de_int=1. semiri1424489471de_int=1. semiri2020571505omplex=1. cOMBB_963856155at_nat=1. finite_finite_int=1. cOMBC_real_nat_real=1. number1226105091de_int=1. arccos=1. produc1935615926l_real=1. produc556554744l_real=1. r=1. sa=1. fequal_int=1. number1443263063umeral=1. arcsin=1. cOMBC_nat_nat_bool=1. sRStar=1. finite_card_nat=1. cOMBB_624415285l_real=1. cOMBC_real_real_real=1. cOMBS_real_bool_bool=1. quadRes=1. cOMBB_599996122l_real=1. cOMBC_real_real_bool=1. fFalse=1. fTrue=1. member_fun_int_bool=1. quickcheck_of_int=1. fequal_real=1. ord_min_nat=1. product_fst_nat_nat=1. cOMBB_real_real_real=1. cOMBC_int_int_int=1. norRRset=1. ord_lessThan_nat=1. s=1. cOMBI_int=1. if_real=1. ord_max_nat=1. t=1. uminus473333897omplex=1. cOMBB_64450052al_nat=1. cOMBB_722610049al_nat=1. invers1449016382omplex=1. negateSnd=1. sgn_sgn_int=1. sin_coeff=1. cOMBB_1480366270al_nat=1. cOMBB_782625832nt_int=1. cOMBB_nat_nat_nat=1. cos_coeff=1. if_nat=1. m1=1. norm_frac_rel=1. sgn_sgn_real=1. twoSqu1770640450sum2sq=1. x=1. y=1. cOMBB_1419615765al_nat=1. cOMBB_1948233853l_real=1. cOMBB_430461877l_real=1. fact=1. if_Pro1731782967nt_int=1. inverse_divide_rat=1. negDivAlg_rel=1. posDivAlg_rel=1. ring_1_of_int_rat=1. big_co1758102091ol_nat=1. cOMBB_1496585939nt_int=1. cOMBB_1934313333l_real=1. cOMBC_2010481033l_real=1. if_int=1. ord_less_eq_bool=1. cOMBB_1015721476ol_nat=1. cOMBB_800536526ol_nat=1. cOMBS_nat_bool_bool=1. dvd_dv174992974umeral=1. field_1210416355s_real=1. norm_norm_real=1. ord_atMost_nat=1. produc282740534nt_int=1. s1=1. twoSqu1907779896sum2sq=1. uminus_uminus_rat=1. zfact=1. big_co387207925at_nat=1. cOMBB_117013006ex_nat=1. cOMBS_337378985l_real=1. d22set=1. fEx_int=1. fNot=1. int_gcd=1. invers1025623611omplex=1. lazy_small_lazy_rel=1. member180897546at_nat=1. nat_gcd_rel=1. ring_1_of_int_real=1. semiri1619134803umeral=1. sgn_sgn_complex=1. upto_rel=1. cOMBB_1533963633al_nat=1. cOMBB_1550063917nt_int=1. cOMBB_182850971nt_int=1. cOMBB_662120866nt_int=1. cOMBB_bool_bool_int=1. cOMBB_int_bool_int=1. cOMBC_226598744l_bool=1. cOMBC_746811033l_real=1. cOMBS_1470252563nt_int=1. finite1395289673t_bool=1. quickc1265749348ro_rel=1. ring_11397209091omplex=1. sgn_sgn_rat=1. cOMBB_1344650379nt_int=1. cOMBB_1418110531ol_int=1. cOMBB_1506897658nt_int=1. cOMBB_1768400739l_real=1. cOMBB_1846624359nt_int=1. cOMBB_1896127834l_real=1. cOMBB_1926947at_nat=1. cOMBB_2092576745l_real=1. cOMBB_320287386nt_int=1. cOMBB_421464275l_real=1. cOMBB_638531587nt_int=1. cOMBC_1329014627t_real=1. cOMBC_1508019322l_real=1. cOMBC_1805044090nt_int=1. cOMBC_1964283556nt_int=1. cOMBC_33652895t_bool=1. cOMBC_707974837nt_int=1. cOMBC_94739984l_bool=1. cOMBI_nat=1. cOMBS_1891347093nt_int=1. cOMBS_400904860l_bool=1. cOMBS_nat_rat_rat=1. div_di1430059507de_int=1. fequal_nat=1. inverse_inverse_rat=1. member2143287562nt_int=1. produc494345619at_nat=1. size_size_nat=1. suminf_complex=1. big_co230513141nt_int=1. cOMBB_1121934354l_real=1. cOMBB_1146692694ol_nat=1. cOMBB_1484610687nt_int=1. cOMBB_1946221070al_int=1. cOMBB_2027861294ol_int=1. cOMBB_2107848501l_real=1. cOMBB_389152643ol_int=1. cOMBB_422423973l_real=1. cOMBB_591320580ol_int=1. cOMBB_793166024ol_int=1. cOMBB_real_bool_int=1. cOMBB_real_bool_real=1. cOMBC_1683390479t_bool=1. cOMBC_983868720t_bool=1. cOMBC_int_nat_int=1. cOMBS_193881978l_real=1. div_di1218280263umeral=1. fequal_rat=1. if_Pro313124157l_real=1. member308914164t_bool=1. minus_1690775515umeral=1. nat_size=1. ratrel=1. size_s945831648umeral=1. sr=1. cOMBB_1102189010nt_int=1. cOMBB_1159313332l_real=1. cOMBB_118231410ol_int=1. cOMBB_1211207212ol_int=1. cOMBB_1247074793l_real=1. cOMBB_1290694363ol_int=1. cOMBB_1317819104ol_int=1. cOMBB_1389251822nt_int=1. cOMBB_1453147152l_real=1. cOMBB_1459832681l_real=1. cOMBB_1593174431ol_int=1. cOMBB_1647910634ol_int=1. cOMBB_1657320178nt_int=1. cOMBB_170828966nt_int=1. cOMBB_1714920583ex_nat=1. cOMBB_172850899l_real=1. cOMBB_1795223523l_real=1. cOMBB_1875755114nt_int=1. cOMBB_188663909l_real=1. cOMBB_2046938128ol_int=1. cOMBB_2050079537l_real=1. cOMBB_2086486755nt_int=1. cOMBB_2087181719ol_int=1. cOMBB_2125215742l_real=1. cOMBB_2129825580al_int=1. cOMBB_2138578194l_real=1. cOMBB_214503962ol_int=1. cOMBB_222539774nt_int=1. cOMBB_237479429l_real=1. cOMBB_270132509l_real=1. cOMBB_348438404at_nat=1. cOMBB_3698607l_real=1. cOMBB_415306361l_real=1. cOMBB_421479067l_real=1. cOMBB_44084907l_real=1. cOMBB_47643171nt_int=1. cOMBB_520243667l_real=1. cOMBB_550562460ol_int=1. cOMBB_557258983l_real=1. cOMBB_562416518ol_int=1. cOMBB_563965291nt_int=1. cOMBB_569260360nt_int=1. cOMBB_595786433nt_int=1. cOMBB_650041899nt_int=1. cOMBB_690122963l_real=1. cOMBB_731595089l_real=1. cOMBB_735549356nt_int=1. cOMBB_773384995nt_int=1. cOMBB_798864284nt_int=1. cOMBB_859312247nt_nat=1. cOMBB_866192175nt_int=1. cOMBB_929094628al_int=1. cOMBB_95956018ol_int=1. cOMBB_984071017l_real=1. cOMBB_bool_bool_nat=1. cOMBB_int_bool_nat=1. cOMBB_int_int_int=1. cOMBB_int_int_nat=1. cOMBB_int_rat_int=1. cOMBB_int_real_int=1. cOMBB_nat_bool_int=1. cOMBB_rat_bool_int=1. cOMBB_rat_rat_nat=1. cOMBB_real_bool_nat=1. cOMBB_real_real_int=1. cOMBC_1113900202t_bool=1. cOMBC_1417754532nt_int=1. cOMBC_1801450329t_bool=1. cOMBC_2037038898nt_int=1. cOMBC_241247961t_bool=1. cOMBC_243356514t_bool=1. cOMBC_295337211t_real=1. cOMBC_412863505t_bool=1. cOMBC_467724138l_real=1. cOMBC_678732631t_bool=1. cOMBC_736024425nt_int=1. cOMBC_824100275t_real=1. cOMBC_int_rat_bool=1. cOMBC_int_real_bool=1. cOMBC_nat_int_int=1. cOMBI_real=1. cOMBS_107495110l_real=1. cOMBS_1610675060t_bool=1. cOMBS_163651580l_real=1. cOMBS_1640193733nt_int=1. cOMBS_1676841879t_bool=1. cOMBS_1720984575t_bool=1. cOMBS_2011279539nt_int=1. cOMBS_2019229620nt_int=1. cOMBS_480885903t_bool=1. cOMBS_511527878l_real=1. cOMBS_549334880nt_rat=1. cOMBS_696648398omplex=1. cOMBS_794971624l_real=1. cOMBS_902912708l_real=1. cOMBS_nat_nat_nat=1. fEx_nat=1. iszero_int=1. iszero_rat=1. pair_leq=1. pair_less=1. produc883642259nt_int=1. ring_1_Ints_real=1. ring_1_of_int_int=1. tn=1. big_co1024481617at_int=1. finite1876863882t_bool=1. frac=1. int=1. minus_534354567de_int=1. ord_atMost_real=1. ord_lessThan_real=1. pred_nat=1. produc1318306967de_int=1. produc2136830103umeral=1. produc713279741t_bool=1. size_size_complex=1. size_size_list_int=1. c1=1. c2=1. c3=1. c4=1. c5=1. c6=1. c7=1. c8=1. c9=1. c10=1. c11=1. c12=1. c13=1. c14=1. c15=1. c16=1. c17=1. c18=1. c19=1. c20=1. c21=1. c22=1. c23=1. hAPP_int_bool=1. hAPP_int_int=1. hAPP_i1948725293t_bool=1. hAPP_int_fun_int_int=1. hAPP_f448129468l_bool=1. hAPP_i2112223885l_bool=1. hAPP_real_real=1. hAPP_nat_nat=1. hAPP_real_bool=1. hAPP_r1134773055l_bool=1. hAPP_nat_bool=1. hAPP_n1699378549t_bool=1. hAPP_nat_fun_nat_nat=1. hAPP_r1250527377l_real=1. hAPP_int_nat=1. hAPP_rat_rat=1. hAPP_rat_bool=1. hAPP_nat_int=1. hAPP_r2115560837t_bool=1. hAPP_nat_real=1. hAPP_rat_fun_rat_rat=1. hAPP_int_real=1. hAPP_r474017924t_real=1. hAPP_complex_complex=1. hAPP_c172932010omplex=1. hAPP_Q2096512830t_bool=1. hAPP_Q1010094925t_bool=1. hAPP_int_fun_nat_int=1. hAPP_n546711566l_real=1. hAPP_Q1762011733de_int=1. hAPP_Q1729056540de_int=1. hAPP_int_rat=1. hAPP_C24501650l_bool=1. hAPP_C10902773l_bool=1. hAPP_f1888375463t_real=1. hAPP_nat_rat=1. hAPP_complex_real=1. hAPP_f54304608l_bool=1. hAPP_f451358433l_real=1. hAPP_f197615720t_real=1. hAPP_f1092742621l_real=1. hAPP_i1584592887nt_int=1. hAPP_i1524277240nt_int=1. deriv_real=1. hAPP_C498520661umeral=1. hAPP_rat_fun_nat_rat=1. hAPP_C1594335432umeral=1. hAPP_n522471361de_int=1. hAPP_r2019696015l_real=1. hAPP_r1195171167l_real=1. hAPP_f103356543l_bool=1. hAPP_nat_complex=1. hAPP_int_complex=1. hAPP_f284875647l_bool=1. hAPP_P1333854989l_bool=1. hAPP_f1566856189t_real=1. hAPP_f1676298106t_real=1. hAPP_int_fun_int_rat=1. hAPP_Q125967240de_int=1. hAPP_P1175774780nt_int=1. hAPP_f1509287679l_real=1. hAPP_f1406902706l_real=1. hAPP_c1088319240omplex=1. hAPP_P603027463t_bool=1. hAPP_r265291036omplex=1. hAPP_real_complex=1. hAPP_n1108039445umeral=1. hAPP_complex_bool=1. hAPP_C471253896umeral=1. hAPP_c1403687985x_bool=1. hAPP_f416620757at_nat=1. hAPP_f42270917t_real=1. hAPP_f1805168059t_bool=1. hAPP_P1654798560at_nat=1. hAPP_f1950183573l_real=1. hAPP_f352196356l_real=1. hAPP_f1734373249l_bool=1. hAPP_f1960414057l_real=1. hAPP_f2144054103l_bool=1. hAPP_f1690286043t_real=1. hAPP_f727283836t_bool=1. hAPP_P1555980039t_bool=1. hAPP_P1975530577nt_int=1. hAPP_f556082547l_real=1. hAPP_i1732201573de_int=1. hAPP_n332733730l_real=1. hAPP_f1501066425l_bool=1. hAPP_n1865633855at_nat=1. hAPP_r1130899993l_bool=1. hAPP_P731461727l_real=1. hAPP_n1289843868at_nat=1. hAPP_f428220345t_bool=1. hAPP_n215258509l_bool=1. divmod_nat_rel=1. posDivAlg=1. divmod_int_rel=1. negDivAlg=1. hAPP_bool_bool=1. hAPP_f2069516469l_real=1. hAPP_f957591787ol_nat=1. hAPP_f1639111240at_nat=1. hAPP_f1731313045at_nat=1. hAPP_f22106695ol_nat=1. image_nat_int=1. hAPP_f248847589l_real=1. image_int_nat=1. hAPP_f215623910l_bool=1. hAPP_i769753017umeral=1. hAPP_f229349961t_bool=1. hAPP_b589554111l_bool=1. rcis=1. hAPP_f1010710745l_bool=1. hAPP_f1485493409l_bool=1. hAPP_f1574055885nt_int=1. hAPP_f1861673493l_bool=1. hAPP_f60716669l_bool=1. big_co1548731110nt_int=1. divmod_int=1. hAPP_f1407095736t_real=1. hAPP_f1796904975t_real=1. hAPP_f628503027l_bool=1. hAPP_f695504873l_bool=1. hAPP_f203520653l_real=1. hAPP_f800510211t_bool=1. divmod_nat=1. hAPP_f1992866895l_real=1. hAPP_f2135977429nt_int=1. bezw=1. hAPP_f1914919701at_nat=1. image_int_int=1. hAPP_f1290670424t_real=1. hAPP_f327026105t_real=1. hAPP_f73570213t_real=1. hAPP_f918606171t_real=1. hAPP_b992065680at_nat=1. hAPP_f1129044203l_real=1. hAPP_f1585078997at_nat=1. hAPP_f1760145644nt_int=1. hAPP_f2054677283nt_int=1. hAPP_f264196187l_real=1. norm_frac=1. setS=1. hAPP_f1393328161l_bool=1. hAPP_f1900646853l_bool=1. hAPP_f1904307534l_real=1. hAPP_f1993765077t_real=1. hAPP_f224989205l_real=1. hAPP_f465821005nt_int=1. hAPP_f521865025ol_nat=1. hAPP_f895854793t_real=1. rRset2norRR=1. hAPP_b1463609396nt_int=1. hAPP_f1067681227l_real=1. hAPP_f1523861863l_real=1. hAPP_f1629352853nt_int=1. hAPP_f1644353557l_real=1. hAPP_f2119584768l_real=1. hAPP_f237669397t_bool=1. hAPP_f59594445l_real=1. hAPP_f93439879ol_nat=1. hAPP_r337325687l_real=1. hAPP_P1145851913nt_int=1. hAPP_P408881810nt_int=1. hAPP_f1080886329l_bool=1. hAPP_f1146629647l_bool=1. hAPP_f115998247l_bool=1. hAPP_f1505651103t_bool=1. hAPP_f561022312t_bool=1. hAPP_f618557131t_bool=1. hAPP_i1730167831nt_int=1. pdivmod=1. hAPP_P1876494581l_bool=1. hAPP_b40753821nt_int=1. hAPP_f1457396693omplex=1. hAPP_f1646117269omplex=1. hAPP_f2062659861at_rat=1. hAPP_f335634176l_real=1. hAPP_f782000547ol_nat=1. hAPP_i345030060t_bool=1. fdisj=1. hAPP_P989584703t_bool=1. hAPP_f1081447804nt_int=1. hAPP_f1607377819t_real=1. hAPP_f1722879237t_bool=1. hAPP_f1791153283t_bool=1. hAPP_f1961001761l_bool=1. hAPP_f2126667875l_real=1. hAPP_f214467655t_bool=1. hAPP_f251264923nt_int=1. hAPP_f279307923nt_int=1. hAPP_f291087797t_bool=1. hAPP_f292751352t_real=1. hAPP_f365317245l_bool=1. hAPP_f526364711l_bool=1. hAPP_f627970963t_bool=1. hAPP_nat_fun_int_int=1. hAPP_r195310020l_real=1. hAPP_r481142414t_bool=1. nat_case_nat=1. hAPP_P1062890741l_bool=1. hAPP_P1668131738nt_int=1. hAPP_P1860904029l_real=1. hAPP_P2110489235nt_int=1. hAPP_P369611207at_nat=1. hAPP_P402336767at_nat=1. hAPP_b646336293l_real=1. hAPP_f1038672605nt_int=1. hAPP_f1073644437at_rat=1. hAPP_f1122023986l_real=1. hAPP_f1131305186at_rat=1. hAPP_f1135648319t_real=1. hAPP_f1163894827t_real=1. hAPP_f1310479593l_real=1. hAPP_f1377453843nt_int=1. hAPP_f142232355nt_int=1. hAPP_f1469637905l_bool=1. hAPP_f1501741374t_real=1. hAPP_f1512942609t_bool=1. hAPP_f1582586579nt_int=1. hAPP_f1591894897l_real=1. hAPP_f1594865479ol_int=1. hAPP_f167782601nt_int=1. hAPP_f1732848529l_real=1. hAPP_f1781059817t_bool=1. hAPP_f1786887945l_real=1. hAPP_f1830834240at_rat=1. hAPP_f1855025978nt_int=1. hAPP_f1874145443l_real=1. hAPP_f197974549nt_int=1. hAPP_f1998755145t_real=1. hAPP_f2018027565t_bool=1. hAPP_f202917053t_bool=1. hAPP_f2119767738t_bool=1. hAPP_f213450978omplex=1. hAPP_f423804115t_bool=1. hAPP_f429935748t_bool=1. hAPP_f461489489t_bool=1. hAPP_f513299955t_bool=1. hAPP_f521271395nt_int=1. hAPP_f617310012nt_int=1. hAPP_f617735571l_real=1. hAPP_f791698359t_bool=1. hAPP_f892584630t_bool=1. hAPP_f960630521nt_int=1. hAPP_i119638232t_bool=1. hAPP_i1216375074nt_int=1. hAPP_i1529485324t_bool=1. hAPP_i1778081398nt_int=1. hAPP_i418383825t_bool=1. hAPP_r373117745t_bool=1. hAPP_r632163725t_bool=1. hAPP_r712953929l_real=1. image_nat_nat=1. xzgcd=1. hAPP_C2129356693al_nat=1. hAPP_P1463840893t_bool=1. hAPP_P1681069257l_bool=1. hAPP_P1982799375l_real=1. hAPP_P252494982t_bool=1. hAPP_P621635040nt_int=1. hAPP_P62333565t_bool=1. hAPP_P880235623l_bool=1. hAPP_P921862901l_bool=1. hAPP_b1482221219l_real=1. hAPP_f101635158l_bool=1. hAPP_f1048215610t_bool=1. hAPP_f1119546819l_real=1. hAPP_f1148171384nt_int=1. hAPP_f1173902867t_bool=1. hAPP_f1251097615t_bool=1. hAPP_f1344929775l_bool=1. hAPP_f1352333353l_real=1. hAPP_f1402056515l_bool=1. hAPP_f1475822739t_bool=1. hAPP_f1545556668t_bool=1. hAPP_f1735705551nt_int=1. hAPP_f1830214065l_bool=1. hAPP_f1925507189l_bool=1. hAPP_f1926459811ol_int=1. hAPP_f2037783381l_bool=1. hAPP_f2105620693nt_int=1. hAPP_f318350629l_real=1. hAPP_f320596107l_real=1. hAPP_f334321603nt_int=1. hAPP_f429384717t_bool=1. hAPP_f472159229t_bool=1. hAPP_f555557763l_real=1. hAPP_f654702867t_bool=1. hAPP_f66927821l_bool=1. hAPP_f735247205t_bool=1. hAPP_f824790735l_bool=1. hAPP_f850851693l_real=1. hAPP_f85237525t_bool=1. hAPP_i1153444985nt_int=1. hAPP_i1836591008nt_int=1. hAPP_i2104874254nt_int=1. hAPP_i371647666l_real=1. hAPP_i68813070l_bool=1. hAPP_n682674106l_real=1. big_co1705425894at_nat=1. hAPP_P1400155668t_bool=1. hAPP_P1853363688nt_rat=1. hAPP_P2053772734t_bool=1. hAPP_P230595783nt_int=1. hAPP_P507092031nt_rat=1. hAPP_P752050971nt_int=1. hAPP_P877703507t_bool=1. hAPP_P941956607nt_int=1. hAPP_f1040184293l_real=1. hAPP_f1052840010omplex=1. hAPP_f1063001928t_bool=1. hAPP_f1065357839nt_rat=1. hAPP_f1076489843l_real=1. hAPP_f1082758869at_rat=1. hAPP_f1083049571t_bool=1. hAPP_f1084724746t_bool=1. hAPP_f1108539711l_real=1. hAPP_f1109256153l_real=1. hAPP_f11151005nt_int=1. hAPP_f1115375056t_bool=1. hAPP_f1120137083nt_rat=1. hAPP_f1121609875t_bool=1. hAPP_f1131424617t_bool=1. hAPP_f1139079189at_int=1. hAPP_f1151807883nt_int=1. hAPP_f1154658180t_bool=1. hAPP_f1161935955l_real=1. hAPP_f1169087445t_real=1. hAPP_f11709033t_bool=1. hAPP_f1177545951t_bool=1. hAPP_f1182713365t_bool=1. hAPP_f1187443546t_bool=1. hAPP_f1198736473t_bool=1. hAPP_f1222178131t_bool=1. hAPP_f1228585167nt_int=1. hAPP_f1241350704t_bool=1. hAPP_f125788821nt_int=1. hAPP_f125837377t_bool=1. hAPP_f1261108921t_bool=1. hAPP_f1270196437nt_rat=1. hAPP_f1273270095t_bool=1. hAPP_f1293550189l_real=1. hAPP_f1295083291t_real=1. hAPP_f1311564564nt_int=1. hAPP_f1315737299t_bool=1. hAPP_f1323932638t_real=1. hAPP_f1325804733t_bool=1. hAPP_f1341793266t_bool=1. hAPP_f134192053t_real=1. hAPP_f1356864277nt_int=1. hAPP_f1361476881t_bool=1. hAPP_f1375227553nt_int=1. hAPP_f1399575567t_bool=1. hAPP_f1408247010at_nat=1. hAPP_f1431025877at_int=1. hAPP_f1448486952t_bool=1. hAPP_f1456711209t_bool=1. hAPP_f1459420095t_bool=1. hAPP_f1480506641t_bool=1. hAPP_f151095827nt_int=1. hAPP_f1547007718nt_rat=1. hAPP_f1576199059t_bool=1. hAPP_f1607917947l_real=1. hAPP_f1613846310t_real=1. hAPP_f1625314454t_bool=1. hAPP_f1663701695l_real=1. hAPP_f1673103573at_int=1. hAPP_f1720068861t_real=1. hAPP_f172451459l_bool=1. hAPP_f1730463541l_real=1. hAPP_f1731308757nt_int=1. hAPP_f1736103445l_real=1. hAPP_f1763467369l_real=1. hAPP_f1769710159nt_int=1. hAPP_f1815301987t_bool=1. hAPP_f1832844496t_bool=1. hAPP_f183342685l_real=1. hAPP_f1840085295t_bool=1. hAPP_f1845081853t_bool=1. hAPP_f1878066172t_bool=1. hAPP_f1879982819l_real=1. hAPP_f1884005689l_bool=1. hAPP_f1900950721nt_rat=1. hAPP_f1902329768t_bool=1. hAPP_f1908301369l_real=1. hAPP_f1914742907nt_int=1. hAPP_f1920544743t_bool=1. hAPP_f1927674023t_bool=1. hAPP_f1935805932nt_int=1. hAPP_f1965248563t_bool=1. hAPP_f1990636069t_bool=1. hAPP_f2002277285nt_int=1. hAPP_f2004554147nt_int=1. hAPP_f2042071795t_bool=1. hAPP_f2050994651t_bool=1. hAPP_f2072293513l_real=1. hAPP_f2084827628t_bool=1. hAPP_f2094497995nt_int=1. hAPP_f2114902373l_real=1. hAPP_f212164053t_real=1. hAPP_f219720582omplex=1. hAPP_f223623489t_bool=1. hAPP_f23851874omplex=1. hAPP_f252743847l_bool=1. hAPP_f269654879t_real=1. hAPP_f271976045t_bool=1. hAPP_f289318786t_bool=1. hAPP_f301246617nt_int=1. hAPP_f324002831l_real=1. hAPP_f326728547t_bool=1. hAPP_f328572565l_real=1. hAPP_f330985925l_bool=1. hAPP_f332806115t_bool=1. hAPP_f340152631t_bool=1. hAPP_f351634443t_bool=1. hAPP_f352226557t_real=1. hAPP_f374502857t_bool=1. hAPP_f381769257l_bool=1. hAPP_f382317021nt_rat=1. hAPP_f389116883t_bool=1. hAPP_f400200275nt_int=1. hAPP_f400538607nt_int=1. hAPP_f403082607t_bool=1. hAPP_f406530075omplex=1. hAPP_f41603166t_bool=1. hAPP_f422375781t_bool=1. hAPP_f434654626nt_int=1. hAPP_f492612297t_bool=1. hAPP_f499530369l_bool=1. hAPP_f512004221l_bool=1. hAPP_f517738447l_real=1. hAPP_f534361363nt_int=1. hAPP_f538565589t_bool=1. hAPP_f55007289nt_int=1. hAPP_f563293398t_real=1. hAPP_f571820015t_bool=1. hAPP_f578362681nt_int=1. hAPP_f601898971t_bool=1. hAPP_f605995867t_bool=1. hAPP_f643658017l_bool=1. hAPP_f644105375l_real=1. hAPP_f659380387ol_int=1. hAPP_f677014621l_real=1. hAPP_f693283245t_bool=1. hAPP_f696844121t_bool=1. hAPP_f704188951t_bool=1. hAPP_f713949180nt_int=1. hAPP_f720654810t_bool=1. hAPP_f725770969t_bool=1. hAPP_f729588221t_bool=1. hAPP_f737665877t_bool=1. hAPP_f74147283t_real=1. hAPP_f743790483t_bool=1. hAPP_f747162069nt_int=1. hAPP_f747475781nt_int=1. hAPP_f766218899t_real=1. hAPP_f77155443t_bool=1. hAPP_f774948957l_real=1. hAPP_f800361423l_real=1. hAPP_f801032407l_bool=1. hAPP_f808647893nt_int=1. hAPP_f823702159l_real=1. hAPP_f834119977l_bool=1. hAPP_f836602773nt_int=1. hAPP_f837737749at_nat=1. hAPP_f847817363t_bool=1. hAPP_f848516149l_bool=1. hAPP_f857296063t_bool=1. hAPP_f866672739t_bool=1. hAPP_f883232462nt_int=1. hAPP_f888249301nt_int=1. hAPP_f894608603t_bool=1. hAPP_f908327869t_bool=1. hAPP_f940061907l_bool=1. hAPP_f956464319t_bool=1. hAPP_f983569339t_bool=1. hAPP_f999082675at_nat=1. hAPP_i157317923t_real=1. hAPP_i16292313t_bool=1. hAPP_i1671359920t_real=1. hAPP_i1796122964t_bool=1. hAPP_i1823918693l_bool=1. hAPP_i447943416t_real=1. hAPP_i61245889nt_int=1. hAPP_i614188481l_bool=1. hAPP_i627165055t_real=1. hAPP_i688218208l_bool=1. hAPP_i744251628nt_int=1. hAPP_i872162435t_bool=1. hAPP_i998473677nt_int=1. hAPP_n1006566506l_bool=1. hAPP_n1983812447omplex=1. hAPP_nat_fun_rat_rat=1. hAPP_r1043989325l_real=1. hAPP_r1150855451l_bool=1. hAPP_r1230691443l_real=1. hAPP_r1392521531t_bool=1. hAPP_r1409264866l_real=1. hAPP_r1562208522t_bool=1. hAPP_r1623423338t_bool=1. hAPP_r1900510169t_bool=1. hAPP_r1938957037l_bool=1. hAPP_r1962736578t_bool=1. hAPP_r2052264067t_bool=1. hAPP_r224952523l_real=1. hAPP_r279757445t_bool=1. hAPP_r285062775l_bool=1. hAPP_r50993370t_real=1. hAPP_r579619957l_real=1. hAPP_r653725362t_bool=1. hAPP_r794344869l_real=1. hAPP_r847664445t_bool=1. hAPP_r931665397l_real=1. hAPP_r938550413l_real=1. multInvPair=1. nat_case_bool=1. code_d418564891umeral=1. hAPP_C1614127039umeral=1. hAPP_C2132160860umeral=1. hAPP_P1586233937at_nat=1. hAPP_Q1256397320de_int=1. hAPP_Q952608535de_int=1. hAPP_complex_nat=1. hAPP_f1418982953t_bool=1. hAPP_f1599440987ol_int=1. hAPP_f538843494t_bool=1. hAPP_list_int_nat=1. image_275383677t_bool=1. quickc495462417de_int=1. f1=1. f6=1. f9=1. f10=1. f18=1. f19=1. f28=1. f29=1. f30=1. f31=1. f33=1. f37=1. f39=1. f43=1. f45=1. f47=1. f49=1. f54=1. f55=1. f59=1. f61=1. f70=1. f71=1. f73=1. f76=1. f80=1. f87=1. f89=1. f90=1. f92=1. f94=1. f95=1. f96=1. f97=1. f100=1. f101=1. f102=1. f107=1. f108=1. f117=1. f118=1. f119=1. f120=1. f137=1. f138=1. f141=1. f142=1. f148=1. f149=1. f150=1. f151=1. f153=1. f160=1. f162=1. f163=1. f164=1. f175=1. f176=1. f180=1. f182=1. f192=1. f193=1. f194=1. f197=1. f198=1. f204=1. f205=1. f206=1. f207=1. f208=1. f210=1. f211=1. f230=1. f231=1. f232=1. f233=1. f241=1. f242=1. f253=1. f254=1. f261=1. f262=1. f263=1. f272=1. f275=1. f280=1. f281=1. f305=1. f306=1. f307=1. f308=1. f311=1. f312=1. f313=1. f317=1. f318=1. f320=1. f321=1. f328=1. f329=1. f332=1. f333=1. f334=1. f343=1. f344=1. f358=1. f359=1. f360=1. f361=1. f369=1. f370=1. f379=1. f380=1. f382=1. f386=1. f387=1. f391=1. f392=1. f408=1. f409=1. f410=1. f419=1. f422=1. f423=1. f424=1. f431=1. f433=1. f437=1. f443=1. f447=1. f448=1. f462=1. f463=1. f464=1. f465=1. f472=1. f478=1. f479=1. f489=1. f493=1. f494=1. f496=1. f500=1. f501=1. f502=1. ord_at875362053st_int=1. div_mod_int=1. zcong=1. legacy_zgcd=1. div_mod_nat=1. ord_at4362885an_nat=1. isCont_real_real=1. accp_P2006205492nt_int=1. archim1246769320r_real=1. natfloor=1. quickcheck_int_of=1. multInv=1. ratreal=1. summable_real=1. log=1. of_real_complex=1. collect_int=1. archim856651990g_real=1. natceiling=1. quotient_of=1. cis=1. standardRes=1. resSet=1. vanishes=1. ord_at238088361st_nat=1. collect_nat=1. is_RRset=1. wset=1. comple124823625p_real=1. normalize=1. sums_real=1. adjust=1. monoseq_real=1. bnorRset=1. ord_gr788844697n_real=1. inv=1. nat_gcd=1. ord_at641636577an_int=1. scaleR1652505878omplex=1. code_S1047413653umeral=1. summable_complex=1. the_real=1. at_real=1. cauchy_real=1. expi=1. frct=1. legendre=1. nat_tsub=1. accp_P490777396at_nat=1. code_int_of=1. produc1518849193nt_int=1. rsetR=1. accp_int=1. archim791455193or_rat=1. cOMBK_1643584600t_bool=1. cauchy_complex=1. diffs_real=1. fimplies=1. phi=1. arg=1. comple219730294t_bool=1. div_mo231679042de_int=1. hilbert_Eps_int=1. produc93441119t_bool=1. sums_complex=1. at_complex=1. bseq_real=1. cOMBK_rat_nat=1. code_c271388182l_size=1. code_nat_of_aux=1. collec1347809874nt_int=1. collect_real=1. div_mo1740067990umeral=1. noXRRset=1. ord_at1589558736t_real=1. ord_gr1297742076an_int=1. ord_gr660468384an_nat=1. produc713050258nt_int=1. scaleR_scaleR_real=1. the_int=1. zcongm=1. cOMBK_bool_nat=1. collec1979865426at_nat=1. collec50511176nt_int=1. isCont_complex_real=1. nat_lcm=1. produc1298267108nt_int=1. z3div=1. z3mod=1. bijR_int_int=1. complex_size=1. dist_dist_complex=1. dist_dist_real=1. hilbert_Eps_real=1. int_ge_less_than=1. int_ge_less_than2=1. int_lcm=1. isCont156215680omplex=1. nat_aux=1. nat_tr160667106at_int=1. nat_tr876908586nt_nat=1. ord_at1496968948n_real=1. produc1391996073at_nat=1. produc1922581855t_bool=1. produc595218619l_real=1. produc973645403t_bool=1. quickc666637781d_zero=1. quickcheck_nat_of=1. the_Pr2103884470nt_int=1. the_Pr588456374at_nat=1. trivial_limit_nat=1. undefined_int=1. f5=1. f20=1. f26=1. f27=1. f34=1. f35=1. f38=1. f46=1. f50=1. f53=1. f56=1. f60=1. f65=1. f69=1. f72=1. f81=1. f82=1. f83=1. f84=1. f85=1. f86=1. f93=1. f121=1. f122=1. f124=1. f143=1. f144=1. f145=1. f155=1. f156=1. f157=1. f158=1. f161=1. f165=1. f166=1. f167=1. f168=1. f174=1. f181=1. f184=1. f188=1. f189=1. f190=1. f191=1. f195=1. f196=1. f199=1. f202=1. f203=1. f209=1. f223=1. f224=1. f228=1. f229=1. f243=1. f252=1. f260=1. f267=1. f270=1. f271=1. f276=1. f278=1. f288=1. f290=1. f291=1. f295=1. f296=1. f300=1. f309=1. f310=1. f319=1. f322=1. f323=1. f324=1. f331=1. f335=1. f336=1. f340=1. f341=1. f342=1. f355=1. f364=1. f366=1. f388=1. f411=1. f412=1. f413=1. f414=1. f416=1. f425=1. f430=1. f432=1. f442=1. f444=1. f449=1. f450=1. f455=1. f466=1. f477=1. f483=1. f485=1. f486=1. f487=1. f488=1. f490=1. f491=1. f495=1. tendsto_nat_real=1. bolzano_bisect=1. tendsto_real_real=1. tendsto_nat_complex=1. tendsto_complex_real=1. tendst1507391555omplex=1. f7=1. f16=1. f17=1. f32=1. f36=1. f44=1. f48=1. f51=1. f52=1. f64=1. f66=1. f74=1. f75=1. f78=1. f79=1. f88=1. f91=1. f103=1. f104=1. f105=1. f106=1. f123=1. f125=1. f126=1. f128=1. f134=1. f135=1. f136=1. f139=1. f140=1. f146=1. f147=1. f154=1. f159=1. f169=1. f170=1. f171=1. f172=1. f173=1. f220=1. f221=1. f222=1. f239=1. f240=1. f244=1. f245=1. f246=1. f248=1. f255=1. f256=1. f259=1. f268=1. f269=1. f273=1. f274=1. f277=1. f279=1. f282=1. f283=1. f284=1. f285=1. f286=1. f287=1. f289=1. f292=1. f304=1. f325=1. f326=1. f327=1. f330=1. f339=1. f347=1. f348=1. f350=1. f351=1. f352=1. f362=1. f365=1. f371=1. f374=1. f375=1. f376=1. f377=1. f378=1. f399=1. f402=1. f403=1. f404=1. f405=1. f428=1. f429=1. f434=1. f435=1. f436=1. f438=1. f439=1. f440=1. f441=1. f445=1. f446=1. f456=1. f457=1. f458=1. f459=1. f460=1. f461=1. f469=1. f470=1. f471=1. f480=1. f481=1. f482=1. f484=1. f2=1. f3=1. f4=1. f11=1. f12=1. f13=1. f14=1. f15=1. f21=1. f22=1. f23=1. f24=1. f25=1. f40=1. f41=1. f42=1. f57=1. f58=1. f67=1. f68=1. f77=1. f110=1. f111=1. f112=1. f113=1. f114=1. f115=1. f116=1. f127=1. f129=1. f130=1. f131=1. f132=1. f133=1. f152=1. f183=1. f185=1. f186=1. f187=1. f200=1. f201=1. f213=1. f214=1. f215=1. f216=1. f217=1. f218=1. f219=1. f225=1. f226=1. f227=1. f237=1. f238=1. f247=1. f249=1. f250=1. f251=1. f264=1. f265=1. f266=1. f293=1. f294=1. f297=1. f298=1. f299=1. f301=1. f302=1. f303=1. f314=1. f315=1. f316=1. f349=1. f353=1. f354=1. f356=1. f357=1. f363=1. f372=1. f373=1. f381=1. f383=1. f384=1. f385=1. f389=1. f390=1. f393=1. f394=1. f395=1. f396=1. f397=1. f398=1. f400=1. f401=1. f406=1. f407=1. f415=1. f420=1. f426=1. f427=1. f451=1. f452=1. f453=1. f454=1. f467=1. f468=1. f473=1. f474=1. f475=1. f476=1. f492=1. f497=1. f498=1. f499=1. f8=1. f98=1. f99=1. f109=1. f177=1. f178=1. f179=1. f212=1. f234=1. f235=1. f236=1. f257=1. f258=1. f337=1. f338=1. f345=1. f346=1. f367=1. f368=1. f417=1. f418=1. f421=1. xzgcda=1. f62=1. f63=1. 21.46/21.58 21.46/21.58 ============================== STATISTICS ============================ 21.46/21.58 21.46/21.58 Given=0. Generated=9497. Kept=8259. proofs=0. 21.46/21.58 Usable=0. Sos=0. Demods=1074. Limbo=8258, Disabled=9497. Hints=0. 21.46/21.58 Kept_by_rule=0, Deleted_by_rule=0. 21.46/21.58 Forward_subsumed=1238. Back_subsumed=0. 21.46/21.58 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 21.46/21.58 New_demodulators=1074 (16 lex), Back_demodulated=0. Back_unit_deleted=0. 21.46/21.58 Demod_attempts=404230. Demod_rewrites=3086. 21.46/21.58 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 21.46/21.58 Nonunit_fsub_feature_tests=6311. Nonunit_bsub_feature_tests=0. 21.46/21.58 Megabytes=419.43. 21.46/21.58 User_CPU=13.68, System_CPU=0.15, Wall_clock=14. 21.46/21.58 21.46/21.58 Megs malloced by palloc(): 400. 21.46/21.58 type (bytes each) gets frees in use bytes 21.46/21.58 chunk ( 104) 225907 225907 0 0.0 K 21.46/21.58 string_buf ( 8) 190488 190488 0 0.0 K 21.46/21.58 token ( 20) 497859 497859 0 0.0 K 21.46/21.58 pterm ( 16) 305261 305261 0 0.0 K 21.46/21.58 hashtab ( 8) 5102 5102 0 0.0 K 21.46/21.58 hashnode ( 8) 19197 19197 0 0.0 K 21.46/21.58 term ( 20) 2612369 1983352 629017 12285.5 K 21.46/21.58 term arg arrays: 3040.8 K 21.46/21.58 attribute ( 12) 48960 1546 47414 555.6 K 21.46/21.58 ilist ( 8) 101806212 101797021 9191 71.8 K 21.46/21.58 plist ( 8) 149195 75845 73350 573.0 K 21.46/21.58 i2list ( 12) 845301 845301 0 0.0 K 21.46/21.58 just ( 12) 34737 8276 26461 310.1 K 21.46/21.58 parajust ( 16) 0 0 0 0.0 K 21.46/21.58 instancejust ( 8) 0 0 0 0.0 K 21.46/21.58 ivyjust ( 24) 0 0 0 0.0 K 21.46/21.58 formula ( 28) 201970 138698 63272 1730.1 K 21.46/21.58 formula arg arrays: 207.3 K 21.46/21.58 topform ( 52) 24903 1238 23665 1201.7 K 21.46/21.58 clist_pos ( 20) 29133 9497 19636 383.5 K 21.46/21.58 clist ( 16) 8 0 8 0.1 K 21.46/21.58 context ( 808) 172220 172220 0 0.0 K 21.46/21.58 trail ( 12) 25542 25542 0 0.0 K 21.46/21.58 ac_match_pos (70044) 0 0 0 0.0 K 21.46/21.58 ac_match_free_vars_pos (20020) 21.46/21.58 0 0 0 0.0 K 21.46/21.58 btm_state ( 60) 0 0 0 0.0 K 21.46/21.58 btu_state ( 60) 0 0 0 0.0 K 21.46/21.58 ac_position (285432) 0 0 0 0.0 K 21.46/21.58 fpa_trie ( 20) 36966 0 36966 722.0 K 21.46/21.58 fpa_state ( 28) 1741 1741 0 0.0 K 21.46/21.58 fpa_index ( 12) 10 0 10 0.1 K 21.46/21.58 fpa_chunk ( 20) 31636 10728 20908 408.4 K 21.46/21.58 fpa_list ( 16) 14359 0 14359 224.4 K 21.46/21.58 fpa_list chunks: 828.1 K 21.46/21.58 discrim ( 12) 27852 0 27852 326.4 K 21.46/21.58 discrim_pos ( 16) 6716 6716 0 0.0 K 21.46/21.58 flat2 ( 32) 561743 561743 0 0.0 K 21.46/21.58 flat ( 48) 0 0 0 0.0 K 21.46/21.58 flatterm ( 32) 711990 711990 0 0.0 K 21.46/21.58 mindex ( 28) 13 0 13 0.4 K 21.46/21.58 mindex_pos ( 56) 138663 138663 0 0.0 K 21.46/21.58 lindex ( 12) 5 0 5 0.1 K 21.46/21.58 clash ( 40) 0 0 0 0.0 K 21.46/21.58 di_tree ( 12) 32888294 0 32888294 385409.7 K 21.46/21.58 avl_node ( 20) 0 0 0 0.0 K 21.46/21.58 21.46/21.58 Memory report, 20 @ 20 = 400 megs (400.00 megs used). 21.46/21.58 List 1, length 2, 0.0 K 21.46/21.58 List 4, length 1, 0.0 K 21.46/21.58 List 5, length 2, 0.0 K 21.46/21.58 List 7, length 34, 0.9 K 21.46/21.58 List 8, length 7200, 225.0 K 21.46/21.58 List 11, length 176, 7.6 K 21.46/21.58 List 12, length 100, 4.7 K 21.46/21.58 List 14, length 4, 0.2 K 21.46/21.58 List 15, length 2, 0.1 K 21.46/21.58 List 16, length 554, 34.6 K 21.46/21.58 List 19, length 2, 0.1 K 21.46/21.58 List 20, length 2, 0.2 K 21.46/21.58 List 26, length 320, 32.5 K 21.46/21.58 List 27, length 1, 0.1 K 21.46/21.58 List 32, length 56, 7.0 K 21.46/21.58 List 49, length 2, 0.4 K 21.46/21.58 List 76, length 2, 0.6 K 21.46/21.58 List 98, length 2, 0.8 K 21.46/21.58 List 101, length 2, 0.8 K 21.46/21.58 List 202, length 2, 1.6 K 21.46/21.58 21.46/21.58 ============================== SELECTOR REPORT ======================= 21.46/21.58 Sos_deleted=0, Sos_displaced=0, Sos_size=0 21.46/21.58 SELECTOR PART PRIORITY ORDER SIZE SELECTED 21.46/21.58 I 2147483647 high age 0 0 21.46/21.58 H 1 high weight 0 0 21.46/21.58 A 1 low age 0 0 21.46/21.58 F 4 low weight 0 0 21.46/21.58 T 4 low weight 0 0 21.46/21.58 ============================== end of selector report ================ 21.46/21.58 21.46/21.58 ============================== end of statistics ===================== 21.46/21.58 21.46/21.58 Exiting with failure. 21.46/21.58 21.46/21.58 Process 25537 exit (max_megs) Tue Aug 9 04:37:26 2022 21.46/21.58 Prover9 interrupted 21.46/21.59 EOF