0.00/0.04 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.00/0.04 % Command : tptp2X_and_run_prover9 %d %s 0.02/0.24 % Computer : n064.star.cs.uiowa.edu 0.02/0.24 % Model : x86_64 x86_64 0.02/0.24 % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz 0.02/0.24 % Memory : 32218.625MB 0.02/0.24 % OS : Linux 3.10.0-693.2.2.el7.x86_64 0.02/0.24 % CPULimit : 300 0.02/0.24 % DateTime : Sat Jul 14 05:24:40 CDT 2018 0.02/0.24 % CPUTime : 1.32/1.56 ============================== Prover9 =============================== 1.32/1.56 Prover9 (32) version 2009-11A, November 2009. 1.32/1.56 Process 48596 was started by sandbox2 on n064.star.cs.uiowa.edu, 1.32/1.56 Sat Jul 14 05:24:41 2018 1.32/1.56 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 300 -f /tmp/Prover9_48564_n064.star.cs.uiowa.edu". 1.32/1.56 ============================== end of head =========================== 1.32/1.56 1.32/1.56 ============================== INPUT ================================= 1.32/1.56 1.32/1.56 % Reading from file /tmp/Prover9_48564_n064.star.cs.uiowa.edu 1.32/1.56 1.32/1.56 set(prolog_style_variables). 1.32/1.56 set(auto2). 1.32/1.56 % set(auto2) -> set(auto). 1.32/1.56 % set(auto) -> set(auto_inference). 1.32/1.56 % set(auto) -> set(auto_setup). 1.32/1.56 % set(auto_setup) -> set(predicate_elim). 1.32/1.56 % set(auto_setup) -> assign(eq_defs, unfold). 1.32/1.56 % set(auto) -> set(auto_limits). 1.32/1.56 % set(auto_limits) -> assign(max_weight, "100.000"). 1.32/1.56 % set(auto_limits) -> assign(sos_limit, 20000). 1.32/1.56 % set(auto) -> set(auto_denials). 1.32/1.56 % set(auto) -> set(auto_process). 1.32/1.56 % set(auto2) -> assign(new_constants, 1). 1.32/1.56 % set(auto2) -> assign(fold_denial_max, 3). 1.32/1.56 % set(auto2) -> assign(max_weight, "200.000"). 1.32/1.56 % set(auto2) -> assign(max_hours, 1). 1.32/1.56 % assign(max_hours, 1) -> assign(max_seconds, 3600). 1.32/1.56 % set(auto2) -> assign(max_seconds, 0). 1.32/1.56 % set(auto2) -> assign(max_minutes, 5). 1.32/1.56 % assign(max_minutes, 5) -> assign(max_seconds, 300). 1.32/1.56 % set(auto2) -> set(sort_initial_sos). 1.32/1.56 % set(auto2) -> assign(sos_limit, -1). 1.32/1.56 % set(auto2) -> assign(lrs_ticks, 3000). 1.32/1.56 % set(auto2) -> assign(max_megs, 400). 1.32/1.56 % set(auto2) -> assign(stats, some). 1.32/1.56 % set(auto2) -> clear(echo_input). 1.32/1.56 % set(auto2) -> set(quiet). 1.32/1.56 % set(auto2) -> clear(print_initial_clauses). 1.32/1.56 % set(auto2) -> clear(print_given). 1.32/1.56 assign(lrs_ticks,-1). 1.32/1.56 assign(sos_limit,10000). 1.32/1.56 assign(order,kbo). 1.32/1.56 set(lex_order_vars). 1.32/1.56 clear(print_given). 1.32/1.56 1.32/1.56 % formulas(sos). % not echoed (1232 formulas) 1.32/1.56 1.32/1.56 ============================== end of input ========================== 1.32/1.56 1.32/1.56 % From the command line: assign(max_seconds, 300). 1.32/1.56 1.32/1.56 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 1.32/1.56 1.32/1.56 % Formulas that are not ordinary clauses: 1.32/1.56 1 (all X_1 all M all Y_1 hAPP_int_int(div_mod_int(hAPP_nat_int(power_power_int(hAPP_int_int(div_mod_int(X_1),M)),Y_1)),M) = hAPP_int_int(div_mod_int(hAPP_nat_int(power_power_int(X_1),Y_1)),M)) # label(fact_1148_zpower__zmod) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 2 (all B_6 all A_7 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_7)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_6),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(A_7),B_6)),zero_zero_nat))))) # label(fact_844_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 3 (all C_12 all B_26 all A_28 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_26),A_28)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_12),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(C_12),A_28)),hAPP_real_real(times_times_real(C_12),B_26)))))) # label(fact_781_mult__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 4 (all Ma all K all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(Ma),K)),hAPP_nat_nat(times_times_nat(N),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),N))))) # label(fact_1042_mult__le__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 5 (all A_69 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(power_power_real(A_69),number_number_of_nat(bit0(bit1(pls))))))) # label(fact_443_zero__le__power2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 6 (all A_81 hAPP_int_int(times_times_int(A_81),zero_zero_int) = zero_zero_int) # label(fact_344_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 7 (all Ma 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),Ma)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(times_times_nat(Ma),N))))) # label(fact_1037_nat__0__less__mult__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 8 (all M all I_1 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(plus_plus_nat(J_1),M))))) # label(fact_977_trans__less__add1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 9 (all X_2 all Y_2 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(B_2),X_2)),hAPP_nat_int(power_power_int(B_2),Y_2)))))) # label(fact_303_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 10 (all X_2 all P_3 (-hBOOL(hAPP_int_bool(zcong(X_2,zero_zero_int),P_3)) <-> standardRes(P_3,X_2) != zero_zero_int)) # label(fact_1190_StandardRes__prop3) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 11 (all X_2 all Y_2 (is_int(Y_2) & is_int(X_2) -> (hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(X_2),X_2)),hAPP_int_int(times_times_int(Y_2),Y_2)) = zero_zero_int <-> zero_zero_int = Y_2 & X_2 = zero_zero_int))) # label(fact_385_sum__squares__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 12 (all D_5 all C all A all B all M (hBOOL(hAPP_int_bool(zcong(A,B),M)) -> (C = B -> (hBOOL(hAPP_int_bool(zcong(C,D_5),M)) -> hBOOL(hAPP_int_bool(zcong(A,D_5),M)))))) # label(fact_638_zcong__eq__trans) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 13 (all A_2 (is_int(A_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(power_power_int(A_2),number_number_of_nat(bit0(bit1(pls)))))) <-> A_2 != zero_zero_int))) # label(fact_454_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 14 (all P_1 (is_int(P_1) -> (hBOOL(hAPP_int_bool(zprime,P_1)) -> (P_1 != number_number_of_int(bit0(bit1(pls))) -> (P_1 != number_number_of_int(bit1(bit1(pls))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(bit1(bit0(bit1(pls))))),P_1))))))) # label(fact_290_prime__g__5) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 15 (all K_1 all L bit0(hAPP_int_int(minus_minus_int(K_1),L)) = hAPP_int_int(minus_minus_int(bit1(K_1)),bit1(L))) # label(fact_625_diff__bin__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 16 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),bit1(K))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K)))) # label(fact_81_rel__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 17 (all A_87 all M_8 all N_35 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_8),N_35)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(A_87),M_8)),hAPP_nat_nat(power_power_nat(A_87),N_35))))) # label(fact_316_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 18 (all A zero_zero_int = hAPP_int_int(div_mod_int(A),number_number_of_int(min))) # label(fact_1155_zmod__minus1__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 19 (all B_48 all Q_4 all R_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_48),Q_4)),R_3))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_48)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_48)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Q_4)))))) # label(fact_602_q__pos__lemma) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 20 (all W all Z (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W),Z)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(W),one_one_int)),Z)))) # label(fact_86_zless__imp__add1__zle) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 21 (all A_48 hAPP_nat_nat(times_times_nat(A_48),zero_zero_nat) = zero_zero_nat) # label(fact_707_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 22 (all N_1 all M all K_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1)) -> (hAPP_nat_nat(times_times_nat(K_1),N_1) = hAPP_nat_nat(times_times_nat(K_1),M) -> M = N_1))) # label(fact_1132_mult__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 23 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(div_mod_nat(M),N_1)),N_1)))) # label(fact_1178_mod__less__divisor) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 24 (all N_1 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_1))) # label(fact_891_less__irrefl__nat) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 25 (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,bit1(K)),min)))) # label(fact_534_rel__simps_I30_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 26 (all P_3 sr(P_3) = collect_int(cOMBS_int_bool_bool(cOMBB_1652995168ol_int(fconj,hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int)),cOMBC_int_int_bool(ord_less_int,P_3)))) # label(fact_1133_SR__def) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 27 (all X_2 all P_2 ((all A_3 (is_int(A_3) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_3)) -> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(minus_minus_int(A_3),one_one_int)))) -> hBOOL(hAPP_int_bool(P_2,A_3))))) -> hBOOL(hAPP_int_bool(P_2,X_2)))) # label(fact_1106_d22set__induct__old) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 28 (all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) <-> N != Ma & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)))) # label(fact_985_nat__less__le) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 29 (all P_2 all A_2 all B_2 (hBOOL(hAPP_nat_bool(P_2,hAPP_nat_nat(minus_minus_nat(A_2),B_2))) <-> -((exists D_2 (A_2 = hAPP_nat_nat(plus_plus_nat(B_2),D_2) & -hBOOL(hAPP_nat_bool(P_2,D_2)))) | -hBOOL(hAPP_nat_bool(P_2,zero_zero_nat)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_2),B_2))))) # label(fact_1032_nat__diff__split__asm) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 30 (all X_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),X_1))) # label(fact_877_dvd_Oorder__refl) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 31 (all N_1 all M M = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(N_1),M)),N_1)) # label(fact_933_diff__add__inverse) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 32 (all W all Z1 all Z2 hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(W),Z1)),hAPP_int_int(times_times_int(W),Z2)) = hAPP_int_int(times_times_int(W),hAPP_int_int(plus_plus_int(Z1),Z2))) # label(fact_211_zadd__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 33 (all C_14 all D_4 all A_30 all B_28 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_30),B_28)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_14),D_4)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_28)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_14)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_30),C_14)),hAPP_int_int(times_times_int(B_28),D_4)))))))) # label(fact_777_mult__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 34 (all X_2 all Y_2 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(B_2),X_2)),hAPP_nat_int(power_power_int(B_2),Y_2))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Y_2))))) # label(fact_496_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 35 (all A_102 all M_13 all B_59 hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_102),B_59)),M_13) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_102),M_13)),hAPP_int_int(times_times_int(B_59),M_13))) # label(fact_173_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 36 (all V_13 all W_12 hAPP_real_real(times_times_real(number267125858f_real(V_13)),number267125858f_real(W_12)) = number267125858f_real(hAPP_int_int(times_times_int(V_13),W_12))) # label(fact_242_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 37 (all X_1 all Y_1 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) -> Y_1 != X_1)) # label(fact_903_dvd_Oless__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 38 (all A all P_1 (hBOOL(hAPP_int_bool(zprime,P_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),P_1)) -> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A),inv(P_1,A)),one_one_int),P_1)))))) # label(fact_1087_inv__is__inv) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 39 (all M all K_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(M),K_1)),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1)))) # label(fact_1003_add__leD2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 40 (all X_2 all Y_2 (Y_2 = zero_zero_real & zero_zero_real = X_2 <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(X_2),X_2)),hAPP_real_real(times_times_real(Y_2),Y_2))),zero_zero_real)))) # label(fact_414_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 41 (all W hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),W))) # label(fact_880_real__le__refl) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 42 (all A hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A))) # label(fact_1124_gcd__lcm__complete__lattice__nat_Obot__least) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 43 (all Z_5 hAPP_real_real(times_times_real(number267125858f_real(bit0(bit1(pls)))),Z_5) = hAPP_real_real(plus_plus_real(Z_5),Z_5)) # label(fact_279_mult__2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 44 (all K (is_int(K) -> (K = min <-> min = bit1(K)))) # label(fact_518_rel__simps_I47_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 45 (all X_12 all Y_9 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(X_12),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_9),number_number_of_nat(bit0(bit1(pls)))))))) # label(fact_475_sum__power2__ge__zero) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 46 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),pls)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K)),pls)))) # label(fact_148_rel__simps_I12_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 47 (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,bit1(K)),pls)))) # label(fact_80_rel__simps_I29_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 48 (all Z_2 all X_3 all Y_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_3),Y_3)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_3),Z_2)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_3),hAPP_real_real(minus_minus_real(Y_3),Z_2)))))) # label(fact_759_dvd__diff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.56 49 (all A_43 all C_21 all B_40 all D_7 hAPP_int_int(plus_plus_int(hAPP_int_int(minus_minus_int(A_43),B_40)),hAPP_int_int(minus_minus_int(C_21),D_7)) = hAPP_int_int(minus_minus_int(hAPP_int_int(plus_plus_int(A_43),C_21)),hAPP_int_int(plus_plus_int(B_40),D_7))) # label(fact_728_add__diff__add) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 50 (all Z all W hAPP_int_int(plus_plus_int(W),Z) = hAPP_int_int(plus_plus_int(Z),W)) # label(fact_147_zadd__commute) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 51 (all X_1 all Y_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) -> Y_1 = X_1))) # label(fact_912_dvd_Oantisym) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 52 (all K_1 all M all N_1 hAPP_nat_nat(minus_minus_nat(M),N_1) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(K_1),M)),hAPP_nat_nat(plus_plus_nat(K_1),N_1))) # label(fact_935_Nat_Odiff__cancel) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 53 (all N_1 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat))) # label(fact_957_not__less0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 54 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(div_mod_int(X_1),Y_1)))))) # label(fact_1139_Divides_Otransfer__nat__int__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 55 (all X_21 all Y_18 all Z_9 hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(X_21),Y_18)),hAPP_int_int(times_times_int(X_21),Z_9)) = hAPP_int_int(times_times_int(X_21),hAPP_int_int(plus_plus_int(Y_18),Z_9))) # label(fact_182_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 56 (all X_1 all P_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP_int_bool(zcong(X_1,number_number_of_int(min)),P_1)) -> -hBOOL(hAPP_int_bool(zcong(X_1,one_one_int),P_1))))) # label(fact_607_zcong__neg__1__impl__ne__1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 57 (all A_44 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_44)) -> zero_zero_nat = A_44)) # label(fact_725_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 58 (all V_6 all V ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls)) -> hAPP_nat_nat(times_times_nat(number_number_of_nat(V)),number_number_of_nat(V_6)) = number_number_of_nat(hAPP_int_int(times_times_int(V),V_6))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls)) -> hAPP_nat_nat(times_times_nat(number_number_of_nat(V)),number_number_of_nat(V_6)) = zero_zero_nat))) # label(fact_567_mult__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 59 (all A_65 all M_5 all N_16 hAPP_real_real(times_times_real(hAPP_nat_real(power_power_real(A_65),M_5)),hAPP_nat_real(power_power_real(A_65),N_16)) = hAPP_nat_real(power_power_real(A_65),hAPP_nat_nat(plus_plus_nat(M_5),N_16))) # label(fact_464_power__add) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 60 (all J_2 all K all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(J_2),K)),I_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),hAPP_nat_nat(plus_plus_nat(I_2),K))))) # label(fact_1007_le__diff__conv) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 61 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K1)),bit0(K2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)))) # label(fact_149_less__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 62 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),bit1(K))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),K)))) # label(fact_535_rel__simps_I26_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 63 (all Q_1 all B all R_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,B),R_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_1),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),hAPP_int_int(div_mod_int(Q_1),C))),R_1)),zero_zero_int)))))) # label(fact_1170_zmult2__lemma__aux2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 64 (all A all P_1 (hBOOL(hAPP_int_bool(zprime,P_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(minus_minus_int(P_1),one_one_int))) -> one_one_int != inv(P_1,A))))) # label(fact_1077_inv__not__1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 65 (all B_10 all A_11 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_11),zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_10)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(A_11),B_10)),zero_zero_nat))))) # label(fact_830_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 66 (all V_21 number267125858f_real(hAPP_int_int(plus_plus_int(V_21),bit1(pls))) = hAPP_real_real(plus_plus_real(number267125858f_real(V_21)),one_one_real)) # label(fact_31_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 67 (all N_6 all A_56 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_56)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_6)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(power_power_nat(A_56),N_6)))))) # label(fact_554_one__less__power) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 68 (all W_1 all Y_2 all X_2 all Z_1 (is_int(W_1) & is_int(Y_2) & is_int(Z_1) & is_int(X_2) -> (hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(W_1),Y_2)),hAPP_int_int(times_times_int(X_2),Z_1)) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(W_1),Z_1)),hAPP_int_int(times_times_int(X_2),Y_2)) <-> X_2 = W_1 | Z_1 = Y_2))) # label(fact_170_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 69 (all C_11 all B_25 all A_27 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_25),A_27)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_11),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_27),C_11)),hAPP_int_int(times_times_int(B_25),C_11)))))) # label(fact_784_mult__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 70 (all N_1 all M hAPP_nat_nat(minus_minus_nat(N_1),hAPP_nat_nat(plus_plus_nat(N_1),M)) = zero_zero_nat) # label(fact_964_diff__add__0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 71 (all N_27 all A_78 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_78)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(power_power_real(A_78),N_27))))) # label(fact_364_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 72 (all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),M)) -> -hBOOL(hAPP_int_bool(zcong(one_one_int,number_number_of_int(min)),M)))) # label(fact_608_one__not__neg__one__mod__m) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 73 (all W_6 number267125858f_real(bit0(W_6)) = hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(one_one_real),one_one_real)),number267125858f_real(W_6))) # label(fact_270_double__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 74 (all A_55 all B_46 all V_3 hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(A_55),number_number_of_int(V_3))),hAPP_int_int(times_times_int(B_46),number_number_of_int(V_3))) = hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(A_55),B_46)),number_number_of_int(V_3))) # label(fact_623_left__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 75 (all C all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C),A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(A),C)),hAPP_nat_nat(minus_minus_nat(B),C)))))) # label(fact_991_diff__less__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 76 (all C_2 all D all A_2 all B_2 (B_2 != A_2 & D != C_2 <-> hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_2),D)),hAPP_nat_nat(times_times_nat(B_2),C_2)) != hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_2),C_2)),hAPP_nat_nat(times_times_nat(B_2),D)))) # label(fact_180_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 77 (all A_35 all B_32 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_35),hAPP_nat_nat(times_times_nat(A_35),B_32)))) # label(fact_751_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 78 (all K_1 all L all I_1 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),L)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(I_1),K_1)),hAPP_nat_nat(plus_plus_nat(J_1),L)))))) # label(fact_1002_add__le__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 79 (all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N_1))) # label(fact_876_le0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 80 (all N_4 one_one_real = hAPP_nat_real(power_power_real(number267125858f_real(min)),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_4))) # label(fact_580_power__m1__even) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 81 (all A_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(A_2),A_2)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_2),zero_zero_real)))) # label(fact_384_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 82 (all A_41 all B_38 all C_19 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(times_times_int(A_41),B_38)),C_19)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_41),C_19)))) # label(fact_734_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 83 (all C_10 all A_26 all B_24 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_26),B_24)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_10)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(C_10),A_26)),hAPP_nat_nat(times_times_nat(C_10),B_24)))))) # label(fact_786_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 84 (all A_9 all B_8 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(A_9),B_8))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_9)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_8))))) # label(fact_837_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 85 (all A_2 all B_2 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (hAPP_int_int(div_mod_int(B_2),Ma) = hAPP_int_int(div_mod_int(A_2),Ma) <-> hBOOL(hAPP_int_bool(zcong(A_2,B_2),Ma))))) # label(fact_1157_zcong__zmod__eq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 86 (all X_1 all Y_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) -> -(-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1))))) # label(fact_897_dvd_Oless__asym) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 87 (all M all N_1 all I_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),I_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(I_1),M)),hAPP_nat_nat(power_power_nat(I_1),N_1))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))))) # label(fact_517_nat__power__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 88 (all C all A all B (A = B -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B),C)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C),B)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),C)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C),A))))) # label(fact_910_dvd_Oord__eq__less__trans) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 89 (all A_75 all N_24 all B_51 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_75),N_24)),hAPP_nat_real(power_power_real(B_51),N_24))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_51)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_75),B_51))))) # label(fact_376_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 90 (all A_80 hAPP_nat_nat(plus_plus_nat(zero_zero_nat),A_80) = A_80) # label(fact_348_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 91 (all I_1 all J_1 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I_1),J_1)),I_1))) # label(fact_974_not__add__less1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 92 (all M_3 all N_9 all A_60 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_60)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_60),M_3)),hAPP_nat_real(power_power_real(A_60),N_9))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_3),N_9))))) # label(fact_497_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 93 (all B_5 all A_6 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_6)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_5)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(A_6),B_5)))))) # label(fact_846_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 94 (all K_1 all I_1 all J_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),J_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(K_1),I_1)),hAPP_int_int(times_times_int(K_1),J_1)))))) # label(fact_404_zmult__zless__mono2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 95 (all J_1 all K_1 all A all P_1 (hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(J_1),K_1),A),P_1)) -> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(multInv(P_1,J_1)),J_1)),K_1),hAPP_int_int(times_times_int(multInv(P_1,J_1)),A)),P_1)))) # label(fact_1074_aux______3) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 96 (all Q_1 all B all R_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,B),R_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_1),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(B),C)),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),hAPP_int_int(div_mod_int(Q_1),C))),R_1))))))) # label(fact_1169_zmult2__lemma__aux1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 97 (all K_1 all N_1 all M hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(K_1),N_1)),M)),N_1) = hAPP_nat_nat(div_mod_nat(M),N_1)) # label(fact_1182_mod__mult__self3) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 98 (all A_2 all W_1 (zero_zero_nat != number_number_of_nat(W_1) & A_2 = zero_zero_nat <-> zero_zero_nat = hAPP_nat_nat(power_power_nat(A_2),number_number_of_nat(W_1)))) # label(fact_583_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 99 (all M all N_1 (N_1 = M | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)))) # label(fact_989_less__or__eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 100 (all N_1 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(power_power_int(X_1),N_1))))) # label(fact_589_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 101 (all A_97 all M_12 hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_97),one_one_nat)),M_12) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_97),M_12)),M_12)) # label(fact_227_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 102 (all A_93 A_93 = hAPP_real_real(times_times_real(number267125858f_real(bit1(pls))),A_93)) # label(fact_257_mult__numeral__1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 103 (all N_25 all A_76 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_76)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(A_76),N_25))))) # label(fact_369_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 104 (all P_2 all N all K (is_int(K) & is_int(N) -> (hBOOL(hAPP_int_bool(P_2,hAPP_int_int(div_mod_int(N),K))) <-> (zero_zero_int = K -> hBOOL(hAPP_int_bool(P_2,N))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),zero_zero_int)) -> (all I all J (is_int(J) & is_int(I) -> (hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(K),I)),J) = N & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),J)) -> hBOOL(hAPP_int_bool(P_2,J)))))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K)) -> (all I all J (is_int(J) & is_int(I) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),J)) & hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(K),I)),J) = N & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,J),K)) -> hBOOL(hAPP_int_bool(P_2,J))))))))) # label(fact_1166_split__zmod) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 105 (all A_102 all M_13 all B_59 hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_102),M_13)),hAPP_nat_nat(times_times_nat(B_59),M_13)) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_102),B_59)),M_13)) # label(fact_174_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 106 (all W hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),W))) # label(fact_35_zle__refl) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 107 (all Z_4 hAPP_real_real(plus_plus_real(Z_4),Z_4) = hAPP_real_real(times_times_real(Z_4),number267125858f_real(bit0(bit1(pls))))) # label(fact_282_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 108 (all A all B hAPP_int_int(minus_minus_int(hAPP_int_int(plus_plus_int(hAPP_int_int(minus_minus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit1(bit1(pls))))),hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(number_number_of_int(bit1(bit1(pls)))),hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls)))))),B))),hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(number_number_of_int(bit1(bit1(pls)))),A)),hAPP_nat_int(power_power_int(B),number_number_of_nat(bit0(bit1(pls))))))),hAPP_nat_int(power_power_int(B),number_number_of_nat(bit1(bit1(pls))))) = hAPP_nat_int(power_power_int(hAPP_int_int(minus_minus_int(A),B)),number_number_of_nat(bit1(bit1(pls))))) # label(fact_651_zdiff__power3) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 109 (all C_2 all D all A_2 all B_2 (D != C_2 & B_2 != A_2 <-> hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_2),D)),hAPP_real_real(times_times_real(B_2),C_2)) != hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_2),C_2)),hAPP_real_real(times_times_real(B_2),D)))) # label(fact_181_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 110 (all I_1 all K_1 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_1)) -> hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(J_1),I_1)),K_1) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(minus_minus_nat(J_1),K_1)),I_1))) # label(fact_1016_diff__add__assoc2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 111 (all N_28 all A_83 all B_53 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_83),B_53)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_83)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_28)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_83),N_28)),hAPP_nat_nat(power_power_nat(B_53),N_28))))))) # label(fact_339_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 112 (all X_2 (is_int(X_2) -> (hBOOL(hAPP_int_bool(twoSqu1154269391sum2sq,X_2)) <-> (exists A_3 exists B_3 (X_2 = twoSqu1241645765sum2sq(product_Pair_int_int(A_3,B_3)) & is_int(B_3) & is_int(A_3)))))) # label(fact_1060_is__sum2sq__def) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 113 (all K all L_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K)),bit0(L_1))))) # label(fact_83_rel__simps_I33_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 114 (all A_2 all B_2 all Ma (hBOOL(hAPP_int_bool(zcong(A_2,B_2),Ma)) <-> hBOOL(hAPP_int_bool(zcong(B_2,A_2),Ma)))) # label(fact_562_zcong__sym) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 115 (all A_1 all B_1 all Q_3 all Y_1 (is_int(Y_1) & is_int(B_1) -> (A_1 = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_1),Q_3)),Y_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,B_1),Y_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_1),zero_zero_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,zero_zero_int),Y_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),B_1))) -> (zero_zero_int != B_1 -> Y_1 = hAPP_int_int(div_mod_int(A_1),B_1)))))) # label(fact_1171_divmod__int__rel__mod__eq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 116 (all B all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),zero_zero_int)) -> hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),hAPP_int_int(div_mod_int(hAPP_int_int(plus_plus_int(B),one_one_int)),A))),one_one_int) = hAPP_int_int(div_mod_int(hAPP_int_int(plus_plus_int(one_one_int),hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),B))),hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),A)))) # label(fact_1134_neg__zmod__mult__2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 117 (all X_1 hAPP_nat_real(power_power_real(hAPP_real_real(times_times_real(number267125858f_real(bit0(bit1(pls)))),X_1)),number_number_of_nat(bit0(bit1(pls)))) = hAPP_real_real(times_times_real(number267125858f_real(bit0(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(X_1),number_number_of_nat(bit0(bit1(pls)))))) # label(fact_293_four__x__squared) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 118 (all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),number_number_of_int(Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(pls)),Y_2)))) # label(fact_164_less__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 119 (all N all Ma (one_one_nat = hAPP_nat_nat(times_times_nat(N),Ma) <-> Ma = one_one_nat & N = one_one_nat)) # label(fact_766_nat__mult__eq__one) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 120 (all Ma all K all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),N)) | K = zero_zero_nat <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(times_times_nat(Ma),K)),hAPP_nat_nat(times_times_nat(N),K))))) # label(fact_860_nat__mult__dvd__cancel__disj_H) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 121 (all N_1 all B all A all P_1 (hBOOL(hAPP_int_bool(zprime,P_1)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),A)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(P_1),N_1)),hAPP_int_int(times_times_int(A),B))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(P_1),N_1)),B)))))) # label(fact_409_zprime__power__zdvd__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 122 (all M_1 all D_1 (is_int(M_1) -> (zero_zero_int = hAPP_int_int(div_mod_int(M_1),D_1) -> (exists Q_2 (is_int(Q_2) & hAPP_int_int(times_times_int(D_1),Q_2) = M_1))))) # label(fact_1175_zmod__eq__0D) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 123 (all V_14 all W_13 all Z_8 hAPP_int_int(times_times_int(number_number_of_int(V_14)),hAPP_int_int(times_times_int(number_number_of_int(W_13)),Z_8)) = hAPP_int_int(times_times_int(number_number_of_int(hAPP_int_int(times_times_int(V_14),W_13))),Z_8)) # label(fact_239_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 124 (all C_2 all A_2 all B_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_2),A_2)),hAPP_real_real(times_times_real(C_2),B_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_2),B_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_2)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_2),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_2),A_2)))) # label(fact_851_mult__less__cancel__left__disj) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 125 (all I_1 all K_1 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_1)) -> hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(I_1),K_1)),J_1) = hAPP_nat_nat(minus_minus_nat(I_1),hAPP_nat_nat(minus_minus_nat(J_1),K_1)))) # label(fact_1006_diff__diff__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 126 (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,bit0(K1)),bit1(K2))))) # label(fact_154_less__eq__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 127 (all X_1 all M all Y_1 (hAPP_int_int(div_mod_int(Y_1),M) = hAPP_int_int(div_mod_int(X_1),M) -> hBOOL(hAPP_int_bool(zcong(X_1,Y_1),M)))) # label(fact_1143_Residues_Oaux) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 128 (all X_13 all Y_10 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(X_13),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_10),number_number_of_nat(bit0(bit1(pls)))))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_10)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_13),Y_10))))) # label(fact_472_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 129 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) -> hAPP_nat_nat(minus_minus_nat(M),N_1) = zero_zero_nat)) # label(fact_967_diff__is__0__eq_H) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 130 (all K all L_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K)),bit0(L_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L_1)))) # label(fact_150_rel__simps_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 131 (exists 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(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int))) & (all Y (is_int(Y) -> (hBOOL(hAPP_int_bool(zcong(s1,Y),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) -> Y = X))) & hBOOL(hAPP_int_bool(zcong(s1,X),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int))) & is_int(X))) # label(fact_308__096EX_B_As_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_A1_A_G_A_091s1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 132 (all I_1 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J_1)) -> (exists K_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2)) & J_1 = hAPP_nat_nat(plus_plus_nat(I_1),K_2))))) # label(fact_1116_less__imp__add__positive) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 133 (all I_1 all K_1 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_1)) -> hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(I_1),J_1)),K_1) = hAPP_nat_nat(plus_plus_nat(I_1),hAPP_nat_nat(minus_minus_nat(J_1),K_1)))) # label(fact_1014_diff__add__assoc) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 134 (all X_17 all Y_14 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(X_17),X_17)),hAPP_int_int(times_times_int(Y_14),Y_14))))) # label(fact_411_sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 135 (all Ma all N all K (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K)) -> (hAPP_nat_nat(times_times_nat(K),N) = hAPP_nat_nat(times_times_nat(K),Ma) <-> Ma = N))) # label(fact_1047_nat__mult__eq__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 136 (all A_75 all N_24 all B_51 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_75),N_24)),hAPP_nat_int(power_power_int(B_51),N_24))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_51)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_75),B_51))))) # label(fact_374_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 137 (all X_1 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),hAPP_nat_int(power_power_int(X_1),number_number_of_nat(bit0(bit1(pls))))))) # label(fact_219_power2__ge__self) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 138 (all Ma all K all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(Ma),K)),hAPP_nat_nat(times_times_nat(N),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),N)))) # label(fact_1035_mult__less__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 139 -(all S_1 (is_int(S_1) -> -(hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,S_1),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int))) & hBOOL(hAPP_int_bool(zcong(s1,S_1),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),S_1))))) # label(fact_309__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]. 1.32/1.57 140 (all A_107 all B_60 all C_33 hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_107),B_60)),C_33) = hAPP_int_int(plus_plus_int(A_107),hAPP_int_int(plus_plus_int(B_60),C_33))) # label(fact_123_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 141 (all B_35 all A_38 all C_17 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_38),C_17)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_38),hAPP_nat_nat(times_times_nat(B_35),C_17))))) # label(fact_742_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 142 (all X_2 all Y_2 (zero_zero_real = X_2 & Y_2 = zero_zero_real <-> zero_zero_real = hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(X_2),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_2),number_number_of_nat(bit0(bit1(pls))))))) # label(fact_669_realpow__two__sum__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 143 (all A_45 all B_41 all C_22 hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_45),C_22)),hAPP_real_real(times_times_real(B_41),C_22)) = hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_45),B_41)),C_22)) # label(fact_721_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 144 (all W_1 all X_2 (X_2 = number267125858f_real(W_1) <-> X_2 = number267125858f_real(W_1))) # label(fact_139_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 145 (all A_69 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(power_power_int(A_69),number_number_of_nat(bit0(bit1(pls))))))) # label(fact_444_zero__le__power2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 146 (all W_1 all X_2 (is_int(X_2) -> (number_number_of_int(W_1) = X_2 <-> number_number_of_int(W_1) = X_2))) # label(fact_138_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 147 (all X_2 all N (hAPP_nat_nat(power_power_nat(X_2),N) = one_one_nat <-> N = zero_zero_nat | one_one_nat = X_2)) # label(fact_869_exp__eq__1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 148 (all B_19 all A_21 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_21)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_19),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(B_19),A_21)),zero_zero_int))))) # label(fact_801_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 149 (all B all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(times_times_int(A),B))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B))))) # label(fact_403_pos__zmult__pos) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 150 (all U_1 all M all N_1 all J_1 all I_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),I_1)) -> hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(I_1),J_1)),U_1)),M)),N_1) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_1),U_1)),M)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_1),U_1)),N_1)))) # label(fact_1053_nat__diff__add__eq1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 151 (all B_2 all A_2 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(bit1(bit0(bit1(pls))))),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),hAPP_int_int(minus_minus_int(P_3),one_one_int))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),hAPP_int_int(minus_minus_int(P_3),one_one_int))) -> (hBOOL(member_int(inv(P_3,B_2),wset(A_2,P_3))) -> hBOOL(member_int(B_2,wset(A_2,P_3)))))))))) # label(fact_1097_wset__inv__mem__mem) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 152 (all Ma all X_2 (hBOOL(hAPP_int_bool(quadRes(Ma),X_2)) <-> (exists Y (is_int(Y) & hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(Y),number_number_of_nat(bit0(bit1(pls)))),X_2),Ma)))))) # label(fact_668_QuadRes__def) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 153 (all P_2 all I_2 all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K)) -> (hBOOL(hAPP_int_bool(P_2,K)) -> ((all I (is_int(I) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I),K)) -> (hBOOL(hAPP_int_bool(P_2,I)) -> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(minus_minus_int(I),one_one_int))))))) -> hBOOL(hAPP_int_bool(P_2,I_2)))))) # label(fact_1104_int__le__induct) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 154 (all C_3 all A_13 all B_12 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_13),B_12)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_3)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_13),C_3)),hAPP_real_real(times_times_real(B_12),C_3)))))) # label(fact_824_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 155 (all U all Ma all N all I_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_2),U)),Ma)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_2),U)),N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(J_2),I_2)),U)),N)))))) # label(fact_1055_nat__le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 156 (all M zero_zero_nat = hAPP_nat_nat(minus_minus_nat(M),M)) # label(fact_886_diff__self__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 157 (all A_2 all B_2 (is_int(A_2) & is_int(B_2) -> (zero_zero_int = hAPP_int_int(times_times_int(A_2),B_2) <-> A_2 = zero_zero_int | B_2 = zero_zero_int))) # label(fact_705_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 158 (all K all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(times_times_nat(K),Ma)),hAPP_nat_nat(times_times_nat(K),N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),N)) | zero_zero_nat = K)) # label(fact_1049_nat__mult__dvd__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 159 (all X_21 all Y_18 all Z_9 hAPP_nat_nat(times_times_nat(X_21),hAPP_nat_nat(plus_plus_nat(Y_18),Z_9)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(X_21),Y_18)),hAPP_nat_nat(times_times_nat(X_21),Z_9))) # label(fact_183_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 160 (all B_2 all A_2 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(bit1(bit0(bit1(pls))))),P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),hAPP_int_int(minus_minus_int(P_3),one_one_int))) -> (hBOOL(member_int(B_2,wset(A_2,P_3))) -> hBOOL(member_int(inv(P_3,B_2),wset(A_2,P_3)))))))) # label(fact_1096_wset__mem__inv__mem) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 161 (all Z1 all Z2 all W hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(Z1),Z2)),W) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(Z1),W)),hAPP_real_real(times_times_real(Z2),W))) # label(fact_767_real__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 162 (all X_1 all Y_1 all Z hAPP_nat_int(power_power_int(X_1),hAPP_nat_nat(times_times_nat(Y_1),Z)) = hAPP_nat_int(power_power_int(hAPP_nat_int(power_power_int(X_1),Y_1)),Z)) # label(fact_47_zpower__zpower) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 163 (all C_4 all A_14 all B_13 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_14),B_13)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_4)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_4),A_14)),hAPP_int_int(times_times_int(C_4),B_13)))))) # label(fact_823_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 164 (all K_1 all I_1 all J_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),J_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(I_1),K_1)),hAPP_int_int(plus_plus_int(J_1),K_1))))) # label(fact_75_zadd__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 165 (all X_11 all Y_8 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(X_11),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_8),number_number_of_nat(bit0(bit1(pls)))))),zero_zero_real))) # label(fact_479_not__sum__power2__lt__zero) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 166 (all Lx_2 all Ly all Rx_2 hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_2),Ly)),Rx_2) = hAPP_nat_nat(times_times_nat(Lx_2),hAPP_nat_nat(times_times_nat(Ly),Rx_2))) # label(fact_104_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 167 (all A_2 (one_one_nat != A_2 <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A_2)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),one_one_nat)))) # label(fact_1125_gcd__lcm__complete__lattice__nat_Obot__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 168 (all A_100 (is_int(A_100) -> A_100 = hAPP_int_int(times_times_int(A_100),one_one_int))) # label(fact_185_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 169 (all W_4 ((number_number_of_nat(W_4) = zero_zero_nat -> one_one_int = hAPP_nat_int(power_power_int(zero_zero_int),number_number_of_nat(W_4))) & (zero_zero_nat != number_number_of_nat(W_4) -> zero_zero_int = hAPP_nat_int(power_power_int(zero_zero_int),number_number_of_nat(W_4))))) # label(fact_593_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 170 (all M all Y_1 all X_1 (is_int(X_1) & is_int(Y_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Y_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M)) -> (hBOOL(hAPP_int_bool(zcong(X_1,Y_1),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),M)) -> Y_1 = X_1)))))))) # label(fact_652_zcong__less__eq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 171 (all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(one_one_int),Z_1)),Z_1)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_1),zero_zero_int)))) # label(fact_428_odd__less__0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 172 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit0(K1)),bit0(K2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)))) # label(fact_71_less__eq__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 173 (all A_111 all B_63 hAPP_real_real(times_times_real(A_111),B_63) = hAPP_real_real(times_times_real(B_63),A_111)) # label(fact_114_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 174 (all W_16 hAPP_nat_real(power_power_real(number267125858f_real(W_16)),number_number_of_nat(bit0(bit1(pls)))) = hAPP_real_real(times_times_real(number267125858f_real(W_16)),number267125858f_real(W_16))) # label(fact_13_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 175 (all A_45 all B_41 all C_22 hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_45),C_22)),hAPP_nat_nat(times_times_nat(B_41),C_22)) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_45),B_41)),C_22)) # label(fact_722_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 176 (all X_14 all Y_11 (hAPP_nat_nat(power_power_nat(Y_11),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_nat(power_power_nat(X_14),number_number_of_nat(bit0(bit1(pls)))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),X_14)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_11)) -> X_14 = Y_11)))) # label(fact_449_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 177 (all X_1 all Y_1 all P_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP_int_bool(zcong(X_1,Y_1),P_1)) -> hBOOL(hAPP_int_bool(zcong(multInv(P_1,X_1),multInv(P_1,Y_1)),P_1))))) # label(fact_1081_MultInv__prop1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 178 (all A_64 all M_4 all N_14 hAPP_nat_int(power_power_int(hAPP_nat_int(power_power_int(A_64),M_4)),N_14) = hAPP_nat_int(power_power_int(A_64),hAPP_nat_nat(times_times_nat(M_4),N_14))) # label(fact_471_power__mult) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 179 (all N all P_2 (-hBOOL(hAPP_nat_bool(P_2,zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(P_2,N)) -> (exists K_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N)) & (all I (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K_2)) -> -hBOOL(hAPP_nat_bool(P_2,I)))) & hBOOL(hAPP_nat_bool(P_2,K_2))))))) # label(fact_1111_ex__least__nat__le) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 180 (all A_52 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_52),A_52))) # label(fact_679_dvd__refl) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 181 (all M all I_1 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),hAPP_nat_nat(plus_plus_nat(J_1),M))))) # label(fact_999_trans__le__add1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 182 (all X_2 all Y_2 (is_int(Y_2) & is_int(X_2) -> (hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X_2),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_2),number_number_of_nat(bit0(bit1(pls))))) = zero_zero_int <-> Y_2 = zero_zero_int & zero_zero_int = X_2))) # label(fact_456_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 183 (all A_99 hAPP_real_real(times_times_real(one_one_real),A_99) = A_99) # label(fact_190_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 184 (all A_79 A_79 = hAPP_real_real(plus_plus_real(A_79),zero_zero_real)) # label(fact_352_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 185 (all K_1 all A all B all M (hBOOL(hAPP_int_bool(zcong(A,B),M)) -> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A),K_1),hAPP_int_int(times_times_int(B),K_1)),M)))) # label(fact_577_zcong__scalar) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 186 (all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> wset(hAPP_int_int(minus_minus_int(P_3),number_number_of_int(bit0(bit1(pls)))),P_3) = d22set(hAPP_int_int(minus_minus_int(P_3),number_number_of_int(bit0(bit1(pls))))))) # label(fact_1110_d22set__eq__wset) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 187 (all X_21 all Y_18 all Z_9 hAPP_real_real(times_times_real(X_21),hAPP_real_real(plus_plus_real(Y_18),Z_9)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(X_21),Y_18)),hAPP_real_real(times_times_real(X_21),Z_9))) # label(fact_184_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 188 (all Z all W hAPP_int_int(times_times_int(Z),W) = hAPP_int_int(times_times_int(W),Z)) # label(fact_143_zmult__commute) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 189 (all A_81 hAPP_real_real(times_times_real(A_81),zero_zero_real) = zero_zero_real) # label(fact_346_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 190 (all K all L_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(K)),number_number_of_int(L_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L_1)))) # label(fact_73_less__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 191 (all P_2 collect_int(P_2) = P_2) # label(fact_119_Collect__def) # label(axiom) # label(non_clause). [assumption]. 1.32/1.57 192 (all M all I_1 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),hAPP_nat_nat(plus_plus_nat(M),J_1))))) # label(fact_1000_trans__le__add2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 193 (all B_2 all P_3 all A_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_2)) -> (hBOOL(member_int(B_2,wset(hAPP_int_int(minus_minus_int(A_2),one_one_int),P_3))) -> hBOOL(member_int(B_2,wset(A_2,P_3)))))) # label(fact_1099_wset__subset) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 194 (all K_1 all L hAPP_int_int(minus_minus_int(bit1(K_1)),bit0(L)) = bit1(hAPP_int_int(minus_minus_int(K_1),L))) # label(fact_624_diff__bin__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 195 (all A_2 all B_2 all C_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_2),B_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_2),A_2)),hAPP_real_real(times_times_real(C_2),B_2)))))) # label(fact_849_mult__less__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 196 (all X_5 all Y_4 (is_int(Y_4) & is_int(X_5) -> (Y_4 != X_5 -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_5),Y_4)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_4),X_5)))))) # label(fact_677_linorder__neqE__linordered__idom) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 197 (all M_11 all A_96 hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_96),one_one_int)),M_11) = hAPP_int_int(plus_plus_int(M_11),hAPP_int_int(times_times_int(A_96),M_11))) # label(fact_229_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 198 (all A_57 one_one_real = hAPP_nat_real(power_power_real(A_57),zero_zero_nat)) # label(fact_544_power__0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 199 (all N_1 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_1)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N_1),M))))) # label(fact_326_zdvd__not__zless) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 200 (all A_104 all C_30 hAPP_real_real(plus_plus_real(C_30),A_104) = hAPP_real_real(plus_plus_real(A_104),C_30)) # label(fact_134_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 201 (all A_59 all N_8 all N_7 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_8),N_7)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_59)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_59),N_8)),hAPP_nat_nat(power_power_nat(A_59),N_7)))))) # label(fact_501_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 202 (all K_1 all I_1 all J_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),J_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(K_1),I_1)),hAPP_int_int(plus_plus_int(K_1),J_1))))) # label(fact_76_zadd__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 203 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> zero_zero_nat != N_1)) # label(fact_954_gr__implies__not0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 204 (all N_1 all M hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_nat_nat(plus_plus_nat(M),N_1)))) # label(fact_995_le__add2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 205 (all A_2 (is_int(A_2) -> (zero_zero_int = hAPP_nat_int(power_power_int(A_2),number_number_of_nat(bit0(bit1(pls)))) <-> A_2 = zero_zero_int))) # label(fact_442_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 206 (all A_108 all B_61 all C_34 hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_108),B_61)),C_34) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_108),C_34)),B_61)) # label(fact_120_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 207 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) -> (M != N_1 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))))) # label(fact_988_le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 208 (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]. 1.32/1.58 209 (all K all L_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(K)),bit0(L_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L_1)))) # label(fact_69_rel__simps_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 210 (all C_3 all A_13 all B_12 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_13),B_12)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(A_13),C_3)),hAPP_nat_nat(times_times_nat(B_12),C_3)))))) # label(fact_825_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 211 (all M all N_1 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> hAPP_nat_nat(div_mod_nat(M),N_1) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(minus_minus_nat(M),N_1)),N_1))) # label(fact_1180_mod__geq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 212 (all N_30 all A_84 (is_int(A_84) -> (A_84 != zero_zero_int -> zero_zero_int != hAPP_nat_int(power_power_int(A_84),N_30)))) # label(fact_332_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 213 (all A_18 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(times_times_real(A_18),A_18)))) # label(fact_812_zero__le__square) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 214 (all U all Ma all N all J_2 all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),I_2)) -> (hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_2),U)),N) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_2),U)),Ma) <-> hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(I_2),J_2)),U)),Ma) = N))) # label(fact_1054_nat__eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 215 (all X_13 all Y_10 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(X_13),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_10),number_number_of_nat(bit0(bit1(pls)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_10)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_13),Y_10))))) # label(fact_474_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 216 (all N_25 all A_76 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_76)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(power_power_real(A_76),N_25))))) # label(fact_370_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 217 one_one_int = t -> (exists X exists Y (is_int(X) & is_int(Y) & hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y),number_number_of_nat(bit0(bit1(pls))))) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int))) # label(fact_1__096t_A_061_A1_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 218 (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]. 1.32/1.58 219 (all X_4 all N_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_3)) -> hAPP_nat_int(power_power_int(X_4),N_3) = hAPP_int_int(times_times_int(hAPP_nat_int(power_power_int(X_4),hAPP_nat_nat(minus_minus_nat(N_3),one_one_nat))),X_4))) # label(fact_694_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 220 (all B_11 all A_12 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_12),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_11),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(times_times_int(A_12),B_11)))))) # label(fact_828_mult__neg__neg) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 221 (all L hAPP_int_int(minus_minus_int(min),bit0(L)) = bit1(hAPP_int_int(minus_minus_int(min),L))) # label(fact_635_diff__bin__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 222 (all A all P_1 (hBOOL(hAPP_int_bool(zprime,P_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(minus_minus_int(P_1),one_one_int))) -> zero_zero_int != inv(P_1,A))))) # label(fact_1082_inv__not__0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 223 (all M_6 all A_86 all N_33 all B_55 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(power_power_real(A_86),N_33)),B_55)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_6),N_33)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(power_power_real(A_86),M_6)),B_55))))) # label(fact_320_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 224 (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),bit0(K))))) # label(fact_549_rel__simps_I25_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 225 (all B_30 all A_32 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_32)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_30),zero_zero_real)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_32),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_30)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_32),B_30)),zero_zero_real)))) # label(fact_770_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 226 (all A all N_1 all P_1 (hBOOL(hAPP_int_bool(zprime,P_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),hAPP_nat_int(power_power_int(A),N_1))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),A))))) # label(fact_390_zprime__zdvd__power) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 227 (all X_1 all Y_1 (is_int(Y_1) & is_int(X_1) -> Y_1 = X_1 | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),X_1)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Y_1)))) # label(fact_38_zless__linear) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 228 (all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),bit1(pls))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(X_2)),one_one_int)))) # label(fact_166_le__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 229 (all K_1 all L hAPP_int_int(plus_plus_int(bit1(K_1)),bit0(L)) = bit1(hAPP_int_int(plus_plus_int(K_1),L))) # label(fact_251_add__Bit1__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 230 (all X_1 all Y_1 all Z hAPP_int_int(plus_plus_int(X_1),hAPP_int_int(plus_plus_int(Y_1),Z)) = hAPP_int_int(plus_plus_int(Y_1),hAPP_int_int(plus_plus_int(X_1),Z))) # label(fact_146_zadd__left__commute) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 231 (all C_2 all A_2 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_2)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),B_2)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_2),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),A_2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_2),A_2)),hAPP_int_int(times_times_int(C_2),B_2))))) # label(fact_852_mult__less__cancel__left__disj) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 232 (all M hAPP_nat_nat(minus_minus_nat(M),zero_zero_nat) = M) # label(fact_885_minus__nat_Odiff__0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 233 (all X_1 all Y_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)))) # label(fact_906_dvd_Oless__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 234 (all W_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(W_1)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),zero_zero_int)))) # label(fact_398_bin__less__0__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 235 (all K_1 all L bit1(hAPP_int_int(plus_plus_int(K_1),L)) = hAPP_int_int(plus_plus_int(bit0(K_1)),bit1(L))) # label(fact_252_add__Bit0__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 236 (all A_2 (A_2 != zero_zero_real <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(power_power_real(A_2),number_number_of_nat(bit0(bit1(pls)))))))) # label(fact_453_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 237 (all B_43 all A_49 (A_49 != zero_zero_nat -> (zero_zero_nat != B_43 -> zero_zero_nat != hAPP_nat_nat(times_times_nat(A_49),B_43)))) # label(fact_702_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 238 (all Lx all Rx all Ry hAPP_nat_nat(times_times_nat(Rx),hAPP_nat_nat(times_times_nat(Lx),Ry)) = hAPP_nat_nat(times_times_nat(Lx),hAPP_nat_nat(times_times_nat(Rx),Ry))) # label(fact_110_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 239 (all J_1 all K_1 all A all P_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP_int_bool(zprime,P_1)) -> (-hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_1)) -> (-hBOOL(hAPP_int_bool(zcong(K_1,zero_zero_int),P_1)) -> (-hBOOL(hAPP_int_bool(zcong(J_1,zero_zero_int),P_1)) -> (hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A),multInv(P_1,J_1)),hAPP_int_int(times_times_int(A),multInv(P_1,K_1))),P_1)) -> hBOOL(hAPP_int_bool(zcong(J_1,K_1),P_1))))))))) # label(fact_1091_MultInv__zcong__prop3) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 240 (all A_52 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_52),A_52))) # label(fact_678_dvd__refl) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 241 (all A_2 all N (zero_zero_nat != N & zero_zero_real = A_2 <-> hAPP_nat_real(power_power_real(A_2),N) = zero_zero_real)) # label(fact_313_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 242 (all A_105 all C_31 all D_9 hAPP_nat_nat(plus_plus_nat(A_105),hAPP_nat_nat(plus_plus_nat(C_31),D_9)) = hAPP_nat_nat(plus_plus_nat(C_31),hAPP_nat_nat(plus_plus_nat(A_105),D_9))) # label(fact_130_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 243 (all C_9 all A_25 all B_23 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_25),B_23)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_9)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(C_9),A_25)),hAPP_int_int(times_times_int(C_9),B_23)))))) # label(fact_790_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 244 (all C_2 all D all A_2 all B_2 all Ma (hBOOL(hAPP_int_bool(zcong(A_2,B_2),Ma)) -> (hBOOL(hAPP_int_bool(zcong(C_2,hAPP_int_int(times_times_int(A_2),D)),Ma)) <-> hBOOL(hAPP_int_bool(zcong(C_2,hAPP_int_int(times_times_int(B_2),D)),Ma))))) # label(fact_644_zcong__zmult__prop1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 245 (all M_10 hAPP_int_int(plus_plus_int(M_10),M_10) = hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(one_one_int),one_one_int)),M_10)) # label(fact_232_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 246 (all B_9 all A_10 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(times_times_int(B_9),A_10))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_10)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_9))))) # label(fact_836_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 247 (all A_45 all B_41 all C_22 hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_45),B_41)),C_22) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_45),C_22)),hAPP_int_int(times_times_int(B_41),C_22))) # label(fact_723_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 248 (all V_1 all U_2 all Y_5 all X_6 all A_54 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_6),A_54)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_5),A_54)) -> (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_1)) -> (hAPP_int_int(plus_plus_int(U_2),V_1) = one_one_int -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(U_2),X_6)),hAPP_int_int(times_times_int(V_1),Y_5))),A_54)))))))) # label(fact_671_convex__bound__lt) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 249 (all Y_1 all X_1 all P_1 (-hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),P_1)) -> (hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(Y_1),number_number_of_nat(bit0(bit1(pls)))),X_1),P_1)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),Y_1))))) # label(fact_504_Euler_Oaux____1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 250 (all A_91 hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(A_91),A_91)),A_91) = hAPP_nat_int(power_power_int(A_91),number_number_of_nat(bit1(bit1(pls))))) # label(fact_271_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 251 (all Lx_3 all Ly_1 all Rx_3 hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_3),Ly_1)),Rx_3) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_3),Rx_3)),Ly_1)) # label(fact_101_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 252 (all K_1 (is_int(K_1) -> hAPP_int_int(minus_minus_int(K_1),pls) = K_1)) # label(fact_614_diff__bin__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 253 (all A all B all C hAPP_int_int(div_mod_int(hAPP_int_int(times_times_int(A),B)),C) = hAPP_int_int(div_mod_int(hAPP_int_int(times_times_int(A),hAPP_int_int(div_mod_int(B),C))),C)) # label(fact_1150_zmod__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 254 (all V_18 all V_17 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_17)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_18)) -> number_number_of_int(hAPP_int_int(plus_plus_int(V_17),V_18)) = hAPP_int_int(plus_plus_int(number_number_of_int(V_17)),number_number_of_int(V_18))))) # label(fact_216_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 255 -hBOOL(hAPP_int_bool(quadRes(hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int)),number_number_of_int(min))) -> one_one_int != legendre(number_number_of_int(min),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int)) # label(fact_610__096_126_AQuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_A_061_061_062_ALegendre_A_N) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 256 (all V_18 all V_17 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_17)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_18)) -> hAPP_real_real(plus_plus_real(number267125858f_real(V_17)),number267125858f_real(V_18)) = number267125858f_real(hAPP_int_int(plus_plus_int(V_17),V_18))))) # label(fact_218_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 257 (all X_25 all N_40 hAPP_nat_nat(power_power_nat(X_25),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_40)) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(power_power_nat(X_25),N_40)),hAPP_nat_nat(power_power_nat(X_25),N_40))) # label(fact_27_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 258 (all A_87 all M_8 all N_35 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_8),N_35)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(power_power_real(A_87),M_8)),hAPP_nat_real(power_power_real(A_87),N_35))))) # label(fact_314_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 259 (all A_9 all B_8 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(times_times_int(A_9),B_8))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_9)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_8))))) # label(fact_839_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 260 (all M all I_1 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(plus_plus_nat(M),J_1))))) # label(fact_978_trans__less__add2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 261 (all A_2 all B_2 all C_2 (zero_zero_real != C_2 -> (hAPP_real_real(times_times_real(B_2),C_2) = hAPP_real_real(times_times_real(A_2),C_2) <-> B_2 = A_2))) # label(fact_761_real__mult__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 262 (all K_1 (is_int(K_1) -> number_number_of_int(K_1) = K_1)) # label(fact_144_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 263 (all I_1 all U_1 all J_1 all K_1 hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_1),U_1)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_1),U_1)),K_1)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(I_1),J_1)),U_1)),K_1)) # label(fact_1046_left__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 264 (all N_29 ((zero_zero_nat != N_29 -> hAPP_nat_nat(power_power_nat(zero_zero_nat),N_29) = zero_zero_nat) & (N_29 = zero_zero_nat -> one_one_nat = hAPP_nat_nat(power_power_nat(zero_zero_nat),N_29)))) # label(fact_335_power__0__left) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 265 (all I_1 all K_1 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_1)) -> hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(J_1),I_1)),K_1) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(minus_minus_nat(J_1),K_1)),I_1))) # label(fact_1015_add__diff__assoc2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 266 (all X_2 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2)) <-> X_2 = Y_2 | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2)))) # label(fact_685_less__eq__real__def) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 267 (all X_2 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(X_2),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_2),number_number_of_nat(bit0(bit1(pls)))))),zero_zero_real)) <-> zero_zero_real = X_2 & zero_zero_real = Y_2)) # label(fact_477_sum__power2__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 268 (all V_1 all U_2 all Y_5 all X_6 all A_54 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_6),A_54)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_5),A_54)) -> (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_1)) -> (one_one_real = hAPP_real_real(plus_plus_real(U_2),V_1) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(U_2),X_6)),hAPP_real_real(times_times_real(V_1),Y_5))),A_54)))))))) # label(fact_670_convex__bound__lt) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 269 (all A all N_1 all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(A),N_1)),hAPP_nat_nat(power_power_nat(B),N_1))) -> (N_1 != zero_zero_nat -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B))))) # label(fact_868_divides__rev) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 270 (all K_1 bit0(K_1) = hAPP_int_int(plus_plus_int(K_1),K_1)) # label(fact_206_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 271 (all V_15 all B_56 all C_28 hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(number_number_of_nat(V_15)),B_56)),hAPP_nat_nat(times_times_nat(number_number_of_nat(V_15)),C_28)) = hAPP_nat_nat(times_times_nat(number_number_of_nat(V_15)),hAPP_nat_nat(plus_plus_nat(B_56),C_28))) # label(fact_224_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 272 (all U_1 all M all N_1 all I_1 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1)) -> hAPP_nat_nat(minus_minus_nat(M),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(J_1),I_1)),U_1)),N_1)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_1),U_1)),M)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_1),U_1)),N_1)))) # label(fact_1056_nat__diff__add__eq2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 273 (all M all N_1 all K_1 hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(M),N_1)),K_1) = hAPP_nat_nat(times_times_nat(M),hAPP_nat_nat(times_times_nat(N_1),K_1))) # label(fact_949_nat__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 274 (all V all W hAPP_int_int(plus_plus_int(number_number_of_int(V)),number_number_of_int(W)) = number_number_of_int(hAPP_int_int(plus_plus_int(V),W))) # label(fact_212_plus__numeral__code_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 275 (all W_1 all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),hAPP_int_int(minus_minus_int(Z_1),one_one_int))))) # label(fact_633_zle__diff1__eq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 276 (all A_46 all E all B_42 all C_23 hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_46),B_42)),E)),C_23) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_46),E)),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_42),E)),C_23))) # label(fact_720_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 277 (all C_4 all A_14 all B_13 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_14),B_13)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_4)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_4),A_14)),hAPP_real_real(times_times_real(C_4),B_13)))))) # label(fact_821_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 278 (all W_5 hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(zero_zero_int),number_number_of_int(W_5))),number_number_of_int(W_5)) = number_number_of_int(bit0(W_5))) # label(fact_421_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 279 (all X_9 all N_5 (one_one_real = X_9 | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_5)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_9),hAPP_nat_real(power_power_real(X_9),N_5))))) # label(fact_556_dvd__power) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 280 (all B_1_1 all B_2_1 (is_int(B_1_1) & is_int(B_2_1) -> is_int(legendre(B_1_1,B_2_1)))) # label(gsy_c_Residues_OLegendre) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 281 (all L all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(L),N_1)),hAPP_nat_nat(minus_minus_nat(L),M))))) # label(fact_943_diff__le__mono2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 282 (all K all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K),N)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K),hAPP_nat_nat(plus_plus_nat(N),K))))) # label(fact_992_dvd__reduce) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 283 (all Lx_4 all Ly_2 all Rx_4 all Ry_2 hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_4),Ly_2)),hAPP_int_int(times_times_int(Rx_4),Ry_2)) = hAPP_int_int(times_times_int(Lx_4),hAPP_int_int(times_times_int(Ly_2),hAPP_int_int(times_times_int(Rx_4),Ry_2)))) # label(fact_97_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 284 (all A all B hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls))))),hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),A)),B))),hAPP_nat_int(power_power_int(B),number_number_of_nat(bit0(bit1(pls))))) = hAPP_nat_int(power_power_int(hAPP_int_int(plus_plus_int(A),B)),number_number_of_nat(bit0(bit1(pls))))) # label(fact_7_zadd__power2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 285 (all N_1 (one_one_int = hAPP_nat_int(power_power_int(number_number_of_int(min)),N_1) | number_number_of_int(min) = hAPP_nat_int(power_power_int(number_number_of_int(min)),N_1))) # label(fact_660_neg__one__power) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 286 (all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),pls)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(X_2)),zero_zero_int)))) # label(fact_436_le__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 287 (all A_46 all E all B_42 all C_23 hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_46),E)),hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(B_42),E)),C_23)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_46),B_42)),E)),C_23)) # label(fact_718_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 288 (all Z all W (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z),W)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),Z)))) # label(fact_881_real__le__linear) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 289 (all Z all X_1 all Y_1 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z),Y_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),Z)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z),X_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Z))))) # label(fact_898_dvd_Oless__trans) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 290 (all A_112 hAPP_int_int(times_times_int(A_112),A_112) = hAPP_nat_int(power_power_int(A_112),number_number_of_nat(bit0(bit1(pls))))) # label(fact_22_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 291 (all V_7 all W_1 (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(W_1)),number_number_of_int(V_7))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(V_7)),number_number_of_int(W_1))))) # label(fact_48_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 292 (all N (zero_zero_nat = N <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),zero_zero_nat)))) # label(fact_965_le__0__eq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 293 (all V_2 all W_2 all C_25 hAPP_real_real(minus_minus_real(number267125858f_real(hAPP_int_int(plus_plus_int(V_2),W_2))),C_25) = hAPP_real_real(plus_plus_real(number267125858f_real(V_2)),hAPP_real_real(minus_minus_real(number267125858f_real(W_2)),C_25))) # label(fact_630_add__number__of__diff1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 294 (all A_79 A_79 = hAPP_nat_nat(plus_plus_nat(A_79),zero_zero_nat)) # label(fact_351_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 295 (all K all Ma all N (Ma = N <-> hAPP_nat_nat(plus_plus_nat(K),Ma) = hAPP_nat_nat(plus_plus_nat(K),N))) # label(fact_930_nat__add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 296 (all Y_1 all X_1 all P_1 (hBOOL(hAPP_int_bool(zprime,P_1)) -> (-hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),P_1)) -> (-hBOOL(hAPP_int_bool(zcong(Y_1,zero_zero_int),P_1)) -> -hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(X_1),Y_1),zero_zero_int),P_1)))))) # label(fact_661_zcong__zmult__prop3) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 297 (all A_48 hAPP_real_real(times_times_real(A_48),zero_zero_real) = zero_zero_real) # label(fact_706_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 298 (all X_23 (is_int(X_23) -> X_23 = hAPP_nat_int(power_power_int(X_23),one_one_nat))) # label(fact_44_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 299 (all N_27 all A_78 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_78)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(power_power_int(A_78),N_27))))) # label(fact_362_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 300 (all C_10 all A_26 all B_24 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_26),B_24)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_10)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(C_10),A_26)),hAPP_int_int(times_times_int(C_10),B_24)))))) # label(fact_787_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 301 (all A_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_2)) <-> A_2 = zero_zero_nat)) # label(fact_1119_gcd__lcm__complete__lattice__nat_Otop__unique) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 302 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),bit1(K))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K)))) # label(fact_529_rel__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 303 (all X_1 all Y_1 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),M)) -> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(standardRes(M,X_1)),standardRes(M,Y_1)),hAPP_int_int(times_times_int(X_1),Y_1)),M)))) # label(fact_1176_StandardRes__prop4) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 304 (all K_1 all I_1 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(K_1),I_1)),hAPP_nat_nat(times_times_nat(K_1),J_1)))))) # label(fact_1033_mult__less__mono2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 305 (all A_2 all B_2 (is_int(B_2) & is_int(A_2) -> (A_2 = B_2 <-> hBOOL(hAPP_int_bool(zcong(A_2,B_2),zero_zero_int))))) # label(fact_573_IntPrimes_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 306 (all A_61 all N_11 all N_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_11),N_10)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),A_61)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(A_61),N_11)),hAPP_nat_nat(power_power_nat(A_61),N_10)))))) # label(fact_489_power__increasing) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 307 (all K_1 all I_1 all J_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),J_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_1),K_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),K_1))))) # label(fact_39_zle__trans) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 308 (all B_5 all A_6 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_6)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_5)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(times_times_int(A_6),B_5)))))) # label(fact_848_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 309 (all B all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(A),B)),zero_zero_int)) -> hAPP_int_int(plus_plus_int(A),B) = hAPP_int_int(div_mod_int(A),B)))) # label(fact_1164_mod__pos__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 310 (all X_1 all P_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP_int_bool(zprime,P_1)) -> (-hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),P_1)) -> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(X_1),multInv(P_1,X_1)),one_one_int),P_1)))))) # label(fact_1095_MultInv__prop2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 311 (all P all Q all R hAPP_i68813070l_bool(cOMBB_1652995168ol_int(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_int_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 312 (all V_9 all W_8 number267125858f_real(hAPP_int_int(plus_plus_int(V_9),W_8)) = hAPP_real_real(plus_plus_real(number267125858f_real(V_9)),number267125858f_real(W_8))) # label(fact_250_number__of__add) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 313 (all M M = hAPP_nat_nat(plus_plus_nat(M),zero_zero_nat)) # label(fact_962_Nat_Oadd__0__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 314 (all B_1_1 all B_2_1 (is_int(B_1_1) & is_int(B_2_1) -> is_int(standardRes(B_1_1,B_2_1)))) # label(gsy_c_Residues_OStandardRes) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 315 (all A_106 all C_32 all D_10 hAPP_int_int(plus_plus_int(A_106),hAPP_int_int(plus_plus_int(C_32),D_10)) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_106),C_32)),D_10)) # label(fact_126_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 316 (all W_1 all Y_2 all X_2 all Z_1 (hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(W_1),Z_1)),hAPP_real_real(times_times_real(X_2),Y_2)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(W_1),Y_2)),hAPP_real_real(times_times_real(X_2),Z_1)) <-> Y_2 = Z_1 | X_2 = W_1)) # label(fact_172_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 317 (all A_42 all B_39 all C_20 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(times_times_int(A_42),B_39)),C_20)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_39),C_20)))) # label(fact_731_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 318 (all Z all X_1 all Y_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),Z)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Z))))) # label(fact_911_dvd_Oorder__trans) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 319 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(div_mod_int(A),B))))) # label(fact_1158_pos__mod__sign) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 320 (all X_1 all Y_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) -> X_1 != Y_1)) # label(fact_904_dvd_Oless__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 321 (all B_7 all A_8 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_8)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_7),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(B_7),A_8)),zero_zero_nat))))) # label(fact_841_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 322 (all A_98 all B_57 all V_16 hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_98),B_57)),number_number_of_int(V_16)) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_98),number_number_of_int(V_16))),hAPP_int_int(times_times_int(B_57),number_number_of_int(V_16)))) # label(fact_220_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 323 (all X_15 all Y_12 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(X_15),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_nat(power_power_nat(Y_12),number_number_of_nat(bit0(bit1(pls)))))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_12)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_15),Y_12))))) # label(fact_446_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 324 (all A_81 hAPP_nat_nat(times_times_nat(A_81),zero_zero_nat) = zero_zero_nat) # label(fact_345_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 325 (all A all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A)) -> (exists R_2 (hAPP_nat_real(power_power_real(R_2),N_1) = A & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_2))))))) # label(fact_872_realpow__pos__nth) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 326 (all K_1 (is_int(K_1) -> K_1 = hAPP_int_int(plus_plus_int(pls),K_1))) # label(fact_204_add__Pls) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 327 (all K all L_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(minus_minus_int(K),L_1)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L_1)))) # label(fact_627_less__bin__lemma) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 328 (all K all I_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_2)) -> (J_2 = hAPP_nat_nat(plus_plus_nat(K),I_2) <-> hAPP_nat_nat(minus_minus_nat(J_2),I_2) = K))) # label(fact_1013_le__imp__diff__is__add) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 329 (all A_105 all C_31 all D_9 hAPP_real_real(plus_plus_real(C_31),hAPP_real_real(plus_plus_real(A_105),D_9)) = hAPP_real_real(plus_plus_real(A_105),hAPP_real_real(plus_plus_real(C_31),D_9))) # label(fact_131_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 330 (all N_25 all A_76 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_76)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(power_power_int(A_76),N_25))))) # label(fact_368_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 331 (all A_74 all N_23 all N_22 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_23),N_22)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_74)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_74),one_one_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_74),N_22)),hAPP_nat_int(power_power_int(A_74),N_23))))))) # label(fact_377_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 332 (all C all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(times_times_nat(C),A)),hAPP_nat_nat(times_times_nat(C),B))))) # label(fact_764_divides__mul__l) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 333 (all C_11 all B_25 all A_27 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_25),A_27)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_11),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_27),C_11)),hAPP_real_real(times_times_real(B_25),C_11)))))) # label(fact_783_mult__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 334 (all N_34 all M_7 all X_19 all Y_16 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_19),Y_16)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_34),M_7)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(X_19),N_34)),hAPP_nat_int(power_power_int(Y_16),M_7)))))) # label(fact_318_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 335 (all K_1 (is_int(K_1) -> hAPP_int_int(plus_plus_int(K_1),pls) = K_1)) # label(fact_203_add__Pls__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 336 (all X_20 all Y_17 all Q_5 hAPP_nat_int(power_power_int(hAPP_int_int(times_times_int(X_20),Y_17)),Q_5) = hAPP_int_int(times_times_int(hAPP_nat_int(power_power_int(X_20),Q_5)),hAPP_nat_int(power_power_int(Y_17),Q_5))) # label(fact_191_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 337 (all B_5 all A_6 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_6)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_5)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(times_times_nat(A_6),B_5)))))) # label(fact_847_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 338 (all K_1 all M all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(minus_minus_int(M),N_1))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),N_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),M))))) # label(fact_619_zdvd__zdiffD) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 339 (all W_1 all Z_1 (is_int(Z_1) & is_int(W_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_1)) | Z_1 = W_1 <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),hAPP_int_int(plus_plus_int(Z_1),one_one_int)))))) # label(fact_158_zless__add1__eq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 340 (all N_1 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_1),zero_zero_int)) -> zfact(N_1) = one_one_int) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N_1),zero_zero_int)) -> zfact(N_1) = hAPP_int_int(times_times_int(N_1),zfact(hAPP_int_int(minus_minus_int(N_1),one_one_int)))))) # label(fact_1071_zfact_Osimps) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 341 (all K_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1))))) # label(fact_1039_dvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 342 (all X_2 all Y_2 hAPP_nat_real(power_power_real(hAPP_real_real(plus_plus_real(X_2),Y_2)),number_number_of_nat(bit0(bit1(pls)))) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(X_2),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_2),number_number_of_nat(bit0(bit1(pls)))))),hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(number267125858f_real(bit0(bit1(pls)))),X_2)),Y_2))) # label(fact_11_power2__sum) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 343 (all I_1 all K_1 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_1)) -> hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(I_1),J_1)),K_1) = hAPP_nat_nat(plus_plus_nat(I_1),hAPP_nat_nat(minus_minus_nat(J_1),K_1)))) # label(fact_1010_add__diff__assoc) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 344 (all N_1 all A all B all P_1 (hBOOL(hAPP_int_bool(zprime,P_1)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(P_1),N_1)),hAPP_int_int(times_times_int(A),B))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(P_1),N_1)),A)))))) # label(fact_410_zprime__power__zdvd__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 345 (all V_20 all V_19 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_19)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_20)) -> hAPP_nat_nat(times_times_nat(number_number_of_nat(V_19)),number_number_of_nat(V_20)) = number_number_of_nat(hAPP_int_int(times_times_int(V_19),V_20))))) # label(fact_214_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 346 (all B_2 all A_2 (zero_zero_nat = A_2 <-> B_2 = hAPP_nat_nat(plus_plus_nat(B_2),A_2))) # label(fact_354_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 347 (all X_2 all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(X_2)),number_number_of_int(Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Y_2)))) # label(fact_51_less__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 348 (all X_1 all M all Y_1 hAPP_int_int(div_mod_int(hAPP_int_int(minus_minus_int(X_1),Y_1)),M) = hAPP_int_int(div_mod_int(hAPP_int_int(minus_minus_int(hAPP_int_int(div_mod_int(X_1),M)),Y_1)),M)) # label(fact_1154_zdiff__zmod__left) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 349 (all K all Ma all N (hAPP_nat_nat(times_times_nat(K),Ma) = hAPP_nat_nat(times_times_nat(K),N) <-> zero_zero_nat = K | Ma = N)) # label(fact_1045_nat__mult__eq__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 350 (all A_63 all N_13 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(power_power_int(A_63),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_13))))) # label(fact_484_zero__le__even__power_H) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 351 (all A all X_1 (is_int(X_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),A)) -> (X_1 != hAPP_int_int(minus_minus_int(A),one_one_int) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),hAPP_int_int(minus_minus_int(A),one_one_int)))))))) # label(fact_632_Euler_Oaux1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 352 (all N_1 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_1)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N_1),M))))) # label(fact_1029_nat__dvd__not__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 353 (all X_2 all Y_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)))) # label(fact_921_dvd_Oless__le__not__le) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 354 (all X_2 all Y_2 (is_int(X_2) & is_int(Y_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X_2),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_2),number_number_of_nat(bit0(bit1(pls)))))),zero_zero_int)) <-> Y_2 = zero_zero_int & X_2 = zero_zero_int))) # label(fact_478_sum__power2__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 355 (all Ma all N all A_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_2)) -> (hAPP_nat_int(power_power_int(A_2),Ma) = hAPP_nat_int(power_power_int(A_2),N) <-> Ma = N))) # label(fact_493_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 356 (all Lx_6 all Ly_4 all Rx_6 all Ry_4 hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_6),Rx_6)),hAPP_nat_nat(times_times_nat(Ly_4),Ry_4)) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_6),Ly_4)),hAPP_nat_nat(times_times_nat(Rx_6),Ry_4))) # label(fact_92_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 357 (all X_2 all A_109 (hBOOL(member_int(X_2,A_109)) <-> hBOOL(hAPP_int_bool(A_109,X_2)))) # label(fact_118_mem__def) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 358 (all X_2 (one_one_int = hAPP_int_int(div_mod_int(X_2),number_number_of_int(bit0(bit1(pls)))) <-> zero_zero_int != hAPP_int_int(div_mod_int(X_2),number_number_of_int(bit0(bit1(pls)))))) # label(fact_1172_neq__one__mod__two) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 359 (all W_4 ((zero_zero_nat != number_number_of_nat(W_4) -> zero_zero_nat = hAPP_nat_nat(power_power_nat(zero_zero_nat),number_number_of_nat(W_4))) & (zero_zero_nat = number_number_of_nat(W_4) -> one_one_nat = hAPP_nat_nat(power_power_nat(zero_zero_nat),number_number_of_nat(W_4))))) # label(fact_592_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 360 (all K all L_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(K)),number_number_of_int(L_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L_1)))) # label(fact_74_less__eq__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 361 (all A all B hBOOL(hAPP_int_bool(zcong(A,B),one_one_int))) # label(fact_574_zcong__1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 362 (all A_82 hAPP_real_real(times_times_real(zero_zero_real),A_82) = zero_zero_real) # label(fact_343_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 363 (all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B),A)) -> -(-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B),A))))) # label(fact_899_dvd_Oless__asym_H) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 364 (all Y_1 all X_1 all P_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP_int_bool(zprime,P_1)) -> (-hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),P_1)) -> (-hBOOL(hAPP_int_bool(zcong(Y_1,zero_zero_int),P_1)) -> (hBOOL(hAPP_int_bool(zcong(multInv(P_1,X_1),multInv(P_1,Y_1)),P_1)) -> hBOOL(hAPP_int_bool(zcong(X_1,Y_1),P_1)))))))) # label(fact_1084_MultInv__prop5) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 365 (all V_7 all V_8 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,number_number_of_nat(V_7)),number_number_of_nat(V_8))) <-> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_7),V_8)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_7),pls))))) # label(fact_420_le__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 366 (all K_1 all A all B all M (hBOOL(hAPP_int_bool(zcong(A,B),M)) -> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(K_1),A),hAPP_int_int(times_times_int(K_1),B)),M)))) # label(fact_576_zcong__scalar2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 367 (all V all W hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),hAPP_int_int(div_mod_int(number_number_of_int(V)),number_number_of_int(W))) = hAPP_int_int(div_mod_int(number_number_of_int(bit0(V))),number_number_of_int(bit0(W)))) # label(fact_1165_zmod__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 368 (all B all A (is_int(A) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),A)) -> A = hAPP_int_int(div_mod_int(A),B))))) # label(fact_1163_mod__neg__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 369 (all N_1 hAPP_nat_nat(minus_minus_nat(zero_zero_nat),N_1) = zero_zero_nat) # label(fact_884_diff__0__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 370 (all B all A (is_int(A) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) -> A = hAPP_int_int(div_mod_int(A),B))))) # label(fact_1160_mod__pos__pos__trivial) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 371 (all A_36 all B_33 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_36),hAPP_int_int(times_times_int(B_33),A_36)))) # label(fact_749_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 372 (all N_1 all M (is_int(M) & is_int(N_1) -> (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_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),N_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N_1),M)) -> M = N_1)))))) # label(fact_327_zdvd__antisym__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 373 (all Ma all N (N != Ma <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),Ma)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)))) # label(fact_893_nat__neq__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 374 (all Lx_6 all Ly_4 all Rx_6 all Ry_4 hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_6),Ly_4)),hAPP_real_real(times_times_real(Rx_6),Ry_4)) = hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_6),Rx_6)),hAPP_real_real(times_times_real(Ly_4),Ry_4))) # label(fact_93_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 375 (all X_25 all N_40 hAPP_nat_real(power_power_real(X_25),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_40)) = hAPP_real_real(times_times_real(hAPP_nat_real(power_power_real(X_25),N_40)),hAPP_nat_real(power_power_real(X_25),N_40))) # label(fact_26_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 376 (all M_3 all N_9 all A_60 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_60)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_60),M_3)),hAPP_nat_int(power_power_int(A_60),N_9))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_3),N_9))))) # label(fact_499_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 377 (all B_35 all A_38 all C_17 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_38),C_17)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_38),hAPP_real_real(times_times_real(B_35),C_17))))) # label(fact_741_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 378 (all N_31 all X_18 all Y_15 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_18),Y_15)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(X_18),N_31)),hAPP_nat_nat(power_power_nat(Y_15),N_31))))) # label(fact_331_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 379 (all K_1 all I_1 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(I_1),K_1)),hAPP_nat_nat(plus_plus_nat(J_1),K_1))))) # label(fact_1001_add__le__mono1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 380 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> M = hAPP_nat_nat(div_mod_nat(M),N_1))) # label(fact_1187_mod__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 381 (all W_16 hAPP_nat_nat(power_power_nat(number_number_of_nat(W_16)),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_nat(times_times_nat(number_number_of_nat(W_16)),number_number_of_nat(W_16))) # label(fact_14_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 382 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(twoSqu1154269391sum2sq,X_1)) -> (hBOOL(hAPP_int_bool(twoSqu1154269391sum2sq,Y_1)) -> hBOOL(hAPP_int_bool(twoSqu1154269391sum2sq,hAPP_int_int(times_times_int(X_1),Y_1)))))) # label(fact_90_is__mult__sum2sq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 383 (all M_9 all N_36 all A_88 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_88)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(A_88),M_9)),hAPP_nat_nat(power_power_nat(A_88),N_36))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_9),N_36))))) # label(fact_301_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 384 (all B_2 all A_2 (hBOOL(member_int(B_2,d22set(A_2))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),A_2)))) # label(fact_1112_d22set__le) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 385 (all A all P_1 (is_int(A) -> (hBOOL(hAPP_int_bool(zprime,P_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),P_1)) -> (hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A),A),one_one_int),P_1)) -> one_one_int = A | A = hAPP_int_int(minus_minus_int(P_1),one_one_int))))))) # label(fact_641_zcong__square__zless) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 386 (all X_2 all Y_2 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),B_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Y_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(B_2),X_2)),hAPP_nat_nat(power_power_nat(B_2),Y_2)))))) # label(fact_495_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 387 (all K_1 all M all N_1 hAPP_nat_nat(times_times_nat(K_1),hAPP_nat_nat(plus_plus_nat(M),N_1)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(K_1),M)),hAPP_nat_nat(times_times_nat(K_1),N_1))) # label(fact_1018_add__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 388 (all A_65 all M_5 all N_16 hAPP_nat_int(power_power_int(A_65),hAPP_nat_nat(plus_plus_nat(M_5),N_16)) = hAPP_int_int(times_times_int(hAPP_nat_int(power_power_int(A_65),M_5)),hAPP_nat_int(power_power_int(A_65),N_16))) # label(fact_465_power__add) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 389 (all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(X_2)),one_one_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),bit1(pls))))) # label(fact_162_less__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 390 (all B_4 all C_1 all A_4 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_4)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_4),C_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_4),hAPP_nat_nat(plus_plus_nat(A_4),C_1)))))) # label(fact_857_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 391 (all N_27 all A_78 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_78)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(A_78),N_27))))) # label(fact_363_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 392 (all X_1 all P_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP_int_bool(zprime,P_1)) -> (-hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),P_1)) -> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(multInv(P_1,X_1)),X_1),one_one_int),P_1)))))) # label(fact_1094_MultInv__prop2a) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 393 (all C_14 all D_4 all A_30 all B_28 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_30),B_28)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_14),D_4)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_28)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_14)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_30),C_14)),hAPP_real_real(times_times_real(B_28),D_4)))))))) # label(fact_775_mult__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.58 394 (all X_1 all P_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP_int_bool(zprime,P_1)) -> (-hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),P_1)) -> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(X_1),multInv(P_1,X_1))),multInv(P_1,multInv(P_1,X_1))),X_1),P_1)))))) # label(fact_1092_Int2_Oaux____2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 395 (all A_101 all B_58 all C_29 hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_101),C_29)),hAPP_int_int(times_times_int(B_58),C_29)) = hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_101),B_58)),C_29)) # label(fact_176_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 396 (all K all N all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),N)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),hAPP_int_int(plus_plus_int(N),hAPP_int_int(times_times_int(K),Ma)))))) # label(fact_372_zdvd__reduce) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 397 (all A_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),one_one_nat)) <-> A_2 = one_one_nat)) # label(fact_1126_gcd__lcm__complete__lattice__nat_Obot__unique) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 398 (all A_2 all B_2 all C_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_2),A_2)),hAPP_int_int(times_times_int(C_2),B_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),B_2))))) # label(fact_850_mult__less__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 399 (all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A)) -> (exists P_4 (is_int(P_4) & hBOOL(hAPP_int_bool(zprime,P_4)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_4),A)))))) # label(fact_1067_zprime__factor__exists) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 400 (all Ma all N all A_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_2)) -> (hAPP_nat_nat(power_power_nat(A_2),Ma) = hAPP_nat_nat(power_power_nat(A_2),N) <-> Ma = N))) # label(fact_492_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 401 (all Ma all N 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(times_times_nat(K),Ma)),hAPP_nat_nat(times_times_nat(K),N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),N))))) # label(fact_1050_nat__mult__dvd__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 402 (all C_27 all D_8 all A_72 all B_50 all R_4 (zero_zero_real != R_4 -> (D_8 != C_27 & A_72 = B_50 -> hAPP_real_real(plus_plus_real(A_72),hAPP_real_real(times_times_real(R_4),C_27)) != hAPP_real_real(plus_plus_real(B_50),hAPP_real_real(times_times_real(R_4),D_8))))) # label(fact_389_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 403 (all B_2 all P_3 all A_2 (is_int(A_2) & is_int(B_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_2)) -> (-hBOOL(member_int(B_2,wset(hAPP_int_int(minus_minus_int(A_2),one_one_int),P_3))) -> (hBOOL(member_int(B_2,wset(A_2,P_3))) -> A_2 = B_2 | inv(P_3,A_2) = B_2))))) # label(fact_1102_wset__mem__imp__or) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 404 (all N (N != zero_zero_nat <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)))) # label(fact_956_neq0__conv) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 405 (all X_10 one_one_int = hAPP_nat_int(power_power_int(X_10),zero_zero_nat)) # label(fact_543_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 406 (all A_2 all B_2 all Ma (is_int(B_2) -> ((exists K_2 (is_int(K_2) & B_2 = hAPP_int_int(plus_plus_int(A_2),hAPP_int_int(times_times_int(Ma),K_2)))) <-> hBOOL(hAPP_int_bool(zcong(A_2,B_2),Ma))))) # label(fact_590_zcong__iff__lin) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 407 (all A all J_1 all K_1 all P_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP_int_bool(zprime,P_1)) -> (-hBOOL(hAPP_int_bool(zcong(K_1,zero_zero_int),P_1)) -> (-hBOOL(hAPP_int_bool(zcong(J_1,zero_zero_int),P_1)) -> (hBOOL(hAPP_int_bool(zcong(J_1,hAPP_int_int(times_times_int(A),multInv(P_1,K_1))),P_1)) -> hBOOL(hAPP_int_bool(zcong(K_1,hAPP_int_int(times_times_int(A),multInv(P_1,J_1))),P_1)))))))) # label(fact_1090_MultInv__zcong__prop2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 408 (all B all M all A (is_int(A) & is_int(B) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),M)) -> (hBOOL(hAPP_int_bool(zcong(A,B),M)) -> A = B))))))) # label(fact_594_zcong__zless__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 409 (all P_2 all N all K (hBOOL(hAPP_nat_bool(P_2,hAPP_nat_nat(div_mod_nat(N),K))) <-> (zero_zero_nat = K -> hBOOL(hAPP_nat_bool(P_2,N))) & (zero_zero_nat != K -> (all I all J (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J),K)) -> (N = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(K),I)),J) -> hBOOL(hAPP_nat_bool(P_2,J)))))))) # label(fact_1194_split__mod) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 410 (all A_18 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(times_times_int(A_18),A_18)))) # label(fact_813_zero__le__square) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 411 (all X_1 -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),X_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),X_1)))) # label(fact_925_dvd_Oless__irrefl) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 412 (all B_21 all A_23 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_23),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_21),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(times_times_real(A_23),B_21)))))) # label(fact_794_mult__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 413 (all A_99 hAPP_nat_nat(times_times_nat(one_one_nat),A_99) = A_99) # label(fact_189_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 414 (all K all P_2 all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D)) -> ((all X (is_int(X) -> (hBOOL(hAPP_int_bool(P_2,X)) -> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(plus_plus_int(X),D)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K)) -> (all X (hBOOL(hAPP_int_bool(P_2,X)) -> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(plus_plus_int(X),hAPP_int_int(times_times_int(K),D)))))))))) # label(fact_1066_incr__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 415 (all V_15 all B_56 all C_28 hAPP_real_real(times_times_real(number267125858f_real(V_15)),hAPP_real_real(plus_plus_real(B_56),C_28)) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(number267125858f_real(V_15)),B_56)),hAPP_real_real(times_times_real(number267125858f_real(V_15)),C_28))) # label(fact_225_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 416 (all X_2 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(X_2)),number267125858f_real(Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),Y_2)))) # label(fact_52_less__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 417 (all Z1 all Z2 all W hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(Z1),W)),hAPP_int_int(times_times_int(Z2),W)) = hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(Z1),Z2)),W)) # label(fact_617_zdiff__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 418 (all A_2 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_2)) | 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_eq_int,A_2),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_2),B_2)),zero_zero_int)))) # label(fact_809_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 419 (all Ma all N (Ma = zero_zero_nat | N = zero_zero_nat <-> zero_zero_nat = hAPP_nat_nat(times_times_nat(Ma),N))) # label(fact_971_mult__is__0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 420 (all K_1 all L all I_1 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_1),L)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I_1),K_1)),hAPP_nat_nat(plus_plus_nat(J_1),L)))))) # label(fact_980_add__less__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 421 (all A_75 all N_24 all B_51 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_75),N_24)),hAPP_nat_nat(power_power_nat(B_51),N_24))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_51)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_75),B_51))))) # label(fact_375_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 422 (all C_5 all A_15 all B_14 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_15),B_14)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_5)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(C_5),A_15)),hAPP_nat_nat(times_times_nat(C_5),B_14)))))) # label(fact_819_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 423 (all Lx_6 all Ly_4 all Rx_6 all Ry_4 hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_6),Rx_6)),hAPP_int_int(times_times_int(Ly_4),Ry_4)) = hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_6),Ly_4)),hAPP_int_int(times_times_int(Rx_6),Ry_4))) # label(fact_91_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 424 (all C_18 all D_6 all A_39 all B_36 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_39),B_36)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,C_18),D_6)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(times_times_int(A_39),C_18)),hAPP_int_int(times_times_int(B_36),D_6)))))) # label(fact_740_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 425 (all P_3 (is_int(P_3) -> (hBOOL(hAPP_int_bool(zprime,P_3)) <-> (all M_2 (is_int(M_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M_2),P_3)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M_2)) -> one_one_int = M_2 | P_3 = M_2))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),P_3))))) # label(fact_508_zprime__def) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 426 (all A_5 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_5),A_5)),zero_zero_real))) # label(fact_855_not__square__less__zero) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 427 (all K_1 all L all I_1 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),L)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(I_1),K_1)),hAPP_nat_nat(times_times_nat(J_1),L)))))) # label(fact_1020_mult__le__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 428 (all A all J_1 all K_1 all P_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP_int_bool(zcong(J_1,K_1),P_1)) -> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A),multInv(P_1,J_1)),hAPP_int_int(times_times_int(A),multInv(P_1,K_1))),P_1))))) # label(fact_1083_MultInv__zcong__prop1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 429 (all U all Ma all N all I_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(J_2),I_2)),U)),N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_2),U)),Ma)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_2),U)),N)))))) # label(fact_1044_nat__less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 430 (all Ma all N (hAPP_nat_nat(minus_minus_nat(Ma),N) = zero_zero_nat <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)))) # label(fact_968_diff__is__0__eq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 431 (all A_50 all B_44 (is_int(A_50) & is_int(B_44) -> (hAPP_int_int(times_times_int(A_50),B_44) = zero_zero_int -> A_50 = zero_zero_int | zero_zero_int = B_44))) # label(fact_700_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 432 (all K all Ma all N (zero_zero_nat = K | N = Ma <-> hAPP_nat_nat(times_times_nat(K),Ma) = hAPP_nat_nat(times_times_nat(K),N))) # label(fact_972_mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 433 (all N_1 all J_1 all K_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),K_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(J_1),N_1)),K_1)))) # label(fact_896_less__imp__diff__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 434 (all K all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(K),Ma)),hAPP_nat_nat(plus_plus_nat(K),N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)))) # label(fact_976_nat__add__left__cancel__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 435 (all X_25 all N_40 hAPP_int_int(times_times_int(hAPP_nat_int(power_power_int(X_25),N_40)),hAPP_nat_int(power_power_int(X_25),N_40)) = hAPP_nat_int(power_power_int(X_25),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_40))) # label(fact_25_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 436 (all Lx_4 all Ly_2 all Rx_4 all Ry_2 hAPP_real_real(times_times_real(Lx_4),hAPP_real_real(times_times_real(Ly_2),hAPP_real_real(times_times_real(Rx_4),Ry_2))) = hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_4),Ly_2)),hAPP_real_real(times_times_real(Rx_4),Ry_2))) # label(fact_99_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 437 (all A_2 all W_1 (is_int(A_2) -> (zero_zero_int = hAPP_nat_int(power_power_int(A_2),number_number_of_nat(W_1)) <-> A_2 = zero_zero_int & number_number_of_nat(W_1) != zero_zero_nat))) # label(fact_584_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 438 (all A_73 all N_21 all N_20 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_21),N_20)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_73)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_73),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_73),N_20)),hAPP_nat_real(power_power_real(A_73),N_21))))))) # label(fact_382_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 439 (all U all Ma all N all I_2 all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_2)) -> (hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_2),U)),Ma) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_2),U)),N) <-> hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(J_2),I_2)),U)),N) = Ma))) # label(fact_1057_nat__eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 440 (all A_103 all N_39 hAPP_nat_nat(power_power_nat(hAPP_nat_nat(power_power_nat(A_103),N_39)),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_nat(power_power_nat(A_103),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_39))) # label(fact_161_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 441 (all K_1 all L hAPP_int_int(minus_minus_int(bit0(K_1)),bit0(L)) = bit0(hAPP_int_int(minus_minus_int(K_1),L))) # label(fact_615_diff__bin__simps_I7_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 442 (all A_41 all B_38 all C_19 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(times_times_real(A_41),B_38)),C_19)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_41),C_19)))) # label(fact_732_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 443 (all X_1 all M hBOOL(hAPP_int_bool(zcong(X_1,standardRes(M,X_1)),M))) # label(fact_1191_StandardRes__prop1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 444 (all K all L_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K)),bit1(L_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L_1)))) # label(fact_64_rel__simps_I17_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 445 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(X_2)),one_one_real)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),bit1(pls))))) # label(fact_163_less__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 446 (all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(plus_plus_nat(Ma),N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)))) # label(fact_1030_add__gr__0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 447 (all C_2 all D all A_2 all B_2 (is_int(A_2) & is_int(B_2) & is_int(D) & is_int(C_2) -> (hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_2),D)),hAPP_int_int(times_times_int(B_2),C_2)) != hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_2),C_2)),hAPP_int_int(times_times_int(B_2),D)) <-> C_2 != D & B_2 != A_2))) # label(fact_179_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 448 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),bit0(K))))) # label(fact_532_rel__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 449 (all N all Ma (is_int(N) & is_int(Ma) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (one_one_int = Ma & N = one_one_int <-> hAPP_int_int(times_times_int(Ma),N) = one_one_int)))) # label(fact_427_pos__zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 450 (all C_27 all D_8 all A_72 all B_50 all R_4 (is_int(D_8) & is_int(R_4) & is_int(C_27) -> (zero_zero_int != R_4 -> (A_72 = B_50 & D_8 != C_27 -> hAPP_int_int(plus_plus_int(B_50),hAPP_int_int(times_times_int(R_4),D_8)) != hAPP_int_int(plus_plus_int(A_72),hAPP_int_int(times_times_int(R_4),C_27)))))) # label(fact_387_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 451 (all A_2 all B_2 all Ma (hBOOL(hAPP_int_bool(zcong(A_2,B_2),Ma)) <-> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(div_mod_int(A_2),Ma),hAPP_int_int(div_mod_int(B_2),Ma)),Ma)))) # label(fact_1145_zcong__zmod) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 452 (all B_2 all A_2 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),hAPP_int_int(minus_minus_int(P_3),one_one_int))) -> (hBOOL(member_int(B_2,wset(A_2,P_3))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_2)))))) # label(fact_1101_wset__g__1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 453 (all B_20 all A_22 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_22),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_20)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_22),B_20)),zero_zero_real))))) # label(fact_796_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 454 (all Z all X_1 all Y_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),Z)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z),Y_1)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z),X_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Z))))) # label(fact_909_dvd_Ole__less__trans) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 455 (all A_94 (is_int(A_94) -> hAPP_int_int(plus_plus_int(A_94),number_number_of_int(pls)) = A_94)) # label(fact_237_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 456 (all K all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(K),Ma)),hAPP_nat_nat(plus_plus_nat(K),N))))) # label(fact_998_nat__add__left__cancel__le) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 457 (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]. 1.32/1.59 458 (all X_2 all Y_2 hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(power_power_nat(X_2),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_nat(power_power_nat(Y_2),number_number_of_nat(bit0(bit1(pls)))))),hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),X_2)),Y_2)) = hAPP_nat_nat(power_power_nat(hAPP_nat_nat(plus_plus_nat(X_2),Y_2)),number_number_of_nat(bit0(bit1(pls))))) # label(fact_10_power2__sum) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 459 (all A all P_1 (hBOOL(hAPP_int_bool(zprime,P_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A)) -> (hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A),A),one_one_int),P_1)) -> hBOOL(hAPP_int_bool(zcong(A,one_one_int),P_1)) | hBOOL(hAPP_int_bool(zcong(A,hAPP_int_int(minus_minus_int(P_1),one_one_int)),P_1)))))) # label(fact_640_zcong__square) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 460 (all Z1 all Z2 all Z3 hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Z1),Z2)),Z3) = hAPP_real_real(times_times_real(Z1),hAPP_real_real(times_times_real(Z2),Z3))) # label(fact_688_real__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 461 (all W_4 ((zero_zero_nat = number_number_of_nat(W_4) -> one_one_real = hAPP_nat_real(power_power_real(zero_zero_real),number_number_of_nat(W_4))) & (number_number_of_nat(W_4) != zero_zero_nat -> hAPP_nat_real(power_power_real(zero_zero_real),number_number_of_nat(W_4)) = zero_zero_real))) # label(fact_591_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 462 (all N_12 all A_62 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),A_62)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),hAPP_nat_nat(power_power_nat(A_62),N_12))))) # label(fact_486_one__le__power) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 463 (all X_2 all Y_2 (is_int(Y_2) & is_int(X_2) -> (Y_2 != zero_zero_int | zero_zero_int != X_2 <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(X_2),X_2)),hAPP_int_int(times_times_int(Y_2),Y_2))))))) # label(fact_418_sum__squares__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 464 (all A_64 all M_4 all N_14 hAPP_nat_real(power_power_real(A_64),hAPP_nat_nat(times_times_nat(M_4),N_14)) = hAPP_nat_real(power_power_real(hAPP_nat_real(power_power_real(A_64),M_4)),N_14)) # label(fact_470_power__mult) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 465 (all C_24 all A_51 all B_45 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_51),B_45)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_45),C_24)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_51),C_24))))) # label(fact_683_dvd__trans) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 466 (all Ma all N (one_one_nat = hAPP_nat_nat(times_times_nat(Ma),N) <-> N = one_one_nat & one_one_nat = Ma)) # label(fact_1025_nat__mult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 467 (all C_9 all A_25 all B_23 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_25),B_23)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_9)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(C_9),A_25)),hAPP_nat_nat(times_times_nat(C_9),B_23)))))) # label(fact_789_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 468 (all N_26 all A_77 all B_52 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_77),B_52)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_77)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_77),N_26)),hAPP_nat_int(power_power_int(B_52),N_26)))))) # label(fact_365_power__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 469 (all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),number_number_of_int(Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Y_2)))) # label(fact_429_less__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 470 (all A_97 all M_12 hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_97),one_one_real)),M_12) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_97),M_12)),M_12)) # label(fact_228_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 471 (all B_1_1 (is_int(B_1_1) -> is_int(zfact(B_1_1)))) # label(gsy_c_IntFact_Ozfact) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 472 (all A -(-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),zero_zero_nat)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A)))) # label(fact_1122_gcd__lcm__complete__lattice__nat_Onot__top__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 473 (all B_18 all A_20 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_20)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_18),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_20),B_18)),zero_zero_nat))))) # label(fact_803_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 474 (all M all N_1 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> M = hAPP_nat_nat(plus_plus_nat(N_1),hAPP_nat_nat(minus_minus_nat(M),N_1)))) # label(fact_983_add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 475 (all X_26 hAPP_int_int(times_times_int(X_26),X_26) = hAPP_nat_int(power_power_int(X_26),number_number_of_nat(bit0(bit1(pls))))) # label(fact_19_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 476 (all V_18 all V_17 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_17)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_18)) -> number_number_of_nat(hAPP_int_int(plus_plus_int(V_17),V_18)) = hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V_17)),number_number_of_nat(V_18))))) # label(fact_217_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 477 (all A_44 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,zero_zero_real),A_44)) -> A_44 = zero_zero_real)) # label(fact_724_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 478 (all Lx_1 all Rx_1 all Ry_1 hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_1),Rx_1)),Ry_1) = hAPP_int_int(times_times_int(Lx_1),hAPP_int_int(times_times_int(Rx_1),Ry_1))) # label(fact_106_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 479 (all Ma all D (is_int(Ma) -> ((exists Q_2 (is_int(Q_2) & Ma = hAPP_int_int(times_times_int(D),Q_2))) <-> hAPP_int_int(div_mod_int(Ma),D) = zero_zero_int))) # label(fact_1141_zmod__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 480 (all V_6 all K_1 all V ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls)) -> hAPP_nat_nat(times_times_nat(number_number_of_nat(V)),hAPP_nat_nat(times_times_nat(number_number_of_nat(V_6)),K_1)) = zero_zero_nat) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls)) -> hAPP_nat_nat(times_times_nat(number_number_of_nat(hAPP_int_int(times_times_int(V),V_6))),K_1) = hAPP_nat_nat(times_times_nat(number_number_of_nat(V)),hAPP_nat_nat(times_times_nat(number_number_of_nat(V_6)),K_1))))) # label(fact_566_nat__number__of__mult__left) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 481 (all A all P_1 (hBOOL(hAPP_int_bool(zprime,P_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(minus_minus_int(P_1),one_one_int))) -> inv(P_1,A) != hAPP_int_int(minus_minus_int(P_1),one_one_int))))) # label(fact_1078_inv__not__p__minus__1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 482 (all Ma all N (hAPP_nat_nat(power_power_nat(Ma),N) = zero_zero_nat <-> Ma = zero_zero_nat & zero_zero_nat != N)) # label(fact_768_nat__power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 483 (all A_80 hAPP_real_real(plus_plus_real(zero_zero_real),A_80) = A_80) # label(fact_349_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 484 (all X_9 all N_5 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_5)) | X_9 = one_one_int -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_9),hAPP_nat_int(power_power_int(X_9),N_5))))) # label(fact_558_dvd__power) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 485 (all Z_2 all X_3 all Y_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_3),Y_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_3),Z_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_3),hAPP_int_int(minus_minus_int(Y_3),Z_2)))))) # label(fact_760_dvd__diff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 486 (all M_11 all A_96 hAPP_nat_nat(plus_plus_nat(M_11),hAPP_nat_nat(times_times_nat(A_96),M_11)) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_96),one_one_nat)),M_11)) # label(fact_230_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 487 (all A_36 all B_33 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_36),hAPP_real_real(times_times_real(B_33),A_36)))) # label(fact_747_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 488 (all C_24 all A_51 all B_45 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_51),B_45)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,B_45),C_24)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_51),C_24))))) # label(fact_681_dvd__trans) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 489 (all K all L_1 (is_int(K) & is_int(L_1) -> (bit1(L_1) = bit1(K) <-> L_1 = K))) # label(fact_140_rel__simps_I51_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 490 (all B_17 all A_19 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_19)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_17)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(times_times_real(A_19),B_17)))))) # label(fact_805_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 491 (all Y_1 all N_1 all P_1 (hBOOL(hAPP_int_bool(zprime,P_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),hAPP_nat_int(power_power_int(Y_1),N_1))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),Y_1)))))) # label(fact_664_zpower__zdvd__prop2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 492 (all P_2 all I_2 all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),K)) -> (hBOOL(hAPP_int_bool(P_2,hAPP_int_int(minus_minus_int(K),one_one_int))) -> ((all I (is_int(I) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I),K)) -> (hBOOL(hAPP_int_bool(P_2,I)) -> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(minus_minus_int(I),one_one_int))))))) -> hBOOL(hAPP_int_bool(P_2,I_2)))))) # label(fact_1105_int__less__induct) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 493 (all Ma all K all F_1 ((all M_2 all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_2),N_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F_1,M_2)),hAPP_nat_nat(F_1,N_2))))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(F_1,Ma)),K)),hAPP_nat_nat(F_1,hAPP_nat_nat(plus_plus_nat(Ma),K)))))) # label(fact_1109_mono__nat__linear__lb) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 494 (all Z_4 hAPP_int_int(plus_plus_int(Z_4),Z_4) = hAPP_int_int(times_times_int(Z_4),number_number_of_int(bit0(bit1(pls))))) # label(fact_280_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 495 (all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),number267125858f_real(Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Y_2)))) # label(fact_433_le__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 496 (all N_1 all X_1 all Y_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(X_1),N_1)),hAPP_nat_nat(power_power_nat(Y_1),N_1))))) # label(fact_769_divides__exp) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 497 (all A_40 all B_37 all K_3 (hAPP_nat_nat(times_times_nat(B_37),K_3) = A_40 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_37),A_40)))) # label(fact_736_dvdI) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 498 (all X_2 (zero_zero_real = X_2 <-> -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(X_2),X_2))))) # label(fact_527_not__real__square__gt__zero) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 499 (all K_1 all I_1 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I_1),K_1)),hAPP_nat_nat(plus_plus_nat(J_1),K_1))))) # label(fact_979_add__less__mono1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 500 (all K_1 all I_1 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(I_1),K_1)),hAPP_nat_nat(times_times_nat(J_1),K_1))))) # label(fact_1022_mult__le__mono1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 501 (all A_50 all B_44 (zero_zero_real = hAPP_real_real(times_times_real(A_50),B_44) -> zero_zero_real = B_44 | A_50 = zero_zero_real)) # label(fact_698_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 502 (all A all B all C hAPP_int_int(div_mod_int(hAPP_int_int(times_times_int(A),B)),C) = hAPP_int_int(div_mod_int(hAPP_int_int(times_times_int(A),hAPP_int_int(div_mod_int(B),C))),C)) # label(fact_1149_zmod__zmult1__eq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 503 (all V_7 all W_1 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,number_number_of_nat(W_1)),number_number_of_nat(V_7))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,number_number_of_nat(V_7)),number_number_of_nat(W_1))))) # label(fact_49_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 504 (all B_9 all A_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(times_times_nat(B_9),A_10))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_10)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_9))))) # label(fact_835_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 505 (all K all L_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit0(K)),bit0(L_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L_1)))) # label(fact_72_rel__simps_I31_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 506 (all B_2 all A_2 (zero_zero_real = A_2 <-> B_2 = hAPP_real_real(plus_plus_real(B_2),A_2))) # label(fact_355_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 507 (all Z (is_int(Z) -> Z = hAPP_int_int(plus_plus_int(Z),zero_zero_int))) # label(fact_361_zadd__0__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 508 (all A_2 all B_2 all C_2 (C_2 != zero_zero_real -> (hAPP_real_real(times_times_real(C_2),A_2) = hAPP_real_real(times_times_real(C_2),B_2) <-> A_2 = B_2))) # label(fact_762_real__mult__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 509 (all X_1 all Y_1 all N_1 (zero_zero_nat != N_1 -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(X_1),N_1)),Y_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))))) # label(fact_867_divides__exp2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 510 (all A_2 all C_2 all B_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_2),C_2)),hAPP_real_real(times_times_real(B_2),C_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_2),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_2),A_2)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_2),B_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_2)))) # label(fact_853_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 511 (all X_7 all Y_6 hAPP_real_real(minus_minus_real(hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(X_7),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_6),number_number_of_nat(bit0(bit1(pls)))))),hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(number267125858f_real(bit0(bit1(pls)))),X_7)),Y_6)) = hAPP_nat_real(power_power_real(hAPP_real_real(minus_minus_real(X_7),Y_6)),number_number_of_nat(bit0(bit1(pls))))) # label(fact_648_power2__diff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 512 (all B_30 all A_32 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_30),zero_zero_nat)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_32)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_32),zero_zero_nat)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_30)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_32),B_30)),zero_zero_nat)))) # label(fact_771_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 513 (all X_20 all Y_17 all Q_5 hAPP_nat_nat(times_times_nat(hAPP_nat_nat(power_power_nat(X_20),Q_5)),hAPP_nat_nat(power_power_nat(Y_17),Q_5)) = hAPP_nat_nat(power_power_nat(hAPP_nat_nat(times_times_nat(X_20),Y_17)),Q_5)) # label(fact_193_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 514 (all A_44 (is_int(A_44) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,zero_zero_int),A_44)) -> zero_zero_int = A_44))) # label(fact_726_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 515 (all C_18 all D_6 all A_39 all B_36 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_39),B_36)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,C_18),D_6)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(times_times_real(A_39),C_18)),hAPP_real_real(times_times_real(B_36),D_6)))))) # label(fact_738_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 516 (all A_59 all N_8 all N_7 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_8),N_7)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_59)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_59),N_8)),hAPP_nat_int(power_power_int(A_59),N_7)))))) # label(fact_502_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 517 (all N_15 hAPP_nat_nat(power_power_nat(one_one_nat),N_15) = one_one_nat) # label(fact_467_power__one) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 518 (all A_2 all B_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(times_times_real(A_2),B_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_2),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),zero_zero_real)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_2)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_2)))) # label(fact_810_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 519 (all A_80 (is_int(A_80) -> A_80 = hAPP_int_int(plus_plus_int(zero_zero_int),A_80))) # label(fact_347_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 520 (all X_2 all Y_2 all B_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),B_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(B_2),X_2)),hAPP_nat_nat(power_power_nat(B_2),Y_2))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_2))))) # label(fact_304_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 521 (all B_35 all A_38 all C_17 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_38),C_17)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_38),hAPP_int_int(times_times_int(B_35),C_17))))) # label(fact_743_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 522 (all Z_6 hAPP_nat_nat(plus_plus_nat(Z_6),Z_6) = hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),Z_6)) # label(fact_276_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 523 (all V all W hAPP_int_int(times_times_int(number_number_of_int(V)),number_number_of_int(W)) = number_number_of_int(hAPP_int_int(times_times_int(V),W))) # label(fact_209_times__numeral__code_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 524 (all Z_6 hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),Z_6) = hAPP_int_int(plus_plus_int(Z_6),Z_6)) # label(fact_275_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 525 (all N_1 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M)) -> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(minus_minus_nat(M),N_1)),N_1) = hAPP_nat_nat(div_mod_nat(M),N_1))) # label(fact_1183_le__mod__geq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 526 (all M_10 hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(one_one_real),one_one_real)),M_10) = hAPP_real_real(plus_plus_real(M_10),M_10)) # label(fact_234_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 527 (all Z_6 hAPP_real_real(times_times_real(number267125858f_real(bit0(bit1(pls)))),Z_6) = hAPP_real_real(plus_plus_real(Z_6),Z_6)) # label(fact_277_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 528 (all L hAPP_int_int(minus_minus_int(min),bit1(L)) = bit0(hAPP_int_int(minus_minus_int(min),L))) # label(fact_636_diff__bin__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 529 (all A_47 hAPP_nat_nat(times_times_nat(zero_zero_nat),A_47) = zero_zero_nat) # label(fact_710_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 530 (all A_2 all C_2 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_2),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),A_2)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_2)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),B_2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_2),C_2)),hAPP_int_int(times_times_int(B_2),C_2))))) # label(fact_854_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 531 (all V all W ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(W))) -> hAPP_int_int(div_mod_int(number_number_of_int(bit1(V))),number_number_of_int(bit0(W))) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),hAPP_int_int(div_mod_int(number_number_of_int(V)),number_number_of_int(W)))),one_one_int)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(W))) -> hAPP_int_int(div_mod_int(number_number_of_int(bit1(V))),number_number_of_int(bit0(W))) = hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),hAPP_int_int(div_mod_int(hAPP_int_int(plus_plus_int(number_number_of_int(V)),one_one_int)),number_number_of_int(W)))),one_one_int)))) # label(fact_1135_zmod__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 532 (all Z1 all Z2 all Z3 hAPP_int_int(times_times_int(Z1),hAPP_int_int(times_times_int(Z2),Z3)) = hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Z1),Z2)),Z3)) # label(fact_142_zmult__assoc) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 533 (all A_101 all B_58 all C_29 hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_101),B_58)),C_29) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_101),C_29)),hAPP_nat_nat(times_times_nat(B_58),C_29))) # label(fact_177_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 534 (all I_1 all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(I_1),M)),hAPP_nat_nat(power_power_nat(I_1),N_1))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),I_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1))))) # label(fact_540_power__dvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 535 (all Z (is_int(Z) -> hAPP_int_int(plus_plus_int(zero_zero_int),Z) = Z)) # label(fact_360_zadd__0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 536 (all A_2 (hAPP_real_real(plus_plus_real(A_2),A_2) = zero_zero_real <-> zero_zero_real = A_2)) # label(fact_357_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 537 (all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Z_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_1)))) # label(fact_426_int__one__le__iff__zero__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 538 (all Z (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(plus_plus_int(one_one_int),Z))))) # label(fact_437_le__imp__0__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 539 (all N_19 all A_71 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_71)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_71),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_71),hAPP_nat_real(power_power_real(A_71),N_19))),hAPP_nat_real(power_power_real(A_71),N_19)))))) # label(fact_408_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 540 (all N_28 all A_83 all B_53 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_83),B_53)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_83)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_28)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_83),N_28)),hAPP_nat_real(power_power_real(B_53),N_28))))))) # label(fact_340_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 541 (all N_37 all A_89 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_89)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(times_times_nat(A_89),hAPP_nat_nat(power_power_nat(A_89),N_37)))))) # label(fact_298_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 542 (all N_1 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N_1),M)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_1)),M)) | N_1 = M | zero_zero_nat = M)) # label(fact_667_divides__cases) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 543 (all N_1 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N_1),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_1),M)))) # label(fact_654_zdvd__bounds) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 544 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),bit0(K))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K)))) # label(fact_157_rel__simps_I21_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 545 (all B_1_1 all B_2_1 (is_int(B_1_1) -> is_bool(member_int(B_1_1,B_2_1)))) # label(gsy_c_member_000tc__Int__Oint) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 546 (all X_1 hAPP_nat_int(power_power_int(X_1),number_number_of_nat(bit0(bit0(bit1(pls))))) = hAPP_nat_int(power_power_int(hAPP_nat_int(power_power_int(X_1),number_number_of_nat(bit0(bit1(pls))))),number_number_of_nat(bit0(bit1(pls))))) # label(fact_274_quartic__square__square) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 547 (all Z all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z),N_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),N_1))))) # label(fact_337_zdvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 548 (all B all Q_4 all R_3 all Q_1 all R_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),Q_4)),R_3)),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),Q_1)),R_1))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_4),Q_1))))))) # label(fact_600_unique__quotient__lemma) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 549 (all A all P_1 (hBOOL(hAPP_int_bool(zprime,P_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(minus_minus_int(P_1),one_one_int))) -> A != inv(P_1,A))))) # label(fact_1076_inv__distinct) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 550 (all A_66 all B_49 all N_17 hAPP_int_int(times_times_int(hAPP_nat_int(power_power_int(A_66),N_17)),hAPP_nat_int(power_power_int(B_49),N_17)) = hAPP_nat_int(power_power_int(hAPP_int_int(times_times_int(A_66),B_49)),N_17)) # label(fact_462_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 551 (all A_2 all B_2 all C_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_2),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_2),A_2)),hAPP_real_real(times_times_real(C_2),B_2))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_2),A_2))))) # label(fact_832_mult__less__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 552 (all W_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(W_1)),zero_zero_int)))) # label(fact_400_bin__less__0__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 553 (all N_1 all M hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_nat_nat(plus_plus_nat(N_1),M)))) # label(fact_996_le__add1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 554 (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]. 1.32/1.59 555 (all N_19 all A_71 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_71)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_71),one_one_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(A_71),hAPP_nat_nat(power_power_nat(A_71),N_19))),hAPP_nat_nat(power_power_nat(A_71),N_19)))))) # label(fact_407_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 556 (all X_16 all Y_13 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(X_16),X_16)),hAPP_real_real(times_times_real(Y_13),Y_13))),zero_zero_real))) # label(fact_417_not__sum__squares__lt__zero) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 557 (all M_6 all A_86 all N_33 all B_55 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(A_86),N_33)),B_55)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_6),N_33)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(A_86),M_6)),B_55))))) # label(fact_321_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 558 (all A all B 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)) -> hAPP_int_int(div_mod_int(hAPP_int_int(div_mod_int(A),B)),M) = hAPP_int_int(div_mod_int(A),M)))) # label(fact_1156_zmod__zdvd__zmod) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 559 (all L_1 (is_int(L_1) -> (pls = bit0(L_1) <-> L_1 = pls))) # label(fact_199_rel__simps_I38_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 560 (all A_95 (is_int(A_95) -> A_95 = hAPP_int_int(plus_plus_int(number_number_of_int(pls)),A_95))) # label(fact_235_add__numeral__0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 561 (all X_8 all Y_7 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_8),Y_7)) -> (Y_7 != X_8 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_8),Y_7))))) # label(fact_569_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 562 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(times_times_int(X_1),Y_1)))))) # label(fact_586_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 563 (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]. 1.32/1.59 564 (all X_12 all Y_9 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X_12),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_9),number_number_of_nat(bit0(bit1(pls)))))))) # label(fact_476_sum__power2__ge__zero) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 565 (all K_1 all M all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),hAPP_int_int(div_mod_int(M),N_1))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),N_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),M))))) # label(fact_1146_zdvd__zmod__imp__zdvd) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 566 (all Lx_5 all Ly_3 all Rx_5 all Ry_3 hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_5),Ly_3)),hAPP_int_int(times_times_int(Rx_5),Ry_3)) = hAPP_int_int(times_times_int(Rx_5),hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_5),Ly_3)),Ry_3))) # label(fact_94_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 567 (all N_1 zero_zero_nat = hAPP_nat_nat(times_times_nat(zero_zero_nat),N_1)) # label(fact_969_mult__0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 568 (all N_1 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M)) -> hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(minus_minus_nat(M),N_1)),N_1) = M)) # label(fact_1012_le__add__diff__inverse2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 569 (all A_33 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,one_one_real),A_33))) # label(fact_756_one__dvd) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 570 (all V_11 all W_10 all Z_7 hAPP_real_real(plus_plus_real(number267125858f_real(V_11)),hAPP_real_real(plus_plus_real(number267125858f_real(W_10)),Z_7)) = hAPP_real_real(plus_plus_real(number267125858f_real(hAPP_int_int(plus_plus_int(V_11),W_10))),Z_7)) # label(fact_246_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 571 (all K_1 all L hAPP_int_int(plus_plus_int(bit0(hAPP_int_int(times_times_int(K_1),L))),L) = hAPP_int_int(times_times_int(bit1(K_1)),L)) # label(fact_268_mult__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 572 (all A_2 all B_2 (hAPP_real_real(times_times_real(A_2),B_2) = zero_zero_real <-> B_2 = zero_zero_real | zero_zero_real = A_2)) # label(fact_704_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 573 (all A_73 all N_21 all N_20 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_21),N_20)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_73)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_73),one_one_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_73),N_20)),hAPP_nat_int(power_power_int(A_73),N_21))))))) # label(fact_380_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 574 (all A_2 (zero_zero_real = A_2 <-> zero_zero_real = hAPP_nat_real(power_power_real(A_2),number_number_of_nat(bit0(bit1(pls)))))) # label(fact_441_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 575 (all B_19 all A_21 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_21)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_19),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(B_19),A_21)),zero_zero_nat))))) # label(fact_800_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 576 (all P_5 all P_2 all X_2 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) -> (hBOOL(P_5) <-> hBOOL(P_2))) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) -> hBOOL(P_2)) <-> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) -> hBOOL(P_5))))) # label(fact_1061_imp__le__cong) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 577 (all Z (is_int(Z) -> hAPP_int_int(times_times_int(Z),one_one_int) = Z)) # label(fact_207_zmult__1__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 578 (all M all N_1 hAPP_nat_nat(times_times_nat(N_1),M) = hAPP_nat_nat(times_times_nat(M),N_1)) # label(fact_950_nat__mult__commute) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 579 (all U all Ma all N all J_2 all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),I_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(I_2),J_2)),U)),Ma)),N)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_2),U)),Ma)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_2),U)),N)))))) # label(fact_1058_nat__less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 580 (all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),number_number_of_int(Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(pls)),Y_2)))) # label(fact_168_le__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 581 (all N_12 all A_62 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),A_62)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_nat_real(power_power_real(A_62),N_12))))) # label(fact_485_one__le__power) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 582 (all W_16 hAPP_nat_int(power_power_int(number_number_of_int(W_16)),number_number_of_nat(bit0(bit1(pls)))) = hAPP_int_int(times_times_int(number_number_of_int(W_16)),number_number_of_int(W_16))) # label(fact_12_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 583 (all A_74 all N_23 all N_22 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_23),N_22)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_74)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_74),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_74),N_22)),hAPP_nat_real(power_power_real(A_74),N_23))))))) # label(fact_379_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 584 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(div_mod_int(A),B))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(div_mod_int(A),B)),B)))) # label(fact_1159_pos__mod__conj) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 585 (all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(pls)),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),number267125858f_real(Y_2))))) # label(fact_165_less__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 586 (all A_70 A_70 = hAPP_nat_nat(power_power_nat(A_70),one_one_nat)) # label(fact_425_power__one__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 587 (all C_16 all A_37 all B_34 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_37),B_34)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_37),hAPP_nat_nat(times_times_nat(B_34),C_16))))) # label(fact_745_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 588 (all A_53 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_53),zero_zero_int))) # label(fact_674_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 589 (all I_2 all J_2 all K (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),hAPP_nat_nat(minus_minus_nat(J_2),K))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I_2),K)),J_2)))) # label(fact_984_less__diff__conv) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 590 (all X_22 all P_6 all Q_6 hAPP_nat_nat(power_power_nat(X_22),hAPP_nat_nat(plus_plus_nat(P_6),Q_6)) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(power_power_nat(X_22),P_6)),hAPP_nat_nat(power_power_nat(X_22),Q_6))) # label(fact_58_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 591 (all A_2 (is_int(A_2) -> (zero_zero_int = hAPP_int_int(plus_plus_int(A_2),A_2) <-> zero_zero_int = A_2))) # label(fact_356_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 592 (all B_1_1 (is_int(B_1_1) -> is_int(bit1(B_1_1)))) # label(gsy_c_Int_OBit1) # label(hypothesis) # label(non_clause). [assumption]. 1.32/1.59 593 (all N_38 all A_90 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_90)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_90),N_38)),hAPP_real_real(times_times_real(A_90),hAPP_nat_real(power_power_real(A_90),N_38)))))) # label(fact_296_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 1.32/1.59 594 (all C_9 all A_25 all B_23 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_25),B_23)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_9)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(C_9),A_25)),hAPP_real_real(times_times_real(C_9),B_23)))))) # label(fact_788_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 595 (all N_1 all F all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,F),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,F),N_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,F),hAPP_int_int(div_mod_int(M),N_1)))))) # label(fact_1147_zdvd__zmod) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 596 (all Ma all N 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,Ma),N)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(K),Ma)),hAPP_nat_nat(times_times_nat(K),N)))))) # label(fact_1048_nat__mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 597 (all A_61 all N_11 all N_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_11),N_10)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A_61)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_61),N_11)),hAPP_nat_int(power_power_int(A_61),N_10)))))) # label(fact_490_power__increasing) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 598 (all C_24 all A_51 all B_45 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_51),B_45)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_45),C_24)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_51),C_24))))) # label(fact_682_dvd__trans) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 599 (all A_99 (is_int(A_99) -> A_99 = hAPP_int_int(times_times_int(one_one_int),A_99))) # label(fact_188_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 600 (all V_7 all W_1 (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(W_1)),number267125858f_real(V_7))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(V_7)),number267125858f_real(W_1))))) # label(fact_50_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 601 (all A (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A)) -> A = zero_zero_nat)) # label(fact_1118_gcd__lcm__complete__lattice__nat_Otop__le) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 602 (all Z Z = hAPP_real_real(times_times_real(one_one_real),Z)) # label(fact_690_real__mult__1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 603 (all J_1 all A all P_1 all K_1 (hBOOL(hAPP_int_bool(zcong(J_1,hAPP_int_int(times_times_int(A),multInv(P_1,K_1))),P_1)) -> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(J_1),K_1),hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(A),multInv(P_1,K_1))),K_1)),P_1)))) # label(fact_1075_aux______1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 604 (all C_4 all A_14 all B_13 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_14),B_13)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_4)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(C_4),A_14)),hAPP_nat_nat(times_times_nat(C_4),B_13)))))) # label(fact_822_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 605 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),bit0(K))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),K)))) # label(fact_152_rel__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 606 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K1)),bit0(K2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)))) # label(fact_82_less__eq__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 607 (all A all R_1 all B all M all C all D_5 all N_1 hAPP_int_int(minus_minus_int(hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A),M)),hAPP_int_int(times_times_int(C),N_1))),hAPP_int_int(times_times_int(R_1),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),M)),hAPP_int_int(times_times_int(D_5),N_1)))) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(A),hAPP_int_int(times_times_int(R_1),B))),M)),hAPP_int_int(times_times_int(hAPP_int_int(minus_minus_int(C),hAPP_int_int(times_times_int(R_1),D_5))),N_1))) # label(fact_628_xzgcda__linear__aux1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 608 (all K_1 all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(times_times_nat(K_1),M)),hAPP_nat_nat(times_times_nat(K_1),N_1))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N_1))))) # label(fact_1040_dvd__mult__cancel) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 609 (all J_1 all I_1 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(J_1),I_1)),I_1))) # label(fact_975_not__add__less2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 610 (all A hAPP_nat_int(power_power_int(A),number_number_of_nat(bit1(bit1(pls)))) = hAPP_int_int(times_times_int(A),hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls)))))) # label(fact_15_cube__square) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 611 (all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),number267125858f_real(Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(pls)),Y_2)))) # label(fact_169_le__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 612 (all A_66 all B_49 all N_17 hAPP_nat_nat(power_power_nat(hAPP_nat_nat(times_times_nat(A_66),B_49)),N_17) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(power_power_nat(A_66),N_17)),hAPP_nat_nat(power_power_nat(B_49),N_17))) # label(fact_460_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 613 (all Lx_4 all Ly_2 all Rx_4 all Ry_2 hAPP_nat_nat(times_times_nat(Lx_4),hAPP_nat_nat(times_times_nat(Ly_2),hAPP_nat_nat(times_times_nat(Rx_4),Ry_2))) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_4),Ly_2)),hAPP_nat_nat(times_times_nat(Rx_4),Ry_2))) # label(fact_98_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 614 (all M all N_1 (hAPP_nat_nat(times_times_nat(M),N_1) = M -> zero_zero_nat = M | one_one_nat = N_1)) # label(fact_1038_mult__eq__self__implies__10) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 615 (all A_33 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,one_one_int),A_33))) # label(fact_758_one__dvd) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 616 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K1)),bit1(K2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)))) # label(fact_65_less__eq__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 617 (all B_6 all A_7 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_7)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_6),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_7),B_6)),zero_zero_int))))) # label(fact_845_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 618 (all A_2 all B_2 all C_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_2),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),A_2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_2),A_2)),hAPP_int_int(times_times_int(C_2),B_2)))))) # label(fact_833_mult__less__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 619 (all B_1_1 all B_2_1 (is_int(B_2_1) & is_int(B_1_1) -> is_int(multInv(B_1_1,B_2_1)))) # label(gsy_c_Int2_OMultInv) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 620 (all A_111 all B_63 hAPP_int_int(times_times_int(B_63),A_111) = hAPP_int_int(times_times_int(A_111),B_63)) # label(fact_112_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 621 (all L hAPP_int_int(minus_minus_int(pls),bit0(L)) = bit0(hAPP_int_int(minus_minus_int(pls),L))) # label(fact_626_diff__bin__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 622 (all N_1 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_1))) # label(fact_894_less__not__refl) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 623 (all J_1 all K_1 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),M)) -> (hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(number_number_of_int(min)),J_1),hAPP_nat_int(power_power_int(number_number_of_int(min)),K_1)),M)) -> hAPP_nat_int(power_power_int(number_number_of_int(min)),J_1) = hAPP_nat_int(power_power_int(number_number_of_int(min)),K_1)))) # label(fact_606_neg__one__power__eq__mod__m) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 624 (all P_1 all Y_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),Y_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),hAPP_nat_int(power_power_int(Y_1),N_1)))))) # label(fact_659_zpower__zdvd__prop1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 625 (all M 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_int,X_1),M)) -> (hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),M)) -> zero_zero_int = X_1))))) # label(fact_658_Int2_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 626 (all A_42 all B_39 all C_20 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(times_times_nat(A_42),B_39)),C_20)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_39),C_20)))) # label(fact_730_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 627 (all B_20 all A_22 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_22),zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_20)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_22),B_20)),zero_zero_nat))))) # label(fact_797_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 628 (all A_2 all B_2 all N (N != zero_zero_nat -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(A_2),N)),hAPP_nat_int(power_power_int(B_2),N))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_2),B_2))))) # label(fact_1117_pow__divides__eq__int) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 629 (all M all X_1 hAPP_int_int(div_mod_int(X_1),M) = standardRes(M,X_1)) # label(fact_1184_StandardRes__def) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 630 (all B_2 all A_2 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),hAPP_int_int(minus_minus_int(P_3),one_one_int))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),A_2)) -> hBOOL(member_int(B_2,wset(A_2,P_3)))))))) # label(fact_1103_wset__mem) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 631 (all B all D_5 all A (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_5),A)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_5),hAPP_nat_nat(plus_plus_nat(A),B))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_5),B))))) # label(fact_763_divides__add__revr) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 632 (all X_2 all Y_2 (is_int(X_2) & is_int(Y_2) -> (X_2 = Y_2 <-> number267125858f_real(Y_2) = number267125858f_real(X_2)))) # label(fact_136_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 633 (all A_110 all B_62 all C_35 all D_11 hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_110),B_62)),hAPP_real_real(plus_plus_real(C_35),D_11)) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_110),C_35)),hAPP_real_real(plus_plus_real(B_62),D_11))) # label(fact_117_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 634 (all B_19 all A_21 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_21)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_19),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(B_19),A_21)),zero_zero_real))))) # label(fact_799_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 635 (all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N_1))) # label(fact_966_less__eq__nat_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 636 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K1)),bit1(K2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)))) # label(fact_63_less__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 637 (all K_1 all L bit0(K_1) != bit1(L)) # label(fact_197_rel__simps_I49_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 638 (all X_1 all P_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP_int_bool(zprime,P_1)) -> (-hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),P_1)) -> hBOOL(hAPP_int_bool(zcong(multInv(P_1,multInv(P_1,X_1)),X_1),P_1)))))) # label(fact_1085_MultInv__prop4) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 639 (all N_34 all M_7 all X_19 all Y_16 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_19),Y_16)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_34),M_7)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(X_19),N_34)),hAPP_nat_nat(power_power_nat(Y_16),M_7)))))) # label(fact_319_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 640 (all A_110 all B_62 all C_35 all D_11 hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_110),B_62)),hAPP_nat_nat(plus_plus_nat(C_35),D_11)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_110),C_35)),hAPP_nat_nat(plus_plus_nat(B_62),D_11))) # label(fact_116_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 641 (all C_7 all B_16 all A_17 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_16),A_17)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_7),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_7),A_17)),hAPP_int_int(times_times_int(C_7),B_16)))))) # label(fact_815_mult__strict__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 642 (all X_4 all N_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_3)) -> hAPP_real_real(times_times_real(hAPP_nat_real(power_power_real(X_4),hAPP_nat_nat(minus_minus_nat(N_3),one_one_nat))),X_4) = hAPP_nat_real(power_power_real(X_4),N_3))) # label(fact_693_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 643 (all M all N_1 all P_1 (hBOOL(hAPP_int_bool(zprime,P_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),hAPP_int_int(times_times_int(M),N_1))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),M)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),N_1))))) # label(fact_657_zprime__zdvd__zmult__better) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 644 (all V_12 all W_11 number_number_of_int(hAPP_int_int(times_times_int(V_12),W_11)) = hAPP_int_int(times_times_int(number_number_of_int(V_12)),number_number_of_int(W_11))) # label(fact_243_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 645 (all C_13 all D_3 all A_29 all B_27 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_29),B_27)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_13),D_3)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_29)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_13)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_29),C_13)),hAPP_real_real(times_times_real(B_27),D_3)))))))) # label(fact_778_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 646 (all B_10 all A_11 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_11),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_10)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_11),B_10)),zero_zero_int))))) # label(fact_831_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 647 (all A_98 all B_57 all V_16 hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_98),number267125858f_real(V_16))),hAPP_real_real(times_times_real(B_57),number267125858f_real(V_16))) = hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_98),B_57)),number267125858f_real(V_16))) # label(fact_222_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 648 (all A_2 all W_1 (hAPP_nat_real(power_power_real(A_2),number_number_of_nat(W_1)) = zero_zero_real <-> number_number_of_nat(W_1) != zero_zero_nat & A_2 = zero_zero_real)) # label(fact_582_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 649 (all A_70 (is_int(A_70) -> hAPP_nat_int(power_power_int(A_70),one_one_nat) = A_70)) # label(fact_423_power__one__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 650 (all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),number267125858f_real(Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Y_2)))) # label(fact_430_less__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 651 (all X_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(X_2)),zero_zero_real)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),pls)))) # label(fact_435_le__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 652 (all V_2 all W_2 all C_25 hAPP_int_int(plus_plus_int(number_number_of_int(V_2)),hAPP_int_int(minus_minus_int(number_number_of_int(W_2)),C_25)) = hAPP_int_int(minus_minus_int(number_number_of_int(hAPP_int_int(plus_plus_int(V_2),W_2))),C_25)) # label(fact_631_add__number__of__diff1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 653 (all Z_5 hAPP_int_int(plus_plus_int(Z_5),Z_5) = hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),Z_5)) # label(fact_278_mult__2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 654 (all V_7 all V_8 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,number_number_of_nat(V_7)),number_number_of_nat(V_8))) <-> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_7),V_8)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V_8))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_7),V_8)))) # label(fact_415_less__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 655 (all A_65 all M_5 all N_16 hAPP_nat_nat(times_times_nat(hAPP_nat_nat(power_power_nat(A_65),M_5)),hAPP_nat_nat(power_power_nat(A_65),N_16)) = hAPP_nat_nat(power_power_nat(A_65),hAPP_nat_nat(plus_plus_nat(M_5),N_16))) # label(fact_463_power__add) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 656 (all A_107 all B_60 all C_33 hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_107),B_60)),C_33) = hAPP_real_real(plus_plus_real(A_107),hAPP_real_real(plus_plus_real(B_60),C_33))) # label(fact_125_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 657 (all K (is_int(K) -> (K = pls <-> bit0(K) = pls))) # label(fact_198_rel__simps_I44_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 658 (all C_5 all A_15 all B_14 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_15),B_14)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_5)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_5),A_15)),hAPP_real_real(times_times_real(C_5),B_14)))))) # label(fact_818_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 659 (all X_2 all P_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),X_2)) <-> hBOOL(hAPP_int_bool(zcong(X_2,zero_zero_int),P_3)))) # label(fact_655_zcong__eq__zdvd__prop) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 660 (all X_2 all Y_2 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_eq_real,X_2),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(Z_1),X_2)),hAPP_real_real(times_times_real(Z_1),Y_2)))))) # label(fact_863_real__mult__le__cancel__iff2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 661 (all A_87 all M_8 all N_35 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_8),N_35)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(A_87),M_8)),hAPP_nat_int(power_power_int(A_87),N_35))))) # label(fact_315_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 662 (all Ma all N 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),N)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(K),Ma)),hAPP_nat_nat(times_times_nat(K),N)))))) # label(fact_1051_nat__mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 663 (all N_12 all A_62 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A_62)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),hAPP_nat_int(power_power_int(A_62),N_12))))) # label(fact_487_one__le__power) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 664 (all Z hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),Z) = hAPP_nat_nat(plus_plus_nat(Z),Z)) # label(fact_60_nat__mult__2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 665 (all B_18 all A_20 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_20)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_18),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_20),B_18)),zero_zero_int))))) # label(fact_804_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 666 (all N_31 all X_18 all Y_15 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_18),Y_15)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(power_power_real(X_18),N_31)),hAPP_nat_real(power_power_real(Y_15),N_31))))) # label(fact_329_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 667 (all M (hBOOL(hAPP_int_bool(zprime,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),M)),one_one_int))) -> one_one_int = legendre(number_number_of_int(min),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),M)),one_one_int)))) # label(fact_665_Legendre__1mod4) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 668 (all A all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M)) -> (exists X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) & (all Y (is_int(Y) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),M)) & hBOOL(hAPP_int_bool(zcong(A,Y),M)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) -> Y = X))) & hBOOL(hAPP_int_bool(zcong(A,X),M)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),M)) & is_int(X))))) # label(fact_1068_zcong__zless__unique) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 669 (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),bit1(K))))) # label(fact_153_rel__simps_I22_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 670 (all M_6 all A_86 all N_33 all B_55 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(A_86),N_33)),B_55)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_6),N_33)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(A_86),M_6)),B_55))))) # label(fact_322_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 671 (all Lx_2 all Ly all Rx_2 hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_2),Ly)),Rx_2) = hAPP_real_real(times_times_real(Lx_2),hAPP_real_real(times_times_real(Ly),Rx_2))) # label(fact_105_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 672 (all N_1 all K_1 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1)) -> hAPP_nat_nat(minus_minus_nat(M),N_1) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(M),K_1)),hAPP_nat_nat(minus_minus_nat(N_1),K_1))))) # label(fact_947_Nat_Odiff__diff__eq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 673 (all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),N_1))) # label(fact_937_le__refl) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 674 (all X_1 all Y_1 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) -> -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))))) # label(fact_905_dvd_Oless__imp__not__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 675 (all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),one_one_nat)) <-> one_one_nat = Ma)) # label(fact_1019_nat__dvd__1__iff__1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 676 (all K_1 all I_1 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),K_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),K_1))))) # label(fact_940_le__trans) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 677 (all X_2 all N (N = zero_zero_nat | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(X_2),N))))) # label(fact_514_nat__zero__less__power__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 678 (all I_1 all J_1 all K_1 hAPP_nat_nat(minus_minus_nat(I_1),hAPP_nat_nat(plus_plus_nat(J_1),K_1)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(I_1),J_1)),K_1)) # label(fact_934_diff__diff__left) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 679 (all P_1 (hBOOL(hAPP_int_bool(zprime,P_1)) -> hBOOL(hAPP_int_bool(zcong(zfact(hAPP_int_int(minus_minus_int(P_1),one_one_int)),number_number_of_int(min)),P_1)))) # label(fact_1070_Wilson__Russ) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 680 (all B_20 all A_22 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_22),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_20)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_22),B_20)),zero_zero_int))))) # label(fact_798_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 681 (all M all K_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(N_1),M)),K_1))))) # label(fact_1008_le__add__diff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 682 (all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B)) -> B = zero_zero_nat | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A),B)))) # label(fact_859_divides__ge) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 683 (all A_85 all N_32 all B_54 (is_int(B_54) & is_int(A_85) -> (hAPP_nat_int(power_power_int(B_54),N_32) = hAPP_nat_int(power_power_int(A_85),N_32) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_85)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_54)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32)) -> A_85 = B_54)))))) # label(fact_323_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 684 (all Z (is_int(Z) -> Z = hAPP_int_int(times_times_int(one_one_int),Z))) # label(fact_208_zmult__1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 685 (all M all N_1 ((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> hAPP_nat_nat(div_mod_nat(M),N_1) = M) & (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(minus_minus_nat(M),N_1)),N_1) = hAPP_nat_nat(div_mod_nat(M),N_1)))) # label(fact_1181_mod__if) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 686 (all A_108 all B_61 all C_34 hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_108),C_34)),B_61) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_108),B_61)),C_34)) # label(fact_122_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 687 (all Ma all N (Ma = zero_zero_nat & N = zero_zero_nat <-> hAPP_nat_nat(plus_plus_nat(Ma),N) = zero_zero_nat)) # label(fact_961_add__is__0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 688 (all C_8 all A_24 all B_22 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_24),B_22)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_8)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_24),C_8)),hAPP_nat_nat(times_times_nat(B_22),C_8)))))) # label(fact_792_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 689 (all Lx_3 all Ly_1 all Rx_3 hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_3),Ly_1)),Rx_3) = hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_3),Rx_3)),Ly_1)) # label(fact_102_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 690 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),t)) -> (exists X exists Y (is_int(X) & hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y),number_number_of_nat(bit0(bit1(pls))))) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int) & is_int(Y))) # label(fact_2__0961_A_060_At_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 691 (all A_9 all B_8 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(times_times_nat(A_9),B_8))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_9)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_8))))) # label(fact_838_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 692 (all A_68 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_68),number_number_of_nat(bit0(bit1(pls))))),zero_zero_int))) # label(fact_452_power2__less__0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 693 (all X_1 all Y_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) -> Y_1 != X_1)) # label(fact_908_dvd_Oless__imp__neq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 694 (all N all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(minus_minus_nat(N),Ma))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)))) # label(fact_959_zero__less__diff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 695 (all A_52 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_52),A_52))) # label(fact_680_dvd__refl) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 696 (all A all B hAPP_nat_int(power_power_int(hAPP_int_int(minus_minus_int(A),B)),number_number_of_nat(bit0(bit1(pls)))) = hAPP_int_int(plus_plus_int(hAPP_int_int(minus_minus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls))))),hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),A)),B))),hAPP_nat_int(power_power_int(B),number_number_of_nat(bit0(bit1(pls)))))) # label(fact_650_zdiff__power2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 697 (all A_103 all N_39 hAPP_nat_real(power_power_real(hAPP_nat_real(power_power_real(A_103),N_39)),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_real(power_power_real(A_103),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_39))) # label(fact_160_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 698 (all B all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A)) -> hAPP_int_int(div_mod_int(hAPP_int_int(plus_plus_int(one_one_int),hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),B))),hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),A)) = hAPP_int_int(plus_plus_int(one_one_int),hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),hAPP_int_int(div_mod_int(B),A))))) # label(fact_1174_pos__zmod__mult__2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 699 (all N_1 all P_1 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M)) -> (hBOOL(hAPP_int_bool(zprime,P_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),hAPP_int_int(times_times_int(M),N_1))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),M)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_1),N_1)))))) # label(fact_603_zprime__zdvd__zmult) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 700 (all B_11 all A_12 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_12),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_11),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(A_12),B_11)))))) # label(fact_827_mult__neg__neg) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 701 (all M all A (is_int(A) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),M)) -> (hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),M)) -> zero_zero_int = A))))) # label(fact_595_zcong__zless__0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 702 (all M all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(M),N_1)),M))) # label(fact_942_Nat_Odiff__le__self) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 703 (all Z1 all Z2 all Z3 hAPP_int_int(plus_plus_int(Z1),hAPP_int_int(plus_plus_int(Z2),Z3)) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(Z1),Z2)),Z3)) # label(fact_145_zadd__assoc) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 704 (all K_1 all A all J_1 all P_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP_int_bool(zprime,P_1)) -> (-hBOOL(hAPP_int_bool(zcong(J_1,zero_zero_int),P_1)) -> (hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(multInv(P_1,J_1)),J_1)),K_1),hAPP_int_int(times_times_int(multInv(P_1,J_1)),A)),P_1)) -> hBOOL(hAPP_int_bool(zcong(K_1,hAPP_int_int(times_times_int(A),multInv(P_1,J_1))),P_1))))))) # label(fact_1088_aux______4) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 705 (all B all Q_1 all R_1 all B_48 all Q_4 all R_3 (hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_48),Q_4)),R_3) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),Q_1)),R_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_48),Q_4)),R_3)),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_48)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_48),B)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_4),Q_1))))))))) # label(fact_597_zdiv__mono2__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 706 (all Y_2 all X_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) -> (Y_2 = X_2 <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2))))) # label(fact_918_dvd_Oantisym__conv) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 707 (all M all N_1 (N_1 = M -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)))) # label(fact_939_eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 708 (all V_21 hAPP_int_int(plus_plus_int(number_number_of_int(V_21)),one_one_int) = number_number_of_int(hAPP_int_int(plus_plus_int(V_21),bit1(pls)))) # label(fact_30_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 709 (all V_5 all W_3 number_number_of_int(hAPP_int_int(minus_minus_int(V_5),W_3)) = hAPP_int_int(minus_minus_int(number_number_of_int(V_5)),number_number_of_int(W_3))) # label(fact_613_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 710 (all Lx all Rx all Ry hAPP_int_int(times_times_int(Rx),hAPP_int_int(times_times_int(Lx),Ry)) = hAPP_int_int(times_times_int(Lx),hAPP_int_int(times_times_int(Rx),Ry))) # label(fact_109_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 711 (all K all P_2 all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D)) -> ((all X (is_int(X) -> (hBOOL(hAPP_int_bool(P_2,X)) -> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(minus_minus_int(X),D)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K)) -> (all X (hBOOL(hAPP_int_bool(P_2,X)) -> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(minus_minus_int(X),hAPP_int_int(times_times_int(K),D)))))))))) # label(fact_1065_decr__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 712 (all K_1 all L bit0(hAPP_int_int(plus_plus_int(K_1),L)) = hAPP_int_int(plus_plus_int(bit0(K_1)),bit0(L))) # label(fact_205_add__Bit0__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 713 (all Z all W hAPP_real_real(times_times_real(W),Z) = hAPP_real_real(times_times_real(Z),W)) # label(fact_689_real__mult__commute) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 714 (all X_15 all Y_12 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(X_15),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_12),number_number_of_nat(bit0(bit1(pls)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_12)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_15),Y_12))))) # label(fact_447_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 715 (all B all M all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),A)) -> -hBOOL(hAPP_int_bool(zcong(A,B),M))))))) # label(fact_588_zcong__not) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 716 (all C all A all B (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B),A)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B)) -> (B = C -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),C)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C),A))))) # label(fact_901_dvd_Oord__less__eq__trans) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 717 (all P_3 all A_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_2)) -> hBOOL(member_int(A_2,wset(A_2,P_3))))) # label(fact_1098_wset__mem__mem) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 718 (all N_30 all A_84 (A_84 != zero_zero_real -> hAPP_nat_real(power_power_real(A_84),N_30) != zero_zero_real)) # label(fact_333_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 719 (all A_73 all N_21 all N_20 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_21),N_20)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_73)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_73),one_one_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_73),N_20)),hAPP_nat_nat(power_power_nat(A_73),N_21))))))) # label(fact_381_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 720 (all A_68 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_68),number_number_of_nat(bit0(bit1(pls))))),zero_zero_real))) # label(fact_451_power2__less__0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 721 (all N_34 all M_7 all X_19 all Y_16 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_19),Y_16)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_34),M_7)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(power_power_real(X_19),N_34)),hAPP_nat_real(power_power_real(Y_16),M_7)))))) # label(fact_317_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 722 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(K1)),bit0(K2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K1),K2)))) # label(fact_68_less__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 723 (all A_53 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_53),zero_zero_real))) # label(fact_672_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 724 (all W_5 hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(zero_zero_real),number267125858f_real(W_5))),number267125858f_real(W_5)) = number267125858f_real(bit0(W_5))) # label(fact_422_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 725 (all M_9 all N_36 all A_88 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_88)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_88),M_9)),hAPP_nat_int(power_power_int(A_88),N_36))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_9),N_36))))) # label(fact_300_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 726 (all K all Ma all N ((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),N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(K),Ma)),hAPP_nat_nat(times_times_nat(K),N))))) # label(fact_1041_mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 727 (all N_26 all A_77 all B_52 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_77),B_52)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_77)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(A_77),N_26)),hAPP_nat_nat(power_power_nat(B_52),N_26)))))) # label(fact_366_power__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 728 (all V_12 all W_11 hAPP_real_real(times_times_real(number267125858f_real(V_12)),number267125858f_real(W_11)) = number267125858f_real(hAPP_int_int(times_times_int(V_12),W_11))) # label(fact_244_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 729 (all X_2 all Y_2 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_2),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(X_2),Z_1)),hAPP_real_real(times_times_real(Y_2),Z_1)))))) # label(fact_861_real__mult__less__iff1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 730 (all X_20 all Y_17 all Q_5 hAPP_real_real(times_times_real(hAPP_nat_real(power_power_real(X_20),Q_5)),hAPP_nat_real(power_power_real(Y_17),Q_5)) = hAPP_nat_real(power_power_real(hAPP_real_real(times_times_real(X_20),Y_17)),Q_5)) # label(fact_192_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 731 (all M all K_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(M),K_1)),N_1)) -> -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1))))) # label(fact_1005_add__leE) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 732 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(div_mod_int(A),B)),B)))) # label(fact_1138_pos__mod__bound) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 733 (all P all Q all R hAPP_int_bool(hAPP_i1948725293t_bool(P,R),Q) = hAPP_int_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]. 1.32/1.60 734 (all A_59 all N_8 all N_7 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_8),N_7)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_59)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_59),N_8)),hAPP_nat_real(power_power_real(A_59),N_7)))))) # label(fact_500_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 735 (all X_24 all P_7 all Q_7 hAPP_nat_nat(power_power_nat(hAPP_nat_nat(power_power_nat(X_24),P_7)),Q_7) = hAPP_nat_nat(power_power_nat(X_24),hAPP_nat_nat(times_times_nat(P_7),Q_7))) # label(fact_43_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 736 (all A -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),one_one_nat)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A)))) # label(fact_1123_gcd__lcm__complete__lattice__nat_Onot__less__bot) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 737 (all N_15 hAPP_nat_int(power_power_int(one_one_int),N_15) = one_one_int) # label(fact_468_power__one) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 738 (all W_1 all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),hAPP_int_int(plus_plus_int(Z_1),one_one_int))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W_1),Z_1)))) # label(fact_88_zle__add1__eq__le) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 739 (all A_67 all N_18 hAPP_nat_nat(times_times_nat(hAPP_nat_nat(power_power_nat(A_67),N_18)),A_67) = hAPP_nat_nat(times_times_nat(A_67),hAPP_nat_nat(power_power_nat(A_67),N_18))) # label(fact_457_power__commutes) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 740 (all A_104 all C_30 hAPP_int_int(plus_plus_int(A_104),C_30) = hAPP_int_int(plus_plus_int(C_30),A_104)) # label(fact_132_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 741 (all K all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(K),Ma)),hAPP_nat_nat(times_times_nat(K),N))) <-> 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),N)))) # label(fact_1036_mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 742 (all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Y_2)))) # label(fact_434_le__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 743 (all A all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A)) -> (exists X ((all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y)) & A = hAPP_nat_real(power_power_real(Y),N_1) -> Y = X)) & A = hAPP_nat_real(power_power_real(X),N_1) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))))))) # label(fact_873_realpow__pos__nth__unique) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 744 (all X_22 all P_6 all Q_6 hAPP_real_real(times_times_real(hAPP_nat_real(power_power_real(X_22),P_6)),hAPP_nat_real(power_power_real(X_22),Q_6)) = hAPP_nat_real(power_power_real(X_22),hAPP_nat_nat(plus_plus_nat(P_6),Q_6))) # label(fact_57_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 745 (all A all P_1 (is_int(A) -> (hBOOL(hAPP_int_bool(zprime,P_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(bit1(bit0(bit1(pls))))),P_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),P_1)) -> A = inv(P_1,inv(P_1,A)))))))) # label(fact_1072_inv__inv) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 746 (all N_29 ((N_29 = zero_zero_nat -> one_one_int = hAPP_nat_int(power_power_int(zero_zero_int),N_29)) & (N_29 != zero_zero_nat -> hAPP_nat_int(power_power_int(zero_zero_int),N_29) = zero_zero_int))) # label(fact_334_power__0__left) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 747 (all B_1_1 is_int(twoSqu1241645765sum2sq(B_1_1))) # label(gsy_c_TwoSquares__Mirabelle__enbualjaop_Osum2sq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 748 (all P_2 all K all I_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),I_2)) -> (hBOOL(hAPP_int_bool(P_2,hAPP_int_int(plus_plus_int(K),one_one_int))) -> ((all I (is_int(I) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),I)) -> (hBOOL(hAPP_int_bool(P_2,I)) -> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(plus_plus_int(I),one_one_int))))))) -> hBOOL(hAPP_int_bool(P_2,I_2)))))) # label(fact_1108_int__gr__induct) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 749 (all B_17 all A_19 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_19)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_17)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(times_times_int(A_19),B_17)))))) # label(fact_807_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 750 (all M_11 all A_96 hAPP_real_real(plus_plus_real(M_11),hAPP_real_real(times_times_real(A_96),M_11)) = hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_96),one_one_real)),M_11)) # label(fact_231_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 751 (all N 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),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(Ma),K)),hAPP_nat_nat(minus_minus_nat(N),K))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)))))) # label(fact_948_le__diff__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 752 (all B_29 all A_31 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_31),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_29),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_29)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_31)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(times_times_int(A_31),B_29))))) # label(fact_774_split__mult__pos__le) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 753 (all N all P_2 (-hBOOL(hAPP_nat_bool(P_2,zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(P_2,N)) -> (exists K_2 (hBOOL(hAPP_nat_bool(P_2,hAPP_nat_nat(plus_plus_nat(K_2),one_one_nat))) & (all I (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),K_2)) -> -hBOOL(hAPP_nat_bool(P_2,I)))) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_2),N))))))) # label(fact_1043_ex__least__nat__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 754 (all Lx_5 all Ly_3 all Rx_5 all Ry_3 hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_5),Ly_3)),hAPP_nat_nat(times_times_nat(Rx_5),Ry_3)) = hAPP_nat_nat(times_times_nat(Rx_5),hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_5),Ly_3)),Ry_3))) # label(fact_95_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 755 (all X_1 all Q_1 all N_1 all R_1 (X_1 = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(Q_1),N_1)),R_1) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),R_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,R_1),N_1)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N_1),X_1)))))) # label(fact_870_divides__div__not) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 756 (all B_1_1 (is_int(B_1_1) -> is_int(bit0(B_1_1)))) # label(gsy_c_Int_OBit0) # label(hypothesis) # label(non_clause). [assumption]. 1.32/1.60 757 (all A_110 all B_62 all C_35 all D_11 hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_110),B_62)),hAPP_int_int(plus_plus_int(C_35),D_11)) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(A_110),C_35)),hAPP_int_int(plus_plus_int(B_62),D_11))) # label(fact_115_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 758 (all Lx all Rx all Ry hAPP_real_real(times_times_real(Lx),hAPP_real_real(times_times_real(Rx),Ry)) = hAPP_real_real(times_times_real(Rx),hAPP_real_real(times_times_real(Lx),Ry))) # label(fact_111_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 759 (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_int,X_1),number_number_of_int(bit0(bit1(pls))))) -> X_1 = zero_zero_int | one_one_int = X_1)))) # label(fact_505_int__pos__lt__two__imp__zero__or__one) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 760 (all I_2 all K all J_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(I_2),K)),J_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),hAPP_nat_nat(minus_minus_nat(J_2),K)))))) # label(fact_1011_le__diff__conv2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 761 (all X_5 all Y_4 (X_5 != Y_4 -> (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_5),Y_4)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_4),X_5))))) # label(fact_676_linorder__neqE__linordered__idom) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 762 (all A_82 hAPP_nat_nat(times_times_nat(zero_zero_nat),A_82) = zero_zero_nat) # label(fact_342_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 763 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(K)),min)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),min)))) # label(fact_550_rel__simps_I11_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 764 (all C_6 all B_15 all A_16 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_15),A_16)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_6),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_16),C_6)),hAPP_real_real(times_times_real(B_15),C_6)))))) # label(fact_816_mult__strict__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 765 (all X_1 all Y_1 (X_1 != Y_1 -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_1),X_1))))) # label(fact_892_linorder__neqE__nat) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 766 (all X_2 all Y_2 (is_int(X_2) & is_int(Y_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X_2),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_2),number_number_of_nat(bit0(bit1(pls))))))) <-> Y_2 != zero_zero_int | zero_zero_int != X_2))) # label(fact_482_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 767 (all B_29 all A_31 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_31)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_29)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_29),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_31),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(times_times_real(A_31),B_29))))) # label(fact_773_split__mult__pos__le) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 768 (all K all L_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(K)),bit1(L_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L_1)))) # label(fact_85_rel__simps_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 769 (all Ma all K all N (hAPP_nat_nat(times_times_nat(Ma),K) = hAPP_nat_nat(times_times_nat(N),K) <-> zero_zero_nat = K | N = Ma)) # label(fact_973_mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 770 (all M hAPP_nat_nat(times_times_nat(M),zero_zero_nat) = zero_zero_nat) # label(fact_970_mult__0__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 771 (all K_1 all M all N_1 hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(times_times_nat(K_1),M)),hAPP_nat_nat(times_times_nat(K_1),N_1)) = hAPP_nat_nat(times_times_nat(K_1),hAPP_nat_nat(minus_minus_nat(M),N_1))) # label(fact_952_diff__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 772 (all X_8 all Y_7 (is_int(Y_7) & is_int(X_8) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_8),Y_7)) -> (Y_7 != X_8 -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_8),Y_7)))))) # label(fact_570_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 773 -(all S1 (is_int(S1) -> -hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(S1),number_number_of_nat(bit0(bit1(pls)))),number_number_of_int(min)),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int))))) # label(fact_512__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]. 1.32/1.60 774 -(all T_1 (is_int(T_1) -> hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(s),number_number_of_nat(bit0(bit1(pls))))),one_one_int) != hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int)),T_1))) # label(fact_34__096_B_Bthesis_O_A_I_B_Bt_O_As_A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A_L_A1_) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 775 (all K_1 bit1(K_1) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(one_one_int),K_1)),K_1)) # label(fact_253_Bit1__def) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 776 (all C_27 all D_8 all A_72 all B_50 all R_4 (R_4 != zero_zero_nat -> (D_8 != C_27 & A_72 = B_50 -> hAPP_nat_nat(plus_plus_nat(A_72),hAPP_nat_nat(times_times_nat(R_4),C_27)) != hAPP_nat_nat(plus_plus_nat(B_50),hAPP_nat_nat(times_times_nat(R_4),D_8))))) # label(fact_388_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 777 (all M all N_1 (hAPP_nat_nat(minus_minus_nat(M),N_1) = zero_zero_nat -> (zero_zero_nat = hAPP_nat_nat(minus_minus_nat(N_1),M) -> M = N_1))) # label(fact_887_diffs0__imp__equal) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 778 (all A_104 all C_30 hAPP_nat_nat(plus_plus_nat(A_104),C_30) = hAPP_nat_nat(plus_plus_nat(C_30),A_104)) # label(fact_133_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 779 (all K_1 bit1(K_1) != pls) # label(fact_194_rel__simps_I46_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 780 (all C_15 all A_34 all B_31 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_34),B_31)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_34),C_15)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_34),hAPP_real_real(plus_plus_real(B_31),C_15)))))) # label(fact_753_dvd__add) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 781 (all A all B hAPP_int_int(minus_minus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(B),number_number_of_nat(bit0(bit1(pls))))) = hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A),B)),hAPP_int_int(minus_minus_int(A),B))) # label(fact_642_zspecial__product) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 782 (all W_15 number_number_of_int(hAPP_int_int(plus_plus_int(bit1(pls)),W_15)) = hAPP_int_int(plus_plus_int(one_one_int),number_number_of_int(W_15))) # label(fact_28_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 783 (all C all A all B all M (hBOOL(hAPP_int_bool(zcong(A,B),M)) -> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(plus_plus_int(A),C),hAPP_int_int(plus_plus_int(B),C)),M)))) # label(fact_646_zcong__shift) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 784 (all V_4 all B_47 all C_26 hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(number_number_of_int(V_4)),B_47)),hAPP_int_int(times_times_int(number_number_of_int(V_4)),C_26)) = hAPP_int_int(times_times_int(number_number_of_int(V_4)),hAPP_int_int(minus_minus_int(B_47),C_26))) # label(fact_621_right__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 785 (all M all N_1 all K_1 all L (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_1),L)) -> (hAPP_nat_nat(plus_plus_nat(M),L) = hAPP_nat_nat(plus_plus_nat(K_1),N_1) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1))))) # label(fact_981_less__add__eq__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 786 (all C_13 all D_3 all A_29 all B_27 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_29),B_27)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_13),D_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_29)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_13)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_29),C_13)),hAPP_nat_nat(times_times_nat(B_27),D_3)))))))) # label(fact_779_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 787 (all U all Ma all N all J_2 all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_2),I_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(I_2),J_2)),U)),Ma)),N)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(I_2),U)),Ma)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(J_2),U)),N)))))) # label(fact_1052_nat__le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 788 (all C_2 all D all A_2 all B_2 all Ma (hBOOL(hAPP_int_bool(zcong(A_2,B_2),Ma)) -> (hBOOL(hAPP_int_bool(zcong(C_2,hAPP_int_int(times_times_int(D),B_2)),Ma)) <-> hBOOL(hAPP_int_bool(zcong(C_2,hAPP_int_int(times_times_int(D),A_2)),Ma))))) # label(fact_645_zcong__zmult__prop2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 789 (all V_13 all W_12 number_number_of_int(hAPP_int_int(times_times_int(V_13),W_12)) = hAPP_int_int(times_times_int(number_number_of_int(V_13)),number_number_of_int(W_12))) # label(fact_241_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 790 (all B_4 all C_1 all A_4 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_4)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_4),C_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_4),hAPP_int_int(plus_plus_int(A_4),C_1)))))) # label(fact_858_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 791 (all C_12 all B_26 all A_28 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_26),A_28)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_12),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(C_12),A_28)),hAPP_int_int(times_times_int(C_12),B_26)))))) # label(fact_782_mult__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 792 (all J_1 all A all P_1 (hBOOL(hAPP_int_bool(zprime,P_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1)) -> (-hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_1)) -> (-hBOOL(hAPP_int_bool(zcong(J_1,zero_zero_int),P_1)) -> (-hBOOL(hAPP_int_bool(quadRes(P_1),A)) -> -hBOOL(hAPP_int_bool(zcong(J_1,hAPP_int_int(times_times_int(A),multInv(P_1,J_1))),P_1)))))))) # label(fact_1073_MultInvPair__distinct) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 793 (all Z_3 hAPP_int_int(plus_plus_int(Z_3),Z_3) = hAPP_int_int(times_times_int(Z_3),number_number_of_int(bit0(bit1(pls))))) # label(fact_283_mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 794 (all N_1 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M)) -> N_1 != M)) # label(fact_890_less__not__refl2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 795 (all B_7 all A_8 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_8)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_7),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(B_7),A_8)),zero_zero_real))))) # label(fact_840_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 796 (all Ma all N (is_int(N) & is_int(Ma) -> (hAPP_int_int(times_times_int(Ma),N) = one_one_int <-> number_number_of_int(min) = N & Ma = number_number_of_int(min) | Ma = one_one_int & N = one_one_int))) # label(fact_552_zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 797 (all A hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),zero_zero_nat))) # label(fact_1121_gcd__lcm__complete__lattice__nat_Otop__greatest) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 798 (all A_57 one_one_nat = hAPP_nat_nat(power_power_nat(A_57),zero_zero_nat)) # label(fact_545_power__0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 799 (all W_1 all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_1),Z_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(W_1),one_one_int)),Z_1)))) # label(fact_87_add1__zle__eq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 800 (all B_10 all A_11 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_11),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_10)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_11),B_10)),zero_zero_real))))) # label(fact_829_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 801 (all A_91 hAPP_nat_real(power_power_real(A_91),number_number_of_nat(bit1(bit1(pls)))) = hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(A_91),A_91)),A_91)) # label(fact_272_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 802 (all Lx_2 all Ly all Rx_2 hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_2),Ly)),Rx_2) = hAPP_int_int(times_times_int(Lx_2),hAPP_int_int(times_times_int(Ly),Rx_2))) # label(fact_103_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.60 803 (all A_92 (is_int(A_92) -> A_92 = hAPP_int_int(times_times_int(A_92),number_number_of_int(bit1(pls))))) # label(fact_258_mult__numeral__1__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 804 (all N_1 N_1 = hAPP_nat_nat(times_times_nat(N_1),one_one_nat)) # label(fact_1026_nat__mult__1__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 805 (all N_1 all K_1 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),hAPP_nat_nat(minus_minus_nat(M),N_1)))))) # label(fact_926_dvd__diff__nat) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 806 (all L_1 (is_int(L_1) -> (L_1 = min <-> min = bit1(L_1)))) # label(fact_519_rel__simps_I43_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 807 (all N_19 all A_71 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_71)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_71),one_one_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_71),hAPP_nat_int(power_power_int(A_71),N_19))),hAPP_nat_int(power_power_int(A_71),N_19)))))) # label(fact_406_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 808 (all V_4 all B_47 all C_26 hAPP_real_real(times_times_real(number267125858f_real(V_4)),hAPP_real_real(minus_minus_real(B_47),C_26)) = hAPP_real_real(minus_minus_real(hAPP_real_real(times_times_real(number267125858f_real(V_4)),B_47)),hAPP_real_real(times_times_real(number267125858f_real(V_4)),C_26))) # label(fact_620_right__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 809 (all X_14 all Y_11 (is_int(X_14) & is_int(Y_11) -> (hAPP_nat_int(power_power_int(Y_11),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_int(power_power_int(X_14),number_number_of_nat(bit0(bit1(pls)))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_14)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_11)) -> X_14 = Y_11))))) # label(fact_450_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 810 (all C_18 all D_6 all A_39 all B_36 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_39),B_36)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_18),D_6)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(times_times_nat(A_39),C_18)),hAPP_nat_nat(times_times_nat(B_36),D_6)))))) # label(fact_739_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 811 (all A_74 all N_23 all N_22 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_23),N_22)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_74)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_74),one_one_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(A_74),N_22)),hAPP_nat_nat(power_power_nat(A_74),N_23))))))) # label(fact_378_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 812 (all Z hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(one_one_int),Z)),Z) != zero_zero_int) # label(fact_405_odd__nonzero) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 813 (all A_2 all N (is_int(A_2) -> (zero_zero_int = hAPP_nat_int(power_power_int(A_2),N) <-> zero_zero_int = A_2 & N != zero_zero_nat))) # label(fact_311_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 814 (all C all A all B (B = A -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B),C)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),C))))) # label(fact_915_dvd_Oord__eq__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 815 (all M all N_1 M = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(M),N_1)),N_1)) # label(fact_932_diff__add__inverse2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 816 (all A_2 all Ma (hBOOL(hAPP_int_bool(zcong(A_2,zero_zero_int),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),A_2)))) # label(fact_656_zcong__zero__equiv__div) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 817 (all N_6 all A_56 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_56)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_6)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_nat_real(power_power_real(A_56),N_6)))))) # label(fact_553_one__less__power) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 818 (all A_93 (is_int(A_93) -> A_93 = hAPP_int_int(times_times_int(number_number_of_int(bit1(pls))),A_93))) # label(fact_256_mult__numeral__1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 819 (all X_2 all N all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),hAPP_int_int(minus_minus_int(X_2),Y_2))) <-> hAPP_int_int(div_mod_int(X_2),N) = hAPP_int_int(div_mod_int(Y_2),N))) # label(fact_1142_zmod__eq__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 820 (all X_10 one_one_real = hAPP_nat_real(power_power_real(X_10),zero_zero_nat)) # label(fact_541_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 821 (all X_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(X_2),N))) <-> N = zero_zero_nat | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X_2)))) # label(fact_515_zero__less__power__nat__eq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 822 (all N_15 one_one_real = hAPP_nat_real(power_power_real(one_one_real),N_15)) # label(fact_466_power__one) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 823 (all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B)) -> (B != A -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B),A))))) # label(fact_916_dvd_Ole__neq__trans) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 824 (all P_1 all M ((zero_zero_nat != M -> hAPP_nat_nat(power_power_nat(P_1),M) = hAPP_nat_nat(times_times_nat(P_1),hAPP_nat_nat(power_power_nat(P_1),hAPP_nat_nat(minus_minus_nat(M),one_one_nat)))) & (M = zero_zero_nat -> hAPP_nat_nat(power_power_nat(P_1),M) = one_one_nat))) # label(fact_696_power__eq__if) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 825 (all A_70 hAPP_nat_real(power_power_real(A_70),one_one_nat) = A_70) # label(fact_424_power__one__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 826 (all X_2 all Y_2 (is_int(Y_2) & is_int(X_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(X_2),X_2)),hAPP_int_int(times_times_int(Y_2),Y_2))),zero_zero_int)) <-> zero_zero_int = Y_2 & zero_zero_int = X_2))) # label(fact_413_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 827 (all A_2 all N (N != zero_zero_nat & A_2 = zero_zero_nat <-> zero_zero_nat = hAPP_nat_nat(power_power_nat(A_2),N))) # label(fact_312_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 828 (all I_1 all J_1 all K_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(I_1),J_1)),K_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),K_1)))) # label(fact_982_add__lessD1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 829 (all A_43 all C_21 all B_40 all D_7 hAPP_real_real(plus_plus_real(hAPP_real_real(minus_minus_real(A_43),B_40)),hAPP_real_real(minus_minus_real(C_21),D_7)) = hAPP_real_real(minus_minus_real(hAPP_real_real(plus_plus_real(A_43),C_21)),hAPP_real_real(plus_plus_real(B_40),D_7))) # label(fact_727_add__diff__add) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 830 (all B_1_1 (is_int(B_1_1) -> is_int(number_number_of_int(B_1_1)))) # label(gsy_c_Int_Onumber__class_Onumber__of_000tc__Int__Oint) # label(hypothesis) # label(non_clause). [assumption]. 1.32/1.61 831 (all V_14 all W_13 all Z_8 hAPP_real_real(times_times_real(number267125858f_real(hAPP_int_int(times_times_int(V_14),W_13))),Z_8) = hAPP_real_real(times_times_real(number267125858f_real(V_14)),hAPP_real_real(times_times_real(number267125858f_real(W_13)),Z_8))) # label(fact_240_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 832 (all X_14 all Y_11 (hAPP_nat_real(power_power_real(X_14),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_real(power_power_real(Y_11),number_number_of_nat(bit0(bit1(pls)))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_14)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_11)) -> Y_11 = X_14)))) # label(fact_448_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 833 (all A_85 all N_32 all B_54 (hAPP_nat_real(power_power_real(B_54),N_32) = hAPP_nat_real(power_power_real(A_85),N_32) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_85)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_54)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32)) -> B_54 = A_85))))) # label(fact_325_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 834 (all C_16 all A_37 all B_34 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_37),B_34)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_37),hAPP_real_real(times_times_real(B_34),C_16))))) # label(fact_744_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 835 (all W_6 hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(one_one_int),one_one_int)),number_number_of_int(W_6)) = number_number_of_int(bit0(W_6))) # label(fact_269_double__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 836 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(K)),pls)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),pls)))) # label(fact_151_rel__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 837 (all A_95 A_95 = hAPP_real_real(plus_plus_real(number267125858f_real(pls)),A_95)) # label(fact_236_add__numeral__0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 838 (all M all K_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(plus_plus_nat(M),K_1)),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)))) # label(fact_1004_add__leD1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 839 (all A_2 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_2),zero_zero_int)) & 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),B_2)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(times_times_int(A_2),B_2))))) # label(fact_811_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 840 (all A_61 all N_11 all N_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_11),N_10)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),A_61)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_61),N_11)),hAPP_nat_real(power_power_real(A_61),N_10)))))) # label(fact_488_power__increasing) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 841 (all L bit1(L) != pls) # label(fact_195_rel__simps_I39_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 842 (all X_1 all P_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),P_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,standardRes(P_1,X_1)),P_1)))) # label(fact_1192_StandardRes__ubound) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 843 (all K_1 all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),hAPP_nat_nat(minus_minus_nat(M),N_1))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N_1)))))) # label(fact_993_dvd__diffD1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 844 (all X_2 all Y_2 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_eq_real,X_2),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(X_2),Z_1)),hAPP_real_real(times_times_real(Y_2),Z_1)))))) # label(fact_862_real__mult__le__cancel__iff1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 845 (all X_22 all P_6 all Q_6 hAPP_nat_int(power_power_int(X_22),hAPP_nat_nat(plus_plus_nat(P_6),Q_6)) = hAPP_int_int(times_times_int(hAPP_nat_int(power_power_int(X_22),P_6)),hAPP_nat_int(power_power_int(X_22),Q_6))) # label(fact_56_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 846 (all B zero_zero_int = hAPP_int_int(div_mod_int(zero_zero_int),B)) # label(fact_1152_zmod__zero) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 847 (all Ma all K all N (Ma = N <-> hAPP_nat_nat(plus_plus_nat(Ma),K) = hAPP_nat_nat(plus_plus_nat(N),K))) # label(fact_931_nat__add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 848 (all C_2 all X_2 all Ta all A_2 all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_2),D)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_2),hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(X_2),hAPP_int_int(times_times_int(C_2),D))),Ta))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_2),hAPP_int_int(plus_plus_int(X_2),Ta)))))) # label(fact_373_zdvd__period) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 849 (all M all N_1 all K_1 hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(M),N_1)),K_1) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(times_times_nat(M),K_1)),hAPP_nat_nat(times_times_nat(N_1),K_1))) # label(fact_951_diff__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 850 (all X_1 all Y_1 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) -> -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1))))) # label(fact_907_dvd_Oless__not__sym) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 851 (all X_1 all P_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP_int_bool(zprime,P_1)) -> (-hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),P_1)) -> hBOOL(hAPP_int_bool(zcong(multInv(P_1,multInv(P_1,X_1)),hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(X_1),multInv(P_1,X_1))),multInv(P_1,multInv(P_1,X_1)))),P_1)))))) # label(fact_1093_Int2_Oaux____1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 852 (all Z1 all Z2 all W hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(Z1),W)),hAPP_int_int(times_times_int(Z2),W)) = hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(Z1),Z2)),W)) # label(fact_210_zadd__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 853 (all A_101 all B_58 all C_29 hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_101),C_29)),hAPP_real_real(times_times_real(B_58),C_29)) = hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_101),B_58)),C_29)) # label(fact_178_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 854 (all A_5 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_5),A_5)),zero_zero_int))) # label(fact_856_not__square__less__zero) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 855 (all B all A all C (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),C)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),C)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B))))) # label(fact_572_Euler_Oaux2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 856 (all K_1 all I_1 all J_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I_1),J_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,J_1),K_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I_1),K_1))))) # label(fact_882_real__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 857 (all M all N_1 (is_int(M) -> (hAPP_int_int(times_times_int(M),N_1) = one_one_int -> M = one_one_int | number_number_of_int(min) = M))) # label(fact_551_pos__zmult__eq__1__iff__lemma) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 858 (all A_35 all B_32 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_35),hAPP_real_real(times_times_real(A_35),B_32)))) # label(fact_750_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 859 (all X_2 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(X_2),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_2),number_number_of_nat(bit0(bit1(pls))))))) <-> Y_2 != zero_zero_real | zero_zero_real != X_2)) # label(fact_481_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 860 (all N all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma)) -> (one_one_nat = N <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(times_times_nat(N),Ma)),Ma))))) # label(fact_874_dvd__mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 861 (all X_2 all Y_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)) <-> X_2 = Y_2 | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)))) # label(fact_923_dvd_Ole__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 862 (all V_5 all W_3 number267125858f_real(hAPP_int_int(minus_minus_int(V_5),W_3)) = hAPP_real_real(minus_minus_real(number267125858f_real(V_5)),number267125858f_real(W_3))) # label(fact_612_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 863 (all Lx_1 all Rx_1 all Ry_1 hAPP_real_real(times_times_real(Lx_1),hAPP_real_real(times_times_real(Rx_1),Ry_1)) = hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_1),Rx_1)),Ry_1)) # label(fact_108_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 864 (all A_2 all P_3 (hBOOL(hAPP_int_bool(zcong(A_2,hAPP_int_int(minus_minus_int(P_3),one_one_int)),P_3)) <-> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A_2),hAPP_int_int(minus_minus_int(P_3),one_one_int)),one_one_int),P_3)))) # label(fact_637_inv__not__p__minus__1__aux) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 865 (all X_1 all Y_1 all Z (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Z)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Y_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(Z),X_1)),hAPP_real_real(times_times_real(Z),Y_1)))))) # label(fact_865_real__mult__less__mono2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 866 (all A_102 all M_13 all B_59 hAPP_real_real(times_times_real(hAPP_real_real(plus_plus_real(A_102),B_59)),M_13) = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(A_102),M_13)),hAPP_real_real(times_times_real(B_59),M_13))) # label(fact_175_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 867 (all B_43 all A_49 (is_int(A_49) & is_int(B_43) -> (zero_zero_int != A_49 -> (B_43 != zero_zero_int -> hAPP_int_int(times_times_int(A_49),B_43) != zero_zero_int)))) # label(fact_703_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 868 (all C_3 all A_13 all B_12 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_13),B_12)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_13),C_3)),hAPP_int_int(times_times_int(B_12),C_3)))))) # label(fact_826_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 869 (all B_7 all A_8 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_8)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_7),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(B_7),A_8)),zero_zero_int))))) # label(fact_842_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 870 (all X_1 all Y_1 all Z hAPP_nat_nat(plus_plus_nat(Y_1),hAPP_nat_nat(plus_plus_nat(X_1),Z)) = hAPP_nat_nat(plus_plus_nat(X_1),hAPP_nat_nat(plus_plus_nat(Y_1),Z))) # label(fact_928_nat__add__left__commute) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 871 (all X_1 all Y_1 (Y_1 = X_1 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)))) # label(fact_919_dvd_Oeq__refl) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 872 (all Z all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),W)))) # label(fact_36_zle__linear) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 873 (all A_47 zero_zero_real = hAPP_real_real(times_times_real(zero_zero_real),A_47)) # label(fact_709_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 874 (all P_2 all Ma all N ((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_2,N),Ma))) -> ((N = Ma -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_2,N),Ma))) -> ((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),Ma)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_2,N),Ma))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_2,N),Ma)))))) # label(fact_888_nat__less__cases) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 875 (all N_4 one_one_int = hAPP_nat_int(power_power_int(number_number_of_int(min)),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_4))) # label(fact_581_power__m1__even) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 876 (all M 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(dvd_dvd_nat,M),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))))) # label(fact_1128_dvd__pos__nat) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 877 (all W all Z1 all Z2 hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(W),Z1)),hAPP_int_int(times_times_int(W),Z2)) = hAPP_int_int(times_times_int(W),hAPP_int_int(minus_minus_int(Z1),Z2))) # label(fact_616_zdiff__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 878 (all N_1 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat))) # label(fact_878_less__zeroE) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 879 (all X_1 all Y_1 hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(X_1),Y_1)),hAPP_nat_nat(minus_minus_nat(X_1),Y_1)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(power_power_nat(X_1),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_nat(power_power_nat(Y_1),number_number_of_nat(bit0(bit1(pls)))))) # label(fact_697_diff__square) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 880 (all M all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(div_mod_nat(M),N_1)),M))) # label(fact_1188_mod__less__eq__dividend) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 881 (all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),pls)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(X_2)),zero_zero_real)))) # label(fact_432_less__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 882 (all A_97 all M_12 hAPP_int_int(times_times_int(hAPP_int_int(plus_plus_int(A_97),one_one_int)),M_12) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A_97),M_12)),M_12)) # label(fact_226_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 883 (all A_42 all B_39 all C_20 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(times_times_real(A_42),B_39)),C_20)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,B_39),C_20)))) # label(fact_729_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 884 (all K_1 all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),hAPP_nat_nat(minus_minus_nat(M),N_1))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),M)))))) # label(fact_994_dvd__diffD) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 885 (all K_1 bit0(K_1) != min) # label(fact_523_rel__simps_I45_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 886 (all A_53 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_53),zero_zero_nat))) # label(fact_673_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 887 (all V_7 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),number_number_of_nat(V_7))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V_7)))) # label(fact_559_less__0__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 888 (all Z all W (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z),W)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),Z)) -> Z = W))) # label(fact_883_real__le__antisym) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 889 (all C all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(times_times_nat(A),C)),hAPP_nat_nat(times_times_nat(B),C))))) # label(fact_765_divides__mul__r) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 890 (all A_40 all B_37 all K_3 (A_40 = hAPP_real_real(times_times_real(B_37),K_3) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,B_37),A_40)))) # label(fact_735_dvdI) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 891 (all A_36 all B_33 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_36),hAPP_nat_nat(times_times_nat(B_33),A_36)))) # label(fact_748_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 892 (all K_1 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_1)),M)))) # label(fact_1140_zmod__le__nonneg__dividend) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 893 (all A_48 hAPP_int_int(times_times_int(A_48),zero_zero_int) = zero_zero_int) # label(fact_708_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 894 (all L min != bit0(L)) # label(fact_524_rel__simps_I42_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 895 (all N_1 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),zero_zero_nat))) # label(fact_955_less__nat__zero__code) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 896 (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]. 1.32/1.61 897 (all X_2 all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),Y_2)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(X_2)),number_number_of_int(Y_2))))) # label(fact_53_le__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 898 (all A_2 all B_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_2),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_2)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_2),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_2),B_2)),zero_zero_real)))) # label(fact_808_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 899 (all A_57 one_one_int = hAPP_nat_int(power_power_int(A_57),zero_zero_nat)) # label(fact_546_power__0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 900 (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]. 1.32/1.61 901 (all A_106 all C_32 all D_10 hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_106),C_32)),D_10) = hAPP_nat_nat(plus_plus_nat(A_106),hAPP_nat_nat(plus_plus_nat(C_32),D_10))) # label(fact_127_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 902 (all C_10 all A_26 all B_24 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_26),B_24)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_10)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(C_10),A_26)),hAPP_real_real(times_times_real(C_10),B_24)))))) # label(fact_785_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 903 (all X_1 all Y_1 all Z hAPP_nat_int(power_power_int(X_1),hAPP_nat_nat(plus_plus_nat(Y_1),Z)) = hAPP_int_int(times_times_int(hAPP_nat_int(power_power_int(X_1),Y_1)),hAPP_nat_int(power_power_int(X_1),Z))) # label(fact_59_zpower__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 904 (all X_1 all M hBOOL(hAPP_int_bool(zcong(X_1,hAPP_int_int(div_mod_int(X_1),M)),M))) # label(fact_1144_mod__mod__is__mod) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 905 (all Z all W (is_int(Z) & is_int(W) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),W)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),Z)) -> W = Z)))) # label(fact_40_zle__antisym) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 906 (all X_2 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(minus_minus_real(X_2),Y_2)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2)))) # label(fact_675_real__le__eq__diff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 907 (all A_63 all N_13 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(power_power_real(A_63),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_13))))) # label(fact_483_zero__le__even__power_H) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 908 (all X_2 all Y_2 all B_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),B_2)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(B_2),X_2)),hAPP_nat_real(power_power_real(B_2),Y_2))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_2),Y_2))))) # label(fact_305_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 909 (all B_30 all A_32 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_32),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_30)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_30),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_32)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_32),B_30)),zero_zero_int)))) # label(fact_772_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 910 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),hAPP_int_int(minus_minus_int(B),one_one_int))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)))) # label(fact_1069_norR__mem__unique__aux) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 911 (all X_2 all W_1 (zero_zero_nat = number_number_of_nat(W_1) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X_2)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(X_2),number_number_of_nat(W_1)))))) # label(fact_516_zero__less__power__nat__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 912 (all N_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(N_1))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(bit0(N_1)))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(bit1(N_1)))))) # label(fact_1059_number__of1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 913 (all X_1 all Y_1 all M hAPP_int_int(div_mod_int(hAPP_int_int(minus_minus_int(X_1),hAPP_int_int(div_mod_int(Y_1),M))),M) = hAPP_int_int(div_mod_int(hAPP_int_int(minus_minus_int(X_1),Y_1)),M)) # label(fact_1153_zdiff__zmod__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 914 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)))) # label(fact_987_less__imp__le__nat) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 915 (all X_2 all Y_2 hAPP_nat_int(power_power_int(hAPP_int_int(plus_plus_int(X_2),Y_2)),number_number_of_nat(bit0(bit1(pls)))) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X_2),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_2),number_number_of_nat(bit0(bit1(pls)))))),hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),X_2)),Y_2))) # label(fact_9_power2__sum) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 916 (all N_28 all A_83 all B_53 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_83),B_53)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_83)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_28)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_83),N_28)),hAPP_nat_int(power_power_int(B_53),N_28))))))) # label(fact_338_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 917 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N_1)) -> N_1 = zero_zero_nat | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)))) # label(fact_1131_divides__le) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 918 (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,bit0(K1)),bit1(K2))))) # label(fact_84_less__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 919 (all Q_1 all B all R_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,zero_zero_int),R_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),hAPP_int_int(div_mod_int(Q_1),C))),R_1))))))) # label(fact_1167_zmult2__lemma__aux3) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 920 (all A all P_1 (hBOOL(hAPP_int_bool(zprime,P_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(minus_minus_int(P_1),one_one_int))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),inv(P_1,A))))))) # label(fact_1079_inv__g__1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 921 (all P all Q all R hAPP_bool_bool(hAPP_i68813070l_bool(P,R),hAPP_int_bool(Q,R)) = hAPP_int_bool(cOMBS_int_bool_bool(P,Q),R)) # label(help_COMBS_1_1_COMBS_000tc__Int__Oint_000tc__HOL__Obool_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 922 (all B_18 all A_20 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_20)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_18),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_20),B_18)),zero_zero_real))))) # label(fact_802_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 923 (all A_2 all B_2 all N (zero_zero_nat != N -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(A_2),N)),hAPP_nat_nat(power_power_nat(B_2),N))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),B_2))))) # label(fact_1129_pow__divides__eq__nat) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 924 (all A_79 (is_int(A_79) -> hAPP_int_int(plus_plus_int(A_79),zero_zero_int) = A_79)) # label(fact_350_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 925 (all M_9 all N_36 all A_88 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_88)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_88),M_9)),hAPP_nat_real(power_power_real(A_88),N_36))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_9),N_36))))) # label(fact_302_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 926 (all N_29 ((N_29 != zero_zero_nat -> hAPP_nat_real(power_power_real(zero_zero_real),N_29) = zero_zero_real) & (N_29 = zero_zero_nat -> one_one_real = hAPP_nat_real(power_power_real(zero_zero_real),N_29)))) # label(fact_336_power__0__left) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 927 (all C all A all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B)) -> (C = B -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),C))))) # label(fact_914_dvd_Oord__le__eq__trans) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 928 (all V_20 all V_19 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_19)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_20)) -> number267125858f_real(hAPP_int_int(times_times_int(V_19),V_20)) = hAPP_real_real(times_times_real(number267125858f_real(V_19)),number267125858f_real(V_20))))) # label(fact_215_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 929 (all X1 all X2 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (standardRes(Ma,X1) = standardRes(Ma,X2) <-> hBOOL(hAPP_int_bool(zcong(X1,X2),Ma))))) # label(fact_1197_StandardRes__prop2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 930 (all A_55 all B_46 all V_3 hAPP_real_real(minus_minus_real(hAPP_real_real(times_times_real(A_55),number267125858f_real(V_3))),hAPP_real_real(times_times_real(B_46),number267125858f_real(V_3))) = hAPP_real_real(times_times_real(hAPP_real_real(minus_minus_real(A_55),B_46)),number267125858f_real(V_3))) # label(fact_622_left__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 931 (all Lx_5 all Ly_3 all Rx_5 all Ry_3 hAPP_real_real(times_times_real(Rx_5),hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_5),Ly_3)),Ry_3)) = hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(Lx_5),Ly_3)),hAPP_real_real(times_times_real(Rx_5),Ry_3))) # label(fact_96_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 932 (all B_48 all Q_4 all R_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_48),Q_4)),R_3)),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_48)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_4),zero_zero_int)))))) # label(fact_601_q__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 933 (all J_1 all A all K_1 all P_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP_int_bool(zprime,P_1)) -> (-hBOOL(hAPP_int_bool(zcong(K_1,zero_zero_int),P_1)) -> (hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(J_1),K_1),hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(A),multInv(P_1,K_1))),K_1)),P_1)) -> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(J_1),K_1),A),P_1))))))) # label(fact_1089_aux______2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 934 (all X_1 all Y_1 hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(X_1),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_1),number_number_of_nat(bit0(bit1(pls)))))),hAPP_real_real(times_times_real(hAPP_real_real(times_times_real(number267125858f_real(bit0(bit1(pls)))),X_1)),Y_1)) = hAPP_nat_real(power_power_real(hAPP_real_real(plus_plus_real(X_1),Y_1)),number_number_of_nat(bit0(bit1(pls))))) # label(fact_292_real__sum__squared__expand) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 935 (all M_10 hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(one_one_nat),one_one_nat)),M_10) = hAPP_nat_nat(plus_plus_nat(M_10),M_10)) # label(fact_233_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 936 (all W pls = hAPP_int_int(times_times_int(pls),W)) # label(fact_201_mult__Pls) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 937 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit0(K)),min)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),min)))) # label(fact_538_rel__simps_I28_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 938 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),hAPP_int_int(div_mod_int(A),B))))) # label(fact_1137_neg__mod__bound) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 939 (all K_1 all M all N_1 hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(times_times_nat(K_1),M)),hAPP_nat_nat(times_times_nat(K_1),N_1)) = hAPP_nat_nat(times_times_nat(K_1),hAPP_nat_nat(div_mod_nat(M),N_1))) # label(fact_1186_mod__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 940 (all C_8 all A_24 all B_22 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_24),B_22)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_8)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_24),C_8)),hAPP_int_int(times_times_int(B_22),C_8)))))) # label(fact_793_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 941 (all X_2 all Y_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,number_number_of_int(X_2)),number_number_of_int(Y_2))) <-> zero_zero_int = hAPP_int_int(div_mod_int(number_number_of_int(Y_2)),number_number_of_int(X_2)))) # label(fact_1136_zdvd__iff__zmod__eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 942 (all A_106 all C_32 all D_10 hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(A_106),C_32)),D_10) = hAPP_real_real(plus_plus_real(A_106),hAPP_real_real(plus_plus_real(C_32),D_10))) # label(fact_128_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 943 (all B_21 all A_23 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_23),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_21),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(times_times_int(A_23),B_21)))))) # label(fact_795_mult__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 944 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),hAPP_int_int(div_mod_int(A),B))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(div_mod_int(A),B)),zero_zero_int)))) # label(fact_1162_neg__mod__conj) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 945 (all A_108 all B_61 all C_34 hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_108),B_61)),C_34) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_108),C_34)),B_61)) # label(fact_121_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 946 (all X_2 all Y_2 (zero_zero_real = hAPP_real_real(plus_plus_real(hAPP_nat_real(power_power_real(X_2),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_2),number_number_of_nat(bit0(bit1(pls))))) <-> Y_2 = zero_zero_real & X_2 = zero_zero_real)) # label(fact_455_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 947 (all A_103 all N_39 hAPP_nat_int(power_power_int(hAPP_nat_int(power_power_int(A_103),N_39)),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_int(power_power_int(A_103),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),N_39))) # label(fact_159_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 948 (all X_2 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_2),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_2),Y_2)) & Y_2 != X_2)) # label(fact_686_real__less__def) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 949 (all N 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),N)) -> (hAPP_nat_nat(minus_minus_nat(Ma),K) = hAPP_nat_nat(minus_minus_nat(N),K) <-> Ma = N)))) # label(fact_946_eq__diff__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 950 (all A_2 all B_2 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),hAPP_int_int(minus_minus_int(A_2),B_2))) <-> hBOOL(hAPP_int_bool(zcong(A_2,B_2),Ma)))) # label(fact_629_zcong__def) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 951 (all N_1 hAPP_nat_nat(plus_plus_nat(zero_zero_nat),N_1) = N_1) # label(fact_963_plus__nat_Oadd__0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 952 (all V_6 all V ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_6),pls)) -> hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),number_number_of_nat(V_6)) = number_number_of_nat(V)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_6),pls)) -> hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),number_number_of_nat(V_6)) = number_number_of_nat(hAPP_int_int(plus_plus_int(V),V_6)))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls)) -> number_number_of_nat(V_6) = hAPP_nat_nat(plus_plus_nat(number_number_of_nat(V)),number_number_of_nat(V_6))))) # label(fact_77_add__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 953 (all K_1 all M hBOOL(hAPP_int_bool(zcong(K_1,K_1),M))) # label(fact_563_zcong__refl) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 954 (all V_20 all V_19 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_19)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_20)) -> hAPP_int_int(times_times_int(number_number_of_int(V_19)),number_number_of_int(V_20)) = number_number_of_int(hAPP_int_int(times_times_int(V_19),V_20))))) # label(fact_213_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 955 (all C all D_5 all A all B all M (hBOOL(hAPP_int_bool(zcong(A,B),M)) -> (hBOOL(hAPP_int_bool(zcong(C,D_5),M)) -> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(plus_plus_int(A),C),hAPP_int_int(plus_plus_int(B),D_5)),M))))) # label(fact_579_zcong__zadd) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 956 (all A all P_1 (hBOOL(hAPP_int_bool(zprime,P_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),hAPP_int_int(minus_minus_int(P_1),one_one_int))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,inv(P_1,A)),hAPP_int_int(minus_minus_int(P_1),one_one_int))))))) # label(fact_1080_inv__less__p__minus__1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 957 (all N_6 all A_56 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_56)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_6)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_nat_int(power_power_int(A_56),N_6)))))) # label(fact_555_one__less__power) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 958 (all I_1 all J_1 all K_1 hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(I_1),J_1)),K_1) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(I_1),K_1)),J_1)) # label(fact_879_diff__commute) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 959 (all A_35 all B_32 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_35),hAPP_int_int(times_times_int(A_35),B_32)))) # label(fact_752_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 960 (all A_107 all B_60 all C_33 hAPP_nat_nat(plus_plus_nat(A_107),hAPP_nat_nat(plus_plus_nat(B_60),C_33)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(A_107),B_60)),C_33)) # label(fact_124_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 961 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit0(K)),pls)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),pls)))) # label(fact_156_rel__simps_I27_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 962 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N_1),M)) -> N_1 = M))) # label(fact_913_dvd__antisym) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 963 (all A_111 all B_63 hAPP_nat_nat(times_times_nat(A_111),B_63) = hAPP_nat_nat(times_times_nat(B_63),A_111)) # label(fact_113_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 964 (all N_1 all M ((M = zero_zero_nat -> hAPP_nat_nat(times_times_nat(M),N_1) = zero_zero_nat) & (zero_zero_nat != M -> hAPP_nat_nat(times_times_nat(M),N_1) = hAPP_nat_nat(plus_plus_nat(N_1),hAPP_nat_nat(times_times_nat(hAPP_nat_nat(minus_minus_nat(M),one_one_nat)),N_1))))) # label(fact_695_mult__eq__if) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 965 (all Lx_1 all Rx_1 all Ry_1 hAPP_nat_nat(times_times_nat(Lx_1),hAPP_nat_nat(times_times_nat(Rx_1),Ry_1)) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(Lx_1),Rx_1)),Ry_1)) # label(fact_107_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 966 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K)),min)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),min)))) # label(fact_528_rel__simps_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 967 (all M all N_1 all K_1 hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(plus_plus_nat(M),N_1)),K_1) = hAPP_nat_nat(plus_plus_nat(M),hAPP_nat_nat(plus_plus_nat(N_1),K_1))) # label(fact_929_nat__add__assoc) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 968 (all S all T (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S),T)) -> S != T)) # label(fact_889_less__not__refl3) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 969 (all X_2 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(X_2),X_2)),hAPP_real_real(times_times_real(Y_2),Y_2)))) <-> zero_zero_real != X_2 | Y_2 != zero_zero_real)) # label(fact_419_sum__squares__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 970 (all B all A all P_1 (hBOOL(hAPP_int_bool(zprime,P_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A)) -> (-hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_1)) & -hBOOL(hAPP_int_bool(zcong(B,zero_zero_int),P_1)) -> -hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A),B),zero_zero_int),P_1)))))) # label(fact_663_zcong__zprime__prod__zero__contra) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 971 (all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) <-> (exists K_2 N = hAPP_nat_nat(plus_plus_nat(Ma),K_2)))) # label(fact_997_le__iff__add) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 972 (all M all N_1 hAPP_nat_nat(plus_plus_nat(M),N_1) = hAPP_nat_nat(plus_plus_nat(N_1),M)) # label(fact_927_nat__add__commute) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 973 (all X_10 one_one_nat = hAPP_nat_nat(power_power_nat(X_10),zero_zero_nat)) # label(fact_542_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 974 (all B all A all P_1 (hBOOL(hAPP_int_bool(zprime,P_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A)) -> (hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A),B),zero_zero_int),P_1)) -> hBOOL(hAPP_int_bool(zcong(B,zero_zero_int),P_1)) | hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_1)))))) # label(fact_662_zcong__zprime__prod__zero) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 975 (all X_4 all N_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_3)) -> hAPP_nat_nat(times_times_nat(hAPP_nat_nat(power_power_nat(X_4),hAPP_nat_nat(minus_minus_nat(N_3),one_one_nat))),X_4) = hAPP_nat_nat(power_power_nat(X_4),N_3))) # label(fact_692_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 976 (all W_7 number_number_of_int(bit1(W_7)) = hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(one_one_int),number_number_of_int(W_7))),number_number_of_int(W_7))) # label(fact_254_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 977 (all M hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(times_times_nat(M),hAPP_nat_nat(times_times_nat(M),M))))) # label(fact_1023_le__cube) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 978 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(div_mod_nat(M),N_1)),N_1)))) # label(fact_1177_mod__le__divisor) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 979 (all A_58 all K_4 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_58),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),K_4))),zero_zero_real)) -> A_58 = zero_zero_real)) # label(fact_506_even__power__le__0__imp__0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 980 (all C_5 all A_15 all B_14 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_15),B_14)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_5)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(C_5),A_15)),hAPP_int_int(times_times_int(C_5),B_14)))))) # label(fact_820_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 981 (all L all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(M),L)),hAPP_nat_nat(minus_minus_nat(N_1),L))))) # label(fact_944_diff__le__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 982 (all M all K_1 all N_1 hAPP_nat_nat(minus_minus_nat(M),N_1) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(plus_plus_nat(M),K_1)),hAPP_nat_nat(plus_plus_nat(N_1),K_1))) # label(fact_936_diff__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 983 (all A all P_1 ((hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_1)) -> legendre(A,P_1) = zero_zero_int) & (-hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_1)) -> (-hBOOL(hAPP_int_bool(quadRes(P_1),A)) -> legendre(A,P_1) = number_number_of_int(min)) & (hBOOL(hAPP_int_bool(quadRes(P_1),A)) -> legendre(A,P_1) = one_one_int)))) # label(fact_666_Legendre__def) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 984 (all Y_1 all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(X_1),Y_1)))))) # label(fact_587_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 985 (all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),bit1(pls))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(X_2)),one_one_real)))) # label(fact_167_le__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 986 (all X_24 all P_7 all Q_7 hAPP_nat_real(power_power_real(X_24),hAPP_nat_nat(times_times_nat(P_7),Q_7)) = hAPP_nat_real(power_power_real(hAPP_nat_real(power_power_real(X_24),P_7)),Q_7)) # label(fact_42_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 987 (all X_2 all Y_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)) & Y_2 != X_2 <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)))) # label(fact_922_dvd_Oless__le) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 988 (all K_1 all M all N_1 (is_int(K_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(times_times_int(K_1),M)),hAPP_int_int(times_times_int(K_1),N_1))) -> (K_1 != zero_zero_int -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),N_1)))))) # label(fact_328_zdvd__mult__cancel) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 989 (all X_23 X_23 = hAPP_nat_real(power_power_real(X_23),one_one_nat)) # label(fact_45_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 990 (all Z_3 hAPP_real_real(plus_plus_real(Z_3),Z_3) = hAPP_real_real(times_times_real(Z_3),number267125858f_real(bit0(bit1(pls))))) # label(fact_284_mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 991 (all M all X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),M)) -> -hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),M))))) # label(fact_653_zcong__not__zero) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 992 (all A_100 A_100 = hAPP_real_real(times_times_real(A_100),one_one_real)) # label(fact_187_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 993 (all A all B hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit1(bit1(pls))))),hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(number_number_of_int(bit1(bit1(pls)))),hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls)))))),B))),hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(number_number_of_int(bit1(bit1(pls)))),A)),hAPP_nat_int(power_power_int(B),number_number_of_nat(bit0(bit1(pls))))))),hAPP_nat_int(power_power_int(B),number_number_of_nat(bit1(bit1(pls))))) = hAPP_nat_int(power_power_int(hAPP_int_int(plus_plus_int(A),B)),number_number_of_nat(bit1(bit1(pls))))) # label(fact_8_zadd__power3) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 994 (all X_2 all Y_2 (is_int(Y_2) & is_int(X_2) -> (number_number_of_int(Y_2) = number_number_of_int(X_2) <-> X_2 = Y_2))) # label(fact_135_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 995 (all C_6 all B_15 all A_16 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_15),A_16)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_6),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(times_times_int(A_16),C_6)),hAPP_int_int(times_times_int(B_15),C_6)))))) # label(fact_817_mult__strict__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 996 (all A_85 all N_32 all B_54 (hAPP_nat_nat(power_power_nat(B_54),N_32) = hAPP_nat_nat(power_power_nat(A_85),N_32) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_85)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_54)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32)) -> A_85 = B_54))))) # label(fact_324_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 997 (all X_2 all Y_2 (zero_zero_real = X_2 & Y_2 = zero_zero_real <-> hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(X_2),X_2)),hAPP_real_real(times_times_real(Y_2),Y_2)) = zero_zero_real)) # label(fact_386_sum__squares__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 998 (all Ma all X_2 (hBOOL(hAPP_int_bool(zcong(X_2,zero_zero_int),Ma)) <-> standardRes(Ma,X_2) = zero_zero_int)) # label(fact_1189_StandardRes__eq__zcong) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 999 (all X_16 all Y_13 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(X_16),X_16)),hAPP_int_int(times_times_int(Y_13),Y_13))),zero_zero_int))) # label(fact_416_not__sum__squares__lt__zero) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 1000 (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]. 1.32/1.61 1001 (all A_94 A_94 = hAPP_real_real(plus_plus_real(A_94),number267125858f_real(pls))) # label(fact_238_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 1002 (all C_14 all D_4 all A_30 all B_28 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_30),B_28)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_14),D_4)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_28)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_14)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(A_30),C_14)),hAPP_nat_nat(times_times_nat(B_28),D_4)))))))) # label(fact_776_mult__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 1003 (all X_15 all Y_12 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(X_15),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_real(power_power_real(Y_12),number_number_of_nat(bit0(bit1(pls)))))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_12)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_15),Y_12))))) # label(fact_445_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 1004 (all R_1 all Q_1 all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A)) -> (hAPP_int_int(plus_plus_int(R_1),hAPP_int_int(times_times_int(A),Q_1)) = A -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),one_one_int)))))) # label(fact_509_self__quotient__aux2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 1005 (all N_1 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_nat_real(power_power_real(number267125858f_real(bit0(bit1(pls)))),N_1)))) # label(fact_871_two__realpow__ge__one) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 1006 (all A_112 hAPP_nat_real(power_power_real(A_112),number_number_of_nat(bit0(bit1(pls)))) = hAPP_real_real(times_times_real(A_112),A_112)) # label(fact_23_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 1007 (all N_38 all A_90 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_90)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_90),N_38)),hAPP_int_int(times_times_int(A_90),hAPP_nat_int(power_power_int(A_90),N_38)))))) # label(fact_294_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 1008 (all C_13 all D_3 all A_29 all B_27 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_29),B_27)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_13),D_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_29)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_13)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(times_times_int(A_29),C_13)),hAPP_int_int(times_times_int(B_27),D_3)))))))) # label(fact_780_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 1009 (all N_37 all A_89 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_89)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(times_times_real(A_89),hAPP_nat_real(power_power_real(A_89),N_37)))))) # label(fact_299_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 1010 (all A_64 all M_4 all N_14 hAPP_nat_nat(power_power_nat(hAPP_nat_nat(power_power_nat(A_64),M_4)),N_14) = hAPP_nat_nat(power_power_nat(A_64),hAPP_nat_nat(times_times_nat(M_4),N_14))) # label(fact_469_power__mult) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 1011 (all X_1 all P_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),P_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),standardRes(P_1,X_1))))) # label(fact_1196_StandardRes__lbound) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 1012 (all B all Q_4 all R_3 all Q_1 all R_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),Q_4)),R_3)),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),Q_1)),R_1))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_1),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),R_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),R_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q_4))))))) # label(fact_598_unique__quotient__lemma__neg) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 1013 (all V_15 all B_56 all C_28 hAPP_int_int(times_times_int(number_number_of_int(V_15)),hAPP_int_int(plus_plus_int(B_56),C_28)) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(V_15)),B_56)),hAPP_int_int(times_times_int(number_number_of_int(V_15)),C_28))) # label(fact_223_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 1014 (all A_2 (A_2 != zero_zero_nat <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_2),zero_zero_nat)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_2)))) # label(fact_1120_gcd__lcm__complete__lattice__nat_Oless__top) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 1015 (all C_8 all A_24 all B_22 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_24),B_22)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_8)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(times_times_real(A_24),C_8)),hAPP_real_real(times_times_real(B_22),C_8)))))) # label(fact_791_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 1016 (all K_1 all I_1 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),J_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(times_times_nat(I_1),K_1)),hAPP_nat_nat(times_times_nat(J_1),K_1)))))) # label(fact_1034_mult__less__mono1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 1017 (all Z_4 hAPP_nat_nat(plus_plus_nat(Z_4),Z_4) = hAPP_nat_nat(times_times_nat(Z_4),number_number_of_nat(bit0(bit1(pls))))) # label(fact_281_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 1018 (all B_6 all A_7 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_7)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_6),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(A_7),B_6)),zero_zero_real))))) # label(fact_843_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 1019 (all Z_10 all Z all W_14 all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_14),W)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_10),Z)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(W_14),Z_10)),hAPP_int_int(plus_plus_int(W),Z)))))) # label(fact_55_zadd__zless__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.61 1020 (all A_66 all B_49 all N_17 hAPP_nat_real(power_power_real(hAPP_real_real(times_times_real(A_66),B_49)),N_17) = hAPP_real_real(times_times_real(hAPP_nat_real(power_power_real(A_66),N_17)),hAPP_nat_real(power_power_real(B_49),N_17))) # label(fact_461_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1021 (all A all N_1 all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(A),N_1)),hAPP_nat_int(power_power_int(B),N_1))) -> (zero_zero_nat != N_1 -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B))))) # label(fact_1130_pow__divides__pow__int) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1022 (all A_82 hAPP_int_int(times_times_int(zero_zero_int),A_82) = zero_zero_int) # label(fact_341_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1023 (all P_2 all A_2 all B_2 ((all D_2 (A_2 = hAPP_nat_nat(plus_plus_nat(B_2),D_2) -> hBOOL(hAPP_nat_bool(P_2,D_2)))) & (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_2),B_2)) -> hBOOL(hAPP_nat_bool(P_2,zero_zero_nat))) <-> hBOOL(hAPP_nat_bool(P_2,hAPP_nat_nat(minus_minus_nat(A_2),B_2))))) # label(fact_1031_nat__diff__split) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1024 (all L bit1(hAPP_int_int(minus_minus_int(min),L)) = hAPP_int_int(minus_minus_int(pls),bit1(L))) # label(fact_634_diff__bin__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1025 (all P_5 all P_2 all X_2 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) -> (hBOOL(P_2) <-> hBOOL(P_5))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) & hBOOL(P_5) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_2)) & hBOOL(P_2)))) # label(fact_1062_conj__le__cong) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1026 (all C_7 all B_16 all A_17 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_16),A_17)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_7),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(times_times_real(C_7),A_17)),hAPP_real_real(times_times_real(C_7),B_16)))))) # label(fact_814_mult__strict__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1027 (all M all N_1 all K_1 hAPP_nat_nat(times_times_nat(hAPP_nat_nat(div_mod_nat(M),N_1)),K_1) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(times_times_nat(M),K_1)),hAPP_nat_nat(times_times_nat(N_1),K_1))) # label(fact_1185_mod__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1028 (all Z all X_1 all Y_1 all M (hBOOL(hAPP_int_bool(zcong(X_1,Y_1),M)) -> hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(X_1),Z),hAPP_nat_int(power_power_int(Y_1),Z)),M)))) # label(fact_647_zcong__zpower) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1029 (all N_1 hAPP_nat_nat(times_times_nat(one_one_nat),N_1) = N_1) # label(fact_1028_nat__mult__1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1030 (all Z all X_1 all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Y_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(plus_plus_real(Z),X_1)),hAPP_real_real(plus_plus_real(Z),Y_1))))) # label(fact_691_real__add__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1031 (all X_13 all Y_10 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(X_13),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_nat(power_power_nat(Y_10),number_number_of_nat(bit0(bit1(pls)))))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_10)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_13),Y_10))))) # label(fact_473_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1032 (all A_50 all B_44 (hAPP_nat_nat(times_times_nat(A_50),B_44) = zero_zero_nat -> zero_zero_nat = B_44 | zero_zero_nat = A_50)) # label(fact_699_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1033 (all A_41 all B_38 all C_19 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(times_times_nat(A_41),B_38)),C_19)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_41),C_19)))) # label(fact_733_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1034 (all B_2 all A_2 all P_3 (hBOOL(hAPP_int_bool(zprime,P_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),hAPP_int_int(minus_minus_int(P_3),one_one_int))) -> (hBOOL(member_int(B_2,wset(A_2,P_3))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),hAPP_int_int(minus_minus_int(P_3),one_one_int))))))) # label(fact_1100_wset__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1035 (all Ma all N (one_one_nat = N & Ma = one_one_nat <-> one_one_nat = hAPP_nat_nat(times_times_nat(Ma),N))) # label(fact_1027_nat__1__eq__mult__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1036 (all W_1 all Y_2 all X_2 all Z_1 (hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(W_1),Z_1)),hAPP_nat_nat(times_times_nat(X_2),Y_2)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(W_1),Y_2)),hAPP_nat_nat(times_times_nat(X_2),Z_1)) <-> Y_2 = Z_1 | W_1 = X_2)) # label(fact_171_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1037 (all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B)) -> hAPP_int_int(div_mod_int(number_number_of_int(min)),B) = hAPP_int_int(minus_minus_int(B),one_one_int))) # label(fact_1173_zmod__minus1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1038 (all A all B all C (is_int(A) -> (hAPP_int_int(minus_minus_int(A),B) = C -> hAPP_int_int(plus_plus_int(C),B) = A))) # label(fact_611_Int2_Oaux1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1039 (all K_1 all I_1 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(times_times_nat(K_1),I_1)),hAPP_nat_nat(times_times_nat(K_1),J_1))))) # label(fact_1021_mult__le__mono2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1040 (all B_2 all A_2 (is_int(B_2) & is_int(A_2) -> (zero_zero_int = A_2 <-> hAPP_int_int(plus_plus_int(B_2),A_2) = B_2))) # label(fact_353_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1041 (all M hBOOL(hAPP_int_bool(zcong(M,zero_zero_int),M))) # label(fact_643_zcong__id) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1042 (all X_26 hAPP_nat_real(power_power_real(X_26),number_number_of_nat(bit0(bit1(pls)))) = hAPP_real_real(times_times_real(X_26),X_26)) # label(fact_20_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1043 (all M all N_1 (hAPP_nat_nat(plus_plus_nat(M),N_1) = M -> N_1 = zero_zero_nat)) # label(fact_960_add__eq__self__zero) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1044 (all A_46 all E all B_42 all C_23 hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_46),E)),hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(B_42),E)),C_23)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_46),B_42)),E)),C_23)) # label(fact_719_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1045 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(M),N_1)),M))))) # label(fact_958_diff__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1046 (all X_2 all Y_2 (X_2 = Y_2 <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)))) # label(fact_687_divides__antisym) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1047 (all X_24 all P_7 all Q_7 hAPP_nat_int(power_power_int(hAPP_nat_int(power_power_int(X_24),P_7)),Q_7) = hAPP_nat_int(power_power_int(X_24),hAPP_nat_nat(times_times_nat(P_7),Q_7))) # label(fact_41_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1048 (all X_2 all Y_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(X_2)),number267125858f_real(Y_2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_2),Y_2)))) # label(fact_54_le__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1049 (all Ma all N all A_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_2)) -> (N = Ma <-> hAPP_nat_real(power_power_real(A_2),Ma) = hAPP_nat_real(power_power_real(A_2),N)))) # label(fact_491_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1050 (all Z all X_1 all Y_1 all P_1 (hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(X_1),Y_1),one_one_int),P_1)) -> hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(X_1),hAPP_nat_nat(times_times_nat(Y_1),Z)),one_one_int),P_1)))) # label(fact_371_zcong__zpower__zmult) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1051 (all V_9 all W_8 number_number_of_int(hAPP_int_int(plus_plus_int(V_9),W_8)) = hAPP_int_int(plus_plus_int(number_number_of_int(V_9)),number_number_of_int(W_8))) # label(fact_249_number__of__add) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1052 (all A_33 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A_33))) # label(fact_757_one__dvd) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1053 (all Ma all Ta all K (is_int(K) -> (zero_zero_int != K -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),Ta)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(times_times_int(K),Ma)),hAPP_int_int(times_times_int(K),Ta))))))) # label(fact_1063_zdvd__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1054 (all N_1 (zero_zero_nat != N_1 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)))) # label(fact_953_gr0I) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1055 (all A_40 all B_37 all K_3 (hAPP_int_int(times_times_int(B_37),K_3) = A_40 -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_37),A_40)))) # label(fact_737_dvdI) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1056 (all A zero_zero_int = hAPP_int_int(div_mod_int(A),A)) # label(fact_1151_zmod__self) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1057 (all X_1 all Y_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) | Y_1 = X_1)) # label(fact_917_dvd_Ole__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1058 (all N_26 all A_77 all B_52 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_77),B_52)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_77)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_77),N_26)),hAPP_nat_real(power_power_real(B_52),N_26)))))) # label(fact_367_power__mono) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1059 (all A_47 hAPP_int_int(times_times_int(zero_zero_int),A_47) = zero_zero_int) # label(fact_711_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1060 (all X_8 all Y_7 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_8),Y_7)) -> (X_8 != Y_7 -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_8),Y_7))))) # label(fact_568_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1061 (all N 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),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(Ma),K)),hAPP_nat_nat(minus_minus_nat(N),K))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)))))) # label(fact_990_less__diff__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1062 (all A_105 all C_31 all D_9 hAPP_int_int(plus_plus_int(A_105),hAPP_int_int(plus_plus_int(C_31),D_9)) = hAPP_int_int(plus_plus_int(C_31),hAPP_int_int(plus_plus_int(A_105),D_9))) # label(fact_129_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1063 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M)))) # label(fact_938_nat__le__linear) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1064 (all Q_1 all B all R_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,zero_zero_int),R_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),hAPP_int_int(div_mod_int(Q_1),C))),R_1)),hAPP_int_int(times_times_int(B),C))))))) # label(fact_1168_zmult2__lemma__aux4) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1065 (all M all N_1 all K_1 hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(M),K_1)),hAPP_nat_nat(times_times_nat(N_1),K_1)) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(M),N_1)),K_1)) # label(fact_1017_add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1066 (all A_100 A_100 = hAPP_nat_nat(times_times_nat(A_100),one_one_nat)) # label(fact_186_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1067 (all Z_1 all W_1 (is_int(Z_1) & is_int(W_1) -> (W_1 != Z_1 & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),W_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_1),W_1))))) # label(fact_37_zless__le) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1068 (all X_2 all Y_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) <-> X_2 = Y_2)) # label(fact_924_dvd_Oeq__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1069 (all M hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(times_times_nat(M),M)))) # label(fact_1024_le__square) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1070 (all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M)) -> M = N_1))) # label(fact_941_le__antisym) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1071 (all K_1 all L hAPP_int_int(times_times_int(bit0(K_1)),L) = bit0(hAPP_int_int(times_times_int(K_1),L))) # label(fact_202_mult__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1072 (all A (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),one_one_nat)) -> one_one_nat = A)) # label(fact_1127_gcd__lcm__complete__lattice__nat_Ole__bot) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1073 (all B_9 all A_10 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(B_9),A_10))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_10)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_9))))) # label(fact_834_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1074 (all I_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),N_1)) -> hAPP_nat_nat(minus_minus_nat(N_1),hAPP_nat_nat(minus_minus_nat(N_1),I_1)) = I_1)) # label(fact_945_diff__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1075 (all Z all X_1 all Y_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Y_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),X_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_1),Z)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_1),Z)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z),X_1))))) # label(fact_900_dvd_Oless__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1076 (all P_2 all X_2 all Y_2 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)) -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_2),Y_2)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y_2),X_2)) -> hBOOL(P_2)))) # label(fact_902_dvd_Oless__imp__triv) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1077 (all P_2 all K all I_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),I_2)) -> (hBOOL(hAPP_int_bool(P_2,K)) -> ((all I (is_int(I) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),I)) -> (hBOOL(hAPP_int_bool(P_2,I)) -> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(plus_plus_int(I),one_one_int))))))) -> hBOOL(hAPP_int_bool(P_2,I_2)))))) # label(fact_1107_int__ge__induct) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1078 (all B_43 all A_49 (A_49 != zero_zero_real -> (zero_zero_real != B_43 -> hAPP_real_real(times_times_real(A_49),B_43) != zero_zero_real))) # label(fact_701_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1079 (all K all L_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit0(K)),bit1(L_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L_1)))) # label(fact_155_rel__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1080 (all X_9 all N_5 (X_9 = one_one_nat | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_5)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_9),hAPP_nat_nat(power_power_nat(X_9),N_5))))) # label(fact_557_dvd__power) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1081 (all X_23 hAPP_nat_nat(power_power_nat(X_23),one_one_nat) = X_23) # label(fact_46_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1082 (all X_11 all Y_8 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X_11),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_8),number_number_of_nat(bit0(bit1(pls)))))),zero_zero_int))) # label(fact_480_not__sum__power2__lt__zero) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1083 (all Lx_3 all Ly_1 all Rx_3 hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_3),Ly_1)),Rx_3) = hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(Lx_3),Rx_3)),Ly_1)) # label(fact_100_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1084 (all C_15 all A_34 all B_31 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_34),B_31)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_34),C_15)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_34),hAPP_nat_nat(plus_plus_nat(B_31),C_15)))))) # label(fact_754_dvd__add) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1085 (all X_2 all Y_2 all B_2 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),B_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_2),Y_2)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(B_2),X_2)),hAPP_nat_real(power_power_real(B_2),Y_2)))))) # label(fact_494_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1086 (all K all L_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K)),bit1(L_1))))) # label(fact_66_rel__simps_I34_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1087 (all V_7 (zero_zero_nat = number_number_of_nat(V_7) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_7),pls)))) # label(fact_561_eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1088 (all Ma all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),N)) <-> Ma = N | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),N)))) # label(fact_986_le__eq__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1089 (all Ma all D (zero_zero_nat = hAPP_nat_nat(div_mod_nat(Ma),D) <-> (exists Q_2 hAPP_nat_nat(times_times_nat(D),Q_2) = Ma))) # label(fact_1179_mod__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1090 (all N_1 all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M)) -> hAPP_nat_nat(plus_plus_nat(N_1),hAPP_nat_nat(minus_minus_nat(M),N_1)) = M)) # label(fact_1009_le__add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1091 (all X_2 all Y_2 (zero_zero_real = hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(X_2),X_2)),hAPP_real_real(times_times_real(Y_2),Y_2)) <-> X_2 = zero_zero_real & zero_zero_real = Y_2)) # label(fact_866_real__two__squares__add__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1092 (all C_15 all A_34 all B_31 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_34),B_31)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_34),C_15)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_34),hAPP_int_int(plus_plus_int(B_31),C_15)))))) # label(fact_755_dvd__add) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1093 (all A_58 all K_4 (is_int(A_58) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_58),hAPP_nat_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls)))),K_4))),zero_zero_int)) -> A_58 = zero_zero_int))) # label(fact_507_even__power__le__0__imp__0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1094 (all V_10 all W_9 hAPP_int_int(plus_plus_int(number_number_of_int(V_10)),number_number_of_int(W_9)) = number_number_of_int(hAPP_int_int(plus_plus_int(V_10),W_9))) # label(fact_247_add__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1095 (all B_2 all A_2 (hBOOL(member_int(B_2,d22set(A_2))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_2)))) # label(fact_1114_d22set__g__1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1096 (all C all D_5 all A all B all M (hBOOL(hAPP_int_bool(zcong(A,B),M)) -> (hBOOL(hAPP_int_bool(zcong(C,D_5),M)) -> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(minus_minus_int(A),C),hAPP_int_int(minus_minus_int(B),D_5)),M))))) # label(fact_618_zcong__zdiff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1097 (all Y_1 all X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(times_times_real(X_1),Y_1)))))) # label(fact_864_real__mult__order) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1098 (all A_112 hAPP_nat_nat(times_times_nat(A_112),A_112) = hAPP_nat_nat(power_power_nat(A_112),number_number_of_nat(bit0(bit1(pls))))) # label(fact_24_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1099 (all X_2 all P_3 (is_int(X_2) -> (hBOOL(member_int(X_2,sr(P_3))) -> X_2 = standardRes(P_3,X_2)))) # label(fact_1193_StandardRes__SR__prop) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1100 (all A all M all B hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A),M),hAPP_int_int(times_times_int(B),M)),M))) # label(fact_578_zcong__zmult__self) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1101 (all V_7 (number_number_of_nat(V_7) = zero_zero_nat <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_7),pls)))) # label(fact_560_eq__number__of__0) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1102 (all A_67 all N_18 hAPP_int_int(times_times_int(hAPP_nat_int(power_power_int(A_67),N_18)),A_67) = hAPP_int_int(times_times_int(A_67),hAPP_nat_int(power_power_int(A_67),N_18))) # label(fact_459_power__commutes) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1103 (all A all B all P_1 all Q_1 twoSqu1241645765sum2sq(product_Pair_int_int(hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(A),P_1)),hAPP_int_int(times_times_int(B),Q_1)),hAPP_int_int(minus_minus_int(hAPP_int_int(times_times_int(A),Q_1)),hAPP_int_int(times_times_int(B),P_1)))) = hAPP_int_int(times_times_int(twoSqu1241645765sum2sq(product_Pair_int_int(A,B))),twoSqu1241645765sum2sq(product_Pair_int_int(P_1,Q_1)))) # label(fact_639_mult__sum2sq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1104 (all N_37 all A_89 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_89)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(times_times_int(A_89),hAPP_nat_int(power_power_int(A_89),N_37)))))) # label(fact_297_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1105 (all N 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(times_times_nat(Ma),N)),Ma)) <-> one_one_nat = N))) # label(fact_875_dvd__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1106 (all N_38 all A_90 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_90)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_90),N_38)),hAPP_nat_nat(times_times_nat(A_90),hAPP_nat_nat(power_power_nat(A_90),N_38)))))) # label(fact_295_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1107 (all A_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(plus_plus_int(A_2),A_2)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),zero_zero_int)))) # label(fact_383_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1108 (all K all L_1 (is_int(K) & is_int(L_1) -> (K = L_1 <-> bit0(L_1) = bit0(K)))) # label(fact_141_rel__simps_I48_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1109 (all K_1 all L bit0(L) != bit1(K_1)) # label(fact_196_rel__simps_I50_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1110 (all A all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(div_mod_int(A),B)),zero_zero_int)))) # label(fact_1161_neg__mod__sign) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1111 (all W_1 all X_2 (number_number_of_nat(W_1) = X_2 <-> number_number_of_nat(W_1) = X_2)) # label(fact_137_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1112 (all A_91 hAPP_nat_nat(power_power_nat(A_91),number_number_of_nat(bit1(bit1(pls)))) = hAPP_nat_nat(times_times_nat(hAPP_nat_nat(times_times_nat(A_91),A_91)),A_91)) # label(fact_273_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1113 (all X_26 hAPP_nat_nat(times_times_nat(X_26),X_26) = hAPP_nat_nat(power_power_nat(X_26),number_number_of_nat(bit0(bit1(pls))))) # label(fact_21_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1114 (all L all M all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),L)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(minus_minus_nat(L),N_1)),hAPP_nat_nat(minus_minus_nat(L),M)))))) # label(fact_895_diff__less__mono2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1115 (all R_1 all Q_1 all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A)) -> (A = hAPP_int_int(plus_plus_int(R_1),hAPP_int_int(times_times_int(A),Q_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),A)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Q_1)))))) # label(fact_510_self__quotient__aux1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1116 (all M_3 all N_9 all A_60 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_60)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_60),M_3)),hAPP_nat_nat(power_power_nat(A_60),N_9))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_3),N_9))))) # label(fact_498_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1117 (all N_31 all X_18 all Y_15 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_18),Y_15)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(X_18),N_31)),hAPP_nat_int(power_power_int(Y_15),N_31))))) # label(fact_330_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1118 (all X_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(X_2)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_2),pls)))) # label(fact_431_less__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1119 (all Q_1 all R_1 all B all C (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,R_1),B)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(B),hAPP_nat_nat(div_mod_nat(Q_1),C))),R_1)),hAPP_nat_nat(times_times_nat(B),C)))))) # label(fact_1195_mod__lemma) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1120 (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]. 1.32/1.62 1121 (all W_15 number267125858f_real(hAPP_int_int(plus_plus_int(bit1(pls)),W_15)) = hAPP_real_real(plus_plus_real(one_one_real),number267125858f_real(W_15))) # label(fact_29_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1122 (all A_92 hAPP_real_real(times_times_real(A_92),number267125858f_real(bit1(pls))) = A_92) # label(fact_259_mult__numeral__1__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1123 (all W_7 number267125858f_real(bit1(W_7)) = hAPP_real_real(plus_plus_real(hAPP_real_real(plus_plus_real(one_one_real),number267125858f_real(W_7))),number267125858f_real(W_7))) # label(fact_255_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1124 (all C_16 all A_37 all B_34 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_37),B_34)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_37),hAPP_int_int(times_times_int(B_34),C_16))))) # label(fact_746_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1125 (all C all A all B all M (hBOOL(hAPP_int_bool(zcong(A,B),M)) -> (hBOOL(hAPP_int_bool(zcong(B,C),M)) -> hBOOL(hAPP_int_bool(zcong(A,C),M))))) # label(fact_564_zcong__trans) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1126 (all V_10 all W_9 hAPP_real_real(plus_plus_real(number267125858f_real(V_10)),number267125858f_real(W_9)) = number267125858f_real(hAPP_int_int(plus_plus_int(V_10),W_9))) # label(fact_248_add__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1127 (all X_1 all P_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP_int_bool(zprime,P_1)) -> (-hBOOL(hAPP_int_bool(zcong(X_1,zero_zero_int),P_1)) -> -hBOOL(hAPP_int_bool(zcong(multInv(P_1,X_1),zero_zero_int),P_1)))))) # label(fact_1086_MultInv__prop3) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1128 (all B_17 all A_19 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_19)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_17)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(times_times_nat(A_19),B_17)))))) # label(fact_806_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1129 (all Z hAPP_nat_nat(plus_plus_nat(Z),Z) = hAPP_nat_nat(times_times_nat(Z),number_number_of_nat(bit0(bit1(pls))))) # label(fact_61_nat__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.32/1.62 1130 (all X_7 all Y_6 hAPP_nat_int(power_power_int(hAPP_int_int(minus_minus_int(X_7),Y_6)),number_number_of_nat(bit0(bit1(pls)))) = hAPP_int_int(minus_minus_int(hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X_7),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y_6),number_number_of_nat(bit0(bit1(pls)))))),hAPP_int_int(times_times_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit1(pls)))),X_7)),Y_6))) # label(fact_649_power2__diff) # label(axiom) # label(non_clause). [assumption]. 1.32/1.63 1131 (all C all D_5 all A all B all M (hBOOL(hAPP_int_bool(zcong(A,B),M)) -> (hBOOL(hAPP_int_bool(zcong(C,D_5),M)) -> hBOOL(hAPP_int_bool(zcong(hAPP_int_int(times_times_int(A),C),hAPP_int_int(times_times_int(B),D_5)),M))))) # label(fact_575_zcong__zmult) # label(axiom) # label(non_clause). [assumption]. 1.32/1.63 1132 (all V_11 all W_10 all Z_7 hAPP_int_int(plus_plus_int(number_number_of_int(V_11)),hAPP_int_int(plus_plus_int(number_number_of_int(W_10)),Z_7)) = hAPP_int_int(plus_plus_int(number_number_of_int(hAPP_int_int(plus_plus_int(V_11),W_10))),Z_7)) # label(fact_245_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 1.32/1.63 1133 (all B all Q_1 all R_1 all B_48 all Q_4 all R_3 (hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B),Q_1)),R_1) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_48),Q_4)),R_3) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(B_48),Q_4)),R_3))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_3),B_48)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_48)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_48),B)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q_4))))))))) # label(fact_599_zdiv__mono2__lemma) # label(axiom) # label(non_clause). [assumption]. 1.32/1.63 1134 (all X_17 all Y_14 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(plus_plus_real(hAPP_real_real(times_times_real(X_17),X_17)),hAPP_real_real(times_times_real(Y_14),Y_14))))) # label(fact_412_sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption]. 1.32/1.63 1135 (all A_2 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),A_2)) -> hBOOL(member_int(B_2,d22set(A_2)))))) # label(fact_1115_d22set__mem) # label(axiom) # label(non_clause). [assumption]. 1.32/1.63 1136 (all A_2 all B_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_2),B_2)) -> -hBOOL(member_int(B_2,d22set(A_2))))) # label(fact_1113_d22set__le__swap) # label(axiom) # label(non_clause). [assumption]. 1.32/1.63 1137 (all A_98 all B_57 all V_16 hAPP_nat_nat(times_times_nat(hAPP_nat_nat(plus_plus_nat(A_98),B_57)),number_number_of_nat(V_16)) = hAPP_nat_nat(plus_plus_nat(hAPP_nat_nat(times_times_nat(A_98),number_number_of_nat(V_16))),hAPP_nat_nat(times_times_nat(B_57),number_number_of_nat(V_16)))) # label(fact_221_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.32/1.63 1138 (all A all B (A != B -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B),A)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B))))) # label(fact_920_dvd_Oneq__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.32/1.63 1139 (all A_67 all N_18 hAPP_real_real(times_times_real(hAPP_nat_real(power_power_real(A_67),N_18)),A_67) = hAPP_real_real(times_times_real(A_67),hAPP_nat_real(power_power_real(A_67),N_18))) # label(fact_458_power__commutes) # label(axiom) # label(non_clause). [assumption]. 1.32/1.63 1140 (all B_1_1 all B_2_1 (is_int(B_1_1) & is_int(B_2_1) -> is_int(inv(B_1_1,B_2_1)))) # label(gsy_c_WilsonRuss_Oinv) # label(axiom) # label(non_clause). [assumption]. 1.32/1.63 1141 -(exists X exists Y hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y),number_number_of_nat(bit0(bit1(pls))))) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int)) # label(conj_0) # label(negated_conjecture) # label(non_clause). [assumption]. 1.32/1.63 1.32/1.63 ============================== end of process non-clausal formulas === 1.32/1.63 1.32/1.63 ============================== PROCESS INITIAL CLAUSES =============== 1.32/1.63 1.32/1.63 ============================== PREDICATE ELIMINATION ================= 1.32/1.63 1.32/1.63 ============================== end predicate elimination ============= 47.59/47.80 47.59/47.80 Auto_denials: (non-Horn, no changes). 47.59/47.80 47.59/47.80 Term ordering decisions: 47.59/47.80 Function symbol KB weights: zero_zero_int=1. ord_less_int=1. pls=1. ord_less_eq_int=1. zero_zero_nat=1. zero_zero_real=1. ord_less_nat=1. ord_less_eq_nat=1. one_one_int=1. dvd_dvd_nat=1. ord_less_real=1. ord_less_eq_real=1. one_one_nat=1. dvd_dvd_int=1. min=1. one_one_real=1. zprime=1. dvd_dvd_real=1. m=1. s=1. t=1. twoSqu1154269391sum2sq=1. s1=1. fconj=1. int=1. c1=1. c2=1. c3=1. c4=1. c5=1. c6=1. c7=1. c8=1. hAPP_int_bool=1. hAPP_int_int=1. hAPP_i1948725293t_bool=1. hAPP_nat_nat=1. hAPP_nat_bool=1. hAPP_n1699378549t_bool=1. hAPP_real_real=1. hAPP_real_bool=1. hAPP_r1134773055l_bool=1. hAPP_nat_int=1. zcong=1. hAPP_nat_real=1. multInv=1. member_int=1. standardRes=1. wset=1. inv=1. legendre=1. product_Pair_int_int=1. hAPP_bool_bool=1. hAPP_b589554111l_bool=1. cOMBB_1652995168ol_int=1. cOMBC_int_int_bool=1. cOMBS_int_bool_bool=1. hAPP_i68813070l_bool=1. f1=1. f9=1. f10=1. f11=1. f13=1. f14=1. f21=1. f25=1. f27=1. f29=1. f30=1. f33=1. times_times_int=1. bit1=1. times_times_nat=1. bit0=1. plus_plus_int=1. times_times_real=1. number_number_of_nat=1. number_number_of_int=1. plus_plus_nat=1. power_power_int=1. div_mod_int=1. power_power_real=1. power_power_nat=1. plus_plus_real=1. minus_minus_nat=1. number267125858f_real=1. minus_minus_int=1. div_mod_nat=1. minus_minus_real=1. quadRes=1. twoSqu1241645765sum2sq=1. zfact=1. d22set=1. collect_int=1. sr=1. undefined_int=1. f7=1. f8=1. f15=1. f20=1. f2=1. f3=1. f4=1. f5=1. f6=1. f12=1. f16=1. f17=1. f18=1. f19=1. f22=1. f23=1. f24=1. f26=1. f28=1. f31=1. f32=1. 47.59/47.80 47.59/47.80 ============================== end of process initial clauses ======== 47.59/47.80 47.59/47.80 ============================== CLAUSES FOR SEARCH ==================== 47.59/47.80 47.59/47.80 ============================== end of clauses for search ============= 47.59/47.80 47.59/47.80 ============================== SEARCH ================================ 47.59/47.80 47.59/47.80 % Starting search at 0.56 seconds. 47.59/47.80 47.59/47.80 Low Water (keep): wt=37.000, iters=3338 47.59/47.80 47.59/47.80 Low Water (keep): wt=36.000, iters=3387 47.59/47.80 47.59/47.80 Low Water (keep): wt=34.000, iters=3343 47.59/47.80 47.59/47.80 Low Water (keep): wt=33.000, iters=3358 47.59/47.80 47.59/47.80 Low Water (keep): wt=32.000, iters=3347 47.59/47.80 47.59/47.80 Low Water (keep): wt=31.000, iters=3418 47.59/47.80 47.59/47.80 Low Water (keep): wt=30.000, iters=3415 47.59/47.80 47.59/47.80 NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 47 (0.00 of 1.32 sec). 47.59/47.80 47.59/47.80 Low Water (keep): wt=29.000, iters=3369 47.59/47.80 47.59/47.80 Low Water (keep): wt=27.000, iters=3368 47.59/47.80 47.59/47.80 Low Water (keep): wt=26.000, iters=3337 47.59/47.80 47.59/47.80 Low Water (keep): wt=25.000, iters=3384 47.59/47.80 47.59/47.80 Low Water (keep): wt=24.000, iters=3336 47.59/47.80 47.59/47.80 Low Water (keep): wt=23.000, iters=3344 47.59/47.80 47.59/47.80 Low Water (keep): wt=22.000, iters=3351 47.59/47.80 47.59/47.80 Low Water (keep): wt=21.000, iters=3337 47.59/47.80 47.59/47.80 Low Water (keep): wt=20.000, iters=3339 47.59/47.80 47.59/47.80 Low Water (keep): wt=19.000, iters=3375 47.59/47.80 47.59/47.80 Low Water (keep): wt=18.000, iters=3353 47.59/47.80 47.59/47.80 Low Water (keep): wt=17.000, iters=3363 47.59/47.80 47.59/47.80 Low Water (keep): wt=16.000, iters=3339 47.59/47.80 47.59/47.80 Low Water (keep): wt=15.000, iters=3366 47.59/47.80 47.59/47.80 Low Water (keep): wt=14.000, iters=3348 47.59/47.80 47.59/47.80 Low Water (keep): wt=13.000, iters=3346 47.59/47.80 47.59/47.80 Low Water (displace): id=15614, wt=10.000 47.59/47.80 47.59/47.80 Low Water (displace): id=16065, wt=8.000 47.59/47.80 47.59/47.80 Low Water (keep): wt=12.000, iters=3641 47.59/47.80 47.59/47.80 Low Water (displace): id=18899, wt=7.000 47.59/47.80 47.59/47.80 Low Water (keep): wt=11.000, iters=3484 47.59/47.80 47.59/47.80 Low Water (displace): id=22807, wt=6.000 47.59/47.80 47.59/47.80 Low Water (keep): wt=10.000, iters=3334 47.59/47.80 47.59/47.80 Low Water (keep): wt=9.000, iters=3349 47.59/47.80 47.59/47.80 Low Water (keep): wt=8.000, iters=3430 47.59/47.80 47.59/47.80 ============================== PROOF ================================= 47.59/47.80 % SZS status Theorem 47.59/47.80 % SZS output start Refutation 47.59/47.80 47.59/47.80 % Proof 1 at 45.34 (+ 0.99) seconds. 47.59/47.80 % Length of proof is 44. 47.59/47.80 % Level of proof is 6. 47.59/47.80 % Maximum clause weight is 50.000. 47.59/47.80 % Given clauses 6413. 47.59/47.80 47.59/47.80 50 (all Z all W hAPP_int_int(plus_plus_int(W),Z) = hAPP_int_int(plus_plus_int(Z),W)) # label(fact_147_zadd__commute) # label(axiom) # label(non_clause). [assumption]. 47.59/47.80 188 (all Z all W hAPP_int_int(times_times_int(Z),W) = hAPP_int_int(times_times_int(W),Z)) # label(fact_143_zmult__commute) # label(axiom) # label(non_clause). [assumption]. 47.59/47.80 196 (all X_5 all Y_4 (is_int(Y_4) & is_int(X_5) -> (Y_4 != X_5 -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_5),Y_4)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_4),X_5)))))) # label(fact_677_linorder__neqE__linordered__idom) # label(axiom) # label(non_clause). [assumption]. 47.64/47.80 199 (all N_1 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_1)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N_1),M))))) # label(fact_326_zdvd__not__zless) # label(axiom) # label(non_clause). [assumption]. 47.64/47.80 217 one_one_int = t -> (exists X exists Y (is_int(X) & is_int(Y) & hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y),number_number_of_nat(bit0(bit1(pls))))) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int))) # label(fact_1__096t_A_061_A1_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06) # label(axiom) # label(non_clause). [assumption]. 47.64/47.80 270 (all K_1 bit0(K_1) = hAPP_int_int(plus_plus_int(K_1),K_1)) # label(fact_206_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 47.64/47.80 537 (all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Z_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_1)))) # label(fact_426_int__one__le__iff__zero__less) # label(axiom) # label(non_clause). [assumption]. 47.64/47.80 615 (all A_33 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,one_one_int),A_33))) # label(fact_758_one__dvd) # label(axiom) # label(non_clause). [assumption]. 47.64/47.80 690 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),t)) -> (exists X exists Y (is_int(X) & hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y),number_number_of_nat(bit0(bit1(pls))))) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int) & is_int(Y))) # label(fact_2__0961_A_060_At_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06) # label(axiom) # label(non_clause). [assumption]. 47.64/47.80 1141 -(exists X exists Y hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(X),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(Y),number_number_of_nat(bit0(bit1(pls))))) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int)) # label(conj_0) # label(negated_conjecture) # label(non_clause). [assumption]. 47.64/47.80 1236 hAPP_int_int(plus_plus_int(A),B) = hAPP_int_int(plus_plus_int(B),A) # label(fact_147_zadd__commute) # label(axiom). [clausify(50)]. 47.64/47.80 1508 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),t)) # label(fact_0_tpos) # label(axiom). [assumption]. 47.64/47.80 1638 hAPP_int_int(times_times_int(A),B) = hAPP_int_int(times_times_int(B),A) # label(fact_143_zmult__commute) # label(axiom). [clausify(188)]. 47.64/47.80 1653 -is_int(A) | -is_int(B) | A = B | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),A)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) # label(fact_677_linorder__neqE__linordered__idom) # label(axiom). [clausify(196)]. 47.64/47.80 1657 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A)) | -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) | -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B),A)) # label(fact_326_zdvd__not__zless) # label(axiom). [clausify(199)]. 47.64/47.80 1678 hAPP_nat_nat(plus_plus_nat(one_one_nat),one_one_nat) = number_number_of_nat(bit0(bit1(pls))) # label(fact_286_semiring__one__add__one__is__two) # label(axiom). [assumption]. 47.64/47.80 1679 number_number_of_nat(bit0(bit1(pls))) = hAPP_nat_nat(plus_plus_nat(one_one_nat),one_one_nat). [copy(1678),flip(a)]. 47.64/47.80 1687 t != one_one_int | hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(c3),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(c4),number_number_of_nat(bit0(bit1(pls))))) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int) # label(fact_1__096t_A_061_A1_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06) # label(axiom). [clausify(217)]. 47.64/47.80 1688 t != one_one_int | hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(c3),hAPP_nat_nat(plus_plus_nat(one_one_nat),one_one_nat))),hAPP_nat_int(power_power_int(c4),hAPP_nat_nat(plus_plus_nat(one_one_nat),one_one_nat))) = hAPP_int_int(plus_plus_int(one_one_int),hAPP_int_int(times_times_int(m),number_number_of_int(bit0(bit0(bit1(pls)))))). [copy(1687),rewrite([1679(9),1679(17),1638(27),1236(30)])]. 47.64/47.80 1767 number_number_of_int(pls) = zero_zero_int # label(fact_395_semiring__numeral__0__eq__0) # label(axiom). [assumption]. 47.64/47.80 1768 zero_zero_int = number_number_of_int(pls). [copy(1767),flip(a)]. 47.64/47.80 1799 bit0(A) = hAPP_int_int(plus_plus_int(A),A) # label(fact_206_Bit0__def) # label(axiom). [clausify(270)]. 47.64/47.80 1833 is_int(one_one_int) # label(gsy_c_Groups_Oone__class_Oone_000tc__Int__Oint) # label(hypothesis). [assumption]. 47.64/47.80 2395 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A)) # label(fact_426_int__one__le__iff__zero__less) # label(axiom). [clausify(537)]. 47.64/47.80 2396 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(pls)),A)). [copy(2395),rewrite([1768(7)])]. 47.64/47.80 2471 zero_zero_int = pls # label(fact_358_Pls__def) # label(axiom). [assumption]. 47.64/47.80 2472 number_number_of_int(pls) = pls. [copy(2471),rewrite([1768(1)])]. 47.64/47.80 2541 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,one_one_int),A)) # label(fact_758_one__dvd) # label(axiom). [clausify(615)]. 47.64/47.80 2678 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),t)) | hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(c5),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(c6),number_number_of_nat(bit0(bit1(pls))))) = hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int) # label(fact_2__0961_A_060_At_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06) # label(axiom). [clausify(690)]. 47.64/47.80 2679 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),t)) | hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(c5),number_number_of_nat(hAPP_int_int(plus_plus_int(bit1(pls)),bit1(pls))))),hAPP_nat_int(power_power_int(c6),number_number_of_nat(hAPP_int_int(plus_plus_int(bit1(pls)),bit1(pls))))) = hAPP_int_int(plus_plus_int(one_one_int),hAPP_int_int(times_times_int(m),number_number_of_int(hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(bit1(pls)),bit1(pls))),hAPP_int_int(plus_plus_int(bit1(pls)),bit1(pls)))))). [copy(2678),rewrite([1799(11),1799(22),1799(31),1799(35),1638(46),1236(49)])]. 47.64/47.80 3240 is_int(t) # label(gsy_v_t____) # label(axiom). [assumption]. 47.64/47.80 3393 number_number_of_nat(bit0(bit1(pls))) = hAPP_nat_nat(plus_plus_nat(one_one_nat),one_one_nat) # label(fact_62_nat__1__add__1) # label(axiom). [assumption]. 47.64/47.80 3394 number_number_of_nat(hAPP_int_int(plus_plus_int(bit1(pls)),bit1(pls))) = hAPP_nat_nat(plus_plus_nat(one_one_nat),one_one_nat). [copy(3393),rewrite([1799(3)])]. 47.64/47.80 3518 hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls))))),hAPP_nat_int(power_power_int(B),number_number_of_nat(bit0(bit1(pls))))) != hAPP_int_int(plus_plus_int(hAPP_int_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls))))),m)),one_one_int) # label(conj_0) # label(negated_conjecture). [clausify(1141)]. 47.64/47.80 3519 hAPP_int_int(plus_plus_int(hAPP_nat_int(power_power_int(A),hAPP_nat_nat(plus_plus_nat(one_one_nat),one_one_nat))),hAPP_nat_int(power_power_int(B),hAPP_nat_nat(plus_plus_nat(one_one_nat),one_one_nat))) != hAPP_int_int(plus_plus_int(one_one_int),hAPP_int_int(times_times_int(m),number_number_of_int(hAPP_int_int(plus_plus_int(hAPP_int_int(plus_plus_int(bit1(pls)),bit1(pls))),hAPP_int_int(plus_plus_int(bit1(pls)),bit1(pls)))))). [copy(3518),rewrite([1799(4),3394(8),1799(11),3394(15),1799(17),1799(21),1638(32),1236(35)])]. 47.64/47.80 3777 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),A)) | -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B)) | -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B),A)). [back_rewrite(1657),rewrite([1768(2),2472(3)])]. 47.64/47.80 3890 t != one_one_int. [back_rewrite(1688),rewrite([1799(26),1799(30)]),flip(b),unit_del(b(flip),3519)]. 47.64/47.80 3949 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),A)). [back_rewrite(2396),rewrite([2472(8)])]. 47.64/47.80 4156 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),t)). [back_rewrite(2679),rewrite([3394(15),3394(23)]),flip(b),unit_del(b(flip),3519)]. 47.64/47.80 4970 -is_int(A) | one_one_int = A | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),one_one_int)). [resolve(1833,a,1653,b),flip(b)]. 47.64/47.80 10733 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),t)). [resolve(3949,a,1508,a)]. 47.64/47.80 168907 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,t),A)) | -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),t)). [resolve(10733,a,3777,a)]. 47.64/47.80 169940 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,t),one_one_int)). [resolve(4970,a,3240,a),flip(a),unit_del(a,3890),unit_del(b,4156)]. 47.64/47.80 172396 $F. [resolve(168907,b,2541,a),unit_del(a,169940)]. 47.64/47.80 47.64/47.80 % SZS output end Refutation 47.64/47.80 ============================== end of proof ========================== 47.64/47.80 47.64/47.80 ============================== STATISTICS ============================ 47.64/47.80 47.64/47.80 Given=6413. Generated=1854670. Kept=170252. proofs=1. 47.64/47.80 Usable=6264. Sos=9998. Demods=1026. Limbo=0, Disabled=155762. Hints=0. 47.64/47.80 Megabytes=98.29. 47.64/47.80 User_CPU=45.34, System_CPU=0.99, Wall_clock=46. 47.64/47.80 47.64/47.80 ============================== end of statistics ===================== 47.64/47.80 47.64/47.80 ============================== end of search ========================= 47.64/47.80 47.64/47.80 THEOREM PROVED 47.64/47.80 % SZS status Theorem 47.64/47.80 47.64/47.80 Exiting with 1 proof. 47.64/47.80 47.64/47.80 Process 48596 exit (max_proofs) Sat Jul 14 05:25:27 2018 47.64/47.80 Prover9 interrupted 47.64/47.81 EOF