0.11/0.13 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.11/0.14 % Command : tptp2X_and_run_prover9 %d %s 0.13/0.34 % Computer : n021.cluster.edu 0.13/0.34 % Model : x86_64 x86_64 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.13/0.34 % Memory : 8042.1875MB 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64 0.13/0.34 % CPULimit : 960 0.13/0.35 % WCLimit : 120 0.13/0.35 % DateTime : Tue Aug 9 02:54:20 EDT 2022 0.13/0.35 % CPUTime : 1.51/1.78 ============================== Prover9 =============================== 1.51/1.78 Prover9 (32) version 2009-11A, November 2009. 1.51/1.78 Process 30752 was started by sandbox2 on n021.cluster.edu, 1.51/1.78 Tue Aug 9 02:54:21 2022 1.51/1.78 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 960 -f /tmp/Prover9_30382_n021.cluster.edu". 1.51/1.78 ============================== end of head =========================== 1.51/1.78 1.51/1.78 ============================== INPUT ================================= 1.51/1.78 1.51/1.78 % Reading from file /tmp/Prover9_30382_n021.cluster.edu 1.51/1.78 1.51/1.78 set(prolog_style_variables). 1.51/1.78 set(auto2). 1.51/1.78 % set(auto2) -> set(auto). 1.51/1.78 % set(auto) -> set(auto_inference). 1.51/1.78 % set(auto) -> set(auto_setup). 1.51/1.78 % set(auto_setup) -> set(predicate_elim). 1.51/1.78 % set(auto_setup) -> assign(eq_defs, unfold). 1.51/1.78 % set(auto) -> set(auto_limits). 1.51/1.78 % set(auto_limits) -> assign(max_weight, "100.000"). 1.51/1.78 % set(auto_limits) -> assign(sos_limit, 20000). 1.51/1.78 % set(auto) -> set(auto_denials). 1.51/1.78 % set(auto) -> set(auto_process). 1.51/1.78 % set(auto2) -> assign(new_constants, 1). 1.51/1.78 % set(auto2) -> assign(fold_denial_max, 3). 1.51/1.78 % set(auto2) -> assign(max_weight, "200.000"). 1.51/1.78 % set(auto2) -> assign(max_hours, 1). 1.51/1.78 % assign(max_hours, 1) -> assign(max_seconds, 3600). 1.51/1.78 % set(auto2) -> assign(max_seconds, 0). 1.51/1.78 % set(auto2) -> assign(max_minutes, 5). 1.51/1.78 % assign(max_minutes, 5) -> assign(max_seconds, 300). 1.51/1.78 % set(auto2) -> set(sort_initial_sos). 1.51/1.78 % set(auto2) -> assign(sos_limit, -1). 1.51/1.78 % set(auto2) -> assign(lrs_ticks, 3000). 1.51/1.78 % set(auto2) -> assign(max_megs, 400). 1.51/1.78 % set(auto2) -> assign(stats, some). 1.51/1.78 % set(auto2) -> clear(echo_input). 1.51/1.78 % set(auto2) -> set(quiet). 1.51/1.78 % set(auto2) -> clear(print_initial_clauses). 1.51/1.78 % set(auto2) -> clear(print_given). 1.51/1.78 assign(lrs_ticks,-1). 1.51/1.78 assign(sos_limit,10000). 1.51/1.78 assign(order,kbo). 1.51/1.78 set(lex_order_vars). 1.51/1.78 clear(print_given). 1.51/1.78 1.51/1.78 % formulas(sos). % not echoed (1236 formulas) 1.51/1.78 1.51/1.78 ============================== end of input ========================== 1.51/1.78 1.51/1.78 % From the command line: assign(max_seconds, 960). 1.51/1.78 1.51/1.78 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 1.51/1.78 1.51/1.78 % Formulas that are not ordinary clauses: 1.51/1.78 1 (all K_1 all L minus_minus_int(bit1(K_1),bit1(L)) = bit0(minus_minus_int(K_1,L))) # label(fact_912_diff__bin__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 2 (all A_71 plus_plus_nat(zero_zero_nat,A_71) = A_71) # label(fact_373_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 3 (all C_23 all A_55 all B_36 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_55),B_36)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_23)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(C_23,A_55)),times_times_real(C_23,B_36)))))) # label(fact_492_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 4 (all Xa all Ya (is_int(Ya) & is_int(Xa) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(hAPP_nat_int(power_power_int(Xa),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_int(power_power_int(Ya),number_number_of_nat(bit0(bit1(pls))))))) <-> Xa != zero_zero_int | Ya != zero_zero_int))) # label(fact_137_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 5 (all Na all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),minus_minus_nat(Na,Ma))))) # label(fact_769_zero__less__diff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 6 (all B all A_1 all C_1 (C_1 = B <-> plus_plus_nat(B,A_1) = plus_plus_nat(C_1,A_1))) # label(fact_329_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 7 (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_int,bit1(K)),bit1(L_1))))) # label(fact_47_rel__simps_I17_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 8 (all Z Z = times_times_real(one_one_real,Z)) # label(fact_901_real__mult__1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 9 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),pls)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit0(K)),pls)))) # label(fact_778_rel__simps_I27_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 10 (all A_1 all E_1 all C_1 all B all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(minus_minus_int(A_1,B),E_1),C_1)),D)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(A_1,E_1),C_1)),plus_plus_int(times_times_int(B,E_1),D))))) # label(fact_518_le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 11 (all I_2 all K_1 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_1)) -> minus_minus_nat(plus_plus_nat(I_2,J_1),K_1) = plus_plus_nat(I_2,minus_minus_nat(J_1,K_1)))) # label(fact_943_diff__add__assoc) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 12 (all Xa all Ya all P_1 ((hBOOL(P_1) -> nat(Xa) = nat(if_int(P_1,Xa,Ya))) & (-hBOOL(P_1) -> nat(if_int(P_1,Xa,Ya)) = nat(Ya)))) # label(fact_476_nat__if__cong) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 13 (all K all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(K,Ma)),plus_plus_nat(K,Na))))) # label(fact_890_nat__add__left__cancel__le) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 14 (all K all Na all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),Na)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),plus_plus_int(Na,times_times_int(K,Ma)))))) # label(fact_998_zdvd__reduce) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 15 (all U all Ma all Na all I_1 all J (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),plus_plus_nat(times_times_nat(minus_minus_nat(J,I_1),U),Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(times_times_nat(I_1,U),Ma)),plus_plus_nat(times_times_nat(J,U),Na)))))) # label(fact_1182_nat__less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 16 (all Xa all Ya (zero_zero_real = Ya & zero_zero_real = Xa <-> zero_zero_real = plus_plus_real(times_times_real(Xa,Xa),times_times_real(Ya,Ya)))) # label(fact_929_real__two__squares__add__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 17 (all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Ya)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(Ya))))) # label(fact_729_le__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 18 (all V_1 nat(number_number_of_int(V_1)) = number_number_of_nat(V_1)) # label(fact_767_nat__number__of__def) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 19 (all C all A all B_1 all M (hBOOL(hAPP_int_bool(zcong(A,B_1),M)) -> hBOOL(hAPP_int_bool(zcong(plus_plus_int(A,C),plus_plus_int(B_1,C)),M)))) # label(fact_1059_zcong__shift) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 20 (all C_24 all B_37 all A_56 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_37),A_56)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_24),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_56,C_24)),times_times_int(B_37,C_24)))))) # label(fact_491_mult__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 21 (all M all N all K_1 times_times_nat(plus_plus_nat(M,N),K_1) = plus_plus_nat(times_times_nat(M,K_1),times_times_nat(N,K_1))) # label(fact_881_add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 22 (all N N = times_times_nat(N,one_one_nat)) # label(fact_896_nat__mult__1__right) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 23 (all A_22 all E all B_11 all C_8 plus_plus_nat(times_times_nat(A_22,E),plus_plus_nat(times_times_nat(B_11,E),C_8)) = plus_plus_nat(times_times_nat(plus_plus_nat(A_22,B_11),E),C_8)) # label(fact_635_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 24 (all A_75 A_75 = plus_plus_nat(A_75,zero_zero_nat)) # label(fact_359_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 25 (all N 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)) -> minus_minus_nat(minus_minus_nat(M,K_1),minus_minus_nat(N,K_1)) = minus_minus_nat(M,N)))) # label(fact_903_Nat_Odiff__diff__eq) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 26 (all Wa all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,nat(Wa)),nat(Z_1))))) # label(fact_792_zless__nat__conj) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 27 (all X_5 all Y_4 all Q_2 times_times_nat(hAPP_nat_nat(power_power_nat(X_5),Q_2),hAPP_nat_nat(power_power_nat(Y_4),Q_2)) = hAPP_nat_nat(power_power_nat(times_times_nat(X_5,Y_4)),Q_2)) # label(fact_701_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 28 (all C_31 all A_66 all B_48 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_66),B_48)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(C_31,A_66)),plus_plus_int(C_31,B_48))))) # label(fact_387_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 29 (all A_1 all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,minus_minus_real(A_1,B)),zero_zero_real)))) # label(fact_758_less__iff__diff__less__0) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 30 (all A_77 all C_36 all D_12 plus_plus_nat(A_77,plus_plus_nat(C_36,D_12)) = plus_plus_nat(C_36,plus_plus_nat(A_77,D_12))) # label(fact_347_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 31 (all B_2 all Q_1 all R_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(B_2,Q_1),R_2)),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),zero_zero_int)))))) # label(fact_832_q__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 32 (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]. 1.51/1.78 33 (all X times_times_real(number267125858f_real(bit0(bit0(bit1(pls)))),hAPP_nat_real(power_power_real(X),number_number_of_nat(bit0(bit1(pls))))) = hAPP_nat_real(power_power_real(times_times_real(number267125858f_real(bit0(bit1(pls))),X)),number_number_of_nat(bit0(bit1(pls))))) # label(fact_962_four__x__squared) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 34 (all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) -> Y != X)) # label(fact_1092_dvd_Oless__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 35 (all Z_4 all Z_1 (is_int(Z_4) & is_int(Z_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_4)) -> (Z_4 = Z_1 <-> nat(Z_4) = nat(Z_1)))))) # label(fact_462_eq__nat__nat__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 36 (all M all K_1 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),minus_minus_nat(plus_plus_nat(N,M),K_1))))) # label(fact_937_le__add__diff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 37 (all L all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,minus_minus_nat(L,N)),minus_minus_nat(L,M))))) # label(fact_907_diff__le__mono2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 38 (all K_1 all I_2 all J_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),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_2),K_1))))) # label(fact_452_zle__trans) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 39 (all A_1 all E_1 all C_1 all B all D (plus_plus_real(times_times_real(B,E_1),D) = plus_plus_real(times_times_real(A_1,E_1),C_1) <-> plus_plus_real(times_times_real(minus_minus_real(B,A_1),E_1),D) = C_1)) # label(fact_523_eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 40 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),M))))) # label(fact_1045_nat__dvd__not__less) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 41 (all A_81 all B_54 all C_40 plus_plus_real(plus_plus_real(A_81,C_40),B_54) = plus_plus_real(plus_plus_real(A_81,B_54),C_40)) # label(fact_336_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 42 (all N_14 all A_92 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_92)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(power_power_int(A_92),N_14))))) # label(fact_266_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 43 (all Lx_4 all Ly_4 all Rx_4 times_times_real(Lx_4,times_times_real(Ly_4,Rx_4)) = times_times_real(times_times_real(Lx_4,Ly_4),Rx_4)) # label(fact_428_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 44 (all B_33 all A_52 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_52),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_33),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),times_times_int(A_52,B_33)))))) # label(fact_502_mult__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 45 (all A_18 (is_int(A_18) -> times_times_int(A_18,one_one_int) = A_18)) # label(fact_658_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 46 (all A_1 all E_1 all C_1 all B all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(minus_minus_int(A_1,B),E_1),C_1)),D)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(A_1,E_1),C_1)),plus_plus_int(times_times_int(B,E_1),D))))) # label(fact_536_less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 47 (all C_1 all A_1 all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(C_1,A_1)),plus_plus_real(C_1,B))))) # label(fact_686_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 48 (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_1139_zcong__zless__0) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 49 (all M all N all P_2 (hBOOL(hAPP_int_bool(zprime,P_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),times_times_int(M,N))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),M)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),N))))) # label(fact_991_zprime__zdvd__zmult__better) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 50 (all A all B_1 hAPP_nat_int(power_power_int(minus_minus_int(A,B_1)),number_number_of_nat(bit0(bit1(pls)))) = plus_plus_int(minus_minus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls)))),times_times_int(times_times_int(number_number_of_int(bit0(bit1(pls))),A),B_1)),hAPP_nat_int(power_power_int(B_1),number_number_of_nat(bit0(bit1(pls)))))) # label(fact_968_zdiff__power2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 51 (all K_1 (is_int(K_1) -> K_1 = plus_plus_int(K_1,pls))) # label(fact_100_add__Pls__right) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 52 (all Z all X all Y (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z),X))))) # label(fact_1089_dvd_Oless__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 53 (all V_1 all W times_times_int(number_number_of_int(V_1),number_number_of_int(W)) = number_number_of_int(times_times_int(V_1,W))) # label(fact_722_times__numeral__code_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 54 (all A all B_1 all C (is_int(A) -> (C = minus_minus_int(A,B_1) -> plus_plus_int(C,B_1) = A))) # label(fact_984_Int2_Oaux1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 55 (all A_35 all N_9 all N_8 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_9),N_8)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),A_35)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_35),N_9)),hAPP_nat_real(power_power_real(A_35),N_8)))))) # label(fact_585_power__increasing) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 56 (all Z all W plus_plus_int(Z,W) = plus_plus_int(W,Z)) # label(fact_67_zadd__commute) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 57 (all V_13 all V_12 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_12)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_13)) -> times_times_int(number_number_of_int(V_12),number_number_of_int(V_13)) = number_number_of_int(times_times_int(V_12,V_13))))) # label(fact_447_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 58 (all A_91 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_91),plus_plus_real(A_91,one_one_real)))) # label(fact_271_less__add__one) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 59 (all Ma all Na (one_one_nat = times_times_nat(Ma,Na) <-> Na = one_one_nat & one_one_nat = Ma)) # label(fact_897_nat__1__eq__mult__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 60 (all Xa all Ya all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(B),Xa)),hAPP_nat_real(power_power_real(B),Ya)))))) # label(fact_277_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 61 (all M hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),times_times_nat(M,M)))) # label(fact_844_le__square) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 62 (all Lx all Ly all Rx all Ry times_times_real(times_times_real(Lx,Rx),times_times_real(Ly,Ry)) = times_times_real(times_times_real(Lx,Ly),times_times_real(Rx,Ry))) # label(fact_438_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 63 (all Lx_1 all Ly_1 all Rx_1 all Ry_1 times_times_int(times_times_int(Lx_1,Ly_1),times_times_int(Rx_1,Ry_1)) = times_times_int(Rx_1,times_times_int(times_times_int(Lx_1,Ly_1),Ry_1))) # label(fact_437_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 64 (all C_3 all A_10 all B_5 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_10),B_5)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(C_3,A_10)),plus_plus_real(C_3,B_5))))) # label(fact_680_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 65 (all K all I_1 all J (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J)) -> (minus_minus_nat(J,I_1) = K <-> J = plus_plus_nat(K,I_1)))) # label(fact_942_le__imp__diff__is__add) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 66 (all M all N plus_plus_nat(M,N) = plus_plus_nat(N,M)) # label(fact_179_nat__add__commute) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 67 (all Z all W times_times_real(W,Z) = times_times_real(Z,W)) # label(fact_978_real__mult__commute) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 68 (all A_70 plus_plus_nat(zero_zero_nat,A_70) = A_70) # label(fact_376_add__0__left) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 69 (all A_95 A_95 = hAPP_nat_real(power_power_real(A_95),one_one_nat)) # label(fact_217_power__one__right) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 70 (all X all P_2 (hBOOL(hAPP_int_bool(zprime,P_2)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),X)) -> hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(X),nat(minus_minus_int(P_2,one_one_int))),one_one_int),P_2))))) # label(fact_1121_Little__Fermat) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 71 (all M all N (N = M -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)))) # label(fact_849_eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 72 (all A_99 A_99 = plus_plus_real(A_99,number267125858f_real(pls))) # label(fact_116_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 73 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,Na))))) # label(fact_474_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 74 (all X hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),X))) # label(fact_1084_dvd_Oorder__refl) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 75 (all A_70 (is_int(A_70) -> A_70 = plus_plus_int(zero_zero_int,A_70))) # label(fact_375_add__0__left) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 76 (all Xa all Ya (Ya != zero_zero_real | Xa != zero_zero_real <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),plus_plus_real(hAPP_nat_real(power_power_real(Xa),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_real(power_power_real(Ya),number_number_of_nat(bit0(bit1(pls))))))))) # label(fact_138_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 77 (all A_8 all N_7 times_times_int(A_8,hAPP_nat_int(power_power_int(A_8),N_7)) = times_times_int(hAPP_nat_int(power_power_int(A_8),N_7),A_8)) # label(fact_696_power__commutes) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 78 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),min)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(K)),min)))) # label(fact_1025_rel__simps_I11_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 79 (all Y all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),times_times_real(X,Y)))))) # label(fact_924_real__mult__order) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 80 (all C_23 all A_55 all B_36 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_55),B_36)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_23)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(C_23,A_55)),times_times_int(C_23,B_36)))))) # label(fact_494_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 81 (all Xa all Ya all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(B),Xa)),hAPP_nat_int(power_power_int(B),Ya))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya))))) # label(fact_737_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 82 (all M all I_2 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),plus_plus_nat(J_1,M))))) # label(fact_189_trans__less__add1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 83 (all N all M zero_zero_nat = minus_minus_nat(N,plus_plus_nat(N,M))) # label(fact_771_diff__add__0) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 84 (all N_19 hAPP_nat_nat(power_power_nat(one_one_nat),N_19) = one_one_nat) # label(fact_234_power__one) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 85 (all B_1_1 all B_2_1 (is_int(B_2_1) & is_int(B_1_1) -> is_int(plus_plus_int(B_1_1,B_2_1)))) # label(gsy_c_Groups_Oplus__class_Oplus_000tc__Int__Oint) # label(hypothesis) # label(non_clause). [assumption]. 1.51/1.78 86 (all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),number_number_of_int(Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(pls)),Ya)))) # label(fact_752_le__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 87 (all V_2 all V_1 ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls)) -> plus_plus_nat(number_number_of_nat(V_1),number_number_of_nat(V_2)) = number_number_of_nat(plus_plus_int(V_1,V_2))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls)) -> plus_plus_nat(number_number_of_nat(V_1),number_number_of_nat(V_2)) = number_number_of_nat(V_1))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls)) -> number_number_of_nat(V_2) = plus_plus_nat(number_number_of_nat(V_1),number_number_of_nat(V_2))))) # label(fact_52_add__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 88 (all Z plus_plus_int(plus_plus_int(one_one_int,Z),Z) != zero_zero_int) # label(fact_129_odd__nonzero) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 89 (all A_72 plus_plus_nat(zero_zero_nat,A_72) = A_72) # label(fact_370_add__0) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 90 (all B_59 all A_86 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_86),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_59),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_86,B_59)),zero_zero_int))))) # label(fact_310_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 91 (all K_1 all L bit1(plus_plus_int(K_1,L)) = plus_plus_int(bit1(K_1),bit0(L))) # label(fact_126_add__Bit1__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 92 (all A all B_1 hBOOL(hAPP_int_bool(zcong(A,B_1),one_one_int))) # label(fact_1127_zcong__1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 93 (all A_17 (is_int(A_17) -> times_times_int(A_17,one_one_int) = A_17)) # label(fact_661_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 94 (all Z1 all Z2 all Z3 plus_plus_int(plus_plus_int(Z1,Z2),Z3) = plus_plus_int(Z1,plus_plus_int(Z2,Z3))) # label(fact_65_zadd__assoc) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 95 (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]. 1.51/1.78 96 (all B_29 all A_48 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_48)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_29)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),times_times_nat(A_48,B_29)))))) # label(fact_515_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 97 (all M all N plus_plus_int(hAPP_nat_int(semiri1621563631at_int,M),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,plus_plus_nat(M,N))) # label(fact_39_zadd__int) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 98 (all K_1 (is_int(K_1) -> K_1 = plus_plus_int(pls,K_1))) # label(fact_101_add__Pls) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 99 (all A_98 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_98),number_number_of_nat(bit0(bit1(pls))))),zero_zero_int))) # label(fact_135_power2__less__0) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 100 (all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) <-> Na != zero_zero_nat)) # label(fact_202_neq0__conv) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 101 (all K_1 all M all N minus_minus_nat(times_times_nat(K_1,M),times_times_nat(K_1,N)) = times_times_nat(K_1,minus_minus_nat(M,N))) # label(fact_900_diff__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 102 (all A_71 A_71 = plus_plus_real(zero_zero_real,A_71)) # label(fact_374_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 103 (all B all A_1 all C_1 (plus_plus_real(C_1,A_1) = plus_plus_real(B,A_1) <-> B = C_1)) # label(fact_330_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 104 (all V_1 all W number_number_of_int(plus_plus_int(V_1,W)) = plus_plus_int(number_number_of_int(V_1),number_number_of_int(W))) # label(fact_32_plus__numeral__code_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 105 (all C_21 all A_53 all B_34 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_53),B_34)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_21)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(A_53,C_21)),times_times_real(B_34,C_21)))))) # label(fact_498_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 106 (all M all I_2 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),plus_plus_nat(M,J_1))))) # label(fact_190_trans__less__add2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 107 (all A_17 times_times_real(A_17,one_one_real) = A_17) # label(fact_659_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 108 (all A_1 all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_1,B)),zero_zero_int)))) # label(fact_528_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 109 (all A all N all B_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(A),N)),hAPP_nat_nat(power_power_nat(B_1),N))) -> (zero_zero_nat != N -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1))))) # label(fact_1050_divides__rev) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 110 (all X_13 one_one_real = hAPP_nat_real(power_power_real(X_13),zero_zero_nat)) # label(fact_302_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 111 -(all R all V (is_int(R) & is_int(V) -> -(hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(number_number_of_int(bit0(bit1(pls))),abs_abs_int(V))),plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n)))) & minus_minus_int(x,times_times_int(R,plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n)))) = V))) # label(fact_1154__096_B_Bthesis_O_A_I_B_Br_Av_O_Av_A_061_Ax_A_N_Ar_A_K_A_I1_A_L_Aint_An) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 112 (all N 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)))) # label(fact_963_two__realpow__ge__one) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 113 (all X_6 all Y_5 all Z_3 plus_plus_int(times_times_int(X_6,Y_5),times_times_int(X_6,Z_3)) = times_times_int(X_6,plus_plus_int(Y_5,Z_3))) # label(fact_651_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 114 (all Va all V_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,number_number_of_nat(Va)),number_number_of_nat(V_3))) <-> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Va),V_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Va),pls))))) # label(fact_950_le__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 115 (all Ma all K all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(Ma,K)),times_times_nat(Na,K))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K)))) # label(fact_919_mult__less__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 116 (all Ma all Na all A_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_1)) -> (hAPP_nat_real(power_power_real(A_1),Na) = hAPP_nat_real(power_power_real(A_1),Ma) <-> Ma = Na))) # label(fact_274_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 117 (all A_81 all B_54 all C_40 plus_plus_int(plus_plus_int(A_81,C_40),B_54) = plus_plus_int(plus_plus_int(A_81,B_54),C_40)) # label(fact_334_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 118 (all C_2 all A_9 all B_4 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_9),B_4)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(A_9,C_2)),plus_plus_real(B_4,C_2))))) # label(fact_683_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 119 (all A_62 all B_43 all V_7 times_times_int(minus_minus_int(A_62,B_43),number_number_of_int(V_7)) = minus_minus_int(times_times_int(A_62,number_number_of_int(V_7)),times_times_int(B_43,number_number_of_int(V_7)))) # label(fact_465_left__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 120 (all K_1 all M all N minus_minus_nat(plus_plus_nat(K_1,M),plus_plus_nat(K_1,N)) = minus_minus_nat(M,N)) # label(fact_706_Nat_Odiff__cancel) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 121 (all Xa all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(Xa),Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Xa)) | zero_zero_nat = Na)) # label(fact_204_zero__less__power__nat__eq) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 122 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(semiri984289939at_nat,Ma)),hAPP_nat_nat(semiri984289939at_nat,Na))))) # label(fact_473_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 123 (all C_24 all B_37 all A_56 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_37),A_56)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_24),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(A_56,C_24)),times_times_real(B_37,C_24)))))) # label(fact_490_mult__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 124 (all C_20 all A_45 all B_27 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(C_20,A_45)),times_times_real(C_20,B_27))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_20)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_45),B_27))))) # label(fact_541_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 125 (all A_15 (is_int(A_15) -> A_15 = times_times_int(one_one_int,A_15))) # label(fact_667_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 126 (all X_11 all Y_9 all A_47 all B_28 plus_plus_real(times_times_real(X_11,minus_minus_real(Y_9,B_28)),times_times_real(minus_minus_real(X_11,A_47),B_28)) = minus_minus_real(times_times_real(X_11,Y_9),times_times_real(A_47,B_28))) # label(fact_525_mult__diff__mult) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 127 (all V_6 all W_3 minus_minus_int(number_number_of_int(V_6),number_number_of_int(W_3)) = number_number_of_int(minus_minus_int(V_6,W_3))) # label(fact_467_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 128 (all X all Y minus_minus_nat(hAPP_nat_nat(power_power_nat(X),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_nat(power_power_nat(Y),number_number_of_nat(bit0(bit1(pls))))) = times_times_nat(plus_plus_nat(X,Y),minus_minus_nat(X,Y))) # label(fact_966_diff__square) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 129 (all Wa all Z_1 (is_int(Z_1) -> ((exists N_1 Z_1 = plus_plus_int(Wa,hAPP_nat_int(semiri1621563631at_int,N_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Wa),Z_1))))) # label(fact_789_zle__iff__zadd) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 130 (all A_1 all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),A_1)) <-> hBOOL(hAPP_int_bool(zcong(A_1,zero_zero_int),Ma)))) # label(fact_1044_zcong__zero__equiv__div) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 131 (all N_26 number267125858f_real(hAPP_nat_int(semiri1621563631at_int,N_26)) = hAPP_nat_real(semiri132038758t_real,N_26)) # label(fact_132_number__of__int) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 132 (all C_3 all A_10 all B_5 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_10),B_5)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(C_3,A_10)),plus_plus_nat(C_3,B_5))))) # label(fact_681_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 133 (all N_25 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,N_25)),hAPP_nat_real(power_power_real(number267125858f_real(bit0(bit1(pls)))),N_25)))) # label(fact_150_of__nat__less__two__power) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 134 (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_76_rel__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 135 (all B_1 all D_1 all A (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_1),A)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_1),plus_plus_nat(A,B_1))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_1),B_1))))) # label(fact_1057_divides__add__revr) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 136 (all D_1 all I_2 (is_int(I_2) -> (zero_zero_int != I_2 -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),I_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(D_1)),abs_abs_int(I_2))))))) # label(fact_1155_dvd__imp__le__int) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 137 (all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),bit1(pls))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(Xa)),one_one_real)))) # label(fact_85_less__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 138 (all C all A all B_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1)) -> (C = B_1 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),C))))) # label(fact_1103_dvd_Oord__le__eq__trans) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 139 (all N all M hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),plus_plus_nat(M,N)))) # label(fact_893_le__add2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 140 (all B_1 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_1),C)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),A)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),B_1))))) # label(fact_964_Euler_Oaux2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.78 141 (all A_1 all C_1 all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(A_1,C_1)),plus_plus_int(B,C_1))))) # label(fact_691_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 142 (all A all B_1 (B_1 != A -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),A))))) # label(fact_1109_dvd_Oneq__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 143 (all A_29 zero_zero_real = times_times_real(zero_zero_real,A_29)) # label(fact_607_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 144 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> zero_zero_nat != N)) # label(fact_197_gr__implies__not0) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 145 (all Wa all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),plus_plus_int(Z_1,one_one_int))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Wa),Z_1)))) # label(fact_811_zle__add1__eq__le) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 146 (all Ya all Xa (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)) <-> Xa = Ya))) # label(fact_1107_dvd_Oantisym__conv) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 147 (all A_80 all B_53 all C_39 plus_plus_int(A_80,plus_plus_int(B_53,C_39)) = plus_plus_int(plus_plus_int(A_80,B_53),C_39)) # label(fact_337_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 148 (all Z all X all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),plus_plus_int(X,times_times_int(plus_plus_int(abs_abs_int(minus_minus_int(X,Z)),one_one_int),D_1)))))) # label(fact_1172_incr__lemma) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 149 (all B_1 all M all A (is_int(B_1) & 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(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),M)) -> (hBOOL(hAPP_int_bool(zcong(A,B_1),M)) -> A = B_1))))))) # label(fact_1140_zcong__zless__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 150 (all K_1 all L times_times_int(bit0(K_1),L) = bit0(times_times_int(K_1,L))) # label(fact_709_mult__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 151 (all W_7 number267125858f_real(bit1(W_7)) = plus_plus_real(plus_plus_real(one_one_real,number267125858f_real(W_7)),number267125858f_real(W_7))) # label(fact_144_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 152 (all N all M (is_int(N) & is_int(M) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),N)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),M)) -> N = M)))))) # label(fact_997_zdvd__antisym__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 153 (all Ma all Z_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),nat(Z_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),Z_1)))) # label(fact_793_zless__nat__eq__int__zless) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 154 (all V_14 all W_9 number_number_of_int(plus_plus_int(V_14,W_9)) = plus_plus_int(number_number_of_int(V_14),number_number_of_int(W_9))) # label(fact_124_number__of__add) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 155 (all A all P_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),P_2)) -> hAPP_nat_int(power_power_int(A),nat(P_2)) = times_times_int(A,hAPP_nat_int(power_power_int(A),minus_minus_nat(nat(P_2),one_one_nat))))) # label(fact_827_Euler_Oaux__1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 156 (all B_1_1 (is_int(B_1_1) -> is_int(succ(B_1_1)))) # label(gsy_c_Int_Osucc) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 157 (all A_32 all C_10 all B_17 all D_4 plus_plus_int(minus_minus_int(A_32,B_17),minus_minus_int(C_10,D_4)) = minus_minus_int(plus_plus_int(A_32,C_10),plus_plus_int(B_17,D_4))) # label(fact_602_add__diff__add) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 158 (all X hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X),hAPP_nat_int(power_power_int(X),number_number_of_nat(bit0(bit1(pls))))))) # label(fact_825_power2__ge__self) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 159 (all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),N))) # label(fact_843_le__refl) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 160 (all I_2 all K_1 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_1)) -> minus_minus_nat(plus_plus_nat(I_2,K_1),J_1) = minus_minus_nat(I_2,minus_minus_nat(J_1,K_1)))) # label(fact_935_diff__diff__right) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 161 (all K all P_1 all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D)) -> ((all X_1 (is_int(X_1) -> (hBOOL(hAPP_int_bool(P_1,X_1)) -> hBOOL(hAPP_int_bool(P_1,plus_plus_int(X_1,D)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K)) -> (all X_1 (hBOOL(hAPP_int_bool(P_1,X_1)) -> hBOOL(hAPP_int_bool(P_1,plus_plus_int(X_1,times_times_int(K,D)))))))))) # label(fact_1180_incr__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 162 (all Wa all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(Wa,one_one_int)),Z_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_1)))) # label(fact_812_add1__zle__eq) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 163 (all A all M all B_1 hBOOL(hAPP_int_bool(zcong(times_times_int(A,M),times_times_int(B_1,M)),M))) # label(fact_1128_zcong__zmult__self) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 164 (all M_2 all N_3 all A_4 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_4)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_4),M_2)),hAPP_nat_real(power_power_real(A_4),N_3))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_3))))) # label(fact_738_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 165 (all N all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(X),N)),hAPP_nat_nat(power_power_nat(Y),N))))) # label(fact_1054_divides__exp) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 166 (all C_1 all D all A_1 all B (B != A_1 & C_1 != D <-> plus_plus_nat(times_times_nat(A_1,D),times_times_nat(B,C_1)) != plus_plus_nat(times_times_nat(A_1,C_1),times_times_nat(B,D)))) # label(fact_647_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 167 (all Ma all Na (zero_zero_nat = Ma | Na = zero_zero_nat <-> zero_zero_nat = times_times_nat(Ma,Na))) # label(fact_867_mult__is__0) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 168 (all Va (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Va),pls)) <-> number_number_of_nat(Va) = zero_zero_nat)) # label(fact_796_eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 169 (all Z (is_int(Z) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z)) -> Z = hAPP_nat_int(semiri1621563631at_int,nat(Z))))) # label(fact_590_nat__0__le) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 170 (all A_99 (is_int(A_99) -> A_99 = plus_plus_int(A_99,number_number_of_int(pls)))) # label(fact_115_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 171 (all A_25 zero_zero_real = times_times_real(A_25,zero_zero_real)) # label(fact_621_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 172 (all Xa (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),times_times_real(Xa,Xa))) <-> Xa = zero_zero_real)) # label(fact_922_not__real__square__gt__zero) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 173 (all A_77 all C_36 all D_12 plus_plus_int(C_36,plus_plus_int(A_77,D_12)) = plus_plus_int(A_77,plus_plus_int(C_36,D_12))) # label(fact_346_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 174 (all N all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(power_power_int(X),N))))) # label(fact_788_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 175 (all C_14 all D_8 all A_39 all B_21 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_39),B_21)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_14),D_8)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_39)) -> (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_real,times_times_real(A_39,C_14)),times_times_real(B_21,D_8)))))))) # label(fact_559_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 176 (all Ma all Na (zero_zero_nat = minus_minus_nat(Ma,Na) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)))) # label(fact_927_diff__is__0__eq) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 177 (all C_30 all A_65 all B_47 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_65),B_47)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(A_65,C_30)),plus_plus_real(B_47,C_30))))) # label(fact_392_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 178 (all A_19 times_times_nat(A_19,one_one_nat) = A_19) # label(fact_654_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 179 (all C_26 all D_9 all A_58 all B_39 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_58),B_39)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_26),D_9)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_58)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_26)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_58,C_26)),times_times_int(B_39,D_9)))))))) # label(fact_487_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 180 (all B_59 all A_86 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_86),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_59),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(A_86,B_59)),zero_zero_real))))) # label(fact_312_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 181 (all A_1 all Wa (is_int(A_1) -> (number_number_of_nat(Wa) != zero_zero_nat & A_1 = zero_zero_int <-> hAPP_nat_int(power_power_int(A_1),number_number_of_nat(Wa)) = zero_zero_int))) # label(fact_117_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 182 (all P_1 ((all X_1 (is_int(X_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> hBOOL(hAPP_int_bool(P_1,X_1))))) <-> (all X_1 hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X_1)))))) # label(fact_786_transfer__int__nat__quantifiers_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 183 (all A_68 all C_33 all B_50 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(A_68,C_33)),plus_plus_real(B_50,C_33))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_68),B_50)))) # label(fact_383_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 184 (all A all P_2 (hBOOL(hAPP_int_bool(zprime,P_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A)) -> (hBOOL(hAPP_int_bool(zcong(times_times_int(A,A),one_one_int),P_2)) -> hBOOL(hAPP_int_bool(zcong(A,one_one_int),P_2)) | hBOOL(hAPP_int_bool(zcong(A,minus_minus_int(P_2,one_one_int)),P_2)))))) # label(fact_1143_zcong__square) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 185 (all B_31 all A_50 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_50)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_31),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(B_31,A_50)),zero_zero_int))))) # label(fact_508_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 186 (all K_1 all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),minus_minus_nat(M,N))) -> (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),M)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N)))))) # label(fact_1049_dvd__diffD1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 187 (all A_1 (A_1 = zero_zero_real <-> zero_zero_real = plus_plus_real(A_1,A_1))) # label(fact_368_double__zero__sym) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 188 (all M_5 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_5)),zero_zero_nat))) # label(fact_288_of__nat__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 189 (all A_1 all E_1 all C_1 all B all D (is_int(D) -> (D = plus_plus_int(times_times_int(minus_minus_int(A_1,B),E_1),C_1) <-> plus_plus_int(times_times_int(B,E_1),D) = plus_plus_int(times_times_int(A_1,E_1),C_1)))) # label(fact_520_eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 190 (all P_1 all I_1 (is_int(I_1) -> (hBOOL(hAPP_nat_bool(P_1,nat(I_1))) <-> (all N_1 (hAPP_nat_int(semiri1621563631at_int,N_1) = I_1 -> hBOOL(hAPP_nat_bool(P_1,N_1)))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),zero_zero_int)) -> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)))))) # label(fact_817_split__nat) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 191 (all Z all X all Y (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(Z,X)),plus_plus_real(Z,Y))))) # label(fact_911_real__add__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 192 (all A_17 times_times_nat(A_17,one_one_nat) = A_17) # label(fact_660_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 193 (all Xa all Ya (Xa = Ya <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)))) # label(fact_1115_divides__antisym) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 194 (all A_75 plus_plus_real(A_75,zero_zero_real) = A_75) # label(fact_360_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 195 (all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),nat(Z_1))))) # label(fact_815_zero__less__nat__eq) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 196 (all U all Ma all Na all I_1 all J (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),plus_plus_nat(times_times_nat(minus_minus_nat(J,I_1),U),Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(times_times_nat(I_1,U),Ma)),plus_plus_nat(times_times_nat(J,U),Na)))))) # label(fact_1193_nat__le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 197 (all P (is_bool(P) -> fTrue = P | P = fFalse)) # label(help_If_3_1_If_000tc__Int__Oint_T) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 198 (all Y all X (hBOOL(hAPP_int_bool(twoSqu1154269391sum2sq,X)) -> (hBOOL(hAPP_int_bool(twoSqu1154269391sum2sq,Y)) -> hBOOL(hAPP_int_bool(twoSqu1154269391sum2sq,times_times_int(X,Y)))))) # label(fact_948_is__mult__sum2sq) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 199 (all Ma all K all Na (Ma = Na | zero_zero_nat = K <-> times_times_nat(Ma,K) = times_times_nat(Na,K))) # label(fact_869_mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 200 (all K_1 bit0(K_1) = plus_plus_int(K_1,K_1)) # label(fact_103_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 201 (all A_30 all B_15 (is_int(A_30) -> A_30 = plus_plus_int(minus_minus_int(A_30,B_15),B_15))) # label(fact_606_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 202 (all A_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_1,A_1)),zero_zero_int)))) # label(fact_63_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 203 (all A_25 zero_zero_int = times_times_int(A_25,zero_zero_int)) # label(fact_623_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 204 (all K_1 all I_2 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(I_2,K_1)),times_times_nat(J_1,K_1))))) # label(fact_850_mult__le__mono1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 205 (all Xa all Ya 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,Xa),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(Z_1,Xa)),times_times_real(Z_1,Ya)))))) # label(fact_859_real__mult__le__cancel__iff2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 206 (all A_18 A_18 = times_times_real(A_18,one_one_real)) # label(fact_656_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 207 (all Ma all Na (is_int(Ma) & is_int(Na) -> (one_one_int = times_times_int(Ma,Na) <-> number_number_of_int(min) = Ma & Na = number_number_of_int(min) | Na = one_one_int & Ma = one_one_int))) # label(fact_1029_zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 208 (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]. 1.51/1.79 209 (all C_26 all D_9 all A_58 all B_39 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_58),B_39)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_26),D_9)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_58)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_26)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(A_58,C_26)),times_times_real(B_39,D_9)))))))) # label(fact_485_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 210 (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_710_rel__simps_I34_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 211 (all C_18 all A_43 all B_25 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(C_18,A_43)),times_times_nat(C_18,B_25))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_18)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_43),B_25))))) # label(fact_548_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 212 (all M all N all Z plus_plus_int(hAPP_nat_int(semiri1621563631at_int,plus_plus_nat(M,N)),Z) = plus_plus_int(hAPP_nat_int(semiri1621563631at_int,M),plus_plus_int(hAPP_nat_int(semiri1621563631at_int,N),Z))) # label(fact_38_zadd__int__left) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 213 (all K_1 all L plus_plus_int(bit0(K_1),bit1(L)) = bit1(plus_plus_int(K_1,L))) # label(fact_127_add__Bit0__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 214 (all N_24 all A_97 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_97)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_24)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_nat_int(power_power_int(A_97),N_24)))))) # label(fact_157_one__less__power) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 215 (all C_12 all D_6 all A_37 all B_19 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_37),B_19)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_12),D_6)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_37)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_12)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(A_37,C_12)),times_times_real(B_19,D_6)))))))) # label(fact_565_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 216 (all I_2 all J_1 all K_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(I_2,J_1)),K_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),K_1)))) # label(fact_194_add__lessD1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 217 (all K all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(K,Ma)),plus_plus_nat(K,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)))) # label(fact_184_nat__add__left__cancel__less) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 218 (all A_1 all E_1 all C_1 all B all D (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(times_times_real(A_1,E_1),C_1)),plus_plus_real(times_times_real(B,E_1),D))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(times_times_real(minus_minus_real(A_1,B),E_1),C_1)),D)))) # label(fact_517_le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 219 (all A_78 all C_37 all D_13 plus_plus_real(plus_plus_real(A_78,C_37),D_13) = plus_plus_real(A_78,plus_plus_real(C_37,D_13))) # label(fact_345_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 220 (all Na (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)))) # label(fact_154_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 221 (all Ma all K all Na (plus_plus_nat(Na,K) = plus_plus_nat(Ma,K) <-> Na = Ma)) # label(fact_183_nat__add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 222 (all A_81 all B_54 all C_40 plus_plus_nat(plus_plus_nat(A_81,C_40),B_54) = plus_plus_nat(plus_plus_nat(A_81,B_54),C_40)) # label(fact_335_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 223 (all Xa all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,Xa)),hAPP_nat_int(semiri1621563631at_int,Ya))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)))) # label(fact_1052_Nat__Transfer_Otransfer__int__nat__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 224 (all V_11 all W_6 times_times_int(number_number_of_int(V_11),number_number_of_int(W_6)) = number_number_of_int(times_times_int(V_11,W_6))) # label(fact_457_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 225 (all N_23 ((zero_zero_nat = N_23 -> one_one_nat = hAPP_nat_nat(power_power_nat(zero_zero_nat),N_23)) & (zero_zero_nat != N_23 -> zero_zero_nat = hAPP_nat_nat(power_power_nat(zero_zero_nat),N_23)))) # label(fact_161_power__0__left) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 226 (all A_12 all C_5 all B_7 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(A_12,C_5)),plus_plus_nat(B_7,C_5))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_12),B_7)))) # label(fact_675_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 227 (all Na all K all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Ma)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Na)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,minus_minus_nat(Ma,K)),minus_minus_nat(Na,K))))))) # label(fact_902_le__diff__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 228 (all A_79 all B_52 all C_38 plus_plus_int(plus_plus_int(A_79,B_52),C_38) = plus_plus_int(A_79,plus_plus_int(B_52,C_38))) # label(fact_340_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 229 (all Ma all Na (hAPP_nat_nat(power_power_nat(Ma),Na) = zero_zero_nat <-> Na != zero_zero_nat & Ma = zero_zero_nat)) # label(fact_402_nat__power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 230 (all A_5 all N_5 all N_4 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_5),N_4)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_5)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_5),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(A_5),N_4)),hAPP_nat_real(power_power_real(A_5),N_5))))))) # label(fact_732_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 231 (all N_23 ((N_23 = zero_zero_nat -> one_one_int = hAPP_nat_int(power_power_int(zero_zero_int),N_23)) & (zero_zero_nat != N_23 -> hAPP_nat_int(power_power_int(zero_zero_int),N_23) = zero_zero_int))) # label(fact_160_power__0__left) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 232 (all A_22 all E all B_11 all C_8 plus_plus_int(times_times_int(A_22,E),plus_plus_int(times_times_int(B_11,E),C_8)) = plus_plus_int(times_times_int(plus_plus_int(A_22,B_11),E),C_8)) # label(fact_636_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 233 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) | Ma = Na <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)))) # label(fact_877_le__eq__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 234 (all Wa all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Wa),minus_minus_int(Z_1,one_one_int))))) # label(fact_949_zle__diff1__eq) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 235 (all A_1 all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),zero_zero_real)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),times_times_real(A_1,B))))) # label(fact_529_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 236 (all B_42 all A_61 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_42),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_61)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_42)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_61),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_61,B_42)),zero_zero_int)))) # label(fact_479_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 237 (all B_30 all A_49 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_49)) -> (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,times_times_int(A_49,B_30)),zero_zero_int))))) # label(fact_511_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 238 (all V_6 all W_3 number267125858f_real(minus_minus_int(V_6,W_3)) = minus_minus_real(number267125858f_real(V_6),number267125858f_real(W_3))) # label(fact_468_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 239 (all K_1 succ(K_1) = plus_plus_int(K_1,one_one_int)) # label(fact_814_succ__def) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 240 (all W_12 ((number_number_of_nat(W_12) != zero_zero_nat -> zero_zero_nat = hAPP_nat_nat(power_power_nat(zero_zero_nat),number_number_of_nat(W_12))) & (zero_zero_nat = number_number_of_nat(W_12) -> one_one_nat = hAPP_nat_nat(power_power_nat(zero_zero_nat),number_number_of_nat(W_12))))) # label(fact_23_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 241 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),M)) -> M = N))) # label(fact_1102_dvd__antisym) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 242 (all X all Y hAPP_nat_int(semiri1621563631at_int,times_times_nat(X,Y)) = times_times_int(hAPP_nat_int(semiri1621563631at_int,X),hAPP_nat_int(semiri1621563631at_int,Y))) # label(fact_932_Nat__Transfer_Otransfer__int__nat__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 243 (all Na all P_1 (-hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(P_1,Na)) -> (exists K_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_2),Na)) & hBOOL(hAPP_nat_bool(P_1,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_1,I))))))))) # label(fact_1179_ex__least__nat__less) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 244 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M)) -> M != N)) # label(fact_187_less__not__refl2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 245 (all A_70 plus_plus_real(zero_zero_real,A_70) = A_70) # label(fact_377_add__0__left) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 246 (all C all D_1 all A all B_1 all M (hBOOL(hAPP_int_bool(zcong(A,B_1),M)) -> (hBOOL(hAPP_int_bool(zcong(C,D_1),M)) -> hBOOL(hAPP_int_bool(zcong(plus_plus_int(A,C),plus_plus_int(B_1,D_1)),M))))) # label(fact_1132_zcong__zadd) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 247 (all K_1 all I_2 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(K_1,I_2)),times_times_nat(K_1,J_1))))) # label(fact_851_mult__le__mono2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 248 (all C_17 all A_42 all B_24 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(C_17,A_42)),times_times_real(C_17,B_24))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_17)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_42),B_24))))) # label(fact_550_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 249 (all A_88 one_one_nat = hAPP_nat_nat(power_power_nat(A_88),zero_zero_nat)) # label(fact_291_power__0) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 250 (all C_11 all D_5 all A_36 all B_18 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_36),B_18)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_11),D_5)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_18)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_11)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_36,C_11)),times_times_nat(B_18,D_5)))))))) # label(fact_569_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 251 (all N all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> nat(hAPP_nat_int(power_power_int(X),N)) = hAPP_nat_nat(power_power_nat(nat(X)),N))) # label(fact_746_Nat__Transfer_Otransfer__nat__int__functions_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 252 (all A_32 all C_10 all B_17 all D_4 minus_minus_real(plus_plus_real(A_32,C_10),plus_plus_real(B_17,D_4)) = plus_plus_real(minus_minus_real(A_32,B_17),minus_minus_real(C_10,D_4))) # label(fact_601_add__diff__add) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 253 (all C_30 all A_65 all B_47 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_65),B_47)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_65,C_30)),plus_plus_int(B_47,C_30))))) # label(fact_390_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 254 (all B_61 all C_45 all A_93 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_93)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_61),C_45)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_61),plus_plus_real(A_93,C_45)))))) # label(fact_259_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 255 (all A_1 all B (is_int(A_1) & is_int(B) -> (zero_zero_int = minus_minus_int(A_1,B) <-> A_1 = B))) # label(fact_592_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 256 (all A_100 A_100 = plus_plus_real(number267125858f_real(pls),A_100)) # label(fact_114_add__numeral__0) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 257 (all Z all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z))))) # label(fact_1100_dvd_Oorder__trans) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 258 (all Z all W (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),Z)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z),W)))) # label(fact_979_real__le__linear) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 259 (all M_9 all N_17 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_9)),hAPP_nat_nat(semiri984289939at_nat,N_17))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_17)))) # label(fact_243_of__nat__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 260 (all A_1 all B (A_1 = B <-> zero_zero_real = minus_minus_real(A_1,B))) # label(fact_593_eq__iff__diff__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 261 (all Z_1 all Wa (is_int(Z_1) & is_int(Wa) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),Wa)) & Z_1 != Wa <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_1),Wa))))) # label(fact_718_zless__le) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 262 (all R_1 all Q all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A)) -> (plus_plus_int(R_1,times_times_int(A,Q)) = 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),one_one_int)))))) # label(fact_828_self__quotient__aux2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 263 (all I_2 all J_1 all K_1 minus_minus_nat(I_2,plus_plus_nat(J_1,K_1)) = minus_minus_nat(minus_minus_nat(I_2,J_1),K_1)) # label(fact_705_diff__diff__left) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 264 (all X all Y (is_int(X) -> X = if_int(fTrue,X,Y))) # label(help_If_1_1_If_000tc__Int__Oint_T) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 265 (all M all I_2 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),plus_plus_nat(M,J_1))))) # label(fact_888_trans__le__add2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 266 (all A_12 all C_5 all B_7 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(A_12,C_5)),plus_plus_int(B_7,C_5))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_12),B_7)))) # label(fact_676_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 267 (all K_1 bit1(K_1) = plus_plus_int(plus_plus_int(one_one_int,K_1),K_1)) # label(fact_128_Bit1__def) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 268 (all X all Y all Z plus_plus_nat(X,plus_plus_nat(Y,Z)) = plus_plus_nat(Y,plus_plus_nat(X,Z))) # label(fact_180_nat__add__left__commute) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 269 (all A_64 all B_46 times_times_int(B_46,A_64) = times_times_int(A_64,B_46)) # label(fact_414_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 270 (all A all N all P_2 (hBOOL(hAPP_int_bool(zprime,P_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),hAPP_nat_int(power_power_int(A),N))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),A))))) # label(fact_994_zprime__zdvd__power) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 271 (all Xa all Ya (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)) <-> Ya != Xa & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)))) # label(fact_1111_dvd_Oless__le) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 272 (all A_91 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_91),plus_plus_int(A_91,one_one_int)))) # label(fact_269_less__add__one) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 273 (all A_73 (is_int(A_73) -> plus_plus_int(A_73,zero_zero_int) = A_73)) # label(fact_364_add__0__right) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 274 (all U_1 all M all N all I_2 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1)) -> minus_minus_nat(M,plus_plus_nat(times_times_nat(minus_minus_nat(J_1,I_2),U_1),N)) = minus_minus_nat(plus_plus_nat(times_times_nat(I_2,U_1),M),plus_plus_nat(times_times_nat(J_1,U_1),N)))) # label(fact_1194_nat__diff__add__eq2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 275 (all A_95 hAPP_nat_nat(power_power_nat(A_95),one_one_nat) = A_95) # label(fact_216_power__one__right) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 276 (all C_31 all A_66 all B_48 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_66),B_48)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(C_31,A_66)),plus_plus_real(C_31,B_48))))) # label(fact_389_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 277 (all X all Y all N (zero_zero_nat != N -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(X),N)),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))))) # label(fact_1051_divides__exp2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 278 (all Z all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z),X))))) # label(fact_1087_dvd_Oless__trans) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 279 (all Lx_5 all Rx_5 all Ry_3 times_times_nat(times_times_nat(Lx_5,Rx_5),Ry_3) = times_times_nat(Lx_5,times_times_nat(Rx_5,Ry_3))) # label(fact_421_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 280 (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_77_rel__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 281 (all A_14 A_14 = times_times_nat(one_one_nat,A_14)) # label(fact_669_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 282 (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_803_less__eq__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 283 (all M minus_minus_nat(M,zero_zero_nat) = M) # label(fact_629_minus__nat_Odiff__0) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 284 (all B_1 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),times_times_int(A,B_1))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))))) # label(fact_779_pos__zmult__pos) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 285 (all Xa all Ya all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(B),Xa)),hAPP_nat_nat(power_power_nat(B),Ya)))))) # label(fact_276_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 286 (all Va (zero_zero_nat = number_number_of_nat(Va) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Va),pls)))) # label(fact_797_eq__number__of__0) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 287 (all Xa (is_int(Xa) -> (one_one_int = Xa <-> one_one_int = Xa))) # label(fact_352_one__reorient) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 288 (all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),pls)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(Xa)),zero_zero_real)))) # label(fact_730_le__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 289 (all N -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N))) # label(fact_186_less__irrefl__nat) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 290 (all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(Xa)),one_one_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),bit1(pls))))) # label(fact_84_less__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 291 (all N_25 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,N_25)),hAPP_nat_int(power_power_int(number_number_of_int(bit0(bit1(pls)))),N_25)))) # label(fact_149_of__nat__less__two__power) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 292 (all V_10 all W_5 times_times_int(number_number_of_int(V_10),number_number_of_int(W_5)) = number_number_of_int(times_times_int(V_10,W_5))) # label(fact_459_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 293 (all M all K_1 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(M,K_1)),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)))) # label(fact_884_add__leD1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 294 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),Ma)) <-> Ma != Na)) # label(fact_178_nat__neq__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 295 (all M all N hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(power_power_nat(M),N)) = hAPP_nat_int(power_power_int(hAPP_nat_int(semiri1621563631at_int,M)),N)) # label(fact_36_zpower__int) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 296 (all C_21 all A_53 all B_34 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_53),B_34)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_21)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_53,C_21)),times_times_nat(B_34,C_21)))))) # label(fact_499_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 297 (all P_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),nat(minus_minus_int(P_2,number_number_of_int(bit0(bit1(pls))))))))) # label(fact_975_Int2_Oaux__2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 298 (all B_32 all A_51 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_51),zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_32)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_51,B_32)),zero_zero_nat))))) # label(fact_504_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 299 (all N all M M = minus_minus_nat(plus_plus_nat(N,M),N)) # label(fact_704_diff__add__inverse) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 300 (all A_79 all B_52 all C_38 plus_plus_nat(plus_plus_nat(A_79,B_52),C_38) = plus_plus_nat(A_79,plus_plus_nat(B_52,C_38))) # label(fact_341_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 301 (all A_1 all C_1 all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_1,C_1)),times_times_int(B,C_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B),A_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_1),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1)))) # label(fact_765_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 302 (all X all Y 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),Y)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(Z,X)),times_times_real(Z,Y)))))) # label(fact_923_real__mult__less__mono2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 303 (all Wa all Ya all Xa all Z_1 (Z_1 = Ya | Xa = Wa <-> plus_plus_nat(times_times_nat(Wa,Ya),times_times_nat(Xa,Z_1)) = plus_plus_nat(times_times_nat(Wa,Z_1),times_times_nat(Xa,Ya)))) # label(fact_638_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 304 (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),bit1(K))))) # label(fact_1000_rel__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 305 (all M all N all K_1 times_times_nat(minus_minus_nat(M,N),K_1) = minus_minus_nat(times_times_nat(M,K_1),times_times_nat(N,K_1))) # label(fact_899_diff__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 306 (all A_41 all C_16 all B_23 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_41,C_16)),times_times_int(B_23,C_16))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_16)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_41),B_23))))) # label(fact_555_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 307 (all K all L_1 (is_int(L_1) & is_int(K) -> (bit0(L_1) = bit0(K) <-> K = L_1))) # label(fact_62_rel__simps_I48_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 308 (all A_1 (is_int(A_1) -> (zero_zero_int = plus_plus_int(A_1,A_1) <-> A_1 = zero_zero_int))) # label(fact_367_double__zero__sym) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 309 (all K_1 all I_2 all J_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),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,times_times_int(K_1,I_2)),times_times_int(K_1,J_1)))))) # label(fact_780_zmult__zless__mono2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 310 (all C_27 all D_10 all A_59 all B_40 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_59),B_40)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_27),D_10)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_40)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_27)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_59,C_27)),times_times_nat(B_40,D_10)))))))) # label(fact_483_mult__mono) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 311 (all C_17 all A_42 all B_24 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(C_17,A_42)),times_times_int(C_17,B_24))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_17)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_42),B_24))))) # label(fact_552_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 312 (all M hAPP_nat_int(semiri1621563631at_int,M) = abs_abs_int(hAPP_nat_int(semiri1621563631at_int,M))) # label(fact_1159_abs__int__eq) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 313 (all A_5 all N_5 all N_4 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_5),N_4)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_5),one_one_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_5),N_4)),hAPP_nat_int(power_power_int(A_5),N_5))))))) # label(fact_734_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 314 (all V_1 ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls)) -> plus_plus_nat(number_number_of_nat(V_1),one_one_nat) = number_number_of_nat(succ(V_1))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls)) -> plus_plus_nat(number_number_of_nat(V_1),one_one_nat) = one_one_nat))) # label(fact_406_nat__number__of__add__1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 315 (all L all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),L)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,minus_minus_nat(L,N)),minus_minus_nat(L,M)))))) # label(fact_693_diff__less__mono2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 316 (all W_7 number_number_of_int(bit1(W_7)) = plus_plus_int(plus_plus_int(one_one_int,number_number_of_int(W_7)),number_number_of_int(W_7))) # label(fact_143_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 317 (all W_8 number267125858f_real(bit0(W_8)) = plus_plus_real(plus_plus_real(zero_zero_real,number267125858f_real(W_8)),number267125858f_real(W_8))) # label(fact_142_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 318 (all Na all K all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Ma)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Na)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,minus_minus_nat(Ma,K)),minus_minus_nat(Na,K))))))) # label(fact_931_less__diff__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 319 (all K_1 all I_2 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(I_2,K_1)),times_times_nat(J_1,K_1)))))) # label(fact_918_mult__less__mono1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 320 (all X all P_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_2)) -> (hBOOL(hAPP_int_bool(zcong(X,number_number_of_int(min)),P_2)) -> -hBOOL(hAPP_int_bool(zcong(X,one_one_int),P_2))))) # label(fact_1078_zcong__neg__1__impl__ne__1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 321 (all I_2 all K_1 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_1)) -> plus_plus_nat(I_2,minus_minus_nat(J_1,K_1)) = minus_minus_nat(plus_plus_nat(I_2,J_1),K_1))) # label(fact_939_add__diff__assoc) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 322 (all N nat(hAPP_nat_int(semiri1621563631at_int,N)) = N) # label(fact_652_nat__int) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 323 (all A_1 all E_1 all C_1 all B all D (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(times_times_real(A_1,E_1),C_1)),plus_plus_real(times_times_real(B,E_1),D))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_1),plus_plus_real(times_times_real(minus_minus_real(B,A_1),E_1),D))))) # label(fact_521_le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 324 (all K_1 all A all B_1 all M (hBOOL(hAPP_int_bool(zcong(A,B_1),M)) -> hBOOL(hAPP_int_bool(zcong(times_times_int(A,K_1),times_times_int(B_1,K_1)),M)))) # label(fact_1129_zcong__scalar) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 325 (all Ma all Va (Ma = nat(number_number_of_int(Va)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(Va))) <-> number_number_of_int(Va) = hAPP_nat_int(semiri1621563631at_int,Ma))) # label(fact_745_int__eq__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 326 (all X_12 X_12 = hAPP_nat_real(power_power_real(X_12),one_one_nat)) # label(fact_401_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 327 (all M all N (zero_zero_nat = minus_minus_nat(M,N) -> (zero_zero_nat = minus_minus_nat(N,M) -> M = N))) # label(fact_627_diffs0__imp__equal) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 328 (all C_32 all D_11 all A_67 all B_49 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_67),B_49)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_32),D_11)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_67,C_32)),plus_plus_int(B_49,D_11)))))) # label(fact_384_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 329 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit0(K1)),bit1(K2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)))) # label(fact_776_less__eq__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 330 (all Xa (zero_zero_real = Xa <-> zero_zero_real = Xa)) # label(fact_315_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 331 (all X (is_int(X) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),number_number_of_int(bit0(bit1(pls))))) -> zero_zero_int = X | X = one_one_int)))) # label(fact_826_int__pos__lt__two__imp__zero__or__one) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 332 -(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)),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))))) # label(fact_1039__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]. 1.51/1.79 333 (all A all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A)) -> (exists X_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_1)) & (all Y_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1)) & A = hAPP_nat_real(power_power_real(Y_1),N) -> Y_1 = X_1)) & hAPP_nat_real(power_power_real(X_1),N) = A))))) # label(fact_403_realpow__pos__nth__unique) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 334 (all A_40 all C_15 all B_22 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(A_40,C_15)),times_times_real(B_22,C_15))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_15)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_40),B_22))))) # label(fact_556_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 335 (all I_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),zero_zero_int)) <-> zero_zero_nat = nat(I_1))) # label(fact_584_nat__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 336 (all N_19 hAPP_nat_real(power_power_real(one_one_real),N_19) = one_one_real) # label(fact_235_power__one) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 337 (all M_2 all N_3 all A_4 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_4)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(A_4),M_2)),hAPP_nat_nat(power_power_nat(A_4),N_3))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_3))))) # label(fact_739_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 338 (all A_41 all C_16 all B_23 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(A_41,C_16)),times_times_real(B_23,C_16))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_16)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_41),B_23))))) # label(fact_553_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 339 (all Ma all Na (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)))) # label(fact_236_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 340 (all C_1 all A_1 all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(C_1,A_1)),plus_plus_real(C_1,B))))) # label(fact_395_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 341 (all A_25 zero_zero_nat = times_times_nat(A_25,zero_zero_nat)) # label(fact_622_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 342 (all C_34 all A_69 all B_51 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(C_34,A_69)),plus_plus_int(C_34,B_51))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_69),B_51)))) # label(fact_378_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 343 (all A all B_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1)) -> (A != B_1 -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),A)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1))))) # label(fact_1105_dvd_Ole__neq__trans) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 344 (all N_24 all A_97 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_97)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_24)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_nat_real(power_power_real(A_97),N_24)))))) # label(fact_159_one__less__power) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 345 (all X_14 all Y_10 (Y_10 != X_14 -> (-hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_14),Y_10)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_10),X_14))))) # label(fact_210_linorder__neqE__linordered__idom) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 346 (all A_82 all B_55 all C_41 all D_14 plus_plus_int(plus_plus_int(A_82,B_55),plus_plus_int(C_41,D_14)) = plus_plus_int(plus_plus_int(A_82,C_41),plus_plus_int(B_55,D_14))) # label(fact_325_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 347 (all C_1 all D all A_1 all B all Ma (hBOOL(hAPP_int_bool(zcong(A_1,B),Ma)) -> (hBOOL(hAPP_int_bool(zcong(C_1,times_times_int(D,A_1)),Ma)) <-> hBOOL(hAPP_int_bool(zcong(C_1,times_times_int(D,B)),Ma))))) # label(fact_1065_zcong__zmult__prop2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 348 (all Ma all Na (zero_zero_nat = Na & Ma = zero_zero_nat <-> zero_zero_nat = plus_plus_nat(Ma,Na))) # label(fact_212_add__is__0) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 349 (all M_4 all N_10 hAPP_nat_real(semiri132038758t_real,times_times_nat(M_4,N_10)) = times_times_real(hAPP_nat_real(semiri132038758t_real,M_4),hAPP_nat_real(semiri132038758t_real,N_10))) # label(fact_469_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 1.51/1.79 350 (all C_20 all A_45 all B_27 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(C_20,A_45)),times_times_nat(C_20,B_27))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_20)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_45),B_27))))) # label(fact_542_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 351 (all I_2 all J_1 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(I_2,J_1)),I_2))) # label(fact_175_not__add__less1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 352 (all M_6 all N_13 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),M_6)),hAPP_nat_real(power_power_real(A_90),N_13))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_6),N_13))))) # label(fact_280_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 353 (all V_16 all W_11 all Z_6 plus_plus_real(number267125858f_real(plus_plus_int(V_16,W_11)),Z_6) = plus_plus_real(number267125858f_real(V_16),plus_plus_real(number267125858f_real(W_11),Z_6))) # label(fact_121_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 354 (all A_19 times_times_real(A_19,one_one_real) = A_19) # label(fact_653_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 355 (all A_84 all B_57 all C_43 (plus_plus_nat(A_84,B_57) = plus_plus_nat(A_84,C_43) -> C_43 = B_57)) # label(fact_320_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 356 (all Z_2 all Z all W_2 all W (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_2),W)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_2),Z)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(W_2,Z_2)),plus_plus_int(W,Z)))))) # label(fact_783_zadd__zless__mono) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 357 (all K_1 all I_2 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),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_2),K_1))))) # label(fact_852_le__trans) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 358 (all K_1 all L bit1(L) != bit0(K_1)) # label(fact_94_rel__simps_I49_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 359 (all C_17 all A_42 all B_24 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(C_17,A_42)),times_times_nat(C_17,B_24))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_17)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_42),B_24))))) # label(fact_551_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 360 (all R_1 all Q all A (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A)) -> (plus_plus_int(R_1,times_times_int(A,Q)) = A -> (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)))))) # label(fact_829_self__quotient__aux1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 361 (all Lx_5 all Rx_5 all Ry_3 times_times_int(Lx_5,times_times_int(Rx_5,Ry_3)) = times_times_int(times_times_int(Lx_5,Rx_5),Ry_3)) # label(fact_420_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 362 (all X all Y all Z hAPP_nat_int(power_power_int(hAPP_nat_int(power_power_int(X),Y)),Z) = hAPP_nat_int(power_power_int(X),times_times_nat(Y,Z))) # label(fact_910_zpower__zpower) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 363 (all M all Y all X (is_int(X) & is_int(Y) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Y)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M)) -> (hBOOL(hAPP_int_bool(zcong(X,Y),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),M)) -> Y = X)))))))) # label(fact_1062_zcong__less__eq) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 364 (all A all X (is_int(X) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),A)) -> (minus_minus_int(A,one_one_int) != X -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),minus_minus_int(A,one_one_int)))))))) # label(fact_915_Euler_Oaux1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 365 (all Y_6 all X_7 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_7)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_6)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_6),one_one_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(Y_6,X_7)),X_7)))))) # label(fact_582_mult__left__le__one__le) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 366 (all B_1 all A all P_2 (hBOOL(hAPP_int_bool(zprime,P_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A)) -> (hBOOL(hAPP_int_bool(zcong(times_times_int(A,B_1),zero_zero_int),P_2)) -> hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_2)) | hBOOL(hAPP_int_bool(zcong(B_1,zero_zero_int),P_2)))))) # label(fact_1070_zcong__zprime__prod__zero) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 367 (all B_1 all Q all R_1 all B_2 all Q_1 all R_2 (plus_plus_int(times_times_int(B_1,Q),R_1) = plus_plus_int(times_times_int(B_2,Q_1),R_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),plus_plus_int(times_times_int(B_2,Q_1),R_2))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),B_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),B_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q),Q_1))))))))) # label(fact_971_zdiv__mono2__lemma) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 368 (all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N))) # label(fact_870_less__eq__nat_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 369 (all B_1_1 all B_2_1 all B_3_1 (is_int(B_3_1) & is_int(B_2_1) & is_bool(B_1_1) -> is_int(if_int(B_1_1,B_2_1,B_3_1)))) # label(gsy_c_If_000tc__Int__Oint) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 370 (all K_1 all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),minus_minus_nat(M,N))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),M)))))) # label(fact_1048_dvd__diffD) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 371 (all W_13 plus_plus_real(one_one_real,number267125858f_real(W_13)) = number267125858f_real(plus_plus_int(bit1(pls),W_13))) # label(fact_13_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 372 (all Ma all Na (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,Ma)),hAPP_nat_real(semiri132038758t_real,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)))) # label(fact_238_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 373 (all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),zero_zero_nat)) <-> zero_zero_nat = Na)) # label(fact_871_le__0__eq) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 374 (all K_1 all I_2 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(I_2,K_1)),plus_plus_nat(J_1,K_1))))) # label(fact_887_add__le__mono1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 375 (all A_20 all B_9 all C_7 plus_plus_real(times_times_real(A_20,C_7),times_times_real(B_9,C_7)) = times_times_real(plus_plus_real(A_20,B_9),C_7)) # label(fact_643_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 376 (all P_1 all Ma all Na ((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Na),Ma))) -> ((Ma = Na -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Na),Ma))) -> ((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),Ma)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Na),Ma))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Na),Ma)))))) # label(fact_195_nat__less__cases) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 377 (all M_10 all N_18 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_18)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_10)),hAPP_nat_int(semiri1621563631at_int,N_18))))) # label(fact_239_less__imp__of__nat__less) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 378 (all A_1 all C_1 all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(A_1,C_1)),plus_plus_nat(B,C_1))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),B)))) # label(fact_690_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 379 (all J all K all I_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,minus_minus_nat(J,K)),I_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),plus_plus_nat(I_1,K))))) # label(fact_936_le__diff__conv) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 380 (all B_58 all A_85 all C_44 (is_int(B_58) & is_int(C_44) -> (plus_plus_int(C_44,A_85) = plus_plus_int(B_58,A_85) -> C_44 = B_58))) # label(fact_316_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 381 (all A_1 all E_1 all C_1 all B all D (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(times_times_real(A_1,E_1),C_1)),plus_plus_real(times_times_real(B,E_1),D))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(times_times_real(minus_minus_real(A_1,B),E_1),C_1)),D)))) # label(fact_535_less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 382 (all N (zero_zero_nat != N -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)))) # label(fact_196_gr0I) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 383 (all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Ya)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),number_number_of_int(Ya))))) # label(fact_71_less__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 384 (all V_5 all U_3 all Y_3 all X_4 all A_6 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_4),A_6)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_3),A_6)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),U_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),V_5)) -> (plus_plus_int(U_3,V_5) = one_one_int -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(U_3,X_4),times_times_int(V_5,Y_3))),A_6)))))))) # label(fact_725_convex__bound__le) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 385 (all A_1 all B all C_1 (is_int(B) & is_int(C_1) -> (C_1 = B <-> plus_plus_int(A_1,C_1) = plus_plus_int(A_1,B)))) # label(fact_331_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 386 (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]. 1.51/1.80 387 (all K_1 (is_int(K_1) -> number_number_of_int(K_1) = K_1)) # label(fact_177_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 388 (all Na all Ma (is_int(Ma) -> (zero_zero_int != Ma -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,times_times_int(Ma,Na)),Ma)) <-> abs_abs_int(Na) = one_one_int)))) # label(fact_1166_zdvd__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 389 (all A_1 (is_int(A_1) -> (A_1 != zero_zero_int <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(power_power_int(A_1),number_number_of_nat(bit0(bit1(pls))))))))) # label(fact_133_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 390 (all M M = plus_plus_nat(M,zero_zero_nat)) # label(fact_213_Nat_Oadd__0__right) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 391 (all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(N))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(bit0(N)))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),number_number_of_int(bit1(N)))))) # label(fact_1152_number__of1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 392 (all A_82 all B_55 all C_41 all D_14 plus_plus_real(plus_plus_real(A_82,B_55),plus_plus_real(C_41,D_14)) = plus_plus_real(plus_plus_real(A_82,C_41),plus_plus_real(B_55,D_14))) # label(fact_327_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 393 (all Ma all Na (hAPP_nat_int(semiri1621563631at_int,Ma) = hAPP_nat_int(semiri1621563631at_int,Na) <-> Ma = Na)) # label(fact_68_int__int__eq) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 394 (all Xa all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Xa)),hAPP_nat_int(semiri1621563631at_int,Ya))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)))) # label(fact_205_Nat__Transfer_Otransfer__int__nat__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 395 (all B_1 all A all P_2 (hBOOL(hAPP_int_bool(zprime,P_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A)) -> (-hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_2)) & -hBOOL(hAPP_int_bool(zcong(B_1,zero_zero_int),P_2)) -> -hBOOL(hAPP_int_bool(zcong(times_times_int(A,B_1),zero_zero_int),P_2)))))) # label(fact_1071_zcong__zprime__prod__zero__contra) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 396 (all L_1 (is_int(L_1) -> (min = L_1 <-> min = bit1(L_1)))) # label(fact_1018_rel__simps_I43_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 397 (all A_84 all B_57 all C_43 (plus_plus_real(A_84,B_57) = plus_plus_real(A_84,C_43) -> C_43 = B_57)) # label(fact_321_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 398 (all A_96 all N_22 all N_21 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_22),N_21)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_96)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_96),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_96),N_21)),hAPP_nat_real(power_power_real(A_96),N_22))))))) # label(fact_165_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 399 (all Z (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),zero_zero_int)) -> nat(Z) = zero_zero_nat)) # label(fact_583_nat__le__0) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 400 (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_777_rel__simps_I21_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 401 (all A_73 A_73 = plus_plus_real(A_73,zero_zero_real)) # label(fact_366_add__0__right) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 402 (all A_1 (zero_zero_real = plus_plus_real(A_1,A_1) <-> zero_zero_real = A_1)) # label(fact_90_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 403 (all A_1 all C_1 all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(A_1,C_1)),times_times_real(B,C_1))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_1),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B),A_1)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B)))) # label(fact_764_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 404 -(hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n))),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))) -> -hBOOL(hAPP_int_bool(twoSqu1154269391sum2sq,times_times_int(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n)))))) # label(fact_830_smaller_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 405 (all P_1 ((exists X1 hBOOL(hAPP_nat_bool(P_1,X1))) <-> (exists X_1 (is_int(X_1) & hBOOL(hAPP_nat_bool(P_1,nat(X_1))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)))))) # label(fact_532_ex__nat) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 406 (all K all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,times_times_nat(K,Ma)),times_times_nat(K,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),Na)) | zero_zero_nat = K)) # label(fact_1187_nat__mult__dvd__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 407 (all N all B_1 all A all P_2 (hBOOL(hAPP_int_bool(zprime,P_2)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),A)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(P_2),N)),times_times_int(A,B_1))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(P_2),N)),B_1)))))) # label(fact_1032_zprime__power__zdvd__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 408 (all N_26 number_number_of_nat(hAPP_nat_int(semiri1621563631at_int,N_26)) = hAPP_nat_nat(semiri984289939at_nat,N_26)) # label(fact_130_number__of__int) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 409 (all A_26 all B_13 (is_int(A_26) & is_int(B_13) -> (zero_zero_int = times_times_int(A_26,B_13) -> A_26 = zero_zero_int | zero_zero_int = B_13))) # label(fact_620_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 410 (all Xa (Xa = one_one_real <-> Xa = one_one_real)) # label(fact_354_one__reorient) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 411 (all C_1 all D all A_1 all B (is_int(C_1) & is_int(D) & is_int(A_1) & is_int(B) -> (B != A_1 & C_1 != D <-> plus_plus_int(times_times_int(A_1,D),times_times_int(B,C_1)) != plus_plus_int(times_times_int(A_1,C_1),times_times_int(B,D))))) # label(fact_648_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 412 (all L pls != bit1(L)) # label(fact_92_rel__simps_I39_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 413 (all A_74 (is_int(A_74) -> plus_plus_int(A_74,zero_zero_int) = A_74)) # label(fact_361_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 414 (all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(Xa)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),pls)))) # label(fact_731_le__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 415 (all A all P_2 ((hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_2)) -> zero_zero_int = legendre(A,P_2)) & (-hBOOL(hAPP_int_bool(zcong(A,zero_zero_int),P_2)) -> (hBOOL(hAPP_int_bool(quadRes(P_2),A)) -> legendre(A,P_2) = one_one_int) & (-hBOOL(hAPP_int_bool(quadRes(P_2),A)) -> legendre(A,P_2) = number_number_of_int(min))))) # label(fact_1119_Legendre__def) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 416 (all A_89 all N_12 all N_11 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_12),N_11)) -> (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,hAPP_nat_nat(power_power_nat(A_89),N_12)),hAPP_nat_nat(power_power_nat(A_89),N_11)))))) # label(fact_282_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 417 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) | N = M -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)))) # label(fact_874_less__or__eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 418 (all M_8 all N_16 hAPP_nat_real(semiri132038758t_real,plus_plus_nat(M_8,N_16)) = plus_plus_real(hAPP_nat_real(semiri132038758t_real,M_8),hAPP_nat_real(semiri132038758t_real,N_16))) # label(fact_247_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 419 (all M_9 all N_17 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_9)),hAPP_nat_int(semiri1621563631at_int,N_17))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_17)))) # label(fact_242_of__nat__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 420 (all A_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(A_1,A_1))))) # label(fact_303_zero__less__double__add__iff__zero__less__single__add) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 421 (all A_31 all B_16 (is_int(A_31) -> A_31 = minus_minus_int(plus_plus_int(A_31,B_16),B_16))) # label(fact_604_add__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 422 (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_800_rel__simps_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 423 (all C_2 all A_9 all B_4 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_9),B_4)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(A_9,C_2)),plus_plus_int(B_4,C_2))))) # label(fact_685_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 424 (all Z1 all Z2 all W plus_plus_int(times_times_int(Z1,W),times_times_int(Z2,W)) = times_times_int(plus_plus_int(Z1,Z2),W)) # label(fact_720_zadd__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 425 (all X_5 all Y_4 all Q_2 hAPP_nat_real(power_power_real(times_times_real(X_5,Y_4)),Q_2) = times_times_real(hAPP_nat_real(power_power_real(X_5),Q_2),hAPP_nat_real(power_power_real(Y_4),Q_2))) # label(fact_700_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 426 (all K all L_1 (is_int(K) & is_int(L_1) -> (K = L_1 <-> bit1(L_1) = bit1(K)))) # label(fact_61_rel__simps_I51_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 427 (all X hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),plus_plus_real(one_one_real,abs_abs_real(X))))) # label(fact_1177_abs__add__one__gt__zero) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 428 (all Ma all Na (Ma = Na <-> hAPP_nat_int(semiri1621563631at_int,Ma) = hAPP_nat_int(semiri1621563631at_int,Na))) # label(fact_220_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 429 (all C_22 all A_54 all B_35 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_54),B_35)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_22)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(C_22,A_54)),times_times_real(C_22,B_35)))))) # label(fact_495_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 430 (all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),pls)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(Xa)),zero_zero_real)))) # label(fact_70_less__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 431 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)) | Ya = Xa <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)))) # label(fact_880_less__eq__real__def) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 432 (all A_1 (is_int(A_1) -> (zero_zero_int = A_1 <-> zero_zero_int = plus_plus_int(A_1,A_1)))) # label(fact_89_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 433 (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_74_less__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 434 (all A all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A)) -> (exists R (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R)) & hAPP_nat_real(power_power_real(R),N) = A))))) # label(fact_404_realpow__pos__nth) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 435 (all L all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,minus_minus_nat(M,L)),minus_minus_nat(N,L))))) # label(fact_906_diff__le__mono) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 436 (all A_16 times_times_real(one_one_real,A_16) = A_16) # label(fact_662_mult__1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 437 (all K_1 all M hBOOL(hAPP_int_bool(zcong(K_1,K_1),M))) # label(fact_1124_zcong__refl) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 438 (all A all B_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),minus_minus_int(B_1,one_one_int))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A),B_1)))) # label(fact_1138_norR__mem__unique__aux) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 439 (all Ma all Na all K (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K)) -> (times_times_nat(K,Ma) = times_times_nat(K,Na) <-> Na = Ma))) # label(fact_1186_nat__mult__eq__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 440 (all Xa all Na (zero_zero_nat = Na | Xa = one_one_nat <-> hAPP_nat_nat(power_power_nat(Xa),Na) = one_one_nat)) # label(fact_299_exp__eq__1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 441 (all X_12 (is_int(X_12) -> X_12 = hAPP_nat_int(power_power_int(X_12),one_one_nat))) # label(fact_399_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 442 (all A_63 all B_45 all C_29 times_times_int(times_times_int(A_63,B_45),C_29) = times_times_int(A_63,times_times_int(B_45,C_29))) # label(fact_423_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 443 (all Wa all Ma (is_int(Wa) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa)) -> Wa = hAPP_nat_int(semiri1621563631at_int,Ma)) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa)) -> Ma = zero_zero_nat) <-> Ma = nat(Wa)))) # label(fact_742_nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 444 (all A_1 all C_1 all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(A_1,C_1)),plus_plus_real(B,C_1))))) # label(fact_689_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 445 (all L bit1(minus_minus_int(min,L)) = minus_minus_int(pls,bit1(L))) # label(fact_1026_diff__bin__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 446 (all A hAPP_nat_int(power_power_int(A),number_number_of_nat(bit1(bit1(pls)))) = times_times_int(A,hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls)))))) # label(fact_824_cube__square) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 447 (all B_30 all A_49 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_49)) -> (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,times_times_real(A_49,B_30)),zero_zero_real))))) # label(fact_509_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 448 (all X all Y (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))))) # label(fact_1096_dvd_Oless__not__sym) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 449 (all N_24 all A_97 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_97)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_24)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_nat_nat(power_power_nat(A_97),N_24)))))) # label(fact_158_one__less__power) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 450 (all A_34 minus_minus_int(A_34,A_34) = zero_zero_int) # label(fact_596_diff__self) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 451 (all V_4 all U_2 all Y_2 all X_3 all A_3 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_3),A_3)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_2),A_3)) -> (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_4)) -> (one_one_real = plus_plus_real(U_2,V_4) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(times_times_real(U_2,X_3),times_times_real(V_4,Y_2))),A_3)))))))) # label(fact_748_convex__bound__lt) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 452 (all Z_2 all Z (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z)) -> times_times_nat(nat(Z),nat(Z_2)) = nat(times_times_int(Z,Z_2)))) # label(fact_953_nat__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 453 (all Z plus_plus_nat(Z,Z) = times_times_nat(number_number_of_nat(bit0(bit1(pls))),Z)) # label(fact_958_nat__mult__2) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 454 (all K_1 all I_2 all J_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,I_2),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_2),K_1))))) # label(fact_981_real__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 455 (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,number_number_of_int(K)),number_number_of_int(L_1))))) # label(fact_723_less__eq__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 456 (all K all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(K,Ma)),times_times_nat(K,Na))))) # label(fact_920_mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 457 (all A_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(A_1,A_1)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),zero_zero_real)))) # label(fact_64_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 458 (all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(Xa)),one_one_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),bit1(pls))))) # label(fact_754_le__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 459 (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]. 1.51/1.80 460 (all B all A_1 (zero_zero_nat = A_1 <-> B = plus_plus_nat(B,A_1))) # label(fact_356_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 461 (all W_12 ((zero_zero_nat != number_number_of_nat(W_12) -> zero_zero_int = hAPP_nat_int(power_power_int(zero_zero_int),number_number_of_nat(W_12))) & (number_number_of_nat(W_12) = zero_zero_nat -> one_one_int = hAPP_nat_int(power_power_int(zero_zero_int),number_number_of_nat(W_12))))) # label(fact_22_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 462 (all Lx_4 all Ly_4 all Rx_4 times_times_nat(times_times_nat(Lx_4,Ly_4),Rx_4) = times_times_nat(Lx_4,times_times_nat(Ly_4,Rx_4))) # label(fact_427_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 1.51/1.80 463 (all A_96 all N_22 all N_21 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_22),N_21)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_96)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_96),one_one_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(A_96),N_21)),hAPP_nat_nat(power_power_nat(A_96),N_22))))))) # label(fact_164_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 464 (all C all A all B_1 all M (hBOOL(hAPP_int_bool(zcong(A,B_1),M)) -> (hBOOL(hAPP_int_bool(zcong(B_1,C),M)) -> hBOOL(hAPP_int_bool(zcong(A,C),M))))) # label(fact_1123_zcong__trans) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 465 (all A_78 all C_37 all D_13 plus_plus_int(A_78,plus_plus_int(C_37,D_13)) = plus_plus_int(plus_plus_int(A_78,C_37),D_13)) # label(fact_343_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 466 (all K_1 all I_2 all J_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),J_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(K_1,I_2)),plus_plus_int(K_1,J_1))))) # label(fact_721_zadd__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 467 (all A_71 (is_int(A_71) -> plus_plus_int(zero_zero_int,A_71) = A_71)) # label(fact_372_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 468 (all A_1 all B all C_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_1)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(C_1,A_1)),times_times_real(C_1,B))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B))))) # label(fact_573_mult__le__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 469 (all Va all V_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Va),V_3)) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Va),V_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V_3))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,number_number_of_nat(Va)),number_number_of_nat(V_3))))) # label(fact_208_less__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 470 (all Y all N all P_2 (hBOOL(hAPP_int_bool(zprime,P_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),hAPP_nat_int(power_power_int(Y),N))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),Y)))))) # label(fact_1033_zpower__zdvd__prop2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 471 (all C_13 all D_7 all A_38 all B_20 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_38),B_20)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_13),D_7)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_38)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_13)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(A_38,C_13)),times_times_real(B_20,D_7)))))))) # label(fact_562_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 472 (all X_13 one_one_int = hAPP_nat_int(power_power_int(X_13),zero_zero_nat)) # label(fact_300_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 473 (all C_27 all D_10 all A_59 all B_40 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_59),B_40)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_27),D_10)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_40)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_27)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_59,C_27)),times_times_int(B_40,D_10)))))))) # label(fact_484_mult__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 474 (all M all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),M)) -> -hBOOL(hAPP_int_bool(zcong(X,zero_zero_int),M))))) # label(fact_1063_zcong__not__zero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 475 (all K all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(K,Ma)),times_times_nat(K,Na))) <-> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na))))) # label(fact_857_mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 476 (all N times_times_nat(one_one_nat,N) = N) # label(fact_898_nat__mult__1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 477 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),K)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),bit1(K))))) # label(fact_1005_rel__simps_I26_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 478 (all I_1 all K all J (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),minus_minus_nat(J,K))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(I_1,K)),J))))) # label(fact_940_le__diff__conv2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 479 (all N_14 all A_92 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_92)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(power_power_real(A_92),N_14))))) # label(fact_268_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 480 (all C_23 all A_55 all B_36 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_55),B_36)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_23)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(C_23,A_55)),times_times_nat(C_23,B_36)))))) # label(fact_493_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 481 (all V_11 all W_6 times_times_real(number267125858f_real(V_11),number267125858f_real(W_6)) = number267125858f_real(times_times_int(V_11,W_6))) # label(fact_456_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 482 (all C_2 all A_9 all B_4 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_9),B_4)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(A_9,C_2)),plus_plus_nat(B_4,C_2))))) # label(fact_684_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 483 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> (N != M -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))))) # label(fact_875_le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 484 (all K_1 bit0(K_1) != min) # label(fact_1014_rel__simps_I45_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 485 (all A_1 (zero_zero_real = A_1 <-> zero_zero_real = hAPP_nat_real(power_power_real(A_1),number_number_of_nat(bit0(bit1(pls)))))) # label(fact_11_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 486 (all C_1 all D all A_1 all B (B != A_1 & C_1 != D <-> plus_plus_real(times_times_real(A_1,C_1),times_times_real(B,D)) != plus_plus_real(times_times_real(A_1,D),times_times_real(B,C_1)))) # label(fact_646_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 487 (all N minus_minus_nat(zero_zero_nat,N) = zero_zero_nat) # label(fact_630_diff__0__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 488 (all A_1 all B all C_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_1),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B),A_1)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(C_1,A_1)),times_times_real(C_1,B)))))) # label(fact_571_mult__le__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 489 (all M all N M = minus_minus_nat(plus_plus_nat(M,N),N)) # label(fact_703_diff__add__inverse2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 490 (all B all A_1 all C_1 (is_int(B) & is_int(C_1) -> (plus_plus_int(C_1,A_1) = plus_plus_int(B,A_1) <-> B = C_1))) # label(fact_328_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 491 (all A_7 all B_3 all N_6 hAPP_nat_nat(power_power_nat(times_times_nat(A_7,B_3)),N_6) = times_times_nat(hAPP_nat_nat(power_power_nat(A_7),N_6),hAPP_nat_nat(power_power_nat(B_3),N_6))) # label(fact_698_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 492 (all Xa all Ya (hAPP_nat_int(semiri1621563631at_int,Xa) = hAPP_nat_int(semiri1621563631at_int,Ya) <-> Ya = Xa)) # label(fact_221_Nat__Transfer_Otransfer__int__nat__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 493 (all Y all X all P_2 (-hBOOL(hAPP_int_bool(zcong(X,zero_zero_int),P_2)) -> (hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(Y),number_number_of_nat(bit0(bit1(pls)))),X),P_2)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),Y))))) # label(fact_1079_Euler_Oaux____1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 494 (all A_1 all B all C_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(C_1,A_1)),times_times_int(C_1,B)))))) # label(fact_574_mult__le__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 495 (all A_77 all C_36 all D_12 plus_plus_real(A_77,plus_plus_real(C_36,D_12)) = plus_plus_real(C_36,plus_plus_real(A_77,D_12))) # label(fact_348_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 496 (all S_1 all T_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S_1),T_1)) -> T_1 != S_1)) # label(fact_188_less__not__refl3) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 497 (all A_89 all N_12 all N_11 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_12),N_11)) -> (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,hAPP_nat_int(power_power_int(A_89),N_12)),hAPP_nat_int(power_power_int(A_89),N_11)))))) # label(fact_281_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 498 (all A_83 all B_56 all C_42 (plus_plus_real(A_83,B_56) = plus_plus_real(A_83,C_42) -> B_56 = C_42)) # label(fact_324_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 499 (all K_1 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N))))) # label(fact_1068_dvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 500 (all Wa all Xa (Xa = number_number_of_nat(Wa) <-> number_number_of_nat(Wa) = Xa)) # label(fact_58_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 501 (all Ma all Na all K (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),Na)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,times_times_nat(K,Ma)),times_times_nat(K,Na)))))) # label(fact_1188_nat__mult__dvd__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 502 (all A_44 all C_19 all B_26 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(A_44,C_19)),times_times_real(B_26,C_19))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_19)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_44),B_26))))) # label(fact_544_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 503 (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_1006_rel__simps_I30_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 504 (all N -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat))) # label(fact_203_not__less0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 505 (all A_63 all B_45 all C_29 times_times_real(A_63,times_times_real(B_45,C_29)) = times_times_real(times_times_real(A_63,B_45),C_29)) # label(fact_425_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 506 (all V_16 all W_11 all Z_6 plus_plus_int(number_number_of_int(plus_plus_int(V_16,W_11)),Z_6) = plus_plus_int(number_number_of_int(V_16),plus_plus_int(number_number_of_int(W_11),Z_6))) # label(fact_120_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 507 (all U all Ma all Na all J all I_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(times_times_nat(minus_minus_nat(I_1,J),U),Ma)),Na)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(times_times_nat(I_1,U),Ma)),plus_plus_nat(times_times_nat(J,U),Na)))))) # label(fact_1190_nat__le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 508 (all A_76 all C_35 plus_plus_int(A_76,C_35) = plus_plus_int(C_35,A_76)) # label(fact_349_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 509 (all X all Y all Z hAPP_nat_int(power_power_int(X),plus_plus_nat(Y,Z)) = times_times_int(hAPP_nat_int(power_power_int(X),Y),hAPP_nat_int(power_power_int(X),Z))) # label(fact_790_zpower__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 510 (all A_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_1)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),plus_plus_real(A_1,A_1))))) # label(fact_304_zero__less__double__add__iff__zero__less__single__add) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 511 (all I_2 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_1)) -> (exists K_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2)) & J_1 = plus_plus_nat(I_2,K_2))))) # label(fact_298_less__imp__add__positive) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 512 (all A_16 (is_int(A_16) -> A_16 = times_times_int(one_one_int,A_16))) # label(fact_664_mult__1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 513 (all P_1 all K all I_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),I_1)) -> (hBOOL(hAPP_int_bool(P_1,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_1,I)) -> hBOOL(hAPP_int_bool(P_1,plus_plus_int(I,one_one_int))))))) -> hBOOL(hAPP_int_bool(P_1,I_1)))))) # label(fact_169_int__gr__induct) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 514 (all N all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,M),N)) -> -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),M))))) # label(fact_995_zdvd__not__zless) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 515 (all Lx_2 all Ly_2 all Rx_2 all Ry_2 times_times_real(Lx_2,times_times_real(Ly_2,times_times_real(Rx_2,Ry_2))) = times_times_real(times_times_real(Lx_2,Ly_2),times_times_real(Rx_2,Ry_2))) # label(fact_434_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 516 (all A_12 all C_5 all B_7 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(A_12,C_5)),plus_plus_real(B_7,C_5))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_12),B_7)))) # label(fact_674_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 517 (all C_4 all D_3 all A_11 all B_6 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_11),B_6)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_4),D_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(A_11,C_4)),plus_plus_nat(B_6,D_3)))))) # label(fact_678_add__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 518 (all Ma all Z_1 (is_int(Z_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) & Ma = nat(Z_1) <-> Z_1 = hAPP_nat_int(semiri1621563631at_int,Ma)))) # label(fact_589_int__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 519 (all Xa all Ya (Ya = zero_zero_real & Xa = zero_zero_real <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(times_times_real(Xa,Xa),times_times_real(Ya,Ya))),zero_zero_real)))) # label(fact_577_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 520 (all A_1 all E_1 all C_1 all B all D (D = plus_plus_real(times_times_real(minus_minus_real(A_1,B),E_1),C_1) <-> plus_plus_real(times_times_real(A_1,E_1),C_1) = plus_plus_real(times_times_real(B,E_1),D))) # label(fact_519_eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 521 (all Z1 all Z2 all W times_times_real(plus_plus_real(Z1,Z2),W) = plus_plus_real(times_times_real(Z1,W),times_times_real(Z2,W))) # label(fact_909_real__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 522 (all A_68 all C_33 all B_50 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_68,C_33)),plus_plus_int(B_50,C_33))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_68),B_50)))) # label(fact_381_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 523 (all N_14 all A_92 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_92)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(A_92),N_14))))) # label(fact_267_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 524 (all X all Y if_nat(fFalse,X,Y) = Y) # label(help_If_2_1_If_000tc__Nat__Onat_T) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 525 (all X_2 all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2)) -> hAPP_nat_int(power_power_int(X_2),N_2) = times_times_int(hAPP_nat_int(power_power_int(X_2),minus_minus_nat(N_2,one_one_nat)),X_2))) # label(fact_757_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 526 (all A_20 all B_9 all C_7 times_times_nat(plus_plus_nat(A_20,B_9),C_7) = plus_plus_nat(times_times_nat(A_20,C_7),times_times_nat(B_9,C_7))) # label(fact_644_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 527 (all C_6 all A_13 all B_8 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(C_6,A_13)),plus_plus_real(C_6,B_8))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_13),B_8)))) # label(fact_671_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 528 (all Xa all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(Xa)),number267125858f_real(Ya))))) # label(fact_454_le__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 529 (all W_13 plus_plus_int(one_one_int,number_number_of_int(W_13)) = number_number_of_int(plus_plus_int(bit1(pls),W_13))) # label(fact_12_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 530 (all Z all X all Y all P_2 (hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(X),Y),one_one_int),P_2)) -> hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(X),times_times_nat(Y,Z)),one_one_int),P_2)))) # label(fact_1151_zcong__zpower__zmult) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 531 (all C all A all B_1 (B_1 = A -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),C)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C),B_1)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C),A)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),C))))) # label(fact_1099_dvd_Oord__eq__less__trans) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 532 (all X all N hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(power_power_int(abs_abs_int(X)),N)))) # label(fact_1161_zero__le__zpower__abs) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 533 (all M all N all K_1 times_times_nat(times_times_nat(M,N),K_1) = times_times_nat(M,times_times_nat(N,K_1))) # label(fact_848_nat__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 534 (all Ma all Na (Na = Ma <-> hAPP_nat_nat(semiri984289939at_nat,Na) = hAPP_nat_nat(semiri984289939at_nat,Ma))) # label(fact_219_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 535 (all A_1 (is_int(A_1) -> (A_1 = zero_zero_int <-> hAPP_nat_int(power_power_int(A_1),number_number_of_nat(bit0(bit1(pls)))) = zero_zero_int))) # label(fact_10_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 536 (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_1116_neg__one__power__eq__mod__m) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 537 (all Ma all Na (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)))) # label(fact_207_zless__int) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 538 (all B_61 all C_45 all A_93 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_93)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_61),C_45)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_61),plus_plus_int(A_93,C_45)))))) # label(fact_257_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 539 (all Va (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),number_number_of_nat(Va))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Va)))) # label(fact_223_less__0__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 540 (all M all N (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> plus_plus_nat(N,minus_minus_nat(M,N)) = M)) # label(fact_773_add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 541 (all A_24 zero_zero_nat = times_times_nat(zero_zero_nat,A_24)) # label(fact_625_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 542 (all K_1 all L bit1(minus_minus_int(K_1,L)) = minus_minus_int(bit1(K_1),bit0(L))) # label(fact_913_diff__bin__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 543 -(all S (is_int(S) -> -(hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,S),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))) & hBOOL(hAPP_int_bool(zcong(s1,S),plus_plus_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))))) # label(fact_1085__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]. 1.53/1.80 544 (all Z (is_int(Z) -> Z = times_times_int(one_one_int,Z))) # label(fact_716_zmult__1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 545 (all C_12 all D_6 all A_37 all B_19 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_37),B_19)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_12),D_6)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_37)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_12)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_37,C_12)),times_times_nat(B_19,D_6)))))))) # label(fact_566_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 546 (all M_4 all N_10 times_times_int(hAPP_nat_int(semiri1621563631at_int,M_4),hAPP_nat_int(semiri1621563631at_int,N_10)) = hAPP_nat_int(semiri1621563631at_int,times_times_nat(M_4,N_10))) # label(fact_471_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 547 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,minus_minus_real(Xa,Ya)),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)))) # label(fact_855_real__le__eq__diff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 548 (all X all Y (is_int(X) & is_int(Y) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Y)) | X = Y | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),X)))) # label(fact_30_zless__linear) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 549 (all A_76 all C_35 plus_plus_real(C_35,A_76) = plus_plus_real(A_76,C_35)) # label(fact_351_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 550 (all K_1 bit1(K_1) = succ(bit0(K_1))) # label(fact_809_succ__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 551 (all M all N hAPP_nat_int(power_power_int(hAPP_nat_int(semiri1621563631at_int,M)),N) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(power_power_nat(M),N))) # label(fact_37_int__power) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 552 (all M all N (abs_abs_int(times_times_int(M,N)) = one_one_int -> abs_abs_int(M) = one_one_int)) # label(fact_1160_abs__zmult__eq__1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 553 (all A_28 times_times_real(A_28,zero_zero_real) = zero_zero_real) # label(fact_610_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 554 (all V_8 all B_44 all C_28 minus_minus_real(times_times_real(number267125858f_real(V_8),B_44),times_times_real(number267125858f_real(V_8),C_28)) = times_times_real(number267125858f_real(V_8),minus_minus_real(B_44,C_28))) # label(fact_464_right__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 555 (all A_26 all B_13 (zero_zero_real = times_times_real(A_26,B_13) -> A_26 = zero_zero_real | zero_zero_real = B_13)) # label(fact_618_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 556 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),min)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(K)),min)))) # label(fact_1001_rel__simps_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 557 (all I_2 all J_1 all K_1 minus_minus_nat(minus_minus_nat(I_2,J_1),K_1) = minus_minus_nat(minus_minus_nat(I_2,K_1),J_1)) # label(fact_475_diff__commute) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 558 (all A_73 plus_plus_nat(A_73,zero_zero_nat) = A_73) # label(fact_365_add__0__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 559 (all L bit1(minus_minus_int(min,L)) = minus_minus_int(min,bit0(L))) # label(fact_1028_diff__bin__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 560 (all V_9 all W_4 all Z_5 times_times_int(number_number_of_int(times_times_int(V_9,W_4)),Z_5) = times_times_int(number_number_of_int(V_9),times_times_int(number_number_of_int(W_4),Z_5))) # label(fact_461_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 561 (all A_63 all B_45 all C_29 times_times_nat(A_63,times_times_nat(B_45,C_29)) = times_times_nat(times_times_nat(A_63,B_45),C_29)) # label(fact_424_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 562 (all Lx_6 all Rx_6 all Ry_4 times_times_nat(Rx_6,times_times_nat(Lx_6,Ry_4)) = times_times_nat(Lx_6,times_times_nat(Rx_6,Ry_4))) # label(fact_418_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 563 (all B_32 all A_51 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_51),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_32)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_51,B_32)),zero_zero_int))))) # label(fact_505_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 564 (all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),number267125858f_real(Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(pls)),Ya)))) # label(fact_751_le__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 565 (all X -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(abs_abs_real(X),one_one_real)),X))) # label(fact_1176_abs__add__one__not__less__self) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 566 (all A_1 all B (hAPP_nat_int(power_power_int(B),number_number_of_nat(bit0(bit1(pls)))) = hAPP_nat_int(power_power_int(A_1),number_number_of_nat(bit0(bit1(pls)))) <-> abs_abs_int(A_1) = abs_abs_int(B))) # label(fact_1170_power2__eq__iff__abs__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 567 (all A_89 all N_12 all N_11 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_12),N_11)) -> (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,hAPP_nat_real(power_power_real(A_89),N_12)),hAPP_nat_real(power_power_real(A_89),N_11)))))) # label(fact_283_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 568 (all P_4 all P_1 all Xa ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) -> (hBOOL(P_4) <-> hBOOL(P_1))) -> (hBOOL(P_1) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) <-> hBOOL(P_4) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))))) # label(fact_1146_conj__le__cong) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 569 -hBOOL(hAPP_int_bool(quadRes(plus_plus_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),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) # label(fact_1036__096_126_AQuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_A_061_061_062_ALegendre_A_) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 570 (all A_14 (is_int(A_14) -> A_14 = times_times_int(one_one_int,A_14))) # label(fact_670_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 571 (all Ma all Ta all K (is_int(K) -> (K != zero_zero_int -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),Ta)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,times_times_int(K,Ma)),times_times_int(K,Ta))))))) # label(fact_1148_zdvd__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 572 (all A_1 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(A_1,A_1)),zero_zero_real)))) # label(fact_306_double__add__less__zero__iff__single__add__less__zero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 573 (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_int,bit0(K)),bit0(L_1))))) # label(fact_50_rel__simps_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 574 (all B_1 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_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),A)) -> -hBOOL(hAPP_int_bool(zcong(A,B_1),M))))))) # label(fact_1134_zcong__not) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 575 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,minus_minus_nat(M,N)),M))))) # label(fact_770_diff__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 576 (all K_1 all I_2 all J_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),J_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(hAPP_nat_int(semiri1621563631at_int,K_1),I_2)),times_times_int(hAPP_nat_int(semiri1621563631at_int,K_1),J_1)))))) # label(fact_820_zmult__zless__mono2__lemma) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 577 (all L_1 (is_int(L_1) -> (L_1 = pls <-> pls = bit0(L_1)))) # label(fact_96_rel__simps_I38_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 578 (all C all A all B_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,times_times_nat(C,A)),times_times_nat(C,B_1))))) # label(fact_1041_divides__mul__l) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 579 (all C_1 all A_1 all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(C_1,A_1)),plus_plus_nat(C_1,B))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B)))) # label(fact_394_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 580 (all A_1 all B all Ma (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),minus_minus_int(A_1,B))) <-> hBOOL(hAPP_int_bool(zcong(A_1,B),Ma)))) # label(fact_1137_zcong__def) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 581 (all Ma all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,Ma)),Z_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),nat(abs_abs_int(Z_1)))))) # label(fact_1167_int__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 582 (all A all B_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A),B_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_1),A)) -> abs_abs_int(B_1) = abs_abs_int(A)))) # label(fact_1157_zdvd__antisym__abs) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 583 (all P_1 all A_1 all B (hBOOL(hAPP_nat_bool(P_1,minus_minus_nat(A_1,B))) <-> -(-hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B)) | (exists D_2 (plus_plus_nat(B,D_2) = A_1 & -hBOOL(hAPP_nat_bool(P_1,D_2))))))) # label(fact_794_nat__diff__split__asm) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 584 (all A_35 all N_9 all N_8 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_9),N_8)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A_35)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_35),N_9)),hAPP_nat_int(power_power_int(A_35),N_8)))))) # label(fact_587_power__increasing) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 585 (all A_1 all E_1 all C_1 all B all D (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(times_times_real(A_1,E_1),C_1)),plus_plus_real(times_times_real(B,E_1),D))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_1),plus_plus_real(times_times_real(minus_minus_real(B,A_1),E_1),D))))) # label(fact_537_less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 586 (all A_31 all B_16 A_31 = minus_minus_real(plus_plus_real(A_31,B_16),B_16)) # label(fact_603_add__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 587 (all A_5 all N_5 all N_4 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_5),N_4)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_5)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_5),one_one_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(A_5),N_4)),hAPP_nat_nat(power_power_nat(A_5),N_5))))))) # label(fact_733_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 588 (all A_40 all C_15 all B_22 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_40,C_15)),times_times_int(B_22,C_15))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_15)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_40),B_22))))) # label(fact_558_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 589 (all A_1 all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,minus_minus_int(A_1,B)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B)))) # label(fact_412_le__iff__diff__le__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 590 (all A_16 times_times_nat(one_one_nat,A_16) = A_16) # label(fact_663_mult__1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 591 (all Xa all Ya 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,times_times_real(Xa,Z_1)),times_times_real(Ya,Z_1))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya))))) # label(fact_858_real__mult__le__cancel__iff1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 592 (all C_3 all A_10 all B_5 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_10),B_5)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(C_3,A_10)),plus_plus_int(C_3,B_5))))) # label(fact_682_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 593 (all K all Ma all Na (times_times_nat(K,Na) = times_times_nat(K,Ma) <-> Ma = Na | zero_zero_nat = K)) # label(fact_868_mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 594 (all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),number267125858f_real(Ya))))) # label(fact_72_less__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 595 (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_73_rel__simps_I12_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 596 (all A_30 all B_15 plus_plus_real(minus_minus_real(A_30,B_15),B_15) = A_30) # label(fact_605_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 597 (all N_20 all A_94 (zero_zero_real != A_94 -> zero_zero_real != hAPP_nat_real(power_power_real(A_94),N_20))) # label(fact_232_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 598 (all U all Ma all Na all J all I_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(times_times_nat(I_1,U),Ma)),plus_plus_nat(times_times_nat(J,U),Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(times_times_nat(minus_minus_nat(I_1,J),U),Ma)),Na))))) # label(fact_1181_nat__less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 599 (all A_98 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(A_98),number_number_of_nat(bit0(bit1(pls))))),zero_zero_real))) # label(fact_136_power2__less__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 600 (all Z (is_int(Z) -> plus_plus_int(Z,zero_zero_int) = Z)) # label(fact_104_zadd__0__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 601 (all B_60 all A_87 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_87)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_60)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),plus_plus_nat(A_87,B_60)))))) # label(fact_308_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 602 (all Ya all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ya)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,nat(Xa)),nat(Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Xa),Ya)))))) # label(fact_1075_transfer__nat__int__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 603 (all A_83 all B_56 all C_42 (is_int(B_56) & is_int(C_42) -> (plus_plus_int(A_83,B_56) = plus_plus_int(A_83,C_42) -> C_42 = B_56))) # label(fact_322_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 604 (all B_58 all A_85 all C_44 (plus_plus_real(B_58,A_85) = plus_plus_real(C_44,A_85) -> C_44 = B_58)) # label(fact_318_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 605 (all M_7 all N_15 hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(power_power_nat(M_7),N_15)) = hAPP_nat_real(power_power_real(hAPP_nat_real(semiri132038758t_real,M_7)),N_15)) # label(fact_253_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 606 (all Z (is_int(Z) -> plus_plus_int(zero_zero_int,Z) = Z)) # label(fact_105_zadd__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 607 (all Wa all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_1)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,nat(Wa)),nat(Z_1)))))) # label(fact_791_nat__mono__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 608 (all A (one_one_int = hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls)))) -> abs_abs_int(A) = one_one_int)) # label(fact_1171_power2__eq1__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 609 (all V_5 all U_3 all Y_3 all X_4 all A_6 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_4),A_6)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_3),A_6)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),U_3)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),V_5)) -> (one_one_real = plus_plus_real(U_3,V_5) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(times_times_real(U_3,X_4),times_times_real(V_5,Y_3))),A_6)))))))) # label(fact_724_convex__bound__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 610 (all K (is_int(K) -> (bit1(K) = min <-> K = min))) # label(fact_1019_rel__simps_I47_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 611 (all C_18 all A_43 all B_25 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(C_18,A_43)),times_times_int(C_18,B_25))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_18)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_43),B_25))))) # label(fact_549_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 612 (all X_10 times_times_real(plus_plus_real(X_10,one_one_real),minus_minus_real(X_10,one_one_real)) = minus_minus_real(times_times_real(X_10,X_10),one_one_real)) # label(fact_539_real__squared__diff__one__factored) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 613 (all Ya all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ya)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,nat(Xa)),nat(Ya))))))) # label(fact_951_transfer__nat__int__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 614 (all A all R_1 all B_1 all M all C all D_1 all N minus_minus_int(plus_plus_int(times_times_int(A,M),times_times_int(C,N)),times_times_int(R_1,plus_plus_int(times_times_int(B_1,M),times_times_int(D_1,N)))) = plus_plus_int(times_times_int(minus_minus_int(A,times_times_int(R_1,B_1)),M),times_times_int(minus_minus_int(C,times_times_int(R_1,D_1)),N))) # label(fact_1135_xzgcda__linear__aux1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 615 (all A_1 all B all C_1 (B = C_1 <-> plus_plus_nat(A_1,C_1) = plus_plus_nat(A_1,B))) # label(fact_332_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 616 (all Xa all Ya (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) <-> Ya = Xa)) # label(fact_1113_dvd_Oeq__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 617 (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)),bit0(K2))))) # label(fact_714_less__eq__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 618 (all Z all X all Y all M (hBOOL(hAPP_int_bool(zcong(X,Y),M)) -> hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(X),Z),hAPP_nat_int(power_power_int(Y),Z)),M)))) # label(fact_1061_zcong__zpower) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 619 (all M all N hAPP_nat_int(semiri1621563631at_int,times_times_nat(M,N)) = times_times_int(hAPP_nat_int(semiri1621563631at_int,M),hAPP_nat_int(semiri1621563631at_int,N))) # label(fact_933_zmult__int) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 620 (all A_64 all B_46 times_times_real(B_46,A_64) = times_times_real(A_64,B_46)) # label(fact_416_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 621 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)))) # label(fact_876_less__imp__le__nat) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 622 (all M_7 all N_15 hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(power_power_nat(M_7),N_15)) = hAPP_nat_nat(power_power_nat(hAPP_nat_nat(semiri984289939at_nat,M_7)),N_15)) # label(fact_252_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 623 (all A_1 all B all C_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_1),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),A_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(C_1,A_1)),times_times_int(C_1,B)))))) # label(fact_572_mult__le__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 624 (all W times_times_int(pls,W) = pls) # label(fact_708_mult__Pls) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 625 (all C_14 all D_8 all A_39 all B_21 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_39),B_21)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_14),D_8)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_39)) -> (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_int,times_times_int(A_39,C_14)),times_times_int(B_21,D_8)))))))) # label(fact_561_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 626 (all N all A all B_1 all P_2 (hBOOL(hAPP_int_bool(zprime,P_2)) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),B_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(P_2),N)),times_times_int(A,B_1))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(power_power_int(P_2),N)),A)))))) # label(fact_1031_zprime__power__zdvd__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 627 (all M all K_1 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(M,K_1)),N)) -> -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N))))) # label(fact_883_add__leE) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 628 (all C all A all B_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C),A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,minus_minus_nat(A,C)),minus_minus_nat(B_1,C)))))) # label(fact_930_diff__less__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 629 (all K_1 all L all I_2 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),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,times_times_nat(I_2,K_1)),times_times_nat(J_1,L)))))) # label(fact_854_mult__le__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 630 (all A_23 all B_12 all C_9 plus_plus_int(times_times_int(A_23,C_9),times_times_int(B_12,C_9)) = times_times_int(plus_plus_int(A_23,B_12),C_9)) # label(fact_633_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 631 (all N 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)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_1),minus_minus_nat(M,N)))))) # label(fact_1042_dvd__diff__nat) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 632 (all X_6 all Y_5 all Z_3 plus_plus_nat(times_times_nat(X_6,Y_5),times_times_nat(X_6,Z_3)) = times_times_nat(X_6,plus_plus_nat(Y_5,Z_3))) # label(fact_650_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 633 (all A_15 times_times_real(one_one_real,A_15) = A_15) # label(fact_665_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 634 (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_46_less__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 635 (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]. 1.53/1.81 636 (all Z_1 (is_int(Z_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,abs_abs_int(Z_1)),one_one_int)) <-> zero_zero_int = Z_1))) # label(fact_1162_zabs__less__one__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 637 (all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) -> Y != X)) # label(fact_1097_dvd_Oless__imp__neq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 638 (all W all Z1 all Z2 minus_minus_int(times_times_int(W,Z1),times_times_int(W,Z2)) = times_times_int(W,minus_minus_int(Z1,Z2))) # label(fact_864_zdiff__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 639 (all Lx_1 all Ly_1 all Rx_1 all Ry_1 times_times_real(times_times_real(Lx_1,Ly_1),times_times_real(Rx_1,Ry_1)) = times_times_real(Rx_1,times_times_real(times_times_real(Lx_1,Ly_1),Ry_1))) # label(fact_435_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 640 (all X all Y all Z plus_plus_int(Y,plus_plus_int(X,Z)) = plus_plus_int(X,plus_plus_int(Y,Z))) # label(fact_66_zadd__left__commute) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 641 (all C_1 all A_1 all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(C_1,A_1)),plus_plus_nat(C_1,B))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),B)))) # label(fact_687_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 642 (all K_1 all L bit1(K_1) != bit0(L)) # label(fact_93_rel__simps_I50_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 643 (all A_79 all B_52 all C_38 plus_plus_real(A_79,plus_plus_real(B_52,C_38)) = plus_plus_real(plus_plus_real(A_79,B_52),C_38)) # label(fact_342_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 644 (all B_30 all A_49 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_49)) -> (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,times_times_nat(A_49,B_30)),zero_zero_nat))))) # label(fact_510_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 645 (all A_1 all B all Ma (is_int(B) -> ((exists K_2 (is_int(K_2) & plus_plus_int(A_1,times_times_int(Ma,K_2)) = B)) <-> hBOOL(hAPP_int_bool(zcong(A_1,B),Ma))))) # label(fact_1136_zcong__iff__lin) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 646 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),plus_plus_int(X,Y)))))) # label(fact_782_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 647 (all M_8 all N_16 hAPP_nat_nat(semiri984289939at_nat,plus_plus_nat(M_8,N_16)) = plus_plus_nat(hAPP_nat_nat(semiri984289939at_nat,M_8),hAPP_nat_nat(semiri984289939at_nat,N_16))) # label(fact_246_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 648 (all Na (Na = zero_zero_nat <-> hAPP_nat_int(semiri1621563631at_int,Na) = zero_zero_int)) # label(fact_43_int__eq__0__conv) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 649 (all K all P_1 all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D)) -> ((all X_1 (is_int(X_1) -> (hBOOL(hAPP_int_bool(P_1,X_1)) -> hBOOL(hAPP_int_bool(P_1,minus_minus_int(X_1,D)))))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K)) -> (all X_1 (hBOOL(hAPP_int_bool(P_1,X_1)) -> hBOOL(hAPP_int_bool(P_1,minus_minus_int(X_1,times_times_int(K,D)))))))))) # label(fact_1175_decr__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 650 (all A_2 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_2,A_2)),zero_zero_int))) # label(fact_763_not__square__less__zero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 651 (all A_88 hAPP_nat_int(power_power_int(A_88),zero_zero_nat) = one_one_int) # label(fact_290_power__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 652 (all N (hAPP_nat_int(power_power_int(number_number_of_int(min)),N) = one_one_int | hAPP_nat_int(power_power_int(number_number_of_int(min)),N) = number_number_of_int(min))) # label(fact_1118_neg__one__power) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 653 (all A_96 all N_22 all N_21 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_22),N_21)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_96)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_96),one_one_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(A_96),N_21)),hAPP_nat_int(power_power_int(A_96),N_22))))))) # label(fact_163_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 654 (all A_1 all B all C_1 all D (minus_minus_int(C_1,D) = minus_minus_int(A_1,B) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_1),D)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B))))) # label(fact_599_diff__eq__diff__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 655 (all Ma all Na all K (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(K,Ma)),times_times_nat(K,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na))))) # label(fact_1185_nat__mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 656 (all A_1 all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,minus_minus_real(A_1,B)),zero_zero_real)))) # label(fact_413_le__iff__diff__le__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 657 (all Wa all Ya all Xa all Z_1 (Ya = Z_1 | Wa = Xa <-> plus_plus_real(times_times_real(Wa,Ya),times_times_real(Xa,Z_1)) = plus_plus_real(times_times_real(Wa,Z_1),times_times_real(Xa,Ya)))) # label(fact_637_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 658 (all V_15 all W_10 number_number_of_int(plus_plus_int(V_15,W_10)) = plus_plus_int(number_number_of_int(V_15),number_number_of_int(W_10))) # label(fact_122_add__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 659 (all M_5 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_5)),zero_zero_int))) # label(fact_287_of__nat__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 660 (all A_62 all B_43 all V_7 minus_minus_real(times_times_real(A_62,number267125858f_real(V_7)),times_times_real(B_43,number267125858f_real(V_7))) = times_times_real(minus_minus_real(A_62,B_43),number267125858f_real(V_7))) # label(fact_466_left__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 661 (all M all N (M = times_times_nat(M,N) -> M = zero_zero_nat | N = one_one_nat)) # label(fact_926_mult__eq__self__implies__10) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 662 (all K1 all K2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(K1)),bit1(K2))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K1),K2)))) # label(fact_801_less__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 663 (all Ya all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ya)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,nat(Xa)),nat(Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)))))) # label(fact_726_transfer__nat__int__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 664 (all X hAPP_nat_int(power_power_int(X),number_number_of_nat(bit0(bit0(bit1(pls))))) = hAPP_nat_int(power_power_int(hAPP_nat_int(power_power_int(X),number_number_of_nat(bit0(bit1(pls))))),number_number_of_nat(bit0(bit1(pls))))) # label(fact_21_quartic__square__square) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 665 (all Z1 all Z2 all Z3 times_times_real(Z1,times_times_real(Z2,Z3)) = times_times_real(times_times_real(Z1,Z2),Z3)) # label(fact_980_real__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 666 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> N = M))) # label(fact_853_le__antisym) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 667 (all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),number267125858f_real(Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(pls)),Ya)))) # label(fact_87_less__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 668 (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_int,bit1(K)),bit0(L_1))))) # label(fact_75_rel__simps_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 669 (all A all B_1 minus_minus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_int(power_power_int(B_1),number_number_of_nat(bit0(bit1(pls))))) = times_times_int(plus_plus_int(A,B_1),minus_minus_int(A,B_1))) # label(fact_965_zspecial__product) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 670 (all P_2 (is_int(P_2) -> (hBOOL(hAPP_int_bool(zprime,P_2)) -> (number_number_of_int(bit0(bit1(pls))) != P_2 -> (P_2 != 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_2))))))) # label(fact_1145_prime__g__5) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 671 (all C all D_1 all A all B_1 all M (hBOOL(hAPP_int_bool(zcong(A,B_1),M)) -> (hBOOL(hAPP_int_bool(zcong(C,D_1),M)) -> hBOOL(hAPP_int_bool(zcong(times_times_int(A,C),times_times_int(B_1,D_1)),M))))) # label(fact_1131_zcong__zmult) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 672 (all Xa all Ya all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(power_power_int(B),Xa)),hAPP_nat_int(power_power_int(B),Ya)))))) # label(fact_275_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 673 (all M all N all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),I_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(power_power_nat(I_2),M)),hAPP_nat_nat(power_power_nat(I_2),N))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))))) # label(fact_198_nat__power__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 674 (all X all Y (is_int(Y) -> if_int(fFalse,X,Y) = Y)) # label(help_If_2_1_If_000tc__Int__Oint_T) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 675 (all X all Y X = if_nat(fTrue,X,Y)) # label(help_If_1_1_If_000tc__Nat__Onat_T) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 676 (all K (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),bit0(K))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K)))) # label(fact_1024_rel__simps_I25_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 677 (all A_1 all E_1 all C_1 all B all D (is_int(C_1) -> (plus_plus_int(times_times_int(A_1,E_1),C_1) = plus_plus_int(times_times_int(B,E_1),D) <-> plus_plus_int(times_times_int(minus_minus_int(B,A_1),E_1),D) = C_1))) # label(fact_524_eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 678 (all C_1 all D all A_1 all B all Ma (hBOOL(hAPP_int_bool(zcong(A_1,B),Ma)) -> (hBOOL(hAPP_int_bool(zcong(C_1,times_times_int(B,D)),Ma)) <-> hBOOL(hAPP_int_bool(zcong(C_1,times_times_int(A_1,D)),Ma))))) # label(fact_1064_zcong__zmult__prop1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 679 (all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) | X = Y)) # label(fact_1106_dvd_Ole__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 680 (all Z_1 all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,nat(abs_abs_int(Z_1))),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_1),hAPP_nat_int(semiri1621563631at_int,Ma))))) # label(fact_1168_dvd__int__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 681 (all X_15 all Y_11 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(hAPP_nat_real(power_power_real(X_15),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_real(power_power_real(Y_11),number_number_of_nat(bit0(bit1(pls)))))),zero_zero_real))) # label(fact_140_not__sum__power2__lt__zero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 682 (all V_1 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls)) -> one_one_nat = plus_plus_nat(one_one_nat,number_number_of_nat(V_1))) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls)) -> number_number_of_nat(succ(V_1)) = plus_plus_nat(one_one_nat,number_number_of_nat(V_1))))) # label(fact_407_nat__1__add__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 683 (all P_1 all Xa all Ya (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)) -> hBOOL(P_1)))) # label(fact_1091_dvd_Oless__imp__triv) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 684 (all A_23 all B_12 all C_9 times_times_nat(plus_plus_nat(A_23,B_12),C_9) = plus_plus_nat(times_times_nat(A_23,C_9),times_times_nat(B_12,C_9))) # label(fact_632_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 685 (all N -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat))) # label(fact_199_less__nat__zero__code) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 686 (all X abs_abs_int(hAPP_nat_int(power_power_int(X),number_number_of_nat(bit1(bit1(pls))))) = hAPP_nat_int(power_power_int(abs_abs_int(X)),number_number_of_nat(bit1(bit1(pls))))) # label(fact_1164_abs__power3__distrib) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 687 (all B_41 all A_60 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_60)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_41)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_60),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_41),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),times_times_int(A_60,B_41))))) # label(fact_481_split__mult__pos__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 688 (all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Z_1)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Z_1)))) # label(fact_806_int__one__le__iff__zero__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 689 (all Ma all Na all K (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(K,Ma)),times_times_nat(K,Na)))))) # label(fact_1189_nat__mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 690 (all C_30 all A_65 all B_47 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_65),B_47)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_65,C_30)),plus_plus_nat(B_47,C_30))))) # label(fact_391_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 691 (all V_8 all B_44 all C_28 minus_minus_int(times_times_int(number_number_of_int(V_8),B_44),times_times_int(number_number_of_int(V_8),C_28)) = times_times_int(number_number_of_int(V_8),minus_minus_int(B_44,C_28))) # label(fact_463_right__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 692 (all C_1 all Xa all Ta all A_1 all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_1),D)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_1),plus_plus_int(Xa,Ta))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_1),plus_plus_int(plus_plus_int(Xa,times_times_int(C_1,D)),Ta)))))) # label(fact_999_zdvd__period) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 693 (all A_1 all Na (is_int(A_1) -> (zero_zero_int = hAPP_nat_int(power_power_int(A_1),Na) <-> Na != zero_zero_nat & A_1 = zero_zero_int))) # label(fact_284_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 694 (all M_2 all N_3 all A_4 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_4)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(power_power_int(A_4),M_2)),hAPP_nat_int(power_power_int(A_4),N_3))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_3))))) # label(fact_740_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 695 (all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),number_number_of_int(Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(pls)),Ya)))) # label(fact_86_less__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 696 (all Z_1 all Wa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,nat(Wa)),nat(Z_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_1))))) # label(fact_727_nat__less__eq__zless) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 697 (all A_100 (is_int(A_100) -> A_100 = plus_plus_int(number_number_of_int(pls),A_100))) # label(fact_113_add__numeral__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 698 (all N all P_2 all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M)) -> (hBOOL(hAPP_int_bool(zprime,P_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),times_times_int(M,N))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),N)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),M)))))) # label(fact_1141_zprime__zdvd__zmult) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 699 (all W_12 ((zero_zero_nat != number_number_of_nat(W_12) -> hAPP_nat_real(power_power_real(zero_zero_real),number_number_of_nat(W_12)) = zero_zero_real) & (zero_zero_nat = number_number_of_nat(W_12) -> hAPP_nat_real(power_power_real(zero_zero_real),number_number_of_nat(W_12)) = one_one_real))) # label(fact_24_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 700 (all B_42 all A_61 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_61),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_42)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_42),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_61)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(A_61,B_42)),zero_zero_real)))) # label(fact_477_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 701 (all Wa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit1(Wa)),zero_zero_int)))) # label(fact_78_bin__less__0__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 702 (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(axiom) # label(non_clause). [assumption]. 1.53/1.81 703 (all A_74 plus_plus_nat(A_74,zero_zero_nat) = A_74) # label(fact_362_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 704 (all A_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_1,A_1)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),zero_zero_int)))) # label(fact_305_double__add__less__zero__iff__single__add__less__zero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 705 (all B_58 all A_85 all C_44 (plus_plus_nat(B_58,A_85) = plus_plus_nat(C_44,A_85) -> B_58 = C_44)) # label(fact_317_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 706 (all A_72 plus_plus_real(zero_zero_real,A_72) = A_72) # label(fact_371_add__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 707 (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,plus_plus_int(W,one_one_int)),Z)))) # label(fact_813_zless__imp__add1__zle) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 708 (all A_24 zero_zero_int = times_times_int(zero_zero_int,A_24)) # label(fact_626_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 709 (all K_1 all M all N plus_plus_nat(times_times_nat(K_1,M),times_times_nat(K_1,N)) = times_times_nat(K_1,plus_plus_nat(M,N))) # label(fact_882_add__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 710 (all B_14 all A_27 (zero_zero_real != A_27 -> (zero_zero_real != B_14 -> times_times_real(A_27,B_14) != zero_zero_real))) # label(fact_615_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 711 (all C_1 all A_1 all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(C_1,A_1)),plus_plus_int(C_1,B))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B)))) # label(fact_688_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 712 (all Xa all Ya (is_int(Xa) & is_int(Ya) -> (Xa = zero_zero_int & zero_zero_int = Ya <-> plus_plus_int(hAPP_nat_int(power_power_int(Xa),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_int(power_power_int(Ya),number_number_of_nat(bit0(bit1(pls))))) = zero_zero_int))) # label(fact_2_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 713 (all C_11 all D_5 all A_36 all B_18 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_36),B_18)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_11),D_5)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_18)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_11)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_36,C_11)),times_times_int(B_18,D_5)))))))) # label(fact_570_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 714 (all W_8 number_number_of_int(bit0(W_8)) = plus_plus_int(plus_plus_int(zero_zero_int,number_number_of_int(W_8)),number_number_of_int(W_8))) # label(fact_141_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 715 (all C_12 all D_6 all A_37 all B_19 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_37),B_19)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_12),D_6)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_37)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_12)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_37,C_12)),times_times_int(B_19,D_6)))))))) # label(fact_567_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 716 -(all X_1 all Y_1 (is_int(X_1) & is_int(Y_1) -> times_times_int(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n))) != plus_plus_int(hAPP_nat_int(power_power_int(X_1),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_int(power_power_int(Y_1),number_number_of_nat(bit0(bit1(pls))))))) # label(fact_839__096_B_Bthesis_O_A_I_B_Bx_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_061_A_I4_A) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 717 (all V_17 plus_plus_int(number_number_of_int(V_17),one_one_int) = number_number_of_int(plus_plus_int(V_17,bit1(pls)))) # label(fact_14_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 718 -(all S all W_1 (is_int(W_1) & is_int(S) -> -(W_1 = minus_minus_int(y,times_times_int(S,plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n)))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(number_number_of_int(bit0(bit1(pls))),abs_abs_int(W_1))),plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n))))))) # label(fact_1153__096_B_Bthesis_O_A_I_B_Bs_Aw_O_Aw_A_061_Ay_A_N_As_A_K_A_I1_A_L_Aint_An) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 719 (all Ma all Na (Na = Ma <-> hAPP_nat_real(semiri132038758t_real,Na) = hAPP_nat_real(semiri132038758t_real,Ma))) # label(fact_218_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 720 (all C_13 all D_7 all A_38 all B_20 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_38),B_20)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_13),D_7)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_38)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_13)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,times_times_int(A_38,C_13)),times_times_int(B_20,D_7)))))))) # label(fact_564_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 721 (all K_1 all L plus_plus_int(bit1(K_1),bit1(L)) = bit0(plus_plus_int(K_1,succ(L)))) # label(fact_821_add__Bit1__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 722 (all C_14 all D_8 all A_39 all B_21 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_39),B_21)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_14),D_8)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_39)) -> (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_nat,times_times_nat(A_39,C_14)),times_times_nat(B_21,D_8)))))))) # label(fact_560_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 723 (all A_1 all B all C_1 all D (minus_minus_real(C_1,D) = minus_minus_real(A_1,B) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_1),D)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B))))) # label(fact_444_diff__eq__diff__less__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 724 (all M all N all K_1 all L (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_1),L)) -> (plus_plus_nat(K_1,N) = plus_plus_nat(M,L) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))))) # label(fact_193_less__add__eq__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 725 (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,bit0(K)),bit1(L_1))))) # label(fact_775_rel__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 726 (all C all D_1 all A all B_1 all M (hBOOL(hAPP_int_bool(zcong(A,B_1),M)) -> (hBOOL(hAPP_int_bool(zcong(C,D_1),M)) -> hBOOL(hAPP_int_bool(zcong(minus_minus_int(A,C),minus_minus_int(B_1,D_1)),M))))) # label(fact_1133_zcong__zdiff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 727 (all X_10 times_times_int(plus_plus_int(X_10,one_one_int),minus_minus_int(X_10,one_one_int)) = minus_minus_int(times_times_int(X_10,X_10),one_one_int)) # label(fact_540_real__squared__diff__one__factored) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 728 (all P_1 all Xa all Ya ((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) -> hBOOL(hAPP_int_bool(P_1,zero_zero_int))) & (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ya),Xa)) -> hBOOL(hAPP_int_bool(P_1,minus_minus_int(hAPP_nat_int(semiri1621563631at_int,Xa),hAPP_nat_int(semiri1621563631at_int,Ya))))) <-> hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,minus_minus_nat(Xa,Ya)))))) # label(fact_1144_zdiff__int__split) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 729 (all A_46 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),times_times_int(A_46,A_46)))) # label(fact_534_zero__le__square) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 730 (all W all Z1 all Z2 times_times_int(W,plus_plus_int(Z1,Z2)) = plus_plus_int(times_times_int(W,Z1),times_times_int(W,Z2))) # label(fact_719_zadd__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 731 (all B_31 all A_50 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_50)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_31),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(B_31,A_50)),zero_zero_nat))))) # label(fact_507_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 732 (all A all B_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A),B_1)) | zero_zero_nat = B_1)) # label(fact_1046_divides__ge) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 733 (all U all Ma all Na all J all I_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I_1)) -> (plus_plus_nat(times_times_nat(J,U),Na) = plus_plus_nat(times_times_nat(I_1,U),Ma) <-> Na = plus_plus_nat(times_times_nat(minus_minus_nat(I_1,J),U),Ma)))) # label(fact_1192_nat__eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 734 (all X_11 all Y_9 all A_47 all B_28 plus_plus_int(times_times_int(X_11,minus_minus_int(Y_9,B_28)),times_times_int(minus_minus_int(X_11,A_47),B_28)) = minus_minus_int(times_times_int(X_11,Y_9),times_times_int(A_47,B_28))) # label(fact_526_mult__diff__mult) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 735 (all N plus_plus_nat(zero_zero_nat,N) = N) # label(fact_214_plus__nat_Oadd__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 736 (all P_3 (is_int(P_3) -> (hBOOL(hAPP_int_bool(zprime,P_3)) <-> (all M_1 (is_int(M_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),M_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M_1),P_3)) -> M_1 = P_3 | one_one_int = M_1))) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),P_3))))) # label(fact_1142_zprime__def) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 737 (all A_1 all B all C_1 all D (minus_minus_real(A_1,B) = minus_minus_real(C_1,D) -> (C_1 = D <-> B = A_1))) # label(fact_442_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 738 (all Xa all Ya all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(B),Xa)),hAPP_nat_nat(power_power_nat(B),Ya)))))) # label(fact_736_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 739 (all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),number267125858f_real(Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Ya)))) # label(fact_728_le__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 740 (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_774_rel__simps_I22_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 741 (all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) -> X = Y))) # label(fact_1101_dvd_Oantisym) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 742 (all X_14 all Y_10 (is_int(Y_10) & is_int(X_14) -> (X_14 != Y_10 -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_14),Y_10)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_10),X_14)))))) # label(fact_209_linorder__neqE__linordered__idom) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 743 (all Xa (Xa = zero_zero_nat <-> zero_zero_nat = Xa)) # label(fact_314_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 744 (all M all I_2 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),J_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),plus_plus_nat(J_1,M))))) # label(fact_889_trans__le__add1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 745 (all C all A all B_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),A)) -> (C = B_1 -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C),A)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),C))))) # label(fact_1090_dvd_Oord__less__eq__trans) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 746 (all X all N hAPP_nat_int(power_power_int(hAPP_nat_int(semiri1621563631at_int,X)),N) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(power_power_nat(X),N))) # label(fact_256_Nat__Transfer_Otransfer__int__nat__functions_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 747 (all X all Q all N all R_1 (plus_plus_nat(times_times_nat(Q,N),R_1) = X -> (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)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),X)))))) # label(fact_1072_divides__div__not) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 748 (all A all B_1 hAPP_nat_int(power_power_int(plus_plus_int(A,B_1)),number_number_of_nat(bit0(bit1(pls)))) = plus_plus_int(plus_plus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls)))),times_times_int(times_times_int(number_number_of_int(bit0(bit1(pls))),A),B_1)),hAPP_nat_int(power_power_int(B_1),number_number_of_nat(bit0(bit1(pls)))))) # label(fact_409_zadd__power2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 749 (all P (is_bool(P) -> fFalse = P | fTrue = P)) # label(help_If_3_1_If_000tc__Nat__Onat_T) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 750 (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]. 1.53/1.81 751 (all M_9 all N_17 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_9)),hAPP_nat_real(semiri132038758t_real,N_17))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_9),N_17)))) # label(fact_244_of__nat__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 752 (all Ma all Wa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,nat(Wa)),Ma)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),hAPP_nat_int(semiri1621563631at_int,Ma)))))) # label(fact_750_nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 753 (all Y_7 all X_8 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_8)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_7)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_7),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(X_8,Y_7)),X_8)))))) # label(fact_579_mult__right__le__one__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 754 (all B_41 all A_60 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_41),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_60),zero_zero_real)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_60)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_41)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),times_times_real(A_60,B_41))))) # label(fact_480_split__mult__pos__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 755 (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_711_less__eq__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 756 (all Lx_6 all Rx_6 all Ry_4 times_times_real(Rx_6,times_times_real(Lx_6,Ry_4)) = times_times_real(Lx_6,times_times_real(Rx_6,Ry_4))) # label(fact_419_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 757 (all A_1 all B all C_1 all D (minus_minus_int(A_1,B) = minus_minus_int(C_1,D) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_1),D))))) # label(fact_443_diff__eq__diff__less__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 758 (all Lx all Ly all Rx all Ry times_times_nat(times_times_nat(Lx,Ly),times_times_nat(Rx,Ry)) = times_times_nat(times_times_nat(Lx,Rx),times_times_nat(Ly,Ry))) # label(fact_439_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 759 (all M (hBOOL(hAPP_int_bool(zprime,plus_plus_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),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),M),one_one_int)))) # label(fact_1081_Legendre__1mod4) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 760 (all A_33 (is_int(A_33) -> minus_minus_int(A_33,zero_zero_int) = A_33)) # label(fact_598_diff__0__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 761 (all K all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K),Na)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K),plus_plus_nat(Na,K))))) # label(fact_1056_dvd__reduce) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 762 (all A_28 zero_zero_int = times_times_int(A_28,zero_zero_int)) # label(fact_612_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 763 (all M all N times_times_nat(M,N) = times_times_nat(N,M)) # label(fact_846_nat__mult__commute) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 764 (all Xa all Ya (is_int(Xa) & is_int(Ya) -> (number267125858f_real(Ya) = number267125858f_real(Xa) <-> Ya = Xa))) # label(fact_57_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 765 (all A_1 all B (is_int(A_1) & is_int(B) -> (zero_zero_int = times_times_int(A_1,B) <-> zero_zero_int = B | A_1 = zero_zero_int))) # label(fact_614_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 766 (all P_2 all Y all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),Y)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_2),hAPP_nat_int(power_power_int(Y),N)))))) # label(fact_1023_zpower__zdvd__prop1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 767 (all X all Y hAPP_nat_real(power_power_real(plus_plus_real(X,Y)),number_number_of_nat(bit0(bit1(pls)))) = plus_plus_real(plus_plus_real(hAPP_nat_real(power_power_real(X),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_real(power_power_real(Y),number_number_of_nat(bit0(bit1(pls))))),times_times_real(times_times_real(number267125858f_real(bit0(bit1(pls))),X),Y))) # label(fact_986_real__sum__squared__expand) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 768 (all A_44 all C_19 all B_26 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_44,C_19)),times_times_nat(B_26,C_19))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_19)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_44),B_26))))) # label(fact_545_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 769 (all B_14 all A_27 (zero_zero_nat != A_27 -> (B_14 != zero_zero_nat -> times_times_nat(A_27,B_14) != zero_zero_nat))) # label(fact_616_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 770 (all A_21 all M_3 all B_10 times_times_nat(plus_plus_nat(A_21,B_10),M_3) = plus_plus_nat(times_times_nat(A_21,M_3),times_times_nat(B_10,M_3))) # label(fact_641_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 771 (all A all B_1 hAPP_nat_int(power_power_int(plus_plus_int(A,B_1)),number_number_of_nat(bit1(bit1(pls)))) = plus_plus_int(plus_plus_int(plus_plus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit1(bit1(pls)))),times_times_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_1)),times_times_int(times_times_int(number_number_of_int(bit1(bit1(pls))),A),hAPP_nat_int(power_power_int(B_1),number_number_of_nat(bit0(bit1(pls)))))),hAPP_nat_int(power_power_int(B_1),number_number_of_nat(bit1(bit1(pls)))))) # label(fact_408_zadd__power3) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 772 (all A_8 all N_7 times_times_nat(hAPP_nat_nat(power_power_nat(A_8),N_7),A_8) = times_times_nat(A_8,hAPP_nat_nat(power_power_nat(A_8),N_7))) # label(fact_695_power__commutes) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 773 (all A_68 all C_33 all B_50 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_68,C_33)),plus_plus_nat(B_50,C_33))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_68),B_50)))) # label(fact_382_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 774 (all Ma all Xa (hBOOL(hAPP_int_bool(quadRes(Ma),Xa)) <-> (exists Y_1 (hBOOL(hAPP_int_bool(zcong(hAPP_nat_int(power_power_int(Y_1),number_number_of_nat(bit0(bit1(pls)))),Xa),Ma)) & is_int(Y_1))))) # label(fact_1120_QuadRes__def) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 775 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),plus_plus_nat(Ma,Na))) <-> 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),Na)))) # label(fact_201_add__gr__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 776 (all A_1 all B all C_1 (C_1 != zero_zero_real -> (times_times_real(B,C_1) = times_times_real(A_1,C_1) <-> B = A_1))) # label(fact_873_real__mult__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 777 (all A_1 all Na (Na != zero_zero_nat & A_1 = zero_zero_nat <-> zero_zero_nat = hAPP_nat_nat(power_power_nat(A_1),Na))) # label(fact_285_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 778 (all Ma all Na ((exists K_2 plus_plus_nat(Ma,K_2) = Na) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)))) # label(fact_891_le__iff__add) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 779 (all C_32 all D_11 all A_67 all B_49 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_67),B_49)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_32),D_11)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(A_67,C_32)),plus_plus_real(B_49,D_11)))))) # label(fact_386_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 780 (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_713_rel__simps_I31_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 781 (all Z all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z),N)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z),N))))) # label(fact_1022_zdvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 782 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),times_times_nat(Ma,Na))) <-> 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),Na)))) # label(fact_921_nat__0__less__mult__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 783 (all Ma all Na (Ma = one_one_nat & one_one_nat = Na <-> times_times_nat(Ma,Na) = one_one_nat)) # label(fact_895_nat__mult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 784 (all P_2 all M ((zero_zero_nat != M -> times_times_nat(P_2,hAPP_nat_nat(power_power_nat(P_2),minus_minus_nat(M,one_one_nat))) = hAPP_nat_nat(power_power_nat(P_2),M)) & (zero_zero_nat = M -> one_one_nat = hAPP_nat_nat(power_power_nat(P_2),M)))) # label(fact_957_power__eq__if) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 785 (all N all M (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),M)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N),M)))) # label(fact_993_zdvd__bounds) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 786 (all A_72 (is_int(A_72) -> A_72 = plus_plus_int(zero_zero_int,A_72))) # label(fact_369_add__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 787 (all Na all Ma (is_int(Na) & is_int(Ma) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma)) -> (Ma = one_one_int & Na = one_one_int <-> times_times_int(Ma,Na) = one_one_int)))) # label(fact_805_pos__zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 788 (all Ma all K all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,times_times_nat(Ma,K)),times_times_nat(Na,K))) <-> K = zero_zero_nat | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),Na)))) # label(fact_1047_nat__mult__dvd__cancel__disj_H) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 789 (all B_33 all A_52 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_52),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_33),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),times_times_real(A_52,B_33)))))) # label(fact_501_mult__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 790 (all K all L_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,bit1(K)),bit0(L_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L_1)))) # label(fact_802_rel__simps_I33_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 791 (all A_1 all C_1 all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(A_1,C_1)),plus_plus_real(B,C_1))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B)))) # label(fact_398_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 792 (all M all K_1 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(M,K_1)),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N)))) # label(fact_885_add__leD2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 793 (all M_6 all N_13 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),M_6)),hAPP_nat_nat(power_power_nat(A_90),N_13))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_6),N_13))))) # label(fact_279_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 794 (all Na all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma)) -> (Na = one_one_nat <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,times_times_nat(Na,Ma)),Ma))))) # label(fact_1074_dvd__mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 795 (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_int,number_number_of_int(K)),number_number_of_int(L_1))))) # label(fact_31_less__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 796 (all K_1 pls != bit1(K_1)) # label(fact_91_rel__simps_I46_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 797 (all Na all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma)) -> (one_one_nat = Na <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,times_times_nat(Ma,Na)),Ma))))) # label(fact_1073_dvd__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 798 (all K_1 (is_int(K_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K_1)) -> (exists N_1 (hAPP_nat_int(semiri1621563631at_int,N_1) = K_1 & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))))))) # label(fact_297_zero__less__imp__eq__int) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 799 (all K_1 (is_int(K_1) -> K_1 = minus_minus_int(K_1,pls))) # label(fact_861_diff__bin__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 800 (all C_6 all A_13 all B_8 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,plus_plus_nat(C_6,A_13)),plus_plus_nat(C_6,B_8))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_13),B_8)))) # label(fact_672_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 801 (all Z1 all Z2 all Z3 times_times_int(times_times_int(Z1,Z2),Z3) = times_times_int(Z1,times_times_int(Z2,Z3))) # label(fact_451_zmult__assoc) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 802 (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,bit0(K)),min)))) # label(fact_1010_rel__simps_I28_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 803 (all A_35 all N_9 all N_8 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_9),N_8)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),A_35)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(power_power_nat(A_35),N_9)),hAPP_nat_nat(power_power_nat(A_35),N_8)))))) # label(fact_586_power__increasing) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 804 (all B_61 all C_45 all A_93 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_93)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_61),C_45)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_61),plus_plus_nat(A_93,C_45)))))) # label(fact_258_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 805 (all Ya all Xa (is_int(Xa) & is_int(Ya) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ya)) -> (Xa = Ya <-> nat(Ya) = nat(Xa)))))) # label(fact_512_transfer__nat__int__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 806 (all Ma all Na all A_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_1)) -> (hAPP_nat_nat(power_power_nat(A_1),Na) = hAPP_nat_nat(power_power_nat(A_1),Ma) <-> Na = Ma))) # label(fact_273_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 807 (all B_29 all A_48 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_48)) -> (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,zero_zero_real),times_times_real(A_48,B_29)))))) # label(fact_514_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 808 (all B_32 all A_51 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_51),zero_zero_real)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_32)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(A_51,B_32)),zero_zero_real))))) # label(fact_503_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 809 (all Xa all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Xa)) | Na = zero_zero_nat <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(Xa),Na))))) # label(fact_200_nat__zero__less__power__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 810 (all I_2 all K_1 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_1)) -> minus_minus_nat(plus_plus_nat(J_1,I_2),K_1) = plus_plus_nat(minus_minus_nat(J_1,K_1),I_2))) # label(fact_944_add__diff__assoc2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 811 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> minus_minus_int(hAPP_nat_int(semiri1621563631at_int,M),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,minus_minus_nat(M,N)))) # label(fact_860_zdiff__int) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 812 (all Ma all K all Na ((hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(Ma,K)),times_times_nat(Na,K))))) # label(fact_856_mult__le__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 813 (all M_10 all N_18 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_18)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_10)),hAPP_nat_real(semiri132038758t_real,N_18))))) # label(fact_241_less__imp__of__nat__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 814 (all C all A all B_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,times_times_nat(A,C)),times_times_nat(B_1,C))))) # label(fact_1040_divides__mul__r) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 815 (all X_15 all Y_11 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(hAPP_nat_int(power_power_int(X_15),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_int(power_power_int(Y_11),number_number_of_nat(bit0(bit1(pls)))))),zero_zero_int))) # label(fact_139_not__sum__power2__lt__zero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 816 (all X all Z all D_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,minus_minus_int(X,times_times_int(plus_plus_int(abs_abs_int(minus_minus_int(X,Z)),one_one_int),D_1))),Z)))) # label(fact_1173_decr__lemma) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 817 (all C_20 all A_45 all B_27 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(C_20,A_45)),times_times_int(C_20,B_27))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_20)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_45),B_27))))) # label(fact_543_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 818 (all C_25 all B_38 all A_57 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_38),A_57)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_25),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(C_25,A_57)),times_times_real(C_25,B_38)))))) # label(fact_488_mult__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 819 (all Ma (Ma = one_one_nat <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),one_one_nat)))) # label(fact_1055_nat__dvd__1__iff__1) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 820 (all M_10 all N_18 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_10),N_18)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_10)),hAPP_nat_nat(semiri984289939at_nat,N_18))))) # label(fact_240_less__imp__of__nat__less) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 821 (all K_1 bit0(succ(K_1)) = succ(bit1(K_1))) # label(fact_810_succ__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 822 (all V_9 all W_4 all Z_5 times_times_real(number267125858f_real(V_9),times_times_real(number267125858f_real(W_4),Z_5)) = times_times_real(number267125858f_real(times_times_int(V_9,W_4)),Z_5)) # label(fact_460_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 823 (all A_1 all B (is_int(B) & is_int(A_1) -> (B = A_1 <-> hBOOL(hAPP_int_bool(zcong(A_1,B),zero_zero_int))))) # label(fact_1126_IntPrimes_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 824 (all Z1 all Z2 all W minus_minus_int(times_times_int(Z1,W),times_times_int(Z2,W)) = times_times_int(minus_minus_int(Z1,Z2),W)) # label(fact_863_zdiff__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 825 (all M zero_zero_nat = times_times_nat(M,zero_zero_nat)) # label(fact_866_mult__0__right) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 826 (all Wa all Xa (is_int(Xa) -> (Xa = number_number_of_int(Wa) <-> Xa = number_number_of_int(Wa)))) # label(fact_59_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 827 (all A_28 times_times_nat(A_28,zero_zero_nat) = zero_zero_nat) # label(fact_611_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 828 (all D_1 all C all A all B_1 all M (hBOOL(hAPP_int_bool(zcong(A,B_1),M)) -> (C = B_1 -> (hBOOL(hAPP_int_bool(zcong(C,D_1),M)) -> hBOOL(hAPP_int_bool(zcong(A,D_1),M)))))) # label(fact_1060_zcong__eq__trans) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 829 (all C_4 all D_3 all A_11 all B_6 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_11),B_6)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_4),D_3)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,plus_plus_real(A_11,C_4)),plus_plus_real(B_6,D_3)))))) # label(fact_677_add__mono) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 830 (all Lx_2 all Ly_2 all Rx_2 all Ry_2 times_times_int(times_times_int(Lx_2,Ly_2),times_times_int(Rx_2,Ry_2)) = times_times_int(Lx_2,times_times_int(Ly_2,times_times_int(Rx_2,Ry_2)))) # label(fact_432_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 831 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) <-> Ma != Na & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)))) # label(fact_878_nat__less__le) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 832 (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_798_rel__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 833 (all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(semiri984289939at_nat,Na))))) # label(fact_155_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 834 (all K_1 all L all I_2 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),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,plus_plus_nat(I_2,K_1)),plus_plus_nat(J_1,L)))))) # label(fact_886_add__le__mono) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 835 (all Lx_3 all Ly_3 all Rx_3 times_times_int(times_times_int(Lx_3,Ly_3),Rx_3) = times_times_int(times_times_int(Lx_3,Rx_3),Ly_3)) # label(fact_429_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 836 (all B_1_1 all B_2_1 (is_int(B_1_1) & is_int(B_2_1) -> is_int(times_times_int(B_1_1,B_2_1)))) # label(gsy_c_Groups_Otimes__class_Otimes_000tc__Int__Oint) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 837 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) -> times_times_nat(nat(X),nat(Y)) = nat(times_times_int(X,Y))))) # label(fact_954_Nat__Transfer_Otransfer__nat__int__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 838 (all A_76 all C_35 plus_plus_nat(A_76,C_35) = plus_plus_nat(C_35,A_76)) # label(fact_350_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 839 (all C_11 all D_5 all A_36 all B_18 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_36),B_18)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_11),D_5)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_18)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_11)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(A_36,C_11)),times_times_real(B_18,D_5)))))))) # label(fact_568_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 840 (all K all F all Na (is_int(K) -> ((all I (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),Na)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(minus_minus_int(hAPP_nat_int(F,plus_plus_nat(I,one_one_nat)),hAPP_nat_int(F,I)))),one_one_int)))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),hAPP_nat_int(F,Na))) -> (exists I (K = hAPP_nat_int(F,I) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),Na))))))))) # label(fact_1196_nat0__intermed__int__val) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 841 (all Xa all Ya (zero_zero_real = Xa & Ya = zero_zero_real <-> zero_zero_real = plus_plus_real(hAPP_nat_real(power_power_real(Xa),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_real(power_power_real(Ya),number_number_of_nat(bit0(bit1(pls))))))) # label(fact_3_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 842 (all A_1 (A_1 != zero_zero_real <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(power_power_real(A_1),number_number_of_nat(bit0(bit1(pls)))))))) # label(fact_134_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 843 (all C_18 all A_43 all B_25 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(C_18,A_43)),times_times_real(C_18,B_25))) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_18)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_43),B_25))))) # label(fact_547_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 844 (all Xa all Wa (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Xa)) | zero_zero_nat = number_number_of_nat(Wa) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(power_power_nat(Xa),number_number_of_nat(Wa)))))) # label(fact_206_zero__less__power__nat__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 845 (all C_4 all D_3 all A_11 all B_6 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_11),B_6)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_4),D_3)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(A_11,C_4)),plus_plus_int(B_6,D_3)))))) # label(fact_679_add__mono) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 846 (all C_31 all A_66 all B_48 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_66),B_48)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(C_31,A_66)),plus_plus_nat(C_31,B_48))))) # label(fact_388_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 847 (all A_46 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),times_times_real(A_46,A_46)))) # label(fact_533_zero__le__square) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 848 (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_49_less__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 849 (all B_1 all Q_1 all R_2 all Q all R_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(B_1,Q_1),R_2)),plus_plus_int(times_times_int(B_1,Q),R_1))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),B_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q))))))) # label(fact_972_unique__quotient__lemma) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 850 (all C_22 all A_54 all B_35 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_54),B_35)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_22)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(C_22,A_54)),times_times_nat(C_22,B_35)))))) # label(fact_496_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 851 (all V_2 all K_1 all V_1 ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls)) -> times_times_nat(number_number_of_nat(V_1),times_times_nat(number_number_of_nat(V_2),K_1)) = zero_zero_nat) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls)) -> times_times_nat(number_number_of_nat(V_1),times_times_nat(number_number_of_nat(V_2),K_1)) = times_times_nat(number_number_of_nat(times_times_int(V_1,V_2)),K_1)))) # label(fact_960_nat__number__of__mult__left) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 852 (all Xa all Ya (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,number267125858f_real(Xa)),number267125858f_real(Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)))) # label(fact_34_less__number__of) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 853 (all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N))) # label(fact_834_le0) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 854 (all K all F all Na (is_int(K) -> ((all I (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),Na)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,abs_abs_int(minus_minus_int(hAPP_nat_int(F,plus_plus_nat(I,one_one_nat)),hAPP_nat_int(F,I)))),one_one_int)))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,zero_zero_nat)),K)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),hAPP_nat_int(F,Na))) -> (exists I (K = hAPP_nat_int(F,I) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),Na))))))))) # label(fact_1197_int__val__lemma) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 855 (all Z_1 all Ma ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_1),hAPP_nat_int(semiri1621563631at_int,Ma)))) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> Ma = zero_zero_nat) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,nat(Z_1)),Ma)))) # label(fact_1077_nat__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 856 (all M all N (plus_plus_nat(M,N) = M -> zero_zero_nat = N)) # label(fact_211_add__eq__self__zero) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 857 (all Z all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z),Y)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z)) -> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z),X)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z))))) # label(fact_1098_dvd_Ole__less__trans) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 858 (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_799_rel__simps_I29_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 859 (all J_1 all I_2 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(J_1,I_2)),I_2))) # label(fact_176_not__add__less2) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 860 (all L min != bit0(L)) # label(fact_1013_rel__simps_I42_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 861 (all A_1 all B all Ma (hBOOL(hAPP_int_bool(zcong(B,A_1),Ma)) <-> hBOOL(hAPP_int_bool(zcong(A_1,B),Ma)))) # label(fact_1125_zcong__sym) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 862 (all A_1 all B all C_1 (B = C_1 <-> plus_plus_real(A_1,C_1) = plus_plus_real(A_1,B))) # label(fact_333_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 863 (all A_20 all B_9 all C_7 plus_plus_int(times_times_int(A_20,C_7),times_times_int(B_9,C_7)) = times_times_int(plus_plus_int(A_20,B_9),C_7)) # label(fact_645_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 864 (all X all Y (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)))) # label(fact_1095_dvd_Oless__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 865 (all K_1 all M all N (is_int(K_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,times_times_int(K_1,M)),times_times_int(K_1,N))) -> (zero_zero_int != K_1 -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),N)))))) # label(fact_996_zdvd__mult__cancel) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 866 (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),plus_plus_int(one_one_int,Z))))) # label(fact_819_le__imp__0__less) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 867 (all A_1 all P_3 (hBOOL(hAPP_int_bool(zcong(times_times_int(A_1,minus_minus_int(P_3,one_one_int)),one_one_int),P_3)) <-> hBOOL(hAPP_int_bool(zcong(A_1,minus_minus_int(P_3,one_one_int)),P_3)))) # label(fact_1150_inv__not__p__minus__1__aux) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 868 (all Wa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,bit0(Wa)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),zero_zero_int)))) # label(fact_80_bin__less__0__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 869 (all N times_times_nat(zero_zero_nat,N) = zero_zero_nat) # label(fact_865_mult__0) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 870 (all A_15 A_15 = times_times_nat(one_one_nat,A_15)) # label(fact_666_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 871 (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_982_real__le__antisym) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 872 (all Ma all Wa (is_int(Wa) -> (nat(Wa) = Ma <-> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa)) -> hAPP_nat_int(semiri1621563631at_int,Ma) = Wa) & (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa)) -> zero_zero_nat = Ma)))) # label(fact_741_nat__eq__iff2) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 873 one_one_int = t -> (exists X_1 exists Y_1 (plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int) = plus_plus_int(hAPP_nat_int(power_power_int(X_1),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_int(power_power_int(Y_1),number_number_of_nat(bit0(bit1(pls))))) & is_int(Y_1) & is_int(X_1))) # label(fact_842__096t_A_061_A1_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 874 (all K_1 all I_2 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(I_2,K_1)),plus_plus_nat(J_1,K_1))))) # label(fact_191_add__less__mono1) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 875 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)))) # label(fact_847_nat__le__linear) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 876 (all N all Z (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z)) -> nat(hAPP_nat_int(power_power_int(Z),N)) = hAPP_nat_nat(power_power_nat(nat(Z)),N))) # label(fact_747_nat__power__eq) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 877 (all C_22 all A_54 all B_35 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_54),B_35)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_22)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(C_22,A_54)),times_times_int(C_22,B_35)))))) # label(fact_497_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 878 (all Lx_2 all Ly_2 all Rx_2 all Ry_2 times_times_nat(times_times_nat(Lx_2,Ly_2),times_times_nat(Rx_2,Ry_2)) = times_times_nat(Lx_2,times_times_nat(Ly_2,times_times_nat(Rx_2,Ry_2)))) # label(fact_433_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 879 (all Z all Z_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_2),Z)) -> minus_minus_nat(nat(Z),nat(Z_2)) = nat(minus_minus_int(Z,Z_2))))) # label(fact_952_nat__diff__distrib) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 880 (all K_1 all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,times_times_nat(K_1,M)),times_times_nat(K_1,N))) -> (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))))) # label(fact_1069_dvd__mult__cancel) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 881 (all Xa (is_int(Xa) -> (Xa = zero_zero_int <-> zero_zero_int = Xa))) # label(fact_313_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 882 (all A_80 all B_53 all C_39 plus_plus_real(A_80,plus_plus_real(B_53,C_39)) = plus_plus_real(plus_plus_real(A_80,B_53),C_39)) # label(fact_339_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 883 (all Xa all Ya (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,Xa)),hAPP_nat_int(semiri1621563631at_int,Ya))))) # label(fact_1053_zdvd__int) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 884 (all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(semiri132038758t_real,Na))))) # label(fact_156_of__nat__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 885 (all A_8 all N_7 times_times_real(hAPP_nat_real(power_power_real(A_8),N_7),A_8) = times_times_real(A_8,hAPP_nat_real(power_power_real(A_8),N_7))) # label(fact_694_power__commutes) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 886 (all K_1 all L all I_2 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_1),L)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(I_2,K_1)),plus_plus_nat(J_1,L)))))) # label(fact_192_add__less__mono) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 887 (all N_19 hAPP_nat_int(power_power_int(one_one_int),N_19) = one_one_int) # label(fact_233_power__one) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 888 (all P_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(bit0(bit1(pls)))),P_2)) -> minus_minus_nat(nat(P_2),number_number_of_nat(bit0(bit1(pls)))) = nat(minus_minus_int(P_2,number_number_of_int(bit0(bit1(pls))))))) # label(fact_974_Int2_Oaux__1) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 889 (all Z (is_int(Z) -> Z = times_times_int(Z,one_one_int))) # label(fact_717_zmult__1__right) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 890 (all X all Y plus_plus_int(hAPP_nat_int(semiri1621563631at_int,X),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,plus_plus_nat(X,Y))) # label(fact_255_Nat__Transfer_Otransfer__int__nat__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 891 (all Lx_5 all Rx_5 all Ry_3 times_times_real(times_times_real(Lx_5,Rx_5),Ry_3) = times_times_real(Lx_5,times_times_real(Rx_5,Ry_3))) # label(fact_422_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 892 (all Wa all Z_1 (is_int(Wa) & is_int(Z_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),plus_plus_int(Z_1,one_one_int))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_1)) | Wa = Z_1))) # label(fact_82_zless__add1__eq) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 893 (all K_1 minus_minus_int(K_1,min) = succ(K_1)) # label(fact_1021_diff__bin__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 894 (all B_29 all A_48 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_48)) -> (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),times_times_int(A_48,B_29)))))) # label(fact_516_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 895 (all V_4 all U_2 all Y_2 all X_3 all A_3 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_3),A_3)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_2),A_3)) -> (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_4)) -> (plus_plus_int(U_2,V_4) = one_one_int -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(U_2,X_3),times_times_int(V_4,Y_2))),A_3)))))))) # label(fact_749_convex__bound__lt) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 896 (all C_1 all A_1 all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(C_1,A_1)),plus_plus_int(C_1,B))))) # label(fact_393_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 897 (all L bit0(minus_minus_int(pls,L)) = minus_minus_int(pls,bit0(L))) # label(fact_914_diff__bin__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 898 (all L minus_minus_int(min,bit1(L)) = bit0(minus_minus_int(min,L))) # label(fact_1027_diff__bin__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 899 (all X all Y (X = Y -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)))) # label(fact_1108_dvd_Oeq__refl) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 900 (all A_1 all E_1 all C_1 all B all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(A_1,E_1),C_1)),plus_plus_int(times_times_int(B,E_1),D))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_1),plus_plus_int(times_times_int(minus_minus_int(B,A_1),E_1),D))))) # label(fact_522_le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 901 (all A_75 (is_int(A_75) -> plus_plus_int(A_75,zero_zero_int) = A_75)) # label(fact_358_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 902 (all C_27 all D_10 all A_59 all B_40 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_59),B_40)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_27),D_10)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_40)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_27)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(A_59,C_27)),times_times_real(B_40,D_10)))))))) # label(fact_482_mult__mono) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 903 (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)) -> Z = W)))) # label(fact_453_zle__antisym) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 904 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),M)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N)),M)) | N = M | zero_zero_nat = M)) # label(fact_1080_divides__cases) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 905 (all M all K_1 all N minus_minus_nat(M,N) = minus_minus_nat(plus_plus_nat(M,K_1),plus_plus_nat(N,K_1))) # label(fact_707_diff__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 906 (all Xa all Ya all P_1 ((-hBOOL(P_1) -> hAPP_nat_int(semiri1621563631at_int,if_nat(P_1,Xa,Ya)) = hAPP_nat_int(semiri1621563631at_int,Ya)) & (hBOOL(P_1) -> hAPP_nat_int(semiri1621563631at_int,if_nat(P_1,Xa,Ya)) = hAPP_nat_int(semiri1621563631at_int,Xa)))) # label(fact_222_int__if__cong) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 907 (all A all P_2 (is_int(A) -> (hBOOL(hAPP_int_bool(zprime,P_2)) -> (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_2)) -> (hBOOL(hAPP_int_bool(zcong(times_times_int(A,A),one_one_int),P_2)) -> one_one_int = A | A = minus_minus_int(P_2,one_one_int))))))) # label(fact_1122_zcong__square__zless) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 908 (all Na (zero_zero_nat = Na <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Na)),zero_zero_int)))) # label(fact_807_int__le__0__conv) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 909 (all B_60 all A_87 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_87)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_60)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),plus_plus_real(A_87,B_60)))))) # label(fact_309_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 910 (all Y_6 all X_7 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_7)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_6)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_6),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(Y_6,X_7)),X_7)))))) # label(fact_581_mult__left__le__one__le) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 911 (all V_15 all W_10 plus_plus_real(number267125858f_real(V_15),number267125858f_real(W_10)) = number267125858f_real(plus_plus_int(V_15,W_10))) # label(fact_123_add__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 912 (all I_2 all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),N)) -> minus_minus_nat(N,minus_minus_nat(N,I_2)) = I_2)) # label(fact_905_diff__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 913 (all V_14 all W_9 plus_plus_real(number267125858f_real(V_14),number267125858f_real(W_9)) = number267125858f_real(plus_plus_int(V_14,W_9))) # label(fact_125_number__of__add) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 914 (all B_60 all A_87 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_87)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_60)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),plus_plus_int(A_87,B_60)))))) # label(fact_307_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 915 (all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) -> X != Y)) # label(fact_1093_dvd_Oless__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 916 (all N_20 all A_94 (is_int(A_94) -> (A_94 != zero_zero_int -> zero_zero_int != hAPP_nat_int(power_power_int(A_94),N_20)))) # label(fact_231_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 917 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X)) -> (exists N_1 (is_int(N_1) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(number_number_of_int(bit0(bit1(pls))),abs_abs_int(minus_minus_int(Y,times_times_int(N_1,X))))),X)))))) # label(fact_1174_best__division__abs) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 918 (all A_22 all E all B_11 all C_8 plus_plus_real(times_times_real(plus_plus_real(A_22,B_11),E),C_8) = plus_plus_real(times_times_real(A_22,E),plus_plus_real(times_times_real(B_11,E),C_8))) # label(fact_634_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 919 (all Xa all P_3 (hBOOL(hAPP_int_bool(zcong(Xa,zero_zero_int),P_3)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),Xa)))) # label(fact_1043_zcong__eq__zdvd__prop) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 920 (all A_44 all C_19 all B_26 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_44,C_19)),times_times_int(B_26,C_19))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_19)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_44),B_26))))) # label(fact_546_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 921 (all V_17 number267125858f_real(plus_plus_int(V_17,bit1(pls))) = plus_plus_real(number267125858f_real(V_17),one_one_real)) # label(fact_15_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 922 (all A_1 all Wa (A_1 = zero_zero_nat & number_number_of_nat(Wa) != zero_zero_nat <-> zero_zero_nat = hAPP_nat_nat(power_power_nat(A_1),number_number_of_nat(Wa)))) # label(fact_118_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 923 (all M all N hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,minus_minus_nat(M,N)),M))) # label(fact_908_Nat_Odiff__le__self) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 924 (all M hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),times_times_nat(M,times_times_nat(M,M))))) # label(fact_845_le__cube) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 925 (all Lx_1 all Ly_1 all Rx_1 all Ry_1 times_times_nat(Rx_1,times_times_nat(times_times_nat(Lx_1,Ly_1),Ry_1)) = times_times_nat(times_times_nat(Lx_1,Ly_1),times_times_nat(Rx_1,Ry_1))) # label(fact_436_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 926 (all W number_number_of_nat(W) = nat(number_number_of_int(W))) # label(fact_766_nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 927 (all Xa all Ya (is_int(Xa) & is_int(Ya) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(Xa,Xa),times_times_int(Ya,Ya))),zero_zero_int)) <-> Xa = zero_zero_int & zero_zero_int = Ya))) # label(fact_578_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 928 (all C_25 all B_38 all A_57 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_38),A_57)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_25),zero_zero_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(C_25,A_57)),times_times_int(C_25,B_38)))))) # label(fact_489_mult__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 929 (all N -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat))) # label(fact_172_less__zeroE) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 930 (all A_80 all B_53 all C_39 plus_plus_nat(plus_plus_nat(A_80,B_53),C_39) = plus_plus_nat(A_80,plus_plus_nat(B_53,C_39))) # label(fact_338_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 931 (all Wa all Ya all Xa all Z_1 (is_int(Ya) & is_int(Xa) & is_int(Z_1) & is_int(Wa) -> (plus_plus_int(times_times_int(Wa,Ya),times_times_int(Xa,Z_1)) = plus_plus_int(times_times_int(Wa,Z_1),times_times_int(Xa,Ya)) <-> Wa = Xa | Z_1 = Ya))) # label(fact_639_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 932 (all M all N hAPP_nat_int(semiri1621563631at_int,times_times_nat(M,N)) = times_times_int(hAPP_nat_int(semiri1621563631at_int,M),hAPP_nat_int(semiri1621563631at_int,N))) # label(fact_934_int__mult) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 933 (all A_1 all B all C_1 (C_1 != zero_zero_real -> (B = A_1 <-> times_times_real(C_1,A_1) = times_times_real(C_1,B)))) # label(fact_872_real__mult__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 934 (all Lx_4 all Ly_4 all Rx_4 times_times_int(Lx_4,times_times_int(Ly_4,Rx_4)) = times_times_int(times_times_int(Lx_4,Ly_4),Rx_4)) # label(fact_426_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 935 (all Z_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_1),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(plus_plus_int(one_one_int,Z_1),Z_1)),zero_zero_int)))) # label(fact_88_odd__less__0) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 936 (all M all N all K_1 plus_plus_nat(plus_plus_nat(M,N),K_1) = plus_plus_nat(M,plus_plus_nat(N,K_1))) # label(fact_181_nat__add__assoc) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 937 (all Xa all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(Xa)),number_number_of_int(Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)))) # label(fact_33_less__number__of) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 938 (all C_34 all A_69 all B_51 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(C_34,A_69)),plus_plus_nat(C_34,B_51))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_69),B_51)))) # label(fact_379_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 939 (all A_84 all B_57 all C_43 (is_int(B_57) & is_int(C_43) -> (plus_plus_int(A_84,C_43) = plus_plus_int(A_84,B_57) -> C_43 = B_57))) # label(fact_319_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 940 (all A_1 all B (is_int(B) & is_int(A_1) -> (zero_zero_int = minus_minus_int(A_1,B) <-> A_1 = B))) # label(fact_594_eq__iff__diff__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 941 (all A_29 times_times_nat(zero_zero_nat,A_29) = zero_zero_nat) # label(fact_608_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 942 (all A_7 all B_3 all N_6 times_times_int(hAPP_nat_int(power_power_int(A_7),N_6),hAPP_nat_int(power_power_int(B_3),N_6)) = hAPP_nat_int(power_power_int(times_times_int(A_7,B_3)),N_6)) # label(fact_699_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 943 (all W hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),W))) # label(fact_977_real__le__refl) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 944 (all M_8 all N_16 hAPP_nat_int(semiri1621563631at_int,plus_plus_nat(M_8,N_16)) = plus_plus_int(hAPP_nat_int(semiri1621563631at_int,M_8),hAPP_nat_int(semiri1621563631at_int,N_16))) # label(fact_245_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 945 (all W all Z times_times_nat(nat(abs_abs_int(W)),nat(abs_abs_int(Z))) = nat(abs_abs_int(times_times_int(W,Z)))) # label(fact_1156_nat__abs__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 946 (all Z_1 (is_int(Z_1) -> (number_number_of_int(min) = Z_1 | Z_1 = one_one_int <-> one_one_int = abs_abs_int(Z_1)))) # label(fact_1163_abs__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 947 (all P_1 ((all X_1 (is_int(X_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) -> hBOOL(hAPP_nat_bool(P_1,nat(X_1)))))) <-> (all X1 hBOOL(hAPP_nat_bool(P_1,X1))))) # label(fact_531_all__nat) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 948 (all B all A_1 (is_int(A_1) & is_int(B) -> (plus_plus_int(B,A_1) = B <-> zero_zero_int = A_1))) # label(fact_355_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 949 (all X -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),X)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),X)))) # label(fact_1114_dvd_Oless__irrefl) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 950 (all A_78 all C_37 all D_13 plus_plus_nat(plus_plus_nat(A_78,C_37),D_13) = plus_plus_nat(A_78,plus_plus_nat(C_37,D_13))) # label(fact_344_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 951 (all A_18 A_18 = times_times_nat(A_18,one_one_nat)) # label(fact_657_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 952 (all A_33 minus_minus_real(A_33,zero_zero_real) = A_33) # label(fact_597_diff__0__right) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 953 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),times_times_int(X,Y)))))) # label(fact_513_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 954 (all A all B_1 (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),A)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1)) -> -(-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),B_1)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),A))))) # label(fact_1088_dvd_Oless__asym_H) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 955 (all B_42 all A_61 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_42),zero_zero_nat)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_61)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_61),zero_zero_nat)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_42)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_61,B_42)),zero_zero_nat)))) # label(fact_478_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 956 (all V_13 all V_12 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_12)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_13)) -> times_times_nat(number_number_of_nat(V_12),number_number_of_nat(V_13)) = number_number_of_nat(times_times_int(V_12,V_13))))) # label(fact_446_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 957 (all M all N (is_int(M) -> (times_times_int(M,N) = one_one_int -> M = number_number_of_int(min) | M = one_one_int))) # label(fact_1030_pos__zmult__eq__1__iff__lemma) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 958 (all Wa all Xa (number267125858f_real(Wa) = Xa <-> number267125858f_real(Wa) = Xa)) # label(fact_60_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 959 (all A_95 (is_int(A_95) -> A_95 = hAPP_nat_int(power_power_int(A_95),one_one_nat))) # label(fact_215_power__one__right) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 960 (all Xa all Ya (Ya != Xa & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)))) # label(fact_879_real__less__def) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 961 (all Xa all Na (is_int(Xa) -> (Xa != zero_zero_int | zero_zero_nat = Na <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(power_power_int(abs_abs_int(Xa)),Na)))))) # label(fact_1165_zero__less__zpower__abs__iff) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 962 (all M_7 all N_15 hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(power_power_nat(M_7),N_15)) = hAPP_nat_int(power_power_int(hAPP_nat_int(semiri1621563631at_int,M_7)),N_15)) # label(fact_251_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 963 (all K_1 all L bit0(minus_minus_int(K_1,L)) = minus_minus_int(bit0(K_1),bit0(L))) # label(fact_862_diff__bin__simps_I7_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 964 (all W hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),W))) # label(fact_448_zle__refl) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 965 (all N all M hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),plus_plus_nat(N,M)))) # label(fact_892_le__add1) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 966 (all Xa all Ya (zero_zero_real = Ya & Xa = zero_zero_real <-> zero_zero_real = plus_plus_real(hAPP_nat_real(power_power_real(Xa),number_number_of_nat(bit0(bit1(pls)))),hAPP_nat_real(power_power_real(Ya),number_number_of_nat(bit0(bit1(pls))))))) # label(fact_153_realpow__two__sum__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 967 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> plus_plus_nat(N,minus_minus_nat(M,N)) = M)) # label(fact_938_le__add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 968 (all A_21 all M_3 all B_10 plus_plus_real(times_times_real(A_21,M_3),times_times_real(B_10,M_3)) = times_times_real(plus_plus_real(A_21,B_10),M_3)) # label(fact_640_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 969 (all X_5 all Y_4 all Q_2 times_times_int(hAPP_nat_int(power_power_int(X_5),Q_2),hAPP_nat_int(power_power_int(Y_4),Q_2)) = hAPP_nat_int(power_power_int(times_times_int(X_5,Y_4)),Q_2)) # label(fact_702_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 970 (all A_1 all E_1 all C_1 all B all D (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(A_1,E_1),C_1)),plus_plus_int(times_times_int(B,E_1),D))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_1),plus_plus_int(times_times_int(minus_minus_int(B,A_1),E_1),D))))) # label(fact_538_less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 971 -(all T (is_int(T) -> times_times_int(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),T) != plus_plus_int(hAPP_nat_int(power_power_int(s),number_number_of_nat(bit0(bit1(pls)))),one_one_int))) # label(fact_976__096_B_Bthesis_O_A_I_B_Bt_O_As_____A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A_) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 972 (all Lx_3 all Ly_3 all Rx_3 times_times_nat(times_times_nat(Lx_3,Rx_3),Ly_3) = times_times_nat(times_times_nat(Lx_3,Ly_3),Rx_3)) # label(fact_430_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 973 (all X (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,abs_abs_real(X)),one_one_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(power_power_real(X),number_number_of_nat(bit0(bit1(pls))))),one_one_real)))) # label(fact_1178_less__one__imp__sqr__less__one) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 974 (all X_12 hAPP_nat_nat(power_power_nat(X_12),one_one_nat) = X_12) # label(fact_400_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 975 (all X_2 all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2)) -> hAPP_nat_real(power_power_real(X_2),N_2) = times_times_real(hAPP_nat_real(power_power_real(X_2),minus_minus_nat(N_2,one_one_nat)),X_2))) # label(fact_755_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 1.55/1.82 976 (all I_2 all K_1 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),J_1)) -> plus_plus_nat(minus_minus_nat(J_1,K_1),I_2) = minus_minus_nat(plus_plus_nat(J_1,I_2),K_1))) # label(fact_945_diff__add__assoc2) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 977 (all K_1 all L plus_plus_int(bit0(times_times_int(K_1,L)),L) = times_times_int(bit1(K_1),L)) # label(fact_804_mult__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 978 (all B_1_1 all B_2_1 (is_int(B_2_1) & is_int(B_1_1) -> is_int(minus_minus_int(B_1_1,B_2_1)))) # label(gsy_c_Groups_Ominus__class_Ominus_000tc__Int__Oint) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 979 (all Ma all Na (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(semiri132038758t_real,Ma)),hAPP_nat_real(semiri132038758t_real,Na))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)))) # label(fact_472_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 980 (all A_40 all C_15 all B_22 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_40,C_15)),times_times_nat(B_22,C_15))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_15)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_40),B_22))))) # label(fact_557_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 981 (all A_1 all C_1 all B (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_1,C_1)),plus_plus_nat(B,C_1))))) # label(fact_397_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 982 (all A_1 all C_1 all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(A_1,C_1)),plus_plus_int(B,C_1))))) # label(fact_396_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 983 (all I_1 all J all K (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),minus_minus_nat(J,K))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(I_1,K)),J)))) # label(fact_772_less__diff__conv) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 984 (all A_1 all B (times_times_real(A_1,B) = zero_zero_real <-> zero_zero_real = B | zero_zero_real = A_1)) # label(fact_613_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 985 (all Xa all Ya (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) | Xa = Ya)) # label(fact_1112_dvd_Ole__less) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 986 (all X_9 all Y_8 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),plus_plus_int(times_times_int(X_9,X_9),times_times_int(Y_8,Y_8))))) # label(fact_576_sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 987 (all Xa all Ya 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,times_times_real(Xa,Z_1)),times_times_real(Ya,Z_1))) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya))))) # label(fact_925_real__mult__less__iff1) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 988 (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_1117_one__not__neg__one__mod__m) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 989 (all P_1 ((exists X_1 hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X_1)))) <-> (exists X_1 (hBOOL(hAPP_int_bool(P_1,X_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) & is_int(X_1))))) # label(fact_785_transfer__int__nat__quantifiers_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 990 (all K (is_int(K) -> (bit0(K) = pls <-> K = pls))) # label(fact_95_rel__simps_I44_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 991 (all A_19 (is_int(A_19) -> A_19 = times_times_int(A_19,one_one_int))) # label(fact_655_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 992 (all B_31 all A_50 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_50)) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_31),zero_zero_real)) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(B_31,A_50)),zero_zero_real))))) # label(fact_506_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 993 (all K_1 all I_2 all J_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),J_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(K_1,I_2)),times_times_nat(K_1,J_1)))))) # label(fact_917_mult__less__mono2) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 994 (all P_4 all P_1 all Xa ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) -> (hBOOL(P_4) <-> hBOOL(P_1))) -> ((hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) -> hBOOL(P_1)) <-> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) -> hBOOL(P_4))))) # label(fact_1147_imp__le__cong) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 995 (exists X_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) & (all Y_1 (is_int(Y_1) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_1),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))) & hBOOL(hAPP_int_bool(zcong(s1,Y_1),plus_plus_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_1)) -> X_1 = Y_1))) & hBOOL(hAPP_int_bool(zcong(s1,X_1),plus_plus_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,X_1),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))) & is_int(X_1))) # label(fact_1083__096EX_B_As_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_A1_A_G_A_091s) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 996 (all C_21 all A_53 all B_34 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_53),B_34)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_21)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(A_53,C_21)),times_times_int(B_34,C_21)))))) # label(fact_500_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 997 (all K all Ma all Na (times_times_nat(K,Na) = times_times_nat(K,Ma) <-> Ma = Na | K = zero_zero_nat)) # label(fact_1183_nat__mult__eq__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 998 (all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Xa),one_one_int)) <-> abs_abs_int(Xa) = one_one_int)) # label(fact_1158_zdvd1__eq) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 999 (all Y all X all P_2 (hBOOL(hAPP_int_bool(zprime,P_2)) -> (-hBOOL(hAPP_int_bool(zcong(X,zero_zero_int),P_2)) -> (-hBOOL(hAPP_int_bool(zcong(Y,zero_zero_int),P_2)) -> -hBOOL(hAPP_int_bool(zcong(times_times_int(X,Y),zero_zero_int),P_2)))))) # label(fact_1067_zcong__zmult__prop3) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1000 (all B_2 all Q_1 all R_2 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),plus_plus_int(times_times_int(B_2,Q_1),R_2))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),B_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Q_1)))))) # label(fact_831_q__pos__lemma) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1001 (all A_1 all B (zero_zero_real = minus_minus_real(A_1,B) <-> B = A_1)) # label(fact_591_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1002 (all N 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,minus_minus_nat(J_1,N)),K_1)))) # label(fact_692_less__imp__diff__less) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1003 (all Xa all Ya all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(power_power_real(B),Xa)),hAPP_nat_real(power_power_real(B),Ya)))))) # label(fact_735_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1004 (all K_1 all I_2 all J_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),J_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(I_2,K_1)),plus_plus_int(J_1,K_1))))) # label(fact_51_zadd__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1005 (all A_21 all M_3 all B_10 times_times_int(plus_plus_int(A_21,B_10),M_3) = plus_plus_int(times_times_int(A_21,M_3),times_times_int(B_10,M_3))) # label(fact_642_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1006 (all M_6 all N_13 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),M_6)),hAPP_nat_int(power_power_int(A_90),N_13))) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_6),N_13))))) # label(fact_278_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1007 (all K_1 all A all B_1 all M (hBOOL(hAPP_int_bool(zcong(A,B_1),M)) -> hBOOL(hAPP_int_bool(zcong(times_times_int(K_1,A),times_times_int(K_1,B_1)),M)))) # label(fact_1130_zcong__scalar2) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1008 (all N_26 hAPP_nat_int(semiri1621563631at_int,N_26) = number_number_of_int(hAPP_nat_int(semiri1621563631at_int,N_26))) # label(fact_131_number__of__int) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1009 (all Z times_times_nat(Z,number_number_of_nat(bit0(bit1(pls)))) = plus_plus_nat(Z,Z)) # label(fact_959_nat__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1010 (all C_26 all D_9 all A_58 all B_39 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_58),B_39)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_26),D_9)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_58)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_26)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,times_times_nat(A_58,C_26)),times_times_nat(B_39,D_9)))))))) # label(fact_486_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1011 (all A_2 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,times_times_real(A_2,A_2)),zero_zero_real))) # label(fact_762_not__square__less__zero) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1012 (all B_59 all A_86 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_86),zero_zero_nat)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_59),zero_zero_nat)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_86,B_59)),zero_zero_nat))))) # label(fact_311_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1013 (all C_13 all D_7 all A_38 all B_20 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_38),B_20)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_13),D_7)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_38)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_13)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_38,C_13)),times_times_nat(B_20,D_7)))))))) # label(fact_563_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1014 (all X all Y (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> -(hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))))) # label(fact_1086_dvd_Oless__asym) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1015 (all Z_2 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_eq_int,zero_zero_int),Z_2)) -> plus_plus_nat(nat(Z),nat(Z_2)) = nat(plus_plus_int(Z,Z_2))))) # label(fact_744_nat__add__distrib) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1016 (all Xa all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,number_number_of_int(Xa)),number_number_of_int(Ya))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)))) # label(fact_455_le__number__of) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1017 (all C_32 all D_11 all A_67 all B_49 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_67),B_49)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_32),D_11)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,plus_plus_nat(A_67,C_32)),plus_plus_nat(B_49,D_11)))))) # label(fact_385_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1018 (all Xa all Ya (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)) & -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) <-> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)))) # label(fact_1110_dvd_Oless__le__not__le) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1019 (all Lx_3 all Ly_3 all Rx_3 times_times_real(times_times_real(Lx_3,Rx_3),Ly_3) = times_times_real(times_times_real(Lx_3,Ly_3),Rx_3)) # label(fact_431_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1020 (all A_1 all Na (zero_zero_nat != Na & A_1 = zero_zero_real <-> zero_zero_real = hAPP_nat_real(power_power_real(A_1),Na))) # label(fact_286_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1021 (all A_24 zero_zero_real = times_times_real(zero_zero_real,A_24)) # label(fact_624_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1022 (all U all Ma all Na all I_1 all J (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J)) -> (Ma = plus_plus_nat(times_times_nat(minus_minus_nat(J,I_1),U),Na) <-> plus_plus_nat(times_times_nat(I_1,U),Ma) = plus_plus_nat(times_times_nat(J,U),Na)))) # label(fact_1195_nat__eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1023 (all K_1 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,K_1)),zero_zero_int))) # label(fact_83_int__less__0__conv) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1024 (all C_6 all A_13 all B_8 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(C_6,A_13)),plus_plus_int(C_6,B_8))) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_13),B_8)))) # label(fact_673_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1025 (all X_2 all N_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_2)) -> times_times_nat(hAPP_nat_nat(power_power_nat(X_2),minus_minus_nat(N_2,one_one_nat)),X_2) = hAPP_nat_nat(power_power_nat(X_2),N_2))) # label(fact_756_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1026 (all V_13 all V_12 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_12)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_13)) -> number267125858f_real(times_times_int(V_12,V_13)) = times_times_real(number267125858f_real(V_12),number267125858f_real(V_13))))) # label(fact_445_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1027 (all A_34 zero_zero_real = minus_minus_real(A_34,A_34)) # label(fact_595_diff__self) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1028 (all M minus_minus_nat(M,M) = zero_zero_nat) # label(fact_628_diff__self__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1029 (all A_1 all Wa (hAPP_nat_real(power_power_real(A_1),number_number_of_nat(Wa)) = zero_zero_real <-> zero_zero_real = A_1 & zero_zero_nat != number_number_of_nat(Wa))) # label(fact_119_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1030 (all Z all W times_times_int(W,Z) = times_times_int(Z,W)) # label(fact_449_zmult__commute) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1031 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) -> minus_minus_nat(M,N) = zero_zero_nat)) # label(fact_928_diff__is__0__eq_H) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1032 (all B_1 all Q all R_1 all B_2 all Q_1 all R_2 (plus_plus_int(times_times_int(B_1,Q),R_1) = plus_plus_int(times_times_int(B_2,Q_1),R_2) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,plus_plus_int(times_times_int(B_2,Q_1),R_2)),zero_zero_int)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_1),B_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_2),B_1)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q))))))))) # label(fact_969_zdiv__mono2__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1033 (all A_1 all B (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),zero_zero_real)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B)) | hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_1)) & hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B),zero_zero_real)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,times_times_real(A_1,B)),zero_zero_real)))) # label(fact_527_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1034 (all N hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N)))) # label(fact_787_zero__zle__int) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1035 (all A_1 all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B),zero_zero_int)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),zero_zero_int)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_1)) & hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),times_times_int(A_1,B))))) # label(fact_530_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1036 (all U_1 all M all N all J_1 all I_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),I_2)) -> minus_minus_nat(plus_plus_nat(times_times_nat(minus_minus_nat(I_2,J_1),U_1),M),N) = minus_minus_nat(plus_plus_nat(times_times_nat(I_2,U_1),M),plus_plus_nat(times_times_nat(J_1,U_1),N)))) # label(fact_1191_nat__diff__add__eq1) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1037 (all A_83 all B_56 all C_42 (plus_plus_nat(A_83,C_42) = plus_plus_nat(A_83,B_56) -> B_56 = C_42)) # label(fact_323_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1038 (all I_2 all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(power_power_nat(I_2),M)),hAPP_nat_nat(power_power_nat(I_2),N))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),I_2)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))))) # label(fact_1076_power__dvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1039 (all M_4 all N_10 hAPP_nat_nat(semiri984289939at_nat,times_times_nat(M_4,N_10)) = times_times_nat(hAPP_nat_nat(semiri984289939at_nat,M_4),hAPP_nat_nat(semiri984289939at_nat,N_10))) # label(fact_470_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1040 (all A_91 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_91),plus_plus_nat(A_91,one_one_nat)))) # label(fact_270_less__add__one) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1041 (all A_82 all B_55 all C_41 all D_14 plus_plus_nat(plus_plus_nat(A_82,C_41),plus_plus_nat(B_55,D_14)) = plus_plus_nat(plus_plus_nat(A_82,B_55),plus_plus_nat(C_41,D_14))) # label(fact_326_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1042 (all M all X (is_int(X) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),M)) -> (hBOOL(hAPP_int_bool(zcong(X,zero_zero_int),M)) -> X = zero_zero_int))))) # label(fact_1066_Int2_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1043 (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_1004_rel__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1044 (all M_5 -hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_5)),zero_zero_real))) # label(fact_289_of__nat__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1045 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,Ma)),hAPP_nat_nat(semiri984289939at_nat,Na))))) # label(fact_237_of__nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1046 (all K_1 all M all N (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),minus_minus_int(M,N))) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),N)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_1),M))))) # label(fact_990_zdvd__zdiffD) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1047 (all A_74 A_74 = plus_plus_real(A_74,zero_zero_real)) # label(fact_363_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1048 (all X all Y (X != Y -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y),X))))) # label(fact_185_linorder__neqE__nat) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1049 (all A_41 all C_16 all B_23 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,times_times_nat(A_41,C_16)),times_times_nat(B_23,C_16))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_16)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_41),B_23))))) # label(fact_554_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1050 (all A_23 all B_12 all C_9 plus_plus_real(times_times_real(A_23,C_9),times_times_real(B_12,C_9)) = times_times_real(plus_plus_real(A_23,B_12),C_9)) # label(fact_631_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1051 (all Z hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Z)))) # label(fact_784_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1052 (all X_6 all Y_5 all Z_3 plus_plus_real(times_times_real(X_6,Y_5),times_times_real(X_6,Z_3)) = times_times_real(X_6,plus_plus_real(Y_5,Z_3))) # label(fact_649_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1053 (all K all Ma all Na (plus_plus_nat(K,Ma) = plus_plus_nat(K,Na) <-> Ma = Na)) # label(fact_182_nat__add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1054 (all Lx_6 all Rx_6 all Ry_4 times_times_int(Lx_6,times_times_int(Rx_6,Ry_4)) = times_times_int(Rx_6,times_times_int(Lx_6,Ry_4))) # label(fact_417_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1055 (all I_2 all U_1 all J_1 all K_1 plus_plus_nat(times_times_nat(plus_plus_nat(I_2,J_1),U_1),K_1) = plus_plus_nat(times_times_nat(I_2,U_1),plus_plus_nat(times_times_nat(J_1,U_1),K_1))) # label(fact_1184_left__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1056 (all Xa (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,number267125858f_real(Xa)),one_one_real)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),bit1(pls))))) # label(fact_753_le__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1057 (all A all B_1 minus_minus_int(plus_plus_int(minus_minus_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit1(bit1(pls)))),times_times_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_1)),times_times_int(times_times_int(number_number_of_int(bit1(bit1(pls))),A),hAPP_nat_int(power_power_int(B_1),number_number_of_nat(bit0(bit1(pls)))))),hAPP_nat_int(power_power_int(B_1),number_number_of_nat(bit1(bit1(pls))))) = hAPP_nat_int(power_power_int(minus_minus_int(A,B_1)),number_number_of_nat(bit1(bit1(pls))))) # label(fact_967_zdiff__power3) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1058 (all Lx all Ly all Rx all Ry times_times_int(times_times_int(Lx,Ly),times_times_int(Rx,Ry)) = times_times_int(times_times_int(Lx,Rx),times_times_int(Ly,Ry))) # label(fact_440_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1059 (all N all M ((M != zero_zero_nat -> plus_plus_nat(N,times_times_nat(minus_minus_nat(M,one_one_nat),N)) = times_times_nat(M,N)) & (M = zero_zero_nat -> zero_zero_nat = times_times_nat(M,N)))) # label(fact_955_mult__eq__if) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1060 (all A_64 all B_46 times_times_nat(B_46,A_64) = times_times_nat(A_64,B_46)) # label(fact_415_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1061 (all V_10 all W_5 times_times_real(number267125858f_real(V_10),number267125858f_real(W_5)) = number267125858f_real(times_times_int(V_10,W_5))) # label(fact_458_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1062 (all A_14 times_times_real(one_one_real,A_14) = A_14) # label(fact_668_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1063 (all N -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N))) # label(fact_174_less__not__refl) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1064 (all N all M (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) -> plus_plus_nat(minus_minus_nat(M,N),N) = M)) # label(fact_941_le__add__diff__inverse2) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1065 (all X_13 hAPP_nat_nat(power_power_nat(X_13),zero_zero_nat) = one_one_nat) # label(fact_301_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1066 (all Xa (one_one_nat = Xa <-> Xa = one_one_nat)) # label(fact_353_one__reorient) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1067 (all Z (is_int(Z) -> (-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z)) -> zero_zero_int = hAPP_nat_int(semiri1621563631at_int,nat(Z))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z)) -> hAPP_nat_int(semiri1621563631at_int,nat(Z)) = Z))) # label(fact_588_int__nat__eq) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1068 (all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Na))))) # label(fact_224_zero__less__int__conv) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1069 (all A_26 all B_13 (times_times_nat(A_26,B_13) = zero_zero_nat -> A_26 = zero_zero_nat | zero_zero_nat = B_13)) # label(fact_619_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1070 (all Na all K all Ma (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Ma)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Na)) -> (minus_minus_nat(Ma,K) = minus_minus_nat(Na,K) <-> Ma = Na)))) # label(fact_904_eq__diff__iff) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1071 (all Xa all Ya (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Xa)),hAPP_nat_int(semiri1621563631at_int,Ya))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)))) # label(fact_946_Nat__Transfer_Otransfer__int__nat__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1072 (all B_1_1 (is_int(B_1_1) -> is_int(abs_abs_int(B_1_1)))) # label(gsy_c_Groups_Oabs__class_Oabs_000tc__Int__Oint) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1073 (all Y_7 all X_8 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_8)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_7)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_7),one_one_int)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,times_times_int(X_8,Y_7)),X_8)))))) # label(fact_580_mult__right__le__one__le) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1074 (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_450_zle__linear) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1075 (all B_1 all Q_1 all R_2 all Q all R_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(times_times_int(B_1,Q_1),R_2)),plus_plus_int(times_times_int(B_1,Q),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_1),R_1)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),R_2)) -> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q),Q_1))))))) # label(fact_970_unique__quotient__lemma__neg) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1076 (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_int,minus_minus_int(K,L_1)),zero_zero_int)))) # label(fact_916_less__bin__lemma) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1077 (all C all A all B_1 (A = B_1 -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_1),C)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A),C))))) # label(fact_1104_dvd_Oord__eq__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1078 (all A_88 hAPP_nat_real(power_power_real(A_88),zero_zero_nat) = one_one_real) # label(fact_292_power__0) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1079 (all K_1 all L bit0(plus_plus_int(K_1,L)) = plus_plus_int(bit0(K_1),bit0(L))) # label(fact_102_add__Bit0__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1080 (all P_1 all A_1 all B ((all D_2 (plus_plus_nat(B,D_2) = A_1 -> hBOOL(hAPP_nat_bool(P_1,D_2)))) & (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B)) -> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))) <-> hBOOL(hAPP_nat_bool(P_1,minus_minus_nat(A_1,B))))) # label(fact_795_nat__diff__split) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1081 (all V_2 all V_1 ((-hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls)) -> times_times_nat(number_number_of_nat(V_1),number_number_of_nat(V_2)) = number_number_of_nat(times_times_int(V_1,V_2))) & (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls)) -> times_times_nat(number_number_of_nat(V_1),number_number_of_nat(V_2)) = zero_zero_nat))) # label(fact_961_mult__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1082 (all A_1 all B (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,minus_minus_int(A_1,B)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B)))) # label(fact_759_less__iff__diff__less__0) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1083 (all Xa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,number_number_of_int(Xa)),zero_zero_int)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),pls)))) # label(fact_69_less__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1084 (all X_9 all Y_8 hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),plus_plus_real(times_times_real(X_9,X_9),times_times_real(Y_8,Y_8))))) # label(fact_575_sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1085 (all Ma all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,Na))))) # label(fact_947_zle__int) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1086 (all A hAPP_nat_int(power_power_int(abs_abs_int(A)),number_number_of_nat(bit0(bit1(pls)))) = abs_abs_int(hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls)))))) # label(fact_1169_abs__power2__distrib) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1087 (all A_7 all B_3 all N_6 times_times_real(hAPP_nat_real(power_power_real(A_7),N_6),hAPP_nat_real(power_power_real(B_3),N_6)) = hAPP_nat_real(power_power_real(times_times_real(A_7,B_3)),N_6)) # label(fact_697_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1088 (all A_1 all B all C_1 all D (minus_minus_real(A_1,B) = minus_minus_real(C_1,D) -> (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_1),D)) <-> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B))))) # label(fact_600_diff__eq__diff__less) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1089 (all Y all X (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X)) -> (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y)) -> nat(plus_plus_int(X,Y)) = plus_plus_nat(nat(X),nat(Y))))) # label(fact_743_Nat__Transfer_Otransfer__nat__int__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1090 (all Ma all Na all A_1 (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_1)) -> (hAPP_nat_int(power_power_int(A_1),Na) = hAPP_nat_int(power_power_int(A_1),Ma) <-> Na = Ma))) # label(fact_272_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1091 (all A_29 zero_zero_int = times_times_int(zero_zero_int,A_29)) # label(fact_609_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1092 (all N_23 ((N_23 != zero_zero_nat -> hAPP_nat_real(power_power_real(zero_zero_real),N_23) = zero_zero_real) & (N_23 = zero_zero_nat -> hAPP_nat_real(power_power_real(zero_zero_real),N_23) = one_one_real))) # label(fact_162_power__0__left) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1093 (all B all A_1 (zero_zero_real = A_1 <-> plus_plus_real(B,A_1) = B)) # label(fact_357_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1094 (all X all Y (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) -> -(-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) & hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X))))) # label(fact_1094_dvd_Oless__imp__not__less) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1095 (all Z_1 all Wa (hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Wa)) | hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,nat(Wa)),nat(Z_1))) <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Wa),Z_1))))) # label(fact_956_nat__le__eq__zle) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1096 (all B_14 all A_27 (is_int(A_27) & is_int(B_14) -> (zero_zero_int != A_27 -> (B_14 != zero_zero_int -> zero_zero_int != times_times_int(A_27,B_14))))) # label(fact_617_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 1.55/1.83 1097 (all Xa all Ya (is_int(Xa) & is_int(Ya) -> (Ya = Xa <-> number_number_of_int(Xa) = number_number_of_int(Ya)))) # label(fact_56_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 1.83/2.11 1098 (all M hBOOL(hAPP_int_bool(zcong(M,zero_zero_int),M))) # label(fact_1058_zcong__id) # label(axiom) # label(non_clause). [assumption]. 1.83/2.11 1099 (all A_1 all B all C_1 all D (is_int(A_1) & is_int(B) & is_int(D) & is_int(C_1) -> (minus_minus_int(A_1,B) = minus_minus_int(C_1,D) -> (B = A_1 <-> C_1 = D)))) # label(fact_441_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 1.83/2.11 1100 (all Na all Ma (one_one_nat = times_times_nat(Na,Ma) <-> Na = one_one_nat & Ma = one_one_nat)) # label(fact_894_nat__mult__eq__one) # label(axiom) # label(non_clause). [assumption]. 1.83/2.11 1101 (all C_34 all A_69 all B_51 (hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,plus_plus_real(C_34,A_69)),plus_plus_real(C_34,B_51))) -> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_69),B_51)))) # label(fact_380_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 1.83/2.11 1102 -(zero_zero_int != hAPP_nat_int(power_power_int(plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n))),number_number_of_nat(bit0(bit1(pls))))) # label(conj_0) # label(negated_conjecture) # label(non_clause). [assumption]. 1.83/2.11 1.83/2.11 ============================== end of process non-clausal formulas === 1.83/2.11 1.83/2.11 ============================== PROCESS INITIAL CLAUSES =============== 1.83/2.11 1.83/2.11 ============================== PREDICATE ELIMINATION ================= 1.83/2.11 1103 -is_bool(A) | A = fTrue | fFalse = A # label(help_If_3_1_If_000tc__Int__Oint_T) # label(axiom). [clausify(197)]. 1.83/2.11 1104 is_bool(fTrue) # label(gsy_c_fTrue) # label(axiom). [assumption]. 1.83/2.11 1105 is_bool(hAPP_real_bool(A,B)) # label(gsy_c_hAPP_000tc__RealDef__Oreal_000tc__HOL__Obool) # label(axiom). [clausify(95)]. 1.83/2.11 Derived: fTrue = fTrue | fFalse = fTrue. [resolve(1103,a,1104,a)]. 1.83/2.11 Derived: hAPP_real_bool(A,B) = fTrue | fFalse = hAPP_real_bool(A,B). [resolve(1103,a,1105,a)]. 1.83/2.11 1106 -is_int(A) | is_bool(hAPP_int_bool(B,A)) # label(gsy_c_hAPP_000tc__Int__Oint_000tc__HOL__Obool) # label(axiom). [clausify(208)]. 1.83/2.11 Derived: -is_int(A) | hAPP_int_bool(B,A) = fTrue | fFalse = hAPP_int_bool(B,A). [resolve(1106,b,1103,a)]. 1.83/2.11 1107 -is_int(A) | -is_int(B) | -is_bool(C) | is_int(if_int(C,B,A)) # label(gsy_c_If_000tc__Int__Oint) # label(axiom). [clausify(369)]. 1.83/2.11 Derived: -is_int(A) | -is_int(B) | is_int(if_int(fTrue,B,A)). [resolve(1107,c,1104,a)]. 1.83/2.11 Derived: -is_int(A) | -is_int(B) | is_int(if_int(hAPP_real_bool(C,D),B,A)). [resolve(1107,c,1105,a)]. 1.83/2.11 Derived: -is_int(A) | -is_int(B) | is_int(if_int(hAPP_int_bool(C,D),B,A)) | -is_int(D). [resolve(1107,c,1106,b)]. 1.83/2.11 1108 is_bool(fFalse) # label(gsy_c_fFalse) # label(axiom). [assumption]. 1.83/2.11 Derived: fFalse = fTrue | fFalse = fFalse. [resolve(1108,a,1103,a)]. 1.83/2.11 Derived: -is_int(A) | -is_int(B) | is_int(if_int(fFalse,B,A)). [resolve(1108,a,1107,c)]. 1.83/2.11 1109 -is_bool(A) | fFalse = A | A = fTrue # label(help_If_3_1_If_000tc__Nat__Onat_T) # label(axiom). [clausify(749)]. 1.83/2.11 1110 is_bool(hAPP_nat_bool(A,B)) # label(gsy_c_hAPP_000tc__Nat__Onat_000tc__HOL__Obool) # label(axiom). [clausify(750)]. 1.83/2.11 Derived: hAPP_nat_bool(A,B) = fTrue | fFalse = hAPP_nat_bool(A,B). [resolve(1110,a,1103,a)]. 1.83/2.11 Derived: -is_int(A) | -is_int(B) | is_int(if_int(hAPP_nat_bool(C,D),B,A)). [resolve(1110,a,1107,c)]. 1.83/2.11 1.83/2.11 ============================== end predicate elimination ============= 1.83/2.11 1.83/2.11 Auto_denials: (non-Horn, no changes). 1.83/2.11 1.83/2.11 Term ordering decisions: 1.83/2.11 Function symbol KB weights: ord_less_eq_int=1. zero_zero_int=1. ord_less_int=1. zero_zero_nat=1. pls=1. ord_less_nat=1. zero_zero_real=1. ord_less_eq_nat=1. one_one_int=1. ord_less_eq_real=1. ord_less_real=1. semiri1621563631at_int=1. dvd_dvd_nat=1. one_one_nat=1. min=1. dvd_dvd_int=1. one_one_real=1. m=1. semiri132038758t_real=1. semiri984289939at_nat=1. zprime=1. n=1. s=1. t=1. fFalse=1. fTrue=1. s1=1. twoSqu1154269391sum2sq=1. x=1. y=1. m1=1. tn=1. int=1. c1=1. c2=1. c3=1. c4=1. c5=1. c6=1. c7=1. c8=1. c9=1. c10=1. c11=1. c12=1. hAPP_int_bool=1. hAPP_i1948725293t_bool=1. hAPP_nat_bool=1. hAPP_n1699378549t_bool=1. times_times_int=1. hAPP_real_bool=1. hAPP_r1134773055l_bool=1. plus_plus_int=1. hAPP_nat_int=1. times_times_nat=1. plus_plus_nat=1. times_times_real=1. plus_plus_real=1. minus_minus_nat=1. minus_minus_int=1. hAPP_nat_nat=1. hAPP_nat_real=1. zcong=1. minus_minus_real=1. legendre=1. f1=1. f5=1. f6=1. f7=1. f10=1. f11=1. f17=1. f18=1. f24=1. bit1=1. bit0=1. number_number_of_int=1. number_number_of_nat=1. power_power_int=1. nat=1. power_power_nat=1. number267125858f_real=1. power_power_real=1. abs_abs_int=1. succ=1. quadRes=1. abs_abs_real=1. undefined_int=1. f3=1. f4=1. f8=1. f9=1. f16=1. f19=1. f25=1. f26=1. f27=1. f28=1. if_int=1. if_nat=1. f2=1. f12=1. f13=1. f14=1. f15=1. f20=1. f21=1. f22=1. f23=1. f29=1. 5.80/6.10 % back CAC tautology: 1848 plus_plus_real(A,plus_plus_real(B,plus_plus_real(C,D))) = plus_plus_real(A,plus_plus_real(C,plus_plus_real(B,D))). [copy(1847),rewrite([1517(3),1517(6)])]. 5.80/6.10 % back CAC tautology: 1241 times_times_int(hAPP_nat_int(power_power_int(A),B),A) = times_times_int(A,hAPP_nat_int(power_power_int(A),B)) # label(fact_696_power__commutes) # label(axiom). [clausify(77)]. 5.80/6.10 % back CAC tautology: 1183 plus_plus_real(plus_plus_real(A,B),C) = plus_plus_real(plus_plus_real(A,C),B) # label(fact_336_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom). [clausify(41)]. 5.80/6.10 5.80/6.10 ============================== end of process initial clauses ======== 5.80/6.10 5.80/6.10 ============================== CLAUSES FOR SEARCH ==================== 5.80/6.10 5.80/6.10 ============================== end of clauses for search ============= 5.80/6.10 5.80/6.10 ============================== SEARCH ================================ 5.80/6.10 5.80/6.10 % Starting search at 0.60 seconds. 5.80/6.10 5.80/6.10 Low Water (keep): wt=41.000, iters=3583 5.80/6.10 5.80/6.10 Low Water (keep): wt=35.000, iters=3396 5.80/6.10 5.80/6.10 Low Water (keep): wt=34.000, iters=3446 5.80/6.10 5.80/6.10 Low Water (keep): wt=32.000, iters=3341 5.80/6.10 5.80/6.10 Low Water (keep): wt=31.000, iters=3346 5.80/6.10 5.80/6.10 Low Water (keep): wt=30.000, iters=3342 5.80/6.10 5.80/6.10 Low Water (keep): wt=29.000, iters=3371 5.80/6.10 5.80/6.10 Low Water (keep): wt=28.000, iters=3341 5.80/6.10 5.80/6.10 Low Water (keep): wt=27.000, iters=3373 5.80/6.10 5.80/6.10 Low Water (keep): wt=26.000, iters=3383 5.80/6.10 5.80/6.10 NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 54 (0.00 of 1.26 sec). 5.80/6.10 5.80/6.10 Low Water (keep): wt=25.000, iters=3379 5.80/6.10 5.80/6.10 Low Water (keep): wt=24.000, iters=3389 5.80/6.10 5.80/6.10 Low Water (keep): wt=23.000, iters=3342 5.80/6.10 5.80/6.10 Low Water (keep): wt=22.000, iters=3338 5.80/6.10 5.80/6.10 Low Water (keep): wt=21.000, iters=3382 5.80/6.10 5.80/6.10 Low Water (keep): wt=20.000, iters=3353 5.80/6.10 5.80/6.10 Low Water (keep): wt=19.000, iters=3363 5.80/6.10 5.80/6.10 Low Water (keep): wt=18.000, iters=3488 5.80/6.10 5.80/6.10 Low Water (keep): wt=17.000, iters=3368 5.80/6.10 5.80/6.10 Low Water (keep): wt=16.000, iters=3377 5.80/6.10 5.80/6.10 Low Water (keep): wt=15.000, iters=3350 5.80/6.10 5.80/6.10 Low Water (keep): wt=14.000, iters=3439 5.80/6.10 5.80/6.10 Low Water (keep): wt=13.000, iters=3338 5.80/6.10 5.80/6.10 Low Water (keep): wt=12.000, iters=3395 5.80/6.10 5.80/6.10 Low Water (keep): wt=11.000, iters=3350 5.80/6.10 5.80/6.10 Low Water (keep): wt=10.000, iters=3360 5.80/6.10 5.80/6.10 Low Water (displace): id=5021, wt=63.000 5.80/6.10 5.80/6.10 Low Water (displace): id=5002, wt=57.000 5.80/6.10 5.80/6.10 Low Water (displace): id=5016, wt=56.000 5.80/6.10 5.80/6.10 Low Water (displace): id=5014, wt=55.000 5.80/6.10 5.80/6.10 Low Water (displace): id=5012, wt=54.000 5.80/6.10 5.80/6.10 Low Water (displace): id=5799, wt=53.000 5.80/6.10 5.80/6.10 Low Water (displace): id=4997, wt=52.000 5.80/6.10 5.80/6.10 Low Water (displace): id=5018, wt=51.000 5.80/6.10 5.80/6.10 Low Water (displace): id=5017, wt=50.000 5.80/6.10 5.80/6.10 Low Water (displace): id=5028, wt=49.000 5.80/6.10 5.80/6.10 Low Water (displace): id=5104, wt=48.000 5.80/6.10 5.80/6.10 Low Water (displace): id=6028, wt=47.000 5.80/6.10 5.80/6.10 Low Water (displace): id=14406, wt=9.000 5.80/6.10 5.80/6.10 Low Water (displace): id=14416, wt=8.000 5.80/6.10 5.80/6.10 Low Water (keep): wt=9.000, iters=3345 5.80/6.10 5.80/6.10 Low Water (displace): id=16850, wt=5.000 5.80/6.10 5.80/6.10 ============================== PROOF ================================= 5.80/6.10 % SZS status Theorem 5.80/6.10 % SZS output start Refutation 5.80/6.10 5.80/6.10 % Proof 1 at 4.31 (+ 0.09) seconds. 5.80/6.10 % Length of proof is 48. 5.80/6.10 % Level of proof is 7. 5.80/6.10 % Maximum clause weight is 16.000. 5.80/6.10 % Given clauses 1882. 5.80/6.10 5.80/6.10 18 (all V_1 nat(number_number_of_int(V_1)) = number_number_of_nat(V_1)) # label(fact_767_nat__number__of__def) # label(axiom) # label(non_clause). [assumption]. 5.80/6.10 56 (all Z all W plus_plus_int(Z,W) = plus_plus_int(W,Z)) # label(fact_67_zadd__commute) # label(axiom) # label(non_clause). [assumption]. 5.80/6.10 66 (all M all N plus_plus_nat(M,N) = plus_plus_nat(N,M)) # label(fact_179_nat__add__commute) # label(axiom) # label(non_clause). [assumption]. 5.80/6.10 97 (all M all N plus_plus_int(hAPP_nat_int(semiri1621563631at_int,M),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,plus_plus_nat(M,N))) # label(fact_39_zadd__int) # label(axiom) # label(non_clause). [assumption]. 5.80/6.10 144 (all M all N (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) -> zero_zero_nat != N)) # label(fact_197_gr__implies__not0) # label(axiom) # label(non_clause). [assumption]. 5.80/6.10 158 (all X hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X),hAPP_nat_int(power_power_int(X),number_number_of_nat(bit0(bit1(pls))))))) # label(fact_825_power2__ge__self) # label(axiom) # label(non_clause). [assumption]. 5.80/6.10 200 (all K_1 bit0(K_1) = plus_plus_int(K_1,K_1)) # label(fact_103_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 5.80/6.10 239 (all K_1 succ(K_1) = plus_plus_int(K_1,one_one_int)) # label(fact_814_succ__def) # label(axiom) # label(non_clause). [assumption]. 5.80/6.10 529 (all W_13 plus_plus_int(one_one_int,number_number_of_int(W_13)) = number_number_of_int(plus_plus_int(bit1(pls),W_13))) # label(fact_12_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.80/6.10 908 (all Na (zero_zero_nat = Na <-> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Na)),zero_zero_int)))) # label(fact_807_int__le__0__conv) # label(axiom) # label(non_clause). [assumption]. 5.80/6.10 1040 (all A_91 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_91),plus_plus_nat(A_91,one_one_nat)))) # label(fact_270_less__add__one) # label(axiom) # label(non_clause). [assumption]. 5.80/6.10 1102 -(zero_zero_int != hAPP_nat_int(power_power_int(plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n))),number_number_of_nat(bit0(bit1(pls))))) # label(conj_0) # label(negated_conjecture) # label(non_clause). [assumption]. 5.80/6.10 1149 nat(number_number_of_int(A)) = number_number_of_nat(A) # label(fact_767_nat__number__of__def) # label(axiom). [clausify(18)]. 5.80/6.10 1150 number_number_of_nat(A) = nat(number_number_of_int(A)). [copy(1149),flip(a)]. 5.80/6.10 1204 hAPP_nat_int(semiri1621563631at_int,one_one_nat) = one_one_int # label(fact_248_of__nat__1) # label(axiom). [assumption]. 5.80/6.10 1209 plus_plus_int(A,B) = plus_plus_int(B,A) # label(fact_67_zadd__commute) # label(axiom). [clausify(56)]. 5.80/6.10 1223 plus_plus_nat(A,B) = plus_plus_nat(B,A) # label(fact_179_nat__add__commute) # label(axiom). [clausify(66)]. 5.80/6.10 1254 number_number_of_int(bit1(pls)) = one_one_int # label(fact_27_numeral__1__eq__1) # label(axiom). [assumption]. 5.80/6.10 1275 plus_plus_int(hAPP_nat_int(semiri1621563631at_int,A),hAPP_nat_int(semiri1621563631at_int,B)) = hAPP_nat_int(semiri1621563631at_int,plus_plus_nat(A,B)) # label(fact_39_zadd__int) # label(axiom). [clausify(97)]. 5.80/6.10 1370 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B)) | B != zero_zero_nat # label(fact_197_gr__implies__not0) # label(axiom). [clausify(144)]. 5.80/6.10 1371 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),B)) | zero_zero_nat != B. [copy(1370),flip(b)]. 5.80/6.10 1391 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),hAPP_nat_int(power_power_int(A),number_number_of_nat(bit0(bit1(pls)))))) # label(fact_825_power2__ge__self) # label(axiom). [clausify(158)]. 5.80/6.10 1392 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),hAPP_nat_int(power_power_int(A),nat(number_number_of_int(bit0(bit1(pls))))))). [copy(1391),rewrite([1150(7)])]. 5.80/6.10 1484 plus_plus_int(A,A) = bit0(A) # label(fact_103_Bit0__def) # label(axiom). [clausify(200)]. 5.80/6.10 1485 bit0(A) = plus_plus_int(A,A). [copy(1484),flip(a)]. 5.80/6.10 1569 succ(A) = plus_plus_int(A,one_one_int) # label(fact_814_succ__def) # label(axiom). [clausify(239)]. 5.80/6.10 1570 plus_plus_int(A,one_one_int) = succ(A). [copy(1569),flip(a)]. 5.80/6.10 2120 number_number_of_int(plus_plus_int(bit1(pls),A)) = plus_plus_int(one_one_int,number_number_of_int(A)) # label(fact_12_add__special_I2_J) # label(axiom). [clausify(529)]. 5.80/6.10 2121 number_number_of_int(plus_plus_int(A,bit1(pls))) = plus_plus_int(one_one_int,number_number_of_int(A)). [copy(2120),rewrite([1209(3)])]. 5.80/6.10 2460 number_number_of_nat(bit0(bit1(pls))) = plus_plus_nat(one_one_nat,one_one_nat) # label(fact_19_semiring__one__add__one__is__two) # label(axiom). [assumption]. 5.80/6.10 2461 nat(succ(one_one_int)) = plus_plus_nat(one_one_nat,one_one_nat). [copy(2460),rewrite([1485(3),1150(6),2121(6),1254(4),1570(3)])]. 5.80/6.10 2802 A = zero_zero_nat | -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,A)),zero_zero_int)) # label(fact_807_int__le__0__conv) # label(axiom). [clausify(908)]. 5.80/6.10 2803 zero_zero_nat = A | -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,A)),zero_zero_int)). [copy(2802),flip(a)]. 5.80/6.10 2961 zero_zero_int = pls # label(fact_98_Pls__def) # label(axiom). [assumption]. 5.80/6.10 2962 pls = zero_zero_int. [copy(2961),flip(a)]. 5.80/6.10 3014 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A),plus_plus_nat(A,one_one_nat))) # label(fact_270_less__add__one) # label(axiom). [clausify(1040)]. 5.80/6.10 3113 hAPP_nat_int(power_power_int(plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n))),number_number_of_nat(bit0(bit1(pls)))) = zero_zero_int # label(conj_0) # label(negated_conjecture). [clausify(1102)]. 5.80/6.10 3114 hAPP_nat_int(power_power_int(plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n))),nat(number_number_of_int(plus_plus_int(bit1(zero_zero_int),bit1(zero_zero_int))))) = zero_zero_int. [copy(3113),rewrite([2962(7),1485(9),1150(12)])]. 5.80/6.10 3176 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),hAPP_nat_int(power_power_int(A),nat(number_number_of_int(plus_plus_int(bit1(zero_zero_int),bit1(zero_zero_int))))))). [back_rewrite(1392),rewrite([2962(4),1485(6)])]. 5.80/6.10 3429 number_number_of_int(plus_plus_int(A,bit1(zero_zero_int))) = plus_plus_int(one_one_int,number_number_of_int(A)). [back_rewrite(2121),rewrite([2962(1)])]. 5.80/6.10 3469 number_number_of_int(bit1(zero_zero_int)) = one_one_int. [back_rewrite(1254),rewrite([2962(1)])]. 5.80/6.10 3545 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A),hAPP_nat_int(power_power_int(A),plus_plus_nat(one_one_nat,one_one_nat)))). [back_rewrite(3176),rewrite([3429(9),3469(7),1570(6),2461(6)])]. 5.80/6.10 3550 hAPP_nat_int(power_power_int(plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n))),plus_plus_nat(one_one_nat,one_one_nat)) = zero_zero_int. [back_rewrite(3114),rewrite([3429(12),3469(10),1570(9),2461(9)])]. 5.80/6.10 3639 hAPP_nat_int(semiri1621563631at_int,plus_plus_nat(A,one_one_nat)) = plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,A)). [para(1204(a,1),1275(a,1,1)),rewrite([1223(7)]),flip(a)]. 5.80/6.10 9595 plus_plus_nat(A,one_one_nat) != zero_zero_nat. [resolve(3014,a,1371,a),flip(a)]. 5.80/6.10 10775 hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,n))),zero_zero_int)). [para(3550(a,1),3545(a,1,2))]. 5.80/6.10 19636 -hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,plus_plus_int(one_one_int,hAPP_nat_int(semiri1621563631at_int,A))),zero_zero_int)). [para(3639(a,1),2803(b,1,1,2)),flip(a),unit_del(a,9595)]. 5.80/6.10 19637 $F. [resolve(19636,a,10775,a)]. 5.80/6.10 5.80/6.10 % SZS output end Refutation 5.80/6.10 ============================== end of proof ========================== 5.80/6.10 5.80/6.10 ============================== STATISTICS ============================ 5.80/6.10 5.80/6.10 Given=1882. Generated=133241. Kept=17755. proofs=1. 5.80/6.10 Usable=1863. Sos=9991. Demods=1258. Limbo=2, Disabled=7649. Hints=0. 5.80/6.10 Megabytes=37.27. 5.80/6.10 User_CPU=4.31, System_CPU=0.09, Wall_clock=5. 5.80/6.10 5.80/6.10 ============================== end of statistics ===================== 5.80/6.10 5.80/6.10 ============================== end of search ========================= 5.80/6.10 5.80/6.10 THEOREM PROVED 5.80/6.10 % SZS status Theorem 5.80/6.10 5.80/6.10 Exiting with 1 proof. 5.80/6.10 5.80/6.10 Process 30752 exit (max_proofs) Tue Aug 9 02:54:26 2022 5.80/6.10 Prover9 interrupted 5.80/6.10 EOF