0.07/0.11 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.07/0.12 % Command : tptp2X_and_run_prover9 %d %s 0.12/0.33 % Computer : n007.cluster.edu 0.12/0.33 % Model : x86_64 x86_64 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.12/0.33 % Memory : 8042.1875MB 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64 0.12/0.33 % CPULimit : 1200 0.12/0.33 % DateTime : Tue Jul 13 16:45:49 EDT 2021 0.12/0.33 % CPUTime : 1.15/1.46 ============================== Prover9 =============================== 1.15/1.46 Prover9 (32) version 2009-11A, November 2009. 1.15/1.46 Process 12013 was started by sandbox on n007.cluster.edu, 1.15/1.46 Tue Jul 13 16:45:50 2021 1.15/1.46 The command was "/export/starexec/sandbox/solver/bin/prover9 -t 1200 -f /tmp/Prover9_11839_n007.cluster.edu". 1.15/1.46 ============================== end of head =========================== 1.15/1.46 1.15/1.46 ============================== INPUT ================================= 1.15/1.46 1.15/1.46 % Reading from file /tmp/Prover9_11839_n007.cluster.edu 1.15/1.46 1.15/1.46 set(prolog_style_variables). 1.15/1.46 set(auto2). 1.15/1.46 % set(auto2) -> set(auto). 1.15/1.46 % set(auto) -> set(auto_inference). 1.15/1.46 % set(auto) -> set(auto_setup). 1.15/1.46 % set(auto_setup) -> set(predicate_elim). 1.15/1.46 % set(auto_setup) -> assign(eq_defs, unfold). 1.15/1.46 % set(auto) -> set(auto_limits). 1.15/1.46 % set(auto_limits) -> assign(max_weight, "100.000"). 1.15/1.46 % set(auto_limits) -> assign(sos_limit, 20000). 1.15/1.46 % set(auto) -> set(auto_denials). 1.15/1.46 % set(auto) -> set(auto_process). 1.15/1.46 % set(auto2) -> assign(new_constants, 1). 1.15/1.46 % set(auto2) -> assign(fold_denial_max, 3). 1.15/1.46 % set(auto2) -> assign(max_weight, "200.000"). 1.15/1.46 % set(auto2) -> assign(max_hours, 1). 1.15/1.46 % assign(max_hours, 1) -> assign(max_seconds, 3600). 1.15/1.46 % set(auto2) -> assign(max_seconds, 0). 1.15/1.46 % set(auto2) -> assign(max_minutes, 5). 1.15/1.46 % assign(max_minutes, 5) -> assign(max_seconds, 300). 1.15/1.46 % set(auto2) -> set(sort_initial_sos). 1.15/1.46 % set(auto2) -> assign(sos_limit, -1). 1.15/1.46 % set(auto2) -> assign(lrs_ticks, 3000). 1.15/1.46 % set(auto2) -> assign(max_megs, 400). 1.15/1.46 % set(auto2) -> assign(stats, some). 1.15/1.46 % set(auto2) -> clear(echo_input). 1.15/1.46 % set(auto2) -> set(quiet). 1.15/1.46 % set(auto2) -> clear(print_initial_clauses). 1.15/1.46 % set(auto2) -> clear(print_given). 1.15/1.46 assign(lrs_ticks,-1). 1.15/1.46 assign(sos_limit,10000). 1.15/1.46 assign(order,kbo). 1.15/1.46 set(lex_order_vars). 1.15/1.46 clear(print_given). 1.15/1.46 1.15/1.46 % formulas(sos). % not echoed (719 formulas) 1.15/1.46 1.15/1.46 ============================== end of input ========================== 1.15/1.46 1.15/1.46 % From the command line: assign(max_seconds, 1200). 1.15/1.46 1.15/1.46 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 1.15/1.46 1.15/1.46 % Formulas that are not ordinary clauses: 1.15/1.46 1 (all M all N (is_int(M) -> (one_one_int = times_times_int(M,N) -> M = number_number_of_int(min) | M = one_one_int))) # label(fact_546_pos__zmult__eq__1__iff__lemma) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 2 (all X_3 all Y_3 (ord_less_eq_real(X_3,Y_3) -> (X_3 != Y_3 -> ord_less_real(X_3,Y_3)))) # label(fact_563_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 3 (all X_20 all N_38 times_times_real(power_power_real(X_20,N_38),power_power_real(X_20,N_38)) = power_power_real(X_20,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_38))) # label(fact_26_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 4 (all N_26 all A_29 all B_7 (ord_less_int(A_29,B_7) -> (ord_less_eq_int(zero_zero_int,A_29) -> (ord_less_nat(zero_zero_nat,N_26) -> ord_less_int(power_power_int(A_29,N_26),power_power_int(B_7,N_26)))))) # label(fact_338_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 5 (all X_2 all Y_2 (X_2 = zero_zero_real & Y_2 = zero_zero_real <-> ord_less_eq_real(plus_plus_real(times_times_real(X_2,X_2),times_times_real(Y_2,Y_2)),zero_zero_real))) # label(fact_411_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 6 (all A_39 A_39 = times_times_real(number267125858f_real(bit1(pls)),A_39)) # label(fact_254_mult__numeral__1) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 7 (all N_35 all A_35 (ord_less_real(one_one_real,A_35) -> ord_less_real(one_one_real,times_times_real(A_35,power_power_real(A_35,N_35))))) # label(fact_295_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 8 (all B_1 all A all P (zprime(P) -> (ord_less_int(zero_zero_int,A) -> (-zcong(A,zero_zero_int,P) & -zcong(B_1,zero_zero_int,P) -> -zcong(times_times_int(A,B_1),zero_zero_int,P))))) # label(fact_655_zcong__zprime__prod__zero__contra) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 9 (all A_10 all M_3 all N_12 power_power_int(power_power_int(A_10,M_3),N_12) = power_power_int(A_10,times_times_nat(M_3,N_12))) # label(fact_469_power__mult) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 10 (all Lx_6 all Ly_4 all Rx_6 all Ry_4 times_times_int(times_times_int(Lx_6,Rx_6),times_times_int(Ly_4,Ry_4)) = times_times_int(times_times_int(Lx_6,Ly_4),times_times_int(Rx_6,Ry_4))) # label(fact_93_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 11 (all C_1 all D_1 all A all B_1 all M (zcong(A,B_1,M) -> (zcong(C_1,D_1,M) -> zcong(plus_plus_int(A,C_1),plus_plus_int(B_1,D_1),M)))) # label(fact_573_zcong__zadd) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 12 (all Y_1 all X_1 (ord_less_real(zero_zero_real,X_1) -> (ord_less_real(zero_zero_real,Y_1) -> ord_less_real(zero_zero_real,times_times_real(X_1,Y_1))))) # label(fact_673_real__mult__order) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 13 (all K_1 (ord_less_int(min,K_1) <-> ord_less_eq_int(min,bit0(K_1)))) # label(fact_544_rel__simps_I25_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 14 (all R_1 all Q all A (ord_less_int(zero_zero_int,A) -> (plus_plus_int(R_1,times_times_int(A,Q)) = A -> (ord_less_eq_int(zero_zero_int,R_1) -> ord_less_eq_int(Q,one_one_int))))) # label(fact_685_self__quotient__aux2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 15 (all A_57 times_times_nat(A_57,A_57) = power_power_nat(A_57,number_number_of_nat(bit0(bit1(pls))))) # label(fact_22_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 16 (all X_2 all Y_2 all Z_1 (ord_less_real(zero_zero_real,Z_1) -> (ord_less_eq_real(X_2,Y_2) <-> ord_less_eq_real(times_times_real(X_2,Z_1),times_times_real(Y_2,Z_1))))) # label(fact_675_real__mult__le__cancel__iff1) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 17 (all K all L minus_minus_int(bit1(K),bit0(L)) = bit1(minus_minus_int(K,L))) # label(fact_599_diff__bin__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 18 (all A all R_1 all B_1 all M all C_1 all D_1 all N minus_minus_int(plus_plus_int(times_times_int(A,M),times_times_int(C_1,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_1,times_times_int(R_1,D_1)),N))) # label(fact_603_xzgcda__linear__aux1) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 19 (all Z Z = times_times_real(one_one_real,Z)) # label(fact_665_real__mult__1) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 20 (all A_56 all B_17 times_times_int(B_17,A_56) = times_times_int(A_56,B_17)) # label(fact_114_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 21 (all X_21 times_times_nat(X_21,X_21) = power_power_nat(X_21,number_number_of_nat(bit0(bit1(pls))))) # label(fact_19_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 22 (all I all M all N (dvd_dvd_nat(power_power_nat(I,M),power_power_nat(I,N)) -> (ord_less_nat(one_one_nat,I) -> ord_less_eq_nat(M,N)))) # label(fact_535_power__dvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 23 (all N_29 all X_13 all Y_11 (dvd_dvd_int(X_13,Y_11) -> dvd_dvd_int(power_power_int(X_13,N_29),power_power_int(Y_11,N_29)))) # label(fact_328_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 24 (all B_1 all M all A (is_int(B_1) & is_int(A) -> (ord_less_eq_int(zero_zero_int,A) -> (ord_less_int(A,M) -> (ord_less_eq_int(zero_zero_int,B_1) -> (ord_less_int(B_1,M) -> (zcong(A,B_1,M) -> B_1 = A))))))) # label(fact_584_zcong__zless__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 25 (all X_2 all Y_2 plus_plus_nat(plus_plus_nat(power_power_nat(X_2,number_number_of_nat(bit0(bit1(pls)))),power_power_nat(Y_2,number_number_of_nat(bit0(bit1(pls))))),times_times_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls))),X_2),Y_2)) = power_power_nat(plus_plus_nat(X_2,Y_2),number_number_of_nat(bit0(bit1(pls))))) # label(fact_10_power2__sum) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 26 (all K_1 all L_1 (ord_less_eq_int(bit1(K_1),bit1(L_1)) <-> ord_less_eq_int(K_1,L_1))) # label(fact_66_rel__simps_I34_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 27 (all K all L times_times_int(bit1(K),L) = plus_plus_int(bit0(times_times_int(K,L)),L)) # label(fact_266_mult__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 28 (all M_9 plus_plus_int(M_9,M_9) = times_times_int(plus_plus_int(one_one_int,one_one_int),M_9)) # label(fact_232_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 29 (all A_45 A_45 = times_times_nat(one_one_nat,A_45)) # label(fact_187_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 30 (all W_1 (ord_less_int(W_1,zero_zero_int) <-> ord_less_int(bit0(W_1),zero_zero_int))) # label(fact_398_bin__less__0__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 31 (all Y_1 all X_1 all P (zprime(P) -> (-zcong(X_1,zero_zero_int,P) -> (-zcong(Y_1,zero_zero_int,P) -> -zcong(times_times_int(X_1,Y_1),zero_zero_int,P))))) # label(fact_653_zcong__zmult__prop3) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 32 (all K min != bit0(K)) # label(fact_518_rel__simps_I45_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 33 (all W_6 plus_plus_int(plus_plus_int(one_one_int,number_number_of_int(W_6)),number_number_of_int(W_6)) = number_number_of_int(bit1(W_6))) # label(fact_253_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 34 (all W_4 number267125858f_real(bit0(W_4)) = plus_plus_real(plus_plus_real(zero_zero_real,number267125858f_real(W_4)),number267125858f_real(W_4))) # label(fact_419_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 35 (all X_1 all P (ord_less_int(number_number_of_int(bit0(bit1(pls))),P) -> (zcong(X_1,number_number_of_int(min),P) -> -zcong(X_1,one_one_int,P)))) # label(fact_620_zcong__neg__1__impl__ne__1) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 36 (all Lx_5 all Ly_3 all Rx_5 all Ry_3 times_times_real(times_times_real(Lx_5,Ly_3),times_times_real(Rx_5,Ry_3)) = times_times_real(Rx_5,times_times_real(times_times_real(Lx_5,Ly_3),Ry_3))) # label(fact_94_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 37 (all N_1 all Ma (one_one_nat = N_1 & Ma = one_one_nat <-> times_times_nat(N_1,Ma) = one_one_nat)) # label(fact_632_nat__mult__eq__one) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 38 (all A times_times_int(A,power_power_int(A,number_number_of_nat(bit0(bit1(pls))))) = power_power_int(A,number_number_of_nat(bit1(bit1(pls))))) # label(fact_15_cube__square) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 39 (all B_1_1 (is_int(B_1_1) -> is_int(number_number_of_int(B_1_1)))) # label(gsy_c_Int_Onumber__class_Onumber__of_000tc__Int__Oint) # label(hypothesis) # label(non_clause). [assumption]. 1.15/1.46 40 (all V_6 all W_8 plus_plus_int(number_number_of_int(V_6),number_number_of_int(W_8)) = number_number_of_int(plus_plus_int(V_6,W_8))) # label(fact_246_add__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 41 (all Z (is_int(Z) -> plus_plus_int(zero_zero_int,Z) = Z)) # label(fact_358_zadd__0) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 42 (all A_41 plus_plus_real(number267125858f_real(pls),A_41) = A_41) # label(fact_233_add__numeral__0) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 43 (all X_1 times_times_real(number267125858f_real(bit0(bit0(bit1(pls)))),power_power_real(X_1,number_number_of_nat(bit0(bit1(pls))))) = power_power_real(times_times_real(number267125858f_real(bit0(bit1(pls))),X_1),number_number_of_nat(bit0(bit1(pls))))) # label(fact_291_four__x__squared) # label(axiom) # label(non_clause). [assumption]. 1.15/1.46 44 (all N (power_power_int(number_number_of_int(min),N) = number_number_of_int(min) | power_power_int(number_number_of_int(min),N) = one_one_int)) # label(fact_617_neg__one__power) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 45 (all X_5 one_one_int = power_power_int(X_5,zero_zero_nat)) # label(fact_538_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 46 (all Y_2 (ord_less_int(bit1(pls),Y_2) <-> ord_less_real(one_one_real,number267125858f_real(Y_2)))) # label(fact_162_less__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 47 (all K_1 all L_1 (ord_less_int(minus_minus_int(K_1,L_1),zero_zero_int) <-> ord_less_int(K_1,L_1))) # label(fact_602_less__bin__lemma) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 48 (all N_13 one_one_real = power_power_real(one_one_real,N_13)) # label(fact_464_power__one) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 49 (all K_1 (ord_less_eq_int(min,bit1(K_1)) <-> ord_less_eq_int(min,K_1))) # label(fact_530_rel__simps_I26_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 50 (all X_20 all N_38 times_times_int(power_power_int(X_20,N_38),power_power_int(X_20,N_38)) = power_power_int(X_20,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_38))) # label(fact_27_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 51 (all A_1 all N_1 (power_power_nat(A_1,N_1) = zero_zero_nat <-> zero_zero_nat != N_1 & zero_zero_nat = A_1)) # label(fact_310_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 52 (all Z (is_int(Z) -> times_times_int(one_one_int,Z) = Z)) # label(fact_206_zmult__1) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 53 (all X_1 ord_less_eq_int(X_1,power_power_int(X_1,number_number_of_nat(bit0(bit1(pls)))))) # label(fact_217_power2__ge__self) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 54 (all A_37 power_power_real(A_37,number_number_of_nat(bit1(bit1(pls)))) = times_times_real(times_times_real(A_37,A_37),A_37)) # label(fact_270_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 55 (all X_2 all Y_2 (ord_less_real(number267125858f_real(X_2),number267125858f_real(Y_2)) <-> ord_less_int(X_2,Y_2))) # label(fact_51_less__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 56 (all K1 all K2 (ord_less_int(K1,K2) <-> ord_less_int(bit0(K1),bit0(K2)))) # label(fact_68_less__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 57 (all Z_3 times_times_int(Z_3,number_number_of_int(bit0(bit1(pls)))) = plus_plus_int(Z_3,Z_3)) # label(fact_280_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 58 (all Z1 all Z2 all W plus_plus_real(times_times_real(Z1,W),times_times_real(Z2,W)) = times_times_real(plus_plus_real(Z1,Z2),W)) # label(fact_671_real__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 59 (all A_9 all N_11 ord_less_eq_real(zero_zero_real,power_power_real(A_9,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_11)))) # label(fact_481_zero__le__even__power_H) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 60 (all X_2 (ord_less_eq_int(X_2,bit1(pls)) <-> ord_less_eq_real(number267125858f_real(X_2),one_one_real))) # label(fact_164_le__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 61 (all W_1 all Z_1 (ord_less_eq_int(plus_plus_int(W_1,one_one_int),Z_1) <-> ord_less_int(W_1,Z_1))) # label(fact_87_add1__zle__eq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 62 (all A_1 all B_2 all Ma (zcong(A_1,B_2,Ma) <-> zcong(B_2,A_1,Ma))) # label(fact_557_zcong__sym) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 63 (all K1 all K2 (ord_less_eq_int(K1,K2) <-> ord_less_eq_int(bit0(K1),bit1(K2)))) # label(fact_152_less__eq__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 64 (all X_1 all Y_1 times_times_nat(plus_plus_nat(X_1,Y_1),minus_minus_nat(X_1,Y_1)) = minus_minus_nat(power_power_nat(X_1,number_number_of_nat(bit0(bit1(pls)))),power_power_nat(Y_1,number_number_of_nat(bit0(bit1(pls)))))) # label(fact_627_diff__square) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 65 (all V_2 all K all V_1 ((-ord_less_int(V_1,pls) -> times_times_nat(number_number_of_nat(times_times_int(V_1,V_2)),K) = times_times_nat(number_number_of_nat(V_1),times_times_nat(number_number_of_nat(V_2),K))) & (ord_less_int(V_1,pls) -> zero_zero_nat = times_times_nat(number_number_of_nat(V_1),times_times_nat(number_number_of_nat(V_2),K))))) # label(fact_561_nat__number__of__mult__left) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 66 (all W_1 all Z_1 (ord_less_int(W_1,plus_plus_int(Z_1,one_one_int)) <-> ord_less_eq_int(W_1,Z_1))) # label(fact_88_zle__add1__eq__le) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 67 (all K all M all N (dvd_dvd_int(K,minus_minus_int(M,N)) -> (dvd_dvd_int(K,N) -> dvd_dvd_int(K,M)))) # label(fact_597_zdvd__zdiffD) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 68 (all X_2 (ord_less_eq_int(X_2,bit1(pls)) <-> ord_less_eq_int(number_number_of_int(X_2),one_one_int))) # label(fact_165_le__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 69 (all B all Q_1 all R_2 (ord_less_int(plus_plus_int(times_times_int(B,Q_1),R_2),zero_zero_int) -> (ord_less_eq_int(zero_zero_int,R_2) -> (ord_less_int(zero_zero_int,B) -> ord_less_eq_int(Q_1,zero_zero_int))))) # label(fact_680_q__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 70 (all B_1 all Q all R_1 all B all Q_1 all R_2 (plus_plus_int(times_times_int(B_1,Q),R_1) = plus_plus_int(times_times_int(B,Q_1),R_2) -> (ord_less_int(plus_plus_int(times_times_int(B,Q_1),R_2),zero_zero_int) -> (ord_less_int(R_1,B_1) -> (ord_less_eq_int(zero_zero_int,R_2) -> (ord_less_int(zero_zero_int,B) -> (ord_less_eq_int(B,B_1) -> ord_less_eq_int(Q_1,Q)))))))) # label(fact_684_zdiv__mono2__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 71 (all A_45 times_times_real(one_one_real,A_45) = A_45) # label(fact_186_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 72 (all Z all X_1 all Y_1 (ord_less_eq_real(X_1,Y_1) -> ord_less_eq_real(plus_plus_real(Z,X_1),plus_plus_real(Z,Y_1)))) # label(fact_668_real__add__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 73 (all Y_1 all X_1 (twoSqu142715416sum2sq(X_1) -> (twoSqu142715416sum2sq(Y_1) -> twoSqu142715416sum2sq(times_times_int(X_1,Y_1))))) # label(fact_90_is__mult__sum2sq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 74 (all A_46 times_times_real(A_46,one_one_real) = A_46) # label(fact_183_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 75 (all A_5 all N_6 all N_5 (ord_less_nat(N_6,N_5) -> (ord_less_real(one_one_real,A_5) -> ord_less_real(power_power_real(A_5,N_6),power_power_real(A_5,N_5))))) # label(fact_498_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 76 (all B_1 all A (ord_less_int(zero_zero_int,A) -> (ord_less_int(zero_zero_int,times_times_int(A,B_1)) -> ord_less_int(zero_zero_int,B_1)))) # label(fact_401_pos__zmult__pos) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 77 (all P (is_int(P) -> (zprime(P) -> (P != number_number_of_int(bit0(bit1(pls))) -> (P != number_number_of_int(bit1(bit1(pls))) -> ord_less_eq_int(number_number_of_int(bit1(bit0(bit1(pls)))),P)))))) # label(fact_288_prime__g__5) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 78 (all A_4 all K_3 (is_int(A_4) -> (ord_less_eq_int(power_power_int(A_4,times_times_nat(number_number_of_nat(bit0(bit1(pls))),K_3)),zero_zero_int) -> A_4 = zero_zero_int))) # label(fact_505_even__power__le__0__imp__0) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 79 (all N_10 all A_8 (ord_less_eq_nat(one_one_nat,A_8) -> ord_less_eq_nat(one_one_nat,power_power_nat(A_8,N_10)))) # label(fact_484_one__le__power) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 80 (all Lx_2 all Ly all Rx_2 times_times_real(times_times_real(Lx_2,Ly),Rx_2) = times_times_real(Lx_2,times_times_real(Ly,Rx_2))) # label(fact_103_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 81 (all X_6 all Y_4 -ord_less_int(plus_plus_int(power_power_int(X_6,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y_4,number_number_of_nat(bit0(bit1(pls))))),zero_zero_int)) # label(fact_478_not__sum__power2__lt__zero) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 82 (all N_24 all A_23 all B_6 (ord_less_eq_int(A_23,B_6) -> (ord_less_eq_int(zero_zero_int,A_23) -> ord_less_eq_int(power_power_int(A_23,N_24),power_power_int(B_6,N_24))))) # label(fact_365_power__mono) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 83 (all A all X_1 (is_int(X_1) -> (ord_less_int(zero_zero_int,X_1) -> (ord_less_int(X_1,A) -> (X_1 != minus_minus_int(A,one_one_int) -> ord_less_int(X_1,minus_minus_int(A,one_one_int))))))) # label(fact_605_Euler_Oaux1) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 84 (all A_13 all N_16 times_times_real(power_power_real(A_13,N_16),A_13) = times_times_real(A_13,power_power_real(A_13,N_16))) # label(fact_456_power__commutes) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 85 (all Z1 all Z2 all W times_times_int(minus_minus_int(Z1,Z2),W) = minus_minus_int(times_times_int(Z1,W),times_times_int(Z2,W))) # label(fact_595_zdiff__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 86 (all A all P ((zcong(A,zero_zero_int,P) -> legendre(A,P) = zero_zero_int) & (-zcong(A,zero_zero_int,P) -> (-quadRes(P,A) -> legendre(A,P) = number_number_of_int(min)) & (quadRes(P,A) -> one_one_int = legendre(A,P))))) # label(fact_621_Legendre__def) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 87 (all K_1 (ord_less_int(pls,K_1) <-> ord_less_int(pls,bit0(K_1)))) # label(fact_150_rel__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 88 (all X_10 all Y_8 (ord_less_eq_real(power_power_real(X_10,number_number_of_nat(bit0(bit1(pls)))),power_power_real(Y_8,number_number_of_nat(bit0(bit1(pls))))) -> (ord_less_eq_real(zero_zero_real,Y_8) -> ord_less_eq_real(X_10,Y_8)))) # label(fact_443_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 89 (all K all I all J (ord_less_eq_int(I,J) -> (ord_less_eq_int(J,K) -> ord_less_eq_int(I,K)))) # label(fact_39_zle__trans) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 90 (all A_43 all M_11 times_times_nat(plus_plus_nat(A_43,one_one_nat),M_11) = plus_plus_nat(times_times_nat(A_43,M_11),M_11)) # label(fact_225_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 91 (all N all B_1 all A all P (zprime(P) -> (-dvd_dvd_int(P,A) -> (dvd_dvd_int(power_power_int(P,N),times_times_int(A,B_1)) -> dvd_dvd_int(power_power_int(P,N),B_1))))) # label(fact_407_zprime__power__zdvd__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 92 (all W_4 number_number_of_int(bit0(W_4)) = plus_plus_int(plus_plus_int(zero_zero_int,number_number_of_int(W_4)),number_number_of_int(W_4))) # label(fact_420_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 93 -quadRes(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),number_number_of_int(min)) -> legendre(number_number_of_int(min),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) != one_one_int # label(fact_591__096_126_AQuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_A_061_061_062_ALegendre_A_N) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 94 (all C all D all A_1 all B_2 (plus_plus_nat(times_times_nat(A_1,D),times_times_nat(B_2,C)) != plus_plus_nat(times_times_nat(A_1,C),times_times_nat(B_2,D)) <-> C != D & A_1 != B_2)) # label(fact_178_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 95 (all V_6 all W_8 number267125858f_real(plus_plus_int(V_6,W_8)) = plus_plus_real(number267125858f_real(V_6),number267125858f_real(W_8))) # label(fact_245_add__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 96 (all N_29 all X_13 all Y_11 (dvd_dvd_nat(X_13,Y_11) -> dvd_dvd_nat(power_power_nat(X_13,N_29),power_power_nat(Y_11,N_29)))) # label(fact_327_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 97 (all X_2 all Y_2 (ord_less_eq_int(X_2,Y_2) <-> ord_less_eq_real(number267125858f_real(X_2),number267125858f_real(Y_2)))) # label(fact_53_le__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 98 (all V_14 all V_13 (ord_less_eq_int(pls,V_13) -> (ord_less_eq_int(pls,V_14) -> number_number_of_int(plus_plus_int(V_13,V_14)) = plus_plus_int(number_number_of_int(V_13),number_number_of_int(V_14))))) # label(fact_216_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 99 (all L bit0(minus_minus_int(min,L)) = minus_minus_int(min,bit1(L))) # label(fact_608_diff__bin__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 100 (all C_2 all D_2 all A_18 all B_4 all R_3 (R_3 != zero_zero_nat -> (B_4 = A_18 & D_2 != C_2 -> plus_plus_nat(A_18,times_times_nat(R_3,C_2)) != plus_plus_nat(B_4,times_times_nat(R_3,D_2))))) # label(fact_386_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 101 (all Z all W times_times_real(Z,W) = times_times_real(W,Z)) # label(fact_666_real__mult__commute) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 102 (all B_1_1 all B_2_1 (is_int(B_2_1) & is_int(B_1_1) -> is_int(times_times_int(B_1_1,B_2_1)))) # label(gsy_c_Groups_Otimes__class_Otimes_000tc__Int__Oint) # label(hypothesis) # label(non_clause). [assumption]. 1.15/1.47 103 (all A_37 times_times_int(times_times_int(A_37,A_37),A_37) = power_power_int(A_37,number_number_of_nat(bit1(bit1(pls))))) # label(fact_271_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 104 (all Z all W times_times_int(W,Z) = times_times_int(Z,W)) # label(fact_141_zmult__commute) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 105 (all A_51 all C_6 all D_3 plus_plus_nat(A_51,plus_plus_nat(C_6,D_3)) = plus_plus_nat(C_6,plus_plus_nat(A_51,D_3))) # label(fact_128_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 106 (all X_8 all Y_6 (ord_less_real(power_power_real(X_8,number_number_of_nat(bit0(bit1(pls)))),power_power_real(Y_6,number_number_of_nat(bit0(bit1(pls))))) -> (ord_less_eq_real(zero_zero_real,Y_6) -> ord_less_real(X_8,Y_6)))) # label(fact_470_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 107 (all X_2 (ord_less_int(X_2,bit1(pls)) <-> ord_less_int(number_number_of_int(X_2),one_one_int))) # label(fact_161_less__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 108 (all Z all W (is_int(W) & is_int(Z) -> (ord_less_eq_int(Z,W) -> (ord_less_eq_int(W,Z) -> W = Z)))) # label(fact_40_zle__antisym) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 109 (all X_2 all Y_2 (is_int(Y_2) & is_int(X_2) -> (Y_2 = zero_zero_int & X_2 = zero_zero_int <-> zero_zero_int = plus_plus_int(power_power_int(X_2,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y_2,number_number_of_nat(bit0(bit1(pls)))))))) # label(fact_454_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 110 (all N all M (ord_less_int(zero_zero_int,M) -> (ord_less_int(M,N) -> -dvd_dvd_int(N,M)))) # label(fact_324_zdvd__not__zless) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 111 (all K_1 all L_1 (ord_less_int(K_1,L_1) <-> ord_less_int(bit1(K_1),bit1(L_1)))) # label(fact_64_rel__simps_I17_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 112 (all Z all W (ord_less_eq_real(Z,W) | ord_less_eq_real(W,Z))) # label(fact_689_real__le__linear) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 113 (all Y_1 all X_1 all P (-zcong(X_1,zero_zero_int,P) -> (zcong(power_power_int(Y_1,number_number_of_nat(bit0(bit1(pls)))),X_1,P) -> -dvd_dvd_int(P,Y_1)))) # label(fact_502_Euler_Oaux____1) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 114 (all Y_2 (ord_less_int(one_one_int,number_number_of_int(Y_2)) <-> ord_less_int(bit1(pls),Y_2))) # label(fact_163_less__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 115 (all V_3 (ord_less_int(pls,V_3) <-> ord_less_nat(zero_zero_nat,number_number_of_nat(V_3)))) # label(fact_554_less__0__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 116 (all X_1 all Y_1 all Z power_power_int(X_1,plus_plus_nat(Y_1,Z)) = times_times_int(power_power_int(X_1,Y_1),power_power_int(X_1,Z))) # label(fact_59_zpower__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 117 (all V_2 all V_1 ((ord_less_int(V_1,pls) -> zero_zero_nat = times_times_nat(number_number_of_nat(V_1),number_number_of_nat(V_2))) & (-ord_less_int(V_1,pls) -> number_number_of_nat(times_times_int(V_1,V_2)) = times_times_nat(number_number_of_nat(V_1),number_number_of_nat(V_2))))) # label(fact_562_mult__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 118 (all A_10 all M_3 all N_12 power_power_nat(A_10,times_times_nat(M_3,N_12)) = power_power_nat(power_power_nat(A_10,M_3),N_12)) # label(fact_467_power__mult) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 119 (all M all N all I (ord_less_nat(zero_zero_nat,I) -> (ord_less_nat(power_power_nat(I,M),power_power_nat(I,N)) -> ord_less_nat(M,N)))) # label(fact_512_nat__power__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 120 (all A_1 (is_int(A_1) -> (zero_zero_int = power_power_int(A_1,number_number_of_nat(bit0(bit1(pls)))) <-> zero_zero_int = A_1))) # label(fact_440_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 121 (all W_1 all Y_2 all X_2 all Z_1 (is_int(X_2) & is_int(Z_1) & is_int(Y_2) & is_int(W_1) -> (W_1 = X_2 | Z_1 = Y_2 <-> plus_plus_int(times_times_int(W_1,Z_1),times_times_int(X_2,Y_2)) = plus_plus_int(times_times_int(W_1,Y_2),times_times_int(X_2,Z_1))))) # label(fact_170_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 122 (all B_1 all Q_1 all R_2 all Q all R_1 (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)) -> (ord_less_eq_int(zero_zero_int,R_2) -> (ord_less_int(R_2,B_1) -> (ord_less_int(R_1,B_1) -> ord_less_eq_int(Q_1,Q)))))) # label(fact_681_unique__quotient__lemma) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 123 (all K all L minus_minus_int(bit1(K),bit1(L)) = bit0(minus_minus_int(K,L))) # label(fact_600_diff__bin__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 124 (all Ma all N_1 (N_1 != zero_zero_nat & zero_zero_nat = Ma <-> power_power_nat(Ma,N_1) = zero_zero_nat)) # label(fact_637_nat__power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 125 (all A_12 all B_3 all N_15 power_power_nat(times_times_nat(A_12,B_3),N_15) = times_times_nat(power_power_nat(A_12,N_15),power_power_nat(B_3,N_15))) # label(fact_458_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 126 (all K1 all K2 (ord_less_eq_int(K1,K2) <-> ord_less_int(bit0(K1),bit1(K2)))) # label(fact_84_less__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 127 (all M all X_1 (is_int(X_1) -> (ord_less_eq_int(zero_zero_int,X_1) -> (ord_less_int(X_1,M) -> (zcong(X_1,zero_zero_int,M) -> X_1 = zero_zero_int))))) # label(fact_651_Int2_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 128 (all W_14 number267125858f_real(plus_plus_int(bit1(pls),W_14)) = plus_plus_real(one_one_real,number267125858f_real(W_14))) # label(fact_28_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 129 (all K all L bit0(L) != bit1(K)) # label(fact_194_rel__simps_I50_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 130 (all A all P (is_int(A) -> (zprime(P) -> (ord_less_int(zero_zero_int,A) -> (ord_less_int(A,P) -> (zcong(times_times_int(A,A),one_one_int,P) -> A = one_one_int | minus_minus_int(P,one_one_int) = A)))))) # label(fact_613_zcong__square__zless) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 131 (all Y_2 (ord_less_eq_real(zero_zero_real,number267125858f_real(Y_2)) <-> ord_less_eq_int(pls,Y_2))) # label(fact_431_le__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 132 (all A_53 all B_14 all C_8 plus_plus_int(plus_plus_int(A_53,B_14),C_8) = plus_plus_int(A_53,plus_plus_int(B_14,C_8))) # label(fact_123_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 133 (all W_1 all Z_1 (ord_less_eq_int(W_1,minus_minus_int(Z_1,one_one_int)) <-> ord_less_int(W_1,Z_1))) # label(fact_606_zle__diff1__eq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 134 (all X_1 all Y_1 all Z (ord_less_real(zero_zero_real,Z) -> (ord_less_real(X_1,Y_1) -> ord_less_real(times_times_real(Z,X_1),times_times_real(Z,Y_1))))) # label(fact_672_real__mult__less__mono2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 135 (all L pls != bit1(L)) # label(fact_193_rel__simps_I39_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 136 (all Z plus_plus_nat(Z,Z) = times_times_nat(number_number_of_nat(bit0(bit1(pls))),Z)) # label(fact_60_nat__mult__2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 137 (all B_1 all M all A (ord_less_int(zero_zero_int,A) -> (ord_less_int(A,M) -> (ord_less_int(zero_zero_int,B_1) -> (ord_less_int(B_1,A) -> -zcong(A,B_1,M)))))) # label(fact_579_zcong__not) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 138 (all Y_2 (ord_less_int(pls,Y_2) <-> ord_less_int(zero_zero_int,number_number_of_int(Y_2)))) # label(fact_428_less__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 139 (all A_41 (is_int(A_41) -> A_41 = plus_plus_int(number_number_of_int(pls),A_41))) # label(fact_234_add__numeral__0) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 140 (all Y_2 (ord_less_eq_int(bit1(pls),Y_2) <-> ord_less_eq_int(one_one_int,number_number_of_int(Y_2)))) # label(fact_167_le__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 141 (all X_2 all Y_2 all B_2 (ord_less_real(one_one_real,B_2) -> (ord_less_real(power_power_real(B_2,X_2),power_power_real(B_2,Y_2)) <-> ord_less_nat(X_2,Y_2)))) # label(fact_492_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 142 (all N all X_1 all Y_1 (dvd_dvd_nat(X_1,Y_1) -> dvd_dvd_nat(power_power_nat(X_1,N),power_power_nat(Y_1,N)))) # label(fact_638_divides__exp) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 143 (all W ord_less_eq_int(W,W)) # label(fact_35_zle__refl) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 144 (all A_26 plus_plus_real(zero_zero_real,A_26) = A_26) # label(fact_345_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 145 (all N all M ((M != zero_zero_nat -> times_times_nat(M,N) = plus_plus_nat(N,times_times_nat(minus_minus_nat(M,one_one_nat),N))) & (zero_zero_nat = M -> times_times_nat(M,N) = zero_zero_nat))) # label(fact_625_mult__eq__if) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 146 (all M_10 all A_42 plus_plus_nat(M_10,times_times_nat(A_42,M_10)) = times_times_nat(plus_plus_nat(A_42,one_one_nat),M_10)) # label(fact_228_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 147 (all B_1 all Q_1 all R_2 all Q all R_1 (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)) -> (ord_less_eq_int(R_1,zero_zero_int) -> (ord_less_int(B_1,R_1) -> (ord_less_int(B_1,R_2) -> ord_less_eq_int(Q,Q_1)))))) # label(fact_683_unique__quotient__lemma__neg) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 148 (all A_20 all N_21 all N_20 (ord_less_eq_nat(N_21,N_20) -> (ord_less_eq_int(zero_zero_int,A_20) -> (ord_less_eq_int(A_20,one_one_int) -> ord_less_eq_int(power_power_int(A_20,N_20),power_power_int(A_20,N_21)))))) # label(fact_377_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 149 (all C_2 all D_2 all A_18 all B_4 all R_3 (is_int(D_2) & is_int(R_3) & is_int(C_2) -> (zero_zero_int != R_3 -> (A_18 = B_4 & D_2 != C_2 -> plus_plus_int(B_4,times_times_int(R_3,D_2)) != plus_plus_int(A_18,times_times_int(R_3,C_2)))))) # label(fact_387_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 150 (all A_7 all N_9 all N_8 (ord_less_eq_nat(N_9,N_8) -> (ord_less_eq_real(one_one_real,A_7) -> ord_less_eq_real(power_power_real(A_7,N_9),power_power_real(A_7,N_8))))) # label(fact_486_power__increasing) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 151 (all A_5 all N_6 all N_5 (ord_less_nat(N_6,N_5) -> (ord_less_int(one_one_int,A_5) -> ord_less_int(power_power_int(A_5,N_6),power_power_int(A_5,N_5))))) # label(fact_500_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 152 (all D_1 all C_1 all A all B_1 all M (zcong(A,B_1,M) -> (C_1 = B_1 -> (zcong(C_1,D_1,M) -> zcong(A,D_1,M))))) # label(fact_624_zcong__eq__trans) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 153 (all A_16 (is_int(A_16) -> power_power_int(A_16,one_one_nat) = A_16)) # label(fact_423_power__one__right) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 154 (all A_31 all N_30 all B_8 (power_power_nat(A_31,N_30) = power_power_nat(B_8,N_30) -> (ord_less_eq_nat(zero_zero_nat,A_31) -> (ord_less_eq_nat(zero_zero_nat,B_8) -> (ord_less_nat(zero_zero_nat,N_30) -> A_31 = B_8))))) # label(fact_322_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 155 (all A_33 all M_7 all N_33 (ord_less_eq_nat(M_7,N_33) -> dvd_dvd_nat(power_power_nat(A_33,M_7),power_power_nat(A_33,N_33)))) # label(fact_312_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 156 (all W pls = times_times_int(pls,W)) # label(fact_199_mult__Pls) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 157 (all X_15 all Y_13 all Q_2 power_power_real(times_times_real(X_15,Y_13),Q_2) = times_times_real(power_power_real(X_15,Q_2),power_power_real(Y_13,Q_2))) # label(fact_190_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 158 (all X_2 all Y_2 (plus_plus_real(times_times_real(X_2,X_2),times_times_real(Y_2,Y_2)) = zero_zero_real <-> X_2 = zero_zero_real & Y_2 = zero_zero_real)) # label(fact_383_sum__squares__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 159 (all X_2 all Y_2 all Z_1 (ord_less_real(zero_zero_real,Z_1) -> (ord_less_eq_real(times_times_real(Z_1,X_2),times_times_real(Z_1,Y_2)) <-> ord_less_eq_real(X_2,Y_2)))) # label(fact_674_real__mult__le__cancel__iff2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 160 (all L_1 (is_int(L_1) -> (pls = bit0(L_1) <-> pls = L_1))) # label(fact_197_rel__simps_I38_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 161 (all A_1 all B_2 all Ma (is_int(B_2) -> (zcong(A_1,B_2,Ma) <-> (exists K_2 (is_int(K_2) & B_2 = plus_plus_int(A_1,times_times_int(Ma,K_2))))))) # label(fact_580_zcong__iff__lin) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 162 (all Y_2 (ord_less_eq_int(pls,Y_2) <-> ord_less_eq_int(zero_zero_int,number_number_of_int(Y_2)))) # label(fact_432_le__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 163 (all Z_9 all Z all W_13 all W (ord_less_int(W_13,W) -> (ord_less_eq_int(Z_9,Z) -> ord_less_int(plus_plus_int(W_13,Z_9),plus_plus_int(W,Z))))) # label(fact_55_zadd__zless__mono) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 164 (all A_3 one_one_real = power_power_real(A_3,zero_zero_nat)) # label(fact_539_power__0) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 165 (all A all B_1 minus_minus_int(power_power_int(A,number_number_of_nat(bit0(bit1(pls)))),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_614_zspecial__product) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 166 (all X_16 all Y_14 all Z_8 times_times_nat(X_16,plus_plus_nat(Y_14,Z_8)) = plus_plus_nat(times_times_nat(X_16,Y_14),times_times_nat(X_16,Z_8))) # label(fact_181_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 167 (all K_1 (ord_less_int(min,K_1) <-> ord_less_int(min,bit0(K_1)))) # label(fact_527_rel__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 168 (all N_1 all Ma (is_int(Ma) & is_int(N_1) -> (ord_less_int(zero_zero_int,Ma) -> (times_times_int(Ma,N_1) = one_one_int <-> one_one_int = N_1 & one_one_int = Ma)))) # label(fact_425_pos__zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 169 (all V_8 all W_10 number_number_of_int(times_times_int(V_8,W_10)) = times_times_int(number_number_of_int(V_8),number_number_of_int(W_10))) # label(fact_242_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 170 (all V_9 all W_11 times_times_int(number_number_of_int(V_9),number_number_of_int(W_11)) = number_number_of_int(times_times_int(V_9,W_11))) # label(fact_240_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 171 (all N all X_1 (ord_less_eq_int(zero_zero_int,X_1) -> ord_less_eq_int(zero_zero_int,power_power_int(X_1,N)))) # label(fact_695_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 172 (all M all N all P (zprime(P) -> (dvd_dvd_int(P,times_times_int(M,N)) -> dvd_dvd_int(P,N) | dvd_dvd_int(P,M)))) # label(fact_650_zprime__zdvd__zmult__better) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 173 (all A_10 all M_3 all N_12 power_power_real(power_power_real(A_10,M_3),N_12) = power_power_real(A_10,times_times_nat(M_3,N_12))) # label(fact_468_power__mult) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 174 (all X_2 all Y_2 (zero_zero_real = plus_plus_real(times_times_real(X_2,X_2),times_times_real(Y_2,Y_2)) <-> zero_zero_real = X_2 & zero_zero_real = Y_2)) # label(fact_677_real__two__squares__add__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 175 (all Y_2 (ord_less_eq_real(one_one_real,number267125858f_real(Y_2)) <-> ord_less_eq_int(bit1(pls),Y_2))) # label(fact_166_le__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 176 (all V_14 all V_13 (ord_less_eq_int(pls,V_13) -> (ord_less_eq_int(pls,V_14) -> plus_plus_real(number267125858f_real(V_13),number267125858f_real(V_14)) = number267125858f_real(plus_plus_int(V_13,V_14))))) # label(fact_214_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 177 (all X_18 (is_int(X_18) -> X_18 = power_power_int(X_18,one_one_nat))) # label(fact_46_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 178 (all A_46 times_times_nat(A_46,one_one_nat) = A_46) # label(fact_184_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 179 (all X_2 all Y_2 all B_2 (ord_less_nat(one_one_nat,B_2) -> (ord_less_eq_nat(power_power_nat(B_2,X_2),power_power_nat(B_2,Y_2)) <-> ord_less_eq_nat(X_2,Y_2)))) # label(fact_302_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 180 (all X_2 all N_1 (N_1 = zero_zero_nat | ord_less_nat(zero_zero_nat,X_2) <-> ord_less_nat(zero_zero_nat,power_power_nat(X_2,N_1)))) # label(fact_510_zero__less__power__nat__eq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 181 (all V_7 all W_9 all Z_6 plus_plus_real(number267125858f_real(plus_plus_int(V_7,W_9)),Z_6) = plus_plus_real(number267125858f_real(V_7),plus_plus_real(number267125858f_real(W_9),Z_6))) # label(fact_243_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 182 (all X_2 all Y_2 (is_int(Y_2) & is_int(X_2) -> (Y_2 != zero_zero_int | zero_zero_int != X_2 <-> ord_less_int(zero_zero_int,plus_plus_int(power_power_int(X_2,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y_2,number_number_of_nat(bit0(bit1(pls))))))))) # label(fact_480_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 183 (all A_49 all N_37 power_power_int(A_49,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_37)) = power_power_int(power_power_int(A_49,N_37),number_number_of_nat(bit0(bit1(pls))))) # label(fact_159_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 184 (all X_2 (zero_zero_real = X_2 <-> -ord_less_real(zero_zero_real,times_times_real(X_2,X_2)))) # label(fact_522_not__real__square__gt__zero) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 185 (all Lx_6 all Ly_4 all Rx_6 all Ry_4 times_times_nat(times_times_nat(Lx_6,Rx_6),times_times_nat(Ly_4,Ry_4)) = times_times_nat(times_times_nat(Lx_6,Ly_4),times_times_nat(Rx_6,Ry_4))) # label(fact_92_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 186 (all A_3 power_power_nat(A_3,zero_zero_nat) = one_one_nat) # label(fact_540_power__0) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 187 (all Z_2 times_times_int(Z_2,number_number_of_int(bit0(bit1(pls)))) = plus_plus_int(Z_2,Z_2)) # label(fact_282_mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 188 (all V_17 plus_plus_real(number267125858f_real(V_17),one_one_real) = number267125858f_real(plus_plus_int(V_17,bit1(pls)))) # label(fact_30_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 189 (all X_11 all Y_9 -ord_less_int(plus_plus_int(times_times_int(X_11,X_11),times_times_int(Y_9,Y_9)),zero_zero_int)) # label(fact_415_not__sum__squares__lt__zero) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 190 (all M all X_1 (ord_less_int(zero_zero_int,X_1) -> (ord_less_int(X_1,M) -> -zcong(X_1,zero_zero_int,M)))) # label(fact_642_zcong__not__zero) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 191 (all Lx_6 all Ly_4 all Rx_6 all Ry_4 times_times_real(times_times_real(Lx_6,Ly_4),times_times_real(Rx_6,Ry_4)) = times_times_real(times_times_real(Lx_6,Rx_6),times_times_real(Ly_4,Ry_4))) # label(fact_91_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.47 192 (all A_3 one_one_int = power_power_int(A_3,zero_zero_nat)) # label(fact_541_power__0) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 193 (all N_17 all A_17 (ord_less_nat(zero_zero_nat,A_17) -> (ord_less_nat(A_17,one_one_nat) -> ord_less_nat(times_times_nat(A_17,power_power_nat(A_17,N_17)),power_power_nat(A_17,N_17))))) # label(fact_405_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 194 (all X_2 all Y_2 (ord_less_eq_real(X_2,Y_2) <-> ord_less_eq_real(minus_minus_real(X_2,Y_2),zero_zero_real))) # label(fact_662_real__le__eq__diff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 195 (all K all I all J (ord_less_int(I,J) -> ord_less_int(plus_plus_int(I,K),plus_plus_int(J,K)))) # label(fact_75_zadd__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 196 (all A_52 all C_7 all D_4 plus_plus_nat(plus_plus_nat(A_52,C_7),D_4) = plus_plus_nat(A_52,plus_plus_nat(C_7,D_4))) # label(fact_125_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 197 (all V_10 all W_12 all Z_7 times_times_int(number_number_of_int(times_times_int(V_10,W_12)),Z_7) = times_times_int(number_number_of_int(V_10),times_times_int(number_number_of_int(W_12),Z_7))) # label(fact_238_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 198 (all A all B_1 (dvd_dvd_nat(A,B_1) -> ord_less_eq_nat(A,B_1) | zero_zero_nat = B_1)) # label(fact_640_divides__ge) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 199 (all Lx_1 all Rx_1 all Ry_1 times_times_nat(Lx_1,times_times_nat(Rx_1,Ry_1)) = times_times_nat(times_times_nat(Lx_1,Rx_1),Ry_1)) # label(fact_107_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 200 (all V_14 all V_13 (ord_less_eq_int(pls,V_13) -> (ord_less_eq_int(pls,V_14) -> plus_plus_nat(number_number_of_nat(V_13),number_number_of_nat(V_14)) = number_number_of_nat(plus_plus_int(V_13,V_14))))) # label(fact_215_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 201 (all W_1 all X_2 (X_2 = number_number_of_nat(W_1) <-> X_2 = number_number_of_nat(W_1))) # label(fact_137_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 202 (all A_1 (ord_less_int(plus_plus_int(A_1,A_1),zero_zero_int) <-> ord_less_int(A_1,zero_zero_int))) # label(fact_382_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 203 (all X_4 all N_3 (one_one_int = X_4 | ord_less_nat(zero_zero_nat,N_3) -> dvd_dvd_int(X_4,power_power_int(X_4,N_3)))) # label(fact_552_dvd__power) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 204 (all A_1 (is_int(A_1) -> (zero_zero_int != A_1 <-> ord_less_int(zero_zero_int,power_power_int(A_1,number_number_of_nat(bit0(bit1(pls)))))))) # label(fact_452_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 205 (all A_1 all B_2 all Ma (dvd_dvd_int(Ma,minus_minus_int(A_1,B_2)) <-> zcong(A_1,B_2,Ma))) # label(fact_604_zcong__def) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 206 (all X_9 all Y_7 (power_power_nat(Y_7,number_number_of_nat(bit0(bit1(pls)))) = power_power_nat(X_9,number_number_of_nat(bit0(bit1(pls)))) -> (ord_less_eq_nat(zero_zero_nat,X_9) -> (ord_less_eq_nat(zero_zero_nat,Y_7) -> X_9 = Y_7)))) # label(fact_447_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 207 (all A_48 all M_12 all B_13 plus_plus_nat(times_times_nat(A_48,M_12),times_times_nat(B_13,M_12)) = times_times_nat(plus_plus_nat(A_48,B_13),M_12)) # label(fact_172_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 208 (all A_40 (is_int(A_40) -> A_40 = plus_plus_int(A_40,number_number_of_int(pls)))) # label(fact_236_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 209 (all A_1 (is_int(A_1) -> (zero_zero_int = A_1 <-> plus_plus_int(A_1,A_1) = zero_zero_int))) # label(fact_355_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 210 (all A_45 (is_int(A_45) -> A_45 = times_times_int(one_one_int,A_45))) # label(fact_188_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 211 (all A_25 A_25 = plus_plus_nat(A_25,zero_zero_nat)) # label(fact_349_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 212 (all N_35 all A_35 (ord_less_int(one_one_int,A_35) -> ord_less_int(one_one_int,times_times_int(A_35,power_power_int(A_35,N_35))))) # label(fact_297_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 213 (all A all P (zprime(P) -> (ord_less_int(zero_zero_int,A) -> (zcong(times_times_int(A,A),one_one_int,P) -> zcong(A,minus_minus_int(P,one_one_int),P) | zcong(A,one_one_int,P))))) # label(fact_612_zcong__square) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 214 (all K1 all K2 (ord_less_eq_int(bit1(K1),bit0(K2)) <-> ord_less_int(K1,K2))) # label(fact_82_less__eq__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 215 (all M_8 all N_34 all A_34 (ord_less_nat(one_one_nat,A_34) -> (ord_less_eq_nat(power_power_nat(A_34,M_8),power_power_nat(A_34,N_34)) -> ord_less_eq_nat(M_8,N_34)))) # label(fact_299_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 216 (all Lx_4 all Ly_2 all Rx_4 all Ry_2 times_times_int(Lx_4,times_times_int(Ly_2,times_times_int(Rx_4,Ry_2))) = times_times_int(times_times_int(Lx_4,Ly_2),times_times_int(Rx_4,Ry_2))) # label(fact_99_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 217 (all Z times_times_nat(Z,number_number_of_nat(bit0(bit1(pls)))) = plus_plus_nat(Z,Z)) # label(fact_61_nat__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 218 (all K_1 (ord_less_int(bit1(K_1),pls) <-> ord_less_int(K_1,pls))) # label(fact_146_rel__simps_I12_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 219 (all Z (is_int(Z) -> Z = plus_plus_int(Z,zero_zero_int))) # label(fact_359_zadd__0__right) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 220 (all K all A all B_1 all M (zcong(A,B_1,M) -> zcong(times_times_int(A,K),times_times_int(B_1,K),M))) # label(fact_571_zcong__scalar) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 221 (all A_43 all M_11 times_times_int(plus_plus_int(A_43,one_one_int),M_11) = plus_plus_int(times_times_int(A_43,M_11),M_11)) # label(fact_226_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 222 (all K_1 all L_1 (is_int(K_1) & is_int(L_1) -> (L_1 = K_1 <-> bit0(L_1) = bit0(K_1)))) # label(fact_139_rel__simps_I48_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 223 (all A_21 all N_22 all B_5 (ord_less_nat(power_power_nat(A_21,N_22),power_power_nat(B_5,N_22)) -> (ord_less_eq_nat(zero_zero_nat,B_5) -> ord_less_nat(A_21,B_5)))) # label(fact_373_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 224 (all K_1 all L_1 (ord_less_eq_int(K_1,L_1) <-> ord_less_eq_int(number_number_of_int(K_1),number_number_of_int(L_1)))) # label(fact_74_less__eq__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 225 (all A_44 all B_11 all V_12 plus_plus_real(times_times_real(A_44,number267125858f_real(V_12)),times_times_real(B_11,number267125858f_real(V_12))) = times_times_real(plus_plus_real(A_44,B_11),number267125858f_real(V_12))) # label(fact_218_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 226 (all N_28 all A_30 (zero_zero_real != A_30 -> power_power_real(A_30,N_28) != zero_zero_real)) # label(fact_330_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 227 (all X_2 all Y_2 all B_2 (ord_less_real(one_one_real,B_2) -> (ord_less_eq_real(power_power_real(B_2,X_2),power_power_real(B_2,Y_2)) <-> ord_less_eq_nat(X_2,Y_2)))) # label(fact_301_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 228 (all A_51 all C_6 all D_3 plus_plus_int(C_6,plus_plus_int(A_51,D_3)) = plus_plus_int(A_51,plus_plus_int(C_6,D_3))) # label(fact_129_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 229 (all Ma all N_1 (is_int(Ma) & is_int(N_1) -> (one_one_int = times_times_int(Ma,N_1) <-> number_number_of_int(min) = N_1 & number_number_of_int(min) = Ma | Ma = one_one_int & N_1 = one_one_int))) # label(fact_547_zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 230 (all Z1 all Z2 all Z3 times_times_int(Z1,times_times_int(Z2,Z3)) = times_times_int(times_times_int(Z1,Z2),Z3)) # label(fact_140_zmult__assoc) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 231 (all K all M zcong(K,K,M)) # label(fact_558_zcong__refl) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 232 (all A_54 all B_15 all C_9 plus_plus_real(plus_plus_real(A_54,C_9),B_15) = plus_plus_real(plus_plus_real(A_54,B_15),C_9)) # label(fact_118_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 233 (all A all M all B_1 zcong(times_times_int(A,M),times_times_int(B_1,M),M)) # label(fact_572_zcong__zmult__self) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 234 (all J all K all M (ord_less_int(number_number_of_int(bit0(bit1(pls))),M) -> (zcong(power_power_int(number_number_of_int(min),J),power_power_int(number_number_of_int(min),K),M) -> power_power_int(number_number_of_int(min),K) = power_power_int(number_number_of_int(min),J)))) # label(fact_589_neg__one__power__eq__mod__m) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 235 (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]. 1.15/1.48 236 (all A_11 all M_4 all N_14 times_times_int(power_power_int(A_11,M_4),power_power_int(A_11,N_14)) = power_power_int(A_11,plus_plus_nat(M_4,N_14))) # label(fact_463_power__add) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 237 (all A_38 A_38 = times_times_real(A_38,number267125858f_real(bit1(pls)))) # label(fact_256_mult__numeral__1__right) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 238 (all N_10 all A_8 (ord_less_eq_real(one_one_real,A_8) -> ord_less_eq_real(one_one_real,power_power_real(A_8,N_10)))) # label(fact_483_one__le__power) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 239 (all X_19 all P_3 all Q_4 power_power_real(power_power_real(X_19,P_3),Q_4) = power_power_real(X_19,times_times_nat(P_3,Q_4))) # label(fact_42_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 240 (all K_1 (ord_less_eq_int(bit1(K_1),min) <-> ord_less_eq_int(K_1,min))) # label(fact_529_rel__simps_I30_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 241 (all K all L times_times_int(bit0(K),L) = bit0(times_times_int(K,L))) # label(fact_200_mult__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 242 (all A_37 times_times_nat(times_times_nat(A_37,A_37),A_37) = power_power_nat(A_37,number_number_of_nat(bit1(bit1(pls))))) # label(fact_269_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 243 (all C_1 all D_1 all A all B_1 all M (zcong(A,B_1,M) -> (zcong(C_1,D_1,M) -> zcong(minus_minus_int(A,C_1),minus_minus_int(B_1,D_1),M)))) # label(fact_592_zcong__zdiff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 244 (all X_2 all Y_2 (ord_less_eq_real(X_2,Y_2) <-> X_2 = Y_2 | ord_less_real(X_2,Y_2))) # label(fact_664_less__eq__real__def) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 245 (all A_47 all B_12 all C_4 plus_plus_int(times_times_int(A_47,C_4),times_times_int(B_12,C_4)) = times_times_int(plus_plus_int(A_47,B_12),C_4)) # label(fact_176_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 246 (all A_27 times_times_real(A_27,zero_zero_real) = zero_zero_real) # label(fact_342_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 247 (all X_4 all N_3 (X_4 = one_one_nat | ord_less_nat(zero_zero_nat,N_3) -> dvd_dvd_nat(X_4,power_power_nat(X_4,N_3)))) # label(fact_551_dvd__power) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 248 (all Lx_4 all Ly_2 all Rx_4 all Ry_2 times_times_nat(Lx_4,times_times_nat(Ly_2,times_times_nat(Rx_4,Ry_2))) = times_times_nat(times_times_nat(Lx_4,Ly_2),times_times_nat(Rx_4,Ry_2))) # label(fact_98_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 249 (all N_4 all A_2 (ord_less_nat(one_one_nat,A_2) -> (ord_less_nat(zero_zero_nat,N_4) -> ord_less_nat(one_one_nat,power_power_nat(A_2,N_4))))) # label(fact_549_one__less__power) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 250 (all N_17 all A_17 (ord_less_int(zero_zero_int,A_17) -> (ord_less_int(A_17,one_one_int) -> ord_less_int(times_times_int(A_17,power_power_int(A_17,N_17)),power_power_int(A_17,N_17))))) # label(fact_406_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 251 (all X_2 all W_1 (number_number_of_nat(W_1) = zero_zero_nat | ord_less_nat(zero_zero_nat,X_2) <-> ord_less_nat(zero_zero_nat,power_power_nat(X_2,number_number_of_nat(W_1))))) # label(fact_511_zero__less__power__nat__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 252 (all M (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_618_Legendre__1mod4) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 253 (all Ma all X_2 (quadRes(Ma,X_2) <-> (exists Y (is_int(Y) & zcong(power_power_int(Y,number_number_of_nat(bit0(bit1(pls)))),X_2,Ma))))) # label(fact_658_QuadRes__def) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 254 (all A_14 -ord_less_int(power_power_int(A_14,number_number_of_nat(bit0(bit1(pls)))),zero_zero_int)) # label(fact_450_power2__less__0) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 255 (all Lx_4 all Ly_2 all Rx_4 all Ry_2 times_times_real(times_times_real(Lx_4,Ly_2),times_times_real(Rx_4,Ry_2)) = times_times_real(Lx_4,times_times_real(Ly_2,times_times_real(Rx_4,Ry_2)))) # label(fact_97_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 256 (all X_1 power_power_int(X_1,number_number_of_nat(bit0(bit0(bit1(pls))))) = power_power_int(power_power_int(X_1,number_number_of_nat(bit0(bit1(pls)))),number_number_of_nat(bit0(bit1(pls))))) # label(fact_272_quartic__square__square) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 257 (all A_27 zero_zero_int = times_times_int(A_27,zero_zero_int)) # label(fact_344_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 258 (all K_1 (ord_less_int(K_1,min) <-> ord_less_int(bit1(K_1),min))) # label(fact_523_rel__simps_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 259 (all M_9 plus_plus_real(M_9,M_9) = times_times_real(plus_plus_real(one_one_real,one_one_real),M_9)) # label(fact_230_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 260 (all Lx_1 all Rx_1 all Ry_1 times_times_int(Lx_1,times_times_int(Rx_1,Ry_1)) = times_times_int(times_times_int(Lx_1,Rx_1),Ry_1)) # label(fact_108_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 261 (all C all D all A_1 all B_2 all Ma (zcong(A_1,B_2,Ma) -> (zcong(C,times_times_int(B_2,D),Ma) <-> zcong(C,times_times_int(A_1,D),Ma)))) # label(fact_635_zcong__zmult__prop1) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 262 (all Z_5 times_times_nat(number_number_of_nat(bit0(bit1(pls))),Z_5) = plus_plus_nat(Z_5,Z_5)) # label(fact_274_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 263 (all A_40 A_40 = plus_plus_real(A_40,number267125858f_real(pls))) # label(fact_235_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 264 (all C_1 all A all B_1 (dvd_dvd_nat(A,B_1) -> dvd_dvd_nat(times_times_nat(A,C_1),times_times_nat(B_1,C_1)))) # label(fact_630_divides__mul__r) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 265 (all W_3 ((zero_zero_nat != number_number_of_nat(W_3) -> power_power_nat(zero_zero_nat,number_number_of_nat(W_3)) = zero_zero_nat) & (zero_zero_nat = number_number_of_nat(W_3) -> power_power_nat(zero_zero_nat,number_number_of_nat(W_3)) = one_one_nat))) # label(fact_582_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 266 (all K1 all K2 (ord_less_eq_int(K1,K2) <-> ord_less_eq_int(bit0(K1),bit0(K2)))) # label(fact_71_less__eq__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 267 (all Lx_5 all Ly_3 all Rx_5 all Ry_3 times_times_nat(times_times_nat(Lx_5,Ly_3),times_times_nat(Rx_5,Ry_3)) = times_times_nat(Rx_5,times_times_nat(times_times_nat(Lx_5,Ly_3),Ry_3))) # label(fact_95_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 268 (all N_13 power_power_int(one_one_int,N_13) = one_one_int) # label(fact_466_power__one) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 269 (all A_49 all N_37 power_power_real(A_49,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_37)) = power_power_real(power_power_real(A_49,N_37),number_number_of_nat(bit0(bit1(pls))))) # label(fact_158_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 270 (all R_1 all Q all A (ord_less_int(zero_zero_int,A) -> (A = plus_plus_int(R_1,times_times_int(A,Q)) -> (ord_less_int(R_1,A) -> ord_less_eq_int(one_one_int,Q))))) # label(fact_660_self__quotient__aux1) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 271 (all V_3 all W_1 (-ord_less_nat(number_number_of_nat(W_1),number_number_of_nat(V_3)) <-> ord_less_eq_nat(number_number_of_nat(V_3),number_number_of_nat(W_1)))) # label(fact_49_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 272 (all X_3 all Y_3 (is_int(Y_3) & is_int(X_3) -> (ord_less_eq_int(X_3,Y_3) -> (X_3 != Y_3 -> ord_less_int(X_3,Y_3))))) # label(fact_565_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 273 (all Z zero_zero_int != plus_plus_int(plus_plus_int(one_one_int,Z),Z)) # label(fact_403_odd__nonzero) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 274 (all N_24 all A_23 all B_6 (ord_less_eq_nat(A_23,B_6) -> (ord_less_eq_nat(zero_zero_nat,A_23) -> ord_less_eq_nat(power_power_nat(A_23,N_24),power_power_nat(B_6,N_24))))) # label(fact_364_power__mono) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 275 (all X_1 all Y_1 plus_plus_real(plus_plus_real(power_power_real(X_1,number_number_of_nat(bit0(bit1(pls)))),power_power_real(Y_1,number_number_of_nat(bit0(bit1(pls))))),times_times_real(times_times_real(number267125858f_real(bit0(bit1(pls))),X_1),Y_1)) = power_power_real(plus_plus_real(X_1,Y_1),number_number_of_nat(bit0(bit1(pls))))) # label(fact_290_real__sum__squared__expand) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 276 (all M_5 all A_32 all N_31 all B_9 (dvd_dvd_nat(power_power_nat(A_32,N_31),B_9) -> (ord_less_eq_nat(M_5,N_31) -> dvd_dvd_nat(power_power_nat(A_32,M_5),B_9)))) # label(fact_318_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 277 (all A all N all B_1 (dvd_dvd_nat(power_power_nat(A,N),power_power_nat(B_1,N)) -> (N != zero_zero_nat -> dvd_dvd_nat(A,B_1)))) # label(fact_646_divides__rev) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 278 (all Z_4 times_times_real(number267125858f_real(bit0(bit1(pls))),Z_4) = plus_plus_real(Z_4,Z_4)) # label(fact_276_mult__2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 279 (all A_1 all B_2 all C (zero_zero_real != C -> (A_1 = B_2 <-> times_times_real(C,A_1) = times_times_real(C,B_2)))) # label(fact_669_real__mult__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 280 (all X_2 all Y_2 (zero_zero_real = plus_plus_real(power_power_real(X_2,number_number_of_nat(bit0(bit1(pls)))),power_power_real(Y_2,number_number_of_nat(bit0(bit1(pls))))) <-> X_2 = zero_zero_real & zero_zero_real = Y_2)) # label(fact_659_realpow__two__sum__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 281 (all K_1 (ord_less_int(bit0(K_1),min) <-> ord_less_eq_int(K_1,min))) # label(fact_545_rel__simps_I11_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 282 (all A_11 all M_4 all N_14 power_power_real(A_11,plus_plus_nat(M_4,N_14)) = times_times_real(power_power_real(A_11,M_4),power_power_real(A_11,N_14))) # label(fact_462_power__add) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 283 t = one_one_int -> (exists X exists Y (plus_plus_int(power_power_int(X,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y,number_number_of_nat(bit0(bit1(pls))))) = plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int) & is_int(Y) & is_int(X))) # label(fact_1__096t_A_061_A1_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 284 (all M_10 all A_42 times_times_int(plus_plus_int(A_42,one_one_int),M_10) = plus_plus_int(M_10,times_times_int(A_42,M_10))) # label(fact_229_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 285 (all A_55 all B_16 all C_10 all D_5 plus_plus_real(plus_plus_real(A_55,B_16),plus_plus_real(C_10,D_5)) = plus_plus_real(plus_plus_real(A_55,C_10),plus_plus_real(B_16,D_5))) # label(fact_115_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 286 (all A all N (ord_less_nat(zero_zero_nat,N) -> (ord_less_real(zero_zero_real,A) -> (exists R (power_power_real(R,N) = A & ord_less_real(zero_zero_real,R)))))) # label(fact_697_realpow__pos__nth) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 287 (all X_2 all Y_2 (is_int(Y_2) & is_int(X_2) -> (number_number_of_int(X_2) = number_number_of_int(Y_2) <-> Y_2 = X_2))) # label(fact_134_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 288 (all A_57 times_times_int(A_57,A_57) = power_power_int(A_57,number_number_of_nat(bit0(bit1(pls))))) # label(fact_24_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 289 (all Z all W plus_plus_int(Z,W) = plus_plus_int(W,Z)) # label(fact_145_zadd__commute) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 290 (all Z all W (ord_less_eq_real(Z,W) -> (ord_less_eq_real(W,Z) -> W = Z))) # label(fact_687_real__le__antisym) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 291 (all A_15 ord_less_eq_int(zero_zero_int,power_power_int(A_15,number_number_of_nat(bit0(bit1(pls)))))) # label(fact_442_zero__le__power2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 292 (all A_16 A_16 = power_power_nat(A_16,one_one_nat)) # label(fact_421_power__one__right) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 293 (all C all D all A_1 all B_2 all Ma (zcong(A_1,B_2,Ma) -> (zcong(C,times_times_int(D,A_1),Ma) <-> zcong(C,times_times_int(D,B_2),Ma)))) # label(fact_634_zcong__zmult__prop2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 294 (all K_1 all L_1 (ord_less_int(K_1,L_1) <-> ord_less_eq_int(bit1(K_1),bit0(L_1)))) # label(fact_83_rel__simps_I33_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 295 (all X_2 all Y_2 (ord_less_eq_real(plus_plus_real(power_power_real(X_2,number_number_of_nat(bit0(bit1(pls)))),power_power_real(Y_2,number_number_of_nat(bit0(bit1(pls))))),zero_zero_real) <-> Y_2 = zero_zero_real & X_2 = zero_zero_real)) # label(fact_475_sum__power2__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 296 (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]. 1.15/1.48 297 (all X_2 all Y_2 (ord_less_eq_int(number_number_of_int(X_2),number_number_of_int(Y_2)) <-> ord_less_eq_int(X_2,Y_2))) # label(fact_54_le__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 298 (all X_2 all P_1 (dvd_dvd_int(P_1,X_2) <-> zcong(X_2,zero_zero_int,P_1))) # label(fact_648_zcong__eq__zdvd__prop) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 299 (all C_2 all D_2 all A_18 all B_4 all R_3 (R_3 != zero_zero_real -> (B_4 = A_18 & C_2 != D_2 -> plus_plus_real(B_4,times_times_real(R_3,D_2)) != plus_plus_real(A_18,times_times_real(R_3,C_2))))) # label(fact_385_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 300 (all N_32 all M_6 all X_14 all Y_12 (dvd_dvd_int(X_14,Y_12) -> (ord_less_eq_nat(N_32,M_6) -> dvd_dvd_int(power_power_int(X_14,N_32),power_power_int(Y_12,M_6))))) # label(fact_316_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 301 (all A_1 (zero_zero_real = A_1 <-> plus_plus_real(A_1,A_1) = zero_zero_real)) # label(fact_354_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 302 (all X_17 all P_2 all Q_3 times_times_real(power_power_real(X_17,P_2),power_power_real(X_17,Q_3)) = power_power_real(X_17,plus_plus_nat(P_2,Q_3))) # label(fact_57_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 303 (all X_2 all Y_2 (X_2 != zero_zero_real | zero_zero_real != Y_2 <-> ord_less_real(zero_zero_real,plus_plus_real(times_times_real(X_2,X_2),times_times_real(Y_2,Y_2))))) # label(fact_416_sum__squares__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 304 (all X_2 all Y_2 (is_int(Y_2) & is_int(X_2) -> (number267125858f_real(Y_2) = number267125858f_real(X_2) <-> X_2 = Y_2))) # label(fact_133_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 305 (all X_12 all Y_10 ord_less_eq_real(zero_zero_real,plus_plus_real(times_times_real(X_12,X_12),times_times_real(Y_10,Y_10)))) # label(fact_409_sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 306 (all V_2 all V_1 ((-ord_less_int(V_1,pls) -> (-ord_less_int(V_2,pls) -> number_number_of_nat(plus_plus_int(V_1,V_2)) = plus_plus_nat(number_number_of_nat(V_1),number_number_of_nat(V_2))) & (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))) & (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_77_add__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 307 (all X_8 all Y_6 (ord_less_int(power_power_int(X_8,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y_6,number_number_of_nat(bit0(bit1(pls))))) -> (ord_less_eq_int(zero_zero_int,Y_6) -> ord_less_int(X_8,Y_6)))) # label(fact_472_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 308 (all M_2 all N_7 all A_6 (ord_less_int(one_one_int,A_6) -> (ord_less_int(power_power_int(A_6,M_2),power_power_int(A_6,N_7)) -> ord_less_nat(M_2,N_7)))) # label(fact_497_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 309 (all A_56 all B_17 times_times_nat(B_17,A_56) = times_times_nat(A_56,B_17)) # label(fact_113_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 310 (all B_1 all Q all R_1 all B all Q_1 all R_2 (plus_plus_int(times_times_int(B_1,Q),R_1) = plus_plus_int(times_times_int(B,Q_1),R_2) -> (ord_less_eq_int(zero_zero_int,plus_plus_int(times_times_int(B,Q_1),R_2)) -> (ord_less_int(R_2,B) -> (ord_less_eq_int(zero_zero_int,R_1) -> (ord_less_int(zero_zero_int,B) -> (ord_less_eq_int(B,B_1) -> ord_less_eq_int(Q,Q_1)))))))) # label(fact_682_zdiv__mono2__lemma) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 311 (all X_7 all Y_5 ord_less_eq_real(zero_zero_real,plus_plus_real(power_power_real(X_7,number_number_of_nat(bit0(bit1(pls)))),power_power_real(Y_5,number_number_of_nat(bit0(bit1(pls))))))) # label(fact_473_sum__power2__ge__zero) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 312 (all Lx_3 all Ly_1 all Rx_3 times_times_int(times_times_int(Lx_3,Ly_1),Rx_3) = times_times_int(times_times_int(Lx_3,Rx_3),Ly_1)) # label(fact_102_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 313 (all N_26 all A_29 all B_7 (ord_less_nat(A_29,B_7) -> (ord_less_eq_nat(zero_zero_nat,A_29) -> (ord_less_nat(zero_zero_nat,N_26) -> ord_less_nat(power_power_nat(A_29,N_26),power_power_nat(B_7,N_26)))))) # label(fact_337_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 314 (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_208_zadd__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 315 (all W_6 number267125858f_real(bit1(W_6)) = plus_plus_real(plus_plus_real(one_one_real,number267125858f_real(W_6)),number267125858f_real(W_6))) # label(fact_252_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 316 (all X_16 all Y_14 all Z_8 times_times_real(X_16,plus_plus_real(Y_14,Z_8)) = plus_plus_real(times_times_real(X_16,Y_14),times_times_real(X_16,Z_8))) # label(fact_180_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 317 (all Y_2 (ord_less_int(pls,Y_2) <-> ord_less_real(zero_zero_real,number267125858f_real(Y_2)))) # label(fact_427_less__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 318 (all K (is_int(K) -> plus_plus_int(pls,K) = K)) # label(fact_202_add__Pls) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 319 (all K1 all K2 (ord_less_int(bit1(K1),bit0(K2)) <-> ord_less_int(K1,K2))) # label(fact_147_less__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 320 (all W all Z1 all Z2 times_times_int(W,minus_minus_int(Z1,Z2)) = minus_minus_int(times_times_int(W,Z1),times_times_int(W,Z2))) # label(fact_596_zdiff__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 321 (all N_25 all A_24 (ord_less_eq_real(zero_zero_real,A_24) -> ord_less_eq_real(zero_zero_real,power_power_real(A_24,N_25)))) # label(fact_360_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 322 (all X_10 all Y_8 (ord_less_eq_nat(power_power_nat(X_10,number_number_of_nat(bit0(bit1(pls)))),power_power_nat(Y_8,number_number_of_nat(bit0(bit1(pls))))) -> (ord_less_eq_nat(zero_zero_nat,Y_8) -> ord_less_eq_nat(X_10,Y_8)))) # label(fact_444_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 323 (all X_12 all Y_10 ord_less_eq_int(zero_zero_int,plus_plus_int(times_times_int(X_12,X_12),times_times_int(Y_10,Y_10)))) # label(fact_410_sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 324 (all L bit1(minus_minus_int(min,L)) = minus_minus_int(min,bit0(L))) # label(fact_609_diff__bin__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 325 (all K all L plus_plus_int(bit1(K),bit0(L)) = bit1(plus_plus_int(K,L))) # label(fact_249_add__Bit1__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 326 (all X_2 all Y_2 (Y_2 != zero_zero_real | zero_zero_real != X_2 <-> ord_less_real(zero_zero_real,plus_plus_real(power_power_real(X_2,number_number_of_nat(bit0(bit1(pls)))),power_power_real(Y_2,number_number_of_nat(bit0(bit1(pls)))))))) # label(fact_479_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 327 (all A_26 A_26 = plus_plus_nat(zero_zero_nat,A_26)) # label(fact_346_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 328 (all A_26 (is_int(A_26) -> A_26 = plus_plus_int(zero_zero_int,A_26))) # label(fact_347_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 329 (all W_1 all Z_1 (is_int(W_1) & is_int(Z_1) -> (ord_less_int(W_1,plus_plus_int(Z_1,one_one_int)) <-> Z_1 = W_1 | ord_less_int(W_1,Z_1)))) # label(fact_156_zless__add1__eq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 330 (all C all D all A_1 all B_2 (plus_plus_real(times_times_real(A_1,D),times_times_real(B_2,C)) != plus_plus_real(times_times_real(A_1,C),times_times_real(B_2,D)) <-> B_2 != A_1 & C != D)) # label(fact_177_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 331 (all N_2 one_one_int = power_power_int(number_number_of_int(min),times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_2))) # label(fact_575_power__m1__even) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 332 (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]. 1.15/1.48 333 (all V_11 all B_10 all C_3 plus_plus_int(times_times_int(number_number_of_int(V_11),B_10),times_times_int(number_number_of_int(V_11),C_3)) = times_times_int(number_number_of_int(V_11),plus_plus_int(B_10,C_3))) # label(fact_223_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 334 (all N_35 all A_35 (ord_less_nat(one_one_nat,A_35) -> ord_less_nat(one_one_nat,times_times_nat(A_35,power_power_nat(A_35,N_35))))) # label(fact_296_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 335 (all X_2 (ord_less_eq_int(X_2,pls) <-> ord_less_eq_int(number_number_of_int(X_2),zero_zero_int))) # label(fact_434_le__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 336 (all X_17 all P_2 all Q_3 power_power_nat(X_17,plus_plus_nat(P_2,Q_3)) = times_times_nat(power_power_nat(X_17,P_2),power_power_nat(X_17,Q_3))) # label(fact_56_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 337 (all Lx_3 all Ly_1 all Rx_3 times_times_real(times_times_real(Lx_3,Ly_1),Rx_3) = times_times_real(times_times_real(Lx_3,Rx_3),Ly_1)) # label(fact_100_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 338 (all K_1 all L_1 (ord_less_int(K_1,L_1) <-> ord_less_int(bit0(K_1),bit0(L_1)))) # label(fact_69_rel__simps_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 339 (all N all P all M (ord_less_eq_int(zero_zero_int,M) -> (zprime(P) -> (dvd_dvd_int(P,times_times_int(M,N)) -> dvd_dvd_int(P,M) | dvd_dvd_int(P,N))))) # label(fact_586_zprime__zdvd__zmult) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 340 (all X_2 all Y_2 all B_2 (ord_less_int(one_one_int,B_2) -> (ord_less_eq_int(power_power_int(B_2,X_2),power_power_int(B_2,Y_2)) <-> ord_less_eq_nat(X_2,Y_2)))) # label(fact_303_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 341 (all B_2 all A_1 (plus_plus_nat(B_2,A_1) = B_2 <-> A_1 = zero_zero_nat)) # label(fact_352_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 342 (all V_3 all W_1 (-ord_less_int(number_number_of_int(W_1),number_number_of_int(V_3)) <-> ord_less_eq_int(number_number_of_int(V_3),number_number_of_int(W_1)))) # label(fact_50_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 343 (all X_1 (is_int(X_1) -> (ord_less_eq_int(zero_zero_int,X_1) -> (ord_less_int(X_1,number_number_of_int(bit0(bit1(pls)))) -> one_one_int = X_1 | zero_zero_int = X_1)))) # label(fact_503_int__pos__lt__two__imp__zero__or__one) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 344 (all A_1 all W_1 (is_int(A_1) -> (zero_zero_int = A_1 & number_number_of_nat(W_1) != zero_zero_nat <-> zero_zero_int = power_power_int(A_1,number_number_of_nat(W_1))))) # label(fact_578_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.48 345 (all Lx_5 all Ly_3 all Rx_5 all Ry_3 times_times_int(Rx_5,times_times_int(times_times_int(Lx_5,Ly_3),Ry_3)) = times_times_int(times_times_int(Lx_5,Ly_3),times_times_int(Rx_5,Ry_3))) # label(fact_96_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 346 (all A_56 all B_17 times_times_real(A_56,B_17) = times_times_real(B_17,A_56)) # label(fact_112_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 347 (all A all B_1 plus_plus_int(minus_minus_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)),power_power_int(B_1,number_number_of_nat(bit0(bit1(pls))))) = power_power_int(minus_minus_int(A,B_1),number_number_of_nat(bit0(bit1(pls))))) # label(fact_615_zdiff__power2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 348 (all A_11 all M_4 all N_14 power_power_nat(A_11,plus_plus_nat(M_4,N_14)) = times_times_nat(power_power_nat(A_11,M_4),power_power_nat(A_11,N_14))) # label(fact_461_power__add) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 349 (all A_20 all N_21 all N_20 (ord_less_eq_nat(N_21,N_20) -> (ord_less_eq_real(zero_zero_real,A_20) -> (ord_less_eq_real(A_20,one_one_real) -> ord_less_eq_real(power_power_real(A_20,N_20),power_power_real(A_20,N_21)))))) # label(fact_375_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 350 (all K1 all K2 (ord_less_eq_int(K1,K2) <-> ord_less_eq_int(bit1(K1),bit1(K2)))) # label(fact_65_less__eq__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 351 (all X_2 all Y_2 plus_plus_real(plus_plus_real(power_power_real(X_2,number_number_of_nat(bit0(bit1(pls)))),power_power_real(Y_2,number_number_of_nat(bit0(bit1(pls))))),times_times_real(times_times_real(number267125858f_real(bit0(bit1(pls))),X_2),Y_2)) = power_power_real(plus_plus_real(X_2,Y_2),number_number_of_nat(bit0(bit1(pls))))) # label(fact_9_power2__sum) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 352 (all X_21 times_times_int(X_21,X_21) = power_power_int(X_21,number_number_of_nat(bit0(bit1(pls))))) # label(fact_21_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 353 (all Y_1 all N all P (zprime(P) -> (dvd_dvd_int(P,power_power_int(Y_1,N)) -> (ord_less_nat(zero_zero_nat,N) -> dvd_dvd_int(P,Y_1))))) # label(fact_657_zpower__zdvd__prop2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 354 ord_less_int(one_one_int,t) -> (exists X exists Y (is_int(Y) & plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int) = plus_plus_int(power_power_int(X,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y,number_number_of_nat(bit0(bit1(pls))))) & is_int(X))) # label(fact_2__0961_A_060_At_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 355 (all X_2 all Y_2 (Y_2 = zero_zero_real & X_2 = zero_zero_real <-> plus_plus_real(power_power_real(X_2,number_number_of_nat(bit0(bit1(pls)))),power_power_real(Y_2,number_number_of_nat(bit0(bit1(pls))))) = zero_zero_real)) # label(fact_453_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 356 (all X_1 all Y_1 (is_int(Y_1) & is_int(X_1) -> ord_less_int(X_1,Y_1) | Y_1 = X_1 | ord_less_int(Y_1,X_1))) # label(fact_38_zless__linear) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 357 (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_210_plus__numeral__code_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 358 (all V_7 all W_9 all Z_6 plus_plus_int(number_number_of_int(V_7),plus_plus_int(number_number_of_int(W_9),Z_6)) = plus_plus_int(number_number_of_int(plus_plus_int(V_7,W_9)),Z_6)) # label(fact_244_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 359 (all X_2 (ord_less_eq_real(number267125858f_real(X_2),zero_zero_real) <-> ord_less_eq_int(X_2,pls))) # label(fact_433_le__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 360 (all W_1 all X_2 (X_2 = number267125858f_real(W_1) <-> X_2 = number267125858f_real(W_1))) # label(fact_135_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 361 (all Ma all K_1 all N_1 (dvd_dvd_nat(times_times_nat(Ma,K_1),times_times_nat(N_1,K_1)) <-> K_1 = zero_zero_nat | dvd_dvd_nat(Ma,N_1))) # label(fact_641_nat__mult__dvd__cancel__disj_H) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 362 (all V_11 all B_10 all C_3 times_times_nat(number_number_of_nat(V_11),plus_plus_nat(B_10,C_3)) = plus_plus_nat(times_times_nat(number_number_of_nat(V_11),B_10),times_times_nat(number_number_of_nat(V_11),C_3))) # label(fact_222_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 363 (all X_2 (ord_less_int(X_2,pls) <-> ord_less_int(number_number_of_int(X_2),zero_zero_int))) # label(fact_430_less__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 364 (all M_5 all A_32 all N_31 all B_9 (dvd_dvd_real(power_power_real(A_32,N_31),B_9) -> (ord_less_eq_nat(M_5,N_31) -> dvd_dvd_real(power_power_real(A_32,M_5),B_9)))) # label(fact_320_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 365 (all A_25 A_25 = plus_plus_real(A_25,zero_zero_real)) # label(fact_348_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 366 (all K (is_int(K) -> plus_plus_int(K,pls) = K)) # label(fact_201_add__Pls__right) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 367 (all A_49 all N_37 power_power_nat(A_49,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_37)) = power_power_nat(power_power_nat(A_49,N_37),number_number_of_nat(bit0(bit1(pls))))) # label(fact_157_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 368 -(all S (is_int(S) -> -(ord_less_eq_int(zero_zero_int,S) & zcong(s1,S,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) & ord_less_int(S,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int))))) # label(fact_307__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]. 1.15/1.49 369 (all X_10 all Y_8 (ord_less_eq_int(power_power_int(X_10,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y_8,number_number_of_nat(bit0(bit1(pls))))) -> (ord_less_eq_int(zero_zero_int,Y_8) -> ord_less_eq_int(X_10,Y_8)))) # label(fact_445_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 370 (all Ma all N_1 all A_1 (ord_less_real(one_one_real,A_1) -> (Ma = N_1 <-> power_power_real(A_1,Ma) = power_power_real(A_1,N_1)))) # label(fact_489_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 371 (all Z all W (ord_less_eq_int(W,Z) | ord_less_eq_int(Z,W))) # label(fact_36_zle__linear) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 372 (all V_3 all W_1 (ord_less_eq_real(number267125858f_real(V_3),number267125858f_real(W_1)) <-> -ord_less_real(number267125858f_real(W_1),number267125858f_real(V_3)))) # label(fact_48_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 373 (all K plus_plus_int(K,K) = bit0(K)) # label(fact_204_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 374 (all A_12 all B_3 all N_15 power_power_int(times_times_int(A_12,B_3),N_15) = times_times_int(power_power_int(A_12,N_15),power_power_int(B_3,N_15))) # label(fact_460_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 375 (all M all Y_1 all X_1 (is_int(X_1) & is_int(Y_1) -> (ord_less_int(zero_zero_int,X_1) -> (ord_less_int(zero_zero_int,Y_1) -> (ord_less_int(zero_zero_int,M) -> (zcong(X_1,Y_1,M) -> (ord_less_int(X_1,M) -> (ord_less_int(Y_1,M) -> X_1 = Y_1)))))))) # label(fact_643_zcong__less__eq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 376 (all K_1 all L_1 (ord_less_int(number_number_of_int(K_1),number_number_of_int(L_1)) <-> ord_less_int(K_1,L_1))) # label(fact_73_less__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 377 (all X_2 all Y_2 (ord_less_real(X_2,Y_2) <-> Y_2 != X_2 & ord_less_eq_real(X_2,Y_2))) # label(fact_663_real__less__def) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 378 (all A all B_1 plus_plus_int(plus_plus_int(plus_plus_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))),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),power_power_int(B_1,number_number_of_nat(bit0(bit1(pls)))))),power_power_int(B_1,number_number_of_nat(bit1(bit1(pls))))) = power_power_int(plus_plus_int(A,B_1),number_number_of_nat(bit1(bit1(pls))))) # label(fact_8_zadd__power3) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 379 (all V_3 (zero_zero_nat = number_number_of_nat(V_3) <-> ord_less_eq_int(V_3,pls))) # label(fact_556_eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 380 (all A_52 all C_7 all D_4 plus_plus_real(plus_plus_real(A_52,C_7),D_4) = plus_plus_real(A_52,plus_plus_real(C_7,D_4))) # label(fact_124_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 381 (all W all Z1 all Z2 plus_plus_int(times_times_int(W,Z1),times_times_int(W,Z2)) = times_times_int(W,plus_plus_int(Z1,Z2))) # label(fact_209_zadd__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 382 (all A_9 all N_11 ord_less_eq_int(zero_zero_int,power_power_int(A_9,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_11)))) # label(fact_482_zero__le__even__power_H) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 383 (all V_3 all V_4 (ord_less_eq_nat(number_number_of_nat(V_3),number_number_of_nat(V_4)) <-> (-ord_less_eq_int(V_3,V_4) -> ord_less_eq_int(V_3,pls)))) # label(fact_418_le__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 384 (all B_1 all A all P (zprime(P) -> (ord_less_int(zero_zero_int,A) -> (zcong(times_times_int(A,B_1),zero_zero_int,P) -> zcong(A,zero_zero_int,P) | zcong(B_1,zero_zero_int,P))))) # label(fact_656_zcong__zprime__prod__zero) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 385 (all Z_1 all W_1 (is_int(Z_1) & is_int(W_1) -> (ord_less_int(Z_1,W_1) <-> ord_less_eq_int(Z_1,W_1) & Z_1 != W_1))) # label(fact_37_zless__le) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 386 (all M zcong(M,zero_zero_int,M)) # label(fact_631_zcong__id) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 387 (all P all Y_1 all N (ord_less_nat(zero_zero_nat,N) -> (dvd_dvd_int(P,Y_1) -> dvd_dvd_int(P,power_power_int(Y_1,N))))) # label(fact_652_zpower__zdvd__prop1) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 388 (all Lx all Rx all Ry times_times_real(Rx,times_times_real(Lx,Ry)) = times_times_real(Lx,times_times_real(Rx,Ry))) # label(fact_109_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 389 (all X_17 all P_2 all Q_3 power_power_int(X_17,plus_plus_nat(P_2,Q_3)) = times_times_int(power_power_int(X_17,P_2),power_power_int(X_17,Q_3))) # label(fact_58_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 390 (all A_52 all C_7 all D_4 plus_plus_int(plus_plus_int(A_52,C_7),D_4) = plus_plus_int(A_52,plus_plus_int(C_7,D_4))) # label(fact_126_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 391 (all Lx all Rx all Ry times_times_nat(Rx,times_times_nat(Lx,Ry)) = times_times_nat(Lx,times_times_nat(Rx,Ry))) # label(fact_110_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 392 (all B_1_1 is_int(twoSqu140629262sum2sq(B_1_1))) # label(gsy_c_TwoSquares__Mirabelle__ccrtsbwhjp_Osum2sq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 393 (all N_25 all A_24 (ord_less_eq_nat(zero_zero_nat,A_24) -> ord_less_eq_nat(zero_zero_nat,power_power_nat(A_24,N_25)))) # label(fact_361_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 394 (all A_20 all N_21 all N_20 (ord_less_eq_nat(N_21,N_20) -> (ord_less_eq_nat(zero_zero_nat,A_20) -> (ord_less_eq_nat(A_20,one_one_nat) -> ord_less_eq_nat(power_power_nat(A_20,N_20),power_power_nat(A_20,N_21)))))) # label(fact_376_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 395 (all K_1 all L_1 (is_int(L_1) & is_int(K_1) -> (bit1(L_1) = bit1(K_1) <-> L_1 = K_1))) # label(fact_138_rel__simps_I51_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 396 (all X_6 all Y_4 -ord_less_real(plus_plus_real(power_power_real(X_6,number_number_of_nat(bit0(bit1(pls)))),power_power_real(Y_4,number_number_of_nat(bit0(bit1(pls))))),zero_zero_real)) # label(fact_477_not__sum__power2__lt__zero) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 397 (all A_1 all W_1 (zero_zero_real = power_power_real(A_1,number_number_of_nat(W_1)) <-> A_1 = zero_zero_real & zero_zero_nat != number_number_of_nat(W_1))) # label(fact_576_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 398 (all K (is_int(K) -> minus_minus_int(K,pls) = K)) # label(fact_593_diff__bin__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 399 (all Z all N (dvd_dvd_int(Z,N) -> (ord_less_int(zero_zero_int,N) -> ord_less_eq_int(Z,N)))) # label(fact_335_zdvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 400 (all N_36 all A_36 (ord_less_int(one_one_int,A_36) -> ord_less_int(power_power_int(A_36,N_36),times_times_int(A_36,power_power_int(A_36,N_36))))) # label(fact_294_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 401 (all A_57 times_times_real(A_57,A_57) = power_power_real(A_57,number_number_of_nat(bit0(bit1(pls))))) # label(fact_23_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 402 (all A_14 -ord_less_real(power_power_real(A_14,number_number_of_nat(bit0(bit1(pls)))),zero_zero_real)) # label(fact_449_power2__less__0) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 403 (all M_10 all A_42 plus_plus_real(M_10,times_times_real(A_42,M_10)) = times_times_real(plus_plus_real(A_42,one_one_real),M_10)) # label(fact_227_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 404 (all X_2 all Y_2 (is_int(Y_2) & is_int(X_2) -> (zero_zero_int = X_2 & Y_2 = zero_zero_int <-> ord_less_eq_int(plus_plus_int(times_times_int(X_2,X_2),times_times_int(Y_2,Y_2)),zero_zero_int)))) # label(fact_412_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 405 (all A_39 (is_int(A_39) -> times_times_int(number_number_of_int(bit1(pls)),A_39) = A_39)) # label(fact_255_mult__numeral__1) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 406 (all B_2 all A_1 (is_int(B_2) & is_int(A_1) -> (A_1 = zero_zero_int <-> plus_plus_int(B_2,A_1) = B_2))) # label(fact_353_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 407 (all N_2 power_power_real(number267125858f_real(min),times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_2)) = one_one_real) # label(fact_574_power__m1__even) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 408 (all W_3 ((zero_zero_nat != number_number_of_nat(W_3) -> zero_zero_real = power_power_real(zero_zero_real,number_number_of_nat(W_3))) & (zero_zero_nat = number_number_of_nat(W_3) -> one_one_real = power_power_real(zero_zero_real,number_number_of_nat(W_3))))) # label(fact_581_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 409 (all A_54 all B_15 all C_9 plus_plus_nat(plus_plus_nat(A_54,C_9),B_15) = plus_plus_nat(plus_plus_nat(A_54,B_15),C_9)) # label(fact_119_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 410 (all X_18 power_power_real(X_18,one_one_nat) = X_18) # label(fact_45_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 411 (all V_3 all V_4 (ord_less_int(V_3,V_4) & (ord_less_int(V_3,V_4) -> ord_less_int(pls,V_4)) <-> ord_less_nat(number_number_of_nat(V_3),number_number_of_nat(V_4)))) # label(fact_413_less__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 412 -(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(power_power_int(s,number_number_of_nat(bit0(bit1(pls)))),one_one_int))) # label(fact_34__096_B_Bthesis_O_A_I_B_Bt_O_As_A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A_L_A1_) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 413 (all Z_4 times_times_int(number_number_of_int(bit0(bit1(pls))),Z_4) = plus_plus_int(Z_4,Z_4)) # label(fact_277_mult__2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 414 (all A_47 all B_12 all C_4 times_times_nat(plus_plus_nat(A_47,B_12),C_4) = plus_plus_nat(times_times_nat(A_47,C_4),times_times_nat(B_12,C_4))) # label(fact_175_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 415 (all Z all X_1 all Y_1 all P (zcong(power_power_int(X_1,Y_1),one_one_int,P) -> zcong(power_power_int(X_1,times_times_nat(Y_1,Z)),one_one_int,P))) # label(fact_369_zcong__zpower__zmult) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 416 (all W_1 all Y_2 all X_2 all Z_1 (Y_2 = Z_1 | X_2 = W_1 <-> plus_plus_nat(times_times_nat(W_1,Z_1),times_times_nat(X_2,Y_2)) = plus_plus_nat(times_times_nat(W_1,Y_2),times_times_nat(X_2,Z_1)))) # label(fact_169_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 417 (all X_15 all Y_13 all Q_2 power_power_nat(times_times_nat(X_15,Y_13),Q_2) = times_times_nat(power_power_nat(X_15,Q_2),power_power_nat(Y_13,Q_2))) # label(fact_189_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 418 (all A_46 (is_int(A_46) -> A_46 = times_times_int(A_46,one_one_int))) # label(fact_185_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 419 (all B_1 all A all C_1 (ord_less_int(A,C_1) -> (ord_less_int(B_1,C_1) -> ord_less_eq_int(A,B_1) | ord_less_eq_int(B_1,A)))) # label(fact_566_Euler_Oaux2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 420 (all A_28 times_times_int(zero_zero_int,A_28) = zero_zero_int) # label(fact_341_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 421 (all X_15 all Y_13 all Q_2 power_power_int(times_times_int(X_15,Y_13),Q_2) = times_times_int(power_power_int(X_15,Q_2),power_power_int(Y_13,Q_2))) # label(fact_191_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 422 (all X_7 all Y_5 ord_less_eq_int(zero_zero_int,plus_plus_int(power_power_int(X_7,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y_5,number_number_of_nat(bit0(bit1(pls))))))) # label(fact_474_sum__power2__ge__zero) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 423 (all M_2 all N_7 all A_6 (ord_less_nat(one_one_nat,A_6) -> (ord_less_nat(power_power_nat(A_6,M_2),power_power_nat(A_6,N_7)) -> ord_less_nat(M_2,N_7)))) # label(fact_496_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 424 (all K all L bit1(plus_plus_int(K,L)) = plus_plus_int(bit0(K),bit1(L))) # label(fact_250_add__Bit0__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 425 (all W_15 power_power_int(number_number_of_int(W_15),number_number_of_nat(bit0(bit1(pls)))) = times_times_int(number_number_of_int(W_15),number_number_of_int(W_15))) # label(fact_14_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 426 (all A_1 all Ma (dvd_dvd_int(Ma,A_1) <-> zcong(A_1,zero_zero_int,Ma))) # label(fact_647_zcong__zero__equiv__div) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 427 (all X_16 all Y_14 all Z_8 plus_plus_int(times_times_int(X_16,Y_14),times_times_int(X_16,Z_8)) = times_times_int(X_16,plus_plus_int(Y_14,Z_8))) # label(fact_182_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 428 (all B_2 all A_1 (A_1 = zero_zero_real <-> plus_plus_real(B_2,A_1) = B_2)) # label(fact_351_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 429 (all A all N all P (zprime(P) -> (dvd_dvd_int(P,power_power_int(A,N)) -> dvd_dvd_int(P,A)))) # label(fact_388_zprime__zdvd__power) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 430 (all X_19 all P_3 all Q_4 power_power_nat(power_power_nat(X_19,P_3),Q_4) = power_power_nat(X_19,times_times_nat(P_3,Q_4))) # label(fact_41_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 431 (all A_4 all K_3 (ord_less_eq_real(power_power_real(A_4,times_times_nat(number_number_of_nat(bit0(bit1(pls))),K_3)),zero_zero_real) -> zero_zero_real = A_4)) # label(fact_504_even__power__le__0__imp__0) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 432 (all K all L minus_minus_int(bit0(K),bit0(L)) = bit0(minus_minus_int(K,L))) # label(fact_594_diff__bin__simps_I7_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 433 (all N_13 power_power_nat(one_one_nat,N_13) = one_one_nat) # label(fact_465_power__one) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 434 (all K_1 all N_1 all Ma (dvd_dvd_int(K_1,plus_plus_int(N_1,times_times_int(K_1,Ma))) <-> dvd_dvd_int(K_1,N_1))) # label(fact_370_zdvd__reduce) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 435 (all K_1 all L_1 (ord_less_int(K_1,L_1) <-> ord_less_int(bit1(K_1),bit0(L_1)))) # label(fact_148_rel__simps_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 436 (all A all B_1 zcong(A,B_1,one_one_int)) # label(fact_568_zcong__1) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 437 (all N_23 all A_22 (ord_less_real(zero_zero_real,A_22) -> ord_less_real(zero_zero_real,power_power_real(A_22,N_23)))) # label(fact_366_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 438 (all W_15 times_times_real(number267125858f_real(W_15),number267125858f_real(W_15)) = power_power_real(number267125858f_real(W_15),number_number_of_nat(bit0(bit1(pls))))) # label(fact_13_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 439 (all K all I all J (ord_less_int(I,J) -> (ord_less_int(zero_zero_int,K) -> ord_less_int(times_times_int(K,I),times_times_int(K,J))))) # label(fact_402_zmult__zless__mono2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 440 (all A all B_1 all C_1 (is_int(A) -> (C_1 = minus_minus_int(A,B_1) -> A = plus_plus_int(C_1,B_1)))) # label(fact_633_Int2_Oaux1) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 441 (all A_31 all N_30 all B_8 (power_power_real(B_8,N_30) = power_power_real(A_31,N_30) -> (ord_less_eq_real(zero_zero_real,A_31) -> (ord_less_eq_real(zero_zero_real,B_8) -> (ord_less_nat(zero_zero_nat,N_30) -> A_31 = B_8))))) # label(fact_321_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 442 (all W_1 all X_2 (is_int(X_2) -> (X_2 = number_number_of_int(W_1) <-> number_number_of_int(W_1) = X_2))) # label(fact_136_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 443 (all K_1 all L_1 (ord_less_int(bit0(K_1),bit1(L_1)) <-> ord_less_eq_int(K_1,L_1))) # label(fact_85_rel__simps_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 444 (all C_1 all D_1 all A all B_1 all M (zcong(A,B_1,M) -> (zcong(C_1,D_1,M) -> zcong(times_times_int(A,C_1),times_times_int(B_1,D_1),M)))) # label(fact_569_zcong__zmult) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 445 (all N all M (dvd_dvd_int(N,M) -> ord_less_eq_int(N,M) | ord_less_eq_int(M,zero_zero_int))) # label(fact_644_zdvd__bounds) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 446 (all A_7 all N_9 all N_8 (ord_less_eq_nat(N_9,N_8) -> (ord_less_eq_int(one_one_int,A_7) -> ord_less_eq_int(power_power_int(A_7,N_9),power_power_int(A_7,N_8))))) # label(fact_488_power__increasing) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 447 (all V_16 all V_15 (ord_less_eq_int(pls,V_15) -> (ord_less_eq_int(pls,V_16) -> number267125858f_real(times_times_int(V_15,V_16)) = times_times_real(number267125858f_real(V_15),number267125858f_real(V_16))))) # label(fact_211_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 448 (all Lx_2 all Ly all Rx_2 times_times_nat(Lx_2,times_times_nat(Ly,Rx_2)) = times_times_nat(times_times_nat(Lx_2,Ly),Rx_2)) # label(fact_104_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 449 (all V_8 all W_10 number267125858f_real(times_times_int(V_8,W_10)) = times_times_real(number267125858f_real(V_8),number267125858f_real(W_10))) # label(fact_241_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 450 (all X_1 all Q all N all R_1 (X_1 = plus_plus_nat(times_times_nat(Q,N),R_1) -> (ord_less_nat(zero_zero_nat,R_1) -> (ord_less_nat(R_1,N) -> -dvd_dvd_nat(N,X_1))))) # label(fact_654_divides__div__not) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 451 (all A_5 all N_6 all N_5 (ord_less_nat(N_6,N_5) -> (ord_less_nat(one_one_nat,A_5) -> ord_less_nat(power_power_nat(A_5,N_6),power_power_nat(A_5,N_5))))) # label(fact_499_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 452 (all A_31 all N_30 all B_8 (is_int(B_8) & is_int(A_31) -> (power_power_int(A_31,N_30) = power_power_int(B_8,N_30) -> (ord_less_eq_int(zero_zero_int,A_31) -> (ord_less_eq_int(zero_zero_int,B_8) -> (ord_less_nat(zero_zero_nat,N_30) -> B_8 = A_31)))))) # label(fact_323_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 453 (all A_50 all C_5 plus_plus_real(A_50,C_5) = plus_plus_real(C_5,A_50)) # label(fact_130_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 454 (all A_44 all B_11 all V_12 times_times_nat(plus_plus_nat(A_44,B_11),number_number_of_nat(V_12)) = plus_plus_nat(times_times_nat(A_44,number_number_of_nat(V_12)),times_times_nat(B_11,number_number_of_nat(V_12)))) # label(fact_219_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 455 (all C all X_2 all Ta all A_1 all D (dvd_dvd_int(A_1,D) -> (dvd_dvd_int(A_1,plus_plus_int(X_2,Ta)) <-> dvd_dvd_int(A_1,plus_plus_int(plus_plus_int(X_2,times_times_int(C,D)),Ta))))) # label(fact_371_zdvd__period) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 456 (all A_48 all M_12 all B_13 plus_plus_int(times_times_int(A_48,M_12),times_times_int(B_13,M_12)) = times_times_int(plus_plus_int(A_48,B_13),M_12)) # label(fact_173_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 457 (all L min != bit0(L)) # label(fact_519_rel__simps_I42_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 458 (all K_1 (ord_less_eq_int(pls,K_1) <-> ord_less_eq_int(pls,bit0(K_1)))) # label(fact_155_rel__simps_I21_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 459 (all A_13 all N_16 times_times_int(A_13,power_power_int(A_13,N_16)) = times_times_int(power_power_int(A_13,N_16),A_13)) # label(fact_457_power__commutes) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 460 (all X_2 all Y_2 (is_int(Y_2) & is_int(X_2) -> (zero_zero_int = X_2 & zero_zero_int = Y_2 <-> ord_less_eq_int(plus_plus_int(power_power_int(X_2,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y_2,number_number_of_nat(bit0(bit1(pls))))),zero_zero_int)))) # label(fact_476_sum__power2__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 461 (all N_4 all A_2 (ord_less_real(one_one_real,A_2) -> (ord_less_nat(zero_zero_nat,N_4) -> ord_less_real(one_one_real,power_power_real(A_2,N_4))))) # label(fact_548_one__less__power) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 462 (all A_1 all W_1 (zero_zero_nat != number_number_of_nat(W_1) & zero_zero_nat = A_1 <-> power_power_nat(A_1,number_number_of_nat(W_1)) = zero_zero_nat)) # label(fact_577_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 463 (all X_2 all Y_2 all Z_1 (ord_less_real(zero_zero_real,Z_1) -> (ord_less_real(times_times_real(X_2,Z_1),times_times_real(Y_2,Z_1)) <-> ord_less_real(X_2,Y_2)))) # label(fact_676_real__mult__less__iff1) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 464 (all X_2 all Y_2 (ord_less_int(X_2,Y_2) <-> ord_less_int(number_number_of_int(X_2),number_number_of_int(Y_2)))) # label(fact_52_less__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 465 (all K1 all K2 (ord_less_int(K1,K2) <-> ord_less_int(bit1(K1),bit1(K2)))) # label(fact_63_less__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 466 (all N_27 ((zero_zero_nat = N_27 -> one_one_nat = power_power_nat(zero_zero_nat,N_27)) & (N_27 != zero_zero_nat -> zero_zero_nat = power_power_nat(zero_zero_nat,N_27)))) # label(fact_333_power__0__left) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 467 (all K bit1(K) != pls) # label(fact_192_rel__simps_I46_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 468 (all A_53 all B_14 all C_8 plus_plus_real(plus_plus_real(A_53,B_14),C_8) = plus_plus_real(A_53,plus_plus_real(B_14,C_8))) # label(fact_121_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 469 (all A_1 all P_1 (zcong(A_1,minus_minus_int(P_1,one_one_int),P_1) <-> zcong(times_times_int(A_1,minus_minus_int(P_1,one_one_int)),one_one_int,P_1))) # label(fact_610_inv__not__p__minus__1__aux) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 470 (all K all A all B_1 all M (zcong(A,B_1,M) -> zcong(times_times_int(K,A),times_times_int(K,B_1),M))) # label(fact_570_zcong__scalar2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 471 (all A all B_1 power_power_int(plus_plus_int(A,B_1),number_number_of_nat(bit0(bit1(pls)))) = plus_plus_int(plus_plus_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)),power_power_int(B_1,number_number_of_nat(bit0(bit1(pls)))))) # label(fact_7_zadd__power2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 472 (all L bit1(minus_minus_int(min,L)) = minus_minus_int(pls,bit1(L))) # label(fact_607_diff__bin__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 473 (all W_1 (ord_less_int(W_1,zero_zero_int) <-> ord_less_int(bit1(W_1),zero_zero_int))) # label(fact_396_bin__less__0__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 474 (all K_1 (ord_less_eq_int(pls,K_1) <-> ord_less_int(pls,bit1(K_1)))) # label(fact_81_rel__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 475 (all X_2 (ord_less_int(X_2,bit1(pls)) <-> ord_less_real(number267125858f_real(X_2),one_one_real))) # label(fact_160_less__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 476 (all B_1 all D_1 all A (dvd_dvd_nat(D_1,A) -> (dvd_dvd_nat(D_1,plus_plus_nat(A,B_1)) -> dvd_dvd_nat(D_1,B_1)))) # label(fact_628_divides__add__revr) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 477 (all M_8 all N_34 all A_34 (ord_less_int(one_one_int,A_34) -> (ord_less_eq_int(power_power_int(A_34,M_8),power_power_int(A_34,N_34)) -> ord_less_eq_nat(M_8,N_34)))) # label(fact_300_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 478 (all V_9 all W_11 times_times_real(number267125858f_real(V_9),number267125858f_real(W_11)) = number267125858f_real(times_times_int(V_9,W_11))) # label(fact_239_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 479 (all N all M (is_int(M) & is_int(N) -> (ord_less_eq_int(zero_zero_int,M) -> (ord_less_eq_int(zero_zero_int,N) -> (dvd_dvd_int(M,N) -> (dvd_dvd_int(N,M) -> M = N)))))) # label(fact_325_zdvd__antisym__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 480 (all N_36 all A_36 (ord_less_nat(one_one_nat,A_36) -> ord_less_nat(power_power_nat(A_36,N_36),times_times_nat(A_36,power_power_nat(A_36,N_36))))) # label(fact_293_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 481 (all Z (ord_less_eq_int(zero_zero_int,Z) -> ord_less_int(zero_zero_int,plus_plus_int(one_one_int,Z)))) # label(fact_435_le__imp__0__less) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 482 (all A_50 all C_5 plus_plus_nat(C_5,A_50) = plus_plus_nat(A_50,C_5)) # label(fact_131_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 483 (all V all W_2 minus_minus_int(number_number_of_int(V),number_number_of_int(W_2)) = number_number_of_int(minus_minus_int(V,W_2))) # label(fact_598_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 484 (all Lx_2 all Ly all Rx_2 times_times_int(Lx_2,times_times_int(Ly,Rx_2)) = times_times_int(times_times_int(Lx_2,Ly),Rx_2)) # label(fact_105_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 485 (all N_32 all M_6 all X_14 all Y_12 (dvd_dvd_real(X_14,Y_12) -> (ord_less_eq_nat(N_32,M_6) -> dvd_dvd_real(power_power_real(X_14,N_32),power_power_real(Y_12,M_6))))) # label(fact_317_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 486 (all A_51 all C_6 all D_3 plus_plus_real(A_51,plus_plus_real(C_6,D_3)) = plus_plus_real(C_6,plus_plus_real(A_51,D_3))) # label(fact_127_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 487 (all W_14 number_number_of_int(plus_plus_int(bit1(pls),W_14)) = plus_plus_int(one_one_int,number_number_of_int(W_14))) # label(fact_29_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.49 488 (all X_2 all Y_2 (is_int(X_2) & is_int(Y_2) -> (zero_zero_int = X_2 & zero_zero_int = Y_2 <-> zero_zero_int = plus_plus_int(times_times_int(X_2,X_2),times_times_int(Y_2,Y_2))))) # label(fact_384_sum__squares__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 489 (all K all I all J (ord_less_eq_int(I,J) -> ord_less_eq_int(plus_plus_int(K,I),plus_plus_int(K,J)))) # label(fact_76_zadd__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 490 (all A_25 (is_int(A_25) -> A_25 = plus_plus_int(A_25,zero_zero_int))) # label(fact_350_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 491 (all A_47 all B_12 all C_4 plus_plus_real(times_times_real(A_47,C_4),times_times_real(B_12,C_4)) = times_times_real(plus_plus_real(A_47,B_12),C_4)) # label(fact_174_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 492 (all Z all X_1 all Y_1 all M (zcong(X_1,Y_1,M) -> zcong(power_power_int(X_1,Z),power_power_int(Y_1,Z),M))) # label(fact_639_zcong__zpower) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 493 (all N_10 all A_8 (ord_less_eq_int(one_one_int,A_8) -> ord_less_eq_int(one_one_int,power_power_int(A_8,N_10)))) # label(fact_485_one__le__power) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 494 (all X_1 all Y_1 all N (N != zero_zero_nat -> (dvd_dvd_nat(power_power_nat(X_1,N),Y_1) -> dvd_dvd_nat(X_1,Y_1)))) # label(fact_645_divides__exp2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 495 (all A_55 all B_16 all C_10 all D_5 plus_plus_nat(plus_plus_nat(A_55,B_16),plus_plus_nat(C_10,D_5)) = plus_plus_nat(plus_plus_nat(A_55,C_10),plus_plus_nat(B_16,D_5))) # label(fact_116_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 496 (all K (is_int(K) -> K = number_number_of_int(K))) # label(fact_142_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 497 (all A_44 all B_11 all V_12 times_times_int(plus_plus_int(A_44,B_11),number_number_of_int(V_12)) = plus_plus_int(times_times_int(A_44,number_number_of_int(V_12)),times_times_int(B_11,number_number_of_int(V_12)))) # label(fact_220_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 498 (all X_2 all N_1 (ord_less_nat(zero_zero_nat,power_power_nat(X_2,N_1)) <-> N_1 = zero_zero_nat | ord_less_nat(zero_zero_nat,X_2))) # label(fact_509_nat__zero__less__power__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 499 (all Z_1 (ord_less_int(Z_1,zero_zero_int) <-> ord_less_int(plus_plus_int(plus_plus_int(one_one_int,Z_1),Z_1),zero_zero_int))) # label(fact_426_odd__less__0) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 500 (all K_1 (ord_less_eq_int(bit0(K_1),pls) <-> ord_less_eq_int(K_1,pls))) # label(fact_154_rel__simps_I27_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 501 (all X_2 (ord_less_int(X_2,pls) <-> ord_less_real(number267125858f_real(X_2),zero_zero_real))) # label(fact_429_less__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 502 (all Ma all N_1 all A_1 (ord_less_nat(one_one_nat,A_1) -> (power_power_nat(A_1,Ma) = power_power_nat(A_1,N_1) <-> Ma = N_1))) # label(fact_490_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 503 (all Lx_3 all Ly_1 all Rx_3 times_times_nat(times_times_nat(Lx_3,Rx_3),Ly_1) = times_times_nat(times_times_nat(Lx_3,Ly_1),Rx_3)) # label(fact_101_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 504 (all A_48 all M_12 all B_13 plus_plus_real(times_times_real(A_48,M_12),times_times_real(B_13,M_12)) = times_times_real(plus_plus_real(A_48,B_13),M_12)) # label(fact_171_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 505 (all N_23 all A_22 (ord_less_int(zero_zero_int,A_22) -> ord_less_int(zero_zero_int,power_power_int(A_22,N_23)))) # label(fact_368_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 506 (all W_5 number267125858f_real(bit0(W_5)) = times_times_real(plus_plus_real(one_one_real,one_one_real),number267125858f_real(W_5))) # label(fact_267_double__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 507 (all X_20 all N_38 times_times_nat(power_power_nat(X_20,N_38),power_power_nat(X_20,N_38)) = power_power_nat(X_20,times_times_nat(number_number_of_nat(bit0(bit1(pls))),N_38))) # label(fact_25_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 508 (all Z_3 plus_plus_nat(Z_3,Z_3) = times_times_nat(Z_3,number_number_of_nat(bit0(bit1(pls))))) # label(fact_279_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 509 (all X_11 all Y_9 -ord_less_real(plus_plus_real(times_times_real(X_11,X_11),times_times_real(Y_9,Y_9)),zero_zero_real)) # label(fact_414_not__sum__squares__lt__zero) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 510 (all A_19 all N_19 all N_18 (ord_less_nat(N_19,N_18) -> (ord_less_int(zero_zero_int,A_19) -> (ord_less_int(A_19,one_one_int) -> ord_less_int(power_power_int(A_19,N_18),power_power_int(A_19,N_19)))))) # label(fact_380_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 511 (all A_33 all M_7 all N_33 (ord_less_eq_nat(M_7,N_33) -> dvd_dvd_real(power_power_real(A_33,M_7),power_power_real(A_33,N_33)))) # label(fact_314_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 512 (all A_27 zero_zero_nat = times_times_nat(A_27,zero_zero_nat)) # label(fact_343_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 513 (all X_19 all P_3 all Q_4 power_power_int(power_power_int(X_19,P_3),Q_4) = power_power_int(X_19,times_times_nat(P_3,Q_4))) # label(fact_43_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 514 (all P all M ((M = zero_zero_nat -> power_power_nat(P,M) = one_one_nat) & (M != zero_zero_nat -> times_times_nat(P,power_power_nat(P,minus_minus_nat(M,one_one_nat))) = power_power_nat(P,M)))) # label(fact_626_power__eq__if) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 515 (all A_55 all B_16 all C_10 all D_5 plus_plus_int(plus_plus_int(A_55,C_10),plus_plus_int(B_16,D_5)) = plus_plus_int(plus_plus_int(A_55,B_16),plus_plus_int(C_10,D_5))) # label(fact_117_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 516 (all M_8 all N_34 all A_34 (ord_less_real(one_one_real,A_34) -> (ord_less_eq_real(power_power_real(A_34,M_8),power_power_real(A_34,N_34)) -> ord_less_eq_nat(M_8,N_34)))) # label(fact_298_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 517 -(all S1 (is_int(S1) -> -zcong(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_507__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]. 1.15/1.50 518 (all A_50 all C_5 plus_plus_int(A_50,C_5) = plus_plus_int(C_5,A_50)) # label(fact_132_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 519 (all X_4 all N_3 (X_4 = one_one_real | ord_less_nat(zero_zero_nat,N_3) -> dvd_dvd_real(X_4,power_power_real(X_4,N_3)))) # label(fact_553_dvd__power) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 520 (all A_15 ord_less_eq_real(zero_zero_real,power_power_real(A_15,number_number_of_nat(bit0(bit1(pls)))))) # label(fact_441_zero__le__power2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 521 (all A_1 (A_1 != zero_zero_real <-> ord_less_real(zero_zero_real,power_power_real(A_1,number_number_of_nat(bit0(bit1(pls))))))) # label(fact_451_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 522 (all Z_2 times_times_real(Z_2,number267125858f_real(bit0(bit1(pls)))) = plus_plus_real(Z_2,Z_2)) # label(fact_281_mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 523 (all X_2 all N_1 (X_2 = one_one_nat | N_1 = zero_zero_nat <-> one_one_nat = power_power_nat(X_2,N_1))) # label(fact_649_exp__eq__1) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 524 (all C all D all A_1 all B_2 (is_int(C) & is_int(B_2) & is_int(A_1) & is_int(D) -> (plus_plus_int(times_times_int(A_1,D),times_times_int(B_2,C)) != plus_plus_int(times_times_int(A_1,C),times_times_int(B_2,D)) <-> A_1 != B_2 & D != C))) # label(fact_179_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 525 (all Lx all Rx all Ry times_times_int(Rx,times_times_int(Lx,Ry)) = times_times_int(Lx,times_times_int(Rx,Ry))) # label(fact_111_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 526 (all A_43 all M_11 plus_plus_real(times_times_real(A_43,M_11),M_11) = times_times_real(plus_plus_real(A_43,one_one_real),M_11)) # label(fact_224_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 527 (all W_5 number_number_of_int(bit0(W_5)) = times_times_int(plus_plus_int(one_one_int,one_one_int),number_number_of_int(W_5))) # label(fact_268_double__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 528 (all N ord_less_eq_real(one_one_real,power_power_real(number267125858f_real(bit0(bit1(pls))),N))) # label(fact_678_two__realpow__ge__one) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 529 (all A_21 all N_22 all B_5 (ord_less_int(power_power_int(A_21,N_22),power_power_int(B_5,N_22)) -> (ord_less_eq_int(zero_zero_int,B_5) -> ord_less_int(A_21,B_5)))) # label(fact_374_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 530 (all X_1 all Y_1 all Z power_power_int(power_power_int(X_1,Y_1),Z) = power_power_int(X_1,times_times_nat(Y_1,Z))) # label(fact_47_zpower__zpower) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 531 (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_207_times__numeral__code_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 532 (all K_1 (ord_less_int(K_1,pls) <-> ord_less_int(bit0(K_1),pls))) # label(fact_149_rel__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 533 (all W all Z (ord_less_int(W,Z) -> ord_less_eq_int(plus_plus_int(W,one_one_int),Z))) # label(fact_86_zless__imp__add1__zle) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 534 (all K_1 (ord_less_int(K_1,pls) <-> ord_less_eq_int(bit1(K_1),pls))) # label(fact_80_rel__simps_I29_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 535 (all Z_1 (ord_less_eq_int(one_one_int,Z_1) <-> ord_less_int(zero_zero_int,Z_1))) # label(fact_424_int__one__le__iff__zero__less) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 536 (all Lx_1 all Rx_1 all Ry_1 times_times_real(Lx_1,times_times_real(Rx_1,Ry_1)) = times_times_real(times_times_real(Lx_1,Rx_1),Ry_1)) # label(fact_106_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 537 (all N_17 all A_17 (ord_less_real(zero_zero_real,A_17) -> (ord_less_real(A_17,one_one_real) -> ord_less_real(times_times_real(A_17,power_power_real(A_17,N_17)),power_power_real(A_17,N_17))))) # label(fact_404_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 538 (all V_5 all W_7 plus_plus_real(number267125858f_real(V_5),number267125858f_real(W_7)) = number267125858f_real(plus_plus_int(V_5,W_7))) # label(fact_247_number__of__add) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 539 (all X_2 all Y_2 all B_2 (ord_less_int(one_one_int,B_2) -> (ord_less_nat(X_2,Y_2) <-> ord_less_int(power_power_int(B_2,X_2),power_power_int(B_2,Y_2))))) # label(fact_494_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 540 (all K_1 all L_1 (ord_less_eq_int(K_1,L_1) <-> ord_less_eq_int(bit0(K_1),bit1(L_1)))) # label(fact_153_rel__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 541 (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_667_real__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 542 (all K all L bit1(L) != bit0(K)) # label(fact_195_rel__simps_I49_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 543 (all N all M (dvd_dvd_nat(N,M) -> ord_less_eq_nat(times_times_nat(number_number_of_nat(bit0(bit1(pls))),N),M) | M = N | zero_zero_nat = M)) # label(fact_622_divides__cases) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 544 (all A_33 all M_7 all N_33 (ord_less_eq_nat(M_7,N_33) -> dvd_dvd_int(power_power_int(A_33,M_7),power_power_int(A_33,N_33)))) # label(fact_313_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 545 (all B_1_1 all B_2_1 (is_int(B_1_1) -> is_int(power_power_int(B_1_1,B_2_1)))) # label(gsy_c_Power_Opower__class_Opower_000tc__Int__Oint) # label(hypothesis) # label(non_clause). [assumption]. 1.15/1.50 546 (all A_1 all B_2 all C (zero_zero_real != C -> (times_times_real(B_2,C) = times_times_real(A_1,C) <-> A_1 = B_2))) # label(fact_670_real__mult__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 547 (all N_32 all M_6 all X_14 all Y_12 (dvd_dvd_nat(X_14,Y_12) -> (ord_less_eq_nat(N_32,M_6) -> dvd_dvd_nat(power_power_nat(X_14,N_32),power_power_nat(Y_12,M_6))))) # label(fact_315_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 548 (all V_16 all V_15 (ord_less_eq_int(pls,V_15) -> (ord_less_eq_int(pls,V_16) -> times_times_nat(number_number_of_nat(V_15),number_number_of_nat(V_16)) = number_number_of_nat(times_times_int(V_15,V_16))))) # label(fact_212_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 549 (all A_16 power_power_real(A_16,one_one_nat) = A_16) # label(fact_422_power__one__right) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 550 (all V_3 (number_number_of_nat(V_3) = zero_zero_nat <-> ord_less_eq_int(V_3,pls))) # label(fact_555_eq__number__of__0) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 551 (all M_5 all A_32 all N_31 all B_9 (dvd_dvd_int(power_power_int(A_32,N_31),B_9) -> (ord_less_eq_nat(M_5,N_31) -> dvd_dvd_int(power_power_int(A_32,M_5),B_9)))) # label(fact_319_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 552 (all A_28 times_times_real(zero_zero_real,A_28) = zero_zero_real) # label(fact_339_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 553 (all K_1 (is_int(K_1) -> (K_1 = pls <-> bit0(K_1) = pls))) # label(fact_196_rel__simps_I44_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 554 (all C_1 all A all B_1 (dvd_dvd_nat(A,B_1) -> dvd_dvd_nat(times_times_nat(C_1,A),times_times_nat(C_1,B_1)))) # label(fact_629_divides__mul__l) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 555 (all L minus_minus_int(pls,bit0(L)) = bit0(minus_minus_int(pls,L))) # label(fact_601_diff__bin__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 556 (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_31_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 557 (all N_23 all A_22 (ord_less_nat(zero_zero_nat,A_22) -> ord_less_nat(zero_zero_nat,power_power_nat(A_22,N_23)))) # label(fact_367_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 558 (all A_1 all B_2 (is_int(A_1) & is_int(B_2) -> (zcong(A_1,B_2,zero_zero_int) <-> A_1 = B_2))) # label(fact_567_IntPrimes_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 559 (all N_4 all A_2 (ord_less_int(one_one_int,A_2) -> (ord_less_nat(zero_zero_nat,N_4) -> ord_less_int(one_one_int,power_power_int(A_2,N_4))))) # label(fact_550_one__less__power) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 560 (all A_7 all N_9 all N_8 (ord_less_eq_nat(N_9,N_8) -> (ord_less_eq_nat(one_one_nat,A_7) -> ord_less_eq_nat(power_power_nat(A_7,N_9),power_power_nat(A_7,N_8))))) # label(fact_487_power__increasing) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 561 (all X_3 all Y_3 (ord_less_eq_nat(X_3,Y_3) -> (Y_3 != X_3 -> ord_less_nat(X_3,Y_3)))) # label(fact_564_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 562 (all Z (is_int(Z) -> Z = times_times_int(Z,one_one_int))) # label(fact_205_zmult__1__right) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 563 (all K_1 (is_int(K_1) -> (min = K_1 <-> bit1(K_1) = min))) # label(fact_513_rel__simps_I47_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 564 (all X_2 all Y_2 (is_int(Y_2) & is_int(X_2) -> (zero_zero_int != X_2 | zero_zero_int != Y_2 <-> ord_less_int(zero_zero_int,plus_plus_int(times_times_int(X_2,X_2),times_times_int(Y_2,Y_2)))))) # label(fact_417_sum__squares__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 565 (all M all A (is_int(A) -> (ord_less_eq_int(zero_zero_int,A) -> (ord_less_int(A,M) -> (zcong(A,zero_zero_int,M) -> A = zero_zero_int))))) # label(fact_585_zcong__zless__0) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 566 (all Y_1 all X_1 (ord_less_eq_int(zero_zero_int,X_1) -> (ord_less_eq_int(zero_zero_int,Y_1) -> ord_less_eq_int(zero_zero_int,times_times_int(X_1,Y_1))))) # label(fact_693_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 567 (all A all B_1 power_power_int(minus_minus_int(A,B_1),number_number_of_nat(bit1(bit1(pls)))) = minus_minus_int(plus_plus_int(minus_minus_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))),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),power_power_int(B_1,number_number_of_nat(bit0(bit1(pls)))))),power_power_int(B_1,number_number_of_nat(bit1(bit1(pls)))))) # label(fact_616_zdiff__power3) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 568 (all M_9 times_times_nat(plus_plus_nat(one_one_nat,one_one_nat),M_9) = plus_plus_nat(M_9,M_9)) # label(fact_231_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 569 (all N_36 all A_36 (ord_less_real(one_one_real,A_36) -> ord_less_real(power_power_real(A_36,N_36),times_times_real(A_36,power_power_real(A_36,N_36))))) # label(fact_292_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 570 (all K bit1(K) = plus_plus_int(plus_plus_int(one_one_int,K),K)) # label(fact_251_Bit1__def) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 571 (exists X (is_int(X) & ord_less_int(X,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) & (all Y (is_int(Y) -> (ord_less_int(Y,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) & zcong(s1,Y,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) & ord_less_eq_int(zero_zero_int,Y) -> Y = X))) & zcong(s1,X,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) & ord_less_eq_int(zero_zero_int,X))) # label(fact_306__096EX_B_As_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_A1_A_G_A_091s1) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 572 (all A_1 (A_1 = zero_zero_real <-> power_power_real(A_1,number_number_of_nat(bit0(bit1(pls)))) = zero_zero_real)) # label(fact_439_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 573 (all N_24 all A_23 all B_6 (ord_less_eq_real(A_23,B_6) -> (ord_less_eq_real(zero_zero_real,A_23) -> ord_less_eq_real(power_power_real(A_23,N_24),power_power_real(B_6,N_24))))) # label(fact_363_power__mono) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 574 (all X_2 all Y_2 all B_2 (ord_less_nat(one_one_nat,B_2) -> (ord_less_nat(X_2,Y_2) <-> ord_less_nat(power_power_nat(B_2,X_2),power_power_nat(B_2,Y_2))))) # label(fact_493_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 575 (all A_53 all B_14 all C_8 plus_plus_nat(A_53,plus_plus_nat(B_14,C_8)) = plus_plus_nat(plus_plus_nat(A_53,B_14),C_8)) # label(fact_122_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 576 (all Z_3 plus_plus_real(Z_3,Z_3) = times_times_real(Z_3,number267125858f_real(bit0(bit1(pls))))) # label(fact_278_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 577 (all N_27 ((N_27 != zero_zero_nat -> zero_zero_real = power_power_real(zero_zero_real,N_27)) & (N_27 = zero_zero_nat -> power_power_real(zero_zero_real,N_27) = one_one_real))) # label(fact_332_power__0__left) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 578 (all C_1 all A all B_1 all M (zcong(A,B_1,M) -> (zcong(B_1,C_1,M) -> zcong(A,C_1,M)))) # label(fact_559_zcong__trans) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 579 (all K all L bit0(plus_plus_int(K,L)) = plus_plus_int(bit0(K),bit0(L))) # label(fact_203_add__Bit0__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 580 (all V_10 all W_12 all Z_7 times_times_real(number267125858f_real(times_times_int(V_10,W_12)),Z_7) = times_times_real(number267125858f_real(V_10),times_times_real(number267125858f_real(W_12),Z_7))) # label(fact_237_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 581 (all V_16 all V_15 (ord_less_eq_int(pls,V_15) -> (ord_less_eq_int(pls,V_16) -> number_number_of_int(times_times_int(V_15,V_16)) = times_times_int(number_number_of_int(V_15),number_number_of_int(V_16))))) # label(fact_213_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 582 (all A_13 all N_16 times_times_nat(power_power_nat(A_13,N_16),A_13) = times_times_nat(A_13,power_power_nat(A_13,N_16))) # label(fact_455_power__commutes) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 583 (all N_25 all A_24 (ord_less_eq_int(zero_zero_int,A_24) -> ord_less_eq_int(zero_zero_int,power_power_int(A_24,N_25)))) # label(fact_362_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 584 (all X_2 all Y_2 power_power_int(plus_plus_int(X_2,Y_2),number_number_of_nat(bit0(bit1(pls)))) = plus_plus_int(plus_plus_int(power_power_int(X_2,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y_2,number_number_of_nat(bit0(bit1(pls))))),times_times_int(times_times_int(number_number_of_int(bit0(bit1(pls))),X_2),Y_2))) # label(fact_11_power2__sum) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 585 (all N_26 all A_29 all B_7 (ord_less_real(A_29,B_7) -> (ord_less_eq_real(zero_zero_real,A_29) -> (ord_less_nat(zero_zero_nat,N_26) -> ord_less_real(power_power_real(A_29,N_26),power_power_real(B_7,N_26)))))) # label(fact_336_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 586 (all X_2 all Y_2 (Y_2 = X_2 <-> dvd_dvd_nat(X_2,Y_2) & dvd_dvd_nat(Y_2,X_2))) # label(fact_623_divides__antisym) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 587 (all X_18 X_18 = power_power_nat(X_18,one_one_nat)) # label(fact_44_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 588 (all C_1 all A all B_1 all M (zcong(A,B_1,M) -> zcong(plus_plus_int(A,C_1),plus_plus_int(B_1,C_1),M))) # label(fact_636_zcong__shift) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 589 (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_143_zadd__assoc) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 590 (all W ord_less_eq_real(W,W)) # label(fact_690_real__le__refl) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 591 (all Z_5 plus_plus_int(Z_5,Z_5) = times_times_int(number_number_of_int(bit0(bit1(pls))),Z_5)) # label(fact_275_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 592 (all A_21 all N_22 all B_5 (ord_less_real(power_power_real(A_21,N_22),power_power_real(B_5,N_22)) -> (ord_less_eq_real(zero_zero_real,B_5) -> ord_less_real(A_21,B_5)))) # label(fact_372_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 593 (all B all Q_1 all R_2 (ord_less_eq_int(zero_zero_int,plus_plus_int(times_times_int(B,Q_1),R_2)) -> (ord_less_int(R_2,B) -> (ord_less_int(zero_zero_int,B) -> ord_less_eq_int(zero_zero_int,Q_1))))) # label(fact_679_q__pos__lemma) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 594 (all X_9 all Y_7 (is_int(Y_7) & is_int(X_9) -> (power_power_int(X_9,number_number_of_nat(bit0(bit1(pls)))) = power_power_int(Y_7,number_number_of_nat(bit0(bit1(pls)))) -> (ord_less_eq_int(zero_zero_int,X_9) -> (ord_less_eq_int(zero_zero_int,Y_7) -> X_9 = Y_7))))) # label(fact_448_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 595 (all K_1 (ord_less_eq_int(K_1,min) <-> ord_less_eq_int(bit0(K_1),min))) # label(fact_533_rel__simps_I28_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 596 (all W_3 ((zero_zero_nat != number_number_of_nat(W_3) -> power_power_int(zero_zero_int,number_number_of_nat(W_3)) = zero_zero_int) & (zero_zero_nat = number_number_of_nat(W_3) -> one_one_int = power_power_int(zero_zero_int,number_number_of_nat(W_3))))) # label(fact_583_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 597 (all X_5 power_power_real(X_5,zero_zero_nat) = one_one_real) # label(fact_536_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 598 (all K_1 (ord_less_eq_int(pls,K_1) <-> ord_less_eq_int(pls,bit1(K_1)))) # label(fact_151_rel__simps_I22_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 599 (all X_1 all Y_1 all Z plus_plus_int(Y_1,plus_plus_int(X_1,Z)) = plus_plus_int(X_1,plus_plus_int(Y_1,Z))) # label(fact_144_zadd__left__commute) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 600 (all X_5 one_one_nat = power_power_nat(X_5,zero_zero_nat)) # label(fact_537_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 601 (all A_1 (ord_less_real(plus_plus_real(A_1,A_1),zero_zero_real) <-> ord_less_real(A_1,zero_zero_real))) # label(fact_381_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 602 (all N_27 ((N_27 = zero_zero_nat -> one_one_int = power_power_int(zero_zero_int,N_27)) & (zero_zero_nat != N_27 -> power_power_int(zero_zero_int,N_27) = zero_zero_int))) # label(fact_334_power__0__left) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 603 (all X_21 times_times_real(X_21,X_21) = power_power_real(X_21,number_number_of_nat(bit0(bit1(pls))))) # label(fact_20_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 604 (all Z_5 plus_plus_real(Z_5,Z_5) = times_times_real(number267125858f_real(bit0(bit1(pls))),Z_5)) # label(fact_273_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 605 (all A_1 all N_1 (is_int(A_1) -> (zero_zero_int = power_power_int(A_1,N_1) <-> zero_zero_int = A_1 & zero_zero_nat != N_1))) # label(fact_311_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 606 (all K_1 all L_1 (ord_less_eq_int(bit0(K_1),bit0(L_1)) <-> ord_less_eq_int(K_1,L_1))) # label(fact_72_rel__simps_I31_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 607 (all M (ord_less_int(number_number_of_int(bit0(bit1(pls))),M) -> -zcong(one_one_int,number_number_of_int(min),M))) # label(fact_619_one__not__neg__one__mod__m) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 608 (all A_38 (is_int(A_38) -> A_38 = times_times_int(A_38,number_number_of_int(bit1(pls))))) # label(fact_257_mult__numeral__1__right) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 609 (all V_5 all W_7 number_number_of_int(plus_plus_int(V_5,W_7)) = plus_plus_int(number_number_of_int(V_5),number_number_of_int(W_7))) # label(fact_248_number__of__add) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 610 (all A_12 all B_3 all N_15 times_times_real(power_power_real(A_12,N_15),power_power_real(B_3,N_15)) = power_power_real(times_times_real(A_12,B_3),N_15)) # label(fact_459_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 611 (all N all A all B_1 all P (zprime(P) -> (-dvd_dvd_int(P,B_1) -> (dvd_dvd_int(power_power_int(P,N),times_times_int(A,B_1)) -> dvd_dvd_int(power_power_int(P,N),A))))) # label(fact_408_zprime__power__zdvd__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 612 (all X_9 all Y_7 (power_power_real(X_9,number_number_of_nat(bit0(bit1(pls)))) = power_power_real(Y_7,number_number_of_nat(bit0(bit1(pls)))) -> (ord_less_eq_real(zero_zero_real,X_9) -> (ord_less_eq_real(zero_zero_real,Y_7) -> X_9 = Y_7)))) # label(fact_446_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 613 (all A_54 all B_15 all C_9 plus_plus_int(plus_plus_int(A_54,C_9),B_15) = plus_plus_int(plus_plus_int(A_54,B_15),C_9)) # label(fact_120_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 614 (all K all I all J (ord_less_eq_real(I,J) -> (ord_less_eq_real(J,K) -> ord_less_eq_real(I,K)))) # label(fact_688_real__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 615 (all A_19 all N_19 all N_18 (ord_less_nat(N_19,N_18) -> (ord_less_real(zero_zero_real,A_19) -> (ord_less_real(A_19,one_one_real) -> ord_less_real(power_power_real(A_19,N_18),power_power_real(A_19,N_19)))))) # label(fact_378_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 616 (all A_19 all N_19 all N_18 (ord_less_nat(N_19,N_18) -> (ord_less_nat(zero_zero_nat,A_19) -> (ord_less_nat(A_19,one_one_nat) -> ord_less_nat(power_power_nat(A_19,N_18),power_power_nat(A_19,N_19)))))) # label(fact_379_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 617 (all W_1 all Y_2 all X_2 all Z_1 (plus_plus_real(times_times_real(W_1,Y_2),times_times_real(X_2,Z_1)) = plus_plus_real(times_times_real(W_1,Z_1),times_times_real(X_2,Y_2)) <-> Z_1 = Y_2 | W_1 = X_2)) # label(fact_168_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 618 (all N_28 all A_30 (is_int(A_30) -> (zero_zero_int != A_30 -> zero_zero_int != power_power_int(A_30,N_28)))) # label(fact_331_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 619 (all A_28 times_times_nat(zero_zero_nat,A_28) = zero_zero_nat) # label(fact_340_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 620 (all W_15 power_power_nat(number_number_of_nat(W_15),number_number_of_nat(bit0(bit1(pls)))) = times_times_nat(number_number_of_nat(W_15),number_number_of_nat(W_15))) # label(fact_12_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 621 (all B_1_1 all B_2_1 (is_int(B_1_1) & is_int(B_2_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]. 1.15/1.50 622 (all K all M all N (is_int(K) -> (dvd_dvd_int(times_times_int(K,M),times_times_int(K,N)) -> (zero_zero_int != K -> dvd_dvd_int(M,N))))) # label(fact_326_zdvd__mult__cancel) # label(axiom) # label(non_clause). [assumption]. 1.15/1.50 623 (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]. 1.15/1.50 624 (all P_1 (is_int(P_1) -> (zprime(P_1) <-> ord_less_int(one_one_int,P_1) & (all M_1 (is_int(M_1) -> (ord_less_eq_int(zero_zero_int,M_1) & dvd_dvd_int(M_1,P_1) -> M_1 = one_one_int | P_1 = M_1)))))) # label(fact_506_zprime__def) # label(axiom) # label(non_clause). [assumption]. 1.15/1.51 625 (all Y_1 all X_1 (ord_less_eq_int(zero_zero_int,X_1) -> (ord_less_eq_int(zero_zero_int,Y_1) -> ord_less_eq_int(zero_zero_int,plus_plus_int(X_1,Y_1))))) # label(fact_694_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.51 626 (all N_29 all X_13 all Y_11 (dvd_dvd_real(X_13,Y_11) -> dvd_dvd_real(power_power_real(X_13,N_29),power_power_real(Y_11,N_29)))) # label(fact_329_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 1.15/1.51 627 (all K_1 (ord_less_int(min,K_1) <-> ord_less_int(min,bit1(K_1)))) # label(fact_524_rel__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.51 628 (all A_1 all N_1 (power_power_real(A_1,N_1) = zero_zero_real <-> N_1 != zero_zero_nat & zero_zero_real = A_1)) # label(fact_309_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.15/1.51 629 (all X_8 all Y_6 (ord_less_nat(power_power_nat(X_8,number_number_of_nat(bit0(bit1(pls)))),power_power_nat(Y_6,number_number_of_nat(bit0(bit1(pls))))) -> (ord_less_eq_nat(zero_zero_nat,Y_6) -> ord_less_nat(X_8,Y_6)))) # label(fact_471_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.15/1.51 630 (all L_1 (is_int(L_1) -> (min = bit1(L_1) <-> L_1 = min))) # label(fact_514_rel__simps_I43_J) # label(axiom) # label(non_clause). [assumption]. 1.15/1.51 631 (all Ma all N_1 all A_1 (ord_less_int(one_one_int,A_1) -> (N_1 = Ma <-> power_power_int(A_1,N_1) = power_power_int(A_1,Ma)))) # label(fact_491_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 1.15/1.51 632 (all V_11 all B_10 all C_3 plus_plus_real(times_times_real(number267125858f_real(V_11),B_10),times_times_real(number267125858f_real(V_11),C_3)) = times_times_real(number267125858f_real(V_11),plus_plus_real(B_10,C_3))) # label(fact_221_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.15/1.51 633 (all M_2 all N_7 all A_6 (ord_less_real(one_one_real,A_6) -> (ord_less_real(power_power_real(A_6,M_2),power_power_real(A_6,N_7)) -> ord_less_nat(M_2,N_7)))) # label(fact_495_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 1.15/1.51 634 (all A all B_1 all P all Q times_times_int(twoSqu140629262sum2sq(product_Pair_int_int(A,B_1)),twoSqu140629262sum2sq(product_Pair_int_int(P,Q))) = twoSqu140629262sum2sq(product_Pair_int_int(plus_plus_int(times_times_int(A,P),times_times_int(B_1,Q)),minus_minus_int(times_times_int(A,Q),times_times_int(B_1,P))))) # label(fact_611_mult__sum2sq) # label(axiom) # label(non_clause). [assumption]. 1.15/1.51 635 -(exists X exists Y plus_plus_int(power_power_int(X,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y,number_number_of_nat(bit0(bit1(pls))))) = plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) # label(conj_0) # label(negated_conjecture) # label(non_clause). [assumption]. 1.15/1.51 1.15/1.51 ============================== end of process non-clausal formulas === 1.15/1.51 1.15/1.51 ============================== PROCESS INITIAL CLAUSES =============== 1.15/1.51 1.15/1.51 ============================== PREDICATE ELIMINATION ================= 1.15/1.51 636 zcong(A,zero_zero_int,B) | -quadRes(B,A) | legendre(A,B) = one_one_int # label(fact_621_Legendre__def) # label(axiom). [clausify(86)]. 1.15/1.51 637 zcong(A,zero_zero_int,B) | quadRes(B,A) | legendre(A,B) = number_number_of_int(min) # label(fact_621_Legendre__def) # label(axiom). [clausify(86)]. 1.15/1.51 Derived: zcong(A,zero_zero_int,B) | legendre(A,B) = one_one_int | zcong(A,zero_zero_int,B) | legendre(A,B) = number_number_of_int(min). [resolve(636,b,637,b)]. 1.15/1.51 638 quadRes(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),number_number_of_int(min)) | legendre(number_number_of_int(min),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) != one_one_int # label(fact_591__096_126_AQuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_A_061_061_062_ALegendre_A_N) # label(axiom). [clausify(93)]. 1.15/1.51 639 -quadRes(A,B) | is_int(f2(A,B)) # label(fact_658_QuadRes__def) # label(axiom). [clausify(253)]. 1.15/1.59 Derived: is_int(f2(A,B)) | zcong(B,zero_zero_int,A) | legendre(B,A) = number_number_of_int(min). [resolve(639,a,637,b)]. 1.15/1.59 Derived: is_int(f2(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),number_number_of_int(min))) | legendre(number_number_of_int(min),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) != one_one_int. [resolve(639,a,638,a)]. 1.15/1.59 640 -quadRes(A,B) | zcong(power_power_int(f2(A,B),number_number_of_nat(bit0(bit1(pls)))),B,A) # label(fact_658_QuadRes__def) # label(axiom). [clausify(253)]. 1.15/1.59 Derived: zcong(power_power_int(f2(A,B),number_number_of_nat(bit0(bit1(pls)))),B,A) | zcong(B,zero_zero_int,A) | legendre(B,A) = number_number_of_int(min). [resolve(640,a,637,b)]. 1.15/1.59 Derived: zcong(power_power_int(f2(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),number_number_of_int(min)),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)) | legendre(number_number_of_int(min),plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) != one_one_int. [resolve(640,a,638,a)]. 1.15/1.59 641 quadRes(A,B) | -is_int(C) | -zcong(power_power_int(C,number_number_of_nat(bit0(bit1(pls)))),B,A) # label(fact_658_QuadRes__def) # label(axiom). [clausify(253)]. 1.15/1.59 Derived: -is_int(A) | -zcong(power_power_int(A,number_number_of_nat(bit0(bit1(pls)))),B,C) | zcong(B,zero_zero_int,C) | legendre(B,C) = one_one_int. [resolve(641,a,636,b)]. 1.15/1.59 Derived: -is_int(A) | -zcong(power_power_int(A,number_number_of_nat(bit0(bit1(pls)))),B,C) | is_int(f2(C,B)). [resolve(641,a,639,a)]. 1.15/1.59 Derived: -is_int(A) | -zcong(power_power_int(A,number_number_of_nat(bit0(bit1(pls)))),B,C) | zcong(power_power_int(f2(C,B),number_number_of_nat(bit0(bit1(pls)))),B,C). [resolve(641,a,640,a)]. 1.15/1.59 642 quadRes(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),number_number_of_int(min)) # label(fact_587__096QuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_096) # label(axiom). [assumption]. 1.15/1.59 Derived: zcong(number_number_of_int(min),zero_zero_int,plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),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)) = one_one_int. [resolve(642,a,636,b)]. 1.15/1.59 Derived: is_int(f2(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),number_number_of_int(min))). [resolve(642,a,639,a)]. 1.15/1.59 Derived: zcong(power_power_int(f2(plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int),number_number_of_int(min)),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)). [resolve(642,a,640,a)]. 1.15/1.59 1.15/1.59 ============================== end predicate elimination ============= 1.15/1.59 1.15/1.59 Auto_denials: (non-Horn, no changes). 1.15/1.59 1.15/1.59 Term ordering decisions: 1.15/1.59 Function symbol KB weights: pls=1. zero_zero_int=1. one_one_int=1. zero_zero_real=1. zero_zero_nat=1. min=1. one_one_nat=1. one_one_real=1. m=1. s=1. t=1. s1=1. int=1. c1=1. c2=1. c3=1. c4=1. c5=1. c6=1. c7=1. c8=1. times_times_int=1. plus_plus_int=1. power_power_int=1. times_times_real=1. times_times_nat=1. power_power_real=1. power_power_nat=1. plus_plus_real=1. plus_plus_nat=1. minus_minus_int=1. legendre=1. minus_minus_nat=1. product_Pair_int_int=1. minus_minus_real=1. f2=1. f3=1. bit1=1. bit0=1. number_number_of_nat=1. number_number_of_int=1. number267125858f_real=1. twoSqu140629262sum2sq=1. undefined_int=1. f4=1. f1=1. 1.15/1.59 % back CAC tautology: 1445 plus_plus_nat(A,plus_plus_nat(B,C)) = plus_plus_nat(A,plus_plus_nat(C,B)). [copy(1444),rewrite([992(2),992(4)])]. 1.15/1.59 % back CAC tautology: 1134 power_power_real(A,times_times_nat(number_number_of_nat(bit0(bit1(pls))),B)) = power_power_real(A,times_times_nat(B,number_number_of_nat(bit0(bit1(pls))))). [copy(1133),rewrite([946(6)]),flip(a)]. 1.15/1.59 % back CAC tautology: 1131 times_times_nat(A,times_times_nat(B,times_times_nat(C,D))) = times_times_nat(C,times_times_nat(A,times_times_nat(B,D))). [copy(1130),rewrite([997(3),997(5)])]. 1.82/2.08 % back CAC tautology: 975 times_times_nat(times_times_nat(A,B),times_times_nat(C,D)) = times_times_nat(times_times_nat(A,C),times_times_nat(B,D)) # label(fact_92_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom). [clausify(185)]. 1.82/2.08 % back CAC tautology: 970 power_power_int(A,times_times_nat(number_number_of_nat(bit0(bit1(pls))),B)) = power_power_int(A,times_times_nat(B,number_number_of_nat(bit0(bit1(pls))))). [copy(969),rewrite([659(6)]),flip(a)]. 1.82/2.08 % back CAC tautology: 779 times_times_real(power_power_real(A,B),A) = times_times_real(A,power_power_real(A,B)) # label(fact_456_power__commutes) # label(axiom). [clausify(84)]. 1.82/2.08 1.82/2.08 ============================== end of process initial clauses ======== 1.82/2.08 1.82/2.08 ============================== CLAUSES FOR SEARCH ==================== 1.82/2.08 1.82/2.08 ============================== end of clauses for search ============= 1.82/2.08 1.82/2.08 ============================== SEARCH ================================ 1.82/2.08 1.82/2.08 % Starting search at 0.26 seconds. 1.82/2.08 1.82/2.08 ============================== PROOF ================================= 1.82/2.08 % SZS status Theorem 1.82/2.08 % SZS output start Refutation 1.82/2.08 1.82/2.08 % Proof 1 at 0.68 (+ 0.01) seconds. 1.82/2.08 % Length of proof is 33. 1.82/2.08 % Level of proof is 6. 1.82/2.08 % Maximum clause weight is 26.000. 1.82/2.08 % Given clauses 379. 1.82/2.08 1.82/2.08 20 (all A_56 all B_17 times_times_int(B_17,A_56) = times_times_int(A_56,B_17)) # label(fact_114_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 1.82/2.08 132 (all A_53 all B_14 all C_8 plus_plus_int(plus_plus_int(A_53,B_14),C_8) = plus_plus_int(A_53,plus_plus_int(B_14,C_8))) # label(fact_123_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 1.82/2.08 228 (all A_51 all C_6 all D_3 plus_plus_int(C_6,plus_plus_int(A_51,D_3)) = plus_plus_int(A_51,plus_plus_int(C_6,D_3))) # label(fact_129_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 1.82/2.08 272 (all X_3 all Y_3 (is_int(Y_3) & is_int(X_3) -> (ord_less_eq_int(X_3,Y_3) -> (X_3 != Y_3 -> ord_less_int(X_3,Y_3))))) # label(fact_565_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 1.82/2.08 283 t = one_one_int -> (exists X exists Y (plus_plus_int(power_power_int(X,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y,number_number_of_nat(bit0(bit1(pls))))) = plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int) & is_int(Y) & is_int(X))) # label(fact_1__096t_A_061_A1_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06) # label(axiom) # label(non_clause). [assumption]. 1.82/2.08 289 (all Z all W plus_plus_int(Z,W) = plus_plus_int(W,Z)) # label(fact_145_zadd__commute) # label(axiom) # label(non_clause). [assumption]. 1.82/2.08 354 ord_less_int(one_one_int,t) -> (exists X exists Y (is_int(Y) & plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int) = plus_plus_int(power_power_int(X,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y,number_number_of_nat(bit0(bit1(pls))))) & is_int(X))) # label(fact_2__0961_A_060_At_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06) # label(axiom) # label(non_clause). [assumption]. 1.82/2.08 373 (all K plus_plus_int(K,K) = bit0(K)) # label(fact_204_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 1.82/2.08 424 (all K all L bit1(plus_plus_int(K,L)) = plus_plus_int(bit0(K),bit1(L))) # label(fact_250_add__Bit0__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.82/2.08 635 -(exists X exists Y plus_plus_int(power_power_int(X,number_number_of_nat(bit0(bit1(pls)))),power_power_int(Y,number_number_of_nat(bit0(bit1(pls))))) = plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int)) # label(conj_0) # label(negated_conjecture) # label(non_clause). [assumption]. 1.82/2.08 676 times_times_int(A,B) = times_times_int(B,A) # label(fact_114_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom). [clausify(20)]. 1.82/2.08 677 is_int(t) # label(gsy_v_t____) # label(axiom). [assumption]. 1.82/2.08 879 plus_plus_int(plus_plus_int(A,B),C) = plus_plus_int(A,plus_plus_int(B,C)) # label(fact_123_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom). [clausify(132)]. 1.82/2.08 1049 plus_plus_int(A,plus_plus_int(B,C)) = plus_plus_int(B,plus_plus_int(A,C)) # label(fact_129_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom). [clausify(228)]. 1.82/2.08 1140 -is_int(A) | -is_int(B) | -ord_less_eq_int(B,A) | A = B | ord_less_int(B,A) # label(fact_565_order__le__neq__implies__less) # label(axiom). [clausify(272)]. 1.82/2.08 1170 t != one_one_int | plus_plus_int(power_power_int(c1,number_number_of_nat(bit0(bit1(pls)))),power_power_int(c2,number_number_of_nat(bit0(bit1(pls))))) = plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int) # label(fact_1__096t_A_061_A1_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06) # label(axiom). [clausify(283)]. 1.82/2.08 1171 t != one_one_int | plus_plus_int(power_power_int(c1,number_number_of_nat(bit0(bit1(pls)))),power_power_int(c2,number_number_of_nat(bit0(bit1(pls))))) = plus_plus_int(times_times_int(m,number_number_of_int(bit0(bit0(bit1(pls))))),one_one_int). [copy(1170),rewrite([676(23)])]. 1.82/2.08 1187 plus_plus_int(A,B) = plus_plus_int(B,A) # label(fact_145_zadd__commute) # label(axiom). [clausify(289)]. 1.82/2.08 1327 -ord_less_int(one_one_int,t) | plus_plus_int(power_power_int(c3,number_number_of_nat(bit0(bit1(pls)))),power_power_int(c4,number_number_of_nat(bit0(bit1(pls))))) = plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int) # label(fact_2__0961_A_060_At_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06) # label(axiom). [clausify(354)]. 1.82/2.08 1328 -ord_less_int(one_one_int,t) | plus_plus_int(power_power_int(c3,number_number_of_nat(bit0(bit1(pls)))),power_power_int(c4,number_number_of_nat(bit0(bit1(pls))))) = plus_plus_int(one_one_int,times_times_int(m,number_number_of_int(bit0(bit0(bit1(pls)))))). [copy(1327),rewrite([676(23),1187(25)])]. 1.82/2.08 1367 plus_plus_int(A,A) = bit0(A) # label(fact_204_Bit0__def) # label(axiom). [clausify(373)]. 1.82/2.08 1368 bit0(A) = plus_plus_int(A,A). [copy(1367),flip(a)]. 1.82/2.08 1475 plus_plus_int(bit0(A),bit1(B)) = bit1(plus_plus_int(A,B)) # label(fact_250_add__Bit0__Bit1) # label(axiom). [clausify(424)]. 1.82/2.08 1476 plus_plus_int(A,plus_plus_int(A,bit1(B))) = bit1(plus_plus_int(A,B)). [copy(1475),rewrite([1368(1),879(3)])]. 1.82/2.08 1551 plus_plus_nat(one_one_nat,one_one_nat) = number_number_of_nat(bit0(bit1(pls))) # label(fact_284_semiring__one__add__one__is__two) # label(axiom). [assumption]. 1.82/2.08 1552 number_number_of_nat(plus_plus_int(bit1(pls),bit1(pls))) = plus_plus_nat(one_one_nat,one_one_nat). [copy(1551),rewrite([1368(6)]),flip(a)]. 1.82/2.08 1707 is_int(one_one_int) # label(gsy_c_Groups_Oone__class_Oone_000tc__Int__Oint) # label(hypothesis). [assumption]. 1.82/2.08 1771 ord_less_eq_int(one_one_int,t) # label(fact_0_tpos) # label(axiom). [assumption]. 1.82/2.08 1932 plus_plus_int(power_power_int(A,number_number_of_nat(bit0(bit1(pls)))),power_power_int(B,number_number_of_nat(bit0(bit1(pls))))) != plus_plus_int(times_times_int(number_number_of_int(bit0(bit0(bit1(pls)))),m),one_one_int) # label(conj_0) # label(negated_conjecture). [clausify(635)]. 1.82/2.08 1933 plus_plus_int(power_power_int(A,plus_plus_nat(one_one_nat,one_one_nat)),power_power_int(B,plus_plus_nat(one_one_nat,one_one_nat))) != plus_plus_int(one_one_int,times_times_int(m,number_number_of_int(plus_plus_int(bit1(pls),bit1(plus_plus_int(pls,bit1(pls))))))). [copy(1932),rewrite([1368(3),1552(6),1368(7),1552(10),1368(12),1368(15),1049(20),1187(19),1476(19),1187(15),676(20),1187(22)])]. 1.82/2.08 2078 t != one_one_int. [back_rewrite(1171),rewrite([1368(7),1552(10),1368(12),1552(15),1368(18),1368(21),1049(26),1187(25),1476(25),1187(21),1187(27)]),flip(b),unit_del(b(flip),1933)]. 1.82/2.08 2111 -ord_less_int(one_one_int,t). [back_rewrite(1328),rewrite([1368(7),1552(10),1368(12),1552(15),1368(19),1368(22),1049(27),1187(26),1476(26),1187(22)]),flip(b),unit_del(b(flip),1933)]. 1.82/2.08 4800 $F. [resolve(1771,a,1140,c),unit_del(a,677),unit_del(b,1707),unit_del(c,2078),unit_del(d,2111)]. 1.82/2.08 1.82/2.08 % SZS output end Refutation 1.82/2.08 ============================== end of proof ========================== 1.82/2.08 1.82/2.08 ============================== STATISTICS ============================ 1.82/2.08 1.82/2.08 Given=379. Generated=7694. Kept=3617. proofs=1. 1.82/2.08 Usable=378. Sos=2646. Demods=326. Limbo=6, Disabled=1555. Hints=0. 1.82/2.08 Megabytes=6.68. 1.82/2.08 User_CPU=0.68, System_CPU=0.01, Wall_clock=1. 1.82/2.08 1.82/2.08 ============================== end of statistics ===================== 1.82/2.08 1.82/2.08 ============================== end of search ========================= 1.82/2.08 1.82/2.08 THEOREM PROVED 1.82/2.08 % SZS status Theorem 1.82/2.08 1.82/2.08 Exiting with 1 proof. 1.82/2.08 1.82/2.08 Process 12013 exit (max_proofs) Tue Jul 13 16:45:51 2021 1.82/2.08 Prover9 interrupted 1.82/2.09 EOF