0.06/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.06/0.12 % Command : tptp2X_and_run_prover9 %d %s 0.12/0.33 % Computer : n012.cluster.edu 0.12/0.33 % Model : x86_64 x86_64 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.12/0.33 % Memory : 8042.1875MB 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64 0.12/0.33 % CPULimit : 960 0.12/0.33 % WCLimit : 120 0.12/0.33 % DateTime : Tue Aug 9 04:19:59 EDT 2022 0.12/0.33 % CPUTime : 1.07/1.35 ============================== Prover9 =============================== 1.07/1.35 Prover9 (32) version 2009-11A, November 2009. 1.07/1.35 Process 18007 was started by sandbox2 on n012.cluster.edu, 1.07/1.35 Tue Aug 9 04:20:00 2022 1.07/1.35 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 960 -f /tmp/Prover9_17854_n012.cluster.edu". 1.07/1.35 ============================== end of head =========================== 1.07/1.35 1.07/1.35 ============================== INPUT ================================= 1.07/1.35 1.07/1.35 % Reading from file /tmp/Prover9_17854_n012.cluster.edu 1.07/1.35 1.07/1.35 set(prolog_style_variables). 1.07/1.35 set(auto2). 1.07/1.35 % set(auto2) -> set(auto). 1.07/1.35 % set(auto) -> set(auto_inference). 1.07/1.35 % set(auto) -> set(auto_setup). 1.07/1.35 % set(auto_setup) -> set(predicate_elim). 1.07/1.35 % set(auto_setup) -> assign(eq_defs, unfold). 1.07/1.35 % set(auto) -> set(auto_limits). 1.07/1.35 % set(auto_limits) -> assign(max_weight, "100.000"). 1.07/1.35 % set(auto_limits) -> assign(sos_limit, 20000). 1.07/1.35 % set(auto) -> set(auto_denials). 1.07/1.35 % set(auto) -> set(auto_process). 1.07/1.35 % set(auto2) -> assign(new_constants, 1). 1.07/1.35 % set(auto2) -> assign(fold_denial_max, 3). 1.07/1.35 % set(auto2) -> assign(max_weight, "200.000"). 1.07/1.35 % set(auto2) -> assign(max_hours, 1). 1.07/1.35 % assign(max_hours, 1) -> assign(max_seconds, 3600). 1.07/1.35 % set(auto2) -> assign(max_seconds, 0). 1.07/1.35 % set(auto2) -> assign(max_minutes, 5). 1.07/1.35 % assign(max_minutes, 5) -> assign(max_seconds, 300). 1.07/1.35 % set(auto2) -> set(sort_initial_sos). 1.07/1.35 % set(auto2) -> assign(sos_limit, -1). 1.07/1.35 % set(auto2) -> assign(lrs_ticks, 3000). 1.07/1.35 % set(auto2) -> assign(max_megs, 400). 1.07/1.35 % set(auto2) -> assign(stats, some). 1.07/1.35 % set(auto2) -> clear(echo_input). 1.07/1.35 % set(auto2) -> set(quiet). 1.07/1.35 % set(auto2) -> clear(print_initial_clauses). 1.07/1.35 % set(auto2) -> clear(print_given). 1.07/1.35 assign(lrs_ticks,-1). 1.07/1.35 assign(sos_limit,10000). 1.07/1.35 assign(order,kbo). 1.07/1.35 set(lex_order_vars). 1.07/1.35 clear(print_given). 1.07/1.35 1.07/1.35 % formulas(sos). % not echoed (655 formulas) 1.07/1.35 1.07/1.35 ============================== end of input ========================== 1.07/1.35 1.07/1.35 % From the command line: assign(max_seconds, 960). 1.07/1.35 1.07/1.35 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 1.07/1.35 1.07/1.35 % Formulas that are not ordinary clauses: 1.07/1.35 1 (all B_1_1 all B_2_1 ti(int,legendre(B_1_1,B_2_1)) = legendre(B_1_1,B_2_1)) # label(tsy_c_Residues_OLegendre_res) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 2 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))) # 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]. 1.07/1.35 3 (all A_1 all B plus_plus(int,plus_plus(int,power_power(int,A_1,number_number_of(nat,bit0(bit1(pls)))),times_times(int,times_times(int,number_number_of(int,bit0(bit1(pls))),A_1),B)),power_power(int,B,number_number_of(nat,bit0(bit1(pls))))) = power_power(int,plus_plus(int,A_1,B),number_number_of(nat,bit0(bit1(pls))))) # label(fact_7_zadd__power2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 4 (all X_a (linorder(X_a) & number(X_a) -> (all V_2 all W_1 (ord_less_eq(X_a,number_number_of(X_a,V_2),number_number_of(X_a,W_1)) <-> -ord_less(X_a,number_number_of(X_a,W_1),number_number_of(X_a,V_2)))))) # label(fact_29_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 5 (all K1 all K2 (ord_less_eq(int,bit1(K1),bit0(K2)) <-> ord_less(int,K1,K2))) # label(fact_57_less__eq__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 6 (all X_a (comm_semiring_1(X_a) -> (all X_1 ti(X_a,X_1) = power_power(X_a,X_1,one_one(nat))))) # label(fact_27_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 7 (all C all A_1 all B (dvd_dvd(nat,A_1,B) -> dvd_dvd(nat,times_times(nat,A_1,C),times_times(nat,B,C)))) # label(fact_391_divides__mul__r) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 8 (all X_2 all Y_2 (zero_zero(real) = X_2 & zero_zero(real) = 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))))))) # label(fact_420_realpow__two__sum__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 9 (all M (zprime(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))) # label(fact_379_Legendre__1mod4) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 10 (all Z zero_zero(int) != plus_plus(int,plus_plus(int,one_one(int),Z),Z)) # label(fact_229_odd__nonzero) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 11 (all N_1 all Ma (Ma = one_one(nat) & one_one(nat) = N_1 <-> times_times(nat,N_1,Ma) = one_one(nat))) # label(fact_393_nat__mult__eq__one) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 12 (all K all L bit1(plus_plus(int,K,L)) = plus_plus(int,bit1(K),bit0(L))) # label(fact_149_add__Bit1__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 13 (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_366_zle__diff1__eq) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 14 (all X_2 all Y_2 (ord_less_eq(real,minus_minus(real,X_2,Y_2),zero_zero(real)) <-> ord_less_eq(real,X_2,Y_2))) # label(fact_423_real__le__eq__diff) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 15 (all K_1 (ord_less(int,K_1,pls) <-> ord_less_eq(int,bit1(K_1),pls))) # label(fact_55_rel__simps_I29_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 16 (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_459_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 17 (all X_a (ordere453448008miring(X_a) -> (all B all A_1 (ord_less_eq(X_a,zero_zero(X_a),A_1) -> (ord_less_eq(X_a,zero_zero(X_a),B) -> ord_less_eq(X_a,zero_zero(X_a),times_times(X_a,A_1,B))))))) # label(fact_493_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 18 (all X_a (number_ring(X_a) -> (all W number_number_of(X_a,plus_plus(int,bit1(pls),W)) = plus_plus(X_a,one_one(X_a),number_number_of(X_a,W))))) # label(fact_16_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 19 (all K all L bit0(minus_minus(int,K,L)) = minus_minus(int,bit1(K),bit1(L))) # label(fact_359_diff__bin__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 20 (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_50_zadd__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 21 (all X_a (number_ring(X_a) -> (all Z plus_plus(X_a,Z,Z) = times_times(X_a,number_number_of(X_a,bit0(bit1(pls))),Z)))) # label(fact_164_mult__2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 22 (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_399_divides__exp) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 23 (all X_a (no_zero_divisors(X_a) -> (all B all A_1 (zero_zero(X_a) != ti(X_a,A_1) -> (ti(X_a,B) != zero_zero(X_a) -> times_times(X_a,A_1,B) != zero_zero(X_a)))))) # label(fact_472_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 24 (all L_1 (bit0(L_1) = pls <-> pls = ti(int,L_1))) # label(fact_119_rel__simps_I38_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 25 (all K plus_plus(int,K,pls) = ti(int,K)) # label(fact_123_add__Pls__right) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 26 (all X_a (comm_semiring_1(X_a) -> (all X_1 all P all Q power_power(X_a,X_1,times_times(nat,P,Q)) = power_power(X_a,power_power(X_a,X_1,P),Q)))) # label(fact_26_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 27 (all K_1 (ord_less(int,K_1,pls) <-> ord_less(int,bit0(K_1),pls))) # label(fact_93_rel__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 28 (all B_1_1 all B_2_1 all X_a (dvd(X_a) -> (dvd_dvd(X_a,B_1_1,ti(X_a,B_2_1)) <-> dvd_dvd(X_a,B_1_1,B_2_1)))) # label(tsy_c_Rings_Odvd__class_Odvd_arg2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 29 (all Z_2 all Z all W_2 all W (ord_less(int,W_2,W) -> (ord_less_eq(int,Z_2,Z) -> ord_less(int,plus_plus(int,W_2,Z_2),plus_plus(int,W,Z))))) # label(fact_32_zadd__zless__mono) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 30 (all C_1 all D_1 all A_2 all B_1 all Ma (zcong(A_2,B_1,Ma) -> (zcong(C_1,times_times(int,B_1,D_1),Ma) <-> zcong(C_1,times_times(int,A_2,D_1),Ma)))) # label(fact_396_zcong__zmult__prop1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 31 (all V_2 (zero_zero(nat) = number_number_of(nat,V_2) <-> ord_less_eq(int,V_2,pls))) # label(fact_319_eq__number__of__0) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 32 (all Ma all X_2 (quadRes(Ma,X_2) <-> (exists Y zcong(power_power(int,Y,number_number_of(nat,bit0(bit1(pls)))),X_2,Ma)))) # label(fact_419_QuadRes__def) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 33 (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_49_less__eq__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 34 (all X_a (number_ring(X_a) -> (all V all W all Z plus_plus(X_a,number_number_of(X_a,plus_plus(int,V,W)),Z) = plus_plus(X_a,number_number_of(X_a,V),plus_plus(X_a,number_number_of(X_a,W),Z))))) # label(fact_146_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 35 (all X_a (number_ring(X_a) -> (all V all W minus_minus(X_a,number_number_of(X_a,V),number_number_of(X_a,W)) = number_number_of(X_a,minus_minus(int,V,W))))) # label(fact_355_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 36 (all A_1 all B plus_plus(int,plus_plus(int,plus_plus(int,power_power(int,A_1,number_number_of(nat,bit1(bit1(pls)))),times_times(int,times_times(int,number_number_of(int,bit1(bit1(pls))),power_power(int,A_1,number_number_of(nat,bit0(bit1(pls))))),B)),times_times(int,times_times(int,number_number_of(int,bit1(bit1(pls))),A_1),power_power(int,B,number_number_of(nat,bit0(bit1(pls)))))),power_power(int,B,number_number_of(nat,bit1(bit1(pls))))) = power_power(int,plus_plus(int,A_1,B),number_number_of(nat,bit1(bit1(pls))))) # label(fact_8_zadd__power3) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 37 (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_275_Euler_Oaux____1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 38 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all Y_2 (ord_less(int,pls,Y_2) <-> ord_less(X_a,zero_zero(X_a),number_number_of(X_a,Y_2)))))) # label(fact_244_less__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 39 (all X_a (comm_semiring_1(X_a) -> (all X_1 all Y_1 all Q times_times(X_a,power_power(X_a,X_1,Q),power_power(X_a,Y_1,Q)) = power_power(X_a,times_times(X_a,X_1,Y_1),Q)))) # label(fact_113_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 40 (all B all Q all R_1 all B_2 all Q_1 all R_2 (plus_plus(int,times_times(int,B,Q),R_1) = plus_plus(int,times_times(int,B_2,Q_1),R_2) -> (ord_less_eq(int,zero_zero(int),plus_plus(int,times_times(int,B_2,Q_1),R_2)) -> (ord_less(int,R_2,B_2) -> (ord_less_eq(int,zero_zero(int),R_1) -> (ord_less(int,zero_zero(int),B_2) -> (ord_less_eq(int,B_2,B) -> ord_less_eq(int,Q,Q_1)))))))) # label(fact_447_zdiv__mono2__lemma) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 41 (all Z times_times(nat,Z,number_number_of(nat,bit0(bit1(pls)))) = plus_plus(nat,Z,Z)) # label(fact_36_nat__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.07/1.35 42 (all V_2 all V_3 (ord_less(nat,number_number_of(nat,V_2),number_number_of(nat,V_3)) <-> (ord_less(int,V_2,V_3) -> ord_less(int,pls,V_3)) & ord_less(int,V_2,V_3))) # label(fact_235_less__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 43 (all X_a (comm_semiring_1(X_a) -> (all Lx all Ly all Rx times_times(X_a,times_times(X_a,Lx,Rx),Ly) = times_times(X_a,times_times(X_a,Lx,Ly),Rx)))) # label(fact_69_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 44 (all B all M all A_1 (ord_less(int,zero_zero(int),A_1) -> (ord_less(int,A_1,M) -> (ord_less(int,zero_zero(int),B) -> (ord_less(int,B,A_1) -> -zcong(A_1,B,M)))))) # label(fact_338_zcong__not) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 45 (all N all M (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) -> ti(int,M) = ti(int,N)))))) # label(fact_189_zdvd__antisym__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 46 (all K_1 (ord_less(int,pls,bit0(K_1)) <-> ord_less(int,pls,K_1))) # label(fact_94_rel__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 47 (all X_a (linord581940658strict(X_a) -> (all X_2 all Y_2 (ord_less(X_a,zero_zero(X_a),plus_plus(X_a,times_times(X_a,X_2,X_2),times_times(X_a,Y_2,Y_2))) <-> zero_zero(X_a) != ti(X_a,X_2) | zero_zero(X_a) != ti(X_a,Y_2))))) # label(fact_237_sum__squares__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 48 (all X_a (monoid_mult(X_a) -> (all X_1 all N (ord_less(nat,zero_zero(nat),N) -> times_times(X_a,power_power(X_a,X_1,minus_minus(nat,N,one_one(nat))),X_1) = power_power(X_a,X_1,N))))) # label(fact_442_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 49 (all X_a (monoid_mult(X_a) -> (all A_1 all N power_power(X_a,A_1,times_times(nat,number_number_of(nat,bit0(bit1(pls))),N)) = power_power(X_a,power_power(X_a,A_1,N),number_number_of(nat,bit0(bit1(pls))))))) # label(fact_101_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 50 (all B_1_1 all B_2_1 all X_a (semiring(X_a) -> plus_plus(X_a,B_1_1,B_2_1) = ti(X_a,plus_plus(X_a,B_1_1,B_2_1)))) # label(tsy_c_Groups_Oplus__class_Oplus_0_res) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 51 (all X_a (semiri456707255roduct(X_a) -> (all C all D all A_1 all B all R_1 (ti(X_a,R_1) != zero_zero(X_a) -> (ti(X_a,C) != ti(X_a,D) & ti(X_a,B) = ti(X_a,A_1) -> plus_plus(X_a,A_1,times_times(X_a,R_1,C)) != plus_plus(X_a,B,times_times(X_a,R_1,D))))))) # label(fact_217_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 52 (all X_a (number_ring(X_a) -> (all W number_number_of(X_a,bit0(W)) = plus_plus(X_a,plus_plus(X_a,zero_zero(X_a),number_number_of(X_a,W)),number_number_of(X_a,W))))) # label(fact_239_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 53 (all X_a (number_semiring(X_a) -> (all V_1 all V (ord_less_eq(int,pls,V) -> (ord_less_eq(int,pls,V_1) -> number_number_of(X_a,times_times(int,V,V_1)) = times_times(X_a,number_number_of(X_a,V),number_number_of(X_a,V_1))))))) # label(fact_133_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 54 (all X_a (ordere453448008miring(X_a) -> (all B all A_1 (ord_less_eq(X_a,A_1,zero_zero(X_a)) -> (ord_less_eq(X_a,zero_zero(X_a),B) -> ord_less_eq(X_a,times_times(X_a,A_1,B),zero_zero(X_a))))))) # label(fact_496_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 55 (all X_a (number_semiring(X_a) -> number_number_of(X_a,bit0(bit1(pls))) = plus_plus(X_a,one_one(X_a),one_one(X_a)))) # label(fact_167_semiring__one__add__one__is__two) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 56 (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_281_nat__zero__less__power__iff) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 57 (all K all L plus_plus(int,bit0(times_times(int,K,L)),L) = times_times(int,bit1(K),L)) # label(fact_159_mult__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 58 (all X_a (semiri456707255roduct(X_a) -> (all W_1 all Y_2 all X_2 all Z_1 (plus_plus(X_a,times_times(X_a,W_1,Z_1),times_times(X_a,X_2,Y_2)) = plus_plus(X_a,times_times(X_a,W_1,Y_2),times_times(X_a,X_2,Z_1)) <-> ti(X_a,Z_1) = ti(X_a,Y_2) | ti(X_a,X_2) = ti(X_a,W_1))))) # label(fact_106_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 59 (all Z all W (ord_less_eq(real,Z,W) -> (ord_less_eq(real,W,Z) -> Z = W))) # label(fact_453_real__le__antisym) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 60 (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) | N = M | zero_zero(nat) = M)) # label(fact_383_divides__cases) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 61 (all X_a (number_ring(X_a) -> (all A_1 ti(X_a,A_1) = times_times(X_a,number_number_of(X_a,bit1(pls)),A_1)))) # label(fact_153_mult__numeral__1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 62 (all X_a (comm_semiring_1(X_a) -> (all A_1 dvd_dvd(X_a,one_one(X_a),A_1)))) # label(fact_488_one__dvd) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 63 (all Z times_times(int,one_one(int),Z) = ti(int,Z)) # label(fact_128_zmult__1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 64 (all W ord_less_eq(real,W,W)) # label(fact_456_real__le__refl) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 65 (all X_a (comm_semiring_1(X_a) -> (all M times_times(X_a,plus_plus(X_a,one_one(X_a),one_one(X_a)),M) = plus_plus(X_a,M,M)))) # label(fact_140_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 66 (all M all N (times_times(int,M,N) = one_one(int) -> ti(int,M) = one_one(int) | number_number_of(int,min) = ti(int,M))) # label(fact_314_pos__zmult__eq__1__iff__lemma) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 67 (all V_1 all K all V ((-ord_less(int,V,pls) -> times_times(nat,number_number_of(nat,V),times_times(nat,number_number_of(nat,V_1),K)) = times_times(nat,number_number_of(nat,times_times(int,V,V_1)),K)) & (ord_less(int,V,pls) -> times_times(nat,number_number_of(nat,V),times_times(nat,number_number_of(nat,V_1),K)) = zero_zero(nat)))) # label(fact_325_nat__number__of__mult__left) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 68 (all B_1_1 all B_2_1 (quadRes(B_1_1,B_2_1) <-> quadRes(B_1_1,ti(int,B_2_1)))) # label(tsy_c_Residues_OQuadRes_arg2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 69 (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_380_one__not__neg__one__mod__m) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 70 (all K bit0(K) = plus_plus(int,K,K)) # label(fact_126_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 71 (all M all Y_1 all X_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) -> ti(int,X_1) = ti(int,Y_1)))))))) # label(fact_404_zcong__less__eq) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 72 (all X_1 all Y_1 all Z times_times(int,power_power(int,X_1,Y_1),power_power(int,X_1,Z)) = power_power(int,X_1,plus_plus(nat,Y_1,Z))) # label(fact_34_zpower__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 73 (all B_1_1 all B_2_1 all X_a (semiring(X_a) -> times_times(X_a,B_1_1,ti(X_a,B_2_1)) = times_times(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_1_arg2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 74 (all K1 all K2 (ord_less(int,K1,K2) <-> ord_less(int,bit1(K1),bit1(K2)))) # label(fact_38_less__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 75 (all X_a (linordered_ring(X_a) -> (all X_1 all Y_1 -ord_less(X_a,plus_plus(X_a,times_times(X_a,X_1,X_1),times_times(X_a,Y_1,Y_1)),zero_zero(X_a))))) # label(fact_236_not__sum__squares__lt__zero) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 76 (all X_a (ring_char_0(X_a) & number_ring(X_a) -> (all X_2 all Y_2 (ti(int,Y_2) = ti(int,X_2) <-> number_number_of(X_a,Y_2) = number_number_of(X_a,X_2))))) # label(fact_80_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 77 (all X_a (number_ring(X_a) -> (all A_1 ti(X_a,A_1) = times_times(X_a,A_1,number_number_of(X_a,bit1(pls)))))) # label(fact_154_mult__numeral__1__right) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 78 (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_429_real__add__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 79 (all V all W number_number_of(int,plus_plus(int,V,W)) = plus_plus(int,number_number_of(int,V),number_number_of(int,W))) # label(fact_132_plus__numeral__code_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 80 (all W_1 (ord_less(int,W_1,zero_zero(int)) <-> ord_less(int,bit1(W_1),zero_zero(int)))) # label(fact_222_bin__less__0__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 81 (all B_1_1 all B_2_1 all X_a (order(X_a) -> (ord_less(X_a,B_1_1,B_2_1) <-> ord_less(X_a,B_1_1,ti(X_a,B_2_1))))) # label(tsy_c_Orderings_Oord__class_Oless_arg2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 82 (all B_1_1 all B_2_1 all X_a (power(X_a) -> times_times(X_a,B_1_1,B_2_1) = ti(X_a,times_times(X_a,B_1_1,B_2_1)))) # label(tsy_c_Groups_Otimes__class_Otimes_4_res) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 83 -(all 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_279__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]. 1.07/1.36 84 (all X_2 (X_2 = zero_zero(real) <-> -ord_less(real,zero_zero(real),times_times(real,X_2,X_2)))) # label(fact_294_not__real__square__gt__zero) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 85 (all B_1_1 all B_2_1 all X_a (ab_group_add(X_a) -> ti(X_a,plus_plus(X_a,B_1_1,B_2_1)) = plus_plus(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Oplus__class_Oplus_1_res) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 86 (all B_1_1 all B_2_1 all X_a (order(X_a) -> (ord_less_eq(X_a,B_1_1,B_2_1) <-> ord_less_eq(X_a,ti(X_a,B_1_1),B_2_1)))) # label(tsy_c_Orderings_Oord__class_Oless__eq_arg1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 87 (all Z all W times_times(int,Z,W) = times_times(int,W,Z)) # label(fact_85_zmult__commute) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 88 (all V_1 all V ((-ord_less(int,V,pls) -> times_times(nat,number_number_of(nat,V),number_number_of(nat,V_1)) = number_number_of(nat,times_times(int,V,V_1))) & (ord_less(int,V,pls) -> zero_zero(nat) = times_times(nat,number_number_of(nat,V),number_number_of(nat,V_1))))) # label(fact_326_mult__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 89 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B all C all D plus_plus(X_a,plus_plus(X_a,A_1,B),plus_plus(X_a,C,D)) = plus_plus(X_a,plus_plus(X_a,A_1,C),plus_plus(X_a,B,D))))) # label(fact_74_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 90 (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_51_zadd__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 91 (all X_a (comm_semiring_1(X_a) -> (all A_1 (dvd_dvd(X_a,zero_zero(X_a),A_1) -> zero_zero(X_a) = ti(X_a,A_1))))) # label(fact_478_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 92 (all X_a (monoid_mult(X_a) -> (all A_1 power_power(X_a,A_1,number_number_of(nat,bit0(bit1(pls)))) = times_times(X_a,A_1,A_1)))) # label(fact_14_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 93 (all X_a (linordered_idom(X_a) -> (all X_1 all Y_1 -ord_less(X_a,plus_plus(X_a,power_power(X_a,X_1,number_number_of(nat,bit0(bit1(pls)))),power_power(X_a,Y_1,number_number_of(nat,bit0(bit1(pls))))),zero_zero(X_a))))) # label(fact_265_not__sum__power2__lt__zero) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 94 (all A_2 all B_1 all C_1 (C_1 != zero_zero(real) -> (B_1 = A_2 <-> times_times(real,A_2,C_1) = times_times(real,B_1,C_1)))) # label(fact_432_real__mult__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 95 (all X_a ti(X_a,undefined(X_a)) = undefined(X_a)) # label(tsy_c_HOL_Oundefined_res) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 96 (all B_1_1 (zprime(ti(int,B_1_1)) <-> zprime(B_1_1))) # label(tsy_c_IntPrimes_Ozprime_arg1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 97 (all B all D all A_1 (dvd_dvd(nat,D,A_1) -> (dvd_dvd(nat,D,plus_plus(nat,A_1,B)) -> dvd_dvd(nat,D,B)))) # label(fact_389_divides__add__revr) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 98 (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),J) = power_power(int,number_number_of(int,min),K)))) # label(fact_346_neg__one__power__eq__mod__m) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 99 (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_210_zdvd__reduce) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 100 (all K_1 all L_1 (ord_less_eq(int,bit1(K_1),bit0(L_1)) <-> ord_less(int,K_1,L_1))) # label(fact_58_rel__simps_I33_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 101 (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_454_real__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 102 (all N_1 all Ma (ord_less(int,zero_zero(int),Ma) -> (ti(int,Ma) = one_one(int) & ti(int,N_1) = one_one(int) <-> times_times(int,Ma,N_1) = one_one(int)))) # label(fact_242_pos__zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 103 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all Y_2 (ord_less(X_a,one_one(X_a),number_number_of(X_a,Y_2)) <-> ord_less(int,bit1(pls),Y_2))))) # label(fact_103_less__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 104 (all A_1 all B zcong(A_1,B,one_one(int))) # label(fact_330_zcong__1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 105 (all B_1_1 all B_2_1 all X_a (power(X_a) -> times_times(X_a,ti(X_a,B_1_1),B_2_1) = times_times(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_4_arg1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 106 (all X_a (comm_semiring_1(X_a) -> (all M all A_1 all N all B (dvd_dvd(X_a,power_power(X_a,A_1,N),B) -> (ord_less_eq(nat,M,N) -> dvd_dvd(X_a,power_power(X_a,A_1,M),B)))))) # label(fact_186_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 107 (all K bit1(K) = plus_plus(int,plus_plus(int,one_one(int),K),K)) # label(fact_151_Bit1__def) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 108 (all B_1_1 ti(int,bit0(B_1_1)) = bit0(B_1_1)) # label(tsy_c_Int_OBit0_res) # label(hypothesis) # label(non_clause). [assumption]. 1.07/1.36 109 (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_384_divides__antisym) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 110 (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_131_zadd__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 111 (all R_1 all Q all A_1 (ord_less(int,zero_zero(int),A_1) -> (plus_plus(int,R_1,times_times(int,A_1,Q)) = ti(int,A_1) -> (ord_less(int,R_1,A_1) -> ord_less_eq(int,one_one(int),Q))))) # label(fact_421_self__quotient__aux1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 112 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B dvd_dvd(X_a,A_1,times_times(X_a,B,A_1))))) # label(fact_480_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 113 (all X_a (linordered_idom(X_a) -> (all A_1 all K (ord_less_eq(X_a,power_power(X_a,A_1,times_times(nat,number_number_of(nat,bit0(bit1(pls))),K)),zero_zero(X_a)) -> ti(X_a,A_1) = zero_zero(X_a))))) # label(fact_277_even__power__le__0__imp__0) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 114 (all X_a (power(X_a) & semiring_0(X_a) -> (all W ((number_number_of(nat,W) != zero_zero(nat) -> power_power(X_a,zero_zero(X_a),number_number_of(nat,W)) = zero_zero(X_a)) & (zero_zero(nat) = number_number_of(nat,W) -> power_power(X_a,zero_zero(X_a),number_number_of(nat,W)) = one_one(X_a)))))) # label(fact_340_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 115 (all B all Q all R_1 all B_2 all Q_1 all R_2 (plus_plus(int,times_times(int,B_2,Q_1),R_2) = plus_plus(int,times_times(int,B,Q),R_1) -> (ord_less(int,plus_plus(int,times_times(int,B_2,Q_1),R_2),zero_zero(int)) -> (ord_less(int,R_1,B) -> (ord_less_eq(int,zero_zero(int),R_2) -> (ord_less(int,zero_zero(int),B_2) -> (ord_less_eq(int,B_2,B) -> ord_less_eq(int,Q_1,Q)))))))) # label(fact_449_zdiv__mono2__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 116 (all X_1 power_power(real,times_times(real,number_number_of(real,bit0(bit1(pls))),X_1),number_number_of(nat,bit0(bit1(pls)))) = times_times(real,number_number_of(real,bit0(bit0(bit1(pls)))),power_power(real,X_1,number_number_of(nat,bit0(bit1(pls)))))) # label(fact_173_four__x__squared) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 117 (all X_a (semiring(X_a) -> (all A_1 all E all B all C plus_plus(X_a,times_times(X_a,A_1,E),plus_plus(X_a,times_times(X_a,B,E),C)) = plus_plus(X_a,times_times(X_a,plus_plus(X_a,A_1,B),E),C)))) # label(fact_477_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 118 (all B_1_1 all B_2_1 all X_a (ab_group_add(X_a) -> minus_minus(X_a,B_1_1,ti(X_a,B_2_1)) = minus_minus(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Ominus__class_Ominus_0_arg2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 119 (all K all A_1 all B all M (zcong(A_1,B,M) -> zcong(times_times(int,K,A_1),times_times(int,K,B),M))) # label(fact_332_zcong__scalar2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 120 (all Z all W (ord_less_eq(int,W,Z) | ord_less_eq(int,Z,W))) # label(fact_21_zle__linear) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 121 (all X_a (number_ring(X_a) -> (all A_1 ti(X_a,A_1) = plus_plus(X_a,A_1,number_number_of(X_a,pls))))) # label(fact_142_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 122 (all P_1 ((all M_1 (ord_less_eq(int,zero_zero(int),M_1) & dvd_dvd(int,M_1,P_1) -> ti(int,M_1) = ti(int,P_1) | ti(int,M_1) = one_one(int))) & ord_less(int,one_one(int),P_1) <-> zprime(P_1))) # label(fact_278_zprime__def) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 123 (all W all Z (ord_less(int,W,Z) -> ord_less_eq(int,plus_plus(int,W,one_one(int)),Z))) # label(fact_61_zless__imp__add1__zle) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 124 (all X_b (ring(X_b) & number(X_b) -> (all A_1 all B all V times_times(X_b,minus_minus(X_b,A_1,B),number_number_of(X_b,V)) = minus_minus(X_b,times_times(X_b,A_1,number_number_of(X_b,V)),times_times(X_b,B,number_number_of(X_b,V)))))) # label(fact_357_left__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 125 (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_414_zcong__zmult__prop3) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 126 (all X_a (number_ring(X_a) -> (all A_1 plus_plus(X_a,number_number_of(X_a,pls),A_1) = ti(X_a,A_1)))) # label(fact_141_add__numeral__0) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 127 (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_248_le__imp__0__less) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 128 (all X_a (comm_semiring_1(X_a) -> (all A_1 dvd_dvd(X_a,A_1,A_1)))) # label(fact_468_dvd__refl) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 129 (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_460_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 130 (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_228_zmult__zless__mono2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 131 (all N (power_power(int,number_number_of(int,min),N) = one_one(int) | power_power(int,number_number_of(int,min),N) = number_number_of(int,min))) # label(fact_378_neg__one__power) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 132 (all K all M all N (dvd_dvd(int,times_times(int,K,M),times_times(int,K,N)) -> (zero_zero(int) != ti(int,K) -> dvd_dvd(int,M,N)))) # label(fact_190_zdvd__mult__cancel) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 133 (all X_a (comm_semiring_1(X_a) -> (all X_1 all N times_times(X_a,power_power(X_a,X_1,N),power_power(X_a,X_1,N)) = power_power(X_a,X_1,times_times(nat,number_number_of(nat,bit0(bit1(pls))),N))))) # label(fact_15_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 134 (all X_a (comm_semiring_1(X_a) -> (all A_1 all C plus_plus(X_a,A_1,C) = plus_plus(X_a,C,A_1)))) # label(fact_79_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 135 (all X_a (linordered_idom(X_a) -> (all X_2 all Y_2 (ti(X_a,X_2) != zero_zero(X_a) | zero_zero(X_a) != ti(X_a,Y_2) <-> ord_less(X_a,zero_zero(X_a),plus_plus(X_a,power_power(X_a,X_2,number_number_of(nat,bit0(bit1(pls)))),power_power(X_a,Y_2,number_number_of(nat,bit0(bit1(pls)))))))))) # label(fact_266_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 136 (all X_a (comm_semiring_1(X_a) -> (all C all A_1 all B (dvd_dvd(X_a,A_1,B) -> dvd_dvd(X_a,A_1,times_times(X_a,B,C)))))) # label(fact_481_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 137 (all K1 all K2 (ord_less(int,bit1(K1),bit0(K2)) <-> ord_less(int,K1,K2))) # label(fact_91_less__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 138 (all X_a (comm_semiring_1(X_a) -> (all X_1 all P all Q times_times(X_a,power_power(X_a,X_1,P),power_power(X_a,X_1,Q)) = power_power(X_a,X_1,plus_plus(nat,P,Q))))) # label(fact_33_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 139 (all X_a (linordered_idom(X_a) -> (all X_1 all Y_1 (ti(X_a,X_1) != ti(X_a,Y_1) -> (-ord_less(X_a,X_1,Y_1) -> ord_less(X_a,Y_1,X_1)))))) # label(fact_466_linorder__neqE__linordered__idom) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 140 (all L bit0(minus_minus(int,pls,L)) = minus_minus(int,pls,bit0(L))) # label(fact_360_diff__bin__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 141 (all X_a (zero_neq_one(X_a) -> zero_zero(X_a) != one_one(X_a))) # label(fact_475_zero__neq__one) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 142 -(all S -(ord_less_eq(int,zero_zero(int),S) & ord_less(int,S,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) & zcong(s1,S,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))))) # label(fact_181__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]. 1.07/1.36 143 (all B_1_1 all B_2_1 all X_a (power(X_a) -> ti(X_a,power_power(X_a,B_1_1,B_2_1)) = power_power(X_a,B_1_1,B_2_1))) # label(tsy_c_Power_Opower__class_Opower_res) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 144 (all X_a (power(X_a) & mult_zero(X_a) & no_zero_divisors(X_a) & zero_neq_one(X_a) -> (all A_2 all W_1 (zero_zero(X_a) = power_power(X_a,A_2,number_number_of(nat,W_1)) <-> ti(X_a,A_2) = zero_zero(X_a) & number_number_of(nat,W_1) != zero_zero(nat))))) # label(fact_337_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 145 (all K ti(int,K) = number_number_of(int,K)) # label(fact_86_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 146 (all X_a (number_ring(X_a) -> (all V plus_plus(X_a,number_number_of(X_a,V),one_one(X_a)) = number_number_of(X_a,plus_plus(int,V,bit1(pls)))))) # label(fact_17_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 147 (all K_1 all L_1 (ord_less(int,K_1,L_1) <-> ord_less(int,bit0(K_1),bit0(L_1)))) # label(fact_44_rel__simps_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 148 (all X_a (power(X_a) -> (all A_1 one_one(X_a) = power_power(X_a,A_1,zero_zero(nat))))) # label(fact_309_power__0) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 149 (all B_1_1 all B_2_1 (quadRes(B_1_1,B_2_1) <-> quadRes(ti(int,B_1_1),B_2_1))) # label(tsy_c_Residues_OQuadRes_arg1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 150 (all L min != bit0(L)) # label(fact_291_rel__simps_I42_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 151 (all K_1 all L_1 (ord_less(int,K_1,L_1) <-> ord_less(int,bit1(K_1),bit0(L_1)))) # label(fact_92_rel__simps_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 152 (all N ord_less_eq(real,one_one(real),power_power(real,number_number_of(real,bit0(bit1(pls))),N))) # label(fact_443_two__realpow__ge__one) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 153 (all A_2 all B_1 all Ma (zcong(B_1,A_2,Ma) <-> zcong(A_2,B_1,Ma))) # label(fact_321_zcong__sym) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 154 (all X_a (comm_semiring_1(X_a) -> (all A_1 plus_plus(X_a,A_1,zero_zero(X_a)) = ti(X_a,A_1)))) # label(fact_199_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 155 (all A_2 all P_1 (zcong(times_times(int,A_2,minus_minus(int,P_1,one_one(int))),one_one(int),P_1) <-> zcong(A_2,minus_minus(int,P_1,one_one(int)),P_1))) # label(fact_370_inv__not__p__minus__1__aux) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 156 (all B_1_1 all B_2_1 product_Pair(int,int,B_1_1,B_2_1) = product_Pair(int,int,B_1_1,ti(int,B_2_1))) # label(tsy_c_Product__Type_OPair_arg2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 157 (all X_a (linordered_semidom(X_a) -> (all A_1 all N all N_2 (ord_less(nat,N,N_2) -> (ord_less(X_a,zero_zero(X_a),A_1) -> (ord_less(X_a,A_1,one_one(X_a)) -> ord_less(X_a,power_power(X_a,A_1,N_2),power_power(X_a,A_1,N)))))))) # label(fact_214_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 158 (all B_1_1 all B_2_1 ti(nat,minus_minus(nat,B_1_1,B_2_1)) = minus_minus(nat,B_1_1,B_2_1)) # label(tsy_c_Groups_Ominus__class_Ominus_1_res) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 159 (all X_a (no_zero_divisors(X_a) -> (all A_1 all B (zero_zero(X_a) = times_times(X_a,A_1,B) -> zero_zero(X_a) = ti(X_a,B) | ti(X_a,A_1) = zero_zero(X_a))))) # label(fact_473_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 160 (all X_a (linordered_semidom(X_a) -> (all A_1 all N all N_2 (ord_less_eq(nat,N,N_2) -> (ord_less_eq(X_a,one_one(X_a),A_1) -> ord_less_eq(X_a,power_power(X_a,A_1,N),power_power(X_a,A_1,N_2))))))) # label(fact_269_power__increasing) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 161 (all X_2 all Y_2 (Y_2 = X_2 | ord_less(real,X_2,Y_2) <-> ord_less_eq(real,X_2,Y_2))) # label(fact_425_less__eq__real__def) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 162 (all X_a (linordered_semidom(X_a) -> (all X_1 all Y_1 (power_power(X_a,X_1,number_number_of(nat,bit0(bit1(pls)))) = power_power(X_a,Y_1,number_number_of(nat,bit0(bit1(pls)))) -> (ord_less_eq(X_a,zero_zero(X_a),X_1) -> (ord_less_eq(X_a,zero_zero(X_a),Y_1) -> ti(X_a,Y_1) = ti(X_a,X_1))))))) # label(fact_253_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 163 (all X_a (zero_neq_one(X_a) -> one_one(X_a) != zero_zero(X_a))) # label(fact_474_one__neq__zero) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 164 (all B_1_1 (twoSqu420862341sum2sq(B_1_1) <-> twoSqu420862341sum2sq(ti(int,B_1_1)))) # label(tsy_c_TwoSquares__Mirabelle__iprzenvtji_Ois__sum2sq_arg1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 165 (all K_1 (ord_less(int,bit1(K_1),min) <-> ord_less(int,K_1,min))) # label(fact_295_rel__simps_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 166 (all X_a (zero_neq_one(X_a) -> one_one(X_a) = ti(X_a,one_one(X_a)))) # label(tsy_c_Groups_Oone__class_Oone_0_res) # label(axiom) # label(non_clause). [assumption]. 1.07/1.36 167 (all X_a (number_semiring(X_a) -> number_number_of(X_a,pls) = zero_zero(X_a))) # label(fact_221_semiring__numeral__0__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 168 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B all C plus_plus(X_a,plus_plus(X_a,A_1,B),C) = plus_plus(X_a,plus_plus(X_a,A_1,C),B)))) # label(fact_75_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 169 (all 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) = ti(int,X_1) | ti(int,X_1) = zero_zero(int)))) # label(fact_276_int__pos__lt__two__imp__zero__or__one) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 170 (all A_1 all M all B zcong(times_times(int,A_1,M),times_times(int,B,M),M)) # label(fact_334_zcong__zmult__self) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 171 (all L minus_minus(int,min,bit1(L)) = bit0(minus_minus(int,min,L))) # label(fact_368_diff__bin__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 172 (all X_a (linord581940658strict(X_a) -> (all A_2 all B_1 (ord_less_eq(X_a,A_2,zero_zero(X_a)) & ord_less_eq(X_a,B_1,zero_zero(X_a)) | ord_less_eq(X_a,zero_zero(X_a),B_1) & ord_less_eq(X_a,zero_zero(X_a),A_2) <-> ord_less_eq(X_a,zero_zero(X_a),times_times(X_a,A_2,B_1)))))) # label(fact_491_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 173 (all W all Z1 all Z2 minus_minus(int,times_times(int,W,Z1),times_times(int,W,Z2)) = times_times(int,W,minus_minus(int,Z1,Z2))) # label(fact_353_zdiff__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 174 (all X_a (comm_ring_1(X_a) -> (all Z all X_1 all Y_1 (dvd_dvd(X_a,X_1,Y_1) -> (dvd_dvd(X_a,X_1,Z) -> dvd_dvd(X_a,X_1,minus_minus(X_a,Y_1,Z))))))) # label(fact_489_dvd__diff) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 175 (all X_2 all N_1 (ord_less(nat,zero_zero(nat),power_power(nat,X_2,N_1)) <-> ord_less(nat,zero_zero(nat),X_2) | N_1 = zero_zero(nat))) # label(fact_282_zero__less__power__nat__eq) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 176 (all X_a (monoid_mult(X_a) -> (all A_1 times_times(X_a,times_times(X_a,A_1,A_1),A_1) = power_power(X_a,A_1,number_number_of(nat,bit1(bit1(pls))))))) # label(fact_161_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 177 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B all C plus_plus(X_a,plus_plus(X_a,A_1,B),C) = plus_plus(X_a,A_1,plus_plus(X_a,B,C))))) # label(fact_76_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 178 (all A_2 all B_1 (zcong(A_2,B_1,zero_zero(int)) <-> ti(int,B_1) = ti(int,A_2))) # label(fact_329_IntPrimes_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 179 (all X_a (linordered_semidom(X_a) -> (all X_1 all Y_1 (ord_less_eq(X_a,power_power(X_a,X_1,number_number_of(nat,bit0(bit1(pls)))),power_power(X_a,Y_1,number_number_of(nat,bit0(bit1(pls))))) -> (ord_less_eq(X_a,zero_zero(X_a),Y_1) -> ord_less_eq(X_a,X_1,Y_1)))))) # label(fact_252_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 180 (all X_a (ordere453448008miring(X_a) -> (all B all A_1 (ord_less_eq(X_a,zero_zero(X_a),A_1) -> (ord_less_eq(X_a,B,zero_zero(X_a)) -> ord_less_eq(X_a,times_times(X_a,A_1,B),zero_zero(X_a))))))) # label(fact_494_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 181 (all X_a (semiring_1(X_a) -> zero_zero(X_a) = power_power(X_a,zero_zero(X_a),number_number_of(nat,bit0(bit1(pls)))))) # label(fact_249_zero__power2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 182 (all Z1 all Z2 all Z3 times_times(real,times_times(real,Z1,Z2),Z3) = times_times(real,Z1,times_times(real,Z2,Z3))) # label(fact_428_real__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 183 (all X_a (ring_11004092258visors(X_a) -> (all N all A_1 (ti(X_a,A_1) != zero_zero(X_a) -> power_power(X_a,A_1,N) != zero_zero(X_a))))) # label(fact_192_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 184 (all Z ti(int,Z) = times_times(int,Z,one_one(int))) # label(fact_127_zmult__1__right) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 185 (all X_a (dvd(X_a) -> (all A_1 all B all K (A_1 = times_times(X_a,B,K) -> dvd_dvd(X_a,B,A_1))))) # label(fact_484_dvdI) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 186 (all A_1 all B (dvd_dvd(nat,A_1,B) -> ord_less_eq(nat,A_1,B) | B = zero_zero(nat))) # label(fact_401_divides__ge) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 187 (all Z plus_plus(int,zero_zero(int),Z) = ti(int,Z)) # label(fact_204_zadd__0) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 188 (all X_a (comm_semiring_1(X_a) -> (all A_1 all M times_times(X_a,plus_plus(X_a,A_1,one_one(X_a)),M) = plus_plus(X_a,times_times(X_a,A_1,M),M)))) # label(fact_138_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 189 (all X_a (linord581940658strict(X_a) -> (all A_2 all B_1 (ord_less_eq(X_a,times_times(X_a,A_2,B_1),zero_zero(X_a)) <-> ord_less_eq(X_a,zero_zero(X_a),B_1) & ord_less_eq(X_a,A_2,zero_zero(X_a)) | ord_less_eq(X_a,zero_zero(X_a),A_2) & ord_less_eq(X_a,B_1,zero_zero(X_a)))))) # label(fact_492_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 190 (all X_a (linordered_idom(X_a) -> (all X_2 all Y_2 (zero_zero(X_a) = ti(X_a,X_2) & zero_zero(X_a) = ti(X_a,Y_2) <-> ord_less_eq(X_a,plus_plus(X_a,power_power(X_a,X_2,number_number_of(nat,bit0(bit1(pls)))),power_power(X_a,Y_2,number_number_of(nat,bit0(bit1(pls))))),zero_zero(X_a)))))) # label(fact_264_sum__power2__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 191 (all X_a (linordered_idom(X_a) -> (all X_1 all Y_1 ord_less_eq(X_a,zero_zero(X_a),plus_plus(X_a,power_power(X_a,X_1,number_number_of(nat,bit0(bit1(pls)))),power_power(X_a,Y_1,number_number_of(nat,bit0(bit1(pls))))))))) # label(fact_263_sum__power2__ge__zero) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 192 (all Y_1 all X_1 (twoSqu420862341sum2sq(X_1) -> (twoSqu420862341sum2sq(Y_1) -> twoSqu420862341sum2sq(times_times(int,X_1,Y_1))))) # label(fact_65_is__mult__sum2sq) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 193 -(all T_1 plus_plus(int,power_power(int,s,number_number_of(nat,bit0(bit1(pls)))),one_one(int)) != times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),T_1)) # label(fact_19__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]. 1.07/1.37 194 (all X_a (linordered_ring(X_a) -> (all A_1 ord_less_eq(X_a,zero_zero(X_a),times_times(X_a,A_1,A_1))))) # label(fact_490_zero__le__square) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 195 (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_403_zcong__not__zero) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 196 (all A_2 all Ma (zcong(A_2,zero_zero(int),Ma) <-> dvd_dvd(int,Ma,A_2))) # label(fact_408_zcong__zero__equiv__div) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 197 (all X_a (number_ring(X_a) -> (all V all W all Z times_times(X_a,number_number_of(X_a,V),times_times(X_a,number_number_of(X_a,W),Z)) = times_times(X_a,number_number_of(X_a,times_times(int,V,W)),Z)))) # label(fact_143_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 198 (all K_1 (ord_less(int,min,bit0(K_1)) <-> ord_less(int,min,K_1))) # label(fact_299_rel__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 199 (all X_a (linordered_semidom(X_a) -> (all N all A_1 (ord_less(X_a,zero_zero(X_a),A_1) -> ord_less(X_a,zero_zero(X_a),power_power(X_a,A_1,N)))))) # label(fact_208_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 200 (all K1 all K2 (ord_less_eq(int,K1,K2) <-> ord_less_eq(int,bit0(K1),bit1(K2)))) # label(fact_96_less__eq__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 201 (all K all L bit1(minus_minus(int,K,L)) = minus_minus(int,bit1(K),bit0(L))) # label(fact_358_diff__bin__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 202 (all B_1_1 all B_2_1 all B_3 (zcong(B_1_1,B_2_1,B_3) <-> zcong(B_1_1,B_2_1,ti(int,B_3)))) # label(tsy_c_IntPrimes_Ozcong_arg3) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 203 (all B_1_1 all B_2_1 ti(product_prod(int,int),product_Pair(int,int,B_1_1,B_2_1)) = product_Pair(int,int,B_1_1,B_2_1)) # label(tsy_c_Product__Type_OPair_res) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 204 (all X_a (number_ring(X_a) -> (all V all W number_number_of(X_a,times_times(int,V,W)) = times_times(X_a,number_number_of(X_a,V),number_number_of(X_a,W))))) # label(fact_145_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 205 (all K_1 all L_1 (bit1(K_1) = bit1(L_1) <-> ti(int,K_1) = ti(int,L_1))) # label(fact_82_rel__simps_I51_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 206 (all Ma all N_1 (N_1 != zero_zero(nat) & Ma = zero_zero(nat) <-> zero_zero(nat) = power_power(nat,Ma,N_1))) # label(fact_398_nat__power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 207 (all B_1_1 all B_2_1 legendre(ti(int,B_1_1),B_2_1) = legendre(B_1_1,B_2_1)) # label(tsy_c_Residues_OLegendre_arg1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 208 (all V_2 (number_number_of(nat,V_2) = zero_zero(nat) <-> ord_less_eq(int,V_2,pls))) # label(fact_320_eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 209 (all B_1_1 all B_2_1 all X_a (dvd(X_a) -> times_times(X_a,B_1_1,B_2_1) = times_times(X_a,B_1_1,ti(X_a,B_2_1)))) # label(tsy_c_Groups_Otimes__class_Otimes_0_arg2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 210 (all B_1_1 bit1(B_1_1) = ti(int,bit1(B_1_1))) # label(tsy_c_Int_OBit1_res) # label(hypothesis) # label(non_clause). [assumption]. 1.07/1.37 211 (all B_1_1 all B_2_1 all B_3 (zcong(B_1_1,B_2_1,B_3) <-> zcong(ti(int,B_1_1),B_2_1,B_3))) # label(tsy_c_IntPrimes_Ozcong_arg1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 212 (all B_1_1 all B_2_1 minus_minus(nat,B_1_1,B_2_1) = minus_minus(nat,ti(nat,B_1_1),B_2_1)) # label(tsy_c_Groups_Ominus__class_Ominus_1_arg1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 213 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B times_times(X_a,B,A_1) = times_times(X_a,A_1,B)))) # label(fact_73_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 214 (all X_a (ordere453448008miring(X_a) -> (all B all A_1 (ord_less_eq(X_a,zero_zero(X_a),A_1) -> (ord_less_eq(X_a,B,zero_zero(X_a)) -> ord_less_eq(X_a,times_times(X_a,B,A_1),zero_zero(X_a))))))) # label(fact_495_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 215 (all X_a (linordered_semidom(X_a) -> (all A_1 all N all N_2 (ord_less(nat,N,N_2) -> (ord_less(X_a,one_one(X_a),A_1) -> ord_less(X_a,power_power(X_a,A_1,N),power_power(X_a,A_1,N_2))))))) # label(fact_273_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 216 (all X_a (linordered_semidom(X_a) -> (all A_1 all N all B (power_power(X_a,A_1,N) = power_power(X_a,B,N) -> (ord_less_eq(X_a,zero_zero(X_a),A_1) -> (ord_less_eq(X_a,zero_zero(X_a),B) -> (ord_less(nat,zero_zero(nat),N) -> ti(X_a,A_1) = ti(X_a,B)))))))) # label(fact_187_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 217 (all X_a (mult_zero(X_a) & no_zero_divisors(X_a) & zero_neq_one(X_a) & power(X_a) -> (all A_2 all N_1 (zero_zero(nat) != N_1 & ti(X_a,A_2) = zero_zero(X_a) <-> zero_zero(X_a) = power_power(X_a,A_2,N_1))))) # label(fact_183_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 218 (all K1 all K2 (ord_less(int,bit0(K1),bit1(K2)) <-> ord_less_eq(int,K1,K2))) # label(fact_59_less__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 219 (all X_a (comm_semiring_1(X_a) -> (all Lx all Ly all Rx all Ry times_times(X_a,Lx,times_times(X_a,Ly,times_times(X_a,Rx,Ry))) = times_times(X_a,times_times(X_a,Lx,Ly),times_times(X_a,Rx,Ry))))) # label(fact_68_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 220 (all X_a (power(X_a) -> (all M all N ((N != zero_zero(nat) -> power_power(X_a,M,N) = times_times(X_a,M,power_power(X_a,M,minus_minus(nat,N,one_one(nat))))) & (zero_zero(nat) = N -> one_one(X_a) = power_power(X_a,M,N)))))) # label(fact_452_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 221 (all X_1 power_power(int,power_power(int,X_1,number_number_of(nat,bit0(bit1(pls)))),number_number_of(nat,bit0(bit1(pls)))) = power_power(int,X_1,number_number_of(nat,bit0(bit0(bit1(pls)))))) # label(fact_162_quartic__square__square) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 222 (all B all Q_1 all R_2 all Q all R_1 (ord_less_eq(int,plus_plus(int,times_times(int,B,Q_1),R_2),plus_plus(int,times_times(int,B,Q),R_1)) -> (ord_less_eq(int,zero_zero(int),R_2) -> (ord_less(int,R_2,B) -> (ord_less(int,R_1,B) -> ord_less_eq(int,Q_1,Q)))))) # label(fact_446_unique__quotient__lemma) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 223 (all B_1_1 bit0(ti(int,B_1_1)) = bit0(B_1_1)) # label(tsy_c_Int_OBit0_arg1) # label(hypothesis) # label(non_clause). [assumption]. 1.07/1.37 224 (all X_a (number_ring(X_a) -> (all V all W number_number_of(X_a,plus_plus(int,V,W)) = plus_plus(X_a,number_number_of(X_a,V),number_number_of(X_a,W))))) # label(fact_147_add__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 225 (all X_a (order(X_a) -> (all X_1 all Y_1 (ord_less_eq(X_a,X_1,Y_1) -> (ti(X_a,Y_1) != ti(X_a,X_1) -> ord_less(X_a,X_1,Y_1)))))) # label(fact_327_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 226 (all B_1_1 all B_2_1 all X_a (semiring(X_a) -> plus_plus(X_a,B_1_1,ti(X_a,B_2_1)) = plus_plus(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Oplus__class_Oplus_0_arg2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 227 (all Z_1 all W_1 (ti(int,W_1) != ti(int,Z_1) & ord_less_eq(int,Z_1,W_1) <-> ord_less(int,Z_1,W_1))) # label(fact_22_zless__le) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 228 (all B_1_1 all B_2_1 all X_a (semiring(X_a) -> ti(X_a,times_times(X_a,B_1_1,B_2_1)) = times_times(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_1_res) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 229 (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_418_zpower__zdvd__prop2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 230 (all B_1_1 all B_2_1 all B_3 (zcong(B_1_1,B_2_1,B_3) <-> zcong(B_1_1,ti(int,B_2_1),B_3))) # label(tsy_c_IntPrimes_Ozcong_arg2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 231 (all X_a (power(X_a) & semiring_0(X_a) -> (all N ((zero_zero(nat) != N -> zero_zero(X_a) = power_power(X_a,zero_zero(X_a),N)) & (zero_zero(nat) = N -> one_one(X_a) = power_power(X_a,zero_zero(X_a),N)))))) # label(fact_193_power__0__left) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 232 (all Z all N (dvd_dvd(int,Z,N) -> (ord_less(int,zero_zero(int),N) -> ord_less_eq(int,Z,N)))) # label(fact_194_zdvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 233 (all B_1_1 all B_2_1 minus_minus(nat,B_1_1,ti(nat,B_2_1)) = minus_minus(nat,B_1_1,B_2_1)) # label(tsy_c_Groups_Ominus__class_Ominus_1_arg2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 234 (all X_a (ring_11004092258visors(X_a) -> (all A_2 (ti(X_a,A_2) = zero_zero(X_a) <-> power_power(X_a,A_2,number_number_of(nat,bit0(bit1(pls)))) = zero_zero(X_a))))) # label(fact_250_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 235 (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_48_less__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 236 (all X_2 all Y_2 (ord_less(real,X_2,Y_2) <-> ord_less_eq(real,X_2,Y_2) & Y_2 != X_2)) # label(fact_424_real__less__def) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 237 (all X_a (number_semiring(X_a) -> (all V_1 all V (ord_less_eq(int,pls,V) -> (ord_less_eq(int,pls,V_1) -> plus_plus(X_a,number_number_of(X_a,V),number_number_of(X_a,V_1)) = number_number_of(X_a,plus_plus(int,V,V_1))))))) # label(fact_134_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 238 (all B all M all A_1 (ord_less_eq(int,zero_zero(int),A_1) -> (ord_less(int,A_1,M) -> (ord_less_eq(int,zero_zero(int),B) -> (ord_less(int,B,M) -> (zcong(A_1,B,M) -> ti(int,A_1) = ti(int,B))))))) # label(fact_341_zcong__zless__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 239 (all A_2 all B_1 all Ma (zcong(A_2,B_1,Ma) <-> (exists K_2 plus_plus(int,A_2,times_times(int,Ma,K_2)) = ti(int,B_1)))) # label(fact_339_zcong__iff__lin) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 240 (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,number_number_of(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_172_real__sum__squared__expand) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 241 (all L bit1(L) != pls) # label(fact_115_rel__simps_I39_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 242 (all Z ti(int,Z) = plus_plus(int,Z,zero_zero(int))) # label(fact_205_zadd__0__right) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 243 (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_461_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 244 (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_343_zprime__zdvd__zmult) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 245 (all Z_1 (ord_less(int,plus_plus(int,plus_plus(int,one_one(int),Z_1),Z_1),zero_zero(int)) <-> ord_less(int,Z_1,zero_zero(int)))) # label(fact_243_odd__less__0) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 246 (all K_1 (ord_less_eq(int,K_1,min) <-> ord_less_eq(int,bit1(K_1),min))) # label(fact_301_rel__simps_I30_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 247 (all B_1_1 all B_2_1 all X_a (mult_zero(X_a) -> times_times(X_a,B_1_1,B_2_1) = times_times(X_a,ti(X_a,B_1_1),B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_3_arg1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 248 (all X_a (linordered_semidom(X_a) -> (all N all A_1 (ord_less(X_a,one_one(X_a),A_1) -> ord_less(X_a,power_power(X_a,A_1,N),times_times(X_a,A_1,power_power(X_a,A_1,N))))))) # label(fact_174_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 249 (all X_2 all Y_2 (zero_zero(real) = Y_2 & X_2 = zero_zero(real) <-> zero_zero(real) = plus_plus(real,times_times(real,X_2,X_2),times_times(real,Y_2,Y_2)))) # label(fact_440_real__two__squares__add__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 250 (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_60_rel__simps_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 251 (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_438_real__mult__le__cancel__iff1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 252 (all Z all W plus_plus(int,Z,W) = plus_plus(int,W,Z)) # label(fact_89_zadd__commute) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 253 (all B_1_1 twoSqu579703163sum2sq(B_1_1) = twoSqu579703163sum2sq(ti(product_prod(int,int),B_1_1))) # label(tsy_c_TwoSquares__Mirabelle__iprzenvtji_Osum2sq_arg1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 254 (all X_a (semiri456707255roduct(X_a) -> (all C_1 all D_1 all A_2 all B_1 (plus_plus(X_a,times_times(X_a,A_2,C_1),times_times(X_a,B_1,D_1)) != plus_plus(X_a,times_times(X_a,A_2,D_1),times_times(X_a,B_1,C_1)) <-> ti(X_a,B_1) != ti(X_a,A_2) & ti(X_a,D_1) != ti(X_a,C_1))))) # label(fact_109_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 255 (all X_a (number_ring(X_a) -> number_number_of(X_a,pls) = zero_zero(X_a))) # label(fact_219_semiring__norm_I112_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 256 (all X_a (linordered_semidom(X_a) -> (all Ma all N_1 all A_2 (ord_less(X_a,one_one(X_a),A_2) -> (power_power(X_a,A_2,N_1) = power_power(X_a,A_2,Ma) <-> N_1 = Ma))))) # label(fact_270_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 257 (all X_a (comm_semiring_1(X_a) -> (all N all X_1 all Y_1 (dvd_dvd(X_a,X_1,Y_1) -> dvd_dvd(X_a,power_power(X_a,X_1,N),power_power(X_a,Y_1,N)))))) # label(fact_191_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 258 (all B all A_1 all P (zprime(P) -> (ord_less(int,zero_zero(int),A_1) -> (zcong(times_times(int,A_1,B),zero_zero(int),P) -> zcong(B,zero_zero(int),P) | zcong(A_1,zero_zero(int),P))))) # label(fact_417_zcong__zprime__prod__zero) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 259 (all N all A_1 all B all P (zprime(P) -> (-dvd_dvd(int,P,B) -> (dvd_dvd(int,power_power(int,P,N),times_times(int,A_1,B)) -> dvd_dvd(int,power_power(int,P,N),A_1))))) # label(fact_232_zprime__power__zdvd__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 260 (all B_1_1 all B_2_1 all X_a (dvd(X_a) -> (dvd_dvd(X_a,ti(X_a,B_1_1),B_2_1) <-> dvd_dvd(X_a,B_1_1,B_2_1)))) # label(tsy_c_Rings_Odvd__class_Odvd_arg1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 261 (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_436_real__mult__order) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 262 (all K1 all K2 (ord_less(int,K1,K2) <-> ord_less(int,bit0(K1),bit0(K2)))) # label(fact_43_less__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 263 (all K all A_1 all B all M (zcong(A_1,B,M) -> zcong(times_times(int,A_1,K),times_times(int,B,K),M))) # label(fact_333_zcong__scalar) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 264 (all K_1 (ord_less_eq(int,pls,K_1) <-> ord_less_eq(int,pls,bit0(K_1)))) # label(fact_99_rel__simps_I21_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 265 (all B_1_1 all X_a (number(X_a) -> number_number_of(X_a,B_1_1) = number_number_of(X_a,ti(int,B_1_1)))) # label(tsy_c_Int_Onumber__class_Onumber__of_arg1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 266 (all X_b (ring(X_b) & number(X_b) -> (all V all B all C minus_minus(X_b,times_times(X_b,number_number_of(X_b,V),B),times_times(X_b,number_number_of(X_b,V),C)) = times_times(X_b,number_number_of(X_b,V),minus_minus(X_b,B,C))))) # label(fact_356_right__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 267 (all X_a (number_ring(X_a) -> (all V all W number_number_of(X_a,times_times(int,V,W)) = times_times(X_a,number_number_of(X_a,V),number_number_of(X_a,W))))) # label(fact_144_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 268 (all Z1 all Z2 all Z3 times_times(int,times_times(int,Z1,Z2),Z3) = times_times(int,Z1,times_times(int,Z2,Z3))) # label(fact_84_zmult__assoc) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 269 (all K_1 (ord_less(int,pls,bit1(K_1)) <-> ord_less_eq(int,pls,K_1))) # label(fact_56_rel__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 270 (all X_a (linordered_idom(X_a) -> (all A_2 (ord_less(X_a,A_2,zero_zero(X_a)) <-> ord_less(X_a,plus_plus(X_a,A_2,A_2),zero_zero(X_a)))))) # label(fact_215_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 271 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all X_2 (ord_less(int,X_2,bit1(pls)) <-> ord_less(X_a,number_number_of(X_a,X_2),one_one(X_a)))))) # label(fact_102_less__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 272 (all B_1_1 ti(int,twoSqu579703163sum2sq(B_1_1)) = twoSqu579703163sum2sq(B_1_1)) # label(tsy_c_TwoSquares__Mirabelle__iprzenvtji_Osum2sq_res) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 273 (all X_a (monoid_mult(X_a) -> (all A_1 ti(X_a,A_1) = power_power(X_a,A_1,one_one(nat))))) # label(fact_240_power__one__right) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 274 (all X_b (monoid_mult(X_b) & number(X_b) -> (all W power_power(X_b,number_number_of(X_b,W),number_number_of(nat,bit0(bit1(pls)))) = times_times(X_b,number_number_of(X_b,W),number_number_of(X_b,W))))) # label(fact_10_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 275 (all X_a (number_ring(X_a) -> (all Z plus_plus(X_a,Z,Z) = times_times(X_a,Z,number_number_of(X_a,bit0(bit1(pls))))))) # label(fact_166_mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 276 (all M all X_1 (ord_less_eq(int,zero_zero(int),X_1) -> (ord_less(int,X_1,M) -> (zcong(X_1,zero_zero(int),M) -> ti(int,X_1) = zero_zero(int))))) # label(fact_412_Int2_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 277 (all C all D all A_1 all B all M (zcong(A_1,B,M) -> (zcong(C,D,M) -> zcong(plus_plus(int,A_1,C),plus_plus(int,B,D),M)))) # label(fact_335_zcong__zadd) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 278 (all X_a (comm_semiring_1(X_a) -> (all C all A_1 all B (dvd_dvd(X_a,A_1,B) -> (dvd_dvd(X_a,B,C) -> dvd_dvd(X_a,A_1,C)))))) # label(fact_467_dvd__trans) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 279 (all A_1 all X_1 (ord_less(int,zero_zero(int),X_1) -> (ord_less(int,X_1,A_1) -> (minus_minus(int,A_1,one_one(int)) != ti(int,X_1) -> ord_less(int,X_1,minus_minus(int,A_1,one_one(int))))))) # label(fact_365_Euler_Oaux1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 280 (all X_a (ring(X_a) -> (all X_1 all Y_1 all A_1 all B plus_plus(X_a,times_times(X_a,X_1,minus_minus(X_a,Y_1,B)),times_times(X_a,minus_minus(X_a,X_1,A_1),B)) = minus_minus(X_a,times_times(X_a,X_1,Y_1),times_times(X_a,A_1,B))))) # label(fact_434_mult__diff__mult) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 281 (all K_1 (ord_less(int,min,K_1) <-> ord_less(int,min,bit1(K_1)))) # label(fact_296_rel__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 282 (all B_1_1 all B_2_1 all X_a (ab_group_add(X_a) -> minus_minus(X_a,B_1_1,B_2_1) = ti(X_a,minus_minus(X_a,B_1_1,B_2_1)))) # label(tsy_c_Groups_Ominus__class_Ominus_0_res) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 283 (all K_1 (ord_less_eq(int,K_1,min) <-> ord_less_eq(int,bit0(K_1),min))) # label(fact_305_rel__simps_I28_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 284 (all X_a (linordered_semidom(X_a) -> (all X_1 all Y_1 (ord_less(X_a,power_power(X_a,X_1,number_number_of(nat,bit0(bit1(pls)))),power_power(X_a,Y_1,number_number_of(nat,bit0(bit1(pls))))) -> (ord_less_eq(X_a,zero_zero(X_a),Y_1) -> ord_less(X_a,X_1,Y_1)))))) # label(fact_262_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 285 (all X_a (semiring_1(X_a) -> one_one(X_a) = power_power(X_a,one_one(X_a),number_number_of(nat,bit0(bit1(pls)))))) # label(fact_12_one__power2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 286 (all X_a (number_semiring(X_a) -> (all X_1 all Y_1 plus_plus(X_a,plus_plus(X_a,power_power(X_a,X_1,number_number_of(nat,bit0(bit1(pls)))),power_power(X_a,Y_1,number_number_of(nat,bit0(bit1(pls))))),times_times(X_a,times_times(X_a,number_number_of(X_a,bit0(bit1(pls))),X_1),Y_1)) = power_power(X_a,plus_plus(X_a,X_1,Y_1),number_number_of(nat,bit0(bit1(pls))))))) # label(fact_9_power2__sum) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 287 (all X_a (comm_semiring_1(X_a) -> (all X_1 power_power(X_a,X_1,number_number_of(nat,bit0(bit1(pls)))) = times_times(X_a,X_1,X_1)))) # label(fact_13_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 288 (all X_a (comm_semiring_1(X_a) -> (all X_1 all N (one_one(X_a) = ti(X_a,X_1) | ord_less(nat,zero_zero(nat),N) -> dvd_dvd(X_a,X_1,power_power(X_a,X_1,N)))))) # label(fact_317_dvd__power) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 289 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all X_2 (ord_less_eq(X_a,number_number_of(X_a,X_2),one_one(X_a)) <-> ord_less_eq(int,X_2,bit1(pls)))))) # label(fact_104_le__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 290 (all A_1 all P ((zcong(A_1,zero_zero(int),P) -> zero_zero(int) = legendre(A_1,P)) & (-zcong(A_1,zero_zero(int),P) -> (-quadRes(P,A_1) -> number_number_of(int,min) = legendre(A_1,P)) & (quadRes(P,A_1) -> one_one(int) = legendre(A_1,P))))) # label(fact_382_Legendre__def) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 291 (all K all L minus_minus(int,bit0(K),bit0(L)) = bit0(minus_minus(int,K,L))) # label(fact_351_diff__bin__simps_I7_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 292 (all X_a (comm_semiring_1(X_a) -> (all C all A_1 all B (dvd_dvd(X_a,A_1,B) -> (dvd_dvd(X_a,A_1,C) -> dvd_dvd(X_a,A_1,plus_plus(X_a,B,C))))))) # label(fact_487_dvd__add) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 293 (all X_a (comm_semiring_1(X_a) -> (all A_1 zero_zero(X_a) = times_times(X_a,A_1,zero_zero(X_a))))) # label(fact_197_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 294 (all B_1_1 all B_2_1 all X_a (power(X_a) -> times_times(X_a,B_1_1,B_2_1) = times_times(X_a,B_1_1,ti(X_a,B_2_1)))) # label(tsy_c_Groups_Otimes__class_Otimes_4_arg2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 295 (all K all L plus_plus(int,bit0(K),bit1(L)) = bit1(plus_plus(int,K,L))) # label(fact_150_add__Bit0__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 296 (all X_a (monoid_mult(X_a) -> (all A_1 all N times_times(X_a,power_power(X_a,A_1,N),A_1) = times_times(X_a,A_1,power_power(X_a,A_1,N))))) # label(fact_257_power__commutes) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 297 (all X_a (linordered_ring(X_a) -> (all X_1 all Y_1 ord_less_eq(X_a,zero_zero(X_a),plus_plus(X_a,times_times(X_a,X_1,X_1),times_times(X_a,Y_1,Y_1)))))) # label(fact_233_sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 298 (all K_1 all L_1 (ord_less(int,K_1,L_1) <-> ord_less(int,minus_minus(int,K_1,L_1),zero_zero(int)))) # label(fact_361_less__bin__lemma) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 299 (all B_2 all Q_1 all R_2 (ord_less(int,plus_plus(int,times_times(int,B_2,Q_1),R_2),zero_zero(int)) -> (ord_less_eq(int,zero_zero(int),R_2) -> (ord_less(int,zero_zero(int),B_2) -> ord_less_eq(int,Q_1,zero_zero(int)))))) # label(fact_445_q__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 300 (all P all M ((zero_zero(nat) != M -> times_times(nat,P,power_power(nat,P,minus_minus(nat,M,one_one(nat)))) = power_power(nat,P,M)) & (zero_zero(nat) = M -> one_one(nat) = power_power(nat,P,M)))) # label(fact_387_power__eq__if) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 301 (all A_2 all B_1 all Ma (dvd_dvd(int,Ma,minus_minus(int,A_2,B_1)) <-> zcong(A_2,B_1,Ma))) # label(fact_363_zcong__def) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 302 (all X_a (comm_semiring_1(X_a) -> (all A_1 all C all D plus_plus(X_a,C,plus_plus(X_a,A_1,D)) = plus_plus(X_a,A_1,plus_plus(X_a,C,D))))) # label(fact_78_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 303 (all X_a (mult_zero(X_a) -> zero_zero(X_a) = ti(X_a,zero_zero(X_a)))) # label(tsy_c_Groups_Ozero__class_Ozero_2_res) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 304 (all X_b (number(X_b) & semiring(X_b) -> (all A_1 all B all V plus_plus(X_b,times_times(X_b,A_1,number_number_of(X_b,V)),times_times(X_b,B,number_number_of(X_b,V))) = times_times(X_b,plus_plus(X_b,A_1,B),number_number_of(X_b,V))))) # label(fact_136_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 305 (all Z Z = times_times(real,one_one(real),Z)) # label(fact_426_real__mult__1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 306 (all B_1_1 bit1(ti(int,B_1_1)) = bit1(B_1_1)) # label(tsy_c_Int_OBit1_arg1) # label(hypothesis) # label(non_clause). [assumption]. 1.07/1.37 307 (all R_1 all Q all A_1 (ord_less(int,zero_zero(int),A_1) -> (plus_plus(int,R_1,times_times(int,A_1,Q)) = ti(int,A_1) -> (ord_less_eq(int,zero_zero(int),R_1) -> ord_less_eq(int,Q,one_one(int)))))) # label(fact_450_self__quotient__aux2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.37 308 (all K_1 (ord_less_eq(int,pls,bit1(K_1)) <-> ord_less_eq(int,pls,K_1))) # label(fact_95_rel__simps_I22_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 309 (all K ti(int,K) = minus_minus(int,K,pls)) # label(fact_350_diff__bin__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 310 (all L bit1(minus_minus(int,min,L)) = minus_minus(int,min,bit0(L))) # label(fact_369_diff__bin__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 311 (all B_2 all Q_1 all R_2 (ord_less_eq(int,zero_zero(int),plus_plus(int,times_times(int,B_2,Q_1),R_2)) -> (ord_less(int,R_2,B_2) -> (ord_less(int,zero_zero(int),B_2) -> ord_less_eq(int,zero_zero(int),Q_1))))) # label(fact_444_q__pos__lemma) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 312 (all X_a (comm_semiring_1(X_a) -> (all M all A_1 plus_plus(X_a,M,times_times(X_a,A_1,M)) = times_times(X_a,plus_plus(X_a,A_1,one_one(X_a)),M)))) # label(fact_139_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 313 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all X_2 (ord_less(int,X_2,pls) <-> ord_less(X_a,number_number_of(X_a,X_2),zero_zero(X_a)))))) # label(fact_245_less__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 314 (all A_2 all B_1 all C_1 (zero_zero(real) != C_1 -> (times_times(real,C_1,B_1) = times_times(real,C_1,A_2) <-> B_1 = A_2))) # label(fact_431_real__mult__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 315 (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_413_zpower__zdvd__prop1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 316 (all K1 all K2 (ord_less_eq(int,K1,K2) <-> ord_less_eq(int,bit0(K1),bit0(K2)))) # label(fact_46_less__eq__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 317 (all X_a (linord219039673up_add(X_a) -> zero_zero(X_a) = ti(X_a,zero_zero(X_a)))) # label(tsy_c_Groups_Ozero__class_Ozero_3_res) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 318 (all A_1 all N (ord_less(nat,zero_zero(nat),N) -> (ord_less(real,zero_zero(real),A_1) -> (exists R (ord_less(real,zero_zero(real),R) & power_power(real,R,N) = A_1))))) # label(fact_464_realpow__pos__nth) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 319 (all Z_1 (ord_less_eq(int,one_one(int),Z_1) <-> ord_less(int,zero_zero(int),Z_1))) # label(fact_241_int__one__le__iff__zero__less) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 320 (all X_a (mult_zero(X_a) -> (all A_1 zero_zero(X_a) = times_times(X_a,zero_zero(X_a),A_1)))) # label(fact_469_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 321 (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_354_zdvd__zdiffD) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 322 (all X_a (comm_semiring_1(X_a) -> (all A_1 times_times(X_a,one_one(X_a),A_1) = ti(X_a,A_1)))) # label(fact_112_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 323 (all A_1 all N all B (dvd_dvd(nat,power_power(nat,A_1,N),power_power(nat,B,N)) -> (zero_zero(nat) != N -> dvd_dvd(nat,A_1,B)))) # label(fact_407_divides__rev) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 324 (all X_a (linordered_semidom(X_a) -> (all N all A_1 (ord_less(X_a,one_one(X_a),A_1) -> ord_less(X_a,one_one(X_a),times_times(X_a,A_1,power_power(X_a,A_1,N))))))) # label(fact_175_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 325 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B all C (dvd_dvd(X_a,times_times(X_a,A_1,B),C) -> dvd_dvd(X_a,B,C))))) # label(fact_486_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 326 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B all C (dvd_dvd(X_a,times_times(X_a,A_1,B),C) -> dvd_dvd(X_a,A_1,C))))) # label(fact_485_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 327 (all V_1 all V ((ord_less(int,V,pls) -> plus_plus(nat,number_number_of(nat,V),number_number_of(nat,V_1)) = number_number_of(nat,V_1)) & (-ord_less(int,V,pls) -> (-ord_less(int,V_1,pls) -> plus_plus(nat,number_number_of(nat,V),number_number_of(nat,V_1)) = number_number_of(nat,plus_plus(int,V,V_1))) & (ord_less(int,V_1,pls) -> plus_plus(nat,number_number_of(nat,V),number_number_of(nat,V_1)) = number_number_of(nat,V))))) # label(fact_52_add__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 328 (all K_1 all L_1 (ord_less(int,bit1(K_1),bit1(L_1)) <-> ord_less(int,K_1,L_1))) # label(fact_39_rel__simps_I17_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 329 (all K_1 all L_1 (bit0(K_1) = bit0(L_1) <-> ti(int,K_1) = ti(int,L_1))) # label(fact_83_rel__simps_I48_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 330 (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_24_zle__trans) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 331 (all X_a (linordered_idom(X_a) -> (all X_2 all Y_2 (zero_zero(X_a) = ti(X_a,Y_2) & zero_zero(X_a) = ti(X_a,X_2) <-> plus_plus(X_a,power_power(X_a,X_2,number_number_of(nat,bit0(bit1(pls)))),power_power(X_a,Y_2,number_number_of(nat,bit0(bit1(pls))))) = zero_zero(X_a))))) # label(fact_256_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 332 (all X_a (number_ring(X_a) -> (all V all W number_number_of(X_a,plus_plus(int,V,W)) = plus_plus(X_a,number_number_of(X_a,V),number_number_of(X_a,W))))) # label(fact_148_number__of__add) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 333 (all L_1 (min = bit1(L_1) <-> min = ti(int,L_1))) # label(fact_286_rel__simps_I43_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 334 (all B_1_1 all B_2_1 all X_a (power(X_a) -> power_power(X_a,B_1_1,B_2_1) = power_power(X_a,ti(X_a,B_1_1),B_2_1))) # label(tsy_c_Power_Opower__class_Opower_arg1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 335 (all X_a (linordered_semidom(X_a) -> (all N all A_1 (ord_less(X_a,one_one(X_a),A_1) -> (ord_less(nat,zero_zero(nat),N) -> ord_less(X_a,one_one(X_a),power_power(X_a,A_1,N))))))) # label(fact_316_one__less__power) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 336 (all X_a (comm_semiring_1(X_a) -> (all Lx all Ly all Rx times_times(X_a,times_times(X_a,Lx,Ly),Rx) = times_times(X_a,Lx,times_times(X_a,Ly,Rx))))) # label(fact_70_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 337 (all X_a (zero_neq_one(X_a) -> ti(X_a,zero_zero(X_a)) = zero_zero(X_a))) # label(tsy_c_Groups_Ozero__class_Ozero_0_res) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 338 (all X_a (mult_zero(X_a) -> (all A_1 zero_zero(X_a) = times_times(X_a,A_1,zero_zero(X_a))))) # label(fact_470_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 339 (exists X (ord_less(int,X,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) & zcong(s1,X,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) & (all Y (ord_less_eq(int,zero_zero(int),Y) & zcong(s1,Y,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) & ord_less(int,Y,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) -> ti(int,Y) = ti(int,X))) & ord_less_eq(int,zero_zero(int),X))) # label(fact_180__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]. 1.07/1.38 340 (all Z all W times_times(real,Z,W) = times_times(real,W,Z)) # label(fact_427_real__mult__commute) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 341 (all A_1 all N all P (zprime(P) -> (dvd_dvd(int,P,power_power(int,A_1,N)) -> dvd_dvd(int,P,A_1)))) # label(fact_218_zprime__zdvd__power) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 342 (all X_a (ring_n68954251visors(X_a) -> (all A_2 all B_1 (zero_zero(X_a) = times_times(X_a,A_2,B_1) <-> ti(X_a,B_1) = zero_zero(X_a) | ti(X_a,A_2) = zero_zero(X_a))))) # label(fact_471_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 343 (all X_a (linord626643107strict(X_a) -> (all V all U all Y_1 all X_1 all A_1 (ord_less(X_a,X_1,A_1) -> (ord_less(X_a,Y_1,A_1) -> (ord_less_eq(X_a,zero_zero(X_a),U) -> (ord_less_eq(X_a,zero_zero(X_a),V) -> (one_one(X_a) = plus_plus(X_a,U,V) -> ord_less(X_a,plus_plus(X_a,times_times(X_a,U,X_1),times_times(X_a,V,Y_1)),A_1))))))))) # label(fact_463_convex__bound__lt) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 344 (all M all A_1 (ord_less_eq(int,zero_zero(int),A_1) -> (ord_less(int,A_1,M) -> (zcong(A_1,zero_zero(int),M) -> zero_zero(int) = ti(int,A_1))))) # label(fact_342_zcong__zless__0) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 345 (all X_a (number_ring(X_a) -> (all W number_number_of(X_a,bit1(W)) = plus_plus(X_a,plus_plus(X_a,one_one(X_a),number_number_of(X_a,W)),number_number_of(X_a,W))))) # label(fact_152_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 346 (all L bit1(minus_minus(int,min,L)) = minus_minus(int,pls,bit1(L))) # label(fact_367_diff__bin__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 347 (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_411_zprime__zdvd__zmult__better) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 348 (all X_a (comm_semiring_1(X_a) -> (all A_1 dvd_dvd(X_a,A_1,zero_zero(X_a))))) # label(fact_465_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 349 (all X_1 all Q all N all R_1 (plus_plus(nat,times_times(nat,Q,N),R_1) = X_1 -> (ord_less(nat,zero_zero(nat),R_1) -> (ord_less(nat,R_1,N) -> -dvd_dvd(nat,N,X_1))))) # label(fact_415_divides__div__not) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 350 (all X_1 all Y_1 (ti(int,Y_1) = ti(int,X_1) | ord_less(int,Y_1,X_1) | ord_less(int,X_1,Y_1))) # label(fact_23_zless__linear) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 351 (all K all L bit0(times_times(int,K,L)) = times_times(int,bit0(K),L)) # label(fact_122_mult__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 352 (all K_1 (ord_less_eq(int,bit0(K_1),pls) <-> ord_less_eq(int,K_1,pls))) # label(fact_98_rel__simps_I27_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 353 (all X_a (comm_semiring_1(X_a) -> (all A_1 times_times(X_a,zero_zero(X_a),A_1) = zero_zero(X_a)))) # label(fact_196_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 354 (all A_1 times_times(int,A_1,power_power(int,A_1,number_number_of(nat,bit0(bit1(pls))))) = power_power(int,A_1,number_number_of(nat,bit1(bit1(pls))))) # label(fact_11_cube__square) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 355 (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_381_zcong__neg__1__impl__ne__1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 356 (all Z all W (ord_less_eq(int,Z,W) -> (ord_less_eq(int,W,Z) -> ti(int,W) = ti(int,Z)))) # label(fact_25_zle__antisym) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 357 (all B_1_1 all B_2_1 all X_a (order(X_a) -> (ord_less(X_a,ti(X_a,B_1_1),B_2_1) <-> ord_less(X_a,B_1_1,B_2_1)))) # label(tsy_c_Orderings_Oord__class_Oless_arg1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 358 (all X_a (linordered_idom(X_a) -> (all A_2 (ord_less(X_a,zero_zero(X_a),power_power(X_a,A_2,number_number_of(nat,bit0(bit1(pls))))) <-> zero_zero(X_a) != ti(X_a,A_2))))) # label(fact_255_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 359 (all V_2 all V_3 ((-ord_less_eq(int,V_2,V_3) -> ord_less_eq(int,V_2,pls)) <-> ord_less_eq(nat,number_number_of(nat,V_2),number_number_of(nat,V_3)))) # label(fact_238_le__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 360 (all X_a (number(X_a) -> (all W_1 all X_2 (number_number_of(X_a,W_1) = ti(X_a,X_2) <-> number_number_of(X_a,W_1) = ti(X_a,X_2))))) # label(fact_81_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 361 (all B_1_1 all B_2_1 all X_a (dvd(X_a) -> times_times(X_a,ti(X_a,B_1_1),B_2_1) = times_times(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_0_arg1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 362 (all C_1 all D_1 all A_2 all B_1 all Ma (zcong(A_2,B_1,Ma) -> (zcong(C_1,times_times(int,D_1,A_2),Ma) <-> zcong(C_1,times_times(int,D_1,B_1),Ma)))) # label(fact_395_zcong__zmult__prop2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 363 (all X_2 all W_1 (ord_less(nat,zero_zero(nat),X_2) | zero_zero(nat) = number_number_of(nat,W_1) <-> ord_less(nat,zero_zero(nat),power_power(nat,X_2,number_number_of(nat,W_1))))) # label(fact_283_zero__less__power__nat__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 364 (all X_a (number_ring(X_a) -> one_one(X_a) = number_number_of(X_a,bit1(pls)))) # label(fact_157_semiring__norm_I110_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 365 (all X_a (number_ring(X_a) -> plus_plus(X_a,one_one(X_a),one_one(X_a)) = number_number_of(X_a,bit0(bit1(pls))))) # label(fact_18_one__add__one__is__two) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 366 (all X_2 all Y_2 all Z_1 (ord_less(real,zero_zero(real),Z_1) -> (ord_less(real,X_2,Y_2) <-> ord_less(real,times_times(real,X_2,Z_1),times_times(real,Y_2,Z_1))))) # label(fact_439_real__mult__less__iff1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 367 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B dvd_dvd(X_a,A_1,times_times(X_a,A_1,B))))) # label(fact_479_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 368 (all X_a (linordered_semidom(X_a) -> (all A_1 all N all N_2 (ord_less_eq(nat,N,N_2) -> (ord_less_eq(X_a,zero_zero(X_a),A_1) -> (ord_less_eq(X_a,A_1,one_one(X_a)) -> ord_less_eq(X_a,power_power(X_a,A_1,N_2),power_power(X_a,A_1,N)))))))) # label(fact_213_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 369 (all W_1 all Z_1 (ord_less_eq(int,W_1,Z_1) <-> ord_less(int,W_1,plus_plus(int,Z_1,one_one(int))))) # label(fact_63_zle__add1__eq__le) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 370 (all A_1 all B minus_minus(int,plus_plus(int,minus_minus(int,power_power(int,A_1,number_number_of(nat,bit1(bit1(pls)))),times_times(int,times_times(int,number_number_of(int,bit1(bit1(pls))),power_power(int,A_1,number_number_of(nat,bit0(bit1(pls))))),B)),times_times(int,times_times(int,number_number_of(int,bit1(bit1(pls))),A_1),power_power(int,B,number_number_of(nat,bit0(bit1(pls)))))),power_power(int,B,number_number_of(nat,bit1(bit1(pls))))) = power_power(int,minus_minus(int,A_1,B),number_number_of(nat,bit1(bit1(pls))))) # label(fact_377_zdiff__power3) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 371 (all X_a (comm_semiring_1(X_a) -> (all Lx all Ly all Rx all Ry times_times(X_a,times_times(X_a,Lx,Ly),times_times(X_a,Rx,Ry)) = times_times(X_a,times_times(X_a,Lx,Rx),times_times(X_a,Ly,Ry))))) # label(fact_66_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 372 (all B all Q_1 all R_2 all Q all R_1 (ord_less_eq(int,plus_plus(int,times_times(int,B,Q_1),R_2),plus_plus(int,times_times(int,B,Q),R_1)) -> (ord_less_eq(int,R_1,zero_zero(int)) -> (ord_less(int,B,R_1) -> (ord_less(int,B,R_2) -> ord_less_eq(int,Q,Q_1)))))) # label(fact_448_unique__quotient__lemma__neg) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 373 (all X_a (linordered_semidom(X_a) -> (all N all A_1 (ord_less(X_a,zero_zero(X_a),A_1) -> (ord_less(X_a,A_1,one_one(X_a)) -> ord_less(X_a,times_times(X_a,A_1,power_power(X_a,A_1,N)),power_power(X_a,A_1,N))))))) # label(fact_230_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 374 (all A_1 all R_1 all B all M all C all D all N plus_plus(int,times_times(int,minus_minus(int,A_1,times_times(int,R_1,B)),M),times_times(int,minus_minus(int,C,times_times(int,R_1,D)),N)) = minus_minus(int,plus_plus(int,times_times(int,A_1,M),times_times(int,C,N)),times_times(int,R_1,plus_plus(int,times_times(int,B,M),times_times(int,D,N))))) # label(fact_362_xzgcda__linear__aux1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 375 (all W ord_less_eq(int,W,W)) # label(fact_20_zle__refl) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 376 (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_406_divides__exp2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 377 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all X_2 all Y_2 (ord_less_eq(int,X_2,Y_2) <-> ord_less_eq(X_a,number_number_of(X_a,X_2),number_number_of(X_a,Y_2)))))) # label(fact_31_le__number__of) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 378 (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_435_real__mult__less__mono2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 379 (all C all D all A_1 all B all M (zcong(A_1,B,M) -> (zcong(C,D,M) -> zcong(times_times(int,A_1,C),times_times(int,B,D),M)))) # label(fact_331_zcong__zmult) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 380 (all B_1_1 all B_2_1 all X_a (mult_zero(X_a) -> ti(X_a,times_times(X_a,B_1_1,B_2_1)) = times_times(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_3_res) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 381 (all K_1 (ord_less_eq(int,min,bit1(K_1)) <-> ord_less_eq(int,min,K_1))) # label(fact_302_rel__simps_I26_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 382 (all W times_times(int,pls,W) = pls) # label(fact_121_mult__Pls) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 383 (all Z1 all Z2 all W times_times(real,plus_plus(real,Z1,Z2),W) = plus_plus(real,times_times(real,Z1,W),times_times(real,Z2,W))) # label(fact_433_real__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 384 (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_400_zcong__zpower) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 385 (all V_2 (ord_less(nat,zero_zero(nat),number_number_of(nat,V_2)) <-> ord_less(int,pls,V_2))) # label(fact_318_less__0__number__of) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 386 (all C all A_1 all B all M (zcong(A_1,B,M) -> zcong(plus_plus(int,A_1,C),plus_plus(int,B,C),M))) # label(fact_397_zcong__shift) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 387 (all X_a (linordered_semidom(X_a) -> (all N all A_1 all B (ord_less(X_a,A_1,B) -> (ord_less_eq(X_a,zero_zero(X_a),A_1) -> (ord_less(nat,zero_zero(nat),N) -> ord_less(X_a,power_power(X_a,A_1,N),power_power(X_a,B,N)))))))) # label(fact_195_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 388 (all C all A_1 all B (dvd_dvd(nat,A_1,B) -> dvd_dvd(nat,times_times(nat,C,A_1),times_times(nat,C,B)))) # label(fact_390_divides__mul__l) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 389 (all V all W times_times(int,number_number_of(int,V),number_number_of(int,W)) = number_number_of(int,times_times(int,V,W))) # label(fact_129_times__numeral__code_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 390 (all B_1_1 all B_2_1 all X_a (ab_group_add(X_a) -> plus_plus(X_a,B_1_1,ti(X_a,B_2_1)) = plus_plus(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Oplus__class_Oplus_1_arg2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 391 (all X_a (comm_semiring_1(X_a) -> (all X_1 one_one(X_a) = power_power(X_a,X_1,zero_zero(nat))))) # label(fact_308_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 392 (all A_1 all B times_times(int,plus_plus(int,A_1,B),minus_minus(int,A_1,B)) = minus_minus(int,power_power(int,A_1,number_number_of(nat,bit0(bit1(pls)))),power_power(int,B,number_number_of(nat,bit0(bit1(pls)))))) # label(fact_374_zspecial__product) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 393 (all X_a (number_ring(X_a) -> (all W number_number_of(X_a,bit0(W)) = times_times(X_a,plus_plus(X_a,one_one(X_a),one_one(X_a)),number_number_of(X_a,W))))) # label(fact_160_double__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 394 (all X_a (number_ring(X_a) -> number_number_of(X_a,bit1(pls)) = one_one(X_a))) # label(fact_156_numeral__1__eq__1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 395 (all W_1 all Z_1 (ord_less(int,W_1,Z_1) <-> ord_less_eq(int,plus_plus(int,W_1,one_one(int)),Z_1))) # label(fact_62_add1__zle__eq) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 396 (all X_a (linordered_idom(X_a) -> (all A_1 ord_less_eq(X_a,zero_zero(X_a),power_power(X_a,A_1,number_number_of(nat,bit0(bit1(pls)))))))) # label(fact_251_zero__le__power2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 397 (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_437_real__mult__le__cancel__iff2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 398 (all K_1 all L_1 (ord_less_eq(int,bit0(K_1),bit1(L_1)) <-> ord_less_eq(int,K_1,L_1))) # label(fact_97_rel__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 399 (all P (zprime(P) -> (number_number_of(int,bit0(bit1(pls))) != ti(int,P) -> (ti(int,P) != number_number_of(int,bit1(bit1(pls))) -> ord_less_eq(int,number_number_of(int,bit1(bit0(bit1(pls)))),P))))) # label(fact_170_prime__g__5) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 400 (all K_1 (ord_less(int,bit0(K_1),min) <-> ord_less_eq(int,K_1,min))) # label(fact_313_rel__simps_I11_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 401 (all X_a (linordered_idom(X_a) -> (all A_1 all N ord_less_eq(X_a,zero_zero(X_a),power_power(X_a,A_1,times_times(nat,number_number_of(nat,bit0(bit1(pls))),N)))))) # label(fact_267_zero__le__even__power_H) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 402 (all B_1_1 all B_2_1 all X_a (semiring(X_a) -> plus_plus(X_a,B_1_1,B_2_1) = plus_plus(X_a,ti(X_a,B_1_1),B_2_1))) # label(tsy_c_Groups_Oplus__class_Oplus_0_arg1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 403 (all X_a (linordered_semidom(X_a) -> (all N all A_1 all B (ord_less_eq(X_a,A_1,B) -> (ord_less_eq(X_a,zero_zero(X_a),A_1) -> ord_less_eq(X_a,power_power(X_a,A_1,N),power_power(X_a,B,N))))))) # label(fact_207_power__mono) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 404 (all X_a (power(X_a) -> ti(X_a,one_one(X_a)) = one_one(X_a))) # label(tsy_c_Groups_Oone__class_Oone_1_res) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 405 (all X_b (semiring(X_b) & number(X_b) -> (all V all B all C times_times(X_b,number_number_of(X_b,V),plus_plus(X_b,B,C)) = plus_plus(X_b,times_times(X_b,number_number_of(X_b,V),B),times_times(X_b,number_number_of(X_b,V),C))))) # label(fact_137_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 406 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all Y_2 (ord_less_eq(int,bit1(pls),Y_2) <-> ord_less_eq(X_a,one_one(X_a),number_number_of(X_a,Y_2)))))) # label(fact_105_le__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 407 (all K bit0(K) != min) # label(fact_290_rel__simps_I45_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 408 (all X_1 ord_less_eq(int,X_1,power_power(int,X_1,number_number_of(nat,bit0(bit1(pls)))))) # label(fact_135_power2__ge__self) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 409 (all X_2 all P_1 (zcong(X_2,zero_zero(int),P_1) <-> dvd_dvd(int,P_1,X_2))) # label(fact_409_zcong__eq__zdvd__prop) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 410 (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_209_zcong__zpower__zmult) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 411 (all K bit1(K) != pls) # label(fact_114_rel__simps_I46_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 412 (all X_a (comm_semiring_1(X_a) -> (all Lx all Rx all Ry times_times(X_a,Lx,times_times(X_a,Rx,Ry)) = times_times(X_a,times_times(X_a,Lx,Rx),Ry)))) # label(fact_71_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 413 (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_402_nat__mult__dvd__cancel__disj_H) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 414 (all X_1 all 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))))) = times_times(nat,plus_plus(nat,X_1,Y_1),minus_minus(nat,X_1,Y_1))) # label(fact_388_diff__square) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 415 (all X_a (number_semiring(X_a) -> (all Z times_times(X_a,Z,number_number_of(X_a,bit0(bit1(pls)))) = plus_plus(X_a,Z,Z)))) # label(fact_165_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 416 (all K_1 (ti(int,K_1) = min <-> min = bit1(K_1))) # label(fact_285_rel__simps_I47_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 417 (all X_2 all N_1 (zero_zero(nat) = N_1 | X_2 = one_one(nat) <-> one_one(nat) = power_power(nat,X_2,N_1))) # label(fact_410_exp__eq__1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 418 (all X_a (comm_semiring_1(X_a) -> (all Lx all Rx all Ry times_times(X_a,Rx,times_times(X_a,Lx,Ry)) = times_times(X_a,Lx,times_times(X_a,Rx,Ry))))) # label(fact_72_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 419 (all X_a (comm_semiring_1(X_a) -> (all A_1 ti(X_a,A_1) = plus_plus(X_a,zero_zero(X_a),A_1)))) # label(fact_198_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 420 (all X_a (comm_semiring_1(X_a) -> (all B all A_1 all C (dvd_dvd(X_a,A_1,C) -> dvd_dvd(X_a,A_1,times_times(X_a,B,C)))))) # label(fact_482_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 421 (all B_1_1 all B_2_1 product_Pair(int,int,B_1_1,B_2_1) = product_Pair(int,int,ti(int,B_1_1),B_2_1)) # label(tsy_c_Product__Type_OPair_arg1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 422 (all X_a (linord581940658strict(X_a) -> (all X_2 all Y_2 (zero_zero(X_a) = ti(X_a,X_2) & zero_zero(X_a) = ti(X_a,Y_2) <-> plus_plus(X_a,times_times(X_a,X_2,X_2),times_times(X_a,Y_2,Y_2)) = zero_zero(X_a))))) # label(fact_216_sum__squares__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 423 (all X_a (linordered_semidom(X_a) -> (all M all N all A_1 (ord_less(X_a,one_one(X_a),A_1) -> (ord_less_eq(X_a,power_power(X_a,A_1,M),power_power(X_a,A_1,N)) -> ord_less_eq(nat,M,N)))))) # label(fact_176_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 424 (all K all M zcong(K,K,M)) # label(fact_322_zcong__refl) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 425 (all K_1 all L_1 (ord_less_eq(int,K_1,L_1) <-> ord_less_eq(int,bit1(K_1),bit1(L_1)))) # label(fact_41_rel__simps_I34_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 426 (all K_1 (ord_less_eq(int,min,bit0(K_1)) <-> ord_less(int,min,K_1))) # label(fact_312_rel__simps_I25_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 427 (all Z all W (ord_less_eq(real,Z,W) | ord_less_eq(real,W,Z))) # label(fact_455_real__le__linear) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 428 (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_307_power__dvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 429 (all X_a (linordered_semidom(X_a) -> (all X_2 all Y_2 all B_1 (ord_less(X_a,one_one(X_a),B_1) -> (ord_less(X_a,power_power(X_a,B_1,X_2),power_power(X_a,B_1,Y_2)) <-> ord_less(nat,X_2,Y_2)))))) # label(fact_271_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 430 (all B_1_1 all B_2_1 all X_a (no_zero_divisors(X_a) -> times_times(X_a,B_1_1,B_2_1) = times_times(X_a,ti(X_a,B_1_1),B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_2_arg1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 431 (all K_1 (ord_less(int,K_1,pls) <-> ord_less(int,bit1(K_1),pls))) # label(fact_90_rel__simps_I12_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 432 (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_284_nat__power__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 433 (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_28_zpower__zpower) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 434 (all K all L bit0(K) != bit1(L)) # label(fact_117_rel__simps_I49_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 435 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all Y_2 (ord_less_eq(int,pls,Y_2) <-> ord_less_eq(X_a,zero_zero(X_a),number_number_of(X_a,Y_2)))))) # label(fact_246_le__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 436 (all B_1_1 all B_2_1 all X_a (no_zero_divisors(X_a) -> times_times(X_a,B_1_1,ti(X_a,B_2_1)) = times_times(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_2_arg2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 437 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all X_2 all Y_2 (ord_less(X_a,number_number_of(X_a,X_2),number_number_of(X_a,Y_2)) <-> ord_less(int,X_2,Y_2))))) # label(fact_30_less__number__of) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 438 (all B_1_1 all B_2_1 all X_a (order(X_a) -> (ord_less_eq(X_a,B_1_1,ti(X_a,B_2_1)) <-> ord_less_eq(X_a,B_1_1,B_2_1)))) # label(tsy_c_Orderings_Oord__class_Oless__eq_arg2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 439 (all C all D all A_1 all B all M (zcong(A_1,B,M) -> (zcong(C,D,M) -> zcong(minus_minus(int,A_1,C),minus_minus(int,B,D),M)))) # label(fact_349_zcong__zdiff) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 440 (all T all A ti(T,A) = ti(T,ti(T,A))) # label(help_ti_idem) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 441 (all B all A_1 (ord_less(int,zero_zero(int),A_1) -> (ord_less(int,zero_zero(int),times_times(int,A_1,B)) -> ord_less(int,zero_zero(int),B)))) # label(fact_227_pos__zmult__pos) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 442 (all X_a (linordered_semidom(X_a) -> (all X_2 all Y_2 all B_1 (ord_less(X_a,one_one(X_a),B_1) -> (ord_less_eq(X_a,power_power(X_a,B_1,X_2),power_power(X_a,B_1,Y_2)) <-> ord_less_eq(nat,X_2,Y_2)))))) # label(fact_177_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 443 -quadRes(plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),number_number_of(int,min)) -> one_one(int) != legendre(number_number_of(int,min),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) # label(fact_348__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]. 1.07/1.38 444 (all X_a (linordered_idom(X_a) -> (all A_1 -ord_less(X_a,power_power(X_a,A_1,number_number_of(nat,bit0(bit1(pls)))),zero_zero(X_a))))) # label(fact_254_power2__less__0) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 445 ord_less(int,one_one(int),t) -> (exists X exists 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)))))) # 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]. 1.07/1.38 446 (all B_1_1 all X_a (number(X_a) -> number_number_of(X_a,B_1_1) = ti(X_a,number_number_of(X_a,B_1_1)))) # label(tsy_c_Int_Onumber__class_Onumber__of_res) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 447 (all B_1_1 all B_2_1 all X_a (dvd(X_a) -> times_times(X_a,B_1_1,B_2_1) = ti(X_a,times_times(X_a,B_1_1,B_2_1)))) # label(tsy_c_Groups_Otimes__class_Otimes_0_res) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 448 (all X_a (linord219039673up_add(X_a) -> (all A_2 (zero_zero(X_a) = plus_plus(X_a,A_2,A_2) <-> zero_zero(X_a) = ti(X_a,A_2))))) # label(fact_201_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.07/1.38 449 (all X_a (number_ring(X_a) -> (all X_1 all Y_1 power_power(X_a,minus_minus(X_a,X_1,Y_1),number_number_of(nat,bit0(bit1(pls)))) = minus_minus(X_a,plus_plus(X_a,power_power(X_a,X_1,number_number_of(nat,bit0(bit1(pls)))),power_power(X_a,Y_1,number_number_of(nat,bit0(bit1(pls))))),times_times(X_a,times_times(X_a,number_number_of(X_a,bit0(bit1(pls))),X_1),Y_1))))) # label(fact_375_power2__diff) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 450 (all K1 all K2 (ord_less_eq(int,K1,K2) <-> ord_less_eq(int,bit1(K1),bit1(K2)))) # label(fact_40_less__eq__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 451 (all X_a (ring_1(X_a) -> (all X_1 minus_minus(X_a,times_times(X_a,X_1,X_1),one_one(X_a)) = times_times(X_a,plus_plus(X_a,X_1,one_one(X_a)),minus_minus(X_a,X_1,one_one(X_a)))))) # label(fact_441_real__squared__diff__one__factored) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 452 (all X_a (linord581940658strict(X_a) -> (all X_2 all Y_2 (ord_less_eq(X_a,plus_plus(X_a,times_times(X_a,X_2,X_2),times_times(X_a,Y_2,Y_2)),zero_zero(X_a)) <-> ti(X_a,X_2) = zero_zero(X_a) & zero_zero(X_a) = ti(X_a,Y_2))))) # label(fact_234_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 453 (all B_1_1 all B_2_1 all X_a (power(X_a) -> power_power(X_a,B_1_1,B_2_1) = power_power(X_a,B_1_1,ti(nat,B_2_1)))) # label(tsy_c_Power_Opower__class_Opower_arg2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 454 (all N all M (ord_less(int,zero_zero(int),M) -> (ord_less(int,M,N) -> -dvd_dvd(int,N,M)))) # label(fact_188_zdvd__not__zless) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 455 (all W_1 all Z_1 (ord_less(int,W_1,plus_plus(int,Z_1,one_one(int))) <-> ti(int,W_1) = ti(int,Z_1) | ord_less(int,W_1,Z_1))) # label(fact_100_zless__add1__eq) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 456 (all C all A_1 all B all M (zcong(A_1,B,M) -> (zcong(B,C,M) -> zcong(A_1,C,M)))) # label(fact_323_zcong__trans) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 457 (all B all A_1 all P (zprime(P) -> (ord_less(int,zero_zero(int),A_1) -> (-zcong(B,zero_zero(int),P) & -zcong(A_1,zero_zero(int),P) -> -zcong(times_times(int,A_1,B),zero_zero(int),P))))) # label(fact_416_zcong__zprime__prod__zero__contra) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 458 (all B_1_1 all B_2_1 all X_a (semiring(X_a) -> times_times(X_a,B_1_1,B_2_1) = times_times(X_a,ti(X_a,B_1_1),B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_1_arg1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 459 (all X_a (semiri456707255roduct(X_a) -> (all B_1 all A_2 (plus_plus(X_a,B_1,A_2) = ti(X_a,B_1) <-> ti(X_a,A_2) = zero_zero(X_a))))) # label(fact_200_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 460 (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_352_zdiff__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 461 (all B_1_1 all B_2_1 all X_a (mult_zero(X_a) -> times_times(X_a,B_1_1,ti(X_a,B_2_1)) = times_times(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_3_arg2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 462 (all A_1 all P (zprime(P) -> (ord_less(int,zero_zero(int),A_1) -> (ord_less(int,A_1,P) -> (zcong(times_times(int,A_1,A_1),one_one(int),P) -> one_one(int) = ti(int,A_1) | minus_minus(int,P,one_one(int)) = ti(int,A_1)))))) # label(fact_373_zcong__square__zless) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 463 (all X_a (comm_semiring_1(X_a) -> (all A_1 all M all B plus_plus(X_a,times_times(X_a,A_1,M),times_times(X_a,B,M)) = times_times(X_a,plus_plus(X_a,A_1,B),M)))) # label(fact_107_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 464 (all K ti(int,K) = plus_plus(int,pls,K)) # label(fact_124_add__Pls) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 465 (all X_a (number_ring(X_a) -> zero_zero(X_a) = number_number_of(X_a,pls))) # label(fact_220_number__of__Pls) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 466 (all X_a (no_zero_divisors(X_a) -> zero_zero(X_a) = ti(X_a,zero_zero(X_a)))) # label(tsy_c_Groups_Ozero__class_Ozero_1_res) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 467 (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_405_zdvd__bounds) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 468 (all X_a (number_semiring(X_a) -> (all Z times_times(X_a,number_number_of(X_a,bit0(bit1(pls))),Z) = plus_plus(X_a,Z,Z)))) # label(fact_163_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 469 (all X_a (comm_semiring_1(X_a) -> (all A_1 times_times(X_a,A_1,one_one(X_a)) = ti(X_a,A_1)))) # label(fact_111_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 470 (all N all M ((M = zero_zero(nat) -> zero_zero(nat) = times_times(nat,M,N)) & (M != zero_zero(nat) -> times_times(nat,M,N) = plus_plus(nat,N,times_times(nat,minus_minus(nat,M,one_one(nat)),N))))) # label(fact_386_mult__eq__if) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 471 (all A_1 all B all C (minus_minus(int,A_1,B) = ti(int,C) -> plus_plus(int,C,B) = ti(int,A_1))) # label(fact_394_Int2_Oaux1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 472 (all K_1 (ti(int,K_1) = pls <-> pls = bit0(K_1))) # label(fact_118_rel__simps_I44_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 473 (all X_a (comm_semiring_1(X_a) -> (all Lx all Ly all Rx all Ry times_times(X_a,times_times(X_a,Lx,Ly),times_times(X_a,Rx,Ry)) = times_times(X_a,Rx,times_times(X_a,times_times(X_a,Lx,Ly),Ry))))) # label(fact_67_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 474 (all C_1 all X_2 all Ta all A_2 all D_1 (dvd_dvd(int,A_2,D_1) -> (dvd_dvd(int,A_2,plus_plus(int,X_2,Ta)) <-> dvd_dvd(int,A_2,plus_plus(int,plus_plus(int,X_2,times_times(int,C_1,D_1)),Ta))))) # label(fact_211_zdvd__period) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 475 (all X_a (comm_monoid_mult(X_a) -> (all A_1 all B all N times_times(X_a,power_power(X_a,A_1,N),power_power(X_a,B,N)) = power_power(X_a,times_times(X_a,A_1,B),N)))) # label(fact_258_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 476 (all X_a (comm_semiring(X_a) -> (all A_1 all B all C plus_plus(X_a,times_times(X_a,A_1,C),times_times(X_a,B,C)) = times_times(X_a,plus_plus(X_a,A_1,B),C)))) # label(fact_476_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 477 (all X_a (monoid_mult(X_a) -> (all N power_power(X_a,one_one(X_a),N) = one_one(X_a)))) # label(fact_260_power__one) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 478 (all K all L bit0(L) != bit1(K)) # label(fact_116_rel__simps_I50_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 479 (all Ma all N_1 (one_one(int) = ti(int,N_1) & ti(int,Ma) = one_one(int) | ti(int,N_1) = number_number_of(int,min) & ti(int,Ma) = number_number_of(int,min) <-> one_one(int) = times_times(int,Ma,N_1))) # label(fact_315_zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 480 (all N all B all A_1 all P (zprime(P) -> (-dvd_dvd(int,P,A_1) -> (dvd_dvd(int,power_power(int,P,N),times_times(int,A_1,B)) -> dvd_dvd(int,power_power(int,P,N),B))))) # label(fact_231_zprime__power__zdvd__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 481 (all A_1 all B power_power(int,minus_minus(int,A_1,B),number_number_of(nat,bit0(bit1(pls)))) = plus_plus(int,minus_minus(int,power_power(int,A_1,number_number_of(nat,bit0(bit1(pls)))),times_times(int,times_times(int,number_number_of(int,bit0(bit1(pls))),A_1),B)),power_power(int,B,number_number_of(nat,bit0(bit1(pls)))))) # label(fact_376_zdiff__power2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 482 (all B_1_1 all B_2_1 legendre(B_1_1,B_2_1) = legendre(B_1_1,ti(int,B_2_1))) # label(tsy_c_Residues_OLegendre_arg2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 483 (all A_1 all B all P all Q twoSqu579703163sum2sq(product_Pair(int,int,plus_plus(int,times_times(int,A_1,P),times_times(int,B,Q)),minus_minus(int,times_times(int,A_1,Q),times_times(int,B,P)))) = times_times(int,twoSqu579703163sum2sq(product_Pair(int,int,A_1,B)),twoSqu579703163sum2sq(product_Pair(int,int,P,Q)))) # label(fact_371_mult__sum2sq) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 484 (all A_1 all P (zprime(P) -> (ord_less(int,zero_zero(int),A_1) -> (zcong(times_times(int,A_1,A_1),one_one(int),P) -> zcong(A_1,minus_minus(int,P,one_one(int)),P) | zcong(A_1,one_one(int),P))))) # label(fact_372_zcong__square) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 485 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all X_2 (ord_less_eq(int,X_2,pls) <-> ord_less_eq(X_a,number_number_of(X_a,X_2),zero_zero(X_a)))))) # label(fact_247_le__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 486 (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_130_zadd__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 487 (all D all C all A_1 all B all M (zcong(A_1,B,M) -> (ti(int,B) = ti(int,C) -> (zcong(C,D,M) -> zcong(A_1,D,M))))) # label(fact_385_zcong__eq__trans) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 488 (all X_a (comm_semiring_1(X_a) -> (all A_1 all M all N (ord_less_eq(nat,M,N) -> dvd_dvd(X_a,power_power(X_a,A_1,M),power_power(X_a,A_1,N)))))) # label(fact_184_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 489 (all X_a (monoid_mult(X_a) -> (all A_1 all M all N times_times(X_a,power_power(X_a,A_1,M),power_power(X_a,A_1,N)) = power_power(X_a,A_1,plus_plus(nat,M,N))))) # label(fact_259_power__add) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 490 (all B all A_1 all C (ord_less(int,A_1,C) -> (ord_less(int,B,C) -> ord_less_eq(int,A_1,B) | ord_less_eq(int,B,A_1)))) # label(fact_328_Euler_Oaux2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 491 (all X_a (number_semiring(X_a) -> number_number_of(X_a,bit1(pls)) = one_one(X_a))) # label(fact_155_semiring__numeral__1__eq__1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 492 (all X_a (comm_semiring_1(X_a) -> (all N all M all X_1 all Y_1 (dvd_dvd(X_a,X_1,Y_1) -> (ord_less_eq(nat,N,M) -> dvd_dvd(X_a,power_power(X_a,X_1,N),power_power(X_a,Y_1,M))))))) # label(fact_185_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 493 (all X_a (comm_semiring_1(X_a) -> (all A_1 all C all D plus_plus(X_a,A_1,plus_plus(X_a,C,D)) = plus_plus(X_a,plus_plus(X_a,A_1,C),D)))) # label(fact_77_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 494 (all X_a (number_ring(X_a) -> (all N power_power(X_a,number_number_of(X_a,min),times_times(nat,number_number_of(nat,bit0(bit1(pls))),N)) = one_one(X_a)))) # label(fact_336_power__m1__even) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 495 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B all C times_times(X_a,plus_plus(X_a,A_1,B),C) = plus_plus(X_a,times_times(X_a,A_1,C),times_times(X_a,B,C))))) # label(fact_108_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 496 (all W_1 (ord_less(int,W_1,zero_zero(int)) <-> ord_less(int,bit0(W_1),zero_zero(int)))) # label(fact_224_bin__less__0__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 497 (all X_a (linordered_semidom(X_a) -> (all A_1 all N all B (ord_less(X_a,power_power(X_a,A_1,N),power_power(X_a,B,N)) -> (ord_less_eq(X_a,zero_zero(X_a),B) -> ord_less(X_a,A_1,B)))))) # label(fact_212_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 498 (all Z plus_plus(nat,Z,Z) = times_times(nat,number_number_of(nat,bit0(bit1(pls))),Z)) # label(fact_35_nat__mult__2) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 499 (all X_a (comm_semiring_1(X_a) -> (all X_1 all Y_1 all Z times_times(X_a,X_1,plus_plus(X_a,Y_1,Z)) = plus_plus(X_a,times_times(X_a,X_1,Y_1),times_times(X_a,X_1,Z))))) # label(fact_110_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 500 (all X_a (number_ring(X_a) -> (all V all W all C plus_plus(X_a,number_number_of(X_a,V),minus_minus(X_a,number_number_of(X_a,W),C)) = minus_minus(X_a,number_number_of(X_a,plus_plus(int,V,W)),C)))) # label(fact_364_add__number__of__diff1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 501 (all K all L bit0(plus_plus(int,K,L)) = plus_plus(int,bit0(K),bit0(L))) # label(fact_125_add__Bit0__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 502 (all X_a (ab_group_add(X_a) -> (all A_1 all C all B all D minus_minus(X_a,plus_plus(X_a,A_1,C),plus_plus(X_a,B,D)) = plus_plus(X_a,minus_minus(X_a,A_1,B),minus_minus(X_a,C,D))))) # label(fact_430_add__diff__add) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 503 (all B_1_1 all B_2_1 all X_a (ab_group_add(X_a) -> plus_plus(X_a,B_1_1,B_2_1) = plus_plus(X_a,ti(X_a,B_1_1),B_2_1))) # label(tsy_c_Groups_Oplus__class_Oplus_1_arg1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 504 (all B_1_1 all B_2_1 all X_a (no_zero_divisors(X_a) -> times_times(X_a,B_1_1,B_2_1) = ti(X_a,times_times(X_a,B_1_1,B_2_1)))) # label(tsy_c_Groups_Otimes__class_Otimes_2_res) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 505 (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_47_rel__simps_I31_J) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 506 (all M zcong(M,zero_zero(int),M)) # label(fact_392_zcong__id) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 507 (all X_a (linordered_semidom(X_a) -> (all N all A_1 (ord_less_eq(X_a,one_one(X_a),A_1) -> ord_less_eq(X_a,one_one(X_a),power_power(X_a,A_1,N)))))) # label(fact_268_one__le__power) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 508 (all X_a (linordered_semidom(X_a) -> (all N all A_1 (ord_less_eq(X_a,zero_zero(X_a),A_1) -> ord_less_eq(X_a,zero_zero(X_a),power_power(X_a,A_1,N)))))) # label(fact_206_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 509 (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_87_zadd__assoc) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 510 (all X_1 all Y_1 all Z plus_plus(int,X_1,plus_plus(int,Y_1,Z)) = plus_plus(int,Y_1,plus_plus(int,X_1,Z))) # label(fact_88_zadd__left__commute) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 511 (all X_a (comm_semiring_1(X_a) -> (all C all D all A_1 all B (dvd_dvd(X_a,A_1,B) -> (dvd_dvd(X_a,C,D) -> dvd_dvd(X_a,times_times(X_a,A_1,C),times_times(X_a,B,D))))))) # label(fact_483_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 512 (all B_1_1 all B_2_1 all X_a (ab_group_add(X_a) -> minus_minus(X_a,ti(X_a,B_1_1),B_2_1) = minus_minus(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Ominus__class_Ominus_0_arg1) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 513 (all X_a (linordered_semidom(X_a) -> (all M all N all A_1 (ord_less(X_a,one_one(X_a),A_1) -> (ord_less(X_a,power_power(X_a,A_1,M),power_power(X_a,A_1,N)) -> ord_less(nat,M,N)))))) # label(fact_272_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 514 (all X_a (monoid_mult(X_a) -> (all A_1 all M all N power_power(X_a,power_power(X_a,A_1,M),N) = power_power(X_a,A_1,times_times(nat,M,N))))) # label(fact_261_power__mult) # label(axiom) # label(non_clause). [assumption]. 1.07/1.39 515 -(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]. 1.07/1.39 1.07/1.39 ============================== end of process non-clausal formulas === 1.07/1.39 1.07/1.39 ============================== PROCESS INITIAL CLAUSES =============== 1.07/1.39 1.07/1.39 ============================== PREDICATE ELIMINATION ================= 1.07/1.39 516 -linord581940658strict(A) | -ord_less(A,zero_zero(A),plus_plus(A,times_times(A,B,B),times_times(A,C,C))) | zero_zero(A) != ti(A,B) | zero_zero(A) != ti(A,C) # label(fact_237_sum__squares__gt__zero__iff) # label(axiom). [clausify(47)]. 1.07/1.39 517 linord581940658strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__ring__strict) # label(axiom). [assumption]. 1.07/1.39 518 linord581940658strict(int) # label(arity_Int_Oint___Rings_Olinordered__ring__strict) # label(axiom). [assumption]. 1.07/1.39 Derived: -ord_less(real,zero_zero(real),plus_plus(real,times_times(real,A,A),times_times(real,B,B))) | zero_zero(real) != ti(real,A) | zero_zero(real) != ti(real,B). [resolve(516,a,517,a)]. 1.07/1.40 Derived: -ord_less(int,zero_zero(int),plus_plus(int,times_times(int,A,A),times_times(int,B,B))) | zero_zero(int) != ti(int,A) | zero_zero(int) != ti(int,B). [resolve(516,a,518,a)]. 1.07/1.40 519 -linord581940658strict(A) | ord_less(A,zero_zero(A),plus_plus(A,times_times(A,B,B),times_times(A,C,C))) | zero_zero(A) = ti(A,B) # label(fact_237_sum__squares__gt__zero__iff) # label(axiom). [clausify(47)]. 1.07/1.40 Derived: ord_less(real,zero_zero(real),plus_plus(real,times_times(real,A,A),times_times(real,B,B))) | zero_zero(real) = ti(real,A). [resolve(519,a,517,a)]. 1.07/1.40 Derived: ord_less(int,zero_zero(int),plus_plus(int,times_times(int,A,A),times_times(int,B,B))) | zero_zero(int) = ti(int,A). [resolve(519,a,518,a)]. 1.07/1.40 520 -linord581940658strict(A) | ord_less(A,zero_zero(A),plus_plus(A,times_times(A,B,B),times_times(A,C,C))) | zero_zero(A) = ti(A,C) # label(fact_237_sum__squares__gt__zero__iff) # label(axiom). [clausify(47)]. 1.07/1.40 Derived: ord_less(real,zero_zero(real),plus_plus(real,times_times(real,A,A),times_times(real,B,B))) | zero_zero(real) = ti(real,B). [resolve(520,a,517,a)]. 1.07/1.40 Derived: ord_less(int,zero_zero(int),plus_plus(int,times_times(int,A,A),times_times(int,B,B))) | zero_zero(int) = ti(int,B). [resolve(520,a,518,a)]. 1.07/1.40 521 -linord581940658strict(A) | -ord_less_eq(A,B,zero_zero(A)) | -ord_less_eq(A,C,zero_zero(A)) | ord_less_eq(A,zero_zero(A),times_times(A,B,C)) # label(fact_491_zero__le__mult__iff) # label(axiom). [clausify(172)]. 1.07/1.40 Derived: -ord_less_eq(real,A,zero_zero(real)) | -ord_less_eq(real,B,zero_zero(real)) | ord_less_eq(real,zero_zero(real),times_times(real,A,B)). [resolve(521,a,517,a)]. 1.07/1.40 Derived: -ord_less_eq(int,A,zero_zero(int)) | -ord_less_eq(int,B,zero_zero(int)) | ord_less_eq(int,zero_zero(int),times_times(int,A,B)). [resolve(521,a,518,a)]. 1.07/1.40 522 -linord581940658strict(A) | -ord_less_eq(A,zero_zero(A),B) | -ord_less_eq(A,zero_zero(A),C) | ord_less_eq(A,zero_zero(A),times_times(A,C,B)) # label(fact_491_zero__le__mult__iff) # label(axiom). [clausify(172)]. 1.07/1.40 Derived: -ord_less_eq(real,zero_zero(real),A) | -ord_less_eq(real,zero_zero(real),B) | ord_less_eq(real,zero_zero(real),times_times(real,B,A)). [resolve(522,a,517,a)]. 1.07/1.40 Derived: -ord_less_eq(int,zero_zero(int),A) | -ord_less_eq(int,zero_zero(int),B) | ord_less_eq(int,zero_zero(int),times_times(int,B,A)). [resolve(522,a,518,a)]. 1.07/1.40 523 -linord581940658strict(A) | ord_less_eq(A,B,zero_zero(A)) | ord_less_eq(A,zero_zero(A),C) | -ord_less_eq(A,zero_zero(A),times_times(A,B,C)) # label(fact_491_zero__le__mult__iff) # label(axiom). [clausify(172)]. 1.07/1.40 Derived: ord_less_eq(real,A,zero_zero(real)) | ord_less_eq(real,zero_zero(real),B) | -ord_less_eq(real,zero_zero(real),times_times(real,A,B)). [resolve(523,a,517,a)]. 1.07/1.40 Derived: ord_less_eq(int,A,zero_zero(int)) | ord_less_eq(int,zero_zero(int),B) | -ord_less_eq(int,zero_zero(int),times_times(int,A,B)). [resolve(523,a,518,a)]. 1.07/1.40 524 -linord581940658strict(A) | ord_less_eq(A,B,zero_zero(A)) | ord_less_eq(A,zero_zero(A),B) | -ord_less_eq(A,zero_zero(A),times_times(A,B,C)) # label(fact_491_zero__le__mult__iff) # label(axiom). [clausify(172)]. 1.07/1.40 Derived: ord_less_eq(int,A,zero_zero(int)) | ord_less_eq(int,zero_zero(int),A) | -ord_less_eq(int,zero_zero(int),times_times(int,A,B)). [resolve(524,a,518,a)]. 1.07/1.40 525 -linord581940658strict(A) | ord_less_eq(A,B,zero_zero(A)) | ord_less_eq(A,zero_zero(A),B) | -ord_less_eq(A,zero_zero(A),times_times(A,C,B)) # label(fact_491_zero__le__mult__iff) # label(axiom). [clausify(172)]. 1.07/1.40 Derived: ord_less_eq(int,A,zero_zero(int)) | ord_less_eq(int,zero_zero(int),A) | -ord_less_eq(int,zero_zero(int),times_times(int,B,A)). [resolve(525,a,518,a)]. 1.07/1.40 526 -linord581940658strict(A) | ord_less_eq(A,B,zero_zero(A)) | ord_less_eq(A,zero_zero(A),C) | -ord_less_eq(A,zero_zero(A),times_times(A,C,B)) # label(fact_491_zero__le__mult__iff) # label(axiom). [clausify(172)]. 1.07/1.40 Derived: ord_less_eq(real,A,zero_zero(real)) | ord_less_eq(real,zero_zero(real),B) | -ord_less_eq(real,zero_zero(real),times_times(real,B,A)). [resolve(526,a,517,a)]. 1.07/1.42 Derived: ord_less_eq(int,A,zero_zero(int)) | ord_less_eq(int,zero_zero(int),B) | -ord_less_eq(int,zero_zero(int),times_times(int,B,A)). [resolve(526,a,518,a)]. 1.07/1.42 527 -linord581940658strict(A) | -ord_less_eq(A,times_times(A,B,C),zero_zero(A)) | ord_less_eq(A,zero_zero(A),C) | ord_less_eq(A,zero_zero(A),B) # label(fact_492_mult__le__0__iff) # label(axiom). [clausify(189)]. 1.07/1.42 Derived: -ord_less_eq(real,times_times(real,A,B),zero_zero(real)) | ord_less_eq(real,zero_zero(real),B) | ord_less_eq(real,zero_zero(real),A). [resolve(527,a,517,a)]. 1.07/1.42 Derived: -ord_less_eq(int,times_times(int,A,B),zero_zero(int)) | ord_less_eq(int,zero_zero(int),B) | ord_less_eq(int,zero_zero(int),A). [resolve(527,a,518,a)]. 1.07/1.42 528 -linord581940658strict(A) | -ord_less_eq(A,times_times(A,B,C),zero_zero(A)) | ord_less_eq(A,zero_zero(A),C) | ord_less_eq(A,C,zero_zero(A)) # label(fact_492_mult__le__0__iff) # label(axiom). [clausify(189)]. 1.07/1.42 Derived: -ord_less_eq(int,times_times(int,A,B),zero_zero(int)) | ord_less_eq(int,zero_zero(int),B) | ord_less_eq(int,B,zero_zero(int)). [resolve(528,a,518,a)]. 1.07/1.42 529 -linord581940658strict(A) | -ord_less_eq(A,times_times(A,B,C),zero_zero(A)) | ord_less_eq(A,B,zero_zero(A)) | ord_less_eq(A,zero_zero(A),B) # label(fact_492_mult__le__0__iff) # label(axiom). [clausify(189)]. 1.07/1.42 Derived: -ord_less_eq(int,times_times(int,A,B),zero_zero(int)) | ord_less_eq(int,A,zero_zero(int)) | ord_less_eq(int,zero_zero(int),A). [resolve(529,a,518,a)]. 1.07/1.42 530 -linord581940658strict(A) | -ord_less_eq(A,times_times(A,B,C),zero_zero(A)) | ord_less_eq(A,B,zero_zero(A)) | ord_less_eq(A,C,zero_zero(A)) # label(fact_492_mult__le__0__iff) # label(axiom). [clausify(189)]. 1.07/1.42 Derived: -ord_less_eq(real,times_times(real,A,B),zero_zero(real)) | ord_less_eq(real,A,zero_zero(real)) | ord_less_eq(real,B,zero_zero(real)). [resolve(530,a,517,a)]. 1.07/1.42 Derived: -ord_less_eq(int,times_times(int,A,B),zero_zero(int)) | ord_less_eq(int,A,zero_zero(int)) | ord_less_eq(int,B,zero_zero(int)). [resolve(530,a,518,a)]. 1.07/1.42 531 -linord581940658strict(A) | ord_less_eq(A,times_times(A,B,C),zero_zero(A)) | -ord_less_eq(A,zero_zero(A),C) | -ord_less_eq(A,B,zero_zero(A)) # label(fact_492_mult__le__0__iff) # label(axiom). [clausify(189)]. 1.07/1.42 Derived: ord_less_eq(real,times_times(real,A,B),zero_zero(real)) | -ord_less_eq(real,zero_zero(real),B) | -ord_less_eq(real,A,zero_zero(real)). [resolve(531,a,517,a)]. 1.07/1.42 Derived: ord_less_eq(int,times_times(int,A,B),zero_zero(int)) | -ord_less_eq(int,zero_zero(int),B) | -ord_less_eq(int,A,zero_zero(int)). [resolve(531,a,518,a)]. 1.07/1.42 532 -linord581940658strict(A) | ord_less_eq(A,times_times(A,B,C),zero_zero(A)) | -ord_less_eq(A,zero_zero(A),B) | -ord_less_eq(A,C,zero_zero(A)) # label(fact_492_mult__le__0__iff) # label(axiom). [clausify(189)]. 1.07/1.42 Derived: ord_less_eq(real,times_times(real,A,B),zero_zero(real)) | -ord_less_eq(real,zero_zero(real),A) | -ord_less_eq(real,B,zero_zero(real)). [resolve(532,a,517,a)]. 1.07/1.42 Derived: ord_less_eq(int,times_times(int,A,B),zero_zero(int)) | -ord_less_eq(int,zero_zero(int),A) | -ord_less_eq(int,B,zero_zero(int)). [resolve(532,a,518,a)]. 1.07/1.42 533 -linord581940658strict(A) | zero_zero(A) != ti(A,B) | zero_zero(A) != ti(A,C) | zero_zero(A) = plus_plus(A,times_times(A,B,B),times_times(A,C,C)) # label(fact_216_sum__squares__eq__zero__iff) # label(axiom). [clausify(422)]. 1.07/1.42 Derived: zero_zero(real) != ti(real,A) | zero_zero(real) != ti(real,B) | zero_zero(real) = plus_plus(real,times_times(real,A,A),times_times(real,B,B)). [resolve(533,a,517,a)]. 1.07/1.42 Derived: zero_zero(int) != ti(int,A) | zero_zero(int) != ti(int,B) | zero_zero(int) = plus_plus(int,times_times(int,A,A),times_times(int,B,B)). [resolve(533,a,518,a)]. 1.07/1.42 534 -linord581940658strict(A) | zero_zero(A) = ti(A,B) | zero_zero(A) != plus_plus(A,times_times(A,B,B),times_times(A,C,C)) # label(fact_216_sum__squares__eq__zero__iff) # label(axiom). [clausify(422)]. 1.07/1.42 Derived: zero_zero(real) = ti(real,A) | zero_zero(real) != plus_plus(real,times_times(real,A,A),times_times(real,B,B)). [resolve(534,a,517,a)]. 1.07/1.42 Derived: zero_zero(int) = ti(int,A) | zero_zero(int) != plus_plus(int,times_times(int,A,A),times_times(int,B,B)). [resolve(534,a,518,a)]. 1.07/1.45 535 -linord581940658strict(A) | zero_zero(A) = ti(A,B) | zero_zero(A) != plus_plus(A,times_times(A,C,C),times_times(A,B,B)) # label(fact_216_sum__squares__eq__zero__iff) # label(axiom). [clausify(422)]. 1.07/1.45 Derived: zero_zero(real) = ti(real,A) | zero_zero(real) != plus_plus(real,times_times(real,B,B),times_times(real,A,A)). [resolve(535,a,517,a)]. 1.07/1.45 Derived: zero_zero(int) = ti(int,A) | zero_zero(int) != plus_plus(int,times_times(int,B,B),times_times(int,A,A)). [resolve(535,a,518,a)]. 1.07/1.45 536 -linord581940658strict(A) | -ord_less_eq(A,plus_plus(A,times_times(A,B,B),times_times(A,C,C)),zero_zero(A)) | zero_zero(A) = ti(A,B) # label(fact_234_sum__squares__le__zero__iff) # label(axiom). [clausify(452)]. 1.07/1.45 Derived: -ord_less_eq(real,plus_plus(real,times_times(real,A,A),times_times(real,B,B)),zero_zero(real)) | zero_zero(real) = ti(real,A). [resolve(536,a,517,a)]. 1.07/1.45 Derived: -ord_less_eq(int,plus_plus(int,times_times(int,A,A),times_times(int,B,B)),zero_zero(int)) | zero_zero(int) = ti(int,A). [resolve(536,a,518,a)]. 1.07/1.45 537 -linord581940658strict(A) | -ord_less_eq(A,plus_plus(A,times_times(A,B,B),times_times(A,C,C)),zero_zero(A)) | zero_zero(A) = ti(A,C) # label(fact_234_sum__squares__le__zero__iff) # label(axiom). [clausify(452)]. 1.07/1.45 Derived: -ord_less_eq(real,plus_plus(real,times_times(real,A,A),times_times(real,B,B)),zero_zero(real)) | zero_zero(real) = ti(real,B). [resolve(537,a,517,a)]. 1.07/1.45 Derived: -ord_less_eq(int,plus_plus(int,times_times(int,A,A),times_times(int,B,B)),zero_zero(int)) | zero_zero(int) = ti(int,B). [resolve(537,a,518,a)]. 1.07/1.45 538 -linord581940658strict(A) | ord_less_eq(A,plus_plus(A,times_times(A,B,B),times_times(A,C,C)),zero_zero(A)) | zero_zero(A) != ti(A,B) | zero_zero(A) != ti(A,C) # label(fact_234_sum__squares__le__zero__iff) # label(axiom). [clausify(452)]. 1.07/1.45 Derived: ord_less_eq(real,plus_plus(real,times_times(real,A,A),times_times(real,B,B)),zero_zero(real)) | zero_zero(real) != ti(real,A) | zero_zero(real) != ti(real,B). [resolve(538,a,517,a)]. 1.07/1.45 Derived: ord_less_eq(int,plus_plus(int,times_times(int,A,A),times_times(int,B,B)),zero_zero(int)) | zero_zero(int) != ti(int,A) | zero_zero(int) != ti(int,B). [resolve(538,a,518,a)]. 1.07/1.45 539 linorder(real) # label(arity_RealDef_Oreal___Orderings_Olinorder) # label(axiom). [assumption]. 1.07/1.45 540 -linorder(A) | -number(A) | -ord_less_eq(A,number_number_of(A,B),number_number_of(A,C)) | -ord_less(A,number_number_of(A,C),number_number_of(A,B)) # label(fact_29_le__number__of__eq__not__less) # label(axiom). [clausify(4)]. 1.07/1.45 541 -linorder(A) | -number(A) | ord_less_eq(A,number_number_of(A,B),number_number_of(A,C)) | ord_less(A,number_number_of(A,C),number_number_of(A,B)) # label(fact_29_le__number__of__eq__not__less) # label(axiom). [clausify(4)]. 1.07/1.45 Derived: -number(real) | -ord_less_eq(real,number_number_of(real,A),number_number_of(real,B)) | -ord_less(real,number_number_of(real,B),number_number_of(real,A)). [resolve(539,a,540,a)]. 1.07/1.45 Derived: -number(real) | ord_less_eq(real,number_number_of(real,A),number_number_of(real,B)) | ord_less(real,number_number_of(real,B),number_number_of(real,A)). [resolve(539,a,541,a)]. 1.07/1.45 542 linorder(nat) # label(arity_Nat_Onat___Orderings_Olinorder) # label(axiom). [assumption]. 1.07/1.45 Derived: -number(nat) | -ord_less_eq(nat,number_number_of(nat,A),number_number_of(nat,B)) | -ord_less(nat,number_number_of(nat,B),number_number_of(nat,A)). [resolve(542,a,540,a)]. 1.07/1.45 Derived: -number(nat) | ord_less_eq(nat,number_number_of(nat,A),number_number_of(nat,B)) | ord_less(nat,number_number_of(nat,B),number_number_of(nat,A)). [resolve(542,a,541,a)]. 1.07/1.45 543 linorder(int) # label(arity_Int_Oint___Orderings_Olinorder) # label(axiom). [assumption]. 1.07/1.45 Derived: -number(int) | -ord_less_eq(int,number_number_of(int,A),number_number_of(int,B)) | -ord_less(int,number_number_of(int,B),number_number_of(int,A)). [resolve(543,a,540,a)]. 1.07/1.45 Derived: -number(int) | ord_less_eq(int,number_number_of(int,A),number_number_of(int,B)) | ord_less(int,number_number_of(int,B),number_number_of(int,A)). [resolve(543,a,541,a)]. 1.07/1.47 544 comm_semiring_1(real) # label(arity_RealDef_Oreal___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 1.07/1.47 545 -comm_semiring_1(A) | power_power(A,B,one_one(nat)) = ti(A,B) # label(fact_27_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom). [clausify(6)]. 1.07/1.47 546 -comm_semiring_1(A) | power_power(A,power_power(A,B,C),D) = power_power(A,B,times_times(nat,C,D)) # label(fact_26_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom). [clausify(26)]. 1.07/1.47 547 -comm_semiring_1(A) | times_times(A,power_power(A,B,C),power_power(A,D,C)) = power_power(A,times_times(A,B,D),C) # label(fact_113_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom). [clausify(39)]. 1.07/1.47 548 -comm_semiring_1(A) | times_times(A,times_times(A,B,C),D) = times_times(A,times_times(A,B,D),C) # label(fact_69_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom). [clausify(43)]. 1.07/1.47 549 -comm_semiring_1(A) | dvd_dvd(A,one_one(A),B) # label(fact_488_one__dvd) # label(axiom). [clausify(62)]. 1.07/1.47 550 -comm_semiring_1(A) | plus_plus(A,B,B) = times_times(A,plus_plus(A,one_one(A),one_one(A)),B) # label(fact_140_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom). [clausify(65)]. 1.07/1.47 551 -comm_semiring_1(A) | plus_plus(A,plus_plus(A,B,C),plus_plus(A,D,E)) = plus_plus(A,plus_plus(A,B,D),plus_plus(A,C,E)) # label(fact_74_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom). [clausify(89)]. 1.07/1.47 552 -comm_semiring_1(A) | -dvd_dvd(A,zero_zero(A),B) | zero_zero(A) = ti(A,B) # label(fact_478_dvd__0__left) # label(axiom). [clausify(91)]. 1.07/1.47 553 -comm_semiring_1(A) | -dvd_dvd(A,power_power(A,B,C),D) | -ord_less_eq(nat,E,C) | dvd_dvd(A,power_power(A,B,E),D) # label(fact_186_power__le__dvd) # label(axiom). [clausify(106)]. 1.07/1.47 554 -comm_semiring_1(A) | dvd_dvd(A,B,times_times(A,C,B)) # label(fact_480_dvd__triv__right) # label(axiom). [clausify(112)]. 1.07/1.47 Derived: power_power(real,A,one_one(nat)) = ti(real,A). [resolve(544,a,545,a)]. 1.07/1.47 Derived: power_power(real,power_power(real,A,B),C) = power_power(real,A,times_times(nat,B,C)). [resolve(544,a,546,a)]. 1.07/1.47 Derived: times_times(real,power_power(real,A,B),power_power(real,C,B)) = power_power(real,times_times(real,A,C),B). [resolve(544,a,547,a)]. 1.07/1.47 Derived: times_times(real,times_times(real,A,B),C) = times_times(real,times_times(real,A,C),B). [resolve(544,a,548,a)]. 1.07/1.47 Derived: dvd_dvd(real,one_one(real),A). [resolve(544,a,549,a)]. 1.07/1.47 Derived: plus_plus(real,A,A) = times_times(real,plus_plus(real,one_one(real),one_one(real)),A). [resolve(544,a,550,a)]. 1.07/1.47 Derived: plus_plus(real,plus_plus(real,A,B),plus_plus(real,C,D)) = plus_plus(real,plus_plus(real,A,C),plus_plus(real,B,D)). [resolve(544,a,551,a)]. 1.07/1.47 Derived: -dvd_dvd(real,zero_zero(real),A) | zero_zero(real) = ti(real,A). [resolve(544,a,552,a)]. 1.07/1.47 Derived: -dvd_dvd(real,power_power(real,A,B),C) | -ord_less_eq(nat,D,B) | dvd_dvd(real,power_power(real,A,D),C). [resolve(544,a,553,a)]. 1.07/1.47 Derived: dvd_dvd(real,A,times_times(real,B,A)). [resolve(544,a,554,a)]. 1.07/1.47 555 comm_semiring_1(int) # label(arity_Int_Oint___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 1.07/1.47 Derived: power_power(int,A,one_one(nat)) = ti(int,A). [resolve(555,a,545,a)]. 1.07/1.47 Derived: times_times(int,power_power(int,A,B),power_power(int,C,B)) = power_power(int,times_times(int,A,C),B). [resolve(555,a,547,a)]. 1.07/1.47 Derived: times_times(int,times_times(int,A,B),C) = times_times(int,times_times(int,A,C),B). [resolve(555,a,548,a)]. 1.07/1.47 Derived: dvd_dvd(int,one_one(int),A). [resolve(555,a,549,a)]. 1.07/1.47 Derived: plus_plus(int,A,A) = times_times(int,plus_plus(int,one_one(int),one_one(int)),A). [resolve(555,a,550,a)]. 1.07/1.47 Derived: plus_plus(int,plus_plus(int,A,B),plus_plus(int,C,D)) = plus_plus(int,plus_plus(int,A,C),plus_plus(int,B,D)). [resolve(555,a,551,a)]. 1.07/1.47 Derived: -dvd_dvd(int,zero_zero(int),A) | zero_zero(int) = ti(int,A). [resolve(555,a,552,a)]. 1.07/1.47 Derived: -dvd_dvd(int,power_power(int,A,B),C) | -ord_less_eq(nat,D,B) | dvd_dvd(int,power_power(int,A,D),C). [resolve(555,a,553,a)]. 1.07/1.47 Derived: dvd_dvd(int,A,times_times(int,B,A)). [resolve(555,a,554,a)]. 1.07/1.47 556 -comm_semiring_1(A) | dvd_dvd(A,B,B) # label(fact_468_dvd__refl) # label(axiom). [clausify(128)]. 1.07/1.47 Derived: dvd_dvd(real,A,A). [resolve(556,a,544,a)]. 1.07/1.47 Derived: dvd_dvd(int,A,A). [resolve(556,a,555,a)]. 1.07/1.47 557 -comm_semiring_1(A) | times_times(A,power_power(A,B,C),power_power(A,B,C)) = power_power(A,B,times_times(nat,number_number_of(nat,bit0(bit1(pls))),C)) # label(fact_15_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom). [clausify(133)]. 1.07/1.47 Derived: times_times(real,power_power(real,A,B),power_power(real,A,B)) = power_power(real,A,times_times(nat,number_number_of(nat,bit0(bit1(pls))),B)). [resolve(557,a,544,a)]. 1.07/1.47 Derived: times_times(int,power_power(int,A,B),power_power(int,A,B)) = power_power(int,A,times_times(nat,number_number_of(nat,bit0(bit1(pls))),B)). [resolve(557,a,555,a)]. 1.07/1.47 558 -comm_semiring_1(A) | plus_plus(A,B,C) = plus_plus(A,C,B) # label(fact_79_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom). [clausify(134)]. 1.07/1.47 Derived: plus_plus(real,A,B) = plus_plus(real,B,A). [resolve(558,a,544,a)]. 1.07/1.47 559 -comm_semiring_1(A) | -dvd_dvd(A,B,C) | dvd_dvd(A,B,times_times(A,C,D)) # label(fact_481_dvd__mult2) # label(axiom). [clausify(136)]. 1.07/1.47 Derived: -dvd_dvd(real,A,B) | dvd_dvd(real,A,times_times(real,B,C)). [resolve(559,a,544,a)]. 1.07/1.47 Derived: -dvd_dvd(int,A,B) | dvd_dvd(int,A,times_times(int,B,C)). [resolve(559,a,555,a)]. 1.07/1.47 560 -comm_semiring_1(A) | times_times(A,power_power(A,B,C),power_power(A,B,D)) = power_power(A,B,plus_plus(nat,C,D)) # label(fact_33_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom). [clausify(138)]. 1.07/1.47 Derived: times_times(real,power_power(real,A,B),power_power(real,A,C)) = power_power(real,A,plus_plus(nat,B,C)). [resolve(560,a,544,a)]. 1.07/1.47 Derived: times_times(int,power_power(int,A,B),power_power(int,A,C)) = power_power(int,A,plus_plus(nat,B,C)). [resolve(560,a,555,a)]. 1.07/1.47 561 comm_semiring_1(nat) # label(arity_Nat_Onat___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 1.07/1.47 Derived: power_power(nat,A,one_one(nat)) = ti(nat,A). [resolve(561,a,545,a)]. 1.07/1.47 Derived: power_power(nat,power_power(nat,A,B),C) = power_power(nat,A,times_times(nat,B,C)). [resolve(561,a,546,a)]. 1.07/1.47 Derived: times_times(nat,power_power(nat,A,B),power_power(nat,C,B)) = power_power(nat,times_times(nat,A,C),B). [resolve(561,a,547,a)]. 1.07/1.47 Derived: times_times(nat,times_times(nat,A,B),C) = times_times(nat,times_times(nat,A,C),B). [resolve(561,a,548,a)]. 1.07/1.47 Derived: dvd_dvd(nat,one_one(nat),A). [resolve(561,a,549,a)]. 1.07/1.47 Derived: plus_plus(nat,A,A) = times_times(nat,plus_plus(nat,one_one(nat),one_one(nat)),A). [resolve(561,a,550,a)]. 1.07/1.47 Derived: plus_plus(nat,plus_plus(nat,A,B),plus_plus(nat,C,D)) = plus_plus(nat,plus_plus(nat,A,C),plus_plus(nat,B,D)). [resolve(561,a,551,a)]. 1.07/1.47 Derived: -dvd_dvd(nat,zero_zero(nat),A) | zero_zero(nat) = ti(nat,A). [resolve(561,a,552,a)]. 1.07/1.47 Derived: -dvd_dvd(nat,power_power(nat,A,B),C) | -ord_less_eq(nat,D,B) | dvd_dvd(nat,power_power(nat,A,D),C). [resolve(561,a,553,a)]. 1.07/1.47 Derived: dvd_dvd(nat,A,times_times(nat,B,A)). [resolve(561,a,554,a)]. 1.07/1.47 Derived: dvd_dvd(nat,A,A). [resolve(561,a,556,a)]. 1.07/1.47 Derived: times_times(nat,power_power(nat,A,B),power_power(nat,A,B)) = power_power(nat,A,times_times(nat,number_number_of(nat,bit0(bit1(pls))),B)). [resolve(561,a,557,a)]. 1.07/1.47 Derived: plus_plus(nat,A,B) = plus_plus(nat,B,A). [resolve(561,a,558,a)]. 1.07/1.47 Derived: -dvd_dvd(nat,A,B) | dvd_dvd(nat,A,times_times(nat,B,C)). [resolve(561,a,559,a)]. 1.07/1.47 Derived: times_times(nat,power_power(nat,A,B),power_power(nat,A,C)) = power_power(nat,A,plus_plus(nat,B,C)). [resolve(561,a,560,a)]. 1.07/1.47 562 -comm_semiring_1(A) | plus_plus(A,B,zero_zero(A)) = ti(A,B) # label(fact_199_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom). [clausify(154)]. 1.07/1.47 Derived: plus_plus(real,A,zero_zero(real)) = ti(real,A). [resolve(562,a,544,a)]. 1.07/1.47 Derived: plus_plus(nat,A,zero_zero(nat)) = ti(nat,A). [resolve(562,a,561,a)]. 1.07/1.47 563 -comm_semiring_1(A) | plus_plus(A,plus_plus(A,B,C),D) = plus_plus(A,plus_plus(A,B,D),C) # label(fact_75_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom). [clausify(168)]. 1.07/1.48 Derived: plus_plus(real,plus_plus(real,A,B),C) = plus_plus(real,plus_plus(real,A,C),B). [resolve(563,a,544,a)]. 1.07/1.48 Derived: plus_plus(int,plus_plus(int,A,B),C) = plus_plus(int,plus_plus(int,A,C),B). [resolve(563,a,555,a)]. 1.07/1.48 Derived: plus_plus(nat,plus_plus(nat,A,B),C) = plus_plus(nat,plus_plus(nat,A,C),B). [resolve(563,a,561,a)]. 1.07/1.48 564 -comm_semiring_1(A) | plus_plus(A,plus_plus(A,B,C),D) = plus_plus(A,B,plus_plus(A,C,D)) # label(fact_76_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom). [clausify(177)]. 1.07/1.48 Derived: plus_plus(real,plus_plus(real,A,B),C) = plus_plus(real,A,plus_plus(real,B,C)). [resolve(564,a,544,a)]. 1.07/1.48 Derived: plus_plus(nat,plus_plus(nat,A,B),C) = plus_plus(nat,A,plus_plus(nat,B,C)). [resolve(564,a,561,a)]. 1.07/1.48 565 -comm_semiring_1(A) | plus_plus(A,times_times(A,B,C),C) = times_times(A,plus_plus(A,B,one_one(A)),C) # label(fact_138_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom). [clausify(188)]. 1.07/1.48 Derived: plus_plus(real,times_times(real,A,B),B) = times_times(real,plus_plus(real,A,one_one(real)),B). [resolve(565,a,544,a)]. 1.07/1.48 Derived: plus_plus(int,times_times(int,A,B),B) = times_times(int,plus_plus(int,A,one_one(int)),B). [resolve(565,a,555,a)]. 1.07/1.48 Derived: plus_plus(nat,times_times(nat,A,B),B) = times_times(nat,plus_plus(nat,A,one_one(nat)),B). [resolve(565,a,561,a)]. 1.07/1.48 566 -comm_semiring_1(A) | times_times(A,B,C) = times_times(A,C,B) # label(fact_73_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom). [clausify(213)]. 1.07/1.48 Derived: times_times(int,A,B) = times_times(int,B,A). [resolve(566,a,555,a)]. 1.07/1.48 Derived: times_times(nat,A,B) = times_times(nat,B,A). [resolve(566,a,561,a)]. 1.07/1.48 567 -comm_semiring_1(A) | times_times(A,times_times(A,B,C),times_times(A,D,E)) = times_times(A,B,times_times(A,C,times_times(A,D,E))) # label(fact_68_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom). [clausify(219)]. 1.07/1.48 Derived: times_times(real,times_times(real,A,B),times_times(real,C,D)) = times_times(real,A,times_times(real,B,times_times(real,C,D))). [resolve(567,a,544,a)]. 1.07/1.48 Derived: times_times(nat,times_times(nat,A,B),times_times(nat,C,D)) = times_times(nat,A,times_times(nat,B,times_times(nat,C,D))). [resolve(567,a,561,a)]. 1.07/1.48 568 -comm_semiring_1(A) | -dvd_dvd(A,B,C) | dvd_dvd(A,power_power(A,B,D),power_power(A,C,D)) # label(fact_191_dvd__power__same) # label(axiom). [clausify(257)]. 1.07/1.48 Derived: -dvd_dvd(real,A,B) | dvd_dvd(real,power_power(real,A,C),power_power(real,B,C)). [resolve(568,a,544,a)]. 1.07/1.48 Derived: -dvd_dvd(int,A,B) | dvd_dvd(int,power_power(int,A,C),power_power(int,B,C)). [resolve(568,a,555,a)]. 1.07/1.48 Derived: -dvd_dvd(nat,A,B) | dvd_dvd(nat,power_power(nat,A,C),power_power(nat,B,C)). [resolve(568,a,561,a)]. 1.07/1.48 569 -comm_semiring_1(A) | -dvd_dvd(A,B,C) | -dvd_dvd(A,C,D) | dvd_dvd(A,B,D) # label(fact_467_dvd__trans) # label(axiom). [clausify(278)]. 1.07/1.48 Derived: -dvd_dvd(real,A,B) | -dvd_dvd(real,B,C) | dvd_dvd(real,A,C). [resolve(569,a,544,a)]. 1.07/1.48 Derived: -dvd_dvd(int,A,B) | -dvd_dvd(int,B,C) | dvd_dvd(int,A,C). [resolve(569,a,555,a)]. 1.07/1.48 Derived: -dvd_dvd(nat,A,B) | -dvd_dvd(nat,B,C) | dvd_dvd(nat,A,C). [resolve(569,a,561,a)]. 1.07/1.48 570 -comm_semiring_1(A) | times_times(A,B,B) = power_power(A,B,number_number_of(nat,bit0(bit1(pls)))) # label(fact_13_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom). [clausify(287)]. 1.07/1.48 Derived: times_times(real,A,A) = power_power(real,A,number_number_of(nat,bit0(bit1(pls)))). [resolve(570,a,544,a)]. 1.07/1.48 Derived: times_times(int,A,A) = power_power(int,A,number_number_of(nat,bit0(bit1(pls)))). [resolve(570,a,555,a)]. 1.07/1.48 Derived: times_times(nat,A,A) = power_power(nat,A,number_number_of(nat,bit0(bit1(pls)))). [resolve(570,a,561,a)]. 1.07/1.48 571 -comm_semiring_1(A) | one_one(A) != ti(A,B) | dvd_dvd(A,B,power_power(A,B,C)) # label(fact_317_dvd__power) # label(axiom). [clausify(288)]. 1.07/1.48 Derived: one_one(real) != ti(real,A) | dvd_dvd(real,A,power_power(real,A,B)). [resolve(571,a,544,a)]. 1.07/1.48 Derived: one_one(int) != ti(int,A) | dvd_dvd(int,A,power_power(int,A,B)). [resolve(571,a,555,a)]. 1.07/1.49 Derived: one_one(nat) != ti(nat,A) | dvd_dvd(nat,A,power_power(nat,A,B)). [resolve(571,a,561,a)]. 1.07/1.49 572 -comm_semiring_1(A) | -ord_less(nat,zero_zero(nat),B) | dvd_dvd(A,C,power_power(A,C,B)) # label(fact_317_dvd__power) # label(axiom). [clausify(288)]. 1.07/1.49 Derived: -ord_less(nat,zero_zero(nat),A) | dvd_dvd(real,B,power_power(real,B,A)). [resolve(572,a,544,a)]. 1.07/1.49 Derived: -ord_less(nat,zero_zero(nat),A) | dvd_dvd(int,B,power_power(int,B,A)). [resolve(572,a,555,a)]. 1.07/1.49 Derived: -ord_less(nat,zero_zero(nat),A) | dvd_dvd(nat,B,power_power(nat,B,A)). [resolve(572,a,561,a)]. 1.07/1.49 573 -comm_semiring_1(A) | -dvd_dvd(A,B,C) | -dvd_dvd(A,B,D) | dvd_dvd(A,B,plus_plus(A,C,D)) # label(fact_487_dvd__add) # label(axiom). [clausify(292)]. 1.07/1.49 Derived: -dvd_dvd(real,A,B) | -dvd_dvd(real,A,C) | dvd_dvd(real,A,plus_plus(real,B,C)). [resolve(573,a,544,a)]. 1.07/1.49 Derived: -dvd_dvd(int,A,B) | -dvd_dvd(int,A,C) | dvd_dvd(int,A,plus_plus(int,B,C)). [resolve(573,a,555,a)]. 1.07/1.49 Derived: -dvd_dvd(nat,A,B) | -dvd_dvd(nat,A,C) | dvd_dvd(nat,A,plus_plus(nat,B,C)). [resolve(573,a,561,a)]. 1.07/1.49 574 -comm_semiring_1(A) | zero_zero(A) = times_times(A,B,zero_zero(A)) # label(fact_197_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom). [clausify(293)]. 1.07/1.49 Derived: zero_zero(real) = times_times(real,A,zero_zero(real)). [resolve(574,a,544,a)]. 1.07/1.49 Derived: zero_zero(int) = times_times(int,A,zero_zero(int)). [resolve(574,a,555,a)]. 1.07/1.49 Derived: zero_zero(nat) = times_times(nat,A,zero_zero(nat)). [resolve(574,a,561,a)]. 1.07/1.49 575 -comm_semiring_1(A) | plus_plus(A,B,plus_plus(A,C,D)) = plus_plus(A,C,plus_plus(A,B,D)) # label(fact_78_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom). [clausify(302)]. 1.07/1.49 Derived: plus_plus(real,A,plus_plus(real,B,C)) = plus_plus(real,B,plus_plus(real,A,C)). [resolve(575,a,544,a)]. 1.07/1.49 Derived: plus_plus(nat,A,plus_plus(nat,B,C)) = plus_plus(nat,B,plus_plus(nat,A,C)). [resolve(575,a,561,a)]. 1.07/1.49 576 -comm_semiring_1(A) | plus_plus(A,B,times_times(A,C,B)) = times_times(A,plus_plus(A,C,one_one(A)),B) # label(fact_139_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom). [clausify(312)]. 1.07/1.49 Derived: plus_plus(real,A,times_times(real,B,A)) = times_times(real,plus_plus(real,B,one_one(real)),A). [resolve(576,a,544,a)]. 1.07/1.49 Derived: plus_plus(int,A,times_times(int,B,A)) = times_times(int,plus_plus(int,B,one_one(int)),A). [resolve(576,a,555,a)]. 1.07/1.49 Derived: plus_plus(nat,A,times_times(nat,B,A)) = times_times(nat,plus_plus(nat,B,one_one(nat)),A). [resolve(576,a,561,a)]. 1.07/1.49 577 -comm_semiring_1(A) | times_times(A,one_one(A),B) = ti(A,B) # label(fact_112_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom). [clausify(322)]. 1.07/1.49 Derived: times_times(real,one_one(real),A) = ti(real,A). [resolve(577,a,544,a)]. 1.07/1.49 Derived: times_times(int,one_one(int),A) = ti(int,A). [resolve(577,a,555,a)]. 1.07/1.49 Derived: times_times(nat,one_one(nat),A) = ti(nat,A). [resolve(577,a,561,a)]. 1.07/1.49 578 -comm_semiring_1(A) | -dvd_dvd(A,times_times(A,B,C),D) | dvd_dvd(A,C,D) # label(fact_486_dvd__mult__right) # label(axiom). [clausify(325)]. 1.07/1.49 Derived: -dvd_dvd(real,times_times(real,A,B),C) | dvd_dvd(real,B,C). [resolve(578,a,544,a)]. 1.07/1.49 Derived: -dvd_dvd(int,times_times(int,A,B),C) | dvd_dvd(int,B,C). [resolve(578,a,555,a)]. 1.07/1.49 Derived: -dvd_dvd(nat,times_times(nat,A,B),C) | dvd_dvd(nat,B,C). [resolve(578,a,561,a)]. 1.07/1.49 579 -comm_semiring_1(A) | -dvd_dvd(A,times_times(A,B,C),D) | dvd_dvd(A,B,D) # label(fact_485_dvd__mult__left) # label(axiom). [clausify(326)]. 1.07/1.49 Derived: -dvd_dvd(real,times_times(real,A,B),C) | dvd_dvd(real,A,C). [resolve(579,a,544,a)]. 1.07/1.49 Derived: -dvd_dvd(int,times_times(int,A,B),C) | dvd_dvd(int,A,C). [resolve(579,a,555,a)]. 1.07/1.49 Derived: -dvd_dvd(nat,times_times(nat,A,B),C) | dvd_dvd(nat,A,C). [resolve(579,a,561,a)]. 1.07/1.49 580 -comm_semiring_1(A) | times_times(A,times_times(A,B,C),D) = times_times(A,B,times_times(A,C,D)) # label(fact_70_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom). [clausify(336)]. 1.07/1.49 Derived: times_times(real,times_times(real,A,B),C) = times_times(real,A,times_times(real,B,C)). [resolve(580,a,544,a)]. 1.07/1.49 Derived: times_times(int,times_times(int,A,B),C) = times_times(int,A,times_times(int,B,C)). [resolve(580,a,555,a)]. 1.07/1.49 Derived: times_times(nat,times_times(nat,A,B),C) = times_times(nat,A,times_times(nat,B,C)). [resolve(580,a,561,a)]. 1.07/1.49 581 -comm_semiring_1(A) | dvd_dvd(A,B,zero_zero(A)) # label(fact_465_dvd__0__right) # label(axiom). [clausify(348)]. 1.07/1.49 Derived: dvd_dvd(real,A,zero_zero(real)). [resolve(581,a,544,a)]. 1.07/1.49 Derived: dvd_dvd(int,A,zero_zero(int)). [resolve(581,a,555,a)]. 1.07/1.49 Derived: dvd_dvd(nat,A,zero_zero(nat)). [resolve(581,a,561,a)]. 1.07/1.49 582 -comm_semiring_1(A) | zero_zero(A) = times_times(A,zero_zero(A),B) # label(fact_196_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom). [clausify(353)]. 1.07/1.49 Derived: zero_zero(real) = times_times(real,zero_zero(real),A). [resolve(582,a,544,a)]. 1.07/1.49 Derived: zero_zero(int) = times_times(int,zero_zero(int),A). [resolve(582,a,555,a)]. 1.07/1.49 Derived: zero_zero(nat) = times_times(nat,zero_zero(nat),A). [resolve(582,a,561,a)]. 1.07/1.49 583 -comm_semiring_1(A) | dvd_dvd(A,B,times_times(A,B,C)) # label(fact_479_dvd__triv__left) # label(axiom). [clausify(367)]. 1.07/1.49 Derived: dvd_dvd(real,A,times_times(real,A,B)). [resolve(583,a,544,a)]. 1.07/1.49 Derived: dvd_dvd(int,A,times_times(int,A,B)). [resolve(583,a,555,a)]. 1.07/1.49 Derived: dvd_dvd(nat,A,times_times(nat,A,B)). [resolve(583,a,561,a)]. 1.07/1.49 584 -comm_semiring_1(A) | times_times(A,times_times(A,B,C),times_times(A,D,E)) = times_times(A,times_times(A,B,D),times_times(A,C,E)) # label(fact_66_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom). [clausify(371)]. 1.07/1.49 Derived: times_times(real,times_times(real,A,B),times_times(real,C,D)) = times_times(real,times_times(real,A,C),times_times(real,B,D)). [resolve(584,a,544,a)]. 1.07/1.49 Derived: times_times(int,times_times(int,A,B),times_times(int,C,D)) = times_times(int,times_times(int,A,C),times_times(int,B,D)). [resolve(584,a,555,a)]. 1.07/1.49 Derived: 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)). [resolve(584,a,561,a)]. 1.07/1.49 585 -comm_semiring_1(A) | one_one(A) = power_power(A,B,zero_zero(nat)) # label(fact_308_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom). [clausify(391)]. 1.07/1.49 Derived: one_one(real) = power_power(real,A,zero_zero(nat)). [resolve(585,a,544,a)]. 1.07/1.49 Derived: one_one(int) = power_power(int,A,zero_zero(nat)). [resolve(585,a,555,a)]. 1.07/1.49 Derived: one_one(nat) = power_power(nat,A,zero_zero(nat)). [resolve(585,a,561,a)]. 1.07/1.49 586 -comm_semiring_1(A) | times_times(A,times_times(A,B,C),D) = times_times(A,B,times_times(A,C,D)) # label(fact_71_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom). [clausify(412)]. 1.07/1.49 587 -comm_semiring_1(A) | times_times(A,B,times_times(A,C,D)) = times_times(A,C,times_times(A,B,D)) # label(fact_72_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom). [clausify(418)]. 1.07/1.49 Derived: times_times(real,A,times_times(real,B,C)) = times_times(real,B,times_times(real,A,C)). [resolve(587,a,544,a)]. 1.07/1.49 Derived: times_times(int,A,times_times(int,B,C)) = times_times(int,B,times_times(int,A,C)). [resolve(587,a,555,a)]. 1.07/1.49 Derived: times_times(nat,A,times_times(nat,B,C)) = times_times(nat,B,times_times(nat,A,C)). [resolve(587,a,561,a)]. 1.07/1.49 588 -comm_semiring_1(A) | plus_plus(A,zero_zero(A),B) = ti(A,B) # label(fact_198_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom). [clausify(419)]. 1.07/1.49 Derived: plus_plus(real,zero_zero(real),A) = ti(real,A). [resolve(588,a,544,a)]. 1.07/1.49 Derived: plus_plus(int,zero_zero(int),A) = ti(int,A). [resolve(588,a,555,a)]. 1.07/1.49 Derived: plus_plus(nat,zero_zero(nat),A) = ti(nat,A). [resolve(588,a,561,a)]. 1.07/1.49 589 -comm_semiring_1(A) | -dvd_dvd(A,B,C) | dvd_dvd(A,B,times_times(A,D,C)) # label(fact_482_dvd__mult) # label(axiom). [clausify(420)]. 1.07/1.49 Derived: -dvd_dvd(real,A,B) | dvd_dvd(real,A,times_times(real,C,B)). [resolve(589,a,544,a)]. 1.07/1.49 Derived: -dvd_dvd(int,A,B) | dvd_dvd(int,A,times_times(int,C,B)). [resolve(589,a,555,a)]. 1.07/1.50 Derived: -dvd_dvd(nat,A,B) | dvd_dvd(nat,A,times_times(nat,C,B)). [resolve(589,a,561,a)]. 1.07/1.50 590 -comm_semiring_1(A) | plus_plus(A,times_times(A,B,C),times_times(A,D,C)) = times_times(A,plus_plus(A,B,D),C) # label(fact_107_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom). [clausify(463)]. 1.07/1.50 Derived: plus_plus(real,times_times(real,A,B),times_times(real,C,B)) = times_times(real,plus_plus(real,A,C),B). [resolve(590,a,544,a)]. 1.07/1.50 Derived: plus_plus(nat,times_times(nat,A,B),times_times(nat,C,B)) = times_times(nat,plus_plus(nat,A,C),B). [resolve(590,a,561,a)]. 1.07/1.50 591 -comm_semiring_1(A) | times_times(A,B,one_one(A)) = ti(A,B) # label(fact_111_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom). [clausify(469)]. 1.07/1.50 Derived: times_times(real,A,one_one(real)) = ti(real,A). [resolve(591,a,544,a)]. 1.07/1.50 Derived: times_times(int,A,one_one(int)) = ti(int,A). [resolve(591,a,555,a)]. 1.07/1.50 Derived: times_times(nat,A,one_one(nat)) = ti(nat,A). [resolve(591,a,561,a)]. 1.07/1.50 592 -comm_semiring_1(A) | times_times(A,times_times(A,B,C),times_times(A,D,E)) = times_times(A,D,times_times(A,times_times(A,B,C),E)) # label(fact_67_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom). [clausify(473)]. 1.07/1.50 593 -comm_semiring_1(A) | -ord_less_eq(nat,B,C) | dvd_dvd(A,power_power(A,D,B),power_power(A,D,C)) # label(fact_184_le__imp__power__dvd) # label(axiom). [clausify(488)]. 1.07/1.50 Derived: -ord_less_eq(nat,A,B) | dvd_dvd(real,power_power(real,C,A),power_power(real,C,B)). [resolve(593,a,544,a)]. 1.07/1.50 Derived: -ord_less_eq(nat,A,B) | dvd_dvd(int,power_power(int,C,A),power_power(int,C,B)). [resolve(593,a,555,a)]. 1.07/1.50 Derived: -ord_less_eq(nat,A,B) | dvd_dvd(nat,power_power(nat,C,A),power_power(nat,C,B)). [resolve(593,a,561,a)]. 1.07/1.50 594 -comm_semiring_1(A) | -dvd_dvd(A,B,C) | -ord_less_eq(nat,D,E) | dvd_dvd(A,power_power(A,B,D),power_power(A,C,E)) # label(fact_185_dvd__power__le) # label(axiom). [clausify(492)]. 1.07/1.50 Derived: -dvd_dvd(real,A,B) | -ord_less_eq(nat,C,D) | dvd_dvd(real,power_power(real,A,C),power_power(real,B,D)). [resolve(594,a,544,a)]. 1.07/1.50 Derived: -dvd_dvd(int,A,B) | -ord_less_eq(nat,C,D) | dvd_dvd(int,power_power(int,A,C),power_power(int,B,D)). [resolve(594,a,555,a)]. 1.07/1.50 Derived: -dvd_dvd(nat,A,B) | -ord_less_eq(nat,C,D) | dvd_dvd(nat,power_power(nat,A,C),power_power(nat,B,D)). [resolve(594,a,561,a)]. 1.07/1.50 595 -comm_semiring_1(A) | plus_plus(A,plus_plus(A,B,C),D) = plus_plus(A,B,plus_plus(A,C,D)) # label(fact_77_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom). [clausify(493)]. 1.07/1.50 596 -comm_semiring_1(A) | plus_plus(A,times_times(A,B,C),times_times(A,D,C)) = times_times(A,plus_plus(A,B,D),C) # label(fact_108_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom). [clausify(495)]. 1.07/1.50 Derived: plus_plus(int,times_times(int,A,B),times_times(int,C,B)) = times_times(int,plus_plus(int,A,C),B). [resolve(596,a,555,a)]. 1.07/1.50 597 -comm_semiring_1(A) | plus_plus(A,times_times(A,B,C),times_times(A,B,D)) = times_times(A,B,plus_plus(A,C,D)) # label(fact_110_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom). [clausify(499)]. 1.07/1.50 Derived: plus_plus(real,times_times(real,A,B),times_times(real,A,C)) = times_times(real,A,plus_plus(real,B,C)). [resolve(597,a,544,a)]. 1.07/1.50 Derived: plus_plus(int,times_times(int,A,B),times_times(int,A,C)) = times_times(int,A,plus_plus(int,B,C)). [resolve(597,a,555,a)]. 1.07/1.50 Derived: plus_plus(nat,times_times(nat,A,B),times_times(nat,A,C)) = times_times(nat,A,plus_plus(nat,B,C)). [resolve(597,a,561,a)]. 1.07/1.50 598 -comm_semiring_1(A) | -dvd_dvd(A,B,C) | -dvd_dvd(A,D,E) | dvd_dvd(A,times_times(A,B,D),times_times(A,C,E)) # label(fact_483_mult__dvd__mono) # label(axiom). [clausify(511)]. 1.07/1.50 Derived: -dvd_dvd(real,A,B) | -dvd_dvd(real,C,D) | dvd_dvd(real,times_times(real,A,C),times_times(real,B,D)). [resolve(598,a,544,a)]. 1.07/1.50 Derived: -dvd_dvd(int,A,B) | -dvd_dvd(int,C,D) | dvd_dvd(int,times_times(int,A,C),times_times(int,B,D)). [resolve(598,a,555,a)]. 1.07/1.50 Derived: -dvd_dvd(nat,A,B) | -dvd_dvd(nat,C,D) | dvd_dvd(nat,times_times(nat,A,C),times_times(nat,B,D)). [resolve(598,a,561,a)]. 1.07/1.55 599 ordere453448008miring(real) # label(arity_RealDef_Oreal___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 1.07/1.55 600 -ordere453448008miring(A) | -ord_less_eq(A,zero_zero(A),B) | -ord_less_eq(A,zero_zero(A),C) | ord_less_eq(A,zero_zero(A),times_times(A,B,C)) # label(fact_493_mult__nonneg__nonneg) # label(axiom). [clausify(17)]. 1.07/1.55 601 -ordere453448008miring(A) | -ord_less_eq(A,B,zero_zero(A)) | -ord_less_eq(A,zero_zero(A),C) | ord_less_eq(A,times_times(A,B,C),zero_zero(A)) # label(fact_496_mult__nonpos__nonneg) # label(axiom). [clausify(54)]. 1.07/1.55 602 -ordere453448008miring(A) | -ord_less_eq(A,zero_zero(A),B) | -ord_less_eq(A,C,zero_zero(A)) | ord_less_eq(A,times_times(A,B,C),zero_zero(A)) # label(fact_494_mult__nonneg__nonpos) # label(axiom). [clausify(180)]. 1.07/1.55 603 -ordere453448008miring(A) | -ord_less_eq(A,zero_zero(A),B) | -ord_less_eq(A,C,zero_zero(A)) | ord_less_eq(A,times_times(A,C,B),zero_zero(A)) # label(fact_495_mult__nonneg__nonpos2) # label(axiom). [clausify(214)]. 1.07/1.55 604 ordere453448008miring(int) # label(arity_Int_Oint___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 1.07/1.55 605 ordere453448008miring(nat) # label(arity_Nat_Onat___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 1.07/1.55 Derived: -ord_less_eq(nat,zero_zero(nat),A) | -ord_less_eq(nat,zero_zero(nat),B) | ord_less_eq(nat,zero_zero(nat),times_times(nat,A,B)). [resolve(605,a,600,a)]. 1.07/1.55 Derived: -ord_less_eq(nat,A,zero_zero(nat)) | -ord_less_eq(nat,zero_zero(nat),B) | ord_less_eq(nat,times_times(nat,A,B),zero_zero(nat)). [resolve(605,a,601,a)]. 1.07/1.55 Derived: -ord_less_eq(nat,zero_zero(nat),A) | -ord_less_eq(nat,B,zero_zero(nat)) | ord_less_eq(nat,times_times(nat,A,B),zero_zero(nat)). [resolve(605,a,602,a)]. 1.07/1.55 606 number_ring(int) # label(arity_Int_Oint___Int_Onumber__ring) # label(axiom). [assumption]. 1.07/1.55 607 -number_ring(A) | plus_plus(A,one_one(A),number_number_of(A,B)) = number_number_of(A,plus_plus(int,bit1(pls),B)) # label(fact_16_add__special_I2_J) # label(axiom). [clausify(18)]. 1.07/1.55 608 -number_ring(A) | plus_plus(A,B,B) = times_times(A,number_number_of(A,bit0(bit1(pls))),B) # label(fact_164_mult__2) # label(axiom). [clausify(21)]. 1.07/1.55 609 -number_ring(A) | plus_plus(A,number_number_of(A,plus_plus(int,B,C)),D) = plus_plus(A,number_number_of(A,B),plus_plus(A,number_number_of(A,C),D)) # label(fact_146_add__number__of__left) # label(axiom). [clausify(34)]. 1.07/1.55 610 -number_ring(A) | minus_minus(A,number_number_of(A,B),number_number_of(A,C)) = number_number_of(A,minus_minus(int,B,C)) # label(fact_355_number__of__diff) # label(axiom). [clausify(35)]. 1.07/1.55 611 -linordered_idom(A) | -number_ring(A) | -ord_less(int,pls,B) | ord_less(A,zero_zero(A),number_number_of(A,B)) # label(fact_244_less__special_I1_J) # label(axiom). [clausify(38)]. 1.07/1.55 612 -linordered_idom(A) | -number_ring(A) | ord_less(int,pls,B) | -ord_less(A,zero_zero(A),number_number_of(A,B)) # label(fact_244_less__special_I1_J) # label(axiom). [clausify(38)]. 1.07/1.55 613 -number_ring(A) | plus_plus(A,plus_plus(A,zero_zero(A),number_number_of(A,B)),number_number_of(A,B)) = number_number_of(A,bit0(B)) # label(fact_239_number__of__Bit0) # label(axiom). [clausify(52)]. 1.07/1.55 614 -number_ring(A) | times_times(A,number_number_of(A,bit1(pls)),B) = ti(A,B) # label(fact_153_mult__numeral__1) # label(axiom). [clausify(61)]. 1.07/1.55 615 -ring_char_0(A) | -number_ring(A) | ti(int,B) != ti(int,C) | number_number_of(A,B) = number_number_of(A,C) # label(fact_80_eq__number__of) # label(axiom). [clausify(76)]. 1.07/1.55 616 -ring_char_0(A) | -number_ring(A) | ti(int,B) = ti(int,C) | number_number_of(A,B) != number_number_of(A,C) # label(fact_80_eq__number__of) # label(axiom). [clausify(76)]. 1.07/1.55 617 -number_ring(A) | times_times(A,B,number_number_of(A,bit1(pls))) = ti(A,B) # label(fact_154_mult__numeral__1__right) # label(axiom). [clausify(77)]. 1.07/1.55 Derived: plus_plus(int,one_one(int),number_number_of(int,A)) = number_number_of(int,plus_plus(int,bit1(pls),A)). [resolve(606,a,607,a)]. 1.07/1.55 Derived: plus_plus(int,A,A) = times_times(int,number_number_of(int,bit0(bit1(pls))),A). [resolve(606,a,608,a)]. 1.07/1.55 Derived: plus_plus(int,number_number_of(int,plus_plus(int,A,B)),C) = plus_plus(int,number_number_of(int,A),plus_plus(int,number_number_of(int,B),C)). [resolve(606,a,609,a)]. 1.07/1.56 Derived: minus_minus(int,number_number_of(int,A),number_number_of(int,B)) = number_number_of(int,minus_minus(int,A,B)). [resolve(606,a,610,a)]. 1.07/1.56 Derived: -linordered_idom(int) | -ord_less(int,pls,A) | ord_less(int,zero_zero(int),number_number_of(int,A)). [resolve(606,a,611,b)]. 1.07/1.56 Derived: -linordered_idom(int) | ord_less(int,pls,A) | -ord_less(int,zero_zero(int),number_number_of(int,A)). [resolve(606,a,612,b)]. 1.07/1.56 Derived: plus_plus(int,plus_plus(int,zero_zero(int),number_number_of(int,A)),number_number_of(int,A)) = number_number_of(int,bit0(A)). [resolve(606,a,613,a)]. 1.07/1.56 Derived: times_times(int,number_number_of(int,bit1(pls)),A) = ti(int,A). [resolve(606,a,614,a)]. 1.07/1.56 Derived: -ring_char_0(int) | ti(int,A) != ti(int,B) | number_number_of(int,A) = number_number_of(int,B). [resolve(606,a,615,b)]. 1.07/1.56 Derived: -ring_char_0(int) | ti(int,A) = ti(int,B) | number_number_of(int,A) != number_number_of(int,B). [resolve(606,a,616,b)]. 1.07/1.56 Derived: times_times(int,A,number_number_of(int,bit1(pls))) = ti(int,A). [resolve(606,a,617,a)]. 1.07/1.56 618 -number_ring(A) | -linordered_idom(A) | -ord_less(A,one_one(A),number_number_of(A,B)) | ord_less(int,bit1(pls),B) # label(fact_103_less__special_I2_J) # label(axiom). [clausify(103)]. 1.07/1.56 Derived: -linordered_idom(int) | -ord_less(int,one_one(int),number_number_of(int,A)) | ord_less(int,bit1(pls),A). [resolve(618,a,606,a)]. 1.07/1.56 619 -number_ring(A) | -linordered_idom(A) | ord_less(A,one_one(A),number_number_of(A,B)) | -ord_less(int,bit1(pls),B) # label(fact_103_less__special_I2_J) # label(axiom). [clausify(103)]. 1.07/1.56 Derived: -linordered_idom(int) | ord_less(int,one_one(int),number_number_of(int,A)) | -ord_less(int,bit1(pls),A). [resolve(619,a,606,a)]. 1.07/1.56 620 -number_ring(A) | plus_plus(A,B,number_number_of(A,pls)) = ti(A,B) # label(fact_142_add__numeral__0__right) # label(axiom). [clausify(121)]. 1.07/1.56 Derived: plus_plus(int,A,number_number_of(int,pls)) = ti(int,A). [resolve(620,a,606,a)]. 1.07/1.56 621 -number_ring(A) | plus_plus(A,number_number_of(A,pls),B) = ti(A,B) # label(fact_141_add__numeral__0) # label(axiom). [clausify(126)]. 1.07/1.56 Derived: plus_plus(int,number_number_of(int,pls),A) = ti(int,A). [resolve(621,a,606,a)]. 1.07/1.56 622 -number_ring(A) | plus_plus(A,number_number_of(A,B),one_one(A)) = number_number_of(A,plus_plus(int,B,bit1(pls))) # label(fact_17_add__special_I3_J) # label(axiom). [clausify(146)]. 1.07/1.56 Derived: plus_plus(int,number_number_of(int,A),one_one(int)) = number_number_of(int,plus_plus(int,A,bit1(pls))). [resolve(622,a,606,a)]. 1.07/1.56 623 -number_ring(A) | times_times(A,number_number_of(A,times_times(int,B,C)),D) = times_times(A,number_number_of(A,B),times_times(A,number_number_of(A,C),D)) # label(fact_143_mult__number__of__left) # label(axiom). [clausify(197)]. 1.07/1.56 Derived: times_times(int,number_number_of(int,times_times(int,A,B)),C) = times_times(int,number_number_of(int,A),times_times(int,number_number_of(int,B),C)). [resolve(623,a,606,a)]. 1.07/1.56 624 number_ring(real) # label(arity_RealDef_Oreal___Int_Onumber__ring) # label(axiom). [assumption]. 1.07/1.56 Derived: plus_plus(real,one_one(real),number_number_of(real,A)) = number_number_of(real,plus_plus(int,bit1(pls),A)). [resolve(624,a,607,a)]. 1.07/1.56 Derived: plus_plus(real,A,A) = times_times(real,number_number_of(real,bit0(bit1(pls))),A). [resolve(624,a,608,a)]. 1.07/1.56 Derived: plus_plus(real,number_number_of(real,plus_plus(int,A,B)),C) = plus_plus(real,number_number_of(real,A),plus_plus(real,number_number_of(real,B),C)). [resolve(624,a,609,a)]. 1.07/1.56 Derived: minus_minus(real,number_number_of(real,A),number_number_of(real,B)) = number_number_of(real,minus_minus(int,A,B)). [resolve(624,a,610,a)]. 1.07/1.56 Derived: -linordered_idom(real) | -ord_less(int,pls,A) | ord_less(real,zero_zero(real),number_number_of(real,A)). [resolve(624,a,611,b)]. 1.07/1.56 Derived: -linordered_idom(real) | ord_less(int,pls,A) | -ord_less(real,zero_zero(real),number_number_of(real,A)). [resolve(624,a,612,b)]. 1.07/1.56 Derived: plus_plus(real,plus_plus(real,zero_zero(real),number_number_of(real,A)),number_number_of(real,A)) = number_number_of(real,bit0(A)). [resolve(624,a,613,a)]. 1.07/1.56 Derived: times_times(real,number_number_of(real,bit1(pls)),A) = ti(real,A). [resolve(624,a,614,a)]. 1.07/1.56 Derived: -ring_char_0(real) | ti(int,A) != ti(int,B) | number_number_of(real,A) = number_number_of(real,B). [resolve(624,a,615,b)]. 1.07/1.56 Derived: -ring_char_0(real) | ti(int,A) = ti(int,B) | number_number_of(real,A) != number_number_of(real,B). [resolve(624,a,616,b)]. 1.07/1.56 Derived: times_times(real,A,number_number_of(real,bit1(pls))) = ti(real,A). [resolve(624,a,617,a)]. 1.07/1.56 Derived: -linordered_idom(real) | -ord_less(real,one_one(real),number_number_of(real,A)) | ord_less(int,bit1(pls),A). [resolve(624,a,618,a)]. 1.07/1.56 Derived: -linordered_idom(real) | ord_less(real,one_one(real),number_number_of(real,A)) | -ord_less(int,bit1(pls),A). [resolve(624,a,619,a)]. 1.07/1.56 Derived: plus_plus(real,A,number_number_of(real,pls)) = ti(real,A). [resolve(624,a,620,a)]. 1.07/1.56 Derived: plus_plus(real,number_number_of(real,pls),A) = ti(real,A). [resolve(624,a,621,a)]. 1.07/1.56 Derived: plus_plus(real,number_number_of(real,A),one_one(real)) = number_number_of(real,plus_plus(int,A,bit1(pls))). [resolve(624,a,622,a)]. 1.07/1.56 Derived: times_times(real,number_number_of(real,times_times(int,A,B)),C) = times_times(real,number_number_of(real,A),times_times(real,number_number_of(real,B),C)). [resolve(624,a,623,a)]. 1.07/1.56 625 -number_ring(A) | times_times(A,number_number_of(A,B),number_number_of(A,C)) = number_number_of(A,times_times(int,B,C)) # label(fact_145_number__of__mult) # label(axiom). [clausify(204)]. 1.07/1.56 Derived: times_times(real,number_number_of(real,A),number_number_of(real,B)) = number_number_of(real,times_times(int,A,B)). [resolve(625,a,624,a)]. 1.07/1.56 626 -number_ring(A) | plus_plus(A,number_number_of(A,B),number_number_of(A,C)) = number_number_of(A,plus_plus(int,B,C)) # label(fact_147_add__number__of__eq) # label(axiom). [clausify(224)]. 1.07/1.56 Derived: plus_plus(int,number_number_of(int,A),number_number_of(int,B)) = number_number_of(int,plus_plus(int,A,B)). [resolve(626,a,606,a)]. 1.07/1.56 Derived: plus_plus(real,number_number_of(real,A),number_number_of(real,B)) = number_number_of(real,plus_plus(int,A,B)). [resolve(626,a,624,a)]. 1.07/1.56 627 -number_ring(A) | zero_zero(A) = number_number_of(A,pls) # label(fact_219_semiring__norm_I112_J) # label(axiom). [clausify(255)]. 1.07/1.56 Derived: zero_zero(int) = number_number_of(int,pls). [resolve(627,a,606,a)]. 1.07/1.56 Derived: zero_zero(real) = number_number_of(real,pls). [resolve(627,a,624,a)]. 1.07/1.56 628 -number_ring(A) | times_times(A,number_number_of(A,B),number_number_of(A,C)) = number_number_of(A,times_times(int,B,C)) # label(fact_144_arith__simps_I32_J) # label(axiom). [clausify(267)]. 1.07/1.56 629 -linordered_idom(A) | -number_ring(A) | -ord_less(int,B,bit1(pls)) | ord_less(A,number_number_of(A,B),one_one(A)) # label(fact_102_less__special_I4_J) # label(axiom). [clausify(271)]. 1.07/1.56 Derived: -linordered_idom(int) | -ord_less(int,A,bit1(pls)) | ord_less(int,number_number_of(int,A),one_one(int)). [resolve(629,b,606,a)]. 1.07/1.56 Derived: -linordered_idom(real) | -ord_less(int,A,bit1(pls)) | ord_less(real,number_number_of(real,A),one_one(real)). [resolve(629,b,624,a)]. 1.07/1.56 630 -linordered_idom(A) | -number_ring(A) | ord_less(int,B,bit1(pls)) | -ord_less(A,number_number_of(A,B),one_one(A)) # label(fact_102_less__special_I4_J) # label(axiom). [clausify(271)]. 1.07/1.56 Derived: -linordered_idom(int) | ord_less(int,A,bit1(pls)) | -ord_less(int,number_number_of(int,A),one_one(int)). [resolve(630,b,606,a)]. 1.07/1.56 Derived: -linordered_idom(real) | ord_less(int,A,bit1(pls)) | -ord_less(real,number_number_of(real,A),one_one(real)). [resolve(630,b,624,a)]. 1.07/1.56 631 -number_ring(A) | plus_plus(A,B,B) = times_times(A,B,number_number_of(A,bit0(bit1(pls)))) # label(fact_166_mult__2__right) # label(axiom). [clausify(275)]. 1.07/1.56 Derived: plus_plus(int,A,A) = times_times(int,A,number_number_of(int,bit0(bit1(pls)))). [resolve(631,a,606,a)]. 1.07/1.56 Derived: plus_plus(real,A,A) = times_times(real,A,number_number_of(real,bit0(bit1(pls)))). [resolve(631,a,624,a)]. 1.07/1.56 632 -number_ring(A) | -linordered_idom(A) | -ord_less_eq(A,number_number_of(A,B),one_one(A)) | ord_less_eq(int,B,bit1(pls)) # label(fact_104_le__special_I4_J) # label(axiom). [clausify(289)]. 1.07/1.57 Derived: -linordered_idom(int) | -ord_less_eq(int,number_number_of(int,A),one_one(int)) | ord_less_eq(int,A,bit1(pls)). [resolve(632,a,606,a)]. 1.07/1.57 Derived: -linordered_idom(real) | -ord_less_eq(real,number_number_of(real,A),one_one(real)) | ord_less_eq(int,A,bit1(pls)). [resolve(632,a,624,a)]. 1.07/1.57 633 -number_ring(A) | -linordered_idom(A) | ord_less_eq(A,number_number_of(A,B),one_one(A)) | -ord_less_eq(int,B,bit1(pls)) # label(fact_104_le__special_I4_J) # label(axiom). [clausify(289)]. 1.07/1.57 Derived: -linordered_idom(int) | ord_less_eq(int,number_number_of(int,A),one_one(int)) | -ord_less_eq(int,A,bit1(pls)). [resolve(633,a,606,a)]. 1.07/1.57 Derived: -linordered_idom(real) | ord_less_eq(real,number_number_of(real,A),one_one(real)) | -ord_less_eq(int,A,bit1(pls)). [resolve(633,a,624,a)]. 1.07/1.57 634 -linordered_idom(A) | -number_ring(A) | -ord_less(int,B,pls) | ord_less(A,number_number_of(A,B),zero_zero(A)) # label(fact_245_less__special_I3_J) # label(axiom). [clausify(313)]. 1.07/1.57 Derived: -linordered_idom(int) | -ord_less(int,A,pls) | ord_less(int,number_number_of(int,A),zero_zero(int)). [resolve(634,b,606,a)]. 1.07/1.57 Derived: -linordered_idom(real) | -ord_less(int,A,pls) | ord_less(real,number_number_of(real,A),zero_zero(real)). [resolve(634,b,624,a)]. 1.07/1.57 635 -linordered_idom(A) | -number_ring(A) | ord_less(int,B,pls) | -ord_less(A,number_number_of(A,B),zero_zero(A)) # label(fact_245_less__special_I3_J) # label(axiom). [clausify(313)]. 1.07/1.57 Derived: -linordered_idom(int) | ord_less(int,A,pls) | -ord_less(int,number_number_of(int,A),zero_zero(int)). [resolve(635,b,606,a)]. 1.07/1.57 Derived: -linordered_idom(real) | ord_less(int,A,pls) | -ord_less(real,number_number_of(real,A),zero_zero(real)). [resolve(635,b,624,a)]. 1.07/1.57 636 -number_ring(A) | plus_plus(A,number_number_of(A,B),number_number_of(A,C)) = number_number_of(A,plus_plus(int,B,C)) # label(fact_148_number__of__add) # label(axiom). [clausify(332)]. 1.07/1.57 637 -number_ring(A) | plus_plus(A,plus_plus(A,one_one(A),number_number_of(A,B)),number_number_of(A,B)) = number_number_of(A,bit1(B)) # label(fact_152_number__of__Bit1) # label(axiom). [clausify(345)]. 1.07/1.57 Derived: plus_plus(int,plus_plus(int,one_one(int),number_number_of(int,A)),number_number_of(int,A)) = number_number_of(int,bit1(A)). [resolve(637,a,606,a)]. 1.07/1.57 Derived: plus_plus(real,plus_plus(real,one_one(real),number_number_of(real,A)),number_number_of(real,A)) = number_number_of(real,bit1(A)). [resolve(637,a,624,a)]. 1.07/1.57 638 -number_ring(A) | one_one(A) = number_number_of(A,bit1(pls)) # label(fact_157_semiring__norm_I110_J) # label(axiom). [clausify(364)]. 1.07/1.57 Derived: one_one(int) = number_number_of(int,bit1(pls)). [resolve(638,a,606,a)]. 1.07/1.57 Derived: one_one(real) = number_number_of(real,bit1(pls)). [resolve(638,a,624,a)]. 1.07/1.57 639 -number_ring(A) | plus_plus(A,one_one(A),one_one(A)) = number_number_of(A,bit0(bit1(pls))) # label(fact_18_one__add__one__is__two) # label(axiom). [clausify(365)]. 1.07/1.57 Derived: plus_plus(int,one_one(int),one_one(int)) = number_number_of(int,bit0(bit1(pls))). [resolve(639,a,606,a)]. 1.07/1.57 Derived: plus_plus(real,one_one(real),one_one(real)) = number_number_of(real,bit0(bit1(pls))). [resolve(639,a,624,a)]. 1.07/1.57 640 -linordered_idom(A) | -number_ring(A) | -ord_less_eq(int,B,C) | ord_less_eq(A,number_number_of(A,B),number_number_of(A,C)) # label(fact_31_le__number__of) # label(axiom). [clausify(377)]. 1.07/1.57 Derived: -linordered_idom(int) | -ord_less_eq(int,A,B) | ord_less_eq(int,number_number_of(int,A),number_number_of(int,B)). [resolve(640,b,606,a)]. 1.07/1.57 Derived: -linordered_idom(real) | -ord_less_eq(int,A,B) | ord_less_eq(real,number_number_of(real,A),number_number_of(real,B)). [resolve(640,b,624,a)]. 1.07/1.57 641 -linordered_idom(A) | -number_ring(A) | ord_less_eq(int,B,C) | -ord_less_eq(A,number_number_of(A,B),number_number_of(A,C)) # label(fact_31_le__number__of) # label(axiom). [clausify(377)]. 1.07/1.57 Derived: -linordered_idom(int) | ord_less_eq(int,A,B) | -ord_less_eq(int,number_number_of(int,A),number_number_of(int,B)). [resolve(641,b,606,a)]. 1.07/1.57 Derived: -linordered_idom(real) | ord_less_eq(int,A,B) | -ord_less_eq(real,number_number_of(real,A),number_number_of(real,B)). [resolve(641,b,624,a)]. 1.07/1.58 642 -number_ring(A) | times_times(A,plus_plus(A,one_one(A),one_one(A)),number_number_of(A,B)) = number_number_of(A,bit0(B)) # label(fact_160_double__number__of__Bit0) # label(axiom). [clausify(393)]. 1.07/1.58 Derived: times_times(int,plus_plus(int,one_one(int),one_one(int)),number_number_of(int,A)) = number_number_of(int,bit0(A)). [resolve(642,a,606,a)]. 1.07/1.58 Derived: times_times(real,plus_plus(real,one_one(real),one_one(real)),number_number_of(real,A)) = number_number_of(real,bit0(A)). [resolve(642,a,624,a)]. 1.07/1.58 643 -number_ring(A) | one_one(A) = number_number_of(A,bit1(pls)) # label(fact_156_numeral__1__eq__1) # label(axiom). [clausify(394)]. 1.07/1.58 644 -number_ring(A) | -linordered_idom(A) | -ord_less_eq(int,bit1(pls),B) | ord_less_eq(A,one_one(A),number_number_of(A,B)) # label(fact_105_le__special_I2_J) # label(axiom). [clausify(406)]. 1.07/1.58 Derived: -linordered_idom(int) | -ord_less_eq(int,bit1(pls),A) | ord_less_eq(int,one_one(int),number_number_of(int,A)). [resolve(644,a,606,a)]. 1.07/1.58 Derived: -linordered_idom(real) | -ord_less_eq(int,bit1(pls),A) | ord_less_eq(real,one_one(real),number_number_of(real,A)). [resolve(644,a,624,a)]. 1.07/1.58 645 -number_ring(A) | -linordered_idom(A) | ord_less_eq(int,bit1(pls),B) | -ord_less_eq(A,one_one(A),number_number_of(A,B)) # label(fact_105_le__special_I2_J) # label(axiom). [clausify(406)]. 1.07/1.58 Derived: -linordered_idom(int) | ord_less_eq(int,bit1(pls),A) | -ord_less_eq(int,one_one(int),number_number_of(int,A)). [resolve(645,a,606,a)]. 1.07/1.58 Derived: -linordered_idom(real) | ord_less_eq(int,bit1(pls),A) | -ord_less_eq(real,one_one(real),number_number_of(real,A)). [resolve(645,a,624,a)]. 1.07/1.58 646 -number_ring(A) | -linordered_idom(A) | -ord_less_eq(int,pls,B) | ord_less_eq(A,zero_zero(A),number_number_of(A,B)) # label(fact_246_le__special_I1_J) # label(axiom). [clausify(435)]. 1.07/1.58 Derived: -linordered_idom(int) | -ord_less_eq(int,pls,A) | ord_less_eq(int,zero_zero(int),number_number_of(int,A)). [resolve(646,a,606,a)]. 1.07/1.58 Derived: -linordered_idom(real) | -ord_less_eq(int,pls,A) | ord_less_eq(real,zero_zero(real),number_number_of(real,A)). [resolve(646,a,624,a)]. 1.07/1.58 647 -number_ring(A) | -linordered_idom(A) | ord_less_eq(int,pls,B) | -ord_less_eq(A,zero_zero(A),number_number_of(A,B)) # label(fact_246_le__special_I1_J) # label(axiom). [clausify(435)]. 1.07/1.58 Derived: -linordered_idom(int) | ord_less_eq(int,pls,A) | -ord_less_eq(int,zero_zero(int),number_number_of(int,A)). [resolve(647,a,606,a)]. 1.07/1.58 Derived: -linordered_idom(real) | ord_less_eq(int,pls,A) | -ord_less_eq(real,zero_zero(real),number_number_of(real,A)). [resolve(647,a,624,a)]. 1.07/1.58 648 -linordered_idom(A) | -number_ring(A) | -ord_less(A,number_number_of(A,B),number_number_of(A,C)) | ord_less(int,B,C) # label(fact_30_less__number__of) # label(axiom). [clausify(437)]. 1.07/1.58 Derived: -linordered_idom(int) | -ord_less(int,number_number_of(int,A),number_number_of(int,B)) | ord_less(int,A,B). [resolve(648,b,606,a)]. 1.07/1.58 Derived: -linordered_idom(real) | -ord_less(real,number_number_of(real,A),number_number_of(real,B)) | ord_less(int,A,B). [resolve(648,b,624,a)]. 1.07/1.58 649 -linordered_idom(A) | -number_ring(A) | ord_less(A,number_number_of(A,B),number_number_of(A,C)) | -ord_less(int,B,C) # label(fact_30_less__number__of) # label(axiom). [clausify(437)]. 1.07/1.58 Derived: -linordered_idom(int) | ord_less(int,number_number_of(int,A),number_number_of(int,B)) | -ord_less(int,A,B). [resolve(649,b,606,a)]. 1.07/1.58 Derived: -linordered_idom(real) | ord_less(real,number_number_of(real,A),number_number_of(real,B)) | -ord_less(int,A,B). [resolve(649,b,624,a)]. 1.07/1.58 650 -number_ring(A) | minus_minus(A,plus_plus(A,power_power(A,B,number_number_of(nat,bit0(bit1(pls)))),power_power(A,C,number_number_of(nat,bit0(bit1(pls))))),times_times(A,times_times(A,number_number_of(A,bit0(bit1(pls))),B),C)) = power_power(A,minus_minus(A,B,C),number_number_of(nat,bit0(bit1(pls)))) # label(fact_375_power2__diff) # label(axiom). [clausify(449)]. 1.07/1.58 Derived: minus_minus(int,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))))),times_times(int,times_times(int,number_number_of(int,bit0(bit1(pls))),A),B)) = power_power(int,minus_minus(int,A,B),number_number_of(nat,bit0(bit1(pls)))). [resolve(650,a,606,a)]. 1.07/1.60 Derived: minus_minus(real,plus_plus(real,power_power(real,A,number_number_of(nat,bit0(bit1(pls)))),power_power(real,B,number_number_of(nat,bit0(bit1(pls))))),times_times(real,times_times(real,number_number_of(real,bit0(bit1(pls))),A),B)) = power_power(real,minus_minus(real,A,B),number_number_of(nat,bit0(bit1(pls)))). [resolve(650,a,624,a)]. 1.07/1.60 651 -number_ring(A) | zero_zero(A) = number_number_of(A,pls) # label(fact_220_number__of__Pls) # label(axiom). [clausify(465)]. 1.07/1.60 652 -number_ring(A) | -linordered_idom(A) | -ord_less_eq(int,B,pls) | ord_less_eq(A,number_number_of(A,B),zero_zero(A)) # label(fact_247_le__special_I3_J) # label(axiom). [clausify(485)]. 1.07/1.60 Derived: -linordered_idom(int) | -ord_less_eq(int,A,pls) | ord_less_eq(int,number_number_of(int,A),zero_zero(int)). [resolve(652,a,606,a)]. 1.07/1.60 Derived: -linordered_idom(real) | -ord_less_eq(int,A,pls) | ord_less_eq(real,number_number_of(real,A),zero_zero(real)). [resolve(652,a,624,a)]. 1.07/1.60 653 -number_ring(A) | -linordered_idom(A) | ord_less_eq(int,B,pls) | -ord_less_eq(A,number_number_of(A,B),zero_zero(A)) # label(fact_247_le__special_I3_J) # label(axiom). [clausify(485)]. 1.07/1.60 Derived: -linordered_idom(int) | ord_less_eq(int,A,pls) | -ord_less_eq(int,number_number_of(int,A),zero_zero(int)). [resolve(653,a,606,a)]. 1.07/1.60 Derived: -linordered_idom(real) | ord_less_eq(int,A,pls) | -ord_less_eq(real,number_number_of(real,A),zero_zero(real)). [resolve(653,a,624,a)]. 1.07/1.60 654 -number_ring(A) | one_one(A) = power_power(A,number_number_of(A,min),times_times(nat,number_number_of(nat,bit0(bit1(pls))),B)) # label(fact_336_power__m1__even) # label(axiom). [clausify(494)]. 1.07/1.60 Derived: one_one(int) = power_power(int,number_number_of(int,min),times_times(nat,number_number_of(nat,bit0(bit1(pls))),A)). [resolve(654,a,606,a)]. 1.07/1.60 Derived: one_one(real) = power_power(real,number_number_of(real,min),times_times(nat,number_number_of(nat,bit0(bit1(pls))),A)). [resolve(654,a,624,a)]. 1.07/1.60 655 -number_ring(A) | minus_minus(A,number_number_of(A,plus_plus(int,B,C)),D) = plus_plus(A,number_number_of(A,B),minus_minus(A,number_number_of(A,C),D)) # label(fact_364_add__number__of__diff1) # label(axiom). [clausify(500)]. 1.07/1.60 Derived: minus_minus(int,number_number_of(int,plus_plus(int,A,B)),C) = plus_plus(int,number_number_of(int,A),minus_minus(int,number_number_of(int,B),C)). [resolve(655,a,606,a)]. 1.07/1.60 Derived: minus_minus(real,number_number_of(real,plus_plus(int,A,B)),C) = plus_plus(real,number_number_of(real,A),minus_minus(real,number_number_of(real,B),C)). [resolve(655,a,624,a)]. 1.07/1.60 656 no_zero_divisors(real) # label(arity_RealDef_Oreal___Rings_Ono__zero__divisors) # label(axiom). [assumption]. 1.07/1.60 657 -no_zero_divisors(A) | zero_zero(A) = ti(A,B) | zero_zero(A) = ti(A,C) | zero_zero(A) != times_times(A,B,C) # label(fact_472_no__zero__divisors) # label(axiom). [clausify(23)]. 1.07/1.60 Derived: zero_zero(real) = ti(real,A) | zero_zero(real) = ti(real,B) | zero_zero(real) != times_times(real,A,B). [resolve(656,a,657,a)]. 1.07/1.60 658 -power(A) | -mult_zero(A) | -no_zero_divisors(A) | -zero_neq_one(A) | zero_zero(A) != power_power(A,B,number_number_of(nat,C)) | zero_zero(A) = ti(A,B) # label(fact_337_power__eq__0__iff__number__of) # label(axiom). [clausify(144)]. 1.07/1.60 Derived: -power(real) | -mult_zero(real) | -zero_neq_one(real) | zero_zero(real) != power_power(real,A,number_number_of(nat,B)) | zero_zero(real) = ti(real,A). [resolve(658,c,656,a)]. 1.07/1.60 659 -power(A) | -mult_zero(A) | -no_zero_divisors(A) | -zero_neq_one(A) | zero_zero(A) != power_power(A,B,number_number_of(nat,C)) | zero_zero(nat) != number_number_of(nat,C) # label(fact_337_power__eq__0__iff__number__of) # label(axiom). [clausify(144)]. 1.07/1.60 Derived: -power(real) | -mult_zero(real) | -zero_neq_one(real) | zero_zero(real) != power_power(real,A,number_number_of(nat,B)) | zero_zero(nat) != number_number_of(nat,B). [resolve(659,c,656,a)]. 1.07/1.60 660 -power(A) | -mult_zero(A) | -no_zero_divisors(A) | -zero_neq_one(A) | zero_zero(A) = power_power(A,B,number_number_of(nat,C)) | zero_zero(A) != ti(A,B) | zero_zero(nat) = number_number_of(nat,C) # label(fact_337_power__eq__0__iff__number__of) # label(axiom). [clausify(144)]. 1.07/1.62 Derived: -power(real) | -mult_zero(real) | -zero_neq_one(real) | zero_zero(real) = power_power(real,A,number_number_of(nat,B)) | zero_zero(real) != ti(real,A) | zero_zero(nat) = number_number_of(nat,B). [resolve(660,c,656,a)]. 1.07/1.62 661 -no_zero_divisors(A) | zero_zero(A) != times_times(A,B,C) | zero_zero(A) = ti(A,C) | zero_zero(A) = ti(A,B) # label(fact_473_divisors__zero) # label(axiom). [clausify(159)]. 1.07/1.62 662 no_zero_divisors(nat) # label(arity_Nat_Onat___Rings_Ono__zero__divisors) # label(axiom). [assumption]. 1.07/1.62 Derived: zero_zero(nat) = ti(nat,A) | zero_zero(nat) = ti(nat,B) | zero_zero(nat) != times_times(nat,A,B). [resolve(662,a,657,a)]. 1.07/1.62 Derived: -power(nat) | -mult_zero(nat) | -zero_neq_one(nat) | zero_zero(nat) != power_power(nat,A,number_number_of(nat,B)) | zero_zero(nat) = ti(nat,A). [resolve(662,a,658,c)]. 1.07/1.62 Derived: -power(nat) | -mult_zero(nat) | -zero_neq_one(nat) | zero_zero(nat) = power_power(nat,A,number_number_of(nat,B)) | zero_zero(nat) != ti(nat,A) | zero_zero(nat) = number_number_of(nat,B). [resolve(662,a,660,c)]. 1.07/1.62 663 -mult_zero(A) | -no_zero_divisors(A) | -zero_neq_one(A) | -power(A) | zero_zero(nat) = B | zero_zero(A) != ti(A,C) | zero_zero(A) = power_power(A,C,B) # label(fact_183_power__eq__0__iff) # label(axiom). [clausify(217)]. 1.07/1.62 Derived: -mult_zero(real) | -zero_neq_one(real) | -power(real) | zero_zero(nat) = A | zero_zero(real) != ti(real,B) | zero_zero(real) = power_power(real,B,A). [resolve(663,b,656,a)]. 1.07/1.62 Derived: -mult_zero(nat) | -zero_neq_one(nat) | -power(nat) | zero_zero(nat) = A | zero_zero(nat) != ti(nat,B) | zero_zero(nat) = power_power(nat,B,A). [resolve(663,b,662,a)]. 1.07/1.62 664 -mult_zero(A) | -no_zero_divisors(A) | -zero_neq_one(A) | -power(A) | zero_zero(nat) != B | zero_zero(A) != power_power(A,C,B) # label(fact_183_power__eq__0__iff) # label(axiom). [clausify(217)]. 1.07/1.62 Derived: -mult_zero(real) | -zero_neq_one(real) | -power(real) | zero_zero(nat) != A | zero_zero(real) != power_power(real,B,A). [resolve(664,b,656,a)]. 1.07/1.62 Derived: -mult_zero(nat) | -zero_neq_one(nat) | -power(nat) | zero_zero(nat) != A | zero_zero(nat) != power_power(nat,B,A). [resolve(664,b,662,a)]. 1.07/1.62 665 -mult_zero(A) | -no_zero_divisors(A) | -zero_neq_one(A) | -power(A) | zero_zero(A) = ti(A,B) | zero_zero(A) != power_power(A,B,C) # label(fact_183_power__eq__0__iff) # label(axiom). [clausify(217)]. 1.07/1.62 Derived: -mult_zero(real) | -zero_neq_one(real) | -power(real) | zero_zero(real) = ti(real,A) | zero_zero(real) != power_power(real,A,B). [resolve(665,b,656,a)]. 1.07/1.62 Derived: -mult_zero(nat) | -zero_neq_one(nat) | -power(nat) | zero_zero(nat) = ti(nat,A) | zero_zero(nat) != power_power(nat,A,B). [resolve(665,b,662,a)]. 1.07/1.62 666 -no_zero_divisors(A) | times_times(A,ti(A,B),C) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_2_arg1) # label(axiom). [clausify(430)]. 1.07/1.62 Derived: times_times(real,ti(real,A),B) = times_times(real,A,B). [resolve(666,a,656,a)]. 1.07/1.62 Derived: times_times(nat,ti(nat,A),B) = times_times(nat,A,B). [resolve(666,a,662,a)]. 1.07/1.62 667 -no_zero_divisors(A) | times_times(A,B,ti(A,C)) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_2_arg2) # label(axiom). [clausify(436)]. 1.07/1.62 Derived: times_times(real,A,ti(real,B)) = times_times(real,A,B). [resolve(667,a,656,a)]. 1.07/1.62 Derived: times_times(nat,A,ti(nat,B)) = times_times(nat,A,B). [resolve(667,a,662,a)]. 1.07/1.62 668 -no_zero_divisors(A) | zero_zero(A) = ti(A,zero_zero(A)) # label(tsy_c_Groups_Ozero__class_Ozero_1_res) # label(axiom). [clausify(466)]. 1.07/1.62 Derived: zero_zero(real) = ti(real,zero_zero(real)). [resolve(668,a,656,a)]. 1.07/1.62 Derived: zero_zero(nat) = ti(nat,zero_zero(nat)). [resolve(668,a,662,a)]. 1.07/1.62 669 no_zero_divisors(int) # label(arity_Int_Oint___Rings_Ono__zero__divisors) # label(axiom). [assumption]. 1.07/1.62 Derived: zero_zero(int) = ti(int,A) | zero_zero(int) = ti(int,B) | zero_zero(int) != times_times(int,A,B). [resolve(669,a,657,a)]. 1.42/1.66 Derived: -power(int) | -mult_zero(int) | -zero_neq_one(int) | zero_zero(int) != power_power(int,A,number_number_of(nat,B)) | zero_zero(int) = ti(int,A). [resolve(669,a,658,c)]. 1.42/1.66 Derived: -power(int) | -mult_zero(int) | -zero_neq_one(int) | zero_zero(int) != power_power(int,A,number_number_of(nat,B)) | zero_zero(nat) != number_number_of(nat,B). [resolve(669,a,659,c)]. 1.42/1.66 Derived: -power(int) | -mult_zero(int) | -zero_neq_one(int) | zero_zero(int) = power_power(int,A,number_number_of(nat,B)) | zero_zero(int) != ti(int,A) | zero_zero(nat) = number_number_of(nat,B). [resolve(669,a,660,c)]. 1.42/1.66 Derived: -mult_zero(int) | -zero_neq_one(int) | -power(int) | zero_zero(nat) = A | zero_zero(int) != ti(int,B) | zero_zero(int) = power_power(int,B,A). [resolve(669,a,663,b)]. 1.42/1.66 Derived: -mult_zero(int) | -zero_neq_one(int) | -power(int) | zero_zero(nat) != A | zero_zero(int) != power_power(int,B,A). [resolve(669,a,664,b)]. 1.42/1.66 Derived: -mult_zero(int) | -zero_neq_one(int) | -power(int) | zero_zero(int) = ti(int,A) | zero_zero(int) != power_power(int,A,B). [resolve(669,a,665,b)]. 1.42/1.66 Derived: times_times(int,ti(int,A),B) = times_times(int,A,B). [resolve(669,a,666,a)]. 1.42/1.66 Derived: times_times(int,A,ti(int,B)) = times_times(int,A,B). [resolve(669,a,667,a)]. 1.42/1.66 Derived: zero_zero(int) = ti(int,zero_zero(int)). [resolve(669,a,668,a)]. 1.42/1.66 670 -no_zero_divisors(A) | times_times(A,B,C) = ti(A,times_times(A,B,C)) # label(tsy_c_Groups_Otimes__class_Otimes_2_res) # label(axiom). [clausify(504)]. 1.42/1.66 Derived: times_times(real,A,B) = ti(real,times_times(real,A,B)). [resolve(670,a,656,a)]. 1.42/1.66 Derived: times_times(nat,A,B) = ti(nat,times_times(nat,A,B)). [resolve(670,a,662,a)]. 1.42/1.66 Derived: times_times(int,A,B) = ti(int,times_times(int,A,B)). [resolve(670,a,669,a)]. 1.42/1.66 671 dvd(real) # label(arity_RealDef_Oreal___Rings_Odvd) # label(axiom). [assumption]. 1.42/1.66 672 -dvd(A) | -dvd_dvd(A,B,ti(A,C)) | dvd_dvd(A,B,C) # label(tsy_c_Rings_Odvd__class_Odvd_arg2) # label(axiom). [clausify(28)]. 1.42/1.66 673 -dvd(A) | dvd_dvd(A,B,ti(A,C)) | -dvd_dvd(A,B,C) # label(tsy_c_Rings_Odvd__class_Odvd_arg2) # label(axiom). [clausify(28)]. 1.42/1.66 Derived: -dvd_dvd(real,A,ti(real,B)) | dvd_dvd(real,A,B). [resolve(671,a,672,a)]. 1.42/1.66 Derived: dvd_dvd(real,A,ti(real,B)) | -dvd_dvd(real,A,B). [resolve(671,a,673,a)]. 1.42/1.66 674 dvd(nat) # label(arity_Nat_Onat___Rings_Odvd) # label(axiom). [assumption]. 1.42/1.66 Derived: -dvd_dvd(nat,A,ti(nat,B)) | dvd_dvd(nat,A,B). [resolve(674,a,672,a)]. 1.42/1.66 Derived: dvd_dvd(nat,A,ti(nat,B)) | -dvd_dvd(nat,A,B). [resolve(674,a,673,a)]. 1.42/1.66 675 -dvd(A) | times_times(A,B,C) != D | dvd_dvd(A,B,D) # label(fact_484_dvdI) # label(axiom). [clausify(185)]. 1.42/1.66 Derived: times_times(real,A,B) != C | dvd_dvd(real,A,C). [resolve(675,a,671,a)]. 1.42/1.66 Derived: times_times(nat,A,B) != C | dvd_dvd(nat,A,C). [resolve(675,a,674,a)]. 1.42/1.66 676 -dvd(A) | times_times(A,B,ti(A,C)) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_0_arg2) # label(axiom). [clausify(209)]. 1.42/1.66 677 -dvd(A) | -dvd_dvd(A,ti(A,B),C) | dvd_dvd(A,B,C) # label(tsy_c_Rings_Odvd__class_Odvd_arg1) # label(axiom). [clausify(260)]. 1.42/1.66 Derived: -dvd_dvd(real,ti(real,A),B) | dvd_dvd(real,A,B). [resolve(677,a,671,a)]. 1.42/1.66 Derived: -dvd_dvd(nat,ti(nat,A),B) | dvd_dvd(nat,A,B). [resolve(677,a,674,a)]. 1.42/1.66 678 -dvd(A) | dvd_dvd(A,ti(A,B),C) | -dvd_dvd(A,B,C) # label(tsy_c_Rings_Odvd__class_Odvd_arg1) # label(axiom). [clausify(260)]. 1.42/1.66 Derived: dvd_dvd(real,ti(real,A),B) | -dvd_dvd(real,A,B). [resolve(678,a,671,a)]. 1.42/1.66 Derived: dvd_dvd(nat,ti(nat,A),B) | -dvd_dvd(nat,A,B). [resolve(678,a,674,a)]. 1.42/1.66 679 -dvd(A) | times_times(A,ti(A,B),C) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_0_arg1) # label(axiom). [clausify(361)]. 1.42/1.66 680 -dvd(A) | times_times(A,B,C) = ti(A,times_times(A,B,C)) # label(tsy_c_Groups_Otimes__class_Otimes_0_res) # label(axiom). [clausify(447)]. 1.42/1.66 681 dvd(int) # label(arity_Int_Oint___Rings_Odvd) # label(axiom). [assumption]. 1.42/1.66 Derived: -dvd_dvd(int,A,ti(int,B)) | dvd_dvd(int,A,B). [resolve(681,a,672,a)]. 1.42/1.66 Derived: dvd_dvd(int,A,ti(int,B)) | -dvd_dvd(int,A,B). [resolve(681,a,673,a)]. 1.42/1.66 Derived: times_times(int,A,B) != C | dvd_dvd(int,A,C). [resolve(681,a,675,a)]. 1.45/1.73 Derived: -dvd_dvd(int,ti(int,A),B) | dvd_dvd(int,A,B). [resolve(681,a,677,a)]. 1.45/1.73 Derived: dvd_dvd(int,ti(int,A),B) | -dvd_dvd(int,A,B). [resolve(681,a,678,a)]. 1.45/1.73 682 -comm_semiring(A) | plus_plus(A,times_times(A,B,C),times_times(A,D,C)) = times_times(A,plus_plus(A,B,D),C) # label(fact_476_comm__semiring__class_Odistrib) # label(axiom). [clausify(476)]. 1.45/1.73 683 comm_semiring(real) # label(arity_RealDef_Oreal___Rings_Ocomm__semiring) # label(axiom). [assumption]. 1.45/1.73 684 comm_semiring(int) # label(arity_Int_Oint___Rings_Ocomm__semiring) # label(axiom). [assumption]. 1.45/1.73 685 comm_semiring(nat) # label(arity_Nat_Onat___Rings_Ocomm__semiring) # label(axiom). [assumption]. 1.45/1.73 686 -number_semiring(A) | -ord_less_eq(int,pls,B) | -ord_less_eq(int,pls,C) | times_times(A,number_number_of(A,B),number_number_of(A,C)) = number_number_of(A,times_times(int,B,C)) # label(fact_133_semiring__mult__number__of) # label(axiom). [clausify(53)]. 1.45/1.73 687 number_semiring(nat) # label(arity_Nat_Onat___Int_Onumber__semiring) # label(axiom). [assumption]. 1.45/1.73 Derived: -ord_less_eq(int,pls,A) | -ord_less_eq(int,pls,B) | times_times(nat,number_number_of(nat,A),number_number_of(nat,B)) = number_number_of(nat,times_times(int,A,B)). [resolve(686,a,687,a)]. 1.45/1.73 688 -number_semiring(A) | plus_plus(A,one_one(A),one_one(A)) = number_number_of(A,bit0(bit1(pls))) # label(fact_167_semiring__one__add__one__is__two) # label(axiom). [clausify(55)]. 1.45/1.73 Derived: plus_plus(nat,one_one(nat),one_one(nat)) = number_number_of(nat,bit0(bit1(pls))). [resolve(688,a,687,a)]. 1.45/1.73 689 -number_semiring(A) | zero_zero(A) = number_number_of(A,pls) # label(fact_221_semiring__numeral__0__eq__0) # label(axiom). [clausify(167)]. 1.45/1.73 Derived: zero_zero(nat) = number_number_of(nat,pls). [resolve(689,a,687,a)]. 1.45/1.73 690 -number_semiring(A) | -ord_less_eq(int,pls,B) | -ord_less_eq(int,pls,C) | plus_plus(A,number_number_of(A,B),number_number_of(A,C)) = number_number_of(A,plus_plus(int,B,C)) # label(fact_134_semiring__add__number__of) # label(axiom). [clausify(237)]. 1.45/1.73 Derived: -ord_less_eq(int,pls,A) | -ord_less_eq(int,pls,B) | plus_plus(nat,number_number_of(nat,A),number_number_of(nat,B)) = number_number_of(nat,plus_plus(int,A,B)). [resolve(690,a,687,a)]. 1.45/1.73 691 number_semiring(real) # label(arity_RealDef_Oreal___Int_Onumber__semiring) # label(axiom). [assumption]. 1.45/1.73 692 -number_semiring(A) | plus_plus(A,plus_plus(A,power_power(A,B,number_number_of(nat,bit0(bit1(pls)))),power_power(A,C,number_number_of(nat,bit0(bit1(pls))))),times_times(A,times_times(A,number_number_of(A,bit0(bit1(pls))),B),C)) = power_power(A,plus_plus(A,B,C),number_number_of(nat,bit0(bit1(pls)))) # label(fact_9_power2__sum) # label(axiom). [clausify(286)]. 1.45/1.73 Derived: plus_plus(nat,plus_plus(nat,power_power(nat,A,number_number_of(nat,bit0(bit1(pls)))),power_power(nat,B,number_number_of(nat,bit0(bit1(pls))))),times_times(nat,times_times(nat,number_number_of(nat,bit0(bit1(pls))),A),B)) = power_power(nat,plus_plus(nat,A,B),number_number_of(nat,bit0(bit1(pls)))). [resolve(692,a,687,a)]. 1.45/1.73 Derived: plus_plus(real,plus_plus(real,power_power(real,A,number_number_of(nat,bit0(bit1(pls)))),power_power(real,B,number_number_of(nat,bit0(bit1(pls))))),times_times(real,times_times(real,number_number_of(real,bit0(bit1(pls))),A),B)) = power_power(real,plus_plus(real,A,B),number_number_of(nat,bit0(bit1(pls)))). [resolve(692,a,691,a)]. 1.45/1.73 693 -number_semiring(A) | plus_plus(A,B,B) = times_times(A,B,number_number_of(A,bit0(bit1(pls)))) # label(fact_165_semiring__mult__2__right) # label(axiom). [clausify(415)]. 1.45/1.73 Derived: plus_plus(nat,A,A) = times_times(nat,A,number_number_of(nat,bit0(bit1(pls)))). [resolve(693,a,687,a)]. 1.45/1.73 694 -number_semiring(A) | plus_plus(A,B,B) = times_times(A,number_number_of(A,bit0(bit1(pls))),B) # label(fact_163_semiring__mult__2) # label(axiom). [clausify(468)]. 1.45/1.73 695 -number_semiring(A) | one_one(A) = number_number_of(A,bit1(pls)) # label(fact_155_semiring__numeral__1__eq__1) # label(axiom). [clausify(491)]. 1.45/1.73 Derived: one_one(nat) = number_number_of(nat,bit1(pls)). [resolve(695,a,687,a)]. 1.45/1.73 696 number_semiring(int) # label(arity_Int_Oint___Int_Onumber__semiring) # label(axiom). [assumption]. 1.54/1.80 Derived: -ord_less_eq(int,pls,A) | -ord_less_eq(int,pls,B) | times_times(int,number_number_of(int,A),number_number_of(int,B)) = number_number_of(int,times_times(int,A,B)). [resolve(696,a,686,a)]. 1.54/1.80 Derived: plus_plus(int,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))))),times_times(int,times_times(int,number_number_of(int,bit0(bit1(pls))),A),B)) = power_power(int,plus_plus(int,A,B),number_number_of(nat,bit0(bit1(pls)))). [resolve(696,a,692,a)]. 1.54/1.80 697 -ring_1(A) | minus_minus(A,times_times(A,B,B),one_one(A)) = times_times(A,plus_plus(A,B,one_one(A)),minus_minus(A,B,one_one(A))) # label(fact_441_real__squared__diff__one__factored) # label(axiom). [clausify(451)]. 1.54/1.80 698 ring_1(int) # label(arity_Int_Oint___Rings_Oring__1) # label(axiom). [assumption]. 1.54/1.80 699 ring_1(real) # label(arity_RealDef_Oreal___Rings_Oring__1) # label(axiom). [assumption]. 1.54/1.80 Derived: minus_minus(int,times_times(int,A,A),one_one(int)) = times_times(int,plus_plus(int,A,one_one(int)),minus_minus(int,A,one_one(int))). [resolve(697,a,698,a)]. 1.54/1.80 Derived: minus_minus(real,times_times(real,A,A),one_one(real)) = times_times(real,plus_plus(real,A,one_one(real)),minus_minus(real,A,one_one(real))). [resolve(697,a,699,a)]. 1.54/1.80 700 -power(A) | -semiring_0(A) | zero_zero(nat) = number_number_of(nat,B) | zero_zero(A) = power_power(A,zero_zero(A),number_number_of(nat,B)) # label(fact_340_power__0__left__number__of) # label(axiom). [clausify(114)]. 1.54/1.80 701 semiring_0(nat) # label(arity_Nat_Onat___Rings_Osemiring__0) # label(axiom). [assumption]. 1.54/1.80 Derived: -power(nat) | zero_zero(nat) = number_number_of(nat,A) | zero_zero(nat) = power_power(nat,zero_zero(nat),number_number_of(nat,A)). [resolve(700,b,701,a)]. 1.54/1.80 702 -power(A) | -semiring_0(A) | zero_zero(nat) != number_number_of(nat,B) | one_one(A) = power_power(A,zero_zero(A),number_number_of(nat,B)) # label(fact_340_power__0__left__number__of) # label(axiom). [clausify(114)]. 1.54/1.80 703 -power(A) | -semiring_0(A) | zero_zero(nat) = B | zero_zero(A) = power_power(A,zero_zero(A),B) # label(fact_193_power__0__left) # label(axiom). [clausify(231)]. 1.54/1.80 Derived: -power(nat) | zero_zero(nat) = A | zero_zero(nat) = power_power(nat,zero_zero(nat),A). [resolve(703,b,701,a)]. 1.54/1.80 704 -power(A) | -semiring_0(A) | zero_zero(nat) != B | one_one(A) = power_power(A,zero_zero(A),B) # label(fact_193_power__0__left) # label(axiom). [clausify(231)]. 1.54/1.80 705 semiring_0(real) # label(arity_RealDef_Oreal___Rings_Osemiring__0) # label(axiom). [assumption]. 1.54/1.80 Derived: -power(real) | zero_zero(nat) = number_number_of(nat,A) | zero_zero(real) = power_power(real,zero_zero(real),number_number_of(nat,A)). [resolve(705,a,700,b)]. 1.54/1.80 Derived: -power(real) | zero_zero(nat) != number_number_of(nat,A) | one_one(real) = power_power(real,zero_zero(real),number_number_of(nat,A)). [resolve(705,a,702,b)]. 1.54/1.80 Derived: -power(real) | zero_zero(nat) = A | zero_zero(real) = power_power(real,zero_zero(real),A). [resolve(705,a,703,b)]. 1.54/1.80 Derived: -power(real) | zero_zero(nat) != A | one_one(real) = power_power(real,zero_zero(real),A). [resolve(705,a,704,b)]. 1.54/1.80 706 semiring_0(int) # label(arity_Int_Oint___Rings_Osemiring__0) # label(axiom). [assumption]. 1.54/1.80 Derived: -power(int) | zero_zero(nat) = number_number_of(nat,A) | zero_zero(int) = power_power(int,zero_zero(int),number_number_of(nat,A)). [resolve(706,a,700,b)]. 1.54/1.80 Derived: -power(int) | zero_zero(nat) != number_number_of(nat,A) | one_one(int) = power_power(int,zero_zero(int),number_number_of(nat,A)). [resolve(706,a,702,b)]. 1.54/1.80 Derived: -power(int) | zero_zero(nat) = A | zero_zero(int) = power_power(int,zero_zero(int),A). [resolve(706,a,703,b)]. 1.54/1.80 Derived: -power(int) | zero_zero(nat) != A | one_one(int) = power_power(int,zero_zero(int),A). [resolve(706,a,704,b)]. 1.54/1.80 707 -semiring_1(A) | zero_zero(A) = power_power(A,zero_zero(A),number_number_of(nat,bit0(bit1(pls)))) # label(fact_249_zero__power2) # label(axiom). [clausify(181)]. 1.54/1.80 708 semiring_1(real) # label(arity_RealDef_Oreal___Rings_Osemiring__1) # label(axiom). [assumption]. 1.59/1.84 709 semiring_1(nat) # label(arity_Nat_Onat___Rings_Osemiring__1) # label(axiom). [assumption]. 1.59/1.84 710 semiring_1(int) # label(arity_Int_Oint___Rings_Osemiring__1) # label(axiom). [assumption]. 1.59/1.84 Derived: zero_zero(real) = power_power(real,zero_zero(real),number_number_of(nat,bit0(bit1(pls)))). [resolve(707,a,708,a)]. 1.59/1.84 Derived: zero_zero(nat) = power_power(nat,zero_zero(nat),number_number_of(nat,bit0(bit1(pls)))). [resolve(707,a,709,a)]. 1.59/1.84 Derived: zero_zero(int) = power_power(int,zero_zero(int),number_number_of(nat,bit0(bit1(pls)))). [resolve(707,a,710,a)]. 1.59/1.84 711 -semiring_1(A) | one_one(A) = power_power(A,one_one(A),number_number_of(nat,bit0(bit1(pls)))) # label(fact_12_one__power2) # label(axiom). [clausify(285)]. 1.59/1.84 Derived: one_one(real) = power_power(real,one_one(real),number_number_of(nat,bit0(bit1(pls)))). [resolve(711,a,708,a)]. 1.59/1.84 Derived: one_one(nat) = power_power(nat,one_one(nat),number_number_of(nat,bit0(bit1(pls)))). [resolve(711,a,709,a)]. 1.59/1.84 Derived: one_one(int) = power_power(int,one_one(int),number_number_of(nat,bit0(bit1(pls)))). [resolve(711,a,710,a)]. 1.59/1.84 712 monoid_mult(nat) # label(arity_Nat_Onat___Groups_Omonoid__mult) # label(axiom). [assumption]. 1.59/1.84 713 -monoid_mult(A) | -ord_less(nat,zero_zero(nat),B) | times_times(A,power_power(A,C,minus_minus(nat,B,one_one(nat))),C) = power_power(A,C,B) # label(fact_442_realpow__minus__mult) # label(axiom). [clausify(48)]. 1.59/1.84 714 -monoid_mult(A) | power_power(A,power_power(A,B,C),number_number_of(nat,bit0(bit1(pls)))) = power_power(A,B,times_times(nat,number_number_of(nat,bit0(bit1(pls))),C)) # label(fact_101_power__even__eq) # label(axiom). [clausify(49)]. 1.59/1.84 715 -monoid_mult(A) | times_times(A,B,B) = power_power(A,B,number_number_of(nat,bit0(bit1(pls)))) # label(fact_14_power2__eq__square) # label(axiom). [clausify(92)]. 1.59/1.84 716 -monoid_mult(A) | times_times(A,times_times(A,B,B),B) = power_power(A,B,number_number_of(nat,bit1(bit1(pls)))) # label(fact_161_power3__eq__cube) # label(axiom). [clausify(176)]. 1.59/1.84 717 -monoid_mult(A) | power_power(A,B,one_one(nat)) = ti(A,B) # label(fact_240_power__one__right) # label(axiom). [clausify(273)]. 1.59/1.84 718 -monoid_mult(A) | -number(A) | times_times(A,number_number_of(A,B),number_number_of(A,B)) = power_power(A,number_number_of(A,B),number_number_of(nat,bit0(bit1(pls)))) # label(fact_10_power2__eq__square__number__of) # label(axiom). [clausify(274)]. 1.59/1.84 719 -monoid_mult(A) | times_times(A,power_power(A,B,C),B) = times_times(A,B,power_power(A,B,C)) # label(fact_257_power__commutes) # label(axiom). [clausify(296)]. 1.59/1.84 Derived: -ord_less(nat,zero_zero(nat),A) | times_times(nat,power_power(nat,B,minus_minus(nat,A,one_one(nat))),B) = power_power(nat,B,A). [resolve(712,a,713,a)]. 1.59/1.84 Derived: power_power(nat,power_power(nat,A,B),number_number_of(nat,bit0(bit1(pls)))) = power_power(nat,A,times_times(nat,number_number_of(nat,bit0(bit1(pls))),B)). [resolve(712,a,714,a)]. 1.59/1.84 Derived: times_times(nat,times_times(nat,A,A),A) = power_power(nat,A,number_number_of(nat,bit1(bit1(pls)))). [resolve(712,a,716,a)]. 1.59/1.84 720 monoid_mult(int) # label(arity_Int_Oint___Groups_Omonoid__mult) # label(axiom). [assumption]. 1.59/1.84 Derived: -ord_less(nat,zero_zero(nat),A) | times_times(int,power_power(int,B,minus_minus(nat,A,one_one(nat))),B) = power_power(int,B,A). [resolve(720,a,713,a)]. 1.59/1.84 Derived: power_power(int,power_power(int,A,B),number_number_of(nat,bit0(bit1(pls)))) = power_power(int,A,times_times(nat,number_number_of(nat,bit0(bit1(pls))),B)). [resolve(720,a,714,a)]. 1.59/1.84 Derived: times_times(int,times_times(int,A,A),A) = power_power(int,A,number_number_of(nat,bit1(bit1(pls)))). [resolve(720,a,716,a)]. 1.59/1.84 721 monoid_mult(real) # label(arity_RealDef_Oreal___Groups_Omonoid__mult) # label(axiom). [assumption]. 1.59/1.84 Derived: -ord_less(nat,zero_zero(nat),A) | times_times(real,power_power(real,B,minus_minus(nat,A,one_one(nat))),B) = power_power(real,B,A). [resolve(721,a,713,a)]. 1.59/1.84 Derived: power_power(real,power_power(real,A,B),number_number_of(nat,bit0(bit1(pls)))) = power_power(real,A,times_times(nat,number_number_of(nat,bit0(bit1(pls))),B)). [resolve(721,a,714,a)]. 1.59/1.84 Derived: times_times(real,times_times(real,A,A),A) = power_power(real,A,number_number_of(nat,bit1(bit1(pls)))). [resolve(721,a,716,a)]. 1.62/1.88 Derived: times_times(real,power_power(real,A,B),A) = times_times(real,A,power_power(real,A,B)). [resolve(721,a,719,a)]. 1.62/1.88 722 -monoid_mult(A) | one_one(A) = power_power(A,one_one(A),B) # label(fact_260_power__one) # label(axiom). [clausify(477)]. 1.62/1.88 Derived: one_one(nat) = power_power(nat,one_one(nat),A). [resolve(722,a,712,a)]. 1.62/1.88 Derived: one_one(int) = power_power(int,one_one(int),A). [resolve(722,a,720,a)]. 1.62/1.88 Derived: one_one(real) = power_power(real,one_one(real),A). [resolve(722,a,721,a)]. 1.62/1.88 723 -monoid_mult(A) | times_times(A,power_power(A,B,C),power_power(A,B,D)) = power_power(A,B,plus_plus(nat,C,D)) # label(fact_259_power__add) # label(axiom). [clausify(489)]. 1.62/1.88 724 -monoid_mult(A) | power_power(A,power_power(A,B,C),D) = power_power(A,B,times_times(nat,C,D)) # label(fact_261_power__mult) # label(axiom). [clausify(514)]. 1.62/1.88 Derived: power_power(int,power_power(int,A,B),C) = power_power(int,A,times_times(nat,B,C)). [resolve(724,a,720,a)]. 1.62/1.88 725 -ab_group_add(A) | plus_plus(A,B,C) = ti(A,plus_plus(A,B,C)) # label(tsy_c_Groups_Oplus__class_Oplus_1_res) # label(axiom). [clausify(85)]. 1.62/1.88 726 ab_group_add(int) # label(arity_Int_Oint___Groups_Oab__group__add) # label(axiom). [assumption]. 1.62/1.88 Derived: plus_plus(int,A,B) = ti(int,plus_plus(int,A,B)). [resolve(725,a,726,a)]. 1.62/1.88 727 -ab_group_add(A) | minus_minus(A,B,ti(A,C)) = minus_minus(A,B,C) # label(tsy_c_Groups_Ominus__class_Ominus_0_arg2) # label(axiom). [clausify(118)]. 1.62/1.88 Derived: minus_minus(int,A,ti(int,B)) = minus_minus(int,A,B). [resolve(727,a,726,a)]. 1.62/1.88 728 ab_group_add(real) # label(arity_RealDef_Oreal___Groups_Oab__group__add) # label(axiom). [assumption]. 1.62/1.88 Derived: plus_plus(real,A,B) = ti(real,plus_plus(real,A,B)). [resolve(728,a,725,a)]. 1.62/1.88 Derived: minus_minus(real,A,ti(real,B)) = minus_minus(real,A,B). [resolve(728,a,727,a)]. 1.62/1.88 729 -ab_group_add(A) | minus_minus(A,B,C) = ti(A,minus_minus(A,B,C)) # label(tsy_c_Groups_Ominus__class_Ominus_0_res) # label(axiom). [clausify(282)]. 1.62/1.88 Derived: minus_minus(int,A,B) = ti(int,minus_minus(int,A,B)). [resolve(729,a,726,a)]. 1.62/1.88 Derived: minus_minus(real,A,B) = ti(real,minus_minus(real,A,B)). [resolve(729,a,728,a)]. 1.62/1.88 730 -ab_group_add(A) | plus_plus(A,B,ti(A,C)) = plus_plus(A,B,C) # label(tsy_c_Groups_Oplus__class_Oplus_1_arg2) # label(axiom). [clausify(390)]. 1.62/1.88 Derived: plus_plus(int,A,ti(int,B)) = plus_plus(int,A,B). [resolve(730,a,726,a)]. 1.62/1.88 Derived: plus_plus(real,A,ti(real,B)) = plus_plus(real,A,B). [resolve(730,a,728,a)]. 1.62/1.88 731 -ab_group_add(A) | minus_minus(A,plus_plus(A,B,C),plus_plus(A,D,E)) = plus_plus(A,minus_minus(A,B,D),minus_minus(A,C,E)) # label(fact_430_add__diff__add) # label(axiom). [clausify(502)]. 1.62/1.88 Derived: minus_minus(int,plus_plus(int,A,B),plus_plus(int,C,D)) = plus_plus(int,minus_minus(int,A,C),minus_minus(int,B,D)). [resolve(731,a,726,a)]. 1.62/1.88 Derived: minus_minus(real,plus_plus(real,A,B),plus_plus(real,C,D)) = plus_plus(real,minus_minus(real,A,C),minus_minus(real,B,D)). [resolve(731,a,728,a)]. 1.62/1.88 732 -ab_group_add(A) | plus_plus(A,ti(A,B),C) = plus_plus(A,B,C) # label(tsy_c_Groups_Oplus__class_Oplus_1_arg1) # label(axiom). [clausify(503)]. 1.62/1.88 Derived: plus_plus(int,ti(int,A),B) = plus_plus(int,A,B). [resolve(732,a,726,a)]. 1.62/1.88 Derived: plus_plus(real,ti(real,A),B) = plus_plus(real,A,B). [resolve(732,a,728,a)]. 1.62/1.88 733 -ab_group_add(A) | minus_minus(A,ti(A,B),C) = minus_minus(A,B,C) # label(tsy_c_Groups_Ominus__class_Ominus_0_arg1) # label(axiom). [clausify(512)]. 1.62/1.88 Derived: minus_minus(int,ti(int,A),B) = minus_minus(int,A,B). [resolve(733,a,726,a)]. 1.62/1.88 Derived: minus_minus(real,ti(real,A),B) = minus_minus(real,A,B). [resolve(733,a,728,a)]. 1.62/1.88 734 semiring(int) # label(arity_Int_Oint___Rings_Osemiring) # label(axiom). [assumption]. 1.62/1.88 735 -semiring(A) | plus_plus(A,B,C) = ti(A,plus_plus(A,B,C)) # label(tsy_c_Groups_Oplus__class_Oplus_0_res) # label(axiom). [clausify(50)]. 1.62/1.88 736 -semiring(A) | times_times(A,B,ti(A,C)) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_1_arg2) # label(axiom). [clausify(73)]. 1.62/1.91 737 -semiring(A) | plus_plus(A,times_times(A,plus_plus(A,B,C),D),E) = plus_plus(A,times_times(A,B,D),plus_plus(A,times_times(A,C,D),E)) # label(fact_477_combine__common__factor) # label(axiom). [clausify(117)]. 1.62/1.91 Derived: plus_plus(int,times_times(int,plus_plus(int,A,B),C),D) = plus_plus(int,times_times(int,A,C),plus_plus(int,times_times(int,B,C),D)). [resolve(737,a,734,a)]. 1.62/1.91 738 semiring(nat) # label(arity_Nat_Onat___Rings_Osemiring) # label(axiom). [assumption]. 1.62/1.91 Derived: plus_plus(nat,A,B) = ti(nat,plus_plus(nat,A,B)). [resolve(738,a,735,a)]. 1.62/1.91 Derived: plus_plus(nat,times_times(nat,plus_plus(nat,A,B),C),D) = plus_plus(nat,times_times(nat,A,C),plus_plus(nat,times_times(nat,B,C),D)). [resolve(738,a,737,a)]. 1.62/1.91 739 -semiring(A) | plus_plus(A,B,ti(A,C)) = plus_plus(A,B,C) # label(tsy_c_Groups_Oplus__class_Oplus_0_arg2) # label(axiom). [clausify(226)]. 1.62/1.91 Derived: plus_plus(nat,A,ti(nat,B)) = plus_plus(nat,A,B). [resolve(739,a,738,a)]. 1.62/1.91 740 -semiring(A) | times_times(A,B,C) = ti(A,times_times(A,B,C)) # label(tsy_c_Groups_Otimes__class_Otimes_1_res) # label(axiom). [clausify(228)]. 1.62/1.91 741 -number(A) | -semiring(A) | plus_plus(A,times_times(A,B,number_number_of(A,C)),times_times(A,D,number_number_of(A,C))) = times_times(A,plus_plus(A,B,D),number_number_of(A,C)) # label(fact_136_left__distrib__number__of) # label(axiom). [clausify(304)]. 1.62/1.91 742 semiring(real) # label(arity_RealDef_Oreal___Rings_Osemiring) # label(axiom). [assumption]. 1.62/1.91 Derived: plus_plus(real,times_times(real,plus_plus(real,A,B),C),D) = plus_plus(real,times_times(real,A,C),plus_plus(real,times_times(real,B,C),D)). [resolve(742,a,737,a)]. 1.62/1.91 743 -semiring(A) | plus_plus(A,ti(A,B),C) = plus_plus(A,B,C) # label(tsy_c_Groups_Oplus__class_Oplus_0_arg1) # label(axiom). [clausify(402)]. 1.62/1.91 Derived: plus_plus(nat,ti(nat,A),B) = plus_plus(nat,A,B). [resolve(743,a,738,a)]. 1.62/1.91 744 -semiring(A) | -number(A) | plus_plus(A,times_times(A,number_number_of(A,B),C),times_times(A,number_number_of(A,B),D)) = times_times(A,number_number_of(A,B),plus_plus(A,C,D)) # label(fact_137_right__distrib__number__of) # label(axiom). [clausify(405)]. 1.62/1.91 745 -semiring(A) | times_times(A,ti(A,B),C) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_1_arg1) # label(axiom). [clausify(458)]. 1.62/1.91 746 semiri456707255roduct(int) # label(arity_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc) # label(axiom). [assumption]. 1.62/1.91 747 -semiri456707255roduct(A) | zero_zero(A) = ti(A,B) | ti(A,C) = ti(A,D) | ti(A,E) != ti(A,F) | plus_plus(A,E,times_times(A,B,C)) != plus_plus(A,F,times_times(A,B,D)) # label(fact_217_add__scale__eq__noteq) # label(axiom). [clausify(51)]. 1.62/1.91 748 -semiri456707255roduct(A) | plus_plus(A,times_times(A,B,C),times_times(A,D,E)) != plus_plus(A,times_times(A,B,E),times_times(A,D,C)) | ti(A,C) = ti(A,E) | ti(A,D) = ti(A,B) # label(fact_106_crossproduct__eq) # label(axiom). [clausify(58)]. 1.62/1.91 749 -semiri456707255roduct(A) | plus_plus(A,times_times(A,B,C),times_times(A,D,E)) = plus_plus(A,times_times(A,B,E),times_times(A,D,C)) | ti(A,C) != ti(A,E) # label(fact_106_crossproduct__eq) # label(axiom). [clausify(58)]. 1.62/1.91 750 -semiri456707255roduct(A) | plus_plus(A,times_times(A,B,C),times_times(A,D,E)) = plus_plus(A,times_times(A,B,E),times_times(A,D,C)) | ti(A,D) != ti(A,B) # label(fact_106_crossproduct__eq) # label(axiom). [clausify(58)]. 1.62/1.91 751 -semiri456707255roduct(A) | plus_plus(A,times_times(A,B,C),times_times(A,D,E)) = plus_plus(A,times_times(A,B,E),times_times(A,D,C)) | ti(A,D) != ti(A,B) # label(fact_109_crossproduct__noteq) # label(axiom). [clausify(254)]. 1.62/1.91 752 -semiri456707255roduct(A) | plus_plus(A,times_times(A,B,C),times_times(A,D,E)) = plus_plus(A,times_times(A,B,E),times_times(A,D,C)) | ti(A,C) != ti(A,E) # label(fact_109_crossproduct__noteq) # label(axiom). [clausify(254)]. 1.62/1.91 753 -semiri456707255roduct(A) | plus_plus(A,times_times(A,B,C),times_times(A,D,E)) != plus_plus(A,times_times(A,B,E),times_times(A,D,C)) | ti(A,D) = ti(A,B) | ti(A,C) = ti(A,E) # label(fact_109_crossproduct__noteq) # label(axiom). [clausify(254)]. 1.62/1.91 Derived: zero_zero(int) = ti(int,A) | ti(int,B) = ti(int,C) | ti(int,D) != ti(int,E) | plus_plus(int,D,times_times(int,A,B)) != plus_plus(int,E,times_times(int,A,C)). [resolve(746,a,747,a)]. 1.70/1.94 Derived: plus_plus(int,times_times(int,A,B),times_times(int,C,D)) != plus_plus(int,times_times(int,A,D),times_times(int,C,B)) | ti(int,B) = ti(int,D) | ti(int,C) = ti(int,A). [resolve(746,a,748,a)]. 1.70/1.94 Derived: plus_plus(int,times_times(int,A,B),times_times(int,C,D)) = plus_plus(int,times_times(int,A,D),times_times(int,C,B)) | ti(int,B) != ti(int,D). [resolve(746,a,749,a)]. 1.70/1.94 Derived: plus_plus(int,times_times(int,A,B),times_times(int,C,D)) = plus_plus(int,times_times(int,A,D),times_times(int,C,B)) | ti(int,C) != ti(int,A). [resolve(746,a,750,a)]. 1.70/1.94 754 semiri456707255roduct(real) # label(arity_RealDef_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossp) # label(axiom). [assumption]. 1.70/1.94 Derived: zero_zero(real) = ti(real,A) | ti(real,B) = ti(real,C) | ti(real,D) != ti(real,E) | plus_plus(real,D,times_times(real,A,B)) != plus_plus(real,E,times_times(real,A,C)). [resolve(754,a,747,a)]. 1.70/1.94 Derived: plus_plus(real,times_times(real,A,B),times_times(real,C,D)) != plus_plus(real,times_times(real,A,D),times_times(real,C,B)) | ti(real,B) = ti(real,D) | ti(real,C) = ti(real,A). [resolve(754,a,748,a)]. 1.70/1.94 Derived: plus_plus(real,times_times(real,A,B),times_times(real,C,D)) = plus_plus(real,times_times(real,A,D),times_times(real,C,B)) | ti(real,B) != ti(real,D). [resolve(754,a,749,a)]. 1.70/1.94 Derived: plus_plus(real,times_times(real,A,B),times_times(real,C,D)) = plus_plus(real,times_times(real,A,D),times_times(real,C,B)) | ti(real,C) != ti(real,A). [resolve(754,a,750,a)]. 1.70/1.94 755 semiri456707255roduct(nat) # label(arity_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc) # label(axiom). [assumption]. 1.70/1.94 Derived: zero_zero(nat) = ti(nat,A) | ti(nat,B) = ti(nat,C) | ti(nat,D) != ti(nat,E) | plus_plus(nat,D,times_times(nat,A,B)) != plus_plus(nat,E,times_times(nat,A,C)). [resolve(755,a,747,a)]. 1.70/1.94 Derived: plus_plus(nat,times_times(nat,A,B),times_times(nat,C,D)) != plus_plus(nat,times_times(nat,A,D),times_times(nat,C,B)) | ti(nat,B) = ti(nat,D) | ti(nat,C) = ti(nat,A). [resolve(755,a,748,a)]. 1.70/1.94 Derived: plus_plus(nat,times_times(nat,A,B),times_times(nat,C,D)) = plus_plus(nat,times_times(nat,A,D),times_times(nat,C,B)) | ti(nat,B) != ti(nat,D). [resolve(755,a,749,a)]. 1.70/1.94 Derived: plus_plus(nat,times_times(nat,A,B),times_times(nat,C,D)) = plus_plus(nat,times_times(nat,A,D),times_times(nat,C,B)) | ti(nat,C) != ti(nat,A). [resolve(755,a,750,a)]. 1.70/1.94 756 -semiri456707255roduct(A) | plus_plus(A,B,C) != ti(A,B) | zero_zero(A) = ti(A,C) # label(fact_200_add__0__iff) # label(axiom). [clausify(459)]. 1.70/1.94 Derived: plus_plus(int,A,B) != ti(int,A) | zero_zero(int) = ti(int,B). [resolve(756,a,746,a)]. 1.70/1.94 Derived: plus_plus(real,A,B) != ti(real,A) | zero_zero(real) = ti(real,B). [resolve(756,a,754,a)]. 1.70/1.94 Derived: plus_plus(nat,A,B) != ti(nat,A) | zero_zero(nat) = ti(nat,B). [resolve(756,a,755,a)]. 1.70/1.94 757 -semiri456707255roduct(A) | plus_plus(A,B,C) = ti(A,B) | zero_zero(A) != ti(A,C) # label(fact_200_add__0__iff) # label(axiom). [clausify(459)]. 1.70/1.94 Derived: plus_plus(int,A,B) = ti(int,A) | zero_zero(int) != ti(int,B). [resolve(757,a,746,a)]. 1.70/1.94 Derived: plus_plus(real,A,B) = ti(real,A) | zero_zero(real) != ti(real,B). [resolve(757,a,754,a)]. 1.70/1.94 Derived: plus_plus(nat,A,B) = ti(nat,A) | zero_zero(nat) != ti(nat,B). [resolve(757,a,755,a)]. 1.70/1.94 758 linordered_ring(real) # label(arity_RealDef_Oreal___Rings_Olinordered__ring) # label(axiom). [assumption]. 1.70/1.94 759 -linordered_ring(A) | -ord_less(A,plus_plus(A,times_times(A,B,B),times_times(A,C,C)),zero_zero(A)) # label(fact_236_not__sum__squares__lt__zero) # label(axiom). [clausify(75)]. 1.70/1.94 760 -linordered_ring(A) | ord_less_eq(A,zero_zero(A),times_times(A,B,B)) # label(fact_490_zero__le__square) # label(axiom). [clausify(194)]. 1.70/1.94 Derived: -ord_less(real,plus_plus(real,times_times(real,A,A),times_times(real,B,B)),zero_zero(real)). [resolve(758,a,759,a)]. 1.70/1.94 Derived: ord_less_eq(real,zero_zero(real),times_times(real,A,A)). [resolve(758,a,760,a)]. 1.70/1.94 761 -linordered_ring(A) | ord_less_eq(A,zero_zero(A),plus_plus(A,times_times(A,B,B),times_times(A,C,C))) # label(fact_233_sum__squares__ge__zero) # label(axiom). [clausify(297)]. 1.70/1.97 Derived: ord_less_eq(real,zero_zero(real),plus_plus(real,times_times(real,A,A),times_times(real,B,B))). [resolve(761,a,758,a)]. 1.70/1.97 762 linordered_ring(int) # label(arity_Int_Oint___Rings_Olinordered__ring) # label(axiom). [assumption]. 1.70/1.97 Derived: -ord_less(int,plus_plus(int,times_times(int,A,A),times_times(int,B,B)),zero_zero(int)). [resolve(762,a,759,a)]. 1.70/1.97 Derived: ord_less_eq(int,zero_zero(int),times_times(int,A,A)). [resolve(762,a,760,a)]. 1.70/1.97 Derived: ord_less_eq(int,zero_zero(int),plus_plus(int,times_times(int,A,A),times_times(int,B,B))). [resolve(762,a,761,a)]. 1.70/1.97 763 order(nat) # label(arity_Nat_Onat___Orderings_Oorder) # label(axiom). [assumption]. 1.70/1.97 764 -order(A) | -ord_less(A,B,C) | ord_less(A,B,ti(A,C)) # label(tsy_c_Orderings_Oord__class_Oless_arg2) # label(axiom). [clausify(81)]. 1.70/1.97 765 -order(A) | ord_less(A,B,C) | -ord_less(A,B,ti(A,C)) # label(tsy_c_Orderings_Oord__class_Oless_arg2) # label(axiom). [clausify(81)]. 1.70/1.97 766 -order(A) | -ord_less_eq(A,B,C) | ord_less_eq(A,ti(A,B),C) # label(tsy_c_Orderings_Oord__class_Oless__eq_arg1) # label(axiom). [clausify(86)]. 1.70/1.97 767 -order(A) | ord_less_eq(A,B,C) | -ord_less_eq(A,ti(A,B),C) # label(tsy_c_Orderings_Oord__class_Oless__eq_arg1) # label(axiom). [clausify(86)]. 1.70/1.97 Derived: -ord_less(nat,A,B) | ord_less(nat,A,ti(nat,B)). [resolve(763,a,764,a)]. 1.70/1.97 Derived: ord_less(nat,A,B) | -ord_less(nat,A,ti(nat,B)). [resolve(763,a,765,a)]. 1.70/1.97 Derived: -ord_less_eq(nat,A,B) | ord_less_eq(nat,ti(nat,A),B). [resolve(763,a,766,a)]. 1.70/1.97 Derived: ord_less_eq(nat,A,B) | -ord_less_eq(nat,ti(nat,A),B). [resolve(763,a,767,a)]. 1.70/1.97 768 order(real) # label(arity_RealDef_Oreal___Orderings_Oorder) # label(axiom). [assumption]. 1.70/1.97 Derived: -ord_less(real,A,B) | ord_less(real,A,ti(real,B)). [resolve(768,a,764,a)]. 1.70/1.97 Derived: ord_less(real,A,B) | -ord_less(real,A,ti(real,B)). [resolve(768,a,765,a)]. 1.70/1.97 Derived: -ord_less_eq(real,A,B) | ord_less_eq(real,ti(real,A),B). [resolve(768,a,766,a)]. 1.70/1.97 Derived: ord_less_eq(real,A,B) | -ord_less_eq(real,ti(real,A),B). [resolve(768,a,767,a)]. 1.70/1.97 769 -order(A) | -ord_less_eq(A,B,C) | ti(A,C) = ti(A,B) | ord_less(A,B,C) # label(fact_327_order__le__neq__implies__less) # label(axiom). [clausify(225)]. 1.70/1.97 Derived: -ord_less_eq(nat,A,B) | ti(nat,B) = ti(nat,A) | ord_less(nat,A,B). [resolve(769,a,763,a)]. 1.70/1.97 Derived: -ord_less_eq(real,A,B) | ti(real,B) = ti(real,A) | ord_less(real,A,B). [resolve(769,a,768,a)]. 1.70/1.97 770 order(bool) # label(arity_HOL_Obool___Orderings_Oorder) # label(axiom). [assumption]. 1.70/1.97 Derived: -ord_less(bool,A,B) | ord_less(bool,A,ti(bool,B)). [resolve(770,a,764,a)]. 1.70/1.97 Derived: ord_less(bool,A,B) | -ord_less(bool,A,ti(bool,B)). [resolve(770,a,765,a)]. 1.70/1.97 Derived: -ord_less_eq(bool,A,B) | ord_less_eq(bool,ti(bool,A),B). [resolve(770,a,766,a)]. 1.70/1.97 Derived: ord_less_eq(bool,A,B) | -ord_less_eq(bool,ti(bool,A),B). [resolve(770,a,767,a)]. 1.70/1.97 Derived: -ord_less_eq(bool,A,B) | ti(bool,B) = ti(bool,A) | ord_less(bool,A,B). [resolve(770,a,769,a)]. 1.70/1.97 771 -order(A) | -ord_less(A,ti(A,B),C) | ord_less(A,B,C) # label(tsy_c_Orderings_Oord__class_Oless_arg1) # label(axiom). [clausify(357)]. 1.70/1.97 Derived: -ord_less(nat,ti(nat,A),B) | ord_less(nat,A,B). [resolve(771,a,763,a)]. 1.70/1.97 Derived: -ord_less(real,ti(real,A),B) | ord_less(real,A,B). [resolve(771,a,768,a)]. 1.70/1.97 Derived: -ord_less(bool,ti(bool,A),B) | ord_less(bool,A,B). [resolve(771,a,770,a)]. 1.70/1.97 772 -order(A) | ord_less(A,ti(A,B),C) | -ord_less(A,B,C) # label(tsy_c_Orderings_Oord__class_Oless_arg1) # label(axiom). [clausify(357)]. 1.70/1.97 Derived: ord_less(nat,ti(nat,A),B) | -ord_less(nat,A,B). [resolve(772,a,763,a)]. 1.70/1.97 Derived: ord_less(real,ti(real,A),B) | -ord_less(real,A,B). [resolve(772,a,768,a)]. 1.70/1.97 Derived: ord_less(bool,ti(bool,A),B) | -ord_less(bool,A,B). [resolve(772,a,770,a)]. 1.70/1.97 773 order(int) # label(arity_Int_Oint___Orderings_Oorder) # label(axiom). [assumption]. 1.70/1.97 Derived: -ord_less(int,A,B) | ord_less(int,A,ti(int,B)). [resolve(773,a,764,a)]. 1.70/1.97 Derived: ord_less(int,A,B) | -ord_less(int,A,ti(int,B)). [resolve(773,a,765,a)]. 1.76/2.04 Derived: -ord_less_eq(int,A,B) | ord_less_eq(int,ti(int,A),B). [resolve(773,a,766,a)]. 1.76/2.04 Derived: ord_less_eq(int,A,B) | -ord_less_eq(int,ti(int,A),B). [resolve(773,a,767,a)]. 1.76/2.04 Derived: -ord_less_eq(int,A,B) | ti(int,B) = ti(int,A) | ord_less(int,A,B). [resolve(773,a,769,a)]. 1.76/2.04 Derived: -ord_less(int,ti(int,A),B) | ord_less(int,A,B). [resolve(773,a,771,a)]. 1.76/2.04 Derived: ord_less(int,ti(int,A),B) | -ord_less(int,A,B). [resolve(773,a,772,a)]. 1.76/2.04 774 -order(A) | -ord_less_eq(A,B,ti(A,C)) | ord_less_eq(A,B,C) # label(tsy_c_Orderings_Oord__class_Oless__eq_arg2) # label(axiom). [clausify(438)]. 1.76/2.04 Derived: -ord_less_eq(nat,A,ti(nat,B)) | ord_less_eq(nat,A,B). [resolve(774,a,763,a)]. 1.76/2.04 Derived: -ord_less_eq(real,A,ti(real,B)) | ord_less_eq(real,A,B). [resolve(774,a,768,a)]. 1.76/2.04 Derived: -ord_less_eq(bool,A,ti(bool,B)) | ord_less_eq(bool,A,B). [resolve(774,a,770,a)]. 1.76/2.04 Derived: -ord_less_eq(int,A,ti(int,B)) | ord_less_eq(int,A,B). [resolve(774,a,773,a)]. 1.76/2.04 775 -order(A) | ord_less_eq(A,B,ti(A,C)) | -ord_less_eq(A,B,C) # label(tsy_c_Orderings_Oord__class_Oless__eq_arg2) # label(axiom). [clausify(438)]. 1.76/2.04 Derived: ord_less_eq(nat,A,ti(nat,B)) | -ord_less_eq(nat,A,B). [resolve(775,a,763,a)]. 1.76/2.04 Derived: ord_less_eq(real,A,ti(real,B)) | -ord_less_eq(real,A,B). [resolve(775,a,768,a)]. 1.76/2.04 Derived: ord_less_eq(bool,A,ti(bool,B)) | -ord_less_eq(bool,A,B). [resolve(775,a,770,a)]. 1.76/2.04 Derived: ord_less_eq(int,A,ti(int,B)) | -ord_less_eq(int,A,B). [resolve(775,a,773,a)]. 1.76/2.04 776 -ring_11004092258visors(A) | zero_zero(A) = ti(A,B) | zero_zero(A) != power_power(A,B,C) # label(fact_192_field__power__not__zero) # label(axiom). [clausify(183)]. 1.76/2.04 777 ring_11004092258visors(int) # label(arity_Int_Oint___Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 1.76/2.04 Derived: zero_zero(int) = ti(int,A) | zero_zero(int) != power_power(int,A,B). [resolve(776,a,777,a)]. 1.76/2.04 778 -ring_11004092258visors(A) | zero_zero(A) != ti(A,B) | zero_zero(A) = power_power(A,B,number_number_of(nat,bit0(bit1(pls)))) # label(fact_250_zero__eq__power2) # label(axiom). [clausify(234)]. 1.76/2.04 Derived: zero_zero(int) != ti(int,A) | zero_zero(int) = power_power(int,A,number_number_of(nat,bit0(bit1(pls)))). [resolve(778,a,777,a)]. 1.76/2.04 779 -ring_11004092258visors(A) | zero_zero(A) = ti(A,B) | zero_zero(A) != power_power(A,B,number_number_of(nat,bit0(bit1(pls)))) # label(fact_250_zero__eq__power2) # label(axiom). [clausify(234)]. 1.76/2.04 780 ring_11004092258visors(real) # label(arity_RealDef_Oreal___Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 1.76/2.04 Derived: zero_zero(real) = ti(real,A) | zero_zero(real) != power_power(real,A,B). [resolve(780,a,776,a)]. 1.76/2.04 Derived: zero_zero(real) != ti(real,A) | zero_zero(real) = power_power(real,A,number_number_of(nat,bit0(bit1(pls)))). [resolve(780,a,778,a)]. 1.76/2.04 781 -comm_monoid_mult(A) | times_times(A,power_power(A,B,C),power_power(A,D,C)) = power_power(A,times_times(A,B,D),C) # label(fact_258_power__mult__distrib) # label(axiom). [clausify(475)]. 1.76/2.04 782 comm_monoid_mult(nat) # label(arity_Nat_Onat___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 1.76/2.04 783 comm_monoid_mult(int) # label(arity_Int_Oint___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 1.76/2.04 784 comm_monoid_mult(real) # label(arity_RealDef_Oreal___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 1.76/2.04 785 ring(int) # label(arity_Int_Oint___Rings_Oring) # label(axiom). [assumption]. 1.76/2.04 786 -ring(A) | -number(A) | minus_minus(A,times_times(A,B,number_number_of(A,C)),times_times(A,D,number_number_of(A,C))) = times_times(A,minus_minus(A,B,D),number_number_of(A,C)) # label(fact_357_left__diff__distrib__number__of) # label(axiom). [clausify(124)]. 1.76/2.04 787 -ring(A) | -number(A) | minus_minus(A,times_times(A,number_number_of(A,B),C),times_times(A,number_number_of(A,B),D)) = times_times(A,number_number_of(A,B),minus_minus(A,C,D)) # label(fact_356_right__diff__distrib__number__of) # label(axiom). [clausify(266)]. 1.76/2.04 Derived: -number(int) | minus_minus(int,times_times(int,number_number_of(int,A),B),times_times(int,number_number_of(int,A),C)) = times_times(int,number_number_of(int,A),minus_minus(int,B,C)). [resolve(785,a,787,a)]. 1.90/2.11 788 -ring(A) | minus_minus(A,times_times(A,B,C),times_times(A,D,E)) = plus_plus(A,times_times(A,B,minus_minus(A,C,E)),times_times(A,minus_minus(A,B,D),E)) # label(fact_434_mult__diff__mult) # label(axiom). [clausify(280)]. 1.90/2.11 Derived: minus_minus(int,times_times(int,A,B),times_times(int,C,D)) = plus_plus(int,times_times(int,A,minus_minus(int,B,D)),times_times(int,minus_minus(int,A,C),D)). [resolve(788,a,785,a)]. 1.90/2.11 789 ring(real) # label(arity_RealDef_Oreal___Rings_Oring) # label(axiom). [assumption]. 1.90/2.11 Derived: -number(real) | minus_minus(real,times_times(real,A,number_number_of(real,B)),times_times(real,C,number_number_of(real,B))) = times_times(real,minus_minus(real,A,C),number_number_of(real,B)). [resolve(789,a,786,a)]. 1.90/2.11 Derived: -number(real) | minus_minus(real,times_times(real,number_number_of(real,A),B),times_times(real,number_number_of(real,A),C)) = times_times(real,number_number_of(real,A),minus_minus(real,B,C)). [resolve(789,a,787,a)]. 1.90/2.11 Derived: minus_minus(real,times_times(real,A,B),times_times(real,C,D)) = plus_plus(real,times_times(real,A,minus_minus(real,B,D)),times_times(real,minus_minus(real,A,C),D)). [resolve(789,a,788,a)]. 1.90/2.11 790 -ring_n68954251visors(A) | zero_zero(A) != times_times(A,B,C) | zero_zero(A) = ti(A,C) | zero_zero(A) = ti(A,B) # label(fact_471_mult__eq__0__iff) # label(axiom). [clausify(342)]. 1.90/2.11 791 ring_n68954251visors(int) # label(arity_Int_Oint___Rings_Oring__no__zero__divisors) # label(axiom). [assumption]. 1.90/2.11 792 ring_n68954251visors(real) # label(arity_RealDef_Oreal___Rings_Oring__no__zero__divisors) # label(axiom). [assumption]. 1.90/2.11 793 -ring_n68954251visors(A) | zero_zero(A) = times_times(A,B,C) | zero_zero(A) != ti(A,C) # label(fact_471_mult__eq__0__iff) # label(axiom). [clausify(342)]. 1.90/2.11 Derived: zero_zero(int) = times_times(int,A,B) | zero_zero(int) != ti(int,B). [resolve(793,a,791,a)]. 1.90/2.11 Derived: zero_zero(real) = times_times(real,A,B) | zero_zero(real) != ti(real,B). [resolve(793,a,792,a)]. 1.90/2.11 794 -ring_n68954251visors(A) | zero_zero(A) = times_times(A,B,C) | zero_zero(A) != ti(A,B) # label(fact_471_mult__eq__0__iff) # label(axiom). [clausify(342)]. 1.90/2.11 Derived: zero_zero(int) = times_times(int,A,B) | zero_zero(int) != ti(int,A). [resolve(794,a,791,a)]. 1.90/2.11 Derived: zero_zero(real) = times_times(real,A,B) | zero_zero(real) != ti(real,A). [resolve(794,a,792,a)]. 1.90/2.11 795 -linord626643107strict(A) | -ord_less(A,B,C) | -ord_less(A,D,C) | -ord_less_eq(A,zero_zero(A),E) | -ord_less_eq(A,zero_zero(A),F) | plus_plus(A,E,F) != one_one(A) | ord_less(A,plus_plus(A,times_times(A,E,B),times_times(A,F,D)),C) # label(fact_463_convex__bound__lt) # label(axiom). [clausify(343)]. 1.90/2.11 796 linord626643107strict(int) # label(arity_Int_Oint___Rings_Olinordered__semiring__1__strict) # label(axiom). [assumption]. 1.90/2.11 Derived: -ord_less(int,A,B) | -ord_less(int,C,B) | -ord_less_eq(int,zero_zero(int),D) | -ord_less_eq(int,zero_zero(int),E) | plus_plus(int,D,E) != one_one(int) | ord_less(int,plus_plus(int,times_times(int,D,A),times_times(int,E,C)),B). [resolve(795,a,796,a)]. 1.90/2.11 797 linord626643107strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semiring__1__strict) # label(axiom). [assumption]. 1.90/2.11 Derived: -ord_less(real,A,B) | -ord_less(real,C,B) | -ord_less_eq(real,zero_zero(real),D) | -ord_less_eq(real,zero_zero(real),E) | plus_plus(real,D,E) != one_one(real) | ord_less(real,plus_plus(real,times_times(real,D,A),times_times(real,E,C)),B). [resolve(797,a,795,a)]. 1.90/2.11 798 linordered_semidom(int) # label(arity_Int_Oint___Rings_Olinordered__semidom) # label(axiom). [assumption]. 1.90/2.11 799 -linordered_semidom(A) | -ord_less(nat,B,C) | -ord_less(A,zero_zero(A),D) | -ord_less(A,D,one_one(A)) | ord_less(A,power_power(A,D,C),power_power(A,D,B)) # label(fact_214_power__strict__decreasing) # label(axiom). [clausify(157)]. 1.90/2.11 800 -linordered_semidom(A) | -ord_less_eq(nat,B,C) | -ord_less_eq(A,one_one(A),D) | ord_less_eq(A,power_power(A,D,B),power_power(A,D,C)) # label(fact_269_power__increasing) # label(axiom). [clausify(160)]. 1.90/2.11 801 -linordered_semidom(A) | power_power(A,B,number_number_of(nat,bit0(bit1(pls)))) != power_power(A,C,number_number_of(nat,bit0(bit1(pls)))) | -ord_less_eq(A,zero_zero(A),C) | -ord_less_eq(A,zero_zero(A),B) | ti(A,B) = ti(A,C) # label(fact_253_power2__eq__imp__eq) # label(axiom). [clausify(162)]. 1.90/2.12 802 -linordered_semidom(A) | -ord_less_eq(A,power_power(A,B,number_number_of(nat,bit0(bit1(pls)))),power_power(A,C,number_number_of(nat,bit0(bit1(pls))))) | -ord_less_eq(A,zero_zero(A),C) | ord_less_eq(A,B,C) # label(fact_252_power2__le__imp__le) # label(axiom). [clausify(179)]. 1.90/2.12 803 -linordered_semidom(A) | -ord_less(A,zero_zero(A),B) | ord_less(A,zero_zero(A),power_power(A,B,C)) # label(fact_208_zero__less__power) # label(axiom). [clausify(199)]. 1.90/2.12 804 -linordered_semidom(A) | -ord_less(nat,B,C) | -ord_less(A,one_one(A),D) | ord_less(A,power_power(A,D,B),power_power(A,D,C)) # label(fact_273_power__strict__increasing) # label(axiom). [clausify(215)]. 1.90/2.12 805 -linordered_semidom(A) | power_power(A,B,C) != power_power(A,D,C) | -ord_less_eq(A,zero_zero(A),D) | -ord_less_eq(A,zero_zero(A),B) | -ord_less(nat,zero_zero(nat),C) | ti(A,B) = ti(A,D) # label(fact_187_power__eq__imp__eq__base) # label(axiom). [clausify(216)]. 1.90/2.12 806 -linordered_semidom(A) | -ord_less(A,one_one(A),B) | ord_less(A,power_power(A,B,C),times_times(A,B,power_power(A,B,C))) # label(fact_174_power__less__power__Suc) # label(axiom). [clausify(248)]. 1.90/2.12 807 -linordered_semidom(A) | -ord_less(A,one_one(A),B) | power_power(A,B,C) != power_power(A,B,D) | C = D # label(fact_270_power__inject__exp) # label(axiom). [clausify(256)]. 1.90/2.12 808 -linordered_semidom(A) | -ord_less(A,one_one(A),B) | power_power(A,B,C) = power_power(A,B,D) | C != D # label(fact_270_power__inject__exp) # label(axiom). [clausify(256)]. 1.90/2.12 Derived: -ord_less(nat,A,B) | -ord_less(int,zero_zero(int),C) | -ord_less(int,C,one_one(int)) | ord_less(int,power_power(int,C,B),power_power(int,C,A)). [resolve(798,a,799,a)]. 1.90/2.12 Derived: -ord_less_eq(nat,A,B) | -ord_less_eq(int,one_one(int),C) | ord_less_eq(int,power_power(int,C,A),power_power(int,C,B)). [resolve(798,a,800,a)]. 1.90/2.12 Derived: power_power(int,A,number_number_of(nat,bit0(bit1(pls)))) != power_power(int,B,number_number_of(nat,bit0(bit1(pls)))) | -ord_less_eq(int,zero_zero(int),B) | -ord_less_eq(int,zero_zero(int),A) | ti(int,A) = ti(int,B). [resolve(798,a,801,a)]. 1.90/2.12 Derived: -ord_less_eq(int,power_power(int,A,number_number_of(nat,bit0(bit1(pls)))),power_power(int,B,number_number_of(nat,bit0(bit1(pls))))) | -ord_less_eq(int,zero_zero(int),B) | ord_less_eq(int,A,B). [resolve(798,a,802,a)]. 1.90/2.12 Derived: -ord_less(int,zero_zero(int),A) | ord_less(int,zero_zero(int),power_power(int,A,B)). [resolve(798,a,803,a)]. 1.90/2.12 Derived: -ord_less(nat,A,B) | -ord_less(int,one_one(int),C) | ord_less(int,power_power(int,C,A),power_power(int,C,B)). [resolve(798,a,804,a)]. 1.90/2.12 Derived: power_power(int,A,B) != power_power(int,C,B) | -ord_less_eq(int,zero_zero(int),C) | -ord_less_eq(int,zero_zero(int),A) | -ord_less(nat,zero_zero(nat),B) | ti(int,A) = ti(int,C). [resolve(798,a,805,a)]. 1.90/2.12 Derived: -ord_less(int,one_one(int),A) | ord_less(int,power_power(int,A,B),times_times(int,A,power_power(int,A,B))). [resolve(798,a,806,a)]. 1.90/2.12 Derived: -ord_less(int,one_one(int),A) | power_power(int,A,B) != power_power(int,A,C) | B = C. [resolve(798,a,807,a)]. 1.90/2.12 Derived: -ord_less(int,one_one(int),A) | power_power(int,A,B) = power_power(int,A,C) | B != C. [resolve(798,a,808,a)]. 1.90/2.12 809 -linordered_semidom(A) | -ord_less(A,power_power(A,B,number_number_of(nat,bit0(bit1(pls)))),power_power(A,C,number_number_of(nat,bit0(bit1(pls))))) | -ord_less_eq(A,zero_zero(A),C) | ord_less(A,B,C) # label(fact_262_power2__less__imp__less) # label(axiom). [clausify(284)]. 1.90/2.12 Derived: -ord_less(int,power_power(int,A,number_number_of(nat,bit0(bit1(pls)))),power_power(int,B,number_number_of(nat,bit0(bit1(pls))))) | -ord_less_eq(int,zero_zero(int),B) | ord_less(int,A,B). [resolve(809,a,798,a)]. 1.90/2.12 810 -linordered_semidom(A) | -ord_less(A,one_one(A),B) | ord_less(A,one_one(A),times_times(A,B,power_power(A,B,C))) # label(fact_175_power__gt1__lemma) # label(axiom). [clausify(324)]. 1.90/2.12 Derived: -ord_less(int,one_one(int),A) | ord_less(int,one_one(int),times_times(int,A,power_power(int,A,B))). [resolve(810,a,798,a)]. 1.90/2.12 811 -linordered_semidom(A) | -ord_less(A,one_one(A),B) | -ord_less(nat,zero_zero(nat),C) | ord_less(A,one_one(A),power_power(A,B,C)) # label(fact_316_one__less__power) # label(axiom). [clausify(335)]. 1.90/2.12 Derived: -ord_less(int,one_one(int),A) | -ord_less(nat,zero_zero(nat),B) | ord_less(int,one_one(int),power_power(int,A,B)). [resolve(811,a,798,a)]. 1.90/2.12 812 -linordered_semidom(A) | -ord_less_eq(nat,B,C) | -ord_less_eq(A,zero_zero(A),D) | -ord_less_eq(A,D,one_one(A)) | ord_less_eq(A,power_power(A,D,C),power_power(A,D,B)) # label(fact_213_power__decreasing) # label(axiom). [clausify(368)]. 1.90/2.12 Derived: -ord_less_eq(nat,A,B) | -ord_less_eq(int,zero_zero(int),C) | -ord_less_eq(int,C,one_one(int)) | ord_less_eq(int,power_power(int,C,B),power_power(int,C,A)). [resolve(812,a,798,a)]. 1.90/2.12 813 -linordered_semidom(A) | -ord_less(A,zero_zero(A),B) | -ord_less(A,B,one_one(A)) | ord_less(A,times_times(A,B,power_power(A,B,C)),power_power(A,B,C)) # label(fact_230_power__Suc__less) # label(axiom). [clausify(373)]. 1.90/2.12 Derived: -ord_less(int,zero_zero(int),A) | -ord_less(int,A,one_one(int)) | ord_less(int,times_times(int,A,power_power(int,A,B)),power_power(int,A,B)). [resolve(813,a,798,a)]. 1.90/2.12 814 -linordered_semidom(A) | -ord_less(A,B,C) | -ord_less_eq(A,zero_zero(A),B) | -ord_less(nat,zero_zero(nat),D) | ord_less(A,power_power(A,B,D),power_power(A,C,D)) # label(fact_195_power__strict__mono) # label(axiom). [clausify(387)]. 1.90/2.12 Derived: -ord_less(int,A,B) | -ord_less_eq(int,zero_zero(int),A) | -ord_less(nat,zero_zero(nat),C) | ord_less(int,power_power(int,A,C),power_power(int,B,C)). [resolve(814,a,798,a)]. 1.90/2.12 815 -linordered_semidom(A) | -ord_less_eq(A,B,C) | -ord_less_eq(A,zero_zero(A),B) | ord_less_eq(A,power_power(A,B,D),power_power(A,C,D)) # label(fact_207_power__mono) # label(axiom). [clausify(403)]. 1.90/2.12 Derived: -ord_less_eq(int,A,B) | -ord_less_eq(int,zero_zero(int),A) | ord_less_eq(int,power_power(int,A,C),power_power(int,B,C)). [resolve(815,a,798,a)]. 1.90/2.12 816 -linordered_semidom(A) | -ord_less(A,one_one(A),B) | -ord_less_eq(A,power_power(A,B,C),power_power(A,B,D)) | ord_less_eq(nat,C,D) # label(fact_176_power__le__imp__le__exp) # label(axiom). [clausify(423)]. 1.90/2.12 Derived: -ord_less(int,one_one(int),A) | -ord_less_eq(int,power_power(int,A,B),power_power(int,A,C)) | ord_less_eq(nat,B,C). [resolve(816,a,798,a)]. 1.90/2.12 817 -linordered_semidom(A) | -ord_less(A,one_one(A),B) | -ord_less(A,power_power(A,B,C),power_power(A,B,D)) | ord_less(nat,C,D) # label(fact_271_power__strict__increasing__iff) # label(axiom). [clausify(429)]. 1.90/2.12 Derived: -ord_less(int,one_one(int),A) | -ord_less(int,power_power(int,A,B),power_power(int,A,C)) | ord_less(nat,B,C). [resolve(817,a,798,a)]. 1.90/2.12 818 -linordered_semidom(A) | -ord_less(A,one_one(A),B) | ord_less(A,power_power(A,B,C),power_power(A,B,D)) | -ord_less(nat,C,D) # label(fact_271_power__strict__increasing__iff) # label(axiom). [clausify(429)]. 1.90/2.12 819 linordered_semidom(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semidom) # label(axiom). [assumption]. 1.90/2.12 Derived: -ord_less(nat,A,B) | -ord_less(real,zero_zero(real),C) | -ord_less(real,C,one_one(real)) | ord_less(real,power_power(real,C,B),power_power(real,C,A)). [resolve(819,a,799,a)]. 1.90/2.12 Derived: -ord_less_eq(nat,A,B) | -ord_less_eq(real,one_one(real),C) | ord_less_eq(real,power_power(real,C,A),power_power(real,C,B)). [resolve(819,a,800,a)]. 1.90/2.12 Derived: power_power(real,A,number_number_of(nat,bit0(bit1(pls)))) != power_power(real,B,number_number_of(nat,bit0(bit1(pls)))) | -ord_less_eq(real,zero_zero(real),B) | -ord_less_eq(real,zero_zero(real),A) | ti(real,A) = ti(real,B). [resolve(819,a,801,a)]. 1.90/2.12 Derived: -ord_less_eq(real,power_power(real,A,number_number_of(nat,bit0(bit1(pls)))),power_power(real,B,number_number_of(nat,bit0(bit1(pls))))) | -ord_less_eq(real,zero_zero(real),B) | ord_less_eq(real,A,B). [resolve(819,a,802,a)]. 1.90/2.12 Derived: -ord_less(real,zero_zero(real),A) | ord_less(real,zero_zero(real),power_power(real,A,B)). [resolve(819,a,803,a)]. 1.90/2.12 Derived: -ord_less(nat,A,B) | -ord_less(real,one_one(real),C) | ord_less(real,power_power(real,C,A),power_power(real,C,B)). [resolve(819,a,804,a)]. 1.90/2.13 Derived: power_power(real,A,B) != power_power(real,C,B) | -ord_less_eq(real,zero_zero(real),C) | -ord_less_eq(real,zero_zero(real),A) | -ord_less(nat,zero_zero(nat),B) | ti(real,A) = ti(real,C). [resolve(819,a,805,a)]. 1.90/2.13 Derived: -ord_less(real,one_one(real),A) | ord_less(real,power_power(real,A,B),times_times(real,A,power_power(real,A,B))). [resolve(819,a,806,a)]. 1.90/2.13 Derived: -ord_less(real,one_one(real),A) | power_power(real,A,B) != power_power(real,A,C) | B = C. [resolve(819,a,807,a)]. 1.90/2.13 Derived: -ord_less(real,one_one(real),A) | power_power(real,A,B) = power_power(real,A,C) | B != C. [resolve(819,a,808,a)]. 1.90/2.13 Derived: -ord_less(real,power_power(real,A,number_number_of(nat,bit0(bit1(pls)))),power_power(real,B,number_number_of(nat,bit0(bit1(pls))))) | -ord_less_eq(real,zero_zero(real),B) | ord_less(real,A,B). [resolve(819,a,809,a)]. 1.90/2.13 Derived: -ord_less(real,one_one(real),A) | ord_less(real,one_one(real),times_times(real,A,power_power(real,A,B))). [resolve(819,a,810,a)]. 1.90/2.13 Derived: -ord_less(real,one_one(real),A) | -ord_less(nat,zero_zero(nat),B) | ord_less(real,one_one(real),power_power(real,A,B)). [resolve(819,a,811,a)]. 1.90/2.13 Derived: -ord_less_eq(nat,A,B) | -ord_less_eq(real,zero_zero(real),C) | -ord_less_eq(real,C,one_one(real)) | ord_less_eq(real,power_power(real,C,B),power_power(real,C,A)). [resolve(819,a,812,a)]. 1.90/2.13 Derived: -ord_less(real,zero_zero(real),A) | -ord_less(real,A,one_one(real)) | ord_less(real,times_times(real,A,power_power(real,A,B)),power_power(real,A,B)). [resolve(819,a,813,a)]. 1.90/2.13 Derived: -ord_less(real,A,B) | -ord_less_eq(real,zero_zero(real),A) | -ord_less(nat,zero_zero(nat),C) | ord_less(real,power_power(real,A,C),power_power(real,B,C)). [resolve(819,a,814,a)]. 1.90/2.13 Derived: -ord_less_eq(real,A,B) | -ord_less_eq(real,zero_zero(real),A) | ord_less_eq(real,power_power(real,A,C),power_power(real,B,C)). [resolve(819,a,815,a)]. 1.90/2.13 Derived: -ord_less(real,one_one(real),A) | -ord_less_eq(real,power_power(real,A,B),power_power(real,A,C)) | ord_less_eq(nat,B,C). [resolve(819,a,816,a)]. 1.90/2.13 Derived: -ord_less(real,one_one(real),A) | -ord_less(real,power_power(real,A,B),power_power(real,A,C)) | ord_less(nat,B,C). [resolve(819,a,817,a)]. 1.90/2.13 820 -linordered_semidom(A) | -ord_less(A,one_one(A),B) | -ord_less_eq(A,power_power(A,B,C),power_power(A,B,D)) | ord_less_eq(nat,C,D) # label(fact_177_power__increasing__iff) # label(axiom). [clausify(442)]. 1.90/2.13 821 -linordered_semidom(A) | -ord_less(A,one_one(A),B) | ord_less_eq(A,power_power(A,B,C),power_power(A,B,D)) | -ord_less_eq(nat,C,D) # label(fact_177_power__increasing__iff) # label(axiom). [clausify(442)]. 1.90/2.13 Derived: -ord_less(int,one_one(int),A) | ord_less_eq(int,power_power(int,A,B),power_power(int,A,C)) | -ord_less_eq(nat,B,C). [resolve(821,a,798,a)]. 1.90/2.13 Derived: -ord_less(real,one_one(real),A) | ord_less_eq(real,power_power(real,A,B),power_power(real,A,C)) | -ord_less_eq(nat,B,C). [resolve(821,a,819,a)]. 1.90/2.13 822 linordered_semidom(nat) # label(arity_Nat_Onat___Rings_Olinordered__semidom) # label(axiom). [assumption]. 1.90/2.13 Derived: -ord_less(nat,A,B) | -ord_less(nat,zero_zero(nat),C) | -ord_less(nat,C,one_one(nat)) | ord_less(nat,power_power(nat,C,B),power_power(nat,C,A)). [resolve(822,a,799,a)]. 1.90/2.13 Derived: -ord_less_eq(nat,A,B) | -ord_less_eq(nat,one_one(nat),C) | ord_less_eq(nat,power_power(nat,C,A),power_power(nat,C,B)). [resolve(822,a,800,a)]. 1.90/2.13 Derived: power_power(nat,A,number_number_of(nat,bit0(bit1(pls)))) != power_power(nat,B,number_number_of(nat,bit0(bit1(pls)))) | -ord_less_eq(nat,zero_zero(nat),B) | -ord_less_eq(nat,zero_zero(nat),A) | ti(nat,A) = ti(nat,B). [resolve(822,a,801,a)]. 1.90/2.13 Derived: -ord_less_eq(nat,power_power(nat,A,number_number_of(nat,bit0(bit1(pls)))),power_power(nat,B,number_number_of(nat,bit0(bit1(pls))))) | -ord_less_eq(nat,zero_zero(nat),B) | ord_less_eq(nat,A,B). [resolve(822,a,802,a)]. 1.90/2.13 Derived: -ord_less(nat,zero_zero(nat),A) | ord_less(nat,zero_zero(nat),power_power(nat,A,B)). [resolve(822,a,803,a)]. 1.90/2.13 Derived: -ord_less(nat,A,B) | -ord_less(nat,one_one(nat),C) | ord_less(nat,power_power(nat,C,A),power_power(nat,C,B)). [resolve(822,a,804,a)]. 1.90/2.13 Derived: power_power(nat,A,B) != power_power(nat,C,B) | -ord_less_eq(nat,zero_zero(nat),C) | -ord_less_eq(nat,zero_zero(nat),A) | -ord_less(nat,zero_zero(nat),B) | ti(nat,A) = ti(nat,C). [resolve(822,a,805,a)]. 1.90/2.13 Derived: -ord_less(nat,one_one(nat),A) | ord_less(nat,power_power(nat,A,B),times_times(nat,A,power_power(nat,A,B))). [resolve(822,a,806,a)]. 1.90/2.13 Derived: -ord_less(nat,one_one(nat),A) | power_power(nat,A,B) != power_power(nat,A,C) | B = C. [resolve(822,a,807,a)]. 1.90/2.13 Derived: -ord_less(nat,one_one(nat),A) | power_power(nat,A,B) = power_power(nat,A,C) | B != C. [resolve(822,a,808,a)]. 1.90/2.13 Derived: -ord_less(nat,power_power(nat,A,number_number_of(nat,bit0(bit1(pls)))),power_power(nat,B,number_number_of(nat,bit0(bit1(pls))))) | -ord_less_eq(nat,zero_zero(nat),B) | ord_less(nat,A,B). [resolve(822,a,809,a)]. 1.90/2.13 Derived: -ord_less(nat,one_one(nat),A) | ord_less(nat,one_one(nat),times_times(nat,A,power_power(nat,A,B))). [resolve(822,a,810,a)]. 1.90/2.13 Derived: -ord_less(nat,one_one(nat),A) | -ord_less(nat,zero_zero(nat),B) | ord_less(nat,one_one(nat),power_power(nat,A,B)). [resolve(822,a,811,a)]. 1.90/2.13 Derived: -ord_less_eq(nat,A,B) | -ord_less_eq(nat,zero_zero(nat),C) | -ord_less_eq(nat,C,one_one(nat)) | ord_less_eq(nat,power_power(nat,C,B),power_power(nat,C,A)). [resolve(822,a,812,a)]. 1.90/2.13 Derived: -ord_less(nat,zero_zero(nat),A) | -ord_less(nat,A,one_one(nat)) | ord_less(nat,times_times(nat,A,power_power(nat,A,B)),power_power(nat,A,B)). [resolve(822,a,813,a)]. 1.90/2.13 Derived: -ord_less(nat,A,B) | -ord_less_eq(nat,zero_zero(nat),A) | -ord_less(nat,zero_zero(nat),C) | ord_less(nat,power_power(nat,A,C),power_power(nat,B,C)). [resolve(822,a,814,a)]. 1.90/2.13 Derived: -ord_less_eq(nat,A,B) | -ord_less_eq(nat,zero_zero(nat),A) | ord_less_eq(nat,power_power(nat,A,C),power_power(nat,B,C)). [resolve(822,a,815,a)]. 1.90/2.13 Derived: -ord_less(nat,one_one(nat),A) | -ord_less_eq(nat,power_power(nat,A,B),power_power(nat,A,C)) | ord_less_eq(nat,B,C). [resolve(822,a,816,a)]. 1.90/2.13 Derived: -ord_less(nat,one_one(nat),A) | -ord_less(nat,power_power(nat,A,B),power_power(nat,A,C)) | ord_less(nat,B,C). [resolve(822,a,817,a)]. 1.90/2.13 Derived: -ord_less(nat,one_one(nat),A) | ord_less_eq(nat,power_power(nat,A,B),power_power(nat,A,C)) | -ord_less_eq(nat,B,C). [resolve(822,a,821,a)]. 1.90/2.13 823 -linordered_semidom(A) | -ord_less(A,power_power(A,B,C),power_power(A,D,C)) | -ord_less_eq(A,zero_zero(A),D) | ord_less(A,B,D) # label(fact_212_power__less__imp__less__base) # label(axiom). [clausify(497)]. 1.90/2.13 Derived: -ord_less(int,power_power(int,A,B),power_power(int,C,B)) | -ord_less_eq(int,zero_zero(int),C) | ord_less(int,A,C). [resolve(823,a,798,a)]. 1.90/2.13 Derived: -ord_less(real,power_power(real,A,B),power_power(real,C,B)) | -ord_less_eq(real,zero_zero(real),C) | ord_less(real,A,C). [resolve(823,a,819,a)]. 1.90/2.13 Derived: -ord_less(nat,power_power(nat,A,B),power_power(nat,C,B)) | -ord_less_eq(nat,zero_zero(nat),C) | ord_less(nat,A,C). [resolve(823,a,822,a)]. 1.90/2.13 824 -linordered_semidom(A) | -ord_less_eq(A,one_one(A),B) | ord_less_eq(A,one_one(A),power_power(A,B,C)) # label(fact_268_one__le__power) # label(axiom). [clausify(507)]. 1.90/2.13 Derived: -ord_less_eq(int,one_one(int),A) | ord_less_eq(int,one_one(int),power_power(int,A,B)). [resolve(824,a,798,a)]. 1.90/2.13 Derived: -ord_less_eq(real,one_one(real),A) | ord_less_eq(real,one_one(real),power_power(real,A,B)). [resolve(824,a,819,a)]. 1.90/2.13 Derived: -ord_less_eq(nat,one_one(nat),A) | ord_less_eq(nat,one_one(nat),power_power(nat,A,B)). [resolve(824,a,822,a)]. 1.90/2.13 825 -linordered_semidom(A) | -ord_less_eq(A,zero_zero(A),B) | ord_less_eq(A,zero_zero(A),power_power(A,B,C)) # label(fact_206_zero__le__power) # label(axiom). [clausify(508)]. 1.90/2.13 Derived: -ord_less_eq(int,zero_zero(int),A) | ord_less_eq(int,zero_zero(int),power_power(int,A,B)). [resolve(825,a,798,a)]. 1.90/2.13 Derived: -ord_less_eq(real,zero_zero(real),A) | ord_less_eq(real,zero_zero(real),power_power(real,A,B)). [resolve(825,a,819,a)]. 1.90/2.13 Derived: -ord_less_eq(nat,zero_zero(nat),A) | ord_less_eq(nat,zero_zero(nat),power_power(nat,A,B)). [resolve(825,a,822,a)]. 2.08/2.30 826 -linordered_semidom(A) | -ord_less(A,one_one(A),B) | -ord_less(A,power_power(A,B,C),power_power(A,B,D)) | ord_less(nat,C,D) # label(fact_272_power__less__imp__less__exp) # label(axiom). [clausify(513)]. 2.08/2.30 827 -linord219039673up_add(A) | zero_zero(A) = ti(A,zero_zero(A)) # label(tsy_c_Groups_Ozero__class_Ozero_3_res) # label(axiom). [clausify(317)]. 2.08/2.30 828 linord219039673up_add(real) # label(arity_RealDef_Oreal___Groups_Olinordered__ab__group__add) # label(axiom). [assumption]. 2.08/2.30 829 linord219039673up_add(int) # label(arity_Int_Oint___Groups_Olinordered__ab__group__add) # label(axiom). [assumption]. 2.08/2.30 830 -linord219039673up_add(A) | zero_zero(A) != plus_plus(A,B,B) | zero_zero(A) = ti(A,B) # label(fact_201_double__eq__0__iff) # label(axiom). [clausify(448)]. 2.08/2.30 Derived: zero_zero(real) != plus_plus(real,A,A) | zero_zero(real) = ti(real,A). [resolve(830,a,828,a)]. 2.08/2.30 Derived: zero_zero(int) != plus_plus(int,A,A) | zero_zero(int) = ti(int,A). [resolve(830,a,829,a)]. 2.08/2.30 831 -linord219039673up_add(A) | zero_zero(A) = plus_plus(A,B,B) | zero_zero(A) != ti(A,B) # label(fact_201_double__eq__0__iff) # label(axiom). [clausify(448)]. 2.08/2.30 Derived: zero_zero(real) = plus_plus(real,A,A) | zero_zero(real) != ti(real,A). [resolve(831,a,828,a)]. 2.08/2.30 Derived: zero_zero(int) = plus_plus(int,A,A) | zero_zero(int) != ti(int,A). [resolve(831,a,829,a)]. 2.08/2.30 832 comm_ring_1(int) # label(arity_Int_Oint___Rings_Ocomm__ring__1) # label(axiom). [assumption]. 2.08/2.30 833 -comm_ring_1(A) | -dvd_dvd(A,B,C) | -dvd_dvd(A,B,D) | dvd_dvd(A,B,minus_minus(A,C,D)) # label(fact_489_dvd__diff) # label(axiom). [clausify(174)]. 2.08/2.30 Derived: -dvd_dvd(int,A,B) | -dvd_dvd(int,A,C) | dvd_dvd(int,A,minus_minus(int,B,C)). [resolve(832,a,833,a)]. 2.08/2.30 834 comm_ring_1(real) # label(arity_RealDef_Oreal___Rings_Ocomm__ring__1) # label(axiom). [assumption]. 2.08/2.30 Derived: -dvd_dvd(real,A,B) | -dvd_dvd(real,A,C) | dvd_dvd(real,A,minus_minus(real,B,C)). [resolve(834,a,833,a)]. 2.08/2.30 2.08/2.30 ============================== end predicate elimination ============= 2.08/2.30 2.08/2.30 Auto_denials: (non-Horn, no changes). 2.08/2.30 2.08/2.30 Term ordering decisions: 2.08/2.30 Function symbol KB weights: int=1. nat=1. real=1. pls=1. min=1. m=1. bool=1. s=1. t=1. s1=1. c1=1. c2=1. c3=1. c4=1. c5=1. c6=1. c7=1. c8=1. number_number_of=1. ti=1. legendre=1. product_prod=1. f1=1. f4=1. zero_zero=1. bit1=1. bit0=1. one_one=1. twoSqu579703163sum2sq=1. undefined=1. f2=1. times_times=1. plus_plus=1. power_power=1. minus_minus=1. f3=1. product_Pair=1. 2.08/2.30 2.08/2.30 ============================== end of process initial clauses ======== 2.08/2.30 2.08/2.30 ============================== CLAUSES FOR SEARCH ==================== 2.08/2.30 2.08/2.30 ============================== end of clauses for search ============= 2.08/2.30 2.08/2.30 ============================== SEARCH ================================ 2.08/2.30 2.08/2.30 % Starting search at 0.99 seconds. 2.08/2.30 2.08/2.30 ============================== PROOF ================================= 2.08/2.30 % SZS status Theorem 2.08/2.30 % SZS output start Refutation 2.08/2.30 2.08/2.30 % Proof 1 at 1.01 (+ 0.00) seconds. 2.08/2.30 % Length of proof is 60. 2.08/2.30 % Level of proof is 7. 2.08/2.30 % Maximum clause weight is 47.000. 2.08/2.30 % Given clauses 186. 2.08/2.30 2.08/2.30 2 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))) # 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]. 2.08/2.30 25 (all K plus_plus(int,K,pls) = ti(int,K)) # label(fact_123_add__Pls__right) # label(axiom) # label(non_clause). [assumption]. 2.08/2.30 70 (all K bit0(K) = plus_plus(int,K,K)) # label(fact_126_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 2.08/2.30 79 (all V all W number_number_of(int,plus_plus(int,V,W)) = plus_plus(int,number_number_of(int,V),number_number_of(int,W))) # label(fact_132_plus__numeral__code_I9_J) # label(axiom) # label(non_clause). [assumption]. 2.08/2.30 87 (all Z all W times_times(int,Z,W) = times_times(int,W,Z)) # label(fact_85_zmult__commute) # label(axiom) # label(non_clause). [assumption]. 2.08/2.30 107 (all K bit1(K) = plus_plus(int,plus_plus(int,one_one(int),K),K)) # label(fact_151_Bit1__def) # label(axiom) # label(non_clause). [assumption]. 2.08/2.30 145 (all K ti(int,K) = number_number_of(int,K)) # label(fact_86_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 2.08/2.30 184 (all Z ti(int,Z) = times_times(int,Z,one_one(int))) # label(fact_127_zmult__1__right) # label(axiom) # label(non_clause). [assumption]. 2.08/2.30 187 (all Z plus_plus(int,zero_zero(int),Z) = ti(int,Z)) # label(fact_204_zadd__0) # label(axiom) # label(non_clause). [assumption]. 2.08/2.30 223 (all B_1_1 bit0(ti(int,B_1_1)) = bit0(B_1_1)) # label(tsy_c_Int_OBit0_arg1) # label(hypothesis) # label(non_clause). [assumption]. 2.08/2.30 227 (all Z_1 all W_1 (ti(int,W_1) != ti(int,Z_1) & ord_less_eq(int,Z_1,W_1) <-> ord_less(int,Z_1,W_1))) # label(fact_22_zless__le) # label(axiom) # label(non_clause). [assumption]. 2.08/2.30 252 (all Z all W plus_plus(int,Z,W) = plus_plus(int,W,Z)) # label(fact_89_zadd__commute) # label(axiom) # label(non_clause). [assumption]. 2.08/2.30 390 (all B_1_1 all B_2_1 all X_a (ab_group_add(X_a) -> plus_plus(X_a,B_1_1,ti(X_a,B_2_1)) = plus_plus(X_a,B_1_1,B_2_1))) # label(tsy_c_Groups_Oplus__class_Oplus_1_arg2) # label(axiom) # label(non_clause). [assumption]. 2.08/2.30 430 (all B_1_1 all B_2_1 all X_a (no_zero_divisors(X_a) -> times_times(X_a,B_1_1,B_2_1) = times_times(X_a,ti(X_a,B_1_1),B_2_1))) # label(tsy_c_Groups_Otimes__class_Otimes_2_arg1) # label(axiom) # label(non_clause). [assumption]. 2.08/2.30 445 ord_less(int,one_one(int),t) -> (exists X exists 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)))))) # 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]. 2.08/2.30 515 -(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]. 2.08/2.30 666 -no_zero_divisors(A) | times_times(A,ti(A,B),C) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_2_arg1) # label(axiom). [clausify(430)]. 2.08/2.30 669 no_zero_divisors(int) # label(arity_Int_Oint___Rings_Ono__zero__divisors) # label(axiom). [assumption]. 2.08/2.30 726 ab_group_add(int) # label(arity_Int_Oint___Groups_Oab__group__add) # label(axiom). [assumption]. 2.08/2.30 730 -ab_group_add(A) | plus_plus(A,B,ti(A,C)) = plus_plus(A,B,C) # label(tsy_c_Groups_Oplus__class_Oplus_1_arg2) # label(axiom). [clausify(390)]. 2.08/2.30 836 one_one(int) != t | plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),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))))) # 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(2)]. 2.08/2.30 837 one_one(int) != t | 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)). [copy(836),flip(b)]. 2.08/2.30 876 plus_plus(int,A,pls) = ti(int,A) # label(fact_123_add__Pls__right) # label(axiom). [clausify(25)]. 2.08/2.30 924 plus_plus(int,A,A) = bit0(A) # label(fact_126_Bit0__def) # label(axiom). [clausify(70)]. 2.08/2.30 925 bit0(A) = plus_plus(int,A,A). [copy(924),flip(a)]. 2.08/2.30 931 plus_plus(int,number_number_of(int,A),number_number_of(int,B)) = number_number_of(int,plus_plus(int,A,B)) # label(fact_132_plus__numeral__code_I9_J) # label(axiom). [clausify(79)]. 2.08/2.30 941 times_times(int,A,B) = times_times(int,B,A) # label(fact_85_zmult__commute) # label(axiom). [clausify(87)]. 2.08/2.30 975 plus_plus(int,plus_plus(int,one_one(int),A),A) = bit1(A) # label(fact_151_Bit1__def) # label(axiom). [clausify(107)]. 2.08/2.30 976 bit1(A) = plus_plus(int,plus_plus(int,one_one(int),A),A). [copy(975),flip(a)]. 2.08/2.30 1042 number_number_of(int,A) = ti(int,A) # label(fact_86_number__of__is__id) # label(axiom). [clausify(145)]. 2.08/2.30 1043 ti(int,A) = number_number_of(int,A). [copy(1042),flip(a)]. 2.08/2.30 1098 times_times(int,A,one_one(int)) = ti(int,A) # label(fact_127_zmult__1__right) # label(axiom). [clausify(184)]. 2.08/2.30 1099 times_times(int,A,one_one(int)) = number_number_of(int,A). [copy(1098),rewrite([1043(6)])]. 2.08/2.30 1102 plus_plus(int,zero_zero(int),A) = ti(int,A) # label(fact_204_zadd__0) # label(axiom). [clausify(187)]. 2.08/2.30 1103 plus_plus(int,zero_zero(int),A) = number_number_of(int,A). [copy(1102),rewrite([1043(6)])]. 2.08/2.30 1169 bit0(ti(int,A)) = bit0(A) # label(tsy_c_Int_OBit0_arg1) # label(hypothesis). [clausify(223)]. 2.08/2.30 1170 number_number_of(int,plus_plus(int,A,A)) = plus_plus(int,A,A). [copy(1169),rewrite([1043(2),925(3),931(6),925(5)])]. 2.08/2.30 1171 ti(int,A) = ti(int,B) | -ord_less_eq(int,A,B) | ord_less(int,A,B) # label(fact_22_zless__le) # label(axiom). [clausify(227)]. 2.08/2.30 1172 number_number_of(int,A) = number_number_of(int,B) | -ord_less_eq(int,A,B) | ord_less(int,A,B). [copy(1171),rewrite([1043(2),1043(4)])]. 2.08/2.30 1181 number_number_of(nat,bit0(bit1(pls))) = plus_plus(nat,one_one(nat),one_one(nat)) # label(fact_37_nat__1__add__1) # label(axiom). [assumption]. 2.08/2.30 1182 number_number_of(nat,plus_plus(int,one_one(int),one_one(int))) = plus_plus(nat,one_one(nat),one_one(nat)). [copy(1181),rewrite([976(3),876(7),1043(6),876(8),1043(7),925(8),931(15),931(12),1170(10),1170(9)])]. 2.08/2.30 1198 one_one(int) = number_number_of(int,bit1(pls)) # label(fact_158_one__is__num__one) # label(axiom). [assumption]. 2.08/2.30 1199 number_number_of(int,number_number_of(int,number_number_of(int,one_one(int)))) = one_one(int). [copy(1198),rewrite([976(5),876(9),1043(8),876(10),1043(9)]),flip(a)]. 2.08/2.30 1224 plus_plus(int,A,B) = plus_plus(int,B,A) # label(fact_89_zadd__commute) # label(axiom). [clausify(252)]. 2.08/2.30 1382 ord_less_eq(int,one_one(int),t) # label(fact_0_tpos) # label(axiom). [assumption]. 2.08/2.30 1401 pls = zero_zero(int) # label(fact_202_Pls__def) # label(axiom). [assumption]. 2.08/2.30 1521 -ord_less(int,one_one(int),t) | plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)) = plus_plus(int,power_power(int,c7,number_number_of(nat,bit0(bit1(pls)))),power_power(int,c8,number_number_of(nat,bit0(bit1(pls))))) # 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(445)]. 2.08/2.30 1522 -ord_less(int,one_one(int),t) | plus_plus(int,one_one(int),times_times(int,m,plus_plus(int,plus_plus(int,one_one(int),one_one(int)),plus_plus(int,one_one(int),one_one(int))))) = plus_plus(int,power_power(int,c7,plus_plus(nat,one_one(nat),one_one(nat))),power_power(int,c8,plus_plus(nat,one_one(nat),one_one(nat)))). [copy(1521),rewrite([1401(9),976(11),1224(15),1103(15),1224(16),1103(16),925(15),931(22),931(19),1170(17),1170(16),925(15),1170(23),941(23),1224(26),1401(31),976(33),1224(37),1103(37),1224(38),1103(38),925(37),931(44),931(41),1170(39),1170(38),1182(37),1401(40),976(42),1224(46),1103(46),1224(47),1103(47),925(46),931(53),931(50),1170(48),1170(47),1182(46)])]. 2.08/2.30 1545 ti(int,t) = t # label(tsy_v_t_____res) # label(axiom). [assumption]. 2.08/2.30 1546 number_number_of(int,t) = t. [copy(1545),rewrite([1043(3)])]. 2.08/2.30 1613 plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)) != 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))))) # label(conj_0) # label(negated_conjecture). [clausify(515)]. 2.08/2.30 1614 plus_plus(int,one_one(int),times_times(int,m,plus_plus(int,plus_plus(int,one_one(int),one_one(int)),plus_plus(int,one_one(int),one_one(int))))) != 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)))). [copy(1613),rewrite([1401(4),976(6),1224(10),1103(10),1224(11),1103(11),925(10),931(17),931(14),1170(12),1170(11),925(10),1170(18),941(18),1224(21),1401(25),976(27),1224(31),1103(31),1224(32),1103(32),925(31),931(38),931(35),1170(33),1170(32),1182(31),1401(33),976(35),1224(39),1103(39),1224(40),1103(40),925(39),931(46),931(43),1170(41),1170(40),1182(39)])]. 2.08/2.31 2052 times_times(int,ti(int,A),B) = times_times(int,A,B). [resolve(669,a,666,a)]. 2.08/2.31 2053 times_times(int,A,number_number_of(int,B)) = times_times(int,A,B). [copy(2052),rewrite([1043(3),941(4)]),rewrite([941(6)])]. 2.08/2.31 2167 plus_plus(int,A,ti(int,B)) = plus_plus(int,A,B). [resolve(730,a,726,a)]. 2.08/2.31 2168 plus_plus(int,A,number_number_of(int,B)) = plus_plus(int,A,B). [copy(2167),rewrite([1043(3)])]. 2.08/2.31 2416 one_one(int) != t. [back_rewrite(837),rewrite([1401(9),976(11),1224(15),1103(15),1224(16),2168(16),1103(14),925(13),2168(18),1224(16),2168(16),1182(15),1401(18),976(20),1224(24),1103(24),1224(25),2168(25),1103(23),925(22),2168(27),1224(25),2168(25),1182(24),1401(28),976(30),1224(34),1103(34),1224(35),2168(35),1103(33),925(32),2168(37),1224(35),2168(35),925(34),1170(42),941(42),1224(45)]),flip(b),unit_del(b,1614)]. 2.08/2.31 2546 -ord_less(int,one_one(int),t). [back_unit_del(1522),unit_del(b,1614)]. 2.08/2.31 2771 number_number_of(int,one_one(int)) = one_one(int). [para(1199(a,1),1099(a,2)),rewrite([941(10),2053(10),2053(8),1099(6)])]. 2.08/2.31 2993 $F. [resolve(1382,a,1172,b),rewrite([2771(4),1546(5)]),unit_del(a,2416),unit_del(b,2546)]. 2.08/2.31 2.08/2.31 % SZS output end Refutation 2.08/2.31 ============================== end of proof ========================== 2.08/2.31 2.08/2.31 ============================== STATISTICS ============================ 2.08/2.31 2.08/2.31 Given=186. Generated=2447. Kept=1543. proofs=1. 2.08/2.31 Usable=184. Sos=979. Demods=172. Limbo=5, Disabled=1756. Hints=0. 2.08/2.31 Megabytes=4.21. 2.08/2.31 User_CPU=1.01, System_CPU=0.00, Wall_clock=1. 2.08/2.31 2.08/2.31 ============================== end of statistics ===================== 2.08/2.31 2.08/2.31 ============================== end of search ========================= 2.08/2.31 2.08/2.31 THEOREM PROVED 2.08/2.31 % SZS status Theorem 2.08/2.31 2.08/2.31 Exiting with 1 proof. 2.08/2.31 2.08/2.31 Process 18007 exit (max_proofs) Tue Aug 9 04:20:01 2022 2.08/2.31 Prover9 interrupted 2.08/2.31 EOF