0.00/0.04 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.00/0.04 % Command : tptp2X_and_run_prover9 %d %s 0.03/0.23 % Computer : n143.star.cs.uiowa.edu 0.03/0.23 % Model : x86_64 x86_64 0.03/0.23 % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz 0.03/0.23 % Memory : 32218.625MB 0.03/0.23 % OS : Linux 3.10.0-693.2.2.el7.x86_64 0.03/0.24 % CPULimit : 300 0.03/0.24 % DateTime : Sat Jul 14 05:24:09 CDT 2018 0.03/0.24 % CPUTime : 0.72/0.92 ============================== Prover9 =============================== 0.72/0.92 Prover9 (32) version 2009-11A, November 2009. 0.72/0.92 Process 5797 was started by sandbox2 on n143.star.cs.uiowa.edu, 0.72/0.92 Sat Jul 14 05:24:10 2018 0.72/0.92 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 300 -f /tmp/Prover9_5765_n143.star.cs.uiowa.edu". 0.72/0.92 ============================== end of head =========================== 0.72/0.92 0.72/0.92 ============================== INPUT ================================= 0.72/0.92 0.72/0.92 % Reading from file /tmp/Prover9_5765_n143.star.cs.uiowa.edu 0.72/0.92 0.72/0.92 set(prolog_style_variables). 0.72/0.92 set(auto2). 0.72/0.92 % set(auto2) -> set(auto). 0.72/0.92 % set(auto) -> set(auto_inference). 0.72/0.92 % set(auto) -> set(auto_setup). 0.72/0.92 % set(auto_setup) -> set(predicate_elim). 0.72/0.92 % set(auto_setup) -> assign(eq_defs, unfold). 0.72/0.92 % set(auto) -> set(auto_limits). 0.72/0.92 % set(auto_limits) -> assign(max_weight, "100.000"). 0.72/0.92 % set(auto_limits) -> assign(sos_limit, 20000). 0.72/0.92 % set(auto) -> set(auto_denials). 0.72/0.92 % set(auto) -> set(auto_process). 0.72/0.92 % set(auto2) -> assign(new_constants, 1). 0.72/0.92 % set(auto2) -> assign(fold_denial_max, 3). 0.72/0.92 % set(auto2) -> assign(max_weight, "200.000"). 0.72/0.92 % set(auto2) -> assign(max_hours, 1). 0.72/0.92 % assign(max_hours, 1) -> assign(max_seconds, 3600). 0.72/0.92 % set(auto2) -> assign(max_seconds, 0). 0.72/0.92 % set(auto2) -> assign(max_minutes, 5). 0.72/0.92 % assign(max_minutes, 5) -> assign(max_seconds, 300). 0.72/0.92 % set(auto2) -> set(sort_initial_sos). 0.72/0.92 % set(auto2) -> assign(sos_limit, -1). 0.72/0.92 % set(auto2) -> assign(lrs_ticks, 3000). 0.72/0.92 % set(auto2) -> assign(max_megs, 400). 0.72/0.92 % set(auto2) -> assign(stats, some). 0.72/0.92 % set(auto2) -> clear(echo_input). 0.72/0.92 % set(auto2) -> set(quiet). 0.72/0.92 % set(auto2) -> clear(print_initial_clauses). 0.72/0.92 % set(auto2) -> clear(print_given). 0.72/0.92 assign(lrs_ticks,-1). 0.72/0.92 assign(sos_limit,10000). 0.72/0.92 assign(order,kbo). 0.72/0.92 set(lex_order_vars). 0.72/0.92 clear(print_given). 0.72/0.92 0.72/0.92 % formulas(sos). % not echoed (655 formulas) 0.72/0.92 0.72/0.92 ============================== end of input ========================== 0.72/0.92 0.72/0.92 % From the command line: assign(max_seconds, 300). 0.72/0.92 0.72/0.92 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.72/0.92 0.72/0.92 % Formulas that are not ordinary clauses: 0.72/0.92 1 (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]. 0.72/0.92 2 (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]. 0.72/0.92 3 -(all T_1 times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),T_1) != plus_plus(int,power_power(int,s,number_number_of(nat,bit0(bit1(pls)))),one_one(int))) # 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]. 0.72/0.92 4 (all X_a (number_ring(X_a) -> (all N one_one(X_a) = power_power(X_a,number_number_of(X_a,min),times_times(nat,number_number_of(nat,bit0(bit1(pls))),N))))) # label(fact_336_power__m1__even) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 5 (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]. 0.72/0.92 6 (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]. 0.72/0.92 7 (all K_1 (ord_less(int,pls,K_1) <-> ord_less(int,pls,bit0(K_1)))) # label(fact_94_rel__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 8 (all V all W number_number_of(int,times_times(int,V,W)) = times_times(int,number_number_of(int,V),number_number_of(int,W))) # label(fact_129_times__numeral__code_I5_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 9 (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(nat,X_2,Y_2) <-> ord_less(X_a,power_power(X_a,B_1,X_2),power_power(X_a,B_1,Y_2))))))) # label(fact_271_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 10 (all Z1 all Z2 all W times_times(int,plus_plus(int,Z1,Z2),W) = plus_plus(int,times_times(int,Z1,W),times_times(int,Z2,W))) # label(fact_130_zadd__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 11 (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]. 0.72/0.92 12 (all X_a (number_ring(X_a) -> one_one(X_a) = number_number_of(X_a,bit1(pls)))) # label(fact_156_numeral__1__eq__1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 13 (all A_1 all B 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))))) = power_power(int,minus_minus(int,A_1,B),number_number_of(nat,bit0(bit1(pls))))) # label(fact_376_zdiff__power2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 14 (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]. 0.72/0.92 15 (all X_1 all Y_1 times_times(nat,plus_plus(nat,X_1,Y_1),minus_minus(nat,X_1,Y_1)) = minus_minus(nat,power_power(nat,X_1,number_number_of(nat,bit0(bit1(pls)))),power_power(nat,Y_1,number_number_of(nat,bit0(bit1(pls)))))) # label(fact_388_diff__square) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 16 (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]. 0.72/0.92 17 (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]. 0.72/0.92 18 (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,C),B) = plus_plus(X_a,plus_plus(X_a,A_1,B),C)))) # label(fact_75_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 19 (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]. 0.72/0.92 20 (all K_1 (pls = ti(int,K_1) <-> pls = bit0(K_1))) # label(fact_118_rel__simps_I44_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 21 (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]. 0.72/0.92 22 (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]. 0.72/0.92 23 (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]. 0.72/0.92 24 (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) | zero_zero(nat) = N_1)) # label(fact_282_zero__less__power__nat__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 25 (all B_1_1 all B_2_1 legendre(B_1_1,ti(int,B_2_1)) = legendre(B_1_1,B_2_1)) # label(tsy_c_Residues_OLegendre_arg2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 26 (all K_1 (ord_less(int,min,K_1) <-> ord_less(int,min,bit0(K_1)))) # label(fact_299_rel__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 27 (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]. 0.72/0.92 28 (all X_2 all Y_2 (zero_zero(real) = plus_plus(real,power_power(real,X_2,number_number_of(nat,bit0(bit1(pls)))),power_power(real,Y_2,number_number_of(nat,bit0(bit1(pls))))) <-> zero_zero(real) = X_2 & zero_zero(real) = Y_2)) # label(fact_420_realpow__two__sum__zero__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 29 (all K_1 all L_1 (ti(int,L_1) = ti(int,K_1) <-> bit0(L_1) = bit0(K_1))) # label(fact_83_rel__simps_I48_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 30 (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]. 0.72/0.92 31 (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]. 0.72/0.92 32 (all A_1 all B all C (ti(int,C) = minus_minus(int,A_1,B) -> plus_plus(int,C,B) = ti(int,A_1))) # label(fact_394_Int2_Oaux1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 33 (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]. 0.72/0.92 34 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all Y_2 (ord_less(X_a,zero_zero(X_a),number_number_of(X_a,Y_2)) <-> ord_less(int,pls,Y_2))))) # label(fact_244_less__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 35 (all L bit0(L) != min) # label(fact_291_rel__simps_I42_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 36 (all X_a (semiring_1(X_a) -> power_power(X_a,one_one(X_a),number_number_of(nat,bit0(bit1(pls)))) = one_one(X_a))) # label(fact_12_one__power2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 37 (all K_1 all L_1 (ord_less_eq(int,K_1,L_1) <-> ord_less(int,bit0(K_1),bit1(L_1)))) # label(fact_60_rel__simps_I15_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 38 (all K all M all N (dvd_dvd(int,times_times(int,K,M),times_times(int,K,N)) -> (ti(int,K) != zero_zero(int) -> dvd_dvd(int,M,N)))) # label(fact_190_zdvd__mult__cancel) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 39 (all Z Z = times_times(real,one_one(real),Z)) # label(fact_426_real__mult__1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 40 (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]. 0.72/0.92 41 (all X_a (comm_semiring_1(X_a) -> (all A_1 zero_zero(X_a) = times_times(X_a,zero_zero(X_a),A_1)))) # label(fact_196_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 42 (all Z plus_plus(int,Z,zero_zero(int)) = ti(int,Z)) # label(fact_205_zadd__0__right) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 43 (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]. 0.72/0.92 44 (all X_a (monoid_mult(X_a) -> (all N one_one(X_a) = power_power(X_a,one_one(X_a),N)))) # label(fact_260_power__one) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 45 (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]. 0.72/0.92 46 (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]. 0.72/0.92 47 (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]. 0.72/0.92 48 (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,B_1_1,ti(X_a,B_2_1))))) # label(tsy_c_Orderings_Oord__class_Oless__eq_arg2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 49 (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]. 0.72/0.92 50 (all X_b (number(X_b) & monoid_mult(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]. 0.72/0.92 51 (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]. 0.72/0.92 52 (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]. 0.72/0.92 53 (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]. 0.72/0.92 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]. 0.72/0.92 55 (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]. 0.72/0.92 56 (all V_2 all V_3 ((ord_less(int,V_2,V_3) -> ord_less(int,pls,V_3)) & ord_less(int,V_2,V_3) <-> ord_less(nat,number_number_of(nat,V_2),number_number_of(nat,V_3)))) # label(fact_235_less__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 57 (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]. 0.72/0.92 58 (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]. 0.72/0.92 59 (all K bit1(K) != pls) # label(fact_114_rel__simps_I46_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 60 (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]. 0.72/0.92 61 (all V_1 all V ((-ord_less(int,V,pls) -> number_number_of(nat,times_times(int,V,V_1)) = times_times(nat,number_number_of(nat,V),number_number_of(nat,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]. 0.72/0.92 62 (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]. 0.72/0.92 63 (all X_a (linordered_semidom(X_a) -> (all A_1 all N all B (power_power(X_a,B,N) = power_power(X_a,A_1,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,B) = ti(X_a,A_1)))))))) # label(fact_187_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 64 (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]. 0.72/0.92 65 (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]. 0.72/0.92 66 (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]. 0.72/0.92 67 (all J all K all M (ord_less(int,number_number_of(int,bit0(bit1(pls))),M) -> (zcong(power_power(int,number_number_of(int,min),J),power_power(int,number_number_of(int,min),K),M) -> power_power(int,number_number_of(int,min),K) = power_power(int,number_number_of(int,min),J)))) # label(fact_346_neg__one__power__eq__mod__m) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 68 (all X_a (power(X_a) -> (all A_1 power_power(X_a,A_1,zero_zero(nat)) = one_one(X_a)))) # label(fact_309_power__0) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 69 (all X_a (linord581940658strict(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_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)))))) # label(fact_234_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 70 (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]. 0.72/0.92 71 (all A_2 all B_1 all Ma (zcong(A_2,B_1,Ma) <-> zcong(B_1,A_2,Ma))) # label(fact_321_zcong__sym) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 72 (all X_2 all Y_2 (ord_less_eq(real,X_2,Y_2) <-> ord_less_eq(real,minus_minus(real,X_2,Y_2),zero_zero(real)))) # label(fact_423_real__le__eq__diff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 73 (all X_a (dvd(X_a) -> (all A_1 all B all K (times_times(X_a,B,K) = A_1 -> dvd_dvd(X_a,B,A_1))))) # label(fact_484_dvdI) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 74 (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]. 0.72/0.92 75 (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]. 0.72/0.92 76 (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]. 0.72/0.92 77 (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]. 0.72/0.92 78 (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]. 0.72/0.92 79 (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]. 0.72/0.92 80 (all Z plus_plus(int,zero_zero(int),Z) = ti(int,Z)) # label(fact_204_zadd__0) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 81 (all A_1 all B (dvd_dvd(nat,A_1,B) -> B = zero_zero(nat) | ord_less_eq(nat,A_1,B))) # label(fact_401_divides__ge) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 82 (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]. 0.72/0.92 83 (all X_a (ring_1(X_a) -> (all X_1 times_times(X_a,plus_plus(X_a,X_1,one_one(X_a)),minus_minus(X_a,X_1,one_one(X_a))) = minus_minus(X_a,times_times(X_a,X_1,X_1),one_one(X_a))))) # label(fact_441_real__squared__diff__one__factored) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 84 (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]. 0.72/0.92 85 (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]. 0.72/0.92 86 (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]. 0.72/0.92 87 (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]. 0.72/0.92 88 (all W_1 all Z_1 (ord_less(int,W_1,Z_1) <-> ord_less_eq(int,W_1,minus_minus(int,Z_1,one_one(int))))) # label(fact_366_zle__diff1__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 89 (all K1 all K2 (ord_less_eq(int,bit1(K1),bit1(K2)) <-> ord_less_eq(int,K1,K2))) # label(fact_40_less__eq__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 90 (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]. 0.72/0.92 91 (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]. 0.72/0.92 92 (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]. 0.72/0.92 93 (all X_1 (ord_less_eq(int,zero_zero(int),X_1) -> (ord_less(int,X_1,number_number_of(int,bit0(bit1(pls)))) -> zero_zero(int) = ti(int,X_1) | ti(int,X_1) = one_one(int)))) # label(fact_276_int__pos__lt__two__imp__zero__or__one) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 94 (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,Y_1) = ti(int,X_1)))))))) # label(fact_404_zcong__less__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 95 (all B_1_1 all B_2_1 all B_3 (zcong(B_1_1,B_2_1,ti(int,B_3)) <-> zcong(B_1_1,B_2_1,B_3))) # label(tsy_c_IntPrimes_Ozcong_arg3) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 96 (all M all N (one_one(int) = times_times(int,M,N) -> one_one(int) = ti(int,M) | ti(int,M) = number_number_of(int,min))) # label(fact_314_pos__zmult__eq__1__iff__lemma) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 97 (all X_a (zero_neq_one(X_a) -> ti(X_a,one_one(X_a)) = one_one(X_a))) # label(tsy_c_Groups_Oone__class_Oone_0_res) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 98 (all X_a (linord581940658strict(X_a) -> (all A_2 all B_1 (ord_less_eq(X_a,zero_zero(X_a),times_times(X_a,A_2,B_1)) <-> 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,A_2,zero_zero(X_a)) & ord_less_eq(X_a,B_1,zero_zero(X_a)))))) # label(fact_491_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 99 (all L minus_minus(int,min,bit0(L)) = bit1(minus_minus(int,min,L))) # label(fact_369_diff__bin__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 100 (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]. 0.72/0.92 101 (all A_1 all X_1 (ord_less(int,zero_zero(int),X_1) -> (ord_less(int,X_1,A_1) -> (ti(int,X_1) != minus_minus(int,A_1,one_one(int)) -> ord_less(int,X_1,minus_minus(int,A_1,one_one(int))))))) # label(fact_365_Euler_Oaux1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 102 (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]. 0.72/0.92 103 (all A_1 all B zcong(A_1,B,one_one(int))) # label(fact_330_zcong__1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 104 (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]. 0.72/0.92 105 (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]. 0.72/0.92 106 (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]. 0.72/0.92 107 (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]. 0.72/0.92 108 (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]. 0.72/0.92 109 (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]. 0.72/0.92 110 (all X_a (number_ring(X_a) -> (all V all W all C minus_minus(X_a,number_number_of(X_a,plus_plus(int,V,W)),C) = plus_plus(X_a,number_number_of(X_a,V),minus_minus(X_a,number_number_of(X_a,W),C))))) # label(fact_364_add__number__of__diff1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 111 (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]. 0.72/0.92 112 (all X_a (comm_semiring_1(X_a) -> (all A_1 ti(X_a,A_1) = times_times(X_a,one_one(X_a),A_1)))) # label(fact_112_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 113 (all X_a (semiri456707255roduct(X_a) -> (all W_1 all Y_2 all X_2 all Z_1 (ti(X_a,X_2) = ti(X_a,W_1) | ti(X_a,Y_2) = ti(X_a,Z_1) <-> plus_plus(X_a,times_times(X_a,W_1,Y_2),times_times(X_a,X_2,Z_1)) = plus_plus(X_a,times_times(X_a,W_1,Z_1),times_times(X_a,X_2,Y_2)))))) # label(fact_106_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 114 (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) <-> A_2 = B_1))) # label(fact_431_real__mult__left__cancel) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 115 (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]. 0.72/0.92 116 (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]. 0.72/0.92 117 (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]. 0.72/0.92 118 (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]. 0.72/0.92 119 (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]. 0.72/0.92 120 (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]. 0.72/0.92 121 (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]. 0.72/0.92 122 (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]. 0.72/0.92 123 (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]. 0.72/0.92 124 (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]. 0.72/0.92 125 (all K_1 (ord_less_eq(int,pls,K_1) <-> ord_less(int,pls,bit1(K_1)))) # label(fact_56_rel__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 126 (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]. 0.72/0.92 127 (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]. 0.72/0.92 128 (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,B_1_1,ti(X_a,B_2_1)))) # label(tsy_c_Groups_Otimes__class_Otimes_1_arg2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 129 (all X_2 all N_1 (one_one(nat) = power_power(nat,X_2,N_1) <-> one_one(nat) = X_2 | N_1 = zero_zero(nat))) # label(fact_410_exp__eq__1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 130 (all X_a (monoid_mult(X_a) -> (all A_1 all M all N power_power(X_a,A_1,plus_plus(nat,M,N)) = times_times(X_a,power_power(X_a,A_1,M),power_power(X_a,A_1,N))))) # label(fact_259_power__add) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 131 (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]. 0.72/0.92 132 (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]. 0.72/0.92 133 (all X_a (comm_semiring_1(X_a) -> (all A_1 times_times(X_a,A_1,zero_zero(X_a)) = zero_zero(X_a)))) # label(fact_197_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 134 (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,Z_1,X_2),times_times(real,Z_1,Y_2))))) # label(fact_437_real__mult__le__cancel__iff2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 135 (all L bit1(L) != pls) # label(fact_115_rel__simps_I39_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 136 (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]. 0.72/0.92 137 (all K_1 (ord_less_eq(int,bit0(K_1),min) <-> ord_less_eq(int,K_1,min))) # label(fact_305_rel__simps_I28_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 138 (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]. 0.72/0.92 139 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all X_2 (ord_less(X_a,number_number_of(X_a,X_2),zero_zero(X_a)) <-> ord_less(int,X_2,pls))))) # label(fact_245_less__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 140 (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]. 0.72/0.92 141 (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]. 0.72/0.92 142 (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]. 0.72/0.92 143 (all X_a (power(X_a) & zero_neq_one(X_a) & no_zero_divisors(X_a) & mult_zero(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]. 0.72/0.92 144 (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]. 0.72/0.92 145 (all T all A ti(T,A) = ti(T,ti(T,A))) # label(help_ti_idem) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 146 (all B_1_1 all B_2_1 all X_a (power(X_a) -> power_power(X_a,ti(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_arg1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 147 (all K_1 (ord_less(int,K_1,min) <-> ord_less(int,bit1(K_1),min))) # label(fact_295_rel__simps_I13_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 148 (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]. 0.72/0.92 149 (all X_b (semiring(X_b) & number(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]. 0.72/0.92 150 (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) -> legendre(A_1,P) = one_one(int)) & (-quadRes(P,A_1) -> legendre(A_1,P) = number_number_of(int,min))))) # label(fact_382_Legendre__def) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 151 (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]. 0.72/0.92 152 (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]. 0.72/0.92 153 (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]. 0.72/0.92 154 (all W_1 all Z_1 (ord_less_eq(int,plus_plus(int,W_1,one_one(int)),Z_1) <-> ord_less(int,W_1,Z_1))) # label(fact_62_add1__zle__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 155 (all A_1 all N (ord_less(nat,zero_zero(nat),N) -> (ord_less(real,zero_zero(real),A_1) -> (exists R (power_power(real,R,N) = A_1 & ord_less(real,zero_zero(real),R)))))) # label(fact_464_realpow__pos__nth) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 156 (all K ti(int,K) = minus_minus(int,K,pls)) # label(fact_350_diff__bin__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 157 (all B all A_1 all P (zprime(P) -> (ord_less(int,zero_zero(int),A_1) -> (-zcong(A_1,zero_zero(int),P) & -zcong(B,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]. 0.72/0.92 158 (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]. 0.72/0.92 159 (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]. 0.72/0.92 160 (all X_a (power(X_a) & semiring_0(X_a) -> (all N ((N != zero_zero(nat) -> power_power(X_a,zero_zero(X_a),N) = zero_zero(X_a)) & (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]. 0.72/0.92 161 (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]. 0.72/0.92 162 (all X_a (comm_semiring_1(X_a) -> (all A_1 ti(X_a,A_1) = times_times(X_a,A_1,one_one(X_a))))) # label(fact_111_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 163 (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]. 0.72/0.92 164 (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]. 0.72/0.92 165 (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]. 0.72/0.92 166 (all X_a (comm_semiring_1(X_a) -> (all A_1 plus_plus(X_a,zero_zero(X_a),A_1) = ti(X_a,A_1)))) # label(fact_198_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 167 (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]. 0.72/0.92 168 (all X_a (linordered_idom(X_a) -> (all X_2 all Y_2 (zero_zero(X_a) != ti(X_a,X_2) | ti(X_a,Y_2) != zero_zero(X_a) <-> 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]. 0.72/0.92 169 (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]. 0.72/0.92 170 (all X_a (comm_semiring(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_476_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 171 (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]. 0.72/0.92 172 (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]. 0.72/0.92 173 (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]. 0.72/0.92 174 (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]. 0.72/0.92 175 (all Ma all N_1 (zero_zero(nat) = Ma & N_1 != zero_zero(nat) <-> power_power(nat,Ma,N_1) = zero_zero(nat))) # label(fact_398_nat__power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 176 (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]. 0.72/0.92 177 (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]. 0.72/0.92 178 (all K_1 (ord_less_eq(int,K_1,min) <-> ord_less(int,bit0(K_1),min))) # label(fact_313_rel__simps_I11_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 179 (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]. 0.72/0.92 180 (all X_a (mult_zero(X_a) -> (all A_1 times_times(X_a,A_1,zero_zero(X_a)) = zero_zero(X_a)))) # label(fact_470_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 181 (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]. 0.72/0.92 182 (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]. 0.72/0.92 183 (all Z plus_plus(nat,Z,Z) = times_times(nat,Z,number_number_of(nat,bit0(bit1(pls))))) # label(fact_36_nat__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 184 (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]. 0.72/0.92 185 (all L minus_minus(int,pls,bit1(L)) = bit1(minus_minus(int,min,L))) # label(fact_367_diff__bin__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 186 (all K all M zcong(K,K,M)) # label(fact_322_zcong__refl) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 187 (all X_a (comm_semiring_1(X_a) -> (all X_1 all P all Q power_power(X_a,power_power(X_a,X_1,P),Q) = power_power(X_a,X_1,times_times(nat,P,Q))))) # label(fact_26_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.92 188 (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]. 0.72/0.92 189 (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]. 0.72/0.92 190 (all Ma all X_2 ((exists Y zcong(power_power(int,Y,number_number_of(nat,bit0(bit1(pls)))),X_2,Ma)) <-> quadRes(Ma,X_2))) # label(fact_419_QuadRes__def) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 191 (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]. 0.72/0.93 192 (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) <-> zero_zero(X_a) = ti(X_a,A_2))))) # label(fact_200_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 193 (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,B_1),Ma) <-> zcong(C_1,times_times(int,D_1,A_2),Ma)))) # label(fact_395_zcong__zmult__prop2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 194 (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]. 0.72/0.93 195 (all X_2 all W_1 (ord_less(nat,zero_zero(nat),power_power(nat,X_2,number_number_of(nat,W_1))) <-> ord_less(nat,zero_zero(nat),X_2) | number_number_of(nat,W_1) = zero_zero(nat))) # label(fact_283_zero__less__power__nat__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 196 (all X_b (semiring(X_b) & number(X_b) -> (all V all B all 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)) = times_times(X_b,number_number_of(X_b,V),plus_plus(X_b,B,C))))) # label(fact_137_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 197 (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]. 0.72/0.93 198 (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]. 0.72/0.93 199 (all K plus_plus(int,pls,K) = ti(int,K)) # label(fact_124_add__Pls) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 200 (all X_a (monoid_mult(X_a) -> (all A_1 all M all N power_power(X_a,A_1,times_times(nat,M,N)) = power_power(X_a,power_power(X_a,A_1,M),N)))) # label(fact_261_power__mult) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 201 (all B_1_1 all B_2_1 legendre(B_1_1,B_2_1) = ti(int,legendre(B_1_1,B_2_1))) # label(tsy_c_Residues_OLegendre_res) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 202 (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,ti(X_a,B_1_1),B_2_1)))) # label(tsy_c_Orderings_Oord__class_Oless_arg1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 203 (all X_b (ring(X_b) & number(X_b) -> (all V all B all C times_times(X_b,number_number_of(X_b,V),minus_minus(X_b,B,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))))) # label(fact_356_right__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 204 (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]. 0.72/0.93 205 (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]. 0.72/0.93 206 (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]. 0.72/0.93 207 (all L_1 (ti(int,L_1) = min <-> min = bit1(L_1))) # label(fact_286_rel__simps_I43_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 208 (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]. 0.72/0.93 209 (all K ti(int,K) = number_number_of(int,K)) # label(fact_86_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 210 (all X_a (number_ring(X_a) -> (all W plus_plus(X_a,one_one(X_a),number_number_of(X_a,W)) = number_number_of(X_a,plus_plus(int,bit1(pls),W))))) # label(fact_16_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 211 (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]. 0.72/0.93 212 (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]. 0.72/0.93 213 (all X_2 all Y_2 (ord_less(real,X_2,Y_2) | 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]. 0.72/0.93 214 (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]. 0.72/0.93 215 (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]. 0.72/0.93 216 -(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]. 0.72/0.93 217 (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]. 0.72/0.93 218 (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,C_1) != ti(X_a,D_1) & ti(X_a,A_2) != ti(X_a,B_1))))) # label(fact_109_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 219 (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]. 0.72/0.93 220 (all B_1_1 all B_2_1 product_Pair(int,int,ti(int,B_1_1),B_2_1) = product_Pair(int,int,B_1_1,B_2_1)) # label(tsy_c_Product__Type_OPair_arg1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 221 (all N all M (dvd_dvd(int,N,M) -> ord_less_eq(int,M,zero_zero(int)) | ord_less_eq(int,N,M))) # label(fact_405_zdvd__bounds) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 222 (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,B_1_1,ti(X_a,B_2_1)))) # label(tsy_c_Groups_Otimes__class_Otimes_3_arg2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 223 (all X_a (linordered_idom(X_a) & number_ring(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]. 0.72/0.93 224 (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]. 0.72/0.93 225 (all X_a (linord219039673up_add(X_a) -> (all A_2 (ti(X_a,A_2) = zero_zero(X_a) <-> plus_plus(X_a,A_2,A_2) = zero_zero(X_a))))) # label(fact_201_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 226 (all K_1 (ord_less_eq(int,bit1(K_1),pls) <-> ord_less(int,K_1,pls))) # label(fact_55_rel__simps_I29_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 227 (all K all L times_times(int,bit1(K),L) = plus_plus(int,bit0(times_times(int,K,L)),L)) # label(fact_159_mult__Bit1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 228 (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]. 0.72/0.93 229 (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]. 0.72/0.93 230 (all B_1_1 ti(int,bit1(B_1_1)) = bit1(B_1_1)) # label(tsy_c_Int_OBit1_res) # label(hypothesis) # label(non_clause). [assumption]. 0.72/0.93 231 (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]. 0.72/0.93 232 (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]. 0.72/0.93 233 (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]. 0.72/0.93 234 (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]. 0.72/0.93 235 (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]. 0.72/0.93 236 (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]. 0.72/0.93 237 (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]. 0.72/0.93 238 (all X_a (ring_11004092258visors(X_a) -> (all A_2 (power_power(X_a,A_2,number_number_of(nat,bit0(bit1(pls)))) = zero_zero(X_a) <-> ti(X_a,A_2) = zero_zero(X_a))))) # label(fact_250_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 239 (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]. 0.72/0.93 240 (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]. 0.72/0.93 241 (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]. 0.72/0.93 242 (all X_a (linordered_idom(X_a) & number_ring(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]. 0.72/0.93 243 (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]. 0.72/0.93 244 (all X_a (comm_semiring_1(X_a) -> (all M plus_plus(X_a,M,M) = times_times(X_a,plus_plus(X_a,one_one(X_a),one_one(X_a)),M)))) # label(fact_140_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 245 (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]. 0.72/0.93 246 (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,C,plus_plus(X_a,A_1,D))))) # label(fact_78_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 247 (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]. 0.72/0.93 248 (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]. 0.72/0.93 249 (all A_1 all B all P all Q times_times(int,twoSqu579703163sum2sq(product_Pair(int,int,A_1,B)),twoSqu579703163sum2sq(product_Pair(int,int,P,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))))) # label(fact_371_mult__sum2sq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 250 (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]. 0.72/0.93 251 (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]. 0.72/0.93 252 (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]. 0.72/0.93 253 (all X_a (comm_semiring_1(X_a) -> (all X_1 all Y_1 all Q power_power(X_a,times_times(X_a,X_1,Y_1),Q) = times_times(X_a,power_power(X_a,X_1,Q),power_power(X_a,Y_1,Q))))) # label(fact_113_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 254 (all A_1 all B power_power(int,plus_plus(int,A_1,B),number_number_of(nat,bit1(bit1(pls)))) = 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)))))) # label(fact_8_zadd__power3) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 255 (all Ma all K_1 all N_1 (K_1 = zero_zero(nat) | dvd_dvd(nat,Ma,N_1) <-> dvd_dvd(nat,times_times(nat,Ma,K_1),times_times(nat,N_1,K_1)))) # label(fact_402_nat__mult__dvd__cancel__disj_H) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 256 (all X_a (comm_semiring_1(X_a) -> (all A_1 all C plus_plus(X_a,C,A_1) = plus_plus(X_a,A_1,C)))) # label(fact_79_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 257 (all B_1_1 bit0(B_1_1) = bit0(ti(int,B_1_1))) # label(tsy_c_Int_OBit0_arg1) # label(hypothesis) # label(non_clause). [assumption]. 0.72/0.93 258 (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,X_1) = ti(X_a,Y_1))))))) # label(fact_253_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 259 (all X_a (number_semiring(X_a) -> one_one(X_a) = number_number_of(X_a,bit1(pls)))) # label(fact_155_semiring__numeral__1__eq__1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 260 (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]. 0.72/0.93 261 (all X_a (monoid_mult(X_a) -> (all X_1 all N (ord_less(nat,zero_zero(nat),N) -> power_power(X_a,X_1,N) = times_times(X_a,power_power(X_a,X_1,minus_minus(nat,N,one_one(nat))),X_1))))) # label(fact_442_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 262 (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]. 0.72/0.93 263 (all B_1_1 all B_2_1 minus_minus(nat,B_1_1,B_2_1) = minus_minus(nat,B_1_1,ti(nat,B_2_1))) # label(tsy_c_Groups_Ominus__class_Ominus_1_arg2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 264 (all K all L minus_minus(int,bit1(K),bit1(L)) = bit0(minus_minus(int,K,L))) # label(fact_359_diff__bin__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 265 (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]. 0.72/0.93 266 (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]. 0.72/0.93 267 (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]. 0.72/0.93 268 (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]. 0.72/0.93 269 (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]. 0.72/0.93 270 (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]. 0.72/0.93 271 (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]. 0.72/0.93 272 (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]. 0.72/0.93 273 (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,Rx,times_times(X_a,Lx,Ry))))) # label(fact_72_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 274 (all Z plus_plus(int,plus_plus(int,one_one(int),Z),Z) != zero_zero(int)) # label(fact_229_odd__nonzero) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 275 (all Ma all N_1 (one_one(int) = times_times(int,Ma,N_1) <-> number_number_of(int,min) = ti(int,Ma) & ti(int,N_1) = number_number_of(int,min) | one_one(int) = ti(int,N_1) & ti(int,Ma) = one_one(int))) # label(fact_315_zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 276 (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]. 0.72/0.93 277 (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]. 0.72/0.93 278 (all N all M ((M = zero_zero(nat) -> times_times(nat,M,N) = zero_zero(nat)) & (M != zero_zero(nat) -> plus_plus(nat,N,times_times(nat,minus_minus(nat,M,one_one(nat)),N)) = times_times(nat,M,N)))) # label(fact_386_mult__eq__if) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 279 (all K1 all K2 (ord_less(int,bit1(K1),bit1(K2)) <-> ord_less(int,K1,K2))) # label(fact_38_less__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 280 (all B_1_1 all B_2_1 all X_a (mult_zero(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_3_arg1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 281 (all K1 all K2 (ord_less(int,K1,K2) <-> ord_less(int,bit1(K1),bit0(K2)))) # label(fact_91_less__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 282 ord_less(int,one_one(int),t) -> (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_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]. 0.72/0.93 283 (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]. 0.72/0.93 284 (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]. 0.72/0.93 285 (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]. 0.72/0.93 286 (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,one_one(int),P) | zcong(A_1,minus_minus(int,P,one_one(int)),P))))) # label(fact_372_zcong__square) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 287 (all X_a (number_ring(X_a) -> (all 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)) = number_number_of(X_a,bit0(W))))) # label(fact_239_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 288 (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]. 0.72/0.93 289 (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]. 0.72/0.93 290 (all X_a (ab_group_add(X_a) -> (all A_1 all C all B all D plus_plus(X_a,minus_minus(X_a,A_1,B),minus_minus(X_a,C,D)) = minus_minus(X_a,plus_plus(X_a,A_1,C),plus_plus(X_a,B,D))))) # label(fact_430_add__diff__add) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 291 (all K_1 all N_1 all Ma (dvd_dvd(int,K_1,N_1) <-> dvd_dvd(int,K_1,plus_plus(int,N_1,times_times(int,K_1,Ma))))) # label(fact_210_zdvd__reduce) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 292 (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]. 0.72/0.93 293 (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,A_2) = zero_zero(X_a) | ti(X_a,B_1) = zero_zero(X_a))))) # label(fact_471_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 294 (all K1 all K2 (ord_less_eq(int,bit0(K1),bit0(K2)) <-> ord_less_eq(int,K1,K2))) # label(fact_46_less__eq__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 295 (all M all N all P (zprime(P) -> (dvd_dvd(int,P,times_times(int,M,N)) -> dvd_dvd(int,P,M) | dvd_dvd(int,P,N)))) # label(fact_411_zprime__zdvd__zmult__better) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 296 (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]. 0.72/0.93 297 (all X_a (number_ring(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_164_mult__2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 298 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all Y_2 (ord_less(int,bit1(pls),Y_2) <-> ord_less(X_a,one_one(X_a),number_number_of(X_a,Y_2)))))) # label(fact_103_less__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 299 (all X_a (linordered_idom(X_a) -> (all X_1 all Y_1 (ti(X_a,Y_1) != ti(X_a,X_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]. 0.72/0.93 300 (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]. 0.72/0.93 301 (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]. 0.72/0.93 302 (all X_a (comm_semiring_1(X_a) -> (all Lx all Ly all Rx all Ry times_times(X_a,Rx,times_times(X_a,times_times(X_a,Lx,Ly),Ry)) = times_times(X_a,times_times(X_a,Lx,Ly),times_times(X_a,Rx,Ry))))) # label(fact_67_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 303 (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]. 0.72/0.93 304 (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]. 0.72/0.93 305 (all X_a (number_ring(X_a) & ring_char_0(X_a) -> (all X_2 all Y_2 (number_number_of(X_a,X_2) = number_number_of(X_a,Y_2) <-> ti(int,Y_2) = ti(int,X_2))))) # label(fact_80_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 306 (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]. 0.72/0.93 307 (all K bit0(K) = plus_plus(int,K,K)) # label(fact_126_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 308 (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]. 0.72/0.93 309 (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]. 0.72/0.93 310 (all W_1 (ord_less(int,bit0(W_1),zero_zero(int)) <-> ord_less(int,W_1,zero_zero(int)))) # label(fact_224_bin__less__0__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 311 (all X_2 all Y_2 all Z_1 (ord_less(real,zero_zero(real),Z_1) -> (ord_less(real,times_times(real,X_2,Z_1),times_times(real,Y_2,Z_1)) <-> ord_less(real,X_2,Y_2)))) # label(fact_439_real__mult__less__iff1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 312 (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]. 0.72/0.93 313 (all X_a (number_ring(X_a) -> (all V all W all Z times_times(X_a,number_number_of(X_a,times_times(int,V,W)),Z) = times_times(X_a,number_number_of(X_a,V),times_times(X_a,number_number_of(X_a,W),Z))))) # label(fact_143_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 314 (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]. 0.72/0.93 315 (all K_1 all L_1 (ord_less(int,bit0(K_1),bit0(L_1)) <-> ord_less(int,K_1,L_1))) # label(fact_44_rel__simps_I14_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 316 (all B_1_1 all B_2_1 legendre(B_1_1,B_2_1) = legendre(ti(int,B_1_1),B_2_1)) # label(tsy_c_Residues_OLegendre_arg1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 317 (all K_1 all L_1 (ti(int,K_1) = ti(int,L_1) <-> bit1(L_1) = bit1(K_1))) # label(fact_82_rel__simps_I51_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 318 (all X_a (number_semiring(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_165_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 319 (all B_1_1 all X_a (number(X_a) -> number_number_of(X_a,ti(int,B_1_1)) = number_number_of(X_a,B_1_1))) # label(tsy_c_Int_Onumber__class_Onumber__of_arg1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 320 (all B_1_1 all B_2_1 all X_a (ab_group_add(X_a) -> ti(X_a,minus_minus(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_res) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 321 (all V_2 all V_3 (ord_less_eq(nat,number_number_of(nat,V_2),number_number_of(nat,V_3)) <-> (-ord_less_eq(int,V_2,V_3) -> ord_less_eq(int,V_2,pls)))) # label(fact_238_le__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 322 (all X_a (power(X_a) -> (all M all N ((zero_zero(nat) = N -> power_power(X_a,M,N) = one_one(X_a)) & (N != zero_zero(nat) -> times_times(X_a,M,power_power(X_a,M,minus_minus(nat,N,one_one(nat)))) = power_power(X_a,M,N)))))) # label(fact_452_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 323 (all K_1 all L_1 (ord_less_eq(int,bit1(K_1),bit1(L_1)) <-> ord_less_eq(int,K_1,L_1))) # label(fact_41_rel__simps_I34_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 324 (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]. 0.72/0.93 325 (all X_a (linordered_idom(X_a) -> (all A_2 (ord_less(X_a,plus_plus(X_a,A_2,A_2),zero_zero(X_a)) <-> ord_less(X_a,A_2,zero_zero(X_a)))))) # label(fact_215_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 326 (all X_a (number_ring(X_a) -> (all X_1 all Y_1 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)) = power_power(X_a,minus_minus(X_a,X_1,Y_1),number_number_of(nat,bit0(bit1(pls))))))) # label(fact_375_power2__diff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 327 (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]. 0.72/0.93 328 (all W times_times(int,pls,W) = pls) # label(fact_121_mult__Pls) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 329 (all W_1 all Z_1 (ord_less(int,W_1,plus_plus(int,Z_1,one_one(int))) <-> ord_less_eq(int,W_1,Z_1))) # label(fact_63_zle__add1__eq__le) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 330 (all K_1 all L_1 (ord_less_eq(int,number_number_of(int,K_1),number_number_of(int,L_1)) <-> ord_less_eq(int,K_1,L_1))) # label(fact_49_less__eq__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 331 (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]. 0.72/0.93 332 (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]. 0.72/0.93 333 (exists X (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) & ord_less(int,Y,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) & zcong(s1,Y,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) -> ti(int,Y) = ti(int,X))) & ord_less(int,X,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) & ord_less_eq(int,zero_zero(int),X))) # label(fact_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]. 0.72/0.93 334 (all K_1 (min = bit1(K_1) <-> ti(int,K_1) = min)) # label(fact_285_rel__simps_I47_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 335 (all X_a (mult_zero(X_a) -> (all A_1 times_times(X_a,zero_zero(X_a),A_1) = zero_zero(X_a)))) # label(fact_469_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 336 (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]. 0.72/0.93 337 (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]. 0.72/0.93 338 (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]. 0.72/0.93 339 (all Z_1 (ord_less(int,Z_1,zero_zero(int)) <-> ord_less(int,plus_plus(int,plus_plus(int,one_one(int),Z_1),Z_1),zero_zero(int)))) # label(fact_243_odd__less__0) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 340 (all B_1_1 all X_a (number(X_a) -> ti(X_a,number_number_of(X_a,B_1_1)) = number_number_of(X_a,B_1_1))) # label(tsy_c_Int_Onumber__class_Onumber__of_res) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 341 (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]. 0.72/0.93 342 (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) -> times_times(X_a,number_number_of(X_a,V),number_number_of(X_a,V_1)) = number_number_of(X_a,times_times(int,V,V_1))))))) # label(fact_133_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 343 (all X_a (comm_semiring_1(X_a) -> (all X_1 all N (ord_less(nat,zero_zero(nat),N) | ti(X_a,X_1) = one_one(X_a) -> dvd_dvd(X_a,X_1,power_power(X_a,X_1,N)))))) # label(fact_317_dvd__power) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 344 (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]. 0.72/0.93 345 (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]. 0.72/0.93 346 (all X_2 all Y_2 (X_2 = zero_zero(real) & zero_zero(real) = Y_2 <-> plus_plus(real,times_times(real,X_2,X_2),times_times(real,Y_2,Y_2)) = zero_zero(real))) # label(fact_440_real__two__squares__add__zero__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 347 (all K_1 all L_1 (ord_less_eq(int,K_1,L_1) <-> ord_less_eq(int,bit0(K_1),bit1(L_1)))) # label(fact_97_rel__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 348 (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]. 0.72/0.93 349 (all Z1 all Z2 all Z3 times_times(real,Z1,times_times(real,Z2,Z3)) = times_times(real,times_times(real,Z1,Z2),Z3)) # label(fact_428_real__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 350 (all X_1 all Y_1 all Z power_power(int,X_1,times_times(nat,Y_1,Z)) = power_power(int,power_power(int,X_1,Y_1),Z)) # label(fact_28_zpower__zpower) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 351 (all A_1 power_power(int,A_1,number_number_of(nat,bit1(bit1(pls)))) = times_times(int,A_1,power_power(int,A_1,number_number_of(nat,bit0(bit1(pls)))))) # label(fact_11_cube__square) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 352 (all X_a (number(X_a) & linorder(X_a) -> (all V_2 all W_1 (-ord_less(X_a,number_number_of(X_a,W_1),number_number_of(X_a,V_2)) <-> ord_less_eq(X_a,number_number_of(X_a,V_2),number_number_of(X_a,W_1)))))) # label(fact_29_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 353 (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]. 0.72/0.93 354 (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]. 0.72/0.93 355 (all X_a (monoid_mult(X_a) -> (all A_1 all N power_power(X_a,power_power(X_a,A_1,N),number_number_of(nat,bit0(bit1(pls)))) = power_power(X_a,A_1,times_times(nat,number_number_of(nat,bit0(bit1(pls))),N))))) # label(fact_101_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 356 (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,N) = ti(int,M)))))) # label(fact_189_zdvd__antisym__nonneg) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 357 (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]. 0.72/0.93 358 (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]. 0.72/0.93 359 (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]. 0.72/0.93 360 (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]. 0.72/0.93 361 (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]. 0.72/0.93 362 (all X_a (linord581940658strict(X_a) -> (all X_2 all Y_2 (ti(X_a,Y_2) != zero_zero(X_a) | zero_zero(X_a) != ti(X_a,X_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))))))) # label(fact_237_sum__squares__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 363 (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]. 0.72/0.93 364 (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]. 0.72/0.93 365 (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]. 0.72/0.93 366 (all D all C all A_1 all B all M (zcong(A_1,B,M) -> (ti(int,C) = ti(int,B) -> (zcong(C,D,M) -> zcong(A_1,D,M))))) # label(fact_385_zcong__eq__trans) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 367 (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]. 0.72/0.93 368 (all K bit0(K) != min) # label(fact_290_rel__simps_I45_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 369 (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]. 0.72/0.93 370 (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]. 0.72/0.93 371 (all L_1 (pls = ti(int,L_1) <-> bit0(L_1) = pls)) # label(fact_119_rel__simps_I38_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 372 (all B_1_1 all B_2_1 all X_a (no_zero_divisors(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_2_arg1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 373 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all X_2 (ord_less_eq(X_a,number_number_of(X_a,X_2),zero_zero(X_a)) <-> ord_less_eq(int,X_2,pls))))) # label(fact_247_le__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 374 (all A_1 all B power_power(int,minus_minus(int,A_1,B),number_number_of(nat,bit1(bit1(pls)))) = 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)))))) # label(fact_377_zdiff__power3) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 375 (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]. 0.72/0.93 376 (all Z ti(int,Z) = times_times(int,one_one(int),Z)) # label(fact_128_zmult__1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 377 (all Z_1 (ord_less(int,zero_zero(int),Z_1) <-> ord_less_eq(int,one_one(int),Z_1))) # label(fact_241_int__one__le__iff__zero__less) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 378 (all X_a (monoid_mult(X_a) -> (all A_1 power_power(X_a,A_1,one_one(nat)) = ti(X_a,A_1)))) # label(fact_240_power__one__right) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 379 (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]. 0.72/0.93 380 (all N_1 all Ma (ord_less(int,zero_zero(int),Ma) -> (times_times(int,Ma,N_1) = one_one(int) <-> ti(int,Ma) = one_one(int) & one_one(int) = ti(int,N_1)))) # label(fact_242_pos__zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 381 (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) -> minus_minus(int,P,one_one(int)) = ti(int,A_1) | ti(int,A_1) = one_one(int)))))) # label(fact_373_zcong__square__zless) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 382 (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]. 0.72/0.93 383 (all B_1_1 all B_2_1 all X_a (power(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_4_arg2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 384 (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]. 0.72/0.93 385 (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]. 0.72/0.93 386 (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]. 0.72/0.93 387 (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]. 0.72/0.93 388 (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]. 0.72/0.93 389 (all M (zprime(plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),M),one_one(int))) -> one_one(int) = legendre(number_number_of(int,min),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),M),one_one(int))))) # label(fact_379_Legendre__1mod4) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 390 (all X_a (linordered_idom(X_a) -> (all A_2 (zero_zero(X_a) != ti(X_a,A_2) <-> ord_less(X_a,zero_zero(X_a),power_power(X_a,A_2,number_number_of(nat,bit0(bit1(pls))))))))) # label(fact_255_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 391 (all X_2 all Y_2 (dvd_dvd(nat,Y_2,X_2) & dvd_dvd(nat,X_2,Y_2) <-> X_2 = Y_2)) # label(fact_384_divides__antisym) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 392 (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]. 0.72/0.93 393 (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]. 0.72/0.93 394 (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]. 0.72/0.93 395 (all X_a (mult_zero(X_a) -> ti(X_a,zero_zero(X_a)) = zero_zero(X_a))) # label(tsy_c_Groups_Ozero__class_Ozero_2_res) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 396 (all X_a (semiring_0(X_a) & power(X_a) -> (all W ((number_number_of(nat,W) = zero_zero(nat) -> one_one(X_a) = power_power(X_a,zero_zero(X_a),number_number_of(nat,W))) & (number_number_of(nat,W) != zero_zero(nat) -> zero_zero(X_a) = power_power(X_a,zero_zero(X_a),number_number_of(nat,W))))))) # label(fact_340_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 397 (all X_1 all Y_1 all Z power_power(int,X_1,plus_plus(nat,Y_1,Z)) = times_times(int,power_power(int,X_1,Y_1),power_power(int,X_1,Z))) # label(fact_34_zpower__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 398 (all X_a (zero_neq_one(X_a) -> one_one(X_a) != zero_zero(X_a))) # label(fact_475_zero__neq__one) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 399 (all X_a (comm_semiring_1(X_a) -> (all A_1 all C all D plus_plus(X_a,plus_plus(X_a,A_1,C),D) = plus_plus(X_a,A_1,plus_plus(X_a,C,D))))) # label(fact_77_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 400 (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]. 0.72/0.93 401 (all K plus_plus(int,plus_plus(int,one_one(int),K),K) = bit1(K)) # label(fact_151_Bit1__def) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 402 (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]. 0.72/0.93 403 (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]. 0.72/0.93 404 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all Y_2 (ord_less_eq(X_a,one_one(X_a),number_number_of(X_a,Y_2)) <-> ord_less_eq(int,bit1(pls),Y_2))))) # label(fact_105_le__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 405 (all X_2 all Y_2 all Z_1 (ord_less(real,zero_zero(real),Z_1) -> (ord_less_eq(real,times_times(real,X_2,Z_1),times_times(real,Y_2,Z_1)) <-> ord_less_eq(real,X_2,Y_2)))) # label(fact_438_real__mult__le__cancel__iff1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 406 (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]. 0.72/0.93 407 (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]. 0.72/0.93 408 (all X_a (ring(X_a) -> (all X_1 all Y_1 all A_1 all B minus_minus(X_a,times_times(X_a,X_1,Y_1),times_times(X_a,A_1,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))))) # label(fact_434_mult__diff__mult) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 409 (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]. 0.72/0.93 410 (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]. 0.72/0.93 411 (all R_1 all Q all A_1 (ord_less(int,zero_zero(int),A_1) -> (ti(int,A_1) = plus_plus(int,R_1,times_times(int,A_1,Q)) -> (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]. 0.72/0.93 412 (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]. 0.72/0.93 413 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all X_2 all Y_2 (ord_less(int,X_2,Y_2) <-> ord_less(X_a,number_number_of(X_a,X_2),number_number_of(X_a,Y_2)))))) # label(fact_30_less__number__of) # label(axiom) # label(non_clause). [assumption]. 0.72/0.93 414 (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_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]. 0.72/0.93 415 (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]. 0.72/0.93 416 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_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]. 0.72/0.94 417 (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]. 0.72/0.94 418 (all B_1_1 twoSqu579703163sum2sq(ti(product_prod(int,int),B_1_1)) = twoSqu579703163sum2sq(B_1_1)) # label(tsy_c_TwoSquares__Mirabelle__iprzenvtji_Osum2sq_arg1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 419 (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]. 0.72/0.94 420 (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]. 0.72/0.94 421 (all W_1 all Z_1 (ord_less(int,W_1,plus_plus(int,Z_1,one_one(int))) <-> ti(int,Z_1) = ti(int,W_1) | ord_less(int,W_1,Z_1))) # label(fact_100_zless__add1__eq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 422 (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]. 0.72/0.94 423 (all X_a (linord581940658strict(X_a) -> (all X_2 all Y_2 (zero_zero(X_a) = plus_plus(X_a,times_times(X_a,X_2,X_2),times_times(X_a,Y_2,Y_2)) <-> ti(X_a,X_2) = zero_zero(X_a) & ti(X_a,Y_2) = zero_zero(X_a))))) # label(fact_216_sum__squares__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 424 (all X_1 all Y_1 power_power(real,plus_plus(real,X_1,Y_1),number_number_of(nat,bit0(bit1(pls)))) = 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))) # label(fact_172_real__sum__squared__expand) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 425 (all A_1 all B power_power(int,plus_plus(int,A_1,B),number_number_of(nat,bit0(bit1(pls)))) = 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)))))) # label(fact_7_zadd__power2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 426 (all X_a (number_ring(X_a) -> number_number_of(X_a,bit0(bit1(pls))) = plus_plus(X_a,one_one(X_a),one_one(X_a)))) # label(fact_18_one__add__one__is__two) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 427 (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]. 0.72/0.94 428 (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]. 0.72/0.94 429 (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]. 0.72/0.94 430 (all Z all W (ord_less_eq(int,Z,W) -> (ord_less_eq(int,W,Z) -> ti(int,Z) = ti(int,W)))) # label(fact_25_zle__antisym) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 431 (all N_1 all Ma (one_one(nat) = Ma & 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]. 0.72/0.94 432 (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]. 0.72/0.94 433 (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) -> zero_zero(X_a) != times_times(X_a,A_1,B)))))) # label(fact_472_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 434 (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]. 0.72/0.94 435 (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]. 0.72/0.94 436 (all B_1_1 all B_2_1 all X_a (ab_group_add(X_a) -> plus_plus(X_a,ti(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_arg1) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 437 (all X_a (linordered_idom(X_a) -> (all X_2 all 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)) <-> ti(X_a,Y_2) = zero_zero(X_a) & ti(X_a,X_2) = zero_zero(X_a))))) # label(fact_264_sum__power2__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 438 (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]. 0.72/0.94 439 (all B_1_1 all B_2_1 all X_a (dvd(X_a) -> (dvd_dvd(X_a,B_1_1,B_2_1) <-> dvd_dvd(X_a,B_1_1,ti(X_a,B_2_1))))) # label(tsy_c_Rings_Odvd__class_Odvd_arg2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 440 (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]. 0.72/0.94 441 (all W ord_less_eq(real,W,W)) # label(fact_456_real__le__refl) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 442 (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]. 0.72/0.94 443 (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]. 0.72/0.94 444 (all X_2 all Y_2 (ord_less(real,X_2,Y_2) <-> ord_less_eq(real,X_2,Y_2) & X_2 != Y_2)) # label(fact_424_real__less__def) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 445 (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]. 0.72/0.94 446 (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]. 0.72/0.94 447 (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]. 0.72/0.94 448 (all K all L bit0(L) != bit1(K)) # label(fact_116_rel__simps_I50_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 449 (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]. 0.72/0.94 450 (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]. 0.72/0.94 451 (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]. 0.72/0.94 452 (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(nat,X_2,Y_2) <-> ord_less_eq(X_a,power_power(X_a,B_1,X_2),power_power(X_a,B_1,Y_2))))))) # label(fact_177_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 453 (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]. 0.72/0.94 454 (all A_2 all B_1 all C_1 (C_1 != zero_zero(real) -> (times_times(real,A_2,C_1) = times_times(real,B_1,C_1) <-> B_1 = A_2))) # label(fact_432_real__mult__right__cancel) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 455 (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]. 0.72/0.94 456 (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)) = zero_zero(nat)) & (-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)))) # label(fact_325_nat__number__of__mult__left) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 457 (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]. 0.72/0.94 458 (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) -> number_number_of(nat,plus_plus(int,V,V_1)) = plus_plus(nat,number_number_of(nat,V),number_number_of(nat,V_1))) & (ord_less(int,V_1,pls) -> number_number_of(nat,V) = plus_plus(nat,number_number_of(nat,V),number_number_of(nat,V_1)))))) # label(fact_52_add__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 459 (all Z times_times(int,Z,one_one(int)) = ti(int,Z)) # label(fact_127_zmult__1__right) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 460 (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]. 0.72/0.94 461 (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) -> (N_1 = Ma <-> power_power(X_a,A_2,Ma) = power_power(X_a,A_2,N_1)))))) # label(fact_270_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 462 (all V_2 (ord_less_eq(int,V_2,pls) <-> number_number_of(nat,V_2) = zero_zero(nat))) # label(fact_320_eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 463 (all Z_1 all W_1 (ord_less(int,Z_1,W_1) <-> ti(int,Z_1) != ti(int,W_1) & ord_less_eq(int,Z_1,W_1))) # label(fact_22_zless__le) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 464 (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]. 0.72/0.94 465 (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]. 0.72/0.94 466 -(all 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))) & ord_less_eq(int,zero_zero(int),S))) # 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]. 0.72/0.94 467 (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]. 0.72/0.94 468 (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]. 0.72/0.94 469 (all X_a (number_ring(X_a) -> (all A_1 times_times(X_a,A_1,number_number_of(X_a,bit1(pls))) = ti(X_a,A_1)))) # label(fact_154_mult__numeral__1__right) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 470 (all L bit0(minus_minus(int,min,L)) = minus_minus(int,min,bit1(L))) # label(fact_368_diff__bin__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 471 (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]. 0.72/0.94 472 (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]. 0.72/0.94 473 (all X_a (ring_11004092258visors(X_a) -> (all N all A_1 (zero_zero(X_a) != ti(X_a,A_1) -> zero_zero(X_a) != power_power(X_a,A_1,N))))) # label(fact_192_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 474 (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]. 0.72/0.94 475 (all X_a (semiri456707255roduct(X_a) -> (all C all D all A_1 all B all R_1 (zero_zero(X_a) != ti(X_a,R_1) -> (ti(X_a,D) != ti(X_a,C) & ti(X_a,A_1) = ti(X_a,B) -> plus_plus(X_a,B,times_times(X_a,R_1,D)) != plus_plus(X_a,A_1,times_times(X_a,R_1,C))))))) # label(fact_217_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 476 (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]. 0.72/0.94 477 (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]. 0.72/0.94 478 (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]. 0.72/0.94 479 -quadRes(plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),number_number_of(int,min)) -> legendre(number_number_of(int,min),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) != one_one(int) # label(fact_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]. 0.72/0.94 480 (all K all L bit1(L) != bit0(K)) # label(fact_117_rel__simps_I49_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 481 (all X_a (monoid_mult(X_a) -> (all A_1 all N times_times(X_a,A_1,power_power(X_a,A_1,N)) = times_times(X_a,power_power(X_a,A_1,N),A_1)))) # label(fact_257_power__commutes) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 482 (all X_a (no_zero_divisors(X_a) -> (all A_1 all B (times_times(X_a,A_1,B) = zero_zero(X_a) -> zero_zero(X_a) = ti(X_a,A_1) | zero_zero(X_a) = ti(X_a,B))))) # label(fact_473_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 483 (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]. 0.72/0.94 484 (all K plus_plus(int,K,pls) = ti(int,K)) # label(fact_123_add__Pls__right) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 485 (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]. 0.72/0.94 486 (all X_a (number_ring(X_a) -> (all V all W times_times(X_a,number_number_of(X_a,V),number_number_of(X_a,W)) = number_number_of(X_a,times_times(int,V,W))))) # label(fact_144_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 487 (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]. 0.72/0.94 488 (all X_a (linordered_idom(X_a) -> (all X_2 all Y_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) <-> zero_zero(X_a) = ti(X_a,Y_2) & zero_zero(X_a) = ti(X_a,X_2))))) # label(fact_256_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 489 (all Z1 all Z2 all W minus_minus(int,times_times(int,Z1,W),times_times(int,Z2,W)) = times_times(int,minus_minus(int,Z1,Z2),W)) # label(fact_352_zdiff__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 490 (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]. 0.72/0.94 491 (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,plus_plus(int,X_2,times_times(int,C_1,D_1)),Ta)) <-> dvd_dvd(int,A_2,plus_plus(int,X_2,Ta))))) # label(fact_211_zdvd__period) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 492 (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]. 0.72/0.94 493 (all A_2 all B_1 all Ma ((exists K_2 ti(int,B_1) = plus_plus(int,A_2,times_times(int,Ma,K_2))) <-> zcong(A_2,B_1,Ma))) # label(fact_339_zcong__iff__lin) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 494 (all K_1 (ord_less_eq(int,pls,bit0(K_1)) <-> ord_less_eq(int,pls,K_1))) # label(fact_99_rel__simps_I21_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 495 (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]. 0.72/0.94 496 (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]. 0.72/0.94 497 (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]. 0.72/0.94 498 (all X_1 all Y_1 (ord_less(int,Y_1,X_1) | ti(int,Y_1) = ti(int,X_1) | ord_less(int,X_1,Y_1))) # label(fact_23_zless__linear) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 499 (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]. 0.72/0.94 500 (all X_a (number_ring(X_a) -> (all V number_number_of(X_a,plus_plus(int,V,bit1(pls))) = plus_plus(X_a,number_number_of(X_a,V),one_one(X_a))))) # label(fact_17_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.94 501 (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]. 0.72/0.95 502 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B times_times(X_a,A_1,B) = times_times(X_a,B,A_1)))) # label(fact_73_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.95 503 (all M zcong(M,zero_zero(int),M)) # label(fact_392_zcong__id) # label(axiom) # label(non_clause). [assumption]. 0.72/0.95 504 (all X_a (number_ring(X_a) -> zero_zero(X_a) = number_number_of(X_a,pls))) # label(fact_219_semiring__norm_I112_J) # label(axiom) # label(non_clause). [assumption]. 0.72/0.95 505 (all X_a ti(X_a,undefined(X_a)) = undefined(X_a)) # label(tsy_c_HOL_Oundefined_res) # label(axiom) # label(non_clause). [assumption]. 0.72/0.95 506 (all W ord_less_eq(int,W,W)) # label(fact_20_zle__refl) # label(axiom) # label(non_clause). [assumption]. 0.72/0.95 507 (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,B_1_1,ti(X_a,B_2_1)))) # label(tsy_c_Groups_Otimes__class_Otimes_2_arg2) # label(axiom) # label(non_clause). [assumption]. 0.72/0.95 508 (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]. 0.72/0.95 509 (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(X_a) = power_power(X_a,A_2,N_1) <-> N_1 != zero_zero(nat) & ti(X_a,A_2) = zero_zero(X_a))))) # label(fact_183_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.95 510 (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]. 0.72/0.95 511 (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) -> zero_zero(int) = ti(int,X_1))))) # label(fact_412_Int2_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 0.72/0.95 512 (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,A_2,zero_zero(X_a)) & ord_less_eq(X_a,zero_zero(X_a),B_1) | ord_less_eq(X_a,B_1,zero_zero(X_a)) & ord_less_eq(X_a,zero_zero(X_a),A_2))))) # label(fact_492_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 0.72/0.95 513 (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]. 0.72/0.95 514 (all Z1 all Z2 all Z3 times_times(int,Z1,times_times(int,Z2,Z3)) = times_times(int,times_times(int,Z1,Z2),Z3)) # label(fact_84_zmult__assoc) # label(axiom) # label(non_clause). [assumption]. 0.72/0.95 515 -(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]. 0.72/0.95 0.72/0.95 ============================== end of process non-clausal formulas === 0.72/0.95 0.72/0.95 ============================== PROCESS INITIAL CLAUSES =============== 0.72/0.95 0.72/0.95 ============================== PREDICATE ELIMINATION ================= 0.72/0.95 516 comm_semiring_1(real) # label(arity_RealDef_Oreal___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 0.72/0.95 517 -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(1)]. 0.72/0.95 518 -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(2)]. 0.72/0.95 519 -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(18)]. 0.72/0.95 520 -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(41)]. 0.72/0.95 521 -comm_semiring_1(A) | times_times(A,B,plus_plus(A,C,D)) = plus_plus(A,times_times(A,B,C),times_times(A,B,D)) # label(fact_110_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom). [clausify(49)]. 0.72/0.95 522 -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(52)]. 0.72/0.95 523 -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(55)]. 0.72/0.95 524 -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(66)]. 0.72/0.95 525 -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(74)]. 0.72/0.95 526 -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(108)]. 0.72/0.95 527 -comm_semiring_1(A) | ti(A,B) = times_times(A,one_one(A),B) # label(fact_112_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom). [clausify(112)]. 0.72/0.95 528 -comm_semiring_1(A) | dvd_dvd(A,one_one(A),B) # label(fact_488_one__dvd) # label(axiom). [clausify(119)]. 0.72/0.95 529 -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(121)]. 0.72/0.95 530 -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(126)]. 0.72/0.95 531 -comm_semiring_1(A) | times_times(A,plus_plus(A,B,C),D) = plus_plus(A,times_times(A,B,D),times_times(A,C,D)) # label(fact_108_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom). [clausify(131)]. 0.72/0.95 532 -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(133)]. 0.72/0.95 533 -comm_semiring_1(A) | ti(A,B) = plus_plus(A,B,zero_zero(A)) # label(fact_199_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom). [clausify(153)]. 0.72/0.95 534 -comm_semiring_1(A) | ti(A,B) = times_times(A,B,one_one(A)) # label(fact_111_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom). [clausify(162)]. 0.72/0.95 535 -comm_semiring_1(A) | times_times(A,plus_plus(A,B,C),D) = plus_plus(A,times_times(A,B,D),times_times(A,C,D)) # label(fact_107_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom). [clausify(164)]. 0.72/0.95 536 -comm_semiring_1(A) | ti(A,B) = plus_plus(A,zero_zero(A),B) # label(fact_198_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom). [clausify(166)]. 0.72/0.95 537 -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(171)]. 0.72/0.95 538 -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(187)]. 0.72/0.95 539 -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(204)]. 0.72/0.95 540 -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(231)]. 0.72/0.95 541 -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(235)]. 0.72/0.95 542 -comm_semiring_1(A) | dvd_dvd(A,B,times_times(A,C,B)) # label(fact_480_dvd__triv__right) # label(axiom). [clausify(237)]. 0.72/0.95 543 -comm_semiring_1(A) | times_times(A,plus_plus(A,one_one(A),one_one(A)),B) = plus_plus(A,B,B) # label(fact_140_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom). [clausify(244)]. 0.72/0.95 544 -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(246)]. 0.72/0.95 545 -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(251)]. 0.72/0.95 546 -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(253)]. 0.72/0.95 547 -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(256)]. 0.72/0.95 548 -comm_semiring_1(A) | times_times(A,plus_plus(A,B,one_one(A)),C) = plus_plus(A,times_times(A,B,C),C) # label(fact_138_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom). [clausify(265)]. 0.72/0.95 549 -comm_semiring_1(A) | dvd_dvd(A,B,times_times(A,B,C)) # label(fact_479_dvd__triv__left) # label(axiom). [clausify(267)]. 0.72/0.95 550 -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(273)]. 0.72/0.95 551 -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(302)]. 0.72/0.95 552 -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(327)]. 0.72/0.95 553 -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(343)]. 0.72/0.95 554 -comm_semiring_1(A) | ti(A,B) != one_one(A) | dvd_dvd(A,B,power_power(A,B,C)) # label(fact_317_dvd__power) # label(axiom). [clausify(343)]. 0.72/0.95 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(516,a,517,a)]. 0.72/0.95 Derived: -dvd_dvd(real,A,B) | dvd_dvd(real,power_power(real,A,C),power_power(real,B,C)). [resolve(516,a,518,a)]. 0.72/0.95 Derived: plus_plus(real,plus_plus(real,A,B),C) = plus_plus(real,plus_plus(real,A,C),B). [resolve(516,a,519,a)]. 0.72/0.95 Derived: zero_zero(real) = times_times(real,zero_zero(real),A). [resolve(516,a,520,a)]. 0.72/0.95 Derived: times_times(real,A,plus_plus(real,B,C)) = plus_plus(real,times_times(real,A,B),times_times(real,A,C)). [resolve(516,a,521,a)]. 0.72/0.95 Derived: one_one(real) = power_power(real,A,zero_zero(nat)). [resolve(516,a,523,a)]. 0.72/0.95 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(516,a,524,a)]. 0.72/0.95 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(516,a,525,a)]. 0.72/0.95 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(516,a,526,a)]. 0.72/0.95 Derived: ti(real,A) = times_times(real,one_one(real),A). [resolve(516,a,527,a)]. 0.72/0.95 Derived: dvd_dvd(real,one_one(real),A). [resolve(516,a,528,a)]. 0.72/0.95 Derived: -dvd_dvd(real,A,B) | dvd_dvd(real,A,times_times(real,C,B)). [resolve(516,a,530,a)]. 0.72/0.95 Derived: times_times(real,plus_plus(real,A,B),C) = plus_plus(real,times_times(real,A,C),times_times(real,B,C)). [resolve(516,a,531,a)]. 0.72/0.96 Derived: zero_zero(real) = times_times(real,A,zero_zero(real)). [resolve(516,a,532,a)]. 0.72/0.96 Derived: ti(real,A) = plus_plus(real,A,zero_zero(real)). [resolve(516,a,533,a)]. 0.72/0.96 Derived: ti(real,A) = times_times(real,A,one_one(real)). [resolve(516,a,534,a)]. 0.72/0.96 Derived: ti(real,A) = plus_plus(real,zero_zero(real),A). [resolve(516,a,536,a)]. 0.72/0.96 Derived: -dvd_dvd(real,times_times(real,A,B),C) | dvd_dvd(real,B,C). [resolve(516,a,537,a)]. 0.72/0.96 Derived: power_power(real,power_power(real,A,B),C) = power_power(real,A,times_times(nat,B,C)). [resolve(516,a,538,a)]. 0.72/0.96 Derived: plus_plus(real,plus_plus(real,A,B),C) = plus_plus(real,A,plus_plus(real,B,C)). [resolve(516,a,539,a)]. 0.72/0.96 Derived: -dvd_dvd(real,A,B) | -dvd_dvd(real,A,C) | dvd_dvd(real,A,plus_plus(real,B,C)). [resolve(516,a,540,a)]. 0.72/0.96 Derived: times_times(real,times_times(real,A,B),C) = times_times(real,times_times(real,A,C),B). [resolve(516,a,541,a)]. 0.72/0.96 Derived: dvd_dvd(real,A,times_times(real,B,A)). [resolve(516,a,542,a)]. 0.72/0.96 Derived: times_times(real,plus_plus(real,one_one(real),one_one(real)),A) = plus_plus(real,A,A). [resolve(516,a,543,a)]. 0.72/0.96 Derived: plus_plus(real,A,plus_plus(real,B,C)) = plus_plus(real,B,plus_plus(real,A,C)). [resolve(516,a,544,a)]. 0.72/0.96 Derived: times_times(real,power_power(real,A,B),power_power(real,A,C)) = power_power(real,A,plus_plus(nat,B,C)). [resolve(516,a,545,a)]. 0.72/0.96 Derived: times_times(real,power_power(real,A,B),power_power(real,C,B)) = power_power(real,times_times(real,A,C),B). [resolve(516,a,546,a)]. 0.72/0.96 Derived: plus_plus(real,A,B) = plus_plus(real,B,A). [resolve(516,a,547,a)]. 0.72/0.96 Derived: times_times(real,plus_plus(real,A,one_one(real)),B) = plus_plus(real,times_times(real,A,B),B). [resolve(516,a,548,a)]. 0.72/0.96 Derived: dvd_dvd(real,A,times_times(real,A,B)). [resolve(516,a,549,a)]. 0.72/0.96 Derived: times_times(real,A,times_times(real,B,C)) = times_times(real,B,times_times(real,A,C)). [resolve(516,a,550,a)]. 0.72/0.96 Derived: -dvd_dvd(real,A,B) | dvd_dvd(real,A,times_times(real,B,C)). [resolve(516,a,552,a)]. 0.72/0.96 Derived: -ord_less(nat,zero_zero(nat),A) | dvd_dvd(real,B,power_power(real,B,A)). [resolve(516,a,553,a)]. 0.72/0.96 Derived: ti(real,A) != one_one(real) | dvd_dvd(real,A,power_power(real,A,B)). [resolve(516,a,554,a)]. 0.72/0.96 555 -comm_semiring_1(A) | -dvd_dvd(A,zero_zero(A),B) | ti(A,B) = zero_zero(A) # label(fact_478_dvd__0__left) # label(axiom). [clausify(354)]. 0.72/0.96 Derived: -dvd_dvd(real,zero_zero(real),A) | ti(real,A) = zero_zero(real). [resolve(555,a,516,a)]. 0.72/0.96 556 -comm_semiring_1(A) | dvd_dvd(A,B,B) # label(fact_468_dvd__refl) # label(axiom). [clausify(357)]. 0.72/0.96 Derived: dvd_dvd(real,A,A). [resolve(556,a,516,a)]. 0.72/0.96 557 -comm_semiring_1(A) | times_times(A,plus_plus(A,B,one_one(A)),C) = plus_plus(A,C,times_times(A,B,C)) # label(fact_139_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom). [clausify(365)]. 0.72/0.96 Derived: times_times(real,plus_plus(real,A,one_one(real)),B) = plus_plus(real,B,times_times(real,A,B)). [resolve(557,a,516,a)]. 0.72/0.96 558 comm_semiring_1(nat) # label(arity_Nat_Onat___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 0.72/0.96 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(558,a,517,a)]. 0.72/0.96 Derived: -dvd_dvd(nat,A,B) | dvd_dvd(nat,power_power(nat,A,C),power_power(nat,B,C)). [resolve(558,a,518,a)]. 0.72/0.96 Derived: plus_plus(nat,plus_plus(nat,A,B),C) = plus_plus(nat,plus_plus(nat,A,C),B). [resolve(558,a,519,a)]. 0.72/0.96 Derived: zero_zero(nat) = times_times(nat,zero_zero(nat),A). [resolve(558,a,520,a)]. 0.72/0.96 Derived: times_times(nat,A,plus_plus(nat,B,C)) = plus_plus(nat,times_times(nat,A,B),times_times(nat,A,C)). [resolve(558,a,521,a)]. 0.72/0.96 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(558,a,522,a)]. 0.72/0.96 Derived: one_one(nat) = power_power(nat,A,zero_zero(nat)). [resolve(558,a,523,a)]. 0.72/0.96 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(558,a,524,a)]. 0.72/0.96 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(558,a,525,a)]. 0.72/0.96 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(558,a,526,a)]. 0.72/0.96 Derived: ti(nat,A) = times_times(nat,one_one(nat),A). [resolve(558,a,527,a)]. 0.72/0.96 Derived: dvd_dvd(nat,one_one(nat),A). [resolve(558,a,528,a)]. 0.72/0.96 Derived: times_times(nat,times_times(nat,A,B),C) = times_times(nat,A,times_times(nat,B,C)). [resolve(558,a,529,a)]. 0.72/0.96 Derived: -dvd_dvd(nat,A,B) | dvd_dvd(nat,A,times_times(nat,C,B)). [resolve(558,a,530,a)]. 0.72/0.96 Derived: times_times(nat,plus_plus(nat,A,B),C) = plus_plus(nat,times_times(nat,A,C),times_times(nat,B,C)). [resolve(558,a,531,a)]. 0.72/0.96 Derived: zero_zero(nat) = times_times(nat,A,zero_zero(nat)). [resolve(558,a,532,a)]. 0.72/0.96 Derived: ti(nat,A) = plus_plus(nat,A,zero_zero(nat)). [resolve(558,a,533,a)]. 0.72/0.96 Derived: ti(nat,A) = times_times(nat,A,one_one(nat)). [resolve(558,a,534,a)]. 0.72/0.96 Derived: ti(nat,A) = plus_plus(nat,zero_zero(nat),A). [resolve(558,a,536,a)]. 0.72/0.96 Derived: -dvd_dvd(nat,times_times(nat,A,B),C) | dvd_dvd(nat,B,C). [resolve(558,a,537,a)]. 0.72/0.96 Derived: power_power(nat,power_power(nat,A,B),C) = power_power(nat,A,times_times(nat,B,C)). [resolve(558,a,538,a)]. 0.72/0.96 Derived: plus_plus(nat,plus_plus(nat,A,B),C) = plus_plus(nat,A,plus_plus(nat,B,C)). [resolve(558,a,539,a)]. 0.72/0.96 Derived: -dvd_dvd(nat,A,B) | -dvd_dvd(nat,A,C) | dvd_dvd(nat,A,plus_plus(nat,B,C)). [resolve(558,a,540,a)]. 0.72/0.96 Derived: times_times(nat,times_times(nat,A,B),C) = times_times(nat,times_times(nat,A,C),B). [resolve(558,a,541,a)]. 0.72/0.96 Derived: dvd_dvd(nat,A,times_times(nat,B,A)). [resolve(558,a,542,a)]. 0.72/0.96 Derived: times_times(nat,plus_plus(nat,one_one(nat),one_one(nat)),A) = plus_plus(nat,A,A). [resolve(558,a,543,a)]. 0.72/0.96 Derived: plus_plus(nat,A,plus_plus(nat,B,C)) = plus_plus(nat,B,plus_plus(nat,A,C)). [resolve(558,a,544,a)]. 0.72/0.96 Derived: times_times(nat,power_power(nat,A,B),power_power(nat,A,C)) = power_power(nat,A,plus_plus(nat,B,C)). [resolve(558,a,545,a)]. 0.72/0.96 Derived: times_times(nat,power_power(nat,A,B),power_power(nat,C,B)) = power_power(nat,times_times(nat,A,C),B). [resolve(558,a,546,a)]. 0.72/0.96 Derived: plus_plus(nat,A,B) = plus_plus(nat,B,A). [resolve(558,a,547,a)]. 0.72/0.96 Derived: times_times(nat,plus_plus(nat,A,one_one(nat)),B) = plus_plus(nat,times_times(nat,A,B),B). [resolve(558,a,548,a)]. 0.72/0.96 Derived: dvd_dvd(nat,A,times_times(nat,A,B)). [resolve(558,a,549,a)]. 0.72/0.96 Derived: times_times(nat,A,times_times(nat,B,C)) = times_times(nat,B,times_times(nat,A,C)). [resolve(558,a,550,a)]. 0.72/0.96 Derived: -dvd_dvd(nat,A,B) | dvd_dvd(nat,A,times_times(nat,B,C)). [resolve(558,a,552,a)]. 0.72/0.96 Derived: -ord_less(nat,zero_zero(nat),A) | dvd_dvd(nat,B,power_power(nat,B,A)). [resolve(558,a,553,a)]. 0.72/0.96 Derived: ti(nat,A) != one_one(nat) | dvd_dvd(nat,A,power_power(nat,A,B)). [resolve(558,a,554,a)]. 0.72/0.96 Derived: -dvd_dvd(nat,zero_zero(nat),A) | ti(nat,A) = zero_zero(nat). [resolve(558,a,555,a)]. 0.72/0.96 Derived: dvd_dvd(nat,A,A). [resolve(558,a,556,a)]. 0.72/0.96 Derived: times_times(nat,plus_plus(nat,A,one_one(nat)),B) = plus_plus(nat,B,times_times(nat,A,B)). [resolve(558,a,557,a)]. 0.72/0.96 559 -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(369)]. 0.72/0.96 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(559,a,516,a)]. 0.72/0.96 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(559,a,558,a)]. 0.72/0.96 560 -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(370)]. 0.72/0.96 Derived: -dvd_dvd(real,A,B) | -dvd_dvd(real,B,C) | dvd_dvd(real,A,C). [resolve(560,a,516,a)]. 0.72/0.96 Derived: -dvd_dvd(nat,A,B) | -dvd_dvd(nat,B,C) | dvd_dvd(nat,A,C). [resolve(560,a,558,a)]. 0.72/0.96 561 -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(379)]. 0.72/0.96 Derived: -dvd_dvd(real,times_times(real,A,B),C) | dvd_dvd(real,A,C). [resolve(561,a,516,a)]. 0.72/0.97 Derived: -dvd_dvd(nat,times_times(nat,A,B),C) | dvd_dvd(nat,A,C). [resolve(561,a,558,a)]. 0.72/0.97 562 -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(399)]. 0.72/0.97 563 -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(402)]. 0.72/0.97 Derived: -ord_less_eq(nat,A,B) | dvd_dvd(real,power_power(real,C,A),power_power(real,C,B)). [resolve(563,a,516,a)]. 0.72/0.97 Derived: -ord_less_eq(nat,A,B) | dvd_dvd(nat,power_power(nat,C,A),power_power(nat,C,B)). [resolve(563,a,558,a)]. 0.72/0.97 564 -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(419)]. 0.72/0.97 Derived: times_times(real,A,A) = power_power(real,A,number_number_of(nat,bit0(bit1(pls)))). [resolve(564,a,516,a)]. 0.72/0.97 Derived: times_times(nat,A,A) = power_power(nat,A,number_number_of(nat,bit0(bit1(pls)))). [resolve(564,a,558,a)]. 0.72/0.97 565 comm_semiring_1(int) # label(arity_Int_Oint___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 0.72/0.97 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(565,a,517,a)]. 0.72/0.97 Derived: -dvd_dvd(int,A,B) | dvd_dvd(int,power_power(int,A,C),power_power(int,B,C)). [resolve(565,a,518,a)]. 0.72/0.97 Derived: plus_plus(int,plus_plus(int,A,B),C) = plus_plus(int,plus_plus(int,A,C),B). [resolve(565,a,519,a)]. 0.72/0.97 Derived: zero_zero(int) = times_times(int,zero_zero(int),A). [resolve(565,a,520,a)]. 0.72/0.97 Derived: times_times(int,A,plus_plus(int,B,C)) = plus_plus(int,times_times(int,A,B),times_times(int,A,C)). [resolve(565,a,521,a)]. 0.72/0.97 Derived: one_one(int) = power_power(int,A,zero_zero(nat)). [resolve(565,a,523,a)]. 0.72/0.97 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(565,a,524,a)]. 0.72/0.97 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(565,a,525,a)]. 0.72/0.97 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(565,a,526,a)]. 0.72/0.97 Derived: ti(int,A) = times_times(int,one_one(int),A). [resolve(565,a,527,a)]. 0.72/0.97 Derived: dvd_dvd(int,one_one(int),A). [resolve(565,a,528,a)]. 0.72/0.97 Derived: -dvd_dvd(int,A,B) | dvd_dvd(int,A,times_times(int,C,B)). [resolve(565,a,530,a)]. 0.72/0.97 Derived: times_times(int,plus_plus(int,A,B),C) = plus_plus(int,times_times(int,A,C),times_times(int,B,C)). [resolve(565,a,531,a)]. 0.72/0.97 Derived: zero_zero(int) = times_times(int,A,zero_zero(int)). [resolve(565,a,532,a)]. 0.72/0.97 Derived: ti(int,A) = plus_plus(int,A,zero_zero(int)). [resolve(565,a,533,a)]. 0.72/0.97 Derived: ti(int,A) = plus_plus(int,zero_zero(int),A). [resolve(565,a,536,a)]. 0.72/0.97 Derived: -dvd_dvd(int,times_times(int,A,B),C) | dvd_dvd(int,B,C). [resolve(565,a,537,a)]. 0.72/0.97 Derived: power_power(int,power_power(int,A,B),C) = power_power(int,A,times_times(nat,B,C)). [resolve(565,a,538,a)]. 0.72/0.97 Derived: plus_plus(int,plus_plus(int,A,B),C) = plus_plus(int,A,plus_plus(int,B,C)). [resolve(565,a,539,a)]. 0.72/0.97 Derived: -dvd_dvd(int,A,B) | -dvd_dvd(int,A,C) | dvd_dvd(int,A,plus_plus(int,B,C)). [resolve(565,a,540,a)]. 0.72/0.97 Derived: times_times(int,times_times(int,A,B),C) = times_times(int,times_times(int,A,C),B). [resolve(565,a,541,a)]. 0.72/0.97 Derived: dvd_dvd(int,A,times_times(int,B,A)). [resolve(565,a,542,a)]. 0.72/0.97 Derived: times_times(int,plus_plus(int,one_one(int),one_one(int)),A) = plus_plus(int,A,A). [resolve(565,a,543,a)]. 0.72/0.97 Derived: plus_plus(int,A,plus_plus(int,B,C)) = plus_plus(int,B,plus_plus(int,A,C)). [resolve(565,a,544,a)]. 0.72/0.97 Derived: times_times(int,power_power(int,A,B),power_power(int,A,C)) = power_power(int,A,plus_plus(nat,B,C)). [resolve(565,a,545,a)]. 0.72/0.97 Derived: times_times(int,power_power(int,A,B),power_power(int,C,B)) = power_power(int,times_times(int,A,C),B). [resolve(565,a,546,a)]. 0.72/0.99 Derived: plus_plus(int,A,B) = plus_plus(int,B,A). [resolve(565,a,547,a)]. 0.72/0.99 Derived: times_times(int,plus_plus(int,A,one_one(int)),B) = plus_plus(int,times_times(int,A,B),B). [resolve(565,a,548,a)]. 0.72/0.99 Derived: dvd_dvd(int,A,times_times(int,A,B)). [resolve(565,a,549,a)]. 0.72/0.99 Derived: times_times(int,A,times_times(int,B,C)) = times_times(int,B,times_times(int,A,C)). [resolve(565,a,550,a)]. 0.72/0.99 Derived: -dvd_dvd(int,A,B) | dvd_dvd(int,A,times_times(int,B,C)). [resolve(565,a,552,a)]. 0.72/0.99 Derived: -ord_less(nat,zero_zero(nat),A) | dvd_dvd(int,B,power_power(int,B,A)). [resolve(565,a,553,a)]. 0.72/0.99 Derived: ti(int,A) != one_one(int) | dvd_dvd(int,A,power_power(int,A,B)). [resolve(565,a,554,a)]. 0.72/0.99 Derived: -dvd_dvd(int,zero_zero(int),A) | ti(int,A) = zero_zero(int). [resolve(565,a,555,a)]. 0.72/0.99 Derived: dvd_dvd(int,A,A). [resolve(565,a,556,a)]. 0.72/0.99 Derived: times_times(int,plus_plus(int,A,one_one(int)),B) = plus_plus(int,B,times_times(int,A,B)). [resolve(565,a,557,a)]. 0.72/0.99 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(565,a,559,a)]. 0.72/0.99 Derived: -dvd_dvd(int,A,B) | -dvd_dvd(int,B,C) | dvd_dvd(int,A,C). [resolve(565,a,560,a)]. 0.72/0.99 Derived: -dvd_dvd(int,times_times(int,A,B),C) | dvd_dvd(int,A,C). [resolve(565,a,561,a)]. 0.72/0.99 Derived: -ord_less_eq(nat,A,B) | dvd_dvd(int,power_power(int,C,A),power_power(int,C,B)). [resolve(565,a,563,a)]. 0.72/0.99 Derived: times_times(int,A,A) = power_power(int,A,number_number_of(nat,bit0(bit1(pls)))). [resolve(565,a,564,a)]. 0.72/0.99 566 -comm_semiring_1(A) | dvd_dvd(A,B,zero_zero(A)) # label(fact_465_dvd__0__right) # label(axiom). [clausify(455)]. 0.72/0.99 Derived: dvd_dvd(real,A,zero_zero(real)). [resolve(566,a,516,a)]. 0.72/0.99 Derived: dvd_dvd(nat,A,zero_zero(nat)). [resolve(566,a,558,a)]. 0.72/0.99 Derived: dvd_dvd(int,A,zero_zero(int)). [resolve(566,a,565,a)]. 0.72/0.99 567 -comm_semiring_1(A) | ti(A,B) = power_power(A,B,one_one(nat)) # label(fact_27_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom). [clausify(490)]. 0.72/0.99 Derived: ti(real,A) = power_power(real,A,one_one(nat)). [resolve(567,a,516,a)]. 0.72/0.99 Derived: ti(nat,A) = power_power(nat,A,one_one(nat)). [resolve(567,a,558,a)]. 0.72/0.99 Derived: ti(int,A) = power_power(int,A,one_one(nat)). [resolve(567,a,565,a)]. 0.72/0.99 568 -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(492)]. 0.72/0.99 Derived: times_times(real,times_times(real,A,B),C) = times_times(real,A,times_times(real,B,C)). [resolve(568,a,516,a)]. 0.72/0.99 569 -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(501)]. 0.72/0.99 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(569,a,516,a)]. 0.72/0.99 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(569,a,558,a)]. 0.72/0.99 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(569,a,565,a)]. 0.72/0.99 570 -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(502)]. 0.72/0.99 Derived: times_times(nat,A,B) = times_times(nat,B,A). [resolve(570,a,558,a)]. 0.72/0.99 Derived: times_times(int,A,B) = times_times(int,B,A). [resolve(570,a,565,a)]. 0.72/0.99 571 number_ring(real) # label(arity_RealDef_Oreal___Int_Onumber__ring) # label(axiom). [assumption]. 0.72/0.99 572 -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(4)]. 0.72/0.99 573 -number_ring(A) | one_one(A) = number_number_of(A,bit1(pls)) # label(fact_156_numeral__1__eq__1) # label(axiom). [clausify(12)]. 0.72/0.99 Derived: one_one(real) = power_power(real,number_number_of(real,min),times_times(nat,number_number_of(nat,bit0(bit1(pls))),A)). [resolve(571,a,572,a)]. 0.72/0.99 Derived: one_one(real) = number_number_of(real,bit1(pls)). [resolve(571,a,573,a)]. 0.72/0.99 574 -number_ring(A) | one_one(A) = number_number_of(A,bit1(pls)) # label(fact_157_semiring__norm_I110_J) # label(axiom). [clausify(21)]. 0.72/0.99 575 -linordered_idom(A) | -number_ring(A) | -ord_less(A,zero_zero(A),number_number_of(A,B)) | ord_less(int,pls,B) # label(fact_244_less__special_I1_J) # label(axiom). [clausify(34)]. 0.72/0.99 Derived: -linordered_idom(real) | -ord_less(real,zero_zero(real),number_number_of(real,A)) | ord_less(int,pls,A). [resolve(575,b,571,a)]. 0.72/0.99 576 -linordered_idom(A) | -number_ring(A) | ord_less(A,zero_zero(A),number_number_of(A,B)) | -ord_less(int,pls,B) # label(fact_244_less__special_I1_J) # label(axiom). [clausify(34)]. 0.72/0.99 Derived: -linordered_idom(real) | ord_less(real,zero_zero(real),number_number_of(real,A)) | -ord_less(int,pls,A). [resolve(576,b,571,a)]. 0.72/0.99 577 -number_ring(A) | zero_zero(A) = number_number_of(A,pls) # label(fact_220_number__of__Pls) # label(axiom). [clausify(78)]. 0.72/0.99 Derived: zero_zero(real) = number_number_of(real,pls). [resolve(577,a,571,a)]. 0.72/0.99 578 -number_ring(A) | number_number_of(A,plus_plus(int,B,C)) = plus_plus(A,number_number_of(A,B),number_number_of(A,C)) # label(fact_148_number__of__add) # label(axiom). [clausify(82)]. 0.72/0.99 Derived: number_number_of(real,plus_plus(int,A,B)) = plus_plus(real,number_number_of(real,A),number_number_of(real,B)). [resolve(578,a,571,a)]. 0.72/0.99 579 -number_ring(A) | number_number_of(A,plus_plus(int,B,C)) = plus_plus(A,number_number_of(A,B),number_number_of(A,C)) # label(fact_147_add__number__of__eq) # label(axiom). [clausify(84)]. 0.72/0.99 580 -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(110)]. 0.72/0.99 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(580,a,571,a)]. 0.72/0.99 581 -number_ring(A) | ti(A,B) = times_times(A,number_number_of(A,bit1(pls)),B) # label(fact_153_mult__numeral__1) # label(axiom). [clausify(115)]. 0.72/0.99 Derived: ti(real,A) = times_times(real,number_number_of(real,bit1(pls)),A). [resolve(581,a,571,a)]. 0.72/0.99 582 -number_ring(A) | -linordered_idom(A) | -ord_less(A,number_number_of(A,B),zero_zero(A)) | ord_less(int,B,pls) # label(fact_245_less__special_I3_J) # label(axiom). [clausify(139)]. 0.72/0.99 Derived: -linordered_idom(real) | -ord_less(real,number_number_of(real,A),zero_zero(real)) | ord_less(int,A,pls). [resolve(582,a,571,a)]. 0.72/0.99 583 -number_ring(A) | -linordered_idom(A) | ord_less(A,number_number_of(A,B),zero_zero(A)) | -ord_less(int,B,pls) # label(fact_245_less__special_I3_J) # label(axiom). [clausify(139)]. 0.72/0.99 Derived: -linordered_idom(real) | ord_less(real,number_number_of(real,A),zero_zero(real)) | -ord_less(int,A,pls). [resolve(583,a,571,a)]. 0.72/0.99 584 -number_ring(A) | number_number_of(A,plus_plus(int,bit1(pls),B)) = plus_plus(A,one_one(A),number_number_of(A,B)) # label(fact_16_add__special_I2_J) # label(axiom). [clausify(210)]. 0.72/0.99 Derived: number_number_of(real,plus_plus(int,bit1(pls),A)) = plus_plus(real,one_one(real),number_number_of(real,A)). [resolve(584,a,571,a)]. 0.72/0.99 585 -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(217)]. 0.72/0.99 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(585,b,571,a)]. 0.72/0.99 586 -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(217)]. 0.72/0.99 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(586,b,571,a)]. 0.72/1.00 587 -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(219)]. 0.72/1.00 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(587,a,571,a)]. 0.72/1.00 588 -linordered_idom(A) | -number_ring(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(223)]. 0.72/1.00 Derived: -linordered_idom(real) | -ord_less_eq(int,pls,A) | ord_less_eq(real,zero_zero(real),number_number_of(real,A)). [resolve(588,b,571,a)]. 0.72/1.00 589 -linordered_idom(A) | -number_ring(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(223)]. 0.72/1.00 Derived: -linordered_idom(real) | ord_less_eq(int,pls,A) | -ord_less_eq(real,zero_zero(real),number_number_of(real,A)). [resolve(589,b,571,a)]. 0.72/1.00 590 -linordered_idom(A) | -number_ring(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(242)]. 0.72/1.00 Derived: -linordered_idom(real) | -ord_less_eq(real,number_number_of(real,A),one_one(real)) | ord_less_eq(int,A,bit1(pls)). [resolve(590,b,571,a)]. 0.72/1.00 591 -linordered_idom(A) | -number_ring(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(242)]. 0.72/1.00 Derived: -linordered_idom(real) | ord_less_eq(real,number_number_of(real,A),one_one(real)) | -ord_less_eq(int,A,bit1(pls)). [resolve(591,b,571,a)]. 0.72/1.00 592 -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(262)]. 0.72/1.00 Derived: -linordered_idom(real) | -ord_less(int,A,bit1(pls)) | ord_less(real,number_number_of(real,A),one_one(real)). [resolve(592,b,571,a)]. 0.72/1.00 593 -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(262)]. 0.72/1.00 Derived: -linordered_idom(real) | ord_less(int,A,bit1(pls)) | -ord_less(real,number_number_of(real,A),one_one(real)). [resolve(593,b,571,a)]. 0.72/1.00 594 -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(277)]. 0.72/1.00 Derived: times_times(real,number_number_of(real,A),number_number_of(real,B)) = number_number_of(real,times_times(int,A,B)). [resolve(594,a,571,a)]. 0.72/1.00 595 -number_ring(A) | number_number_of(A,bit1(B)) = plus_plus(A,plus_plus(A,one_one(A),number_number_of(A,B)),number_number_of(A,B)) # label(fact_152_number__of__Bit1) # label(axiom). [clausify(285)]. 0.72/1.00 Derived: number_number_of(real,bit1(A)) = plus_plus(real,plus_plus(real,one_one(real),number_number_of(real,A)),number_number_of(real,A)). [resolve(595,a,571,a)]. 0.72/1.00 596 -number_ring(A) | number_number_of(A,bit0(B)) = plus_plus(A,plus_plus(A,zero_zero(A),number_number_of(A,B)),number_number_of(A,B)) # label(fact_239_number__of__Bit0) # label(axiom). [clausify(287)]. 0.72/1.00 Derived: number_number_of(real,bit0(A)) = plus_plus(real,plus_plus(real,zero_zero(real),number_number_of(real,A)),number_number_of(real,A)). [resolve(596,a,571,a)]. 0.72/1.00 597 -number_ring(A) | times_times(A,number_number_of(A,bit0(bit1(pls))),B) = plus_plus(A,B,B) # label(fact_164_mult__2) # label(axiom). [clausify(297)]. 0.72/1.00 Derived: times_times(real,number_number_of(real,bit0(bit1(pls))),A) = plus_plus(real,A,A). [resolve(597,a,571,a)]. 0.72/1.00 598 -linordered_idom(A) | -number_ring(A) | -ord_less(int,bit1(pls),B) | ord_less(A,one_one(A),number_number_of(A,B)) # label(fact_103_less__special_I2_J) # label(axiom). [clausify(298)]. 0.72/1.00 Derived: -linordered_idom(real) | -ord_less(int,bit1(pls),A) | ord_less(real,one_one(real),number_number_of(real,A)). [resolve(598,b,571,a)]. 0.72/1.00 599 -linordered_idom(A) | -number_ring(A) | ord_less(int,bit1(pls),B) | -ord_less(A,one_one(A),number_number_of(A,B)) # label(fact_103_less__special_I2_J) # label(axiom). [clausify(298)]. 0.72/1.00 Derived: -linordered_idom(real) | ord_less(int,bit1(pls),A) | -ord_less(real,one_one(real),number_number_of(real,A)). [resolve(599,b,571,a)]. 0.72/1.00 600 -number_ring(A) | -ring_char_0(A) | number_number_of(A,B) != number_number_of(A,C) | ti(int,B) = ti(int,C) # label(fact_80_eq__number__of) # label(axiom). [clausify(305)]. 0.72/1.00 Derived: -ring_char_0(real) | number_number_of(real,A) != number_number_of(real,B) | ti(int,A) = ti(int,B). [resolve(600,a,571,a)]. 0.72/1.00 601 -number_ring(A) | -ring_char_0(A) | number_number_of(A,B) = number_number_of(A,C) | ti(int,B) != ti(int,C) # label(fact_80_eq__number__of) # label(axiom). [clausify(305)]. 0.72/1.00 Derived: -ring_char_0(real) | number_number_of(real,A) = number_number_of(real,B) | ti(int,A) != ti(int,B). [resolve(601,a,571,a)]. 0.72/1.00 602 -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(313)]. 0.72/1.00 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(602,a,571,a)]. 0.72/1.00 603 -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(326)]. 0.72/1.00 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(603,a,571,a)]. 0.72/1.00 604 -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(336)]. 0.72/1.00 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(604,a,571,a)]. 0.72/1.00 605 -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(358)]. 0.72/1.00 Derived: minus_minus(real,number_number_of(real,A),number_number_of(real,B)) = number_number_of(real,minus_minus(int,A,B)). [resolve(605,a,571,a)]. 0.72/1.00 606 -linordered_idom(A) | -number_ring(A) | -ord_less_eq(A,number_number_of(A,B),zero_zero(A)) | ord_less_eq(int,B,pls) # label(fact_247_le__special_I3_J) # label(axiom). [clausify(373)]. 0.72/1.00 Derived: -linordered_idom(real) | -ord_less_eq(real,number_number_of(real,A),zero_zero(real)) | ord_less_eq(int,A,pls). [resolve(606,b,571,a)]. 0.72/1.01 607 -linordered_idom(A) | -number_ring(A) | ord_less_eq(A,number_number_of(A,B),zero_zero(A)) | -ord_less_eq(int,B,pls) # label(fact_247_le__special_I3_J) # label(axiom). [clausify(373)]. 0.72/1.01 Derived: -linordered_idom(real) | ord_less_eq(real,number_number_of(real,A),zero_zero(real)) | -ord_less_eq(int,A,pls). [resolve(607,b,571,a)]. 0.72/1.01 608 -number_ring(A) | ti(A,B) = plus_plus(A,number_number_of(A,pls),B) # label(fact_141_add__numeral__0) # label(axiom). [clausify(384)]. 0.72/1.01 Derived: ti(real,A) = plus_plus(real,number_number_of(real,pls),A). [resolve(608,a,571,a)]. 0.72/1.01 609 number_ring(int) # label(arity_Int_Oint___Int_Onumber__ring) # label(axiom). [assumption]. 0.72/1.01 Derived: one_one(int) = power_power(int,number_number_of(int,min),times_times(nat,number_number_of(nat,bit0(bit1(pls))),A)). [resolve(609,a,572,a)]. 0.72/1.01 Derived: one_one(int) = number_number_of(int,bit1(pls)). [resolve(609,a,573,a)]. 0.72/1.01 Derived: -linordered_idom(int) | -ord_less(int,zero_zero(int),number_number_of(int,A)) | ord_less(int,pls,A). [resolve(609,a,575,b)]. 0.72/1.01 Derived: -linordered_idom(int) | ord_less(int,zero_zero(int),number_number_of(int,A)) | -ord_less(int,pls,A). [resolve(609,a,576,b)]. 0.72/1.01 Derived: zero_zero(int) = number_number_of(int,pls). [resolve(609,a,577,a)]. 0.72/1.01 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(609,a,580,a)]. 0.72/1.01 Derived: ti(int,A) = times_times(int,number_number_of(int,bit1(pls)),A). [resolve(609,a,581,a)]. 0.72/1.01 Derived: -linordered_idom(int) | -ord_less(int,number_number_of(int,A),zero_zero(int)) | ord_less(int,A,pls). [resolve(609,a,582,a)]. 0.72/1.01 Derived: -linordered_idom(int) | ord_less(int,number_number_of(int,A),zero_zero(int)) | -ord_less(int,A,pls). [resolve(609,a,583,a)]. 0.72/1.01 Derived: number_number_of(int,plus_plus(int,bit1(pls),A)) = plus_plus(int,one_one(int),number_number_of(int,A)). [resolve(609,a,584,a)]. 0.72/1.01 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(609,a,585,b)]. 0.72/1.01 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(609,a,586,b)]. 0.72/1.01 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(609,a,587,a)]. 0.72/1.01 Derived: -linordered_idom(int) | -ord_less_eq(int,pls,A) | ord_less_eq(int,zero_zero(int),number_number_of(int,A)). [resolve(609,a,588,b)]. 0.72/1.01 Derived: -linordered_idom(int) | ord_less_eq(int,pls,A) | -ord_less_eq(int,zero_zero(int),number_number_of(int,A)). [resolve(609,a,589,b)]. 0.72/1.01 Derived: -linordered_idom(int) | -ord_less_eq(int,number_number_of(int,A),one_one(int)) | ord_less_eq(int,A,bit1(pls)). [resolve(609,a,590,b)]. 0.72/1.01 Derived: -linordered_idom(int) | ord_less_eq(int,number_number_of(int,A),one_one(int)) | -ord_less_eq(int,A,bit1(pls)). [resolve(609,a,591,b)]. 0.72/1.01 Derived: -linordered_idom(int) | -ord_less(int,A,bit1(pls)) | ord_less(int,number_number_of(int,A),one_one(int)). [resolve(609,a,592,b)]. 0.72/1.01 Derived: -linordered_idom(int) | ord_less(int,A,bit1(pls)) | -ord_less(int,number_number_of(int,A),one_one(int)). [resolve(609,a,593,b)]. 0.72/1.01 Derived: times_times(int,number_number_of(int,A),number_number_of(int,B)) = number_number_of(int,times_times(int,A,B)). [resolve(609,a,594,a)]. 0.72/1.01 Derived: number_number_of(int,bit1(A)) = plus_plus(int,plus_plus(int,one_one(int),number_number_of(int,A)),number_number_of(int,A)). [resolve(609,a,595,a)]. 0.72/1.01 Derived: number_number_of(int,bit0(A)) = plus_plus(int,plus_plus(int,zero_zero(int),number_number_of(int,A)),number_number_of(int,A)). [resolve(609,a,596,a)]. 0.72/1.01 Derived: times_times(int,number_number_of(int,bit0(bit1(pls))),A) = plus_plus(int,A,A). [resolve(609,a,597,a)]. 0.72/1.01 Derived: -linordered_idom(int) | -ord_less(int,bit1(pls),A) | ord_less(int,one_one(int),number_number_of(int,A)). [resolve(609,a,598,b)]. 0.72/1.01 Derived: -linordered_idom(int) | ord_less(int,bit1(pls),A) | -ord_less(int,one_one(int),number_number_of(int,A)). [resolve(609,a,599,b)]. 0.72/1.01 Derived: -ring_char_0(int) | number_number_of(int,A) != number_number_of(int,B) | ti(int,A) = ti(int,B). [resolve(609,a,600,a)]. 0.72/1.01 Derived: -ring_char_0(int) | number_number_of(int,A) = number_number_of(int,B) | ti(int,A) != ti(int,B). [resolve(609,a,601,a)]. 0.72/1.01 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(609,a,602,a)]. 0.72/1.01 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(609,a,603,a)]. 0.72/1.01 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(609,a,604,a)]. 0.72/1.02 Derived: minus_minus(int,number_number_of(int,A),number_number_of(int,B)) = number_number_of(int,minus_minus(int,A,B)). [resolve(609,a,605,a)]. 0.72/1.02 Derived: -linordered_idom(int) | -ord_less_eq(int,number_number_of(int,A),zero_zero(int)) | ord_less_eq(int,A,pls). [resolve(609,a,606,b)]. 0.72/1.02 Derived: -linordered_idom(int) | ord_less_eq(int,number_number_of(int,A),zero_zero(int)) | -ord_less_eq(int,A,pls). [resolve(609,a,607,b)]. 0.72/1.02 Derived: ti(int,A) = plus_plus(int,number_number_of(int,pls),A). [resolve(609,a,608,a)]. 0.72/1.02 610 -linordered_idom(A) | -number_ring(A) | -ord_less_eq(A,one_one(A),number_number_of(A,B)) | ord_less_eq(int,bit1(pls),B) # label(fact_105_le__special_I2_J) # label(axiom). [clausify(404)]. 0.72/1.02 Derived: -linordered_idom(real) | -ord_less_eq(real,one_one(real),number_number_of(real,A)) | ord_less_eq(int,bit1(pls),A). [resolve(610,b,571,a)]. 0.72/1.02 Derived: -linordered_idom(int) | -ord_less_eq(int,one_one(int),number_number_of(int,A)) | ord_less_eq(int,bit1(pls),A). [resolve(610,b,609,a)]. 0.72/1.02 611 -linordered_idom(A) | -number_ring(A) | ord_less_eq(A,one_one(A),number_number_of(A,B)) | -ord_less_eq(int,bit1(pls),B) # label(fact_105_le__special_I2_J) # label(axiom). [clausify(404)]. 0.72/1.02 Derived: -linordered_idom(real) | ord_less_eq(real,one_one(real),number_number_of(real,A)) | -ord_less_eq(int,bit1(pls),A). [resolve(611,b,571,a)]. 0.72/1.02 Derived: -linordered_idom(int) | ord_less_eq(int,one_one(int),number_number_of(int,A)) | -ord_less_eq(int,bit1(pls),A). [resolve(611,b,609,a)]. 0.72/1.02 612 -linordered_idom(A) | -number_ring(A) | -ord_less(int,B,C) | ord_less(A,number_number_of(A,B),number_number_of(A,C)) # label(fact_30_less__number__of) # label(axiom). [clausify(413)]. 0.72/1.02 Derived: -linordered_idom(real) | -ord_less(int,A,B) | ord_less(real,number_number_of(real,A),number_number_of(real,B)). [resolve(612,b,571,a)]. 0.72/1.02 Derived: -linordered_idom(int) | -ord_less(int,A,B) | ord_less(int,number_number_of(int,A),number_number_of(int,B)). [resolve(612,b,609,a)]. 0.72/1.02 613 -linordered_idom(A) | -number_ring(A) | ord_less(int,B,C) | -ord_less(A,number_number_of(A,B),number_number_of(A,C)) # label(fact_30_less__number__of) # label(axiom). [clausify(413)]. 0.72/1.02 Derived: -linordered_idom(real) | ord_less(int,A,B) | -ord_less(real,number_number_of(real,A),number_number_of(real,B)). [resolve(613,b,571,a)]. 0.72/1.02 Derived: -linordered_idom(int) | ord_less(int,A,B) | -ord_less(int,number_number_of(int,A),number_number_of(int,B)). [resolve(613,b,609,a)]. 0.72/1.02 614 -number_ring(A) | number_number_of(A,bit0(bit1(pls))) = plus_plus(A,one_one(A),one_one(A)) # label(fact_18_one__add__one__is__two) # label(axiom). [clausify(426)]. 0.72/1.02 Derived: number_number_of(real,bit0(bit1(pls))) = plus_plus(real,one_one(real),one_one(real)). [resolve(614,a,571,a)]. 0.72/1.02 Derived: number_number_of(int,bit0(bit1(pls))) = plus_plus(int,one_one(int),one_one(int)). [resolve(614,a,609,a)]. 0.72/1.02 615 -number_ring(A) | times_times(A,B,number_number_of(A,bit0(bit1(pls)))) = plus_plus(A,B,B) # label(fact_166_mult__2__right) # label(axiom). [clausify(451)]. 0.72/1.02 Derived: times_times(real,A,number_number_of(real,bit0(bit1(pls)))) = plus_plus(real,A,A). [resolve(615,a,571,a)]. 0.72/1.02 Derived: times_times(int,A,number_number_of(int,bit0(bit1(pls)))) = plus_plus(int,A,A). [resolve(615,a,609,a)]. 0.72/1.02 616 -number_ring(A) | ti(A,B) = times_times(A,B,number_number_of(A,bit1(pls))) # label(fact_154_mult__numeral__1__right) # label(axiom). [clausify(469)]. 0.72/1.02 Derived: ti(real,A) = times_times(real,A,number_number_of(real,bit1(pls))). [resolve(616,a,571,a)]. 0.72/1.02 Derived: ti(int,A) = times_times(int,A,number_number_of(int,bit1(pls))). [resolve(616,a,609,a)]. 0.72/1.02 617 -number_ring(A) | ti(A,B) = plus_plus(A,B,number_number_of(A,pls)) # label(fact_142_add__numeral__0__right) # label(axiom). [clausify(471)]. 0.72/1.02 Derived: ti(real,A) = plus_plus(real,A,number_number_of(real,pls)). [resolve(617,a,571,a)]. 0.72/1.02 Derived: ti(int,A) = plus_plus(int,A,number_number_of(int,pls)). [resolve(617,a,609,a)]. 0.72/1.02 618 -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(486)]. 0.72/1.04 619 -number_ring(A) | number_number_of(A,plus_plus(int,B,bit1(pls))) = plus_plus(A,number_number_of(A,B),one_one(A)) # label(fact_17_add__special_I3_J) # label(axiom). [clausify(500)]. 0.72/1.04 Derived: number_number_of(real,plus_plus(int,A,bit1(pls))) = plus_plus(real,number_number_of(real,A),one_one(real)). [resolve(619,a,571,a)]. 0.72/1.04 Derived: number_number_of(int,plus_plus(int,A,bit1(pls))) = plus_plus(int,number_number_of(int,A),one_one(int)). [resolve(619,a,609,a)]. 0.72/1.04 620 -number_ring(A) | zero_zero(A) = number_number_of(A,pls) # label(fact_219_semiring__norm_I112_J) # label(axiom). [clausify(504)]. 0.72/1.04 621 zero_neq_one(real) # label(arity_RealDef_Oreal___Rings_Ozero__neq__one) # label(axiom). [assumption]. 0.72/1.04 622 -zero_neq_one(A) | zero_zero(A) != one_one(A) # label(fact_474_one__neq__zero) # label(axiom). [clausify(6)]. 0.72/1.04 623 -zero_neq_one(A) | ti(A,zero_zero(A)) = zero_zero(A) # label(tsy_c_Groups_Ozero__class_Ozero_0_res) # label(axiom). [clausify(45)]. 0.72/1.04 624 -zero_neq_one(A) | ti(A,one_one(A)) = one_one(A) # label(tsy_c_Groups_Oone__class_Oone_0_res) # label(axiom). [clausify(97)]. 0.72/1.04 625 -power(A) | -zero_neq_one(A) | -no_zero_divisors(A) | -mult_zero(A) | zero_zero(A) != power_power(A,B,number_number_of(nat,C)) | ti(A,B) = zero_zero(A) # label(fact_337_power__eq__0__iff__number__of) # label(axiom). [clausify(143)]. 0.72/1.04 626 -power(A) | -zero_neq_one(A) | -no_zero_divisors(A) | -mult_zero(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(143)]. 0.72/1.04 627 -power(A) | -zero_neq_one(A) | -no_zero_divisors(A) | -mult_zero(A) | zero_zero(A) = power_power(A,B,number_number_of(nat,C)) | ti(A,B) != zero_zero(A) | zero_zero(nat) = number_number_of(nat,C) # label(fact_337_power__eq__0__iff__number__of) # label(axiom). [clausify(143)]. 0.72/1.04 Derived: zero_zero(real) != one_one(real). [resolve(621,a,622,a)]. 0.72/1.04 Derived: ti(real,zero_zero(real)) = zero_zero(real). [resolve(621,a,623,a)]. 0.72/1.04 Derived: ti(real,one_one(real)) = one_one(real). [resolve(621,a,624,a)]. 0.72/1.04 Derived: -power(real) | -no_zero_divisors(real) | -mult_zero(real) | zero_zero(real) != power_power(real,A,number_number_of(nat,B)) | ti(real,A) = zero_zero(real). [resolve(621,a,625,b)]. 0.72/1.04 Derived: -power(real) | -no_zero_divisors(real) | -mult_zero(real) | zero_zero(real) != power_power(real,A,number_number_of(nat,B)) | zero_zero(nat) != number_number_of(nat,B). [resolve(621,a,626,b)]. 0.72/1.04 Derived: -power(real) | -no_zero_divisors(real) | -mult_zero(real) | zero_zero(real) = power_power(real,A,number_number_of(nat,B)) | ti(real,A) != zero_zero(real) | zero_zero(nat) = number_number_of(nat,B). [resolve(621,a,627,b)]. 0.72/1.04 628 zero_neq_one(int) # label(arity_Int_Oint___Rings_Ozero__neq__one) # label(axiom). [assumption]. 0.72/1.04 Derived: zero_zero(int) != one_one(int). [resolve(628,a,622,a)]. 0.72/1.04 Derived: ti(int,zero_zero(int)) = zero_zero(int). [resolve(628,a,623,a)]. 0.72/1.04 Derived: ti(int,one_one(int)) = one_one(int). [resolve(628,a,624,a)]. 0.72/1.04 Derived: -power(int) | -no_zero_divisors(int) | -mult_zero(int) | zero_zero(int) != power_power(int,A,number_number_of(nat,B)) | ti(int,A) = zero_zero(int). [resolve(628,a,625,b)]. 0.72/1.04 Derived: -power(int) | -no_zero_divisors(int) | -mult_zero(int) | zero_zero(int) != power_power(int,A,number_number_of(nat,B)) | zero_zero(nat) != number_number_of(nat,B). [resolve(628,a,626,b)]. 0.72/1.04 Derived: -power(int) | -no_zero_divisors(int) | -mult_zero(int) | zero_zero(int) = power_power(int,A,number_number_of(nat,B)) | ti(int,A) != zero_zero(int) | zero_zero(nat) = number_number_of(nat,B). [resolve(628,a,627,b)]. 0.72/1.04 629 zero_neq_one(nat) # label(arity_Nat_Onat___Rings_Ozero__neq__one) # label(axiom). [assumption]. 0.72/1.04 Derived: zero_zero(nat) != one_one(nat). [resolve(629,a,622,a)]. 0.72/1.04 Derived: ti(nat,zero_zero(nat)) = zero_zero(nat). [resolve(629,a,623,a)]. 0.72/1.04 Derived: ti(nat,one_one(nat)) = one_one(nat). [resolve(629,a,624,a)]. 0.72/1.04 Derived: -power(nat) | -no_zero_divisors(nat) | -mult_zero(nat) | zero_zero(nat) != power_power(nat,A,number_number_of(nat,B)) | ti(nat,A) = zero_zero(nat). [resolve(629,a,625,b)]. 0.72/1.07 Derived: -power(nat) | -no_zero_divisors(nat) | -mult_zero(nat) | zero_zero(nat) != power_power(nat,A,number_number_of(nat,B)) | zero_zero(nat) != number_number_of(nat,B). [resolve(629,a,626,b)]. 0.72/1.07 Derived: -power(nat) | -no_zero_divisors(nat) | -mult_zero(nat) | zero_zero(nat) = power_power(nat,A,number_number_of(nat,B)) | ti(nat,A) != zero_zero(nat) | zero_zero(nat) = number_number_of(nat,B). [resolve(629,a,627,b)]. 0.72/1.07 630 -zero_neq_one(A) | zero_zero(A) != one_one(A) # label(fact_475_zero__neq__one) # label(axiom). [clausify(398)]. 0.72/1.07 631 -mult_zero(A) | -no_zero_divisors(A) | -zero_neq_one(A) | -power(A) | zero_zero(A) != power_power(A,B,C) | zero_zero(nat) != C # label(fact_183_power__eq__0__iff) # label(axiom). [clausify(509)]. 0.72/1.07 Derived: -mult_zero(real) | -no_zero_divisors(real) | -power(real) | zero_zero(real) != power_power(real,A,B) | zero_zero(nat) != B. [resolve(631,c,621,a)]. 0.72/1.07 Derived: -mult_zero(int) | -no_zero_divisors(int) | -power(int) | zero_zero(int) != power_power(int,A,B) | zero_zero(nat) != B. [resolve(631,c,628,a)]. 0.72/1.07 Derived: -mult_zero(nat) | -no_zero_divisors(nat) | -power(nat) | zero_zero(nat) != power_power(nat,A,B) | zero_zero(nat) != B. [resolve(631,c,629,a)]. 0.72/1.07 632 -mult_zero(A) | -no_zero_divisors(A) | -zero_neq_one(A) | -power(A) | zero_zero(A) != power_power(A,B,C) | ti(A,B) = zero_zero(A) # label(fact_183_power__eq__0__iff) # label(axiom). [clausify(509)]. 0.72/1.07 Derived: -mult_zero(real) | -no_zero_divisors(real) | -power(real) | zero_zero(real) != power_power(real,A,B) | ti(real,A) = zero_zero(real). [resolve(632,c,621,a)]. 0.72/1.07 Derived: -mult_zero(int) | -no_zero_divisors(int) | -power(int) | zero_zero(int) != power_power(int,A,B) | ti(int,A) = zero_zero(int). [resolve(632,c,628,a)]. 0.72/1.07 Derived: -mult_zero(nat) | -no_zero_divisors(nat) | -power(nat) | zero_zero(nat) != power_power(nat,A,B) | ti(nat,A) = zero_zero(nat). [resolve(632,c,629,a)]. 0.72/1.07 633 -mult_zero(A) | -no_zero_divisors(A) | -zero_neq_one(A) | -power(A) | zero_zero(A) = power_power(A,B,C) | zero_zero(nat) = C | ti(A,B) != zero_zero(A) # label(fact_183_power__eq__0__iff) # label(axiom). [clausify(509)]. 0.72/1.07 Derived: -mult_zero(real) | -no_zero_divisors(real) | -power(real) | zero_zero(real) = power_power(real,A,B) | zero_zero(nat) = B | ti(real,A) != zero_zero(real). [resolve(633,c,621,a)]. 0.72/1.07 Derived: -mult_zero(int) | -no_zero_divisors(int) | -power(int) | zero_zero(int) = power_power(int,A,B) | zero_zero(nat) = B | ti(int,A) != zero_zero(int). [resolve(633,c,628,a)]. 0.72/1.07 Derived: -mult_zero(nat) | -no_zero_divisors(nat) | -power(nat) | zero_zero(nat) = power_power(nat,A,B) | zero_zero(nat) = B | ti(nat,A) != zero_zero(nat). [resolve(633,c,629,a)]. 0.72/1.07 634 linordered_semidom(nat) # label(arity_Nat_Onat___Rings_Olinordered__semidom) # label(axiom). [assumption]. 0.72/1.07 635 -linordered_semidom(A) | -ord_less(A,one_one(A),B) | -ord_less(nat,C,D) | ord_less(A,power_power(A,B,C),power_power(A,B,D)) # label(fact_271_power__strict__increasing__iff) # label(axiom). [clausify(9)]. 0.72/1.07 636 -linordered_semidom(A) | -ord_less(A,one_one(A),B) | ord_less(nat,C,D) | -ord_less(A,power_power(A,B,C),power_power(A,B,D)) # label(fact_271_power__strict__increasing__iff) # label(axiom). [clausify(9)]. 0.72/1.07 637 -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(17)]. 0.72/1.07 638 -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(33)]. 0.72/1.07 Derived: -ord_less(nat,one_one(nat),A) | -ord_less(nat,B,C) | ord_less(nat,power_power(nat,A,B),power_power(nat,A,C)). [resolve(634,a,635,a)]. 0.72/1.07 Derived: -ord_less(nat,one_one(nat),A) | ord_less(nat,B,C) | -ord_less(nat,power_power(nat,A,B),power_power(nat,A,C)). [resolve(634,a,636,a)]. 0.72/1.07 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(634,a,637,a)]. 0.72/1.08 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(634,a,638,a)]. 0.72/1.08 639 linordered_semidom(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semidom) # label(axiom). [assumption]. 0.72/1.08 Derived: -ord_less(real,one_one(real),A) | -ord_less(nat,B,C) | ord_less(real,power_power(real,A,B),power_power(real,A,C)). [resolve(639,a,635,a)]. 0.72/1.08 Derived: -ord_less(real,one_one(real),A) | ord_less(nat,B,C) | -ord_less(real,power_power(real,A,B),power_power(real,A,C)). [resolve(639,a,636,a)]. 0.72/1.08 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(639,a,637,a)]. 0.72/1.08 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(639,a,638,a)]. 0.72/1.08 640 -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(63)]. 0.72/1.08 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(640,a,634,a)]. 0.72/1.08 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(640,a,639,a)]. 0.72/1.08 641 -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(92)]. 0.72/1.08 Derived: -ord_less(nat,one_one(nat),A) | ord_less(nat,one_one(nat),times_times(nat,A,power_power(nat,A,B))). [resolve(641,a,634,a)]. 0.72/1.08 Derived: -ord_less(real,one_one(real),A) | ord_less(real,one_one(real),times_times(real,A,power_power(real,A,B))). [resolve(641,a,639,a)]. 0.72/1.08 642 -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(106)]. 0.72/1.08 643 -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(123)]. 0.72/1.08 Derived: -ord_less(real,zero_zero(real),A) | ord_less(real,zero_zero(real),power_power(real,A,B)). [resolve(643,a,639,a)]. 0.72/1.08 644 -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(127)]. 0.72/1.08 Derived: -ord_less_eq(nat,zero_zero(nat),A) | ord_less_eq(nat,zero_zero(nat),power_power(nat,A,B)). [resolve(644,a,634,a)]. 0.72/1.08 Derived: -ord_less_eq(real,zero_zero(real),A) | ord_less_eq(real,zero_zero(real),power_power(real,A,B)). [resolve(644,a,639,a)]. 0.72/1.08 645 -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(144)]. 0.72/1.08 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(645,a,634,a)]. 0.72/1.08 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(645,a,639,a)]. 0.72/1.08 646 -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(181)]. 0.92/1.09 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(646,a,634,a)]. 0.92/1.09 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(646,a,639,a)]. 0.92/1.09 647 -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(239)]. 0.92/1.09 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(647,a,634,a)]. 0.92/1.09 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(647,a,639,a)]. 0.92/1.09 648 -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(258)]. 0.92/1.09 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(648,a,634,a)]. 0.92/1.09 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(648,a,639,a)]. 0.92/1.09 649 -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(269)]. 0.92/1.09 Derived: -ord_less_eq(nat,one_one(nat),A) | ord_less_eq(nat,one_one(nat),power_power(nat,A,B)). [resolve(649,a,634,a)]. 0.92/1.09 Derived: -ord_less_eq(real,one_one(real),A) | ord_less_eq(real,one_one(real),power_power(real,A,B)). [resolve(649,a,639,a)]. 0.92/1.09 650 -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(341)]. 0.92/1.09 651 -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(344)]. 0.92/1.09 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(651,a,634,a)]. 0.92/1.09 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(651,a,639,a)]. 0.92/1.09 652 -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(382)]. 0.92/1.09 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(652,a,634,a)]. 0.92/1.09 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(652,a,639,a)]. 0.92/1.09 653 -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(392)]. 0.92/1.09 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(653,a,634,a)]. 0.92/1.09 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(653,a,639,a)]. 0.92/1.09 654 -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(422)]. 0.92/1.09 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(654,a,634,a)]. 0.92/1.09 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(654,a,639,a)]. 0.92/1.09 655 -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(428)]. 0.92/1.09 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(655,a,634,a)]. 0.92/1.09 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(655,a,639,a)]. 0.92/1.09 656 linordered_semidom(int) # label(arity_Int_Oint___Rings_Olinordered__semidom) # label(axiom). [assumption]. 0.92/1.09 Derived: -ord_less(int,one_one(int),A) | -ord_less(nat,B,C) | ord_less(int,power_power(int,A,B),power_power(int,A,C)). [resolve(656,a,635,a)]. 0.92/1.09 Derived: -ord_less(int,one_one(int),A) | ord_less(nat,B,C) | -ord_less(int,power_power(int,A,B),power_power(int,A,C)). [resolve(656,a,636,a)]. 0.92/1.09 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(656,a,637,a)]. 0.92/1.09 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(656,a,638,a)]. 0.92/1.09 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(656,a,640,a)]. 0.92/1.09 Derived: -ord_less(int,one_one(int),A) | ord_less(int,one_one(int),times_times(int,A,power_power(int,A,B))). [resolve(656,a,641,a)]. 0.92/1.09 Derived: -ord_less(int,zero_zero(int),A) | ord_less(int,zero_zero(int),power_power(int,A,B)). [resolve(656,a,643,a)]. 0.92/1.09 Derived: -ord_less_eq(int,zero_zero(int),A) | ord_less_eq(int,zero_zero(int),power_power(int,A,B)). [resolve(656,a,644,a)]. 0.92/1.09 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(656,a,645,a)]. 0.92/1.09 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(656,a,646,a)]. 0.92/1.09 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(656,a,647,a)]. 0.92/1.09 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(656,a,648,a)]. 0.92/1.09 Derived: -ord_less_eq(int,one_one(int),A) | ord_less_eq(int,one_one(int),power_power(int,A,B)). [resolve(656,a,649,a)]. 0.92/1.09 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(656,a,651,a)]. 0.92/1.09 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(656,a,652,a)]. 0.92/1.09 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(656,a,653,a)]. 0.92/1.09 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(656,a,654,a)]. 0.92/1.09 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(656,a,655,a)]. 0.92/1.12 657 -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(445)]. 0.92/1.12 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(657,a,634,a)]. 0.92/1.12 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(657,a,639,a)]. 0.92/1.12 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(657,a,656,a)]. 0.92/1.12 658 -linordered_semidom(A) | -ord_less(A,one_one(A),B) | -ord_less_eq(nat,C,D) | ord_less_eq(A,power_power(A,B,C),power_power(A,B,D)) # label(fact_177_power__increasing__iff) # label(axiom). [clausify(452)]. 0.92/1.12 Derived: -ord_less(nat,one_one(nat),A) | -ord_less_eq(nat,B,C) | ord_less_eq(nat,power_power(nat,A,B),power_power(nat,A,C)). [resolve(658,a,634,a)]. 0.92/1.12 Derived: -ord_less(real,one_one(real),A) | -ord_less_eq(nat,B,C) | ord_less_eq(real,power_power(real,A,B),power_power(real,A,C)). [resolve(658,a,639,a)]. 0.92/1.12 Derived: -ord_less(int,one_one(int),A) | -ord_less_eq(nat,B,C) | ord_less_eq(int,power_power(int,A,B),power_power(int,A,C)). [resolve(658,a,656,a)]. 0.92/1.12 659 -linordered_semidom(A) | -ord_less(A,one_one(A),B) | ord_less_eq(nat,C,D) | -ord_less_eq(A,power_power(A,B,C),power_power(A,B,D)) # label(fact_177_power__increasing__iff) # label(axiom). [clausify(452)]. 0.92/1.12 660 -linordered_semidom(A) | -ord_less(A,one_one(A),B) | C != D | power_power(A,B,C) = power_power(A,B,D) # label(fact_270_power__inject__exp) # label(axiom). [clausify(461)]. 0.92/1.12 Derived: -ord_less(nat,one_one(nat),A) | B != C | power_power(nat,A,B) = power_power(nat,A,C). [resolve(660,a,634,a)]. 0.92/1.12 Derived: -ord_less(real,one_one(real),A) | B != C | power_power(real,A,B) = power_power(real,A,C). [resolve(660,a,639,a)]. 0.92/1.12 Derived: -ord_less(int,one_one(int),A) | B != C | power_power(int,A,B) = power_power(int,A,C). [resolve(660,a,656,a)]. 0.92/1.12 661 -linordered_semidom(A) | -ord_less(A,one_one(A),B) | C = D | power_power(A,B,C) != power_power(A,B,D) # label(fact_270_power__inject__exp) # label(axiom). [clausify(461)]. 0.92/1.12 Derived: -ord_less(nat,one_one(nat),A) | B = C | power_power(nat,A,B) != power_power(nat,A,C). [resolve(661,a,634,a)]. 0.92/1.12 Derived: -ord_less(real,one_one(real),A) | B = C | power_power(real,A,B) != power_power(real,A,C). [resolve(661,a,639,a)]. 0.92/1.12 Derived: -ord_less(int,one_one(int),A) | B = C | power_power(int,A,B) != power_power(int,A,C). [resolve(661,a,656,a)]. 0.92/1.12 662 -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(495)]. 0.92/1.12 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(662,a,634,a)]. 0.92/1.12 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(662,a,639,a)]. 0.92/1.12 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(662,a,656,a)]. 0.92/1.12 663 -linord219039673up_add(A) | ti(A,B) != zero_zero(A) | zero_zero(A) = plus_plus(A,B,B) # label(fact_201_double__eq__0__iff) # label(axiom). [clausify(225)]. 0.92/1.12 664 linord219039673up_add(int) # label(arity_Int_Oint___Groups_Olinordered__ab__group__add) # label(axiom). [assumption]. 0.92/1.12 Derived: ti(int,A) != zero_zero(int) | zero_zero(int) = plus_plus(int,A,A). [resolve(663,a,664,a)]. 0.92/1.12 665 -linord219039673up_add(A) | ti(A,B) = zero_zero(A) | zero_zero(A) != plus_plus(A,B,B) # label(fact_201_double__eq__0__iff) # label(axiom). [clausify(225)]. 0.92/1.12 Derived: ti(int,A) = zero_zero(int) | zero_zero(int) != plus_plus(int,A,A). [resolve(665,a,664,a)]. 0.92/1.12 666 linord219039673up_add(real) # label(arity_RealDef_Oreal___Groups_Olinordered__ab__group__add) # label(axiom). [assumption]. 1.04/1.22 Derived: ti(real,A) != zero_zero(real) | zero_zero(real) = plus_plus(real,A,A). [resolve(666,a,663,a)]. 1.04/1.22 Derived: ti(real,A) = zero_zero(real) | zero_zero(real) != plus_plus(real,A,A). [resolve(666,a,665,a)]. 1.04/1.22 667 -linord219039673up_add(A) | ti(A,zero_zero(A)) = zero_zero(A) # label(tsy_c_Groups_Ozero__class_Ozero_3_res) # label(axiom). [clausify(270)]. 1.04/1.22 668 -number(A) | -linorder(A) | ord_less(A,number_number_of(A,B),number_number_of(A,C)) | ord_less_eq(A,number_number_of(A,C),number_number_of(A,B)) # label(fact_29_le__number__of__eq__not__less) # label(axiom). [clausify(352)]. 1.04/1.22 669 linorder(nat) # label(arity_Nat_Onat___Orderings_Olinorder) # label(axiom). [assumption]. 1.04/1.22 670 linorder(real) # label(arity_RealDef_Oreal___Orderings_Olinorder) # label(axiom). [assumption]. 1.04/1.22 Derived: -number(nat) | ord_less(nat,number_number_of(nat,A),number_number_of(nat,B)) | ord_less_eq(nat,number_number_of(nat,B),number_number_of(nat,A)). [resolve(668,b,669,a)]. 1.04/1.22 Derived: -number(real) | ord_less(real,number_number_of(real,A),number_number_of(real,B)) | ord_less_eq(real,number_number_of(real,B),number_number_of(real,A)). [resolve(668,b,670,a)]. 1.04/1.22 671 -number(A) | -linorder(A) | -ord_less(A,number_number_of(A,B),number_number_of(A,C)) | -ord_less_eq(A,number_number_of(A,C),number_number_of(A,B)) # label(fact_29_le__number__of__eq__not__less) # label(axiom). [clausify(352)]. 1.04/1.22 Derived: -number(nat) | -ord_less(nat,number_number_of(nat,A),number_number_of(nat,B)) | -ord_less_eq(nat,number_number_of(nat,B),number_number_of(nat,A)). [resolve(671,b,669,a)]. 1.04/1.22 Derived: -number(real) | -ord_less(real,number_number_of(real,A),number_number_of(real,B)) | -ord_less_eq(real,number_number_of(real,B),number_number_of(real,A)). [resolve(671,b,670,a)]. 1.04/1.22 672 linorder(int) # label(arity_Int_Oint___Orderings_Olinorder) # label(axiom). [assumption]. 1.04/1.22 Derived: -number(int) | ord_less(int,number_number_of(int,A),number_number_of(int,B)) | ord_less_eq(int,number_number_of(int,B),number_number_of(int,A)). [resolve(672,a,668,b)]. 1.04/1.22 Derived: -number(int) | -ord_less(int,number_number_of(int,A),number_number_of(int,B)) | -ord_less_eq(int,number_number_of(int,B),number_number_of(int,A)). [resolve(672,a,671,b)]. 1.04/1.22 673 -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(36)]. 1.04/1.22 674 semiring_1(real) # label(arity_RealDef_Oreal___Rings_Osemiring__1) # label(axiom). [assumption]. 1.04/1.22 Derived: one_one(real) = power_power(real,one_one(real),number_number_of(nat,bit0(bit1(pls)))). [resolve(673,a,674,a)]. 1.04/1.22 675 -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(60)]. 1.04/1.22 Derived: zero_zero(real) = power_power(real,zero_zero(real),number_number_of(nat,bit0(bit1(pls)))). [resolve(675,a,674,a)]. 1.04/1.22 676 semiring_1(nat) # label(arity_Nat_Onat___Rings_Osemiring__1) # label(axiom). [assumption]. 1.04/1.22 Derived: one_one(nat) = power_power(nat,one_one(nat),number_number_of(nat,bit0(bit1(pls)))). [resolve(676,a,673,a)]. 1.04/1.22 Derived: zero_zero(nat) = power_power(nat,zero_zero(nat),number_number_of(nat,bit0(bit1(pls)))). [resolve(676,a,675,a)]. 1.04/1.22 677 semiring_1(int) # label(arity_Int_Oint___Rings_Osemiring__1) # label(axiom). [assumption]. 1.04/1.22 Derived: one_one(int) = power_power(int,one_one(int),number_number_of(nat,bit0(bit1(pls)))). [resolve(677,a,673,a)]. 1.04/1.22 Derived: zero_zero(int) = power_power(int,zero_zero(int),number_number_of(nat,bit0(bit1(pls)))). [resolve(677,a,675,a)]. 1.04/1.22 678 -comm_semiring(A) | times_times(A,plus_plus(A,B,C),D) = plus_plus(A,times_times(A,B,D),times_times(A,C,D)) # label(fact_476_comm__semiring__class_Odistrib) # label(axiom). [clausify(170)]. 1.04/1.22 679 comm_semiring(real) # label(arity_RealDef_Oreal___Rings_Ocomm__semiring) # label(axiom). [assumption]. 1.04/1.22 680 comm_semiring(nat) # label(arity_Nat_Onat___Rings_Ocomm__semiring) # label(axiom). [assumption]. 1.04/1.22 681 comm_semiring(int) # label(arity_Int_Oint___Rings_Ocomm__semiring) # label(axiom). [assumption]. 1.09/1.29 682 monoid_mult(nat) # label(arity_Nat_Onat___Groups_Omonoid__mult) # label(axiom). [assumption]. 1.09/1.29 683 -monoid_mult(A) | one_one(A) = power_power(A,one_one(A),B) # label(fact_260_power__one) # label(axiom). [clausify(44)]. 1.09/1.29 684 -number(A) | -monoid_mult(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(50)]. 1.09/1.29 685 -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(130)]. 1.09/1.29 686 -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(200)]. 1.09/1.29 687 -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(250)]. 1.09/1.29 688 -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(261)]. 1.09/1.29 689 -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(332)]. 1.09/1.29 Derived: one_one(nat) = power_power(nat,one_one(nat),A). [resolve(682,a,683,a)]. 1.09/1.29 Derived: times_times(nat,times_times(nat,A,A),A) = power_power(nat,A,number_number_of(nat,bit1(bit1(pls)))). [resolve(682,a,687,a)]. 1.09/1.29 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(682,a,688,a)]. 1.09/1.29 690 -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(355)]. 1.09/1.29 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(690,a,682,a)]. 1.09/1.29 691 -monoid_mult(A) | ti(A,B) = power_power(A,B,one_one(nat)) # label(fact_240_power__one__right) # label(axiom). [clausify(378)]. 1.09/1.29 692 monoid_mult(real) # label(arity_RealDef_Oreal___Groups_Omonoid__mult) # label(axiom). [assumption]. 1.09/1.29 Derived: one_one(real) = power_power(real,one_one(real),A). [resolve(692,a,683,a)]. 1.09/1.29 Derived: times_times(real,times_times(real,A,A),A) = power_power(real,A,number_number_of(nat,bit1(bit1(pls)))). [resolve(692,a,687,a)]. 1.09/1.29 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(692,a,688,a)]. 1.09/1.29 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(692,a,690,a)]. 1.09/1.29 693 -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(481)]. 1.09/1.29 694 monoid_mult(int) # label(arity_Int_Oint___Groups_Omonoid__mult) # label(axiom). [assumption]. 1.09/1.29 Derived: one_one(int) = power_power(int,one_one(int),A). [resolve(694,a,683,a)]. 1.09/1.29 Derived: times_times(int,times_times(int,A,A),A) = power_power(int,A,number_number_of(nat,bit1(bit1(pls)))). [resolve(694,a,687,a)]. 1.09/1.29 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(694,a,688,a)]. 1.09/1.29 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(694,a,690,a)]. 1.09/1.29 695 -ring_11004092258visors(A) | zero_zero(A) != power_power(A,B,number_number_of(nat,bit0(bit1(pls)))) | ti(A,B) = zero_zero(A) # label(fact_250_zero__eq__power2) # label(axiom). [clausify(238)]. 1.09/1.33 696 ring_11004092258visors(real) # label(arity_RealDef_Oreal___Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 1.09/1.33 697 ring_11004092258visors(int) # label(arity_Int_Oint___Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 1.09/1.33 Derived: zero_zero(real) != power_power(real,A,number_number_of(nat,bit0(bit1(pls)))) | ti(real,A) = zero_zero(real). [resolve(695,a,696,a)]. 1.09/1.33 Derived: zero_zero(int) != power_power(int,A,number_number_of(nat,bit0(bit1(pls)))) | ti(int,A) = zero_zero(int). [resolve(695,a,697,a)]. 1.09/1.33 698 -ring_11004092258visors(A) | zero_zero(A) = power_power(A,B,number_number_of(nat,bit0(bit1(pls)))) | ti(A,B) != zero_zero(A) # label(fact_250_zero__eq__power2) # label(axiom). [clausify(238)]. 1.09/1.33 Derived: zero_zero(real) = power_power(real,A,number_number_of(nat,bit0(bit1(pls)))) | ti(real,A) != zero_zero(real). [resolve(698,a,696,a)]. 1.09/1.33 Derived: zero_zero(int) = power_power(int,A,number_number_of(nat,bit0(bit1(pls)))) | ti(int,A) != zero_zero(int). [resolve(698,a,697,a)]. 1.09/1.33 699 -ring_11004092258visors(A) | ti(A,B) = zero_zero(A) | zero_zero(A) != power_power(A,B,C) # label(fact_192_field__power__not__zero) # label(axiom). [clausify(473)]. 1.09/1.33 Derived: ti(real,A) = zero_zero(real) | zero_zero(real) != power_power(real,A,B). [resolve(699,a,696,a)]. 1.09/1.33 Derived: ti(int,A) = zero_zero(int) | zero_zero(int) != power_power(int,A,B). [resolve(699,a,697,a)]. 1.09/1.33 700 order(int) # label(arity_Int_Oint___Orderings_Oorder) # label(axiom). [assumption]. 1.09/1.33 701 -order(A) | -ord_less_eq(A,B,C) | ord_less_eq(A,B,ti(A,C)) # label(tsy_c_Orderings_Oord__class_Oless__eq_arg2) # label(axiom). [clausify(48)]. 1.09/1.33 702 -order(A) | ord_less_eq(A,B,C) | -ord_less_eq(A,B,ti(A,C)) # label(tsy_c_Orderings_Oord__class_Oless__eq_arg2) # label(axiom). [clausify(48)]. 1.09/1.33 703 -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(136)]. 1.09/1.33 704 -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(138)]. 1.09/1.33 705 -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(138)]. 1.09/1.33 706 -order(A) | -ord_less(A,B,C) | ord_less(A,ti(A,B),C) # label(tsy_c_Orderings_Oord__class_Oless_arg1) # label(axiom). [clausify(202)]. 1.09/1.33 707 -order(A) | ord_less(A,B,C) | -ord_less(A,ti(A,B),C) # label(tsy_c_Orderings_Oord__class_Oless_arg1) # label(axiom). [clausify(202)]. 1.09/1.33 Derived: -ord_less_eq(int,A,B) | ord_less_eq(int,A,ti(int,B)). [resolve(700,a,701,a)]. 1.09/1.33 Derived: ord_less_eq(int,A,B) | -ord_less_eq(int,A,ti(int,B)). [resolve(700,a,702,a)]. 1.09/1.33 Derived: -ord_less_eq(int,A,B) | ti(int,B) = ti(int,A) | ord_less(int,A,B). [resolve(700,a,703,a)]. 1.09/1.33 Derived: -ord_less(int,A,B) | ord_less(int,A,ti(int,B)). [resolve(700,a,704,a)]. 1.09/1.33 Derived: ord_less(int,A,B) | -ord_less(int,A,ti(int,B)). [resolve(700,a,705,a)]. 1.09/1.33 Derived: -ord_less(int,A,B) | ord_less(int,ti(int,A),B). [resolve(700,a,706,a)]. 1.09/1.33 Derived: ord_less(int,A,B) | -ord_less(int,ti(int,A),B). [resolve(700,a,707,a)]. 1.09/1.33 708 order(bool) # label(arity_HOL_Obool___Orderings_Oorder) # label(axiom). [assumption]. 1.09/1.33 Derived: -ord_less_eq(bool,A,B) | ord_less_eq(bool,A,ti(bool,B)). [resolve(708,a,701,a)]. 1.09/1.33 Derived: ord_less_eq(bool,A,B) | -ord_less_eq(bool,A,ti(bool,B)). [resolve(708,a,702,a)]. 1.09/1.33 Derived: -ord_less_eq(bool,A,B) | ti(bool,B) = ti(bool,A) | ord_less(bool,A,B). [resolve(708,a,703,a)]. 1.09/1.33 Derived: -ord_less(bool,A,B) | ord_less(bool,A,ti(bool,B)). [resolve(708,a,704,a)]. 1.09/1.33 Derived: ord_less(bool,A,B) | -ord_less(bool,A,ti(bool,B)). [resolve(708,a,705,a)]. 1.09/1.33 Derived: -ord_less(bool,A,B) | ord_less(bool,ti(bool,A),B). [resolve(708,a,706,a)]. 1.09/1.33 Derived: ord_less(bool,A,B) | -ord_less(bool,ti(bool,A),B). [resolve(708,a,707,a)]. 1.09/1.33 709 order(real) # label(arity_RealDef_Oreal___Orderings_Oorder) # label(axiom). [assumption]. 1.09/1.33 Derived: -ord_less_eq(real,A,B) | ord_less_eq(real,A,ti(real,B)). [resolve(709,a,701,a)]. 1.09/1.33 Derived: ord_less_eq(real,A,B) | -ord_less_eq(real,A,ti(real,B)). [resolve(709,a,702,a)]. 1.21/1.40 Derived: -ord_less_eq(real,A,B) | ti(real,B) = ti(real,A) | ord_less(real,A,B). [resolve(709,a,703,a)]. 1.21/1.40 Derived: -ord_less(real,A,B) | ord_less(real,A,ti(real,B)). [resolve(709,a,704,a)]. 1.21/1.40 Derived: ord_less(real,A,B) | -ord_less(real,A,ti(real,B)). [resolve(709,a,705,a)]. 1.21/1.40 Derived: -ord_less(real,A,B) | ord_less(real,ti(real,A),B). [resolve(709,a,706,a)]. 1.21/1.40 Derived: ord_less(real,A,B) | -ord_less(real,ti(real,A),B). [resolve(709,a,707,a)]. 1.21/1.40 710 order(nat) # label(arity_Nat_Onat___Orderings_Oorder) # label(axiom). [assumption]. 1.21/1.40 Derived: -ord_less_eq(nat,A,B) | ord_less_eq(nat,A,ti(nat,B)). [resolve(710,a,701,a)]. 1.21/1.40 Derived: ord_less_eq(nat,A,B) | -ord_less_eq(nat,A,ti(nat,B)). [resolve(710,a,702,a)]. 1.21/1.40 Derived: -ord_less_eq(nat,A,B) | ti(nat,B) = ti(nat,A) | ord_less(nat,A,B). [resolve(710,a,703,a)]. 1.21/1.40 Derived: -ord_less(nat,A,B) | ord_less(nat,A,ti(nat,B)). [resolve(710,a,704,a)]. 1.21/1.40 Derived: ord_less(nat,A,B) | -ord_less(nat,A,ti(nat,B)). [resolve(710,a,705,a)]. 1.21/1.40 Derived: -ord_less(nat,A,B) | ord_less(nat,ti(nat,A),B). [resolve(710,a,706,a)]. 1.21/1.40 Derived: ord_less(nat,A,B) | -ord_less(nat,ti(nat,A),B). [resolve(710,a,707,a)]. 1.21/1.40 711 -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(508)]. 1.21/1.40 Derived: -ord_less_eq(int,A,B) | ord_less_eq(int,ti(int,A),B). [resolve(711,a,700,a)]. 1.21/1.40 Derived: -ord_less_eq(bool,A,B) | ord_less_eq(bool,ti(bool,A),B). [resolve(711,a,708,a)]. 1.21/1.40 Derived: -ord_less_eq(real,A,B) | ord_less_eq(real,ti(real,A),B). [resolve(711,a,709,a)]. 1.21/1.40 Derived: -ord_less_eq(nat,A,B) | ord_less_eq(nat,ti(nat,A),B). [resolve(711,a,710,a)]. 1.21/1.40 712 -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(508)]. 1.21/1.40 Derived: ord_less_eq(int,A,B) | -ord_less_eq(int,ti(int,A),B). [resolve(712,a,700,a)]. 1.21/1.40 Derived: ord_less_eq(bool,A,B) | -ord_less_eq(bool,ti(bool,A),B). [resolve(712,a,708,a)]. 1.21/1.40 Derived: ord_less_eq(real,A,B) | -ord_less_eq(real,ti(real,A),B). [resolve(712,a,709,a)]. 1.21/1.40 Derived: ord_less_eq(nat,A,B) | -ord_less_eq(nat,ti(nat,A),B). [resolve(712,a,710,a)]. 1.21/1.40 713 linord626643107strict(int) # label(arity_Int_Oint___Rings_Olinordered__semiring__1__strict) # label(axiom). [assumption]. 1.21/1.40 714 -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) | one_one(A) != plus_plus(A,E,F) | 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(53)]. 1.21/1.40 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) | one_one(int) != plus_plus(int,D,E) | ord_less(int,plus_plus(int,times_times(int,D,A),times_times(int,E,C)),B). [resolve(713,a,714,a)]. 1.21/1.40 715 linord626643107strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semiring__1__strict) # label(axiom). [assumption]. 1.21/1.40 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) | one_one(real) != plus_plus(real,D,E) | ord_less(real,plus_plus(real,times_times(real,D,A),times_times(real,E,C)),B). [resolve(715,a,714,a)]. 1.21/1.40 716 ordere453448008miring(int) # label(arity_Int_Oint___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 1.21/1.40 717 -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)]. 1.21/1.40 718 -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(120)]. 1.21/1.40 Derived: -ord_less_eq(int,A,zero_zero(int)) | -ord_less_eq(int,zero_zero(int),B) | ord_less_eq(int,times_times(int,A,B),zero_zero(int)). [resolve(716,a,717,a)]. 1.21/1.40 719 ordere453448008miring(nat) # label(arity_Nat_Onat___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 1.21/1.43 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(719,a,717,a)]. 1.21/1.43 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(719,a,718,a)]. 1.21/1.43 720 ordere453448008miring(real) # label(arity_RealDef_Oreal___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 1.21/1.43 Derived: -ord_less_eq(real,A,zero_zero(real)) | -ord_less_eq(real,zero_zero(real),B) | ord_less_eq(real,times_times(real,A,B),zero_zero(real)). [resolve(720,a,717,a)]. 1.21/1.43 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,A,B)). [resolve(720,a,718,a)]. 1.21/1.43 721 -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(450)]. 1.21/1.43 Derived: -ord_less_eq(int,zero_zero(int),A) | -ord_less_eq(int,B,zero_zero(int)) | ord_less_eq(int,times_times(int,A,B),zero_zero(int)). [resolve(721,a,716,a)]. 1.21/1.43 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(721,a,719,a)]. 1.21/1.43 Derived: -ord_less_eq(real,zero_zero(real),A) | -ord_less_eq(real,B,zero_zero(real)) | ord_less_eq(real,times_times(real,A,B),zero_zero(real)). [resolve(721,a,720,a)]. 1.21/1.43 722 -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(477)]. 1.21/1.43 723 linord581940658strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__ring__strict) # label(axiom). [assumption]. 1.21/1.43 724 -linord581940658strict(A) | ti(A,B) != zero_zero(A) | ti(A,C) != zero_zero(A) | ord_less_eq(A,plus_plus(A,times_times(A,B,B),times_times(A,C,C)),zero_zero(A)) # label(fact_234_sum__squares__le__zero__iff) # label(axiom). [clausify(69)]. 1.21/1.43 725 -linord581940658strict(A) | ti(A,B) = zero_zero(A) | -ord_less_eq(A,plus_plus(A,times_times(A,B,B),times_times(A,C,C)),zero_zero(A)) # label(fact_234_sum__squares__le__zero__iff) # label(axiom). [clausify(69)]. 1.21/1.43 726 -linord581940658strict(A) | ti(A,B) = zero_zero(A) | -ord_less_eq(A,plus_plus(A,times_times(A,C,C),times_times(A,B,B)),zero_zero(A)) # label(fact_234_sum__squares__le__zero__iff) # label(axiom). [clausify(69)]. 1.21/1.43 727 -linord581940658strict(A) | -ord_less_eq(A,zero_zero(A),times_times(A,B,C)) | ord_less_eq(A,zero_zero(A),C) | ord_less_eq(A,B,zero_zero(A)) # label(fact_491_zero__le__mult__iff) # label(axiom). [clausify(98)]. 1.21/1.43 728 -linord581940658strict(A) | -ord_less_eq(A,zero_zero(A),times_times(A,B,C)) | ord_less_eq(A,zero_zero(A),C) | ord_less_eq(A,C,zero_zero(A)) # label(fact_491_zero__le__mult__iff) # label(axiom). [clausify(98)]. 1.21/1.43 729 -linord581940658strict(A) | -ord_less_eq(A,zero_zero(A),times_times(A,B,C)) | ord_less_eq(A,zero_zero(A),B) | ord_less_eq(A,B,zero_zero(A)) # label(fact_491_zero__le__mult__iff) # label(axiom). [clausify(98)]. 1.21/1.43 730 -linord581940658strict(A) | -ord_less_eq(A,zero_zero(A),times_times(A,B,C)) | ord_less_eq(A,zero_zero(A),B) | ord_less_eq(A,C,zero_zero(A)) # label(fact_491_zero__le__mult__iff) # label(axiom). [clausify(98)]. 1.21/1.43 731 -linord581940658strict(A) | ord_less_eq(A,zero_zero(A),times_times(A,B,C)) | -ord_less_eq(A,zero_zero(A),C) | -ord_less_eq(A,zero_zero(A),B) # label(fact_491_zero__le__mult__iff) # label(axiom). [clausify(98)]. 1.21/1.43 732 -linord581940658strict(A) | ord_less_eq(A,zero_zero(A),times_times(A,B,C)) | -ord_less_eq(A,B,zero_zero(A)) | -ord_less_eq(A,C,zero_zero(A)) # label(fact_491_zero__le__mult__iff) # label(axiom). [clausify(98)]. 1.21/1.43 Derived: ti(real,A) != zero_zero(real) | ti(real,B) != zero_zero(real) | ord_less_eq(real,plus_plus(real,times_times(real,A,A),times_times(real,B,B)),zero_zero(real)). [resolve(723,a,724,a)]. 1.21/1.43 Derived: ti(real,A) = zero_zero(real) | -ord_less_eq(real,plus_plus(real,times_times(real,A,A),times_times(real,B,B)),zero_zero(real)). [resolve(723,a,725,a)]. 1.27/1.44 Derived: ti(real,A) = zero_zero(real) | -ord_less_eq(real,plus_plus(real,times_times(real,B,B),times_times(real,A,A)),zero_zero(real)). [resolve(723,a,726,a)]. 1.27/1.44 Derived: -ord_less_eq(real,zero_zero(real),times_times(real,A,B)) | ord_less_eq(real,zero_zero(real),B) | ord_less_eq(real,A,zero_zero(real)). [resolve(723,a,727,a)]. 1.27/1.44 Derived: -ord_less_eq(real,zero_zero(real),times_times(real,A,B)) | ord_less_eq(real,zero_zero(real),B) | ord_less_eq(real,B,zero_zero(real)). [resolve(723,a,728,a)]. 1.27/1.44 Derived: -ord_less_eq(real,zero_zero(real),times_times(real,A,B)) | ord_less_eq(real,zero_zero(real),A) | ord_less_eq(real,A,zero_zero(real)). [resolve(723,a,729,a)]. 1.27/1.44 Derived: -ord_less_eq(real,zero_zero(real),times_times(real,A,B)) | ord_less_eq(real,zero_zero(real),A) | ord_less_eq(real,B,zero_zero(real)). [resolve(723,a,730,a)]. 1.27/1.44 Derived: ord_less_eq(real,zero_zero(real),times_times(real,A,B)) | -ord_less_eq(real,A,zero_zero(real)) | -ord_less_eq(real,B,zero_zero(real)). [resolve(723,a,732,a)]. 1.27/1.44 733 -linord581940658strict(A) | ti(A,B) = zero_zero(A) | ord_less(A,zero_zero(A),plus_plus(A,times_times(A,C,C),times_times(A,B,B))) # label(fact_237_sum__squares__gt__zero__iff) # label(axiom). [clausify(362)]. 1.27/1.44 Derived: ti(real,A) = zero_zero(real) | ord_less(real,zero_zero(real),plus_plus(real,times_times(real,B,B),times_times(real,A,A))). [resolve(733,a,723,a)]. 1.27/1.44 734 -linord581940658strict(A) | ti(A,B) = zero_zero(A) | ord_less(A,zero_zero(A),plus_plus(A,times_times(A,B,B),times_times(A,C,C))) # label(fact_237_sum__squares__gt__zero__iff) # label(axiom). [clausify(362)]. 1.27/1.44 Derived: ti(real,A) = zero_zero(real) | ord_less(real,zero_zero(real),plus_plus(real,times_times(real,A,A),times_times(real,B,B))). [resolve(734,a,723,a)]. 1.27/1.44 735 -linord581940658strict(A) | ti(A,B) != zero_zero(A) | ti(A,C) != zero_zero(A) | -ord_less(A,zero_zero(A),plus_plus(A,times_times(A,C,C),times_times(A,B,B))) # label(fact_237_sum__squares__gt__zero__iff) # label(axiom). [clausify(362)]. 1.27/1.44 Derived: ti(real,A) != zero_zero(real) | ti(real,B) != zero_zero(real) | -ord_less(real,zero_zero(real),plus_plus(real,times_times(real,B,B),times_times(real,A,A))). [resolve(735,a,723,a)]. 1.27/1.44 736 linord581940658strict(int) # label(arity_Int_Oint___Rings_Olinordered__ring__strict) # label(axiom). [assumption]. 1.27/1.44 Derived: ti(int,A) != zero_zero(int) | ti(int,B) != zero_zero(int) | ord_less_eq(int,plus_plus(int,times_times(int,A,A),times_times(int,B,B)),zero_zero(int)). [resolve(736,a,724,a)]. 1.27/1.44 Derived: ti(int,A) = zero_zero(int) | -ord_less_eq(int,plus_plus(int,times_times(int,A,A),times_times(int,B,B)),zero_zero(int)). [resolve(736,a,725,a)]. 1.27/1.44 Derived: ti(int,A) = zero_zero(int) | -ord_less_eq(int,plus_plus(int,times_times(int,B,B),times_times(int,A,A)),zero_zero(int)). [resolve(736,a,726,a)]. 1.27/1.44 Derived: -ord_less_eq(int,zero_zero(int),times_times(int,A,B)) | ord_less_eq(int,zero_zero(int),B) | ord_less_eq(int,A,zero_zero(int)). [resolve(736,a,727,a)]. 1.27/1.44 Derived: -ord_less_eq(int,zero_zero(int),times_times(int,A,B)) | ord_less_eq(int,zero_zero(int),B) | ord_less_eq(int,B,zero_zero(int)). [resolve(736,a,728,a)]. 1.27/1.44 Derived: -ord_less_eq(int,zero_zero(int),times_times(int,A,B)) | ord_less_eq(int,zero_zero(int),A) | ord_less_eq(int,A,zero_zero(int)). [resolve(736,a,729,a)]. 1.27/1.44 Derived: -ord_less_eq(int,zero_zero(int),times_times(int,A,B)) | ord_less_eq(int,zero_zero(int),A) | ord_less_eq(int,B,zero_zero(int)). [resolve(736,a,730,a)]. 1.27/1.44 Derived: ord_less_eq(int,zero_zero(int),times_times(int,A,B)) | -ord_less_eq(int,zero_zero(int),B) | -ord_less_eq(int,zero_zero(int),A). [resolve(736,a,731,a)]. 1.27/1.44 Derived: ord_less_eq(int,zero_zero(int),times_times(int,A,B)) | -ord_less_eq(int,A,zero_zero(int)) | -ord_less_eq(int,B,zero_zero(int)). [resolve(736,a,732,a)]. 1.27/1.44 Derived: ti(int,A) = zero_zero(int) | ord_less(int,zero_zero(int),plus_plus(int,times_times(int,B,B),times_times(int,A,A))). [resolve(736,a,733,a)]. 1.27/1.44 Derived: ti(int,A) = zero_zero(int) | ord_less(int,zero_zero(int),plus_plus(int,times_times(int,A,A),times_times(int,B,B))). [resolve(736,a,734,a)]. 1.27/1.45 Derived: ti(int,A) != zero_zero(int) | ti(int,B) != zero_zero(int) | -ord_less(int,zero_zero(int),plus_plus(int,times_times(int,B,B),times_times(int,A,A))). [resolve(736,a,735,a)]. 1.27/1.45 737 -linord581940658strict(A) | zero_zero(A) != plus_plus(A,times_times(A,B,B),times_times(A,C,C)) | ti(A,B) = zero_zero(A) # label(fact_216_sum__squares__eq__zero__iff) # label(axiom). [clausify(423)]. 1.27/1.45 Derived: zero_zero(real) != plus_plus(real,times_times(real,A,A),times_times(real,B,B)) | ti(real,A) = zero_zero(real). [resolve(737,a,723,a)]. 1.27/1.45 Derived: zero_zero(int) != plus_plus(int,times_times(int,A,A),times_times(int,B,B)) | ti(int,A) = zero_zero(int). [resolve(737,a,736,a)]. 1.27/1.45 738 -linord581940658strict(A) | zero_zero(A) != plus_plus(A,times_times(A,B,B),times_times(A,C,C)) | ti(A,C) = zero_zero(A) # label(fact_216_sum__squares__eq__zero__iff) # label(axiom). [clausify(423)]. 1.27/1.45 Derived: zero_zero(real) != plus_plus(real,times_times(real,A,A),times_times(real,B,B)) | ti(real,B) = zero_zero(real). [resolve(738,a,723,a)]. 1.27/1.45 Derived: zero_zero(int) != plus_plus(int,times_times(int,A,A),times_times(int,B,B)) | ti(int,B) = zero_zero(int). [resolve(738,a,736,a)]. 1.27/1.45 739 -linord581940658strict(A) | zero_zero(A) = plus_plus(A,times_times(A,B,B),times_times(A,C,C)) | ti(A,B) != zero_zero(A) | ti(A,C) != zero_zero(A) # label(fact_216_sum__squares__eq__zero__iff) # label(axiom). [clausify(423)]. 1.27/1.45 Derived: zero_zero(real) = plus_plus(real,times_times(real,A,A),times_times(real,B,B)) | ti(real,A) != zero_zero(real) | ti(real,B) != zero_zero(real). [resolve(739,a,723,a)]. 1.27/1.45 Derived: zero_zero(int) = plus_plus(int,times_times(int,A,A),times_times(int,B,B)) | ti(int,A) != zero_zero(int) | ti(int,B) != zero_zero(int). [resolve(739,a,736,a)]. 1.27/1.45 740 -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(512)]. 1.27/1.45 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(740,a,723,a)]. 1.27/1.45 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(740,a,736,a)]. 1.27/1.45 741 -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(512)]. 1.27/1.45 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,zero_zero(real),A). [resolve(741,a,723,a)]. 1.27/1.45 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(741,a,736,a)]. 1.27/1.45 742 -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(512)]. 1.27/1.45 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,B,zero_zero(real)). [resolve(742,a,723,a)]. 1.27/1.45 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(742,a,736,a)]. 1.27/1.45 743 -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(512)]. 1.27/1.45 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(743,a,723,a)]. 1.27/1.45 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(743,a,736,a)]. 1.27/1.45 744 -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),C) # label(fact_492_mult__le__0__iff) # label(axiom). [clausify(512)]. 1.34/1.51 745 -linord581940658strict(A) | ord_less_eq(A,times_times(A,B,C),zero_zero(A)) | -ord_less_eq(A,C,zero_zero(A)) | -ord_less_eq(A,zero_zero(A),B) # label(fact_492_mult__le__0__iff) # label(axiom). [clausify(512)]. 1.34/1.51 746 dvd(nat) # label(arity_Nat_Onat___Rings_Odvd) # label(axiom). [assumption]. 1.34/1.51 747 -dvd(A) | times_times(A,B,C) != D | dvd_dvd(A,B,D) # label(fact_484_dvdI) # label(axiom). [clausify(73)]. 1.34/1.51 Derived: times_times(nat,A,B) != C | dvd_dvd(nat,A,C). [resolve(746,a,747,a)]. 1.34/1.51 748 -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(173)]. 1.34/1.51 Derived: -dvd_dvd(nat,ti(nat,A),B) | dvd_dvd(nat,A,B). [resolve(748,a,746,a)]. 1.34/1.51 749 -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(173)]. 1.34/1.51 Derived: dvd_dvd(nat,ti(nat,A),B) | -dvd_dvd(nat,A,B). [resolve(749,a,746,a)]. 1.34/1.51 750 -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(229)]. 1.34/1.51 Derived: times_times(nat,A,ti(nat,B)) = times_times(nat,A,B). [resolve(750,a,746,a)]. 1.34/1.51 751 dvd(int) # label(arity_Int_Oint___Rings_Odvd) # label(axiom). [assumption]. 1.34/1.51 Derived: times_times(int,A,B) != C | dvd_dvd(int,A,C). [resolve(751,a,747,a)]. 1.34/1.51 Derived: -dvd_dvd(int,ti(int,A),B) | dvd_dvd(int,A,B). [resolve(751,a,748,a)]. 1.34/1.51 Derived: dvd_dvd(int,ti(int,A),B) | -dvd_dvd(int,A,B). [resolve(751,a,749,a)]. 1.34/1.51 Derived: times_times(int,A,ti(int,B)) = times_times(int,A,B). [resolve(751,a,750,a)]. 1.34/1.51 752 -dvd(A) | ti(A,times_times(A,B,C)) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_0_res) # label(axiom). [clausify(361)]. 1.34/1.51 Derived: ti(nat,times_times(nat,A,B)) = times_times(nat,A,B). [resolve(752,a,746,a)]. 1.34/1.51 Derived: ti(int,times_times(int,A,B)) = times_times(int,A,B). [resolve(752,a,751,a)]. 1.34/1.51 753 dvd(real) # label(arity_RealDef_Oreal___Rings_Odvd) # label(axiom). [assumption]. 1.34/1.51 Derived: times_times(real,A,B) != C | dvd_dvd(real,A,C). [resolve(753,a,747,a)]. 1.34/1.51 Derived: -dvd_dvd(real,ti(real,A),B) | dvd_dvd(real,A,B). [resolve(753,a,748,a)]. 1.34/1.51 Derived: dvd_dvd(real,ti(real,A),B) | -dvd_dvd(real,A,B). [resolve(753,a,749,a)]. 1.34/1.51 Derived: times_times(real,A,ti(real,B)) = times_times(real,A,B). [resolve(753,a,750,a)]. 1.34/1.51 Derived: ti(real,times_times(real,A,B)) = times_times(real,A,B). [resolve(753,a,752,a)]. 1.34/1.51 754 -dvd(A) | -dvd_dvd(A,B,C) | dvd_dvd(A,B,ti(A,C)) # label(tsy_c_Rings_Odvd__class_Odvd_arg2) # label(axiom). [clausify(439)]. 1.34/1.51 Derived: -dvd_dvd(nat,A,B) | dvd_dvd(nat,A,ti(nat,B)). [resolve(754,a,746,a)]. 1.34/1.51 Derived: -dvd_dvd(int,A,B) | dvd_dvd(int,A,ti(int,B)). [resolve(754,a,751,a)]. 1.34/1.51 Derived: -dvd_dvd(real,A,B) | dvd_dvd(real,A,ti(real,B)). [resolve(754,a,753,a)]. 1.34/1.51 755 -dvd(A) | dvd_dvd(A,B,C) | -dvd_dvd(A,B,ti(A,C)) # label(tsy_c_Rings_Odvd__class_Odvd_arg2) # label(axiom). [clausify(439)]. 1.34/1.51 Derived: dvd_dvd(nat,A,B) | -dvd_dvd(nat,A,ti(nat,B)). [resolve(755,a,746,a)]. 1.34/1.51 Derived: dvd_dvd(int,A,B) | -dvd_dvd(int,A,ti(int,B)). [resolve(755,a,751,a)]. 1.34/1.51 Derived: dvd_dvd(real,A,B) | -dvd_dvd(real,A,ti(real,B)). [resolve(755,a,753,a)]. 1.34/1.51 756 -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(460)]. 1.34/1.51 Derived: times_times(nat,ti(nat,A),B) = times_times(nat,A,B). [resolve(756,a,746,a)]. 1.34/1.51 Derived: times_times(int,ti(int,A),B) = times_times(int,A,B). [resolve(756,a,751,a)]. 1.34/1.51 Derived: times_times(real,ti(real,A),B) = times_times(real,A,B). [resolve(756,a,753,a)]. 1.34/1.51 757 linordered_ring(real) # label(arity_RealDef_Oreal___Rings_Olinordered__ring) # label(axiom). [assumption]. 1.34/1.51 758 -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(76)]. 1.34/1.51 759 -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(132)]. 1.34/1.51 Derived: ord_less_eq(real,zero_zero(real),plus_plus(real,times_times(real,A,A),times_times(real,B,B))). [resolve(757,a,758,a)]. 1.38/1.56 Derived: -ord_less(real,plus_plus(real,times_times(real,A,A),times_times(real,B,B)),zero_zero(real)). [resolve(757,a,759,a)]. 1.38/1.56 760 linordered_ring(int) # label(arity_Int_Oint___Rings_Olinordered__ring) # label(axiom). [assumption]. 1.38/1.56 Derived: ord_less_eq(int,zero_zero(int),plus_plus(int,times_times(int,A,A),times_times(int,B,B))). [resolve(760,a,758,a)]. 1.38/1.56 Derived: -ord_less(int,plus_plus(int,times_times(int,A,A),times_times(int,B,B)),zero_zero(int)). [resolve(760,a,759,a)]. 1.38/1.56 761 -linordered_ring(A) | ord_less_eq(A,zero_zero(A),times_times(A,B,B)) # label(fact_490_zero__le__square) # label(axiom). [clausify(443)]. 1.38/1.56 Derived: ord_less_eq(real,zero_zero(real),times_times(real,A,A)). [resolve(761,a,757,a)]. 1.38/1.56 Derived: ord_less_eq(int,zero_zero(int),times_times(int,A,A)). [resolve(761,a,760,a)]. 1.38/1.56 762 number_semiring(real) # label(arity_RealDef_Oreal___Int_Onumber__semiring) # label(axiom). [assumption]. 1.38/1.56 763 -number_semiring(A) | -ord_less_eq(int,pls,B) | -ord_less_eq(int,pls,C) | number_number_of(A,plus_plus(int,B,C)) = plus_plus(A,number_number_of(A,B),number_number_of(A,C)) # label(fact_134_semiring__add__number__of) # label(axiom). [clausify(77)]. 1.38/1.56 764 -number_semiring(A) | zero_zero(A) = number_number_of(A,pls) # label(fact_221_semiring__numeral__0__eq__0) # label(axiom). [clausify(100)]. 1.38/1.56 765 -number_semiring(A) | times_times(A,number_number_of(A,bit0(bit1(pls))),B) = plus_plus(A,B,B) # label(fact_163_semiring__mult__2) # label(axiom). [clausify(243)]. 1.38/1.56 766 -number_semiring(A) | one_one(A) = number_number_of(A,bit1(pls)) # label(fact_155_semiring__numeral__1__eq__1) # label(axiom). [clausify(259)]. 1.38/1.56 767 number_semiring(int) # label(arity_Int_Oint___Int_Onumber__semiring) # label(axiom). [assumption]. 1.38/1.56 768 -number_semiring(A) | number_number_of(A,bit0(bit1(pls))) = plus_plus(A,one_one(A),one_one(A)) # label(fact_167_semiring__one__add__one__is__two) # label(axiom). [clausify(304)]. 1.38/1.56 769 -number_semiring(A) | times_times(A,B,number_number_of(A,bit0(bit1(pls)))) = plus_plus(A,B,B) # label(fact_165_semiring__mult__2__right) # label(axiom). [clausify(318)]. 1.38/1.56 770 -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(342)]. 1.38/1.56 771 number_semiring(nat) # label(arity_Nat_Onat___Int_Onumber__semiring) # label(axiom). [assumption]. 1.38/1.56 Derived: -ord_less_eq(int,pls,A) | -ord_less_eq(int,pls,B) | number_number_of(nat,plus_plus(int,A,B)) = plus_plus(nat,number_number_of(nat,A),number_number_of(nat,B)). [resolve(771,a,763,a)]. 1.38/1.56 Derived: one_one(nat) = number_number_of(nat,bit1(pls)). [resolve(771,a,766,a)]. 1.38/1.56 Derived: number_number_of(nat,bit0(bit1(pls))) = plus_plus(nat,one_one(nat),one_one(nat)). [resolve(771,a,768,a)]. 1.38/1.56 Derived: times_times(nat,A,number_number_of(nat,bit0(bit1(pls)))) = plus_plus(nat,A,A). [resolve(771,a,769,a)]. 1.38/1.56 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(771,a,770,a)]. 1.38/1.56 772 -number_semiring(A) | power_power(A,plus_plus(A,B,C),number_number_of(nat,bit0(bit1(pls)))) = 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)) # label(fact_9_power2__sum) # label(axiom). [clausify(410)]. 1.38/1.56 Derived: power_power(int,plus_plus(int,A,B),number_number_of(nat,bit0(bit1(pls)))) = 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)). [resolve(772,a,767,a)]. 1.38/1.56 Derived: power_power(nat,plus_plus(nat,A,B),number_number_of(nat,bit0(bit1(pls)))) = 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)). [resolve(772,a,771,a)]. 1.47/1.66 773 ring_1(real) # label(arity_RealDef_Oreal___Rings_Oring__1) # label(axiom). [assumption]. 1.47/1.66 774 -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(83)]. 1.47/1.66 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(773,a,774,a)]. 1.47/1.66 775 ring_1(int) # label(arity_Int_Oint___Rings_Oring__1) # label(axiom). [assumption]. 1.47/1.66 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(775,a,774,a)]. 1.47/1.66 776 -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(128)]. 1.47/1.66 777 semiring(nat) # label(arity_Nat_Onat___Rings_Osemiring) # label(axiom). [assumption]. 1.47/1.66 778 -semiring(A) | -number(A) | times_times(A,plus_plus(A,B,C),number_number_of(A,D)) = plus_plus(A,times_times(A,B,number_number_of(A,D)),times_times(A,C,number_number_of(A,D))) # label(fact_136_left__distrib__number__of) # label(axiom). [clausify(149)]. 1.47/1.66 779 -semiring(A) | ti(A,plus_plus(A,B,C)) = plus_plus(A,B,C) # label(tsy_c_Groups_Oplus__class_Oplus_0_res) # label(axiom). [clausify(184)]. 1.47/1.66 Derived: ti(nat,plus_plus(nat,A,B)) = plus_plus(nat,A,B). [resolve(779,a,777,a)]. 1.47/1.66 780 -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(189)]. 1.47/1.66 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(780,a,777,a)]. 1.47/1.66 781 -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(191)]. 1.47/1.66 782 semiring(real) # label(arity_RealDef_Oreal___Rings_Osemiring) # label(axiom). [assumption]. 1.47/1.66 Derived: ti(real,plus_plus(real,A,B)) = plus_plus(real,A,B). [resolve(782,a,779,a)]. 1.47/1.66 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(782,a,780,a)]. 1.47/1.66 783 -semiring(A) | -number(A) | times_times(A,number_number_of(A,B),plus_plus(A,C,D)) = plus_plus(A,times_times(A,number_number_of(A,B),C),times_times(A,number_number_of(A,B),D)) # label(fact_137_right__distrib__number__of) # label(axiom). [clausify(196)]. 1.47/1.66 784 -semiring(A) | ti(A,times_times(A,B,C)) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_1_res) # label(axiom). [clausify(417)]. 1.47/1.66 785 -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(453)]. 1.47/1.66 Derived: plus_plus(nat,ti(nat,A),B) = plus_plus(nat,A,B). [resolve(785,a,777,a)]. 1.47/1.66 Derived: plus_plus(real,ti(real,A),B) = plus_plus(real,A,B). [resolve(785,a,782,a)]. 1.47/1.66 786 -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(496)]. 1.47/1.66 Derived: plus_plus(nat,A,ti(nat,B)) = plus_plus(nat,A,B). [resolve(786,a,777,a)]. 1.47/1.66 Derived: plus_plus(real,A,ti(real,B)) = plus_plus(real,A,B). [resolve(786,a,782,a)]. 1.47/1.66 787 semiring(int) # label(arity_Int_Oint___Rings_Osemiring) # label(axiom). [assumption]. 1.47/1.66 Derived: ti(int,plus_plus(int,A,B)) = plus_plus(int,A,B). [resolve(787,a,779,a)]. 1.47/1.66 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(787,a,780,a)]. 1.47/1.66 Derived: plus_plus(int,ti(int,A),B) = plus_plus(int,A,B). [resolve(787,a,785,a)]. 1.47/1.66 Derived: plus_plus(int,A,ti(int,B)) = plus_plus(int,A,B). [resolve(787,a,786,a)]. 1.47/1.66 788 semiri456707255roduct(nat) # label(arity_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc) # label(axiom). [assumption]. 1.47/1.67 789 -semiri456707255roduct(A) | ti(A,B) != ti(A,C) | plus_plus(A,times_times(A,B,D),times_times(A,C,E)) = plus_plus(A,times_times(A,B,E),times_times(A,C,D)) # label(fact_106_crossproduct__eq) # label(axiom). [clausify(113)]. 1.47/1.67 790 -semiri456707255roduct(A) | ti(A,B) != ti(A,C) | plus_plus(A,times_times(A,D,B),times_times(A,E,C)) = plus_plus(A,times_times(A,D,C),times_times(A,E,B)) # label(fact_106_crossproduct__eq) # label(axiom). [clausify(113)]. 1.47/1.67 791 -semiri456707255roduct(A) | ti(A,B) = ti(A,C) | ti(A,D) = ti(A,E) | plus_plus(A,times_times(A,B,D),times_times(A,C,E)) != plus_plus(A,times_times(A,B,E),times_times(A,C,D)) # label(fact_106_crossproduct__eq) # label(axiom). [clausify(113)]. 1.47/1.67 Derived: ti(nat,A) != ti(nat,B) | plus_plus(nat,times_times(nat,A,C),times_times(nat,B,D)) = plus_plus(nat,times_times(nat,A,D),times_times(nat,B,C)). [resolve(788,a,789,a)]. 1.47/1.67 Derived: ti(nat,A) != ti(nat,B) | plus_plus(nat,times_times(nat,C,A),times_times(nat,D,B)) = plus_plus(nat,times_times(nat,C,B),times_times(nat,D,A)). [resolve(788,a,790,a)]. 1.47/1.67 Derived: ti(nat,A) = ti(nat,B) | ti(nat,C) = ti(nat,D) | plus_plus(nat,times_times(nat,A,C),times_times(nat,B,D)) != plus_plus(nat,times_times(nat,A,D),times_times(nat,B,C)). [resolve(788,a,791,a)]. 1.47/1.67 792 -semiri456707255roduct(A) | ti(A,B) != plus_plus(A,B,C) | ti(A,C) = zero_zero(A) # label(fact_200_add__0__iff) # label(axiom). [clausify(192)]. 1.47/1.67 Derived: ti(nat,A) != plus_plus(nat,A,B) | ti(nat,B) = zero_zero(nat). [resolve(792,a,788,a)]. 1.47/1.67 793 -semiri456707255roduct(A) | ti(A,B) = plus_plus(A,B,C) | ti(A,C) != zero_zero(A) # label(fact_200_add__0__iff) # label(axiom). [clausify(192)]. 1.47/1.67 Derived: ti(nat,A) = plus_plus(nat,A,B) | ti(nat,B) != zero_zero(nat). [resolve(793,a,788,a)]. 1.47/1.67 794 -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(218)]. 1.47/1.67 795 -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,B) != ti(A,D) # label(fact_109_crossproduct__noteq) # label(axiom). [clausify(218)]. 1.47/1.67 796 -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,B) = ti(A,D) # label(fact_109_crossproduct__noteq) # label(axiom). [clausify(218)]. 1.47/1.67 797 semiri456707255roduct(int) # label(arity_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc) # label(axiom). [assumption]. 1.47/1.67 Derived: ti(int,A) != ti(int,B) | plus_plus(int,times_times(int,A,C),times_times(int,B,D)) = plus_plus(int,times_times(int,A,D),times_times(int,B,C)). [resolve(797,a,789,a)]. 1.47/1.67 Derived: ti(int,A) != ti(int,B) | plus_plus(int,times_times(int,C,A),times_times(int,D,B)) = plus_plus(int,times_times(int,C,B),times_times(int,D,A)). [resolve(797,a,790,a)]. 1.47/1.67 Derived: ti(int,A) = ti(int,B) | ti(int,C) = ti(int,D) | plus_plus(int,times_times(int,A,C),times_times(int,B,D)) != plus_plus(int,times_times(int,A,D),times_times(int,B,C)). [resolve(797,a,791,a)]. 1.47/1.67 Derived: ti(int,A) != plus_plus(int,A,B) | ti(int,B) = zero_zero(int). [resolve(797,a,792,a)]. 1.47/1.67 Derived: ti(int,A) = plus_plus(int,A,B) | ti(int,B) != zero_zero(int). [resolve(797,a,793,a)]. 1.47/1.67 798 semiri456707255roduct(real) # label(arity_RealDef_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossp) # label(axiom). [assumption]. 1.47/1.67 Derived: ti(real,A) != ti(real,B) | plus_plus(real,times_times(real,A,C),times_times(real,B,D)) = plus_plus(real,times_times(real,A,D),times_times(real,B,C)). [resolve(798,a,789,a)]. 1.47/1.67 Derived: ti(real,A) != ti(real,B) | plus_plus(real,times_times(real,C,A),times_times(real,D,B)) = plus_plus(real,times_times(real,C,B),times_times(real,D,A)). [resolve(798,a,790,a)]. 1.47/1.67 Derived: ti(real,A) = ti(real,B) | ti(real,C) = ti(real,D) | plus_plus(real,times_times(real,A,C),times_times(real,B,D)) != plus_plus(real,times_times(real,A,D),times_times(real,B,C)). [resolve(798,a,791,a)]. 1.51/1.74 Derived: ti(real,A) != plus_plus(real,A,B) | ti(real,B) = zero_zero(real). [resolve(798,a,792,a)]. 1.51/1.74 Derived: ti(real,A) = plus_plus(real,A,B) | ti(real,B) != zero_zero(real). [resolve(798,a,793,a)]. 1.51/1.74 799 -semiri456707255roduct(A) | ti(A,B) = zero_zero(A) | 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(475)]. 1.51/1.74 Derived: ti(nat,A) = zero_zero(nat) | 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(799,a,788,a)]. 1.51/1.74 Derived: ti(int,A) = zero_zero(int) | 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(799,a,797,a)]. 1.51/1.74 Derived: ti(real,A) = zero_zero(real) | 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(799,a,798,a)]. 1.51/1.74 800 -ab_group_add(A) | ti(A,plus_plus(A,B,C)) = plus_plus(A,B,C) # label(tsy_c_Groups_Oplus__class_Oplus_1_res) # label(axiom). [clausify(142)]. 1.51/1.74 801 ab_group_add(real) # label(arity_RealDef_Oreal___Groups_Oab__group__add) # label(axiom). [assumption]. 1.51/1.74 802 -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(198)]. 1.51/1.74 Derived: minus_minus(real,A,ti(real,B)) = minus_minus(real,A,B). [resolve(802,a,801,a)]. 1.51/1.74 803 -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(236)]. 1.51/1.74 Derived: minus_minus(real,ti(real,A),B) = minus_minus(real,A,B). [resolve(803,a,801,a)]. 1.51/1.74 804 -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(290)]. 1.51/1.74 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(804,a,801,a)]. 1.51/1.74 805 -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(320)]. 1.51/1.74 Derived: minus_minus(real,A,B) = ti(real,minus_minus(real,A,B)). [resolve(805,a,801,a)]. 1.51/1.74 806 -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(436)]. 1.51/1.74 807 -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(440)]. 1.51/1.74 808 ab_group_add(int) # label(arity_Int_Oint___Groups_Oab__group__add) # label(axiom). [assumption]. 1.51/1.74 Derived: minus_minus(int,A,ti(int,B)) = minus_minus(int,A,B). [resolve(808,a,802,a)]. 1.51/1.74 Derived: minus_minus(int,ti(int,A),B) = minus_minus(int,A,B). [resolve(808,a,803,a)]. 1.51/1.74 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(808,a,804,a)]. 1.51/1.74 Derived: minus_minus(int,A,B) = ti(int,minus_minus(int,A,B)). [resolve(808,a,805,a)]. 1.51/1.74 809 -ring_n68954251visors(A) | zero_zero(A) != times_times(A,B,C) | ti(A,B) = zero_zero(A) | ti(A,C) = zero_zero(A) # label(fact_471_mult__eq__0__iff) # label(axiom). [clausify(293)]. 1.51/1.74 810 ring_n68954251visors(int) # label(arity_Int_Oint___Rings_Oring__no__zero__divisors) # label(axiom). [assumption]. 1.51/1.74 Derived: zero_zero(int) != times_times(int,A,B) | ti(int,A) = zero_zero(int) | ti(int,B) = zero_zero(int). [resolve(809,a,810,a)]. 1.51/1.74 811 -ring_n68954251visors(A) | zero_zero(A) = times_times(A,B,C) | ti(A,B) != zero_zero(A) # label(fact_471_mult__eq__0__iff) # label(axiom). [clausify(293)]. 1.51/1.74 Derived: zero_zero(int) = times_times(int,A,B) | ti(int,A) != zero_zero(int). [resolve(811,a,810,a)]. 1.51/1.74 812 -ring_n68954251visors(A) | zero_zero(A) = times_times(A,B,C) | ti(A,C) != zero_zero(A) # label(fact_471_mult__eq__0__iff) # label(axiom). [clausify(293)]. 1.65/1.84 Derived: zero_zero(int) = times_times(int,A,B) | ti(int,B) != zero_zero(int). [resolve(812,a,810,a)]. 1.65/1.84 813 ring_n68954251visors(real) # label(arity_RealDef_Oreal___Rings_Oring__no__zero__divisors) # label(axiom). [assumption]. 1.65/1.84 Derived: zero_zero(real) != times_times(real,A,B) | ti(real,A) = zero_zero(real) | ti(real,B) = zero_zero(real). [resolve(813,a,809,a)]. 1.65/1.84 Derived: zero_zero(real) = times_times(real,A,B) | ti(real,A) != zero_zero(real). [resolve(813,a,811,a)]. 1.65/1.84 Derived: zero_zero(real) = times_times(real,A,B) | ti(real,B) != zero_zero(real). [resolve(813,a,812,a)]. 1.65/1.84 814 -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(387)]. 1.65/1.84 815 comm_monoid_mult(real) # label(arity_RealDef_Oreal___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 1.65/1.84 816 comm_monoid_mult(nat) # label(arity_Nat_Onat___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 1.65/1.84 817 comm_monoid_mult(int) # label(arity_Int_Oint___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 1.65/1.84 818 semiring_0(real) # label(arity_RealDef_Oreal___Rings_Osemiring__0) # label(axiom). [assumption]. 1.65/1.84 819 -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(160)]. 1.65/1.84 820 -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(160)]. 1.65/1.84 Derived: -power(real) | zero_zero(nat) = A | zero_zero(real) = power_power(real,zero_zero(real),A). [resolve(818,a,819,b)]. 1.65/1.84 Derived: -power(real) | zero_zero(nat) != A | one_one(real) = power_power(real,zero_zero(real),A). [resolve(818,a,820,b)]. 1.65/1.84 821 semiring_0(nat) # label(arity_Nat_Onat___Rings_Osemiring__0) # label(axiom). [assumption]. 1.65/1.84 Derived: -power(nat) | zero_zero(nat) = A | zero_zero(nat) = power_power(nat,zero_zero(nat),A). [resolve(821,a,819,b)]. 1.65/1.84 Derived: -power(nat) | zero_zero(nat) != A | one_one(nat) = power_power(nat,zero_zero(nat),A). [resolve(821,a,820,b)]. 1.65/1.84 822 -semiring_0(A) | -power(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(396)]. 1.65/1.84 823 -semiring_0(A) | -power(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(396)]. 1.65/1.84 824 semiring_0(int) # label(arity_Int_Oint___Rings_Osemiring__0) # label(axiom). [assumption]. 1.65/1.84 Derived: -power(int) | zero_zero(nat) = A | zero_zero(int) = power_power(int,zero_zero(int),A). [resolve(824,a,819,b)]. 1.65/1.84 Derived: -power(int) | zero_zero(nat) != A | one_one(int) = power_power(int,zero_zero(int),A). [resolve(824,a,820,b)]. 1.65/1.84 825 ring(real) # label(arity_RealDef_Oreal___Rings_Oring) # label(axiom). [assumption]. 1.65/1.84 826 -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(203)]. 1.65/1.84 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(825,a,826,a)]. 1.65/1.84 827 -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(241)]. 1.65/1.84 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(827,a,825,a)]. 1.65/1.84 828 -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(408)]. 1.89/2.09 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(828,a,825,a)]. 1.89/2.09 829 ring(int) # label(arity_Int_Oint___Rings_Oring) # label(axiom). [assumption]. 1.89/2.09 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(829,a,826,a)]. 1.89/2.09 Derived: -number(int) | minus_minus(int,times_times(int,A,number_number_of(int,B)),times_times(int,C,number_number_of(int,B))) = times_times(int,minus_minus(int,A,C),number_number_of(int,B)). [resolve(829,a,827,a)]. 1.89/2.09 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(829,a,828,a)]. 1.89/2.09 830 comm_ring_1(int) # label(arity_Int_Oint___Rings_Ocomm__ring__1) # label(axiom). [assumption]. 1.89/2.09 831 -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(233)]. 1.89/2.09 Derived: -dvd_dvd(int,A,B) | -dvd_dvd(int,A,C) | dvd_dvd(int,A,minus_minus(int,B,C)). [resolve(830,a,831,a)]. 1.89/2.09 832 comm_ring_1(real) # label(arity_RealDef_Oreal___Rings_Ocomm__ring__1) # label(axiom). [assumption]. 1.89/2.09 Derived: -dvd_dvd(real,A,B) | -dvd_dvd(real,A,C) | dvd_dvd(real,A,minus_minus(real,B,C)). [resolve(832,a,831,a)]. 1.89/2.09 1.89/2.09 ============================== end predicate elimination ============= 1.89/2.09 1.89/2.09 Auto_denials: (non-Horn, no changes). 1.89/2.09 1.89/2.09 Term ordering decisions: 1.89/2.09 Function symbol KB weights: int=1. real=1. nat=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. f2=1. zero_zero=1. bit1=1. bit0=1. one_one=1. twoSqu579703163sum2sq=1. undefined=1. f3=1. times_times=1. plus_plus=1. power_power=1. minus_minus=1. f4=1. product_Pair=1. 1.89/2.09 1.89/2.09 ============================== end of process initial clauses ======== 1.89/2.09 1.89/2.09 ============================== CLAUSES FOR SEARCH ==================== 1.89/2.09 1.89/2.09 ============================== end of clauses for search ============= 1.89/2.09 1.89/2.09 ============================== SEARCH ================================ 1.89/2.09 1.89/2.09 % Starting search at 1.14 seconds. 1.89/2.09 1.89/2.09 ============================== PROOF ================================= 1.89/2.09 % SZS status Theorem 1.89/2.09 % SZS output start Refutation 1.89/2.09 1.89/2.09 % Proof 1 at 1.21 (+ 0.01) seconds. 1.89/2.09 % Length of proof is 63. 1.89/2.09 % Level of proof is 8. 1.89/2.09 % Maximum clause weight is 115.000. 1.89/2.09 % Given clauses 250. 1.89/2.09 1.89/2.09 22 (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]. 1.89/2.09 80 (all Z plus_plus(int,zero_zero(int),Z) = ti(int,Z)) # label(fact_204_zadd__0) # label(axiom) # label(non_clause). [assumption]. 1.89/2.09 97 (all X_a (zero_neq_one(X_a) -> ti(X_a,one_one(X_a)) = one_one(X_a))) # label(tsy_c_Groups_Oone__class_Oone_0_res) # label(axiom) # label(non_clause). [assumption]. 1.89/2.09 167 (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]. 1.89/2.09 184 (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]. 1.89/2.09 209 (all K ti(int,K) = number_number_of(int,K)) # label(fact_86_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 1.89/2.09 230 (all B_1_1 ti(int,bit1(B_1_1)) = bit1(B_1_1)) # label(tsy_c_Int_OBit1_res) # label(hypothesis) # label(non_clause). [assumption]. 1.89/2.09 282 ord_less(int,one_one(int),t) -> (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_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]. 1.89/2.09 307 (all K bit0(K) = plus_plus(int,K,K)) # label(fact_126_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 1.89/2.09 401 (all K plus_plus(int,plus_plus(int,one_one(int),K),K) = bit1(K)) # label(fact_151_Bit1__def) # label(axiom) # label(non_clause). [assumption]. 1.89/2.09 416 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_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]. 1.89/2.09 463 (all Z_1 all W_1 (ord_less(int,Z_1,W_1) <-> ti(int,Z_1) != ti(int,W_1) & ord_less_eq(int,Z_1,W_1))) # label(fact_22_zless__le) # label(axiom) # label(non_clause). [assumption]. 1.89/2.09 468 (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]. 1.89/2.09 515 -(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]. 1.89/2.09 624 -zero_neq_one(A) | ti(A,one_one(A)) = one_one(A) # label(tsy_c_Groups_Oone__class_Oone_0_res) # label(axiom). [clausify(97)]. 1.89/2.09 628 zero_neq_one(int) # label(arity_Int_Oint___Rings_Ozero__neq__one) # label(axiom). [assumption]. 1.89/2.09 779 -semiring(A) | ti(A,plus_plus(A,B,C)) = plus_plus(A,B,C) # label(tsy_c_Groups_Oplus__class_Oplus_0_res) # label(axiom). [clausify(184)]. 1.89/2.09 787 semiring(int) # label(arity_Int_Oint___Rings_Osemiring) # label(axiom). [assumption]. 1.89/2.09 835 t = ti(int,t) # label(tsy_v_t_____res) # label(axiom). [assumption]. 1.89/2.09 836 ti(int,t) = t. [copy(835),flip(a)]. 1.89/2.09 837 zero_zero(int) = pls # label(fact_202_Pls__def) # label(axiom). [assumption]. 1.89/2.09 838 pls = zero_zero(int). [copy(837),flip(a)]. 1.89/2.09 859 times_times(int,A,B) = times_times(int,B,A) # label(fact_85_zmult__commute) # label(axiom). [clausify(22)]. 1.89/2.09 932 ti(int,A) = plus_plus(int,zero_zero(int),A) # label(fact_204_zadd__0) # label(axiom). [clausify(80)]. 1.89/2.09 933 plus_plus(int,zero_zero(int),A) = ti(int,A). [copy(932),flip(a)]. 1.89/2.09 974 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]. 1.89/2.09 975 plus_plus(nat,one_one(nat),one_one(nat)) = number_number_of(nat,bit0(bit1(zero_zero(int)))). [copy(974),rewrite([838(2)]),flip(a)]. 1.89/2.09 1035 plus_plus(int,A,B) = plus_plus(int,B,A) # label(fact_89_zadd__commute) # label(axiom). [clausify(167)]. 1.89/2.09 1096 ti(int,A) = number_number_of(int,A) # label(fact_86_number__of__is__id) # label(axiom). [clausify(209)]. 1.89/2.09 1133 ti(int,bit1(A)) = bit1(A) # label(tsy_c_Int_OBit1_res) # label(hypothesis). [clausify(230)]. 1.89/2.09 1134 number_number_of(int,bit1(A)) = bit1(A). [copy(1133),rewrite([1096(3)])]. 1.89/2.09 1143 ord_less_eq(int,one_one(int),t) # label(fact_0_tpos) # label(axiom). [assumption]. 1.89/2.09 1210 -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,c3,number_number_of(nat,bit0(bit1(pls)))),power_power(int,c4,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(282)]. 1.89/2.09 1211 -ord_less(int,one_one(int),t) | plus_plus(int,power_power(int,c3,number_number_of(nat,bit0(bit1(zero_zero(int))))),power_power(int,c4,number_number_of(nat,bit0(bit1(zero_zero(int)))))) = plus_plus(int,one_one(int),times_times(int,m,number_number_of(int,bit0(bit0(bit1(zero_zero(int))))))). [copy(1210),rewrite([838(9),859(16),1035(19),838(24),838(33)]),flip(b)]. 1.89/2.09 1241 bit0(A) = plus_plus(int,A,A) # label(fact_126_Bit0__def) # label(axiom). [clausify(307)]. 1.89/2.09 1347 number_number_of(int,bit1(pls)) = one_one(int) # label(fact_158_one__is__num__one) # label(axiom). [assumption]. 1.89/2.09 1348 bit1(zero_zero(int)) = one_one(int). [copy(1347),rewrite([838(2),1134(5)])]. 1.89/2.09 1398 bit1(A) = plus_plus(int,plus_plus(int,one_one(int),A),A) # label(fact_151_Bit1__def) # label(axiom). [clausify(401)]. 1.89/2.09 1399 bit1(A) = plus_plus(int,A,plus_plus(int,A,one_one(int))). [copy(1398),rewrite([1035(6),1035(7)])]. 1.89/2.09 1421 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,c6,number_number_of(nat,bit0(bit1(pls)))),power_power(int,c7,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(416)]. 1.89/2.09 1422 one_one(int) != t | plus_plus(int,power_power(int,c6,number_number_of(nat,plus_plus(int,number_number_of(int,number_number_of(int,one_one(int))),number_number_of(int,number_number_of(int,one_one(int)))))),power_power(int,c7,number_number_of(nat,plus_plus(int,number_number_of(int,number_number_of(int,one_one(int))),number_number_of(int,number_number_of(int,one_one(int))))))) = plus_plus(int,one_one(int),times_times(int,m,number_number_of(int,plus_plus(int,plus_plus(int,number_number_of(int,number_number_of(int,one_one(int))),number_number_of(int,number_number_of(int,one_one(int)))),plus_plus(int,number_number_of(int,number_number_of(int,one_one(int))),number_number_of(int,number_number_of(int,one_one(int)))))))). [copy(1421),rewrite([838(8),1399(10),933(16),1096(14),933(15),1096(13),1241(14),1241(22),859(40),1035(43),838(48),1399(50),933(56),1096(54),933(55),1096(53),1241(54),838(67),1399(69),933(75),1096(73),933(74),1096(72),1241(73)]),flip(b)]. 1.89/2.09 1506 ord_less(int,A,B) | ti(int,A) = ti(int,B) | -ord_less_eq(int,A,B) # label(fact_22_zless__le) # label(axiom). [clausify(463)]. 1.89/2.09 1507 ord_less(int,A,B) | number_number_of(int,A) = number_number_of(int,B) | -ord_less_eq(int,A,B). [copy(1506),rewrite([1096(4),1096(6)])]. 1.89/2.09 1523 number_number_of(int,plus_plus(int,A,B)) = plus_plus(int,number_number_of(int,A),number_number_of(int,B)) # label(fact_132_plus__numeral__code_I9_J) # label(axiom). [clausify(468)]. 1.89/2.09 1524 plus_plus(int,number_number_of(int,A),number_number_of(int,B)) = number_number_of(int,plus_plus(int,A,B)). [copy(1523),flip(a)]. 1.89/2.09 1589 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)]. 1.89/2.09 1590 plus_plus(int,power_power(int,A,number_number_of(nat,number_number_of(int,number_number_of(int,plus_plus(int,one_one(int),one_one(int)))))),power_power(int,B,number_number_of(nat,number_number_of(int,number_number_of(int,plus_plus(int,one_one(int),one_one(int))))))) != plus_plus(int,one_one(int),times_times(int,m,number_number_of(int,number_number_of(int,number_number_of(int,plus_plus(int,plus_plus(int,one_one(int),one_one(int)),plus_plus(int,one_one(int),one_one(int)))))))). [copy(1589),rewrite([838(4),1399(6),933(12),1096(10),933(11),1096(9),1241(10),1524(17),1524(14),1241(14),1524(25),1524(22),859(24),1035(27),838(31),1399(33),933(39),1096(37),933(38),1096(36),1241(37),1524(44),1524(41),838(45),1399(47),933(53),1096(51),933(52),1096(50),1241(51),1524(58),1524(55)]),flip(a)]. 1.89/2.09 1926 ti(int,one_one(int)) = one_one(int). [resolve(628,a,624,a)]. 1.89/2.09 1927 number_number_of(int,one_one(int)) = one_one(int). [copy(1926),rewrite([1096(4)])]. 1.89/2.09 2259 ti(int,plus_plus(int,A,B)) = plus_plus(int,A,B). [resolve(787,a,779,a)]. 1.89/2.09 2260 number_number_of(int,plus_plus(int,A,B)) = plus_plus(int,A,B). [copy(2259),rewrite([1096(4)])]. 1.89/2.09 2394 number_number_of(int,t) = t. [back_rewrite(836),rewrite([1096(3)])]. 1.89/2.09 2416 -ord_less(int,one_one(int),t) | plus_plus(int,power_power(int,c3,number_number_of(nat,plus_plus(int,plus_plus(int,zero_zero(int),plus_plus(int,zero_zero(int),one_one(int))),plus_plus(int,zero_zero(int),plus_plus(int,zero_zero(int),one_one(int)))))),power_power(int,c4,number_number_of(nat,plus_plus(int,plus_plus(int,zero_zero(int),plus_plus(int,zero_zero(int),one_one(int))),plus_plus(int,zero_zero(int),plus_plus(int,zero_zero(int),one_one(int))))))) = plus_plus(int,one_one(int),times_times(int,m,plus_plus(int,plus_plus(int,plus_plus(int,zero_zero(int),plus_plus(int,zero_zero(int),one_one(int))),plus_plus(int,zero_zero(int),plus_plus(int,zero_zero(int),one_one(int)))),plus_plus(int,plus_plus(int,zero_zero(int),plus_plus(int,zero_zero(int),one_one(int))),plus_plus(int,zero_zero(int),plus_plus(int,zero_zero(int),one_one(int))))))). [back_rewrite(1211),rewrite([1399(12),1241(20),1399(39),1241(47),1399(70),1241(78),1241(90),2260(114)])]. 1.89/2.09 2443 number_number_of(nat,plus_plus(int,plus_plus(int,zero_zero(int),plus_plus(int,zero_zero(int),one_one(int))),plus_plus(int,zero_zero(int),plus_plus(int,zero_zero(int),one_one(int))))) = plus_plus(nat,one_one(nat),one_one(nat)). [back_rewrite(975),rewrite([1399(10),1241(18)]),flip(a)]. 1.89/2.09 2471 plus_plus(int,zero_zero(int),plus_plus(int,zero_zero(int),one_one(int))) = one_one(int). [back_rewrite(1348),rewrite([1399(3)])]. 1.89/2.09 2511 one_one(int) != t | plus_plus(int,power_power(int,c6,number_number_of(nat,plus_plus(int,one_one(int),one_one(int)))),power_power(int,c7,number_number_of(nat,plus_plus(int,one_one(int),one_one(int))))) = 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))))). [back_rewrite(1422),rewrite([1927(14),1927(13),1927(16),1927(15),1927(25),1927(24),1927(27),1927(26),1927(41),1927(40),1927(43),1927(42),1927(47),1927(46),1927(49),1927(48),2260(49)])]. 1.89/2.09 2583 plus_plus(int,power_power(int,A,number_number_of(nat,plus_plus(int,one_one(int),one_one(int)))),power_power(int,B,number_number_of(nat,plus_plus(int,one_one(int),one_one(int))))) != 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))))). [back_rewrite(1590),rewrite([2260(12),2260(11),2260(22),2260(21),2260(45),2260(44),2260(43)])]. 1.89/2.09 2646 -ord_less(int,one_one(int),t). [back_rewrite(2416),rewrite([2471(20),2471(22),2471(31),2471(33),2471(46),2471(48),2471(52),2471(54)]),unit_del(b,2583)]. 1.89/2.09 2658 number_number_of(nat,plus_plus(int,one_one(int),one_one(int))) = plus_plus(nat,one_one(nat),one_one(nat)). [back_rewrite(2443),rewrite([2471(12),2471(14)])]. 1.89/2.09 2677 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,c6,plus_plus(nat,one_one(nat),one_one(nat))),power_power(int,c7,plus_plus(nat,one_one(nat),one_one(nat)))). [back_unit_del(2511),rewrite([2658(15),2658(24)]),flip(b)]. 1.89/2.09 2732 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)))). [back_rewrite(2583),rewrite([2658(10),2658(18)]),flip(a)]. 1.89/2.09 2766 one_one(int) != t. [back_unit_del(2677),unit_del(b,2732)]. 1.89/2.09 3204 $F. [resolve(1507,c,1143,a),rewrite([1927(9),2394(10)]),unit_del(a,2646),unit_del(b,2766)]. 1.89/2.09 1.89/2.09 % SZS output end Refutation 1.89/2.09 ============================== end of proof ========================== 1.89/2.09 1.89/2.09 ============================== STATISTICS ============================ 1.89/2.09 1.89/2.09 Given=250. Generated=3041. Kept=1799. proofs=1. 1.89/2.09 Usable=249. Sos=1019. Demods=187. Limbo=2, Disabled=1915. Hints=0. 1.89/2.09 Megabytes=4.68. 1.89/2.09 User_CPU=1.21, System_CPU=0.01, Wall_clock=1. 1.89/2.09 1.89/2.09 ============================== end of statistics ===================== 1.89/2.09 1.89/2.09 ============================== end of search ========================= 1.89/2.09 1.89/2.09 THEOREM PROVED 1.89/2.09 % SZS status Theorem 1.89/2.09 1.89/2.09 Exiting with 1 proof. 1.89/2.09 1.89/2.09 Process 5797 exit (max_proofs) Sat Jul 14 05:24:11 2018 1.89/2.09 Prover9 interrupted 1.89/2.10 EOF