0.07/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.07/0.13 % Command : tptp2X_and_run_prover9 %d %s 0.12/0.34 % Computer : n021.cluster.edu 0.12/0.34 % Model : x86_64 x86_64 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.12/0.34 % Memory : 8042.1875MB 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64 0.12/0.34 % CPULimit : 960 0.12/0.34 % WCLimit : 120 0.12/0.34 % DateTime : Tue Aug 9 04:29:35 EDT 2022 0.12/0.34 % CPUTime : 1.59/1.95 ============================== Prover9 =============================== 1.59/1.95 Prover9 (32) version 2009-11A, November 2009. 1.59/1.95 Process 3775 was started by sandbox on n021.cluster.edu, 1.59/1.95 Tue Aug 9 04:29:36 2022 1.59/1.95 The command was "/export/starexec/sandbox/solver/bin/prover9 -t 960 -f /tmp/Prover9_3622_n021.cluster.edu". 1.59/1.95 ============================== end of head =========================== 1.59/1.95 1.59/1.95 ============================== INPUT ================================= 1.59/1.95 1.59/1.95 % Reading from file /tmp/Prover9_3622_n021.cluster.edu 1.59/1.95 1.59/1.95 set(prolog_style_variables). 1.59/1.95 set(auto2). 1.59/1.95 % set(auto2) -> set(auto). 1.59/1.95 % set(auto) -> set(auto_inference). 1.59/1.95 % set(auto) -> set(auto_setup). 1.59/1.95 % set(auto_setup) -> set(predicate_elim). 1.59/1.95 % set(auto_setup) -> assign(eq_defs, unfold). 1.59/1.95 % set(auto) -> set(auto_limits). 1.59/1.95 % set(auto_limits) -> assign(max_weight, "100.000"). 1.59/1.95 % set(auto_limits) -> assign(sos_limit, 20000). 1.59/1.95 % set(auto) -> set(auto_denials). 1.59/1.95 % set(auto) -> set(auto_process). 1.59/1.95 % set(auto2) -> assign(new_constants, 1). 1.59/1.95 % set(auto2) -> assign(fold_denial_max, 3). 1.59/1.95 % set(auto2) -> assign(max_weight, "200.000"). 1.59/1.95 % set(auto2) -> assign(max_hours, 1). 1.59/1.95 % assign(max_hours, 1) -> assign(max_seconds, 3600). 1.59/1.95 % set(auto2) -> assign(max_seconds, 0). 1.59/1.95 % set(auto2) -> assign(max_minutes, 5). 1.59/1.95 % assign(max_minutes, 5) -> assign(max_seconds, 300). 1.59/1.95 % set(auto2) -> set(sort_initial_sos). 1.59/1.95 % set(auto2) -> assign(sos_limit, -1). 1.59/1.95 % set(auto2) -> assign(lrs_ticks, 3000). 1.59/1.95 % set(auto2) -> assign(max_megs, 400). 1.59/1.95 % set(auto2) -> assign(stats, some). 1.59/1.95 % set(auto2) -> clear(echo_input). 1.59/1.95 % set(auto2) -> set(quiet). 1.59/1.95 % set(auto2) -> clear(print_initial_clauses). 1.59/1.95 % set(auto2) -> clear(print_given). 1.59/1.95 assign(lrs_ticks,-1). 1.59/1.95 assign(sos_limit,10000). 1.59/1.95 assign(order,kbo). 1.59/1.95 set(lex_order_vars). 1.59/1.95 clear(print_given). 1.59/1.95 1.59/1.95 % formulas(sos). % not echoed (1257 formulas) 1.59/1.95 1.59/1.95 ============================== end of input ========================== 1.59/1.95 1.59/1.95 % From the command line: assign(max_seconds, 960). 1.59/1.95 1.59/1.95 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 1.59/1.95 1.59/1.95 % Formulas that are not ordinary clauses: 1.59/1.95 1 (all A_3 all B_2 all Ma (hBOOL(hAPP(int,bool,zcong(B_2,A_3),Ma)) <-> hBOOL(hAPP(int,bool,zcong(A_3,B_2),Ma)))) # label(fact_326_zcong__sym) # label(axiom) # label(non_clause). [assumption]. 1.59/1.95 2 (all X_a (comm_semiring_1(X_a) -> (all Lx all Rx all Ry hAPP(X_a,X_a,times_times(X_a,Lx),hAPP(X_a,X_a,times_times(X_a,Rx),Ry)) = hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,times_times(X_a,Lx),Rx)),Ry)))) # label(fact_71_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.95 3 (all X_a (comm_semiring_1(X_a) -> (all Lx all Ly all Rx hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,times_times(X_a,Lx),Ly)),Rx) = hAPP(X_a,X_a,times_times(X_a,Lx),hAPP(X_a,X_a,times_times(X_a,Ly),Rx))))) # label(fact_70_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.95 4 (all P_2 all X_2 all Y_2 (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_2),X_2)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_2),Y_2)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_2),X_2)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_2),Y_2)) -> hBOOL(P_2)))) # label(fact_581_dvd_Oless__imp__triv) # label(axiom) # label(non_clause). [assumption]. 1.59/1.95 5 (all Ma all K all N (hAPP(nat,nat,plus_plus(nat,Ma),K) = hAPP(nat,nat,plus_plus(nat,N),K) <-> Ma = N)) # label(fact_610_nat__add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.59/1.95 6 (all V_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),V_2),pls)) <-> number_number_of(nat,V_2) = zero_zero(nat))) # label(fact_324_eq__number__of__0) # label(axiom) # label(non_clause). [assumption]. 1.59/1.95 7 (all K_1 all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,div_mod(int,M),K_1)),M)))) # label(fact_909_zmod__le__nonneg__dividend) # label(axiom) # label(non_clause). [assumption]. 1.59/1.95 8 (all X_a (ordere223160158up_add(X_a) -> (all C all D_2 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_2)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),C)),hAPP(X_a,X_a,plus_plus(X_a,B),D_2)))))))) # label(fact_782_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/1.95 9 (all X_a (linord581940658strict(X_a) -> (all C all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),C)))))))) # label(fact_489_mult__strict__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.59/1.95 10 (all B_2 all A_3 all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,bit1(bit0(bit1(pls))))),P_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(int,int,minus_minus(int,P_3),one_one(int)))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),B_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_2),hAPP(int,int,minus_minus(int,P_3),one_one(int)))) -> (hBOOL(hAPP(fun(int,bool),bool,member(int,inv(P_3,B_2)),wset(A_3,P_3))) -> hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),wset(A_3,P_3)))))))))) # label(fact_842_wset__inv__mem__mem) # label(axiom) # label(non_clause). [assumption]. 1.59/1.95 11 (all K_1 all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,times_times(nat,I_1),K_1)),hAPP(nat,nat,times_times(nat,J_1),K_1)))))) # label(fact_713_mult__less__mono1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.95 12 (all K_1 all L_1 bit0(hAPP(int,int,plus_plus(int,K_1),L_1)) = hAPP(int,int,plus_plus(int,bit0(K_1)),bit0(L_1))) # label(fact_127_add__Bit0__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.59/1.95 13 (all X_2 all Y_2 all Z_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Z_2)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,X_2),Z_2)),hAPP(real,real,times_times(real,Y_2),Z_2))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),Y_2))))) # label(fact_520_real__mult__le__cancel__iff1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.95 14 (all X_a (comm_semiring_1(X_a) -> (all N_1 all M all X_1 all Y_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X_1),Y_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(nat,X_a,power_power(X_a,X_1),N_1)),hAPP(nat,X_a,power_power(X_a,Y_1),M)))))))) # label(fact_187_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 1.59/1.95 15 (all X_a (number_ring(X_a) -> (all X_1 all Y_1 hAPP(nat,X_a,power_power(X_a,hAPP(X_a,X_a,minus_minus(X_a,X_1),Y_1)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(X_a,X_a,minus_minus(X_a,hAPP(X_a,X_a,plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,X_1),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,X_a,power_power(X_a,Y_1),number_number_of(nat,bit0(bit1(pls)))))),hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,times_times(X_a,number_number_of(X_a,bit0(bit1(pls)))),X_1)),Y_1))))) # label(fact_401_power2__diff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.95 16 (all X_a (ordered_ring(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,times_times(X_a,A_1),B))))))) # label(fact_472_split__mult__pos__le) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 17 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),B)) -> ti(int,A_1) = hAPP(int,int,div_mod(int,A_1),B)))) # label(fact_957_mod__pos__pos__trivial) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 18 (all X_a (comm_semiring_1(X_a) -> (all A_1 all M all B hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),M)),hAPP(X_a,X_a,times_times(X_a,B),M)) = hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),M)))) # label(fact_109_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 19 (all K_1 all I_1 all J_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_1),J_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),J_1),K_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_1),K_1))))) # label(fact_24_zle__trans) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 20 (all X_a (linord219039673up_add(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,A_3),A_3))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_3)))))) # label(fact_797_zero__le__double__add__iff__zero__le__single__add) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 21 (all X_a (linordered_semidom(X_a) -> (all N_1 all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),hAPP(nat,X_a,power_power(X_a,A_1),N_1))))))) # label(fact_270_one__le__power) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 22 (all X_a (linordered_semidom(X_a) -> (all A_1 all N_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N_1)),hAPP(nat,X_a,power_power(X_a,B),N_1))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))))))) # label(fact_214_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 23 (all N all P_2 (-hBOOL(hAPP(nat,bool,P_2,zero_zero(nat))) -> (hBOOL(hAPP(nat,bool,P_2,N)) -> (exists K_2 ((all I (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),K_2)) -> -hBOOL(hAPP(nat,bool,P_2,I)))) & hBOOL(hAPP(nat,bool,P_2,hAPP(nat,nat,plus_plus(nat,K_2),one_one(nat)))) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_2),N))))))) # label(fact_723_ex__least__nat__less) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 24 (all M all Y_1 all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),X_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Y_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M)) -> (hBOOL(hAPP(int,bool,zcong(X_1,Y_1),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),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]. 1.59/1.96 25 (all B_1_1 all X_a (no_zero_divisors(X_a) -> ti(fun(X_a,X_a),times_times(X_a,B_1_1)) = times_times(X_a,B_1_1))) # label(tsy_c_Groups_Otimes__class_Otimes_5_res) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 26 (all X_a (ordere779506340up_add(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,C),A_1)),hAPP(X_a,X_a,plus_plus(X_a,C),B))))))) # label(fact_774_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 27 (all X_2 all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,power_power(nat,X_2),N))) <-> zero_zero(nat) = N | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X_2)))) # label(fact_286_nat__zero__less__power__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 28 (all X_a all X_c all B_1_1 all B_2_1 hAPP(X_a,X_c,B_1_1,B_2_1) = hAPP(X_a,X_c,ti(fun(X_a,X_c),B_1_1),B_2_1)) # label(tsy_c_hAPP_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 29 (all K_1 pls != bit1(K_1)) # label(fact_116_rel__simps_I46_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 30 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),pls)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K)),pls)))) # label(fact_100_rel__simps_I27_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 31 (all I_1 all K_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),J_1)) -> hAPP(nat,nat,plus_plus(nat,I_1),hAPP(nat,nat,minus_minus(nat,J_1),K_1)) = hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,plus_plus(nat,I_1),J_1)),K_1))) # label(fact_689_add__diff__assoc) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 32 -hBOOL(hAPP(int,bool,quadRes(hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int))),number_number_of(int,min))) -> legendre(number_number_of(int,min),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int))) != one_one(int) # label(fact_367__096_126_AQuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_A_061_061_062_ALegendre_A_N) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 33 (all X_a (semiring_div(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,div_mod(X_a,A_1),zero_zero(X_a))))) # label(fact_943_mod__by__0) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 34 (all K_1 all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),hAPP(nat,nat,minus_minus(nat,M),N_1))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),N_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),M)))))) # label(fact_673_dvd__diffD) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 35 (all X_a all X_b all X_c all B_1_1 all B_2_1 combc(X_a,X_b,X_c,B_1_1,B_2_1) = combc(X_a,X_b,X_c,ti(fun(X_a,fun(X_b,X_c)),B_1_1),B_2_1)) # label(tsy_c_COMBC_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 36 (all X_a (dvd(X_a) & comm_ring(X_a) -> (all Ta all D all D_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),D_3)) -> (all X all K_2 (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,minus_minus(X_a,X),hAPP(X_a,X_a,times_times(X_a,K_2),D_3))),Ta))) <-> -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),hAPP(X_a,X_a,plus_plus(X_a,X),Ta))))))))) # label(fact_745_inf__period_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 37 (all B_1_1 all B_2_1 ti(fun(int,bool),wset(B_1_1,B_2_1)) = wset(B_1_1,B_2_1)) # label(tsy_c_WilsonRuss_Owset_res) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 38 (all X_b (semiring(X_b) & number(X_b) -> (all A_1 all B all V hAPP(X_b,X_b,plus_plus(X_b,hAPP(X_b,X_b,times_times(X_b,A_1),number_number_of(X_b,V))),hAPP(X_b,X_b,times_times(X_b,B),number_number_of(X_b,V))) = hAPP(X_b,X_b,times_times(X_b,hAPP(X_b,X_b,plus_plus(X_b,A_1),B)),number_number_of(X_b,V))))) # label(fact_138_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 39 (all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),A_1)) -> -(-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),A_1))))) # label(fact_578_dvd_Oless__asym_H) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 40 (all X_a (linordered_semidom(X_a) -> (all A_1 all N_1 all N_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),N_3)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),one_one(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N_3)),hAPP(nat,X_a,power_power(X_a,A_1),N_1))))))))) # label(fact_215_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 41 (all X_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),P_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),standardRes(P_1,X_1))))) # label(fact_993_StandardRes__lbound) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 42 (all A_1 all R_1 all B all M all C all D_2 all N_1 hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,A_1),hAPP(int,int,times_times(int,R_1),B))),M)),hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,C),hAPP(int,int,times_times(int,R_1),D_2))),N_1)) = hAPP(int,int,minus_minus(int,hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A_1),M)),hAPP(int,int,times_times(int,C),N_1))),hAPP(int,int,times_times(int,R_1),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),M)),hAPP(int,int,times_times(int,D_2),N_1))))) # label(fact_382_xzgcda__linear__aux1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 43 (all X_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),P_1)) -> hBOOL(hAPP(int,bool,zcong(multInv(P_1,multInv(P_1,X_1)),X_1),P_1)))))) # label(fact_830_MultInv__prop4) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 44 (all N all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Ma)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,times_times(nat,Ma),N)),Ma)) <-> N = one_one(nat)))) # label(fact_554_dvd__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 45 (all N_1 -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),zero_zero(nat)))) # label(fact_634_less__nat__zero__code) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 46 (all A_3 all B_2 all C_1 (zero_zero(real) != C_1 -> (hAPP(real,real,times_times(real,C_1),B_2) = hAPP(real,real,times_times(real,C_1),A_3) <-> A_3 = B_2))) # label(fact_463_real__mult__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 47 (all C all D_2 all A_1 all B all M (hBOOL(hAPP(int,bool,zcong(A_1,B),M)) -> (hBOOL(hAPP(int,bool,zcong(C,D_2),M)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,minus_minus(int,A_1),C),hAPP(int,int,minus_minus(int,B),D_2)),M))))) # label(fact_374_zcong__zdiff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 48 (all M all N_1 M = hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,plus_plus(nat,M),N_1)),N_1)) # label(fact_611_diff__add__inverse2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 49 (all X_2 all Y_2 (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_2),X_2)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_2),Y_2)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_2),Y_2)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_2),X_2)))) # label(fact_600_dvd_Oless__le__not__le) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 50 (all X_a (real_normed_algebra(X_a) -> (all X_1 hAPP(X_a,X_a,times_times(X_a,X_1),zero_zero(X_a)) = zero_zero(X_a)))) # label(fact_884_mult__right_Ozero) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 51 (all X_1 all Y_1 hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,power_power(nat,X_1),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,nat,power_power(nat,Y_1),number_number_of(nat,bit0(bit1(pls))))) = hAPP(nat,nat,times_times(nat,hAPP(nat,nat,plus_plus(nat,X_1),Y_1)),hAPP(nat,nat,minus_minus(nat,X_1),Y_1))) # label(fact_439_diff__square) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 52 (all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1)) -> zero_zero(nat) != N_1)) # label(fact_633_gr__implies__not0) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 53 (all Z1 all Z2 all Z3 hAPP(int,int,times_times(int,Z1),hAPP(int,int,times_times(int,Z2),Z3)) = hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,Z1),Z2)),Z3)) # label(fact_86_zmult__assoc) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 54 (all P_2 all K all I_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),I_2)) -> (hBOOL(hAPP(int,bool,P_2,K)) -> ((all I (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),I)) -> (hBOOL(hAPP(int,bool,P_2,I)) -> hBOOL(hAPP(int,bool,P_2,hAPP(int,int,plus_plus(int,I),one_one(int))))))) -> hBOOL(hAPP(int,bool,P_2,I_2)))))) # label(fact_852_int__ge__induct) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 55 (all X_a (comm_semiring_1(X_a) -> (all C all D_2 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),C),D_2)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),D_2)))))))) # label(fact_454_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 56 (all X_2 (-hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,times_times(real,X_2),X_2))) <-> X_2 = zero_zero(real))) # label(fact_299_not__real__square__gt__zero) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 57 (all Z_3 all Z_1 all W_2 all W (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_2),W)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_3),Z_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,W_2),Z_3)),hAPP(int,int,plus_plus(int,W),Z_1)))))) # label(fact_32_zadd__zless__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 58 (all B_2 all P_3 all A_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_3)) -> (-hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),wset(hAPP(int,int,minus_minus(int,A_3),one_one(int)),P_3))) -> (hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),wset(A_3,P_3))) -> ti(int,B_2) = inv(P_3,A_3) | ti(int,B_2) = ti(int,A_3))))) # label(fact_847_wset__mem__imp__or) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 59 (all A_1 all N_1 all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,power_power(int,A_1),N_1)),hAPP(nat,int,power_power(int,B),N_1))) -> (N_1 != zero_zero(nat) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_1),B))))) # label(fact_879_pow__divides__pow__int) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 60 (all A_3 (zero_zero(nat) != A_3 <-> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A_3)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_3),zero_zero(nat))))) # label(fact_869_gcd__lcm__complete__lattice__nat_Oless__top) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 61 (all I_1 all U_1 all J_1 all K_1 hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,hAPP(nat,nat,plus_plus(nat,I_1),J_1)),U_1)),K_1) = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,I_1),U_1)),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,J_1),U_1)),K_1))) # label(fact_726_left__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 62 (all Ma all N all K (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K)) -> (N = Ma <-> hAPP(nat,nat,times_times(nat,K),N) = hAPP(nat,nat,times_times(nat,K),Ma)))) # label(fact_727_nat__mult__eq__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 63 (all K all Ma all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,times_times(nat,K),Ma)),hAPP(nat,nat,times_times(nat,K),N))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),N)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K)))) # label(fact_715_mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 64 (all N_1 all P_1 all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M)) -> (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),hAPP(int,int,times_times(int,M),N_1))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),N_1)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),M)))))) # label(fact_360_zprime__zdvd__zmult) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 65 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all Y_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),Y_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),number_number_of(X_a,Y_2))))))) # label(fact_248_le__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 66 (all K_1 all M all N_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_1),hAPP(int,int,minus_minus(int,M),N_1))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_1),N_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_1),M))))) # label(fact_375_zdvd__zdiffD) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 67 (all X_a (ordere236663937imp_le(X_a) -> (all C_1 all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,C_1),A_3)),hAPP(X_a,X_a,plus_plus(X_a,C_1),B_2))))))) # label(fact_779_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 68 (all X_a (semiring_div(X_a) -> (all B all A_1 hAPP(X_a,X_a,div_mod(X_a,A_1),B) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,B),A_1)),B)))) # label(fact_928_mod__add__self1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 69 (all A_1 hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit1(bit1(pls)))) = hAPP(int,int,times_times(int,A_1),hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls)))))) # label(fact_11_cube__square) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 70 (all X_a (comm_semiring_1(X_a) -> (all A_1 hAPP(X_a,X_a,plus_plus(X_a,A_1),zero_zero(X_a)) = ti(X_a,A_1)))) # label(fact_201_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 71 (all N_1 all F all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),F),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),F),N_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),F),hAPP(int,int,div_mod(int,M),N_1)))))) # label(fact_916_zdvd__zmod) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 72 (all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,minus_minus(int,P_1),one_one(int)))) -> one_one(int) != inv(P_1,A_1))))) # label(fact_822_inv__not__1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 73 (all P_2 all N all K (hBOOL(hAPP(nat,bool,P_2,hAPP(nat,nat,div_mod(nat,N),K))) <-> (zero_zero(nat) != K -> (all I all J (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J),K)) -> (N = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,K),I)),J) -> hBOOL(hAPP(nat,bool,P_2,J)))))) & (zero_zero(nat) = K -> hBOOL(hAPP(nat,bool,P_2,N))))) # label(fact_991_split__mod) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 74 (all P_1 all Y_1 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),Y_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),hAPP(nat,int,power_power(int,Y_1),N_1)))))) # label(fact_411_zpower__zdvd__prop1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 75 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),K)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),bit1(K))))) # label(fact_301_rel__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 76 (all X_a (linordered_semidom(X_a) -> (all M all N_1 all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,A_1),M)),hAPP(nat,X_a,power_power(X_a,A_1),N_1))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1))))))) # label(fact_178_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 77 (all W_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),zero_zero(int))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(W_1)),zero_zero(int))))) # label(fact_224_bin__less__0__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 78 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B hAPP(X_a,X_a,times_times(X_a,A_1),B) = hAPP(X_a,X_a,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]. 1.59/1.96 79 (all V_2 all V_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),number_number_of(nat,V_2)),number_number_of(nat,V_3))) <-> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),V_2),V_3)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),V_2),pls))))) # label(fact_240_le__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 80 (all K_1 all L_1 hAPP(int,int,times_times(int,bit0(K_1)),L_1) = bit0(hAPP(int,int,times_times(int,K_1),L_1))) # label(fact_124_mult__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 81 (all Z_1 all X_1 all Y_1 all P_1 (hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,X_1),Y_1),one_one(int)),P_1)) -> hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,X_1),hAPP(nat,nat,times_times(nat,Y_1),Z_1)),one_one(int)),P_1)))) # label(fact_211_zcong__zpower__zmult) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 82 (all X_a (monoid_mult(X_a) -> (all A_1 all M all N_1 hAPP(nat,X_a,power_power(X_a,A_1),hAPP(nat,nat,plus_plus(nat,M),N_1)) = hAPP(X_a,X_a,times_times(X_a,hAPP(nat,X_a,power_power(X_a,A_1),M)),hAPP(nat,X_a,power_power(X_a,A_1),N_1))))) # label(fact_261_power__add) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 83 (all X_a (comm_semiring_1(X_a) -> (all A_1 all C all D_2 hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),C)),D_2) = hAPP(X_a,X_a,plus_plus(X_a,A_1),hAPP(X_a,X_a,plus_plus(X_a,C),D_2))))) # label(fact_79_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 84 (all N_1 hAPP(nat,nat,times_times(nat,zero_zero(nat)),N_1) = zero_zero(nat)) # label(fact_648_mult__0) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 85 (all J_2 all K all I_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_2),hAPP(nat,nat,plus_plus(nat,I_2),K))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,minus_minus(nat,J_2),K)),I_2)))) # label(fact_686_le__diff__conv) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 86 (all X_a (linordered_semidom(X_a) -> (all N_1 all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,power_power(X_a,A_1),N_1))))))) # label(fact_210_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 87 (all X_a (linord20386208strict(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,C),A_1)),hAPP(X_a,X_a,times_times(X_a,C),B))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))))))) # label(fact_539_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 88 (all K_1 all L_1 bit1(hAPP(int,int,plus_plus(int,K_1),L_1)) = hAPP(int,int,plus_plus(int,bit1(K_1)),bit0(L_1))) # label(fact_151_add__Bit1__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 89 (all K_1 all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,I_1),K_1)),hAPP(nat,nat,plus_plus(nat,J_1),K_1))))) # label(fact_680_add__le__mono1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 90 (all X_a (comm_semiring_1(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),hAPP(X_a,X_a,plus_plus(X_a,B),C)))))))) # label(fact_459_dvd__add) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 91 (all B all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (-hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_1)) & -hBOOL(hAPP(int,bool,zcong(B,zero_zero(int)),P_1)) -> -hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_1),B),zero_zero(int)),P_1)))))) # label(fact_415_zcong__zprime__prod__zero__contra) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 92 (all B_1_1 all X_a (cancel_semigroup_add(X_a) -> plus_plus(X_a,ti(X_a,B_1_1)) = plus_plus(X_a,B_1_1))) # label(tsy_c_Groups_Oplus__class_Oplus_2_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 93 (all Y_1 all X_1 (hBOOL(hAPP(int,bool,twoSqu658283162sum2sq,X_1)) -> (hBOOL(hAPP(int,bool,twoSqu658283162sum2sq,Y_1)) -> hBOOL(hAPP(int,bool,twoSqu658283162sum2sq,hAPP(int,int,times_times(int,X_1),Y_1)))))) # label(fact_65_is__mult__sum2sq) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 94 (all Z_1 all X_1 all Y_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),Z_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Z_1))))) # label(fact_590_dvd_Oorder__trans) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 95 (all X_a (number_ring(X_a) -> number_number_of(X_a,bit0(bit1(pls))) = hAPP(X_a,X_a,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]. 1.59/1.96 96 (all X_a (comm_semiring_1(X_a) -> (all X_1 all P_1 all Q_1 hAPP(nat,X_a,power_power(X_a,hAPP(nat,X_a,power_power(X_a,X_1),P_1)),Q_1) = hAPP(nat,X_a,power_power(X_a,X_1),hAPP(nat,nat,times_times(nat,P_1),Q_1))))) # label(fact_26_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 97 (all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,minus_minus(int,P_1),one_one(int)))) -> hAPP(int,int,minus_minus(int,P_1),one_one(int)) != inv(P_1,A_1))))) # label(fact_823_inv__not__p__minus__1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 98 (all B_1_1 quadRes(ti(int,B_1_1)) = quadRes(B_1_1)) # label(tsy_c_Residues_OQuadRes_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 99 (all Y_1 all X_1 all P_1 (-hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),P_1)) -> (hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,Y_1),number_number_of(nat,bit0(bit1(pls)))),X_1),P_1)) -> -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),Y_1))))) # label(fact_277_Euler_Oaux____1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 100 (all X_a (ring_11004092258visors(X_a) -> (all A_3 (zero_zero(X_a) = hAPP(nat,X_a,power_power(X_a,A_3),number_number_of(nat,bit0(bit1(pls)))) <-> zero_zero(X_a) = ti(X_a,A_3))))) # label(fact_252_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 101 (all M all N_1 (one_one(int) = hAPP(int,int,times_times(int,M),N_1) -> one_one(int) = ti(int,M) | number_number_of(int,min) = ti(int,M))) # label(fact_319_pos__zmult__eq__1__iff__lemma) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 102 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K)),bit1(L))))) # label(fact_60_rel__simps_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 103 (all L_1 all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),L_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,minus_minus(nat,L_1),N_1)),hAPP(nat,nat,minus_minus(nat,L_1),M)))))) # label(fact_574_diff__less__mono2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 104 (all X_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),P_1)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,multInv(P_1,X_1)),X_1),one_one(int)),P_1)))))) # label(fact_839_MultInv__prop2a) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 105 (all W hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),W),W))) # label(fact_559_real__le__refl) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 106 (all X_a (comm_semiring_1(X_a) -> (all A_1 all M hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),M)),M) = hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),one_one(X_a))),M)))) # label(fact_140_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 107 (all X_a (real_normed_algebra(X_a) -> (all X_1 all Y_1 all A_1 all B hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,minus_minus(X_a,X_1),A_1)),hAPP(X_a,X_a,minus_minus(X_a,Y_1),B))),hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,minus_minus(X_a,X_1),A_1)),B))),hAPP(X_a,X_a,times_times(X_a,A_1),hAPP(X_a,X_a,minus_minus(X_a,Y_1),B))) = hAPP(X_a,X_a,minus_minus(X_a,hAPP(X_a,X_a,times_times(X_a,X_1),Y_1)),hAPP(X_a,X_a,times_times(X_a,A_1),B))))) # label(fact_882_mult_Oprod__diff__prod) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 108 (all K all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),hAPP(nat,nat,plus_plus(nat,N),K))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),N)))) # label(fact_671_dvd__reduce) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 109 (all X_a (ordere216010020id_add(X_a) -> (all Y_2 all X_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X_2)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y_2)) -> (hAPP(X_a,X_a,plus_plus(X_a,X_2),Y_2) = zero_zero(X_a) <-> ti(X_a,Y_2) = zero_zero(X_a) & zero_zero(X_a) = ti(X_a,X_2))))))) # label(fact_800_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 110 (all Ma all K all N ((hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),N))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,times_times(nat,Ma),K)),hAPP(nat,nat,times_times(nat,N),K))))) # label(fact_721_mult__le__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 111 (all X_a (ordere236663937imp_le(X_a) -> (all A_3 all C_1 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_3),C_1)),hAPP(X_a,X_a,plus_plus(X_a,B_2),C_1))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2)))))) # label(fact_771_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 112 (all X_a (comm_semiring_1(X_a) -> (all A_1 hAPP(X_a,X_a,plus_plus(X_a,zero_zero(X_a)),A_1) = ti(X_a,A_1)))) # label(fact_200_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 113 (all B_1_1 all X_a (plus(X_a) & dvd(X_a) & linorder(X_a) -> plus_plus(X_a,B_1_1) = ti(fun(X_a,X_a),plus_plus(X_a,B_1_1)))) # label(tsy_c_Groups_Oplus__class_Oplus_0_res) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 114 (all X_a (linordered_semidom(X_a) -> (all M all N_1 all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),M)),hAPP(nat,X_a,power_power(X_a,A_1),N_1))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1))))))) # label(fact_274_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 115 (all X_a all X_b all X_c all B_1_1 all B_2_1 combs(X_a,X_b,X_c,ti(fun(X_a,fun(X_b,X_c)),B_1_1),B_2_1) = combs(X_a,X_b,X_c,B_1_1,B_2_1)) # label(tsy_c_COMBS_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 116 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B all C (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),B)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),B),C)))))) # label(fact_451_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 117 (all X_a (ordered_ring(X_a) -> (all C all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),C)))))))) # label(fact_476_mult__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 118 (all X_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),P_1)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,X_1),multInv(P_1,X_1))),multInv(P_1,multInv(P_1,X_1))),X_1),P_1)))))) # label(fact_837_Int2_Oaux____2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 119 (all B_1_1 all B_2_1 legendre(ti(int,B_1_1),B_2_1) = legendre(B_1_1,B_2_1)) # label(tsy_c_Residues_OLegendre_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 120 (all X_a (comm_semiring_1(X_a) -> (all M all A_1 all N_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N_1)),B)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(nat,X_a,power_power(X_a,A_1),M)),B))))))) # label(fact_188_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 121 (all X_a all X_c all X_b all P all Q all R hAPP(X_a,X_c,combs(X_a,X_b,X_c,P,Q),R) = hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),P,R),hAPP(X_a,X_b,Q,R))) # label(help_COMBS_1_1_U) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 122 (all M all N_1 all K_1 hAPP(nat,nat,times_times(nat,hAPP(nat,nat,times_times(nat,M),N_1)),K_1) = hAPP(nat,nat,times_times(nat,M),hAPP(nat,nat,times_times(nat,N_1),K_1))) # label(fact_628_nat__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 123 (all N_1 all X_1 all Y_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,power_power(nat,X_1),N_1)),hAPP(nat,nat,power_power(nat,Y_1),N_1))))) # label(fact_470_divides__exp) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 124 (all A_1 all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),hAPP(int,int,minus_minus(int,B),one_one(int)))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),B)))) # label(fact_752_norR__mem__unique__aux) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 125 (all X_a (linordered_semiring(X_a) -> (all A_1 all C all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),C))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))))))) # label(fact_534_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 126 (all X_1 all Y_1 (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) -> -(hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1))))) # label(fact_576_dvd_Oless__asym) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 127 (all N all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),N)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,minus_minus(nat,N),Ma))))) # label(fact_638_zero__less__diff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 128 (all Z_1 all X_1 all Y_1 (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),Z_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Z_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z_1),X_1))))) # label(fact_579_dvd_Oless__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 129 (all M_1 all D_1 (hAPP(int,int,div_mod(int,M_1),D_1) = zero_zero(int) -> (exists Q_2 hAPP(int,int,times_times(int,D_1),Q_2) = ti(int,M_1)))) # label(fact_972_zmod__eq__0D) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 130 (all X_a (linord219039673up_add(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,A_3),A_3))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_3)))))) # label(fact_804_zero__less__double__add__iff__zero__less__single__add) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 131 (all X_a (monoid_mult(X_a) -> (all N_1 hAPP(nat,X_a,power_power(X_a,one_one(X_a)),N_1) = one_one(X_a)))) # label(fact_262_power__one) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 132 (all K_1 hAPP(int,int,plus_plus(int,K_1),pls) = ti(int,K_1)) # label(fact_125_add__Pls__right) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 133 (all X_a (one(X_a) -> ti(X_a,one_one(X_a)) = one_one(X_a))) # label(tsy_c_Groups_Oone__class_Oone_res) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 134 (all B_1_1 all B_2_1 multInv(B_1_1,B_2_1) = multInv(ti(int,B_1_1),B_2_1)) # label(tsy_c_Int2_OMultInv_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 135 (all C all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_1),B)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),A_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,minus_minus(nat,A_1),C)),hAPP(nat,nat,minus_minus(nat,B),C)))))) # label(fact_670_diff__less__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 136 (all B_1_1 all X_a (ab_semigroup_mult(X_a) -> ti(fun(X_a,X_a),times_times(X_a,B_1_1)) = times_times(X_a,B_1_1))) # label(tsy_c_Groups_Otimes__class_Otimes_0_res) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 137 (all X_2 all P_3 (hBOOL(hAPP(fun(int,bool),bool,member(int,X_2),sr(P_3))) -> ti(int,X_2) = standardRes(P_3,X_2))) # label(fact_990_StandardRes__SR__prop) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 138 (all B_1_1 all X_a (ab_semigroup_add(X_a) -> plus_plus(X_a,B_1_1) = ti(fun(X_a,X_a),plus_plus(X_a,B_1_1)))) # label(tsy_c_Groups_Oplus__class_Oplus_3_res) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 139 (all M hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),hAPP(nat,nat,times_times(nat,M),M)))) # label(fact_703_le__square) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 140 (all X_a (semiri456707255roduct(X_a) -> (all W_1 all Y_2 all X_2 all Z_2 (ti(X_a,Y_2) = ti(X_a,Z_2) | ti(X_a,X_2) = ti(X_a,W_1) <-> hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,W_1),Y_2)),hAPP(X_a,X_a,times_times(X_a,X_2),Z_2)) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,W_1),Z_2)),hAPP(X_a,X_a,times_times(X_a,X_2),Y_2)))))) # label(fact_108_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 141 (all B_2 all A_3 (hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),d22set(A_3))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),A_3)))) # label(fact_857_d22set__le) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 142 (all X_a (linord20386208strict(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),B)),zero_zero(X_a)))))))) # label(fact_494_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 143 (all A_1 all B all C (hAPP(int,int,minus_minus(int,A_1),B) = ti(int,C) -> hAPP(int,int,plus_plus(int,C),B) = ti(int,A_1))) # label(fact_368_Int2_Oaux1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 144 (all N_1 all K_1 all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),M)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),N_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),hAPP(nat,nat,minus_minus(nat,M),N_1)))))) # label(fact_605_dvd__diff__nat) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 145 (all J_1 all K_1 all A_1 all P_1 (hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,J_1),K_1),A_1),P_1)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,multInv(P_1,J_1)),J_1)),K_1),hAPP(int,int,times_times(int,multInv(P_1,J_1)),A_1)),P_1)))) # label(fact_819_aux______3) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 146 (all Z_1 all W hAPP(real,real,times_times(real,Z_1),W) = hAPP(real,real,times_times(real,W),Z_1)) # label(fact_433_real__mult__commute) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 147 (all N_1 (hAPP(nat,int,power_power(int,number_number_of(int,min)),N_1) = number_number_of(int,min) | one_one(int) = hAPP(nat,int,power_power(int,number_number_of(int,min)),N_1))) # label(fact_412_neg__one__power) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 148 (all I_1 all J_1 all K_1 hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,minus_minus(nat,I_1),J_1)),K_1) = hAPP(nat,nat,minus_minus(nat,I_1),hAPP(nat,nat,plus_plus(nat,J_1),K_1))) # label(fact_613_diff__diff__left) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 149 (all X_a (real_normed_algebra(X_a) -> (all X_1 all Y_1 all Ya hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,X_1),Ya)),hAPP(X_a,X_a,times_times(X_a,Y_1),Ya)) = hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,plus_plus(X_a,X_1),Y_1)),Ya)))) # label(fact_888_mult__left_Oadd) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 150 (all V_1 all K_1 all V ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls)) -> hAPP(nat,nat,times_times(nat,number_number_of(nat,V)),hAPP(nat,nat,times_times(nat,number_number_of(nat,V_1)),K_1)) = zero_zero(nat)) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls)) -> hAPP(nat,nat,times_times(nat,number_number_of(nat,hAPP(int,int,times_times(int,V),V_1))),K_1) = hAPP(nat,nat,times_times(nat,number_number_of(nat,V)),hAPP(nat,nat,times_times(nat,number_number_of(nat,V_1)),K_1))))) # label(fact_330_nat__number__of__mult__left) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 151 (all X_a (number_ring(X_a) -> (all W number_number_of(X_a,bit0(W)) = hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,plus_plus(X_a,one_one(X_a)),one_one(X_a))),number_number_of(X_a,W))))) # label(fact_162_double__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 152 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,A_1),B)),zero_zero(int))) -> hAPP(int,int,div_mod(int,A_1),B) = hAPP(int,int,plus_plus(int,A_1),B)))) # label(fact_961_mod__pos__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 153 (all X_a (dvd(X_a) -> (all A_1 all B all K_1 (A_1 = hAPP(X_a,X_a,times_times(X_a,B),K_1) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),B),A_1)))))) # label(fact_453_dvdI) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 154 (all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,minus_minus(nat,M),N_1)),M))))) # label(fact_637_diff__less) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 155 (all Ma all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Ma)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,times_times(nat,Ma),N))))) # label(fact_716_nat__0__less__mult__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 156 (all X_a (linordered_semidom(X_a) -> (all X_1 all Y_1 (hAPP(nat,X_a,power_power(X_a,X_1),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,X_a,power_power(X_a,Y_1),number_number_of(nat,bit0(bit1(pls)))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y_1)) -> ti(X_a,X_1) = ti(X_a,Y_1))))))) # label(fact_255_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 157 (all B_1_1 all B_2_1 ti(fun(int,bool),zcong(B_1_1,B_2_1)) = zcong(B_1_1,B_2_1)) # label(tsy_c_IntPrimes_Ozcong_res) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 158 (all X_a (linordered_semidom(X_a) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),one_one(X_a))))) # label(fact_506_zero__le__one) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 159 (all U all Ma all N all J_2 all I_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_2),I_2)) -> (hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,J_2),U)),N) = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,I_2),U)),Ma) <-> hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,hAPP(nat,nat,minus_minus(nat,I_2),J_2)),U)),Ma) = N))) # label(fact_734_nat__eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 160 (all X_a (linord581940658strict(X_a) -> (all C all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,C),A_1)),hAPP(X_a,X_a,times_times(X_a,C),B)))))))) # label(fact_488_mult__strict__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 161 (all Y_1 all X_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,times_times(real,X_1),Y_1)))))) # label(fact_522_real__mult__order) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 162 (all X_a (monoid_mult(X_a) -> (all A_1 all N_1 hAPP(X_a,X_a,times_times(X_a,hAPP(nat,X_a,power_power(X_a,A_1),N_1)),A_1) = hAPP(X_a,X_a,times_times(X_a,A_1),hAPP(nat,X_a,power_power(X_a,A_1),N_1))))) # label(fact_259_power__commutes) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 163 (all X_a (number_ring(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,times_times(X_a,number_number_of(X_a,bit1(pls))),A_1)))) # label(fact_155_mult__numeral__1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 164 (all X_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),P_1)) -> hBOOL(hAPP(int,bool,zcong(multInv(P_1,multInv(P_1,X_1)),hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,X_1),multInv(P_1,X_1))),multInv(P_1,multInv(P_1,X_1)))),P_1)))))) # label(fact_838_Int2_Oaux____1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 165 (all X_a (linorder(X_a) & dvd(X_a) & plus(X_a) -> (all D all Sa exists Z all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),X)) -> (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),hAPP(X_a,X_a,plus_plus(X_a,X),Sa))) <-> -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),hAPP(X_a,X_a,plus_plus(X_a,X),Sa)))))))) # label(fact_864_pinf_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 166 (all X_a (linord581940658strict(X_a) -> (all A_3 all B_2 all C_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),A_3)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,C_1),A_3)),hAPP(X_a,X_a,times_times(X_a,C_1),B_2)))))))) # label(fact_495_mult__less__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 167 (all K_1 all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,times_times(nat,K_1),I_1)),hAPP(nat,nat,times_times(nat,K_1),J_1))))) # label(fact_700_mult__le__mono2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 168 (all X_a (comm_semiring_1(X_a) -> (all X_1 all Y_1 all Q_1 hAPP(X_a,X_a,times_times(X_a,hAPP(nat,X_a,power_power(X_a,X_1),Q_1)),hAPP(nat,X_a,power_power(X_a,Y_1),Q_1)) = hAPP(nat,X_a,power_power(X_a,hAPP(X_a,X_a,times_times(X_a,X_1),Y_1)),Q_1)))) # label(fact_115_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 169 (all X_a (comm_semiring_1(X_a) -> (all Lx all Ly all Rx all Ry hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,times_times(X_a,Lx),Rx)),hAPP(X_a,X_a,times_times(X_a,Ly),Ry)) = hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,times_times(X_a,Lx),Ly)),hAPP(X_a,X_a,times_times(X_a,Rx),Ry))))) # label(fact_66_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 170 (all X_a (semiring_1(X_a) -> hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(X_a))) # label(fact_251_zero__power2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 171 (all X_a (semiring_div(X_a) -> (all A_1 all C all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),C),B)) -> hAPP(X_a,X_a,div_mod(X_a,A_1),C) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,div_mod(X_a,A_1),B)),C))))) # label(fact_925_mod__mod__cancel) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 172 (all X_1 all M all Y_1 hAPP(int,int,div_mod(int,hAPP(int,int,minus_minus(int,X_1),Y_1)),M) = hAPP(int,int,div_mod(int,hAPP(int,int,minus_minus(int,hAPP(int,int,div_mod(int,X_1),M)),Y_1)),M)) # label(fact_946_zdiff__zmod__left) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 173 (all K_1 all L_1 hAPP(int,int,plus_plus(int,bit0(K_1)),bit1(L_1)) = bit1(hAPP(int,int,plus_plus(int,K_1),L_1))) # label(fact_152_add__Bit0__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 174 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(K)),pls)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),pls)))) # label(fact_55_rel__simps_I29_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 175 (all N all K all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),Ma)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),N)) -> (hAPP(nat,nat,minus_minus(nat,Ma),K) = hAPP(nat,nat,minus_minus(nat,N),K) <-> Ma = N)))) # label(fact_625_eq__diff__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 176 (all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1)) | N_1 = M -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)))) # label(fact_668_less__or__eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 177 (all X_1 all Y_1 all Z_1 hAPP(nat,int,power_power(int,hAPP(nat,int,power_power(int,X_1),Y_1)),Z_1) = hAPP(nat,int,power_power(int,X_1),hAPP(nat,nat,times_times(nat,Y_1),Z_1))) # label(fact_28_zpower__zpower) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 178 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K1)),bit0(K2))))) # label(fact_46_less__eq__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.96 179 (all X_a (ordere216010020id_add(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,A_1),B)))))))) # label(fact_799_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 180 (all X_a (linordered_semidom(X_a) -> (all A_1 all N_1 all N_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),N_3)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N_1)),hAPP(nat,X_a,power_power(X_a,A_1),N_3)))))))) # label(fact_275_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 181 (all X_1 all Y_1 all M hAPP(int,int,div_mod(int,hAPP(int,int,minus_minus(int,X_1),Y_1)),M) = hAPP(int,int,div_mod(int,hAPP(int,int,minus_minus(int,X_1),hAPP(int,int,div_mod(int,Y_1),M))),M)) # label(fact_945_zdiff__zmod__right) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 182 (all A_1 all B hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit1(bit1(pls))))),hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,number_number_of(int,bit1(bit1(pls)))),hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls)))))),B))),hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,number_number_of(int,bit1(bit1(pls)))),A_1)),hAPP(nat,int,power_power(int,B),number_number_of(nat,bit0(bit1(pls))))))),hAPP(nat,int,power_power(int,B),number_number_of(nat,bit1(bit1(pls))))) = hAPP(nat,int,power_power(int,hAPP(int,int,plus_plus(int,A_1),B)),number_number_of(nat,bit1(bit1(pls))))) # label(fact_8_zadd__power3) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 183 (all X_a (ordered_semiring(X_a) -> (all C all D_2 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_2)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),D_2)))))))))) # label(fact_473_mult__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 184 (all M all N_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,minus_minus(nat,M),N_1)),M))) # label(fact_621_Nat_Odiff__le__self) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 185 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(K1)),bit0(K2))))) # label(fact_93_less__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 186 (all M all K_1 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,M),K_1)),N_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)))) # label(fact_683_add__leD1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 187 (all X_2 all P_2 ((all A_5 ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_5)) -> hBOOL(hAPP(int,bool,P_2,hAPP(int,int,minus_minus(int,A_5),one_one(int))))) -> hBOOL(hAPP(int,bool,P_2,A_5)))) -> hBOOL(hAPP(int,bool,P_2,X_2)))) # label(fact_851_d22set__induct__old) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 188 (all M all N_1 all K_1 all L_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_1),L_1)) -> (hAPP(nat,nat,plus_plus(nat,K_1),N_1) = hAPP(nat,nat,plus_plus(nat,M),L_1) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1))))) # label(fact_660_less__add__eq__less) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 189 (all X_a (idom(X_a) -> (all C_1 all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,times_times(X_a,C_1),A_3)),hAPP(X_a,X_a,times_times(X_a,C_1),B_2))) <-> zero_zero(X_a) = ti(X_a,C_1) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_3),B_2)))))) # label(fact_512_dvd__mult__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 190 (all B_1_1 all X_a (power(X_a) -> power_power(X_a,ti(X_a,B_1_1)) = power_power(X_a,B_1_1))) # label(tsy_c_Power_Opower__class_Opower_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 191 (all X_a (linordered_semidom(X_a) -> -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),zero_zero(X_a))))) # label(fact_509_not__one__less__zero) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 192 (all R_1 all Q_1 all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hAPP(int,int,plus_plus(int,R_1),hAPP(int,int,times_times(int,A_1),Q_1)) = ti(int,A_1) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_1),one_one(int))))))) # label(fact_281_self__quotient__aux2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 193 (all X_a (linord581940658strict(X_a) -> (all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,times_times(X_a,A_3),B_2))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_3)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B_2)) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),zero_zero(X_a))) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),zero_zero(X_a))))))) # label(fact_486_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 194 (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]. 1.59/1.97 195 (all M all N_1 all K_1 hAPP(nat,nat,times_times(nat,hAPP(nat,nat,plus_plus(nat,M),N_1)),K_1) = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,M),K_1)),hAPP(nat,nat,times_times(nat,N_1),K_1))) # label(fact_696_add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 196 (all X_a (number_semiring(X_a) -> hAPP(X_a,X_a,plus_plus(X_a,one_one(X_a)),one_one(X_a)) = number_number_of(X_a,bit0(bit1(pls))))) # label(fact_169_semiring__one__add__one__is__two) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 197 (all K_1 all L_1 all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_1),L_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,I_1),K_1)),hAPP(nat,nat,plus_plus(nat,J_1),L_1)))))) # label(fact_659_add__less__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 198 (all X_a ti(X_a,undefined(X_a)) = undefined(X_a)) # label(tsy_c_HOL_Oundefined_res) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 199 (all M all N_1 (M = hAPP(nat,nat,plus_plus(nat,M),N_1) -> N_1 = zero_zero(nat))) # label(fact_639_add__eq__self__zero) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 200 (all C all A_1 all B (A_1 = B -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),C)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C))))) # label(fact_594_dvd_Oord__eq__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 201 (all M all N_1 hAPP(nat,nat,plus_plus(nat,M),N_1) = hAPP(nat,nat,plus_plus(nat,N_1),M)) # label(fact_606_nat__add__commute) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 202 (all A_1 all B all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M),B)) -> hAPP(int,int,div_mod(int,A_1),M) = hAPP(int,int,div_mod(int,hAPP(int,int,div_mod(int,A_1),B)),M)))) # label(fact_953_zmod__zdvd__zmod) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 203 (all N_1 N_1 = hAPP(nat,nat,plus_plus(nat,zero_zero(nat)),N_1)) # label(fact_642_plus__nat_Oadd__0) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 204 (all B_1_1 all B_2_1 inv(B_1_1,B_2_1) = inv(ti(int,B_1_1),B_2_1)) # label(tsy_c_WilsonRuss_Oinv_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 205 (all X_a all B_1_1 collect(X_a,ti(fun(X_a,bool),B_1_1)) = collect(X_a,B_1_1)) # label(tsy_c_Set_OCollect_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 206 (all X_1 all Y_1 all N_1 (zero_zero(nat) != N_1 -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,power_power(nat,X_1),N_1)),Y_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1))))) # label(fact_525_divides__exp2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 207 (all B_2 all A_3 all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(int,int,minus_minus(int,P_3),one_one(int)))) -> (hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),wset(A_3,P_3))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),B_2)))))) # label(fact_846_wset__g__1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 208 (all K all L (bit1(L) = bit1(K) <-> ti(int,L) = ti(int,K))) # label(fact_84_rel__simps_I51_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 209 (all X_a (semiring_div(X_a) -> (all A_1 all B all C hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),C) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),hAPP(X_a,X_a,div_mod(X_a,B),C))),C)))) # label(fact_932_zmod__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 210 (all X_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),P_1)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,X_1),multInv(P_1,X_1)),one_one(int)),P_1)))))) # label(fact_840_MultInv__prop2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 211 (all B all Q_1 all R_1 all B_3 all Q_4 all R_3 (hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B_3),Q_4)),R_3) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),Q_1)),R_1) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B_3),Q_4)),R_3)),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_1),B)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_3),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_4),Q_1))))))))) # label(fact_354_zdiv__mono2__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 212 (all X_a (order(X_a) -> ti(fun(X_a,fun(X_a,bool)),ord_less_eq(X_a)) = ord_less_eq(X_a))) # label(tsy_c_Orderings_Oord__class_Oless__eq_res) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 213 (all X_1 hAPP(nat,real,power_power(real,hAPP(real,real,times_times(real,number_number_of(real,bit0(bit1(pls)))),X_1)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(real,real,times_times(real,number_number_of(real,bit0(bit0(bit1(pls))))),hAPP(nat,real,power_power(real,X_1),number_number_of(nat,bit0(bit1(pls)))))) # label(fact_175_four__x__squared) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 214 (all A_3 all B_2 all N (N != zero_zero(nat) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,power_power(nat,A_3),N)),hAPP(nat,nat,power_power(nat,B_2),N))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_3),B_2))))) # label(fact_878_pow__divides__eq__nat) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 215 (all X_a (ordere223160158up_add(X_a) -> (all C all D_2 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_2)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),C)),hAPP(X_a,X_a,plus_plus(X_a,B),D_2)))))))) # label(fact_808_add__less__le__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 216 (all X_a (ordered_ab_group_add(X_a) -> (all A_3 all B_2 all C_1 all D (hAPP(X_a,X_a,minus_minus(X_a,C_1),D) = hAPP(X_a,X_a,minus_minus(X_a,A_3),B_2) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),D))))))) # label(fact_790_diff__eq__diff__less) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 217 (all X_a (semiring_div(X_a) -> (all A_1 all B all C hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,div_mod(X_a,A_1),C)),hAPP(X_a,X_a,div_mod(X_a,B),C))),C) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),C)))) # label(fact_931_mod__add__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 218 (all M all N_1 (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1)) -> hAPP(nat,nat,div_mod(nat,M),N_1) = hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,minus_minus(nat,M),N_1)),N_1))) # label(fact_977_mod__geq) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 219 (all B all M all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),M)) -> (hBOOL(hAPP(int,bool,zcong(A_1,B),M)) -> ti(int,A_1) = ti(int,B))))))) # label(fact_351_zcong__zless__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 220 (all I_1 all K_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),J_1)) -> hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,plus_plus(nat,I_1),K_1)),J_1) = hAPP(nat,nat,minus_minus(nat,I_1),hAPP(nat,nat,minus_minus(nat,J_1),K_1)))) # label(fact_685_diff__diff__right) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 221 (all B_1_1 all B_2_1 ti(int,standardRes(B_1_1,B_2_1)) = standardRes(B_1_1,B_2_1)) # label(tsy_c_Residues_OStandardRes_res) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 222 (all Z_1 ti(int,Z_1) = hAPP(int,int,times_times(int,Z_1),one_one(int))) # label(fact_129_zmult__1__right) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 223 (all X_a (linord20386208strict(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,C),A_1)),hAPP(X_a,X_a,times_times(X_a,C),B)))))))) # label(fact_491_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 224 (all W hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W),W))) # label(fact_20_zle__refl) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 225 (all X_a (zero_neq_one(X_a) -> one_one(X_a) != zero_zero(X_a))) # label(fact_445_zero__neq__one) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 226 (all B_1_1 minus_minus(nat,ti(nat,B_1_1)) = minus_minus(nat,B_1_1)) # label(tsy_c_Groups_Ominus__class_Ominus_1_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 227 (all X_b (ring(X_b) & number(X_b) -> (all A_1 all B all V hAPP(X_b,X_b,minus_minus(X_b,hAPP(X_b,X_b,times_times(X_b,A_1),number_number_of(X_b,V))),hAPP(X_b,X_b,times_times(X_b,B),number_number_of(X_b,V))) = hAPP(X_b,X_b,times_times(X_b,hAPP(X_b,X_b,minus_minus(X_b,A_1),B)),number_number_of(X_b,V))))) # label(fact_377_left__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 228 t = one_one(int) -> (exists X exists Y hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,X),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,Y),number_number_of(nat,bit0(bit1(pls))))) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int))) # label(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]. 1.59/1.97 229 (all X_1 hAPP(nat,int,power_power(int,hAPP(nat,int,power_power(int,X_1),number_number_of(nat,bit0(bit1(pls))))),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,int,power_power(int,X_1),number_number_of(nat,bit0(bit0(bit1(pls)))))) # label(fact_164_quartic__square__square) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 230 (all B all A_1 all C (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),C)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),A_1)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),B))))) # label(fact_334_Euler_Oaux2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 231 (all U all Ma all N all I_2 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_2)) -> (hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,J_2),U)),N) = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,I_2),U)),Ma) <-> Ma = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,hAPP(nat,nat,minus_minus(nat,J_2),I_2)),U)),N)))) # label(fact_737_nat__eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 232 (all X_a (monoid_mult(X_a) -> (all A_1 hAPP(X_a,X_a,times_times(X_a,A_1),one_one(X_a)) = ti(X_a,A_1)))) # label(fact_793_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 233 (all K_1 all A_1 all B all M (hBOOL(hAPP(int,bool,zcong(A_1,B),M)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_1),K_1),hAPP(int,int,times_times(int,B),K_1)),M)))) # label(fact_339_zcong__scalar) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 234 (all X_a (linord581940658strict(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,times_times(X_a,A_1),B)))))))) # label(fact_493_mult__neg__neg) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 235 (all N_1 all M hAPP(nat,nat,minus_minus(nat,N_1),hAPP(nat,nat,plus_plus(nat,N_1),M)) = zero_zero(nat)) # label(fact_643_diff__add__0) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 236 (all A_3 all B_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),B_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),A_3)) -> hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),d22set(A_3)))))) # label(fact_860_d22set__mem) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 237 (all X_a (real_normed_algebra(X_a) -> (all Xa all X_1 all Y_1 hAPP(X_a,X_a,times_times(X_a,Xa),hAPP(X_a,X_a,minus_minus(X_a,X_1),Y_1)) = hAPP(X_a,X_a,minus_minus(X_a,hAPP(X_a,X_a,times_times(X_a,Xa),X_1)),hAPP(X_a,X_a,times_times(X_a,Xa),Y_1))))) # label(fact_893_mult__right_Odiff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 238 (all N_1 all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,power_power(int,X_1),N_1))))) # label(fact_348_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 239 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1)) -> hAPP(int,int,plus_plus(int,one_one(int)),hAPP(int,int,times_times(int,number_number_of(int,bit0(bit1(pls)))),hAPP(int,int,div_mod(int,B),A_1))) = hAPP(int,int,div_mod(int,hAPP(int,int,plus_plus(int,one_one(int)),hAPP(int,int,times_times(int,number_number_of(int,bit0(bit1(pls)))),B))),hAPP(int,int,times_times(int,number_number_of(int,bit0(bit1(pls)))),A_1)))) # label(fact_971_pos__zmod__mult__2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 240 (all X_a (ab_semigroup_add(X_a) -> (all A_1 all B all C hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),C) = hAPP(X_a,X_a,plus_plus(X_a,A_1),hAPP(X_a,X_a,plus_plus(X_a,B),C))))) # label(fact_758_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 241 (all X_a (number_ring(X_a) -> one_one(X_a) = number_number_of(X_a,bit1(pls)))) # label(fact_159_semiring__norm_I110_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 242 (all X_a (linordered_semidom(X_a) -> (all N_1 all M (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),M)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),N_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(X_a,X_a,times_times(X_a,M),N_1)))))))) # label(fact_510_less__1__mult) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 243 (all A_1 -(-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),one_one(nat)),A_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),one_one(nat))))) # label(fact_872_gcd__lcm__complete__lattice__nat_Onot__less__bot) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 244 (all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),M)) -> -hBOOL(hAPP(int,bool,zcong(one_one(int),number_number_of(int,min)),M)))) # label(fact_365_one__not__neg__one__mod__m) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 245 (all X_a (linordered_idom(X_a) -> (all Y_1 all X_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_1),one_one(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,Y_1),X_1)),X_1)))))))) # label(fact_540_mult__left__le__one__le) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 246 (all M all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),M)) -> (hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),M)) -> ti(int,A_1) = zero_zero(int))))) # label(fact_352_zcong__zless__0) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 247 (all A_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),one_one(nat))) -> one_one(nat) = A_1)) # label(fact_876_gcd__lcm__complete__lattice__nat_Ole__bot) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 248 (all M all N_1 (hAPP(nat,nat,times_times(nat,M),N_1) = M -> N_1 = one_one(nat) | M = zero_zero(nat))) # label(fact_717_mult__eq__self__implies__10) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 249 (all X_a (linordered_ring(X_a) -> (all X_1 all Y_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,X_1),X_1)),hAPP(X_a,X_a,times_times(X_a,Y_1),Y_1))))))) # label(fact_235_sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 250 (all B zero_zero(int) = hAPP(int,int,div_mod(int,zero_zero(int)),B)) # label(fact_922_zmod__zero) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 251 (all X_a (power(X_a) & semiring_0(X_a) -> (all W ((zero_zero(nat) != number_number_of(nat,W) -> zero_zero(X_a) = hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),number_number_of(nat,W))) & (number_number_of(nat,W) = zero_zero(nat) -> one_one(X_a) = hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),number_number_of(nat,W))))))) # label(fact_350_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 252 (all A_1 all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),hAPP(int,int,div_mod(int,A_1),B))))) # label(fact_906_neg__mod__bound) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 253 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(K)),bit0(L))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L)))) # label(fact_94_rel__simps_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 254 (all K_1 all I_1 all J_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_1),J_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,K_1),I_1)),hAPP(int,int,times_times(int,K_1),J_1)))))) # label(fact_230_zmult__zless__mono2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 255 (all A_1 all B hBOOL(hAPP(int,bool,zcong(A_1,B),one_one(int)))) # label(fact_336_zcong__1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 256 (all Ma all Ta all K (ti(int,K) != zero_zero(int) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Ma),Ta)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,times_times(int,K),Ma)),hAPP(int,int,times_times(int,K),Ta)))))) # label(fact_743_zdvd__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 257 (all B_1_1 all B_2_1 wset(B_1_1,B_2_1) = wset(B_1_1,ti(int,B_2_1))) # label(tsy_c_WilsonRuss_Owset_arg2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 258 (all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),P_1)) -> (hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_1),A_1),one_one(int)),P_1)) -> ti(int,A_1) = one_one(int) | hAPP(int,int,minus_minus(int,P_1),one_one(int)) = ti(int,A_1)))))) # label(fact_394_zcong__square__zless) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 259 (all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)) -> hAPP(nat,nat,minus_minus(nat,M),N_1) = zero_zero(nat))) # label(fact_646_diff__is__0__eq_H) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 260 (exists X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int)))) & hBOOL(hAPP(int,bool,zcong(s1,X),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int)))) & (all Y (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int)))) & hBOOL(hAPP(int,bool,zcong(s1,Y),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int)))) -> ti(int,Y) = ti(int,X))))) # label(fact_182__096EX_B_As_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_A1_A_G_A_091s1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 261 (all C all D_2 all A_1 all B all M (hBOOL(hAPP(int,bool,zcong(A_1,B),M)) -> (hBOOL(hAPP(int,bool,zcong(C,D_2),M)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_1),C),hAPP(int,int,times_times(int,B),D_2)),M))))) # label(fact_337_zcong__zmult) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 262 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),bit0(K))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),K)))) # label(fact_304_rel__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 263 (all X_a (group_add(X_a) -> (all A_3 all B_2 (zero_zero(X_a) = hAPP(X_a,X_a,minus_minus(X_a,A_3),B_2) <-> ti(X_a,B_2) = ti(X_a,A_3))))) # label(fact_785_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 264 (all C all A_1 all B (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),A_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) -> (B = C -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),C),A_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C))))) # label(fact_580_dvd_Oord__less__eq__trans) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 265 (all X_a (monoid_add(X_a) -> (all A_1 hAPP(X_a,X_a,plus_plus(X_a,A_1),zero_zero(X_a)) = ti(X_a,A_1)))) # label(fact_769_add__0__right) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 266 (all X_a (number_ring(X_a) -> zero_zero(X_a) = number_number_of(X_a,pls))) # label(fact_221_semiring__norm_I112_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 267 (all X_a (linordered_semidom(X_a) -> -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),zero_zero(X_a))))) # label(fact_507_not__one__le__zero) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 268 (all B_1_1 ti(int,bit0(B_1_1)) = bit0(B_1_1)) # label(tsy_c_Int_OBit0_res) # label(hypothesis) # label(non_clause). [assumption]. 1.59/1.97 269 (all N_1 -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),N_1))) # label(fact_570_less__irrefl__nat) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 270 (all Ma all N (one_one(nat) = hAPP(nat,nat,times_times(nat,Ma),N) <-> one_one(nat) = N & Ma = one_one(nat))) # label(fact_704_nat__mult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 271 (all X_a (linord219039673up_add(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_3),A_3)),zero_zero(X_a))))))) # label(fact_798_double__add__le__zero__iff__single__add__le__zero) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 272 (all X_a (group_add(X_a) -> (all A_1 all B hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,minus_minus(X_a,A_1),B)),B) = ti(X_a,A_1)))) # label(fact_796_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 273 (all X_2 all Y_2 all Z_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Z_2)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,X_2),Z_2)),hAPP(real,real,times_times(real,Y_2),Z_2))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_2),Y_2))))) # label(fact_519_real__mult__less__iff1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 274 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all X_2 all Y_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),number_number_of(X_a,X_2)),number_number_of(X_a,Y_2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_2),Y_2)))))) # label(fact_30_less__number__of) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 275 (all X_a (number_ring(X_a) -> (all V all W all C hAPP(X_a,X_a,plus_plus(X_a,number_number_of(X_a,V)),hAPP(X_a,X_a,minus_minus(X_a,number_number_of(X_a,W)),C)) = hAPP(X_a,X_a,minus_minus(X_a,number_number_of(X_a,hAPP(int,int,plus_plus(int,V),W))),C)))) # label(fact_384_add__number__of__diff1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 276 (all Ma all N (Ma = zero_zero(nat) & zero_zero(nat) = N <-> hAPP(nat,nat,plus_plus(nat,Ma),N) = zero_zero(nat))) # label(fact_640_add__is__0) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 277 (all X_a (monoid_mult(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,times_times(X_a,one_one(X_a)),A_1)))) # label(fact_791_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 278 (all X_1 hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X_1),hAPP(nat,int,power_power(int,X_1),number_number_of(nat,bit0(bit1(pls))))))) # label(fact_137_power2__ge__self) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 279 (all X_a (dvd(X_a) & linorder(X_a) & plus(X_a) -> (all D all Sa exists Z all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Z)) -> (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),hAPP(X_a,X_a,plus_plus(X_a,X),Sa))) <-> -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),hAPP(X_a,X_a,plus_plus(X_a,X),Sa)))))))) # label(fact_863_minf_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 280 (all X_a (ordere453448008miring(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,B),A_1)),zero_zero(X_a)))))))) # label(fact_482_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 281 (all X_a (linord20386208strict(X_a) -> (all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,times_times(X_a,A_1),B))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B))))))) # label(fact_497_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 282 (all P_3 all A_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_3)) -> hBOOL(hAPP(fun(int,bool),bool,member(int,A_3),wset(A_3,P_3))))) # label(fact_843_wset__mem__mem) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 283 (all X_a (power(X_a) & zero_neq_one(X_a) & no_zero_divisors(X_a) & mult_zero(X_a) -> (all A_3 all W_1 (number_number_of(nat,W_1) != zero_zero(nat) & zero_zero(X_a) = ti(X_a,A_3) <-> zero_zero(X_a) = hAPP(nat,X_a,power_power(X_a,A_3),number_number_of(nat,W_1)))))) # label(fact_343_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 284 (all X_a (number_ring(X_a) -> (all V all W all Z_1 hAPP(X_a,X_a,times_times(X_a,number_number_of(X_a,V)),hAPP(X_a,X_a,times_times(X_a,number_number_of(X_a,W)),Z_1)) = hAPP(X_a,X_a,times_times(X_a,number_number_of(X_a,hAPP(int,int,times_times(int,V),W))),Z_1)))) # label(fact_145_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 285 (all X_a (linordered_idom(X_a) -> (all Y_1 all X_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_1),one_one(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,X_1),Y_1)),X_1)))))))) # label(fact_541_mult__right__le__one__le) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 286 (all X_1 all Y_1 all Z_1 hAPP(nat,int,power_power(int,X_1),hAPP(nat,nat,plus_plus(nat,Y_1),Z_1)) = hAPP(int,int,times_times(int,hAPP(nat,int,power_power(int,X_1),Y_1)),hAPP(nat,int,power_power(int,X_1),Z_1))) # label(fact_34_zpower__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 287 (all K_1 all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,times_times(nat,K_1),M)),hAPP(nat,nat,times_times(nat,K_1),N_1))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),N_1))))) # label(fact_719_dvd__mult__cancel) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 288 (all X_a (comm_semiring(X_a) -> (all A_1 all B all C hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),C) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),C))))) # label(fact_448_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 289 (all B_1_1 ti(fun(int,bool),sr(B_1_1)) = sr(B_1_1)) # label(tsy_c_Residues_OSR_res) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 290 (all X_a (linord20386208strict(X_a) -> (all C all D_2 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_2)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),D_2)))))))))) # label(fact_533_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 291 (all X_a (semiring_div(X_a) -> (all A_1 hAPP(X_a,X_a,div_mod(X_a,zero_zero(X_a)),A_1) = zero_zero(X_a)))) # label(fact_944_mod__0) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 292 (all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> hAPP(int,int,div_mod(int,number_number_of(int,min)),B) = hAPP(int,int,minus_minus(int,B),one_one(int)))) # label(fact_970_zmod__minus1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 293 (all K_1 all L_1 bit1(hAPP(int,int,minus_minus(int,K_1),L_1)) = hAPP(int,int,minus_minus(int,bit1(K_1)),bit0(L_1))) # label(fact_378_diff__bin__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 294 (all X_a (comm_semiring_1(X_a) -> (all X_1 all Y_1 all Z_1 hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,X_1),Y_1)),hAPP(X_a,X_a,times_times(X_a,X_1),Z_1)) = hAPP(X_a,X_a,times_times(X_a,X_1),hAPP(X_a,X_a,plus_plus(X_a,Y_1),Z_1))))) # label(fact_112_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 295 (all X_2 all Y_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_2),X_2)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_2),Y_2)) <-> X_2 = Y_2)) # label(fact_431_divides__antisym) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 296 (all N all P_2 (-hBOOL(hAPP(nat,bool,P_2,zero_zero(nat))) -> (hBOOL(hAPP(nat,bool,P_2,N)) -> (exists K_2 (hBOOL(hAPP(nat,bool,P_2,K_2)) & (all I (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),K_2)) -> -hBOOL(hAPP(nat,bool,P_2,I)))) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_2),N))))))) # label(fact_856_ex__least__nat__le) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 297 (all N_1 all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N_1),M)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),N_1)),M)) | M = N_1 | zero_zero(nat) = M)) # label(fact_419_divides__cases) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 298 (all M all K_1 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,M),K_1)),N_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N_1)))) # label(fact_682_add__leD2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 299 (all B_1_1 all X_a (no_zero_divisors(X_a) -> times_times(X_a,B_1_1) = times_times(X_a,ti(X_a,B_1_1)))) # label(tsy_c_Groups_Otimes__class_Otimes_5_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 300 (all N_1 all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1)) -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N_1),M))))) # label(fact_708_nat__dvd__not__less) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 301 (all X_a (linordered_semidom(X_a) -> (all X_1 all Y_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,X_1),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,X_a,power_power(X_a,Y_1),number_number_of(nat,bit0(bit1(pls)))))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1))))))) # label(fact_254_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 302 (all K all Ma all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,K),Ma)),hAPP(nat,nat,plus_plus(nat,K),N))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),N)))) # label(fact_655_nat__add__left__cancel__less) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 303 (all X_a (group_add(X_a) -> (all A_1 all B ti(X_a,A_1) = hAPP(X_a,X_a,minus_minus(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),B)))) # label(fact_795_add__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 304 (all X_a (semiring_div(X_a) -> (all A_1 all B zero_zero(X_a) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),B)),B)))) # label(fact_900_mod__mult__self2__is__0) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 305 (all X_a (linordered_semidom(X_a) -> (all N_1 all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N_1)),hAPP(X_a,X_a,times_times(X_a,A_1),hAPP(nat,X_a,power_power(X_a,A_1),N_1)))))))) # label(fact_176_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 306 (all V_2 all V_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),number_number_of(nat,V_2)),number_number_of(nat,V_3))) <-> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_2),V_3)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),V_3))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_2),V_3)))) # label(fact_237_less__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 307 (all N_1 ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),N_1),zero_zero(int))) -> zfact(N_1) = hAPP(int,int,times_times(int,N_1),zfact(hAPP(int,int,minus_minus(int,N_1),one_one(int))))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),N_1),zero_zero(int))) -> one_one(int) = zfact(N_1)))) # label(fact_754_zfact_Osimps) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 308 (all B_1_1 all B_2_1 multInv(B_1_1,B_2_1) = multInv(B_1_1,ti(int,B_2_1))) # label(tsy_c_Int2_OMultInv_arg2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 309 (all X_a (semiring_div(X_a) -> (all A_1 all B hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,div_mod(X_a,A_1),B)),B) = hAPP(X_a,X_a,div_mod(X_a,A_1),B)))) # label(fact_918_mod__mod__trivial) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 310 (all X_a (number_ring(X_a) -> (all V all W hAPP(X_a,X_a,minus_minus(X_a,number_number_of(X_a,V)),number_number_of(X_a,W)) = number_number_of(X_a,hAPP(int,int,minus_minus(int,V),W))))) # label(fact_369_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 311 (all A_1 all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,div_mod(int,A_1),B)),zero_zero(int))))) # label(fact_958_neg__mod__sign) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 312 (all K_1 all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,times_times(nat,K_1),I_1)),hAPP(nat,nat,times_times(nat,K_1),J_1)))))) # label(fact_712_mult__less__mono2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 313 (all X_a (ordered_semiring(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,C),A_1)),hAPP(X_a,X_a,times_times(X_a,C),B)))))))) # label(fact_478_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 314 (all A_1 all B hAPP(nat,int,power_power(int,hAPP(int,int,minus_minus(int,A_1),B)),number_number_of(nat,bit1(bit1(pls)))) = hAPP(int,int,minus_minus(int,hAPP(int,int,plus_plus(int,hAPP(int,int,minus_minus(int,hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit1(bit1(pls))))),hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,number_number_of(int,bit1(bit1(pls)))),hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls)))))),B))),hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,number_number_of(int,bit1(bit1(pls)))),A_1)),hAPP(nat,int,power_power(int,B),number_number_of(nat,bit0(bit1(pls))))))),hAPP(nat,int,power_power(int,B),number_number_of(nat,bit1(bit1(pls)))))) # label(fact_403_zdiff__power3) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 315 (all M all K_1 all N_1 hAPP(nat,nat,minus_minus(nat,M),N_1) = hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,plus_plus(nat,M),K_1)),hAPP(nat,nat,plus_plus(nat,N_1),K_1))) # label(fact_615_diff__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 316 (all Z1 all Z2 all W hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,Z1),Z2)),W) = hAPP(int,int,minus_minus(int,hAPP(int,int,times_times(int,Z1),W)),hAPP(int,int,times_times(int,Z2),W))) # label(fact_373_zdiff__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 317 (all X_1 all M hBOOL(hAPP(int,bool,zcong(X_1,hAPP(int,int,div_mod(int,X_1),M)),M))) # label(fact_913_mod__mod__is__mod) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 318 (all X_a (real_normed_algebra(X_a) -> (all B zero_zero(X_a) = hAPP(X_a,X_a,times_times(X_a,zero_zero(X_a)),B)))) # label(fact_887_mult_Ozero__left) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 319 (all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1)) -> (exists P_4 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_4),A_1)) & hBOOL(hAPP(int,bool,zprime,P_4)))))) # label(fact_750_zprime__factor__exists) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 320 (all X_a (comm_semiring_1(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),B),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),C))))))) # label(fact_427_dvd__trans) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 321 (all X_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),X_1))) # label(fact_556_dvd_Oorder__refl) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 322 (all C all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,times_times(nat,C),A_1)),hAPP(nat,nat,times_times(nat,C),B))))) # label(fact_465_divides__mul__l) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 323 (all X_a (ab_semigroup_mult(X_a) -> (all A_1 all B all C hAPP(X_a,X_a,times_times(X_a,A_1),hAPP(X_a,X_a,times_times(X_a,B),C)) = hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),B)),C)))) # label(fact_757_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 324 (all X_a (ordere236663937imp_le(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,C),A_1)),hAPP(X_a,X_a,plus_plus(X_a,C),B))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)))))) # label(fact_784_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 325 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),K)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),bit0(K))))) # label(fact_96_rel__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 326 (all A_1 all B hAPP(int,int,plus_plus(int,hAPP(int,int,minus_minus(int,hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls))))),hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit1(pls)))),A_1)),B))),hAPP(nat,int,power_power(int,B),number_number_of(nat,bit0(bit1(pls))))) = hAPP(nat,int,power_power(int,hAPP(int,int,minus_minus(int,A_1),B)),number_number_of(nat,bit0(bit1(pls))))) # label(fact_402_zdiff__power2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 327 (all X_a (number_ring(X_a) -> (all N_1 one_one(X_a) = hAPP(nat,X_a,power_power(X_a,number_number_of(X_a,min)),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),N_1))))) # label(fact_342_power__m1__even) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 328 (all X_a (linord581940658strict(X_a) -> (all X_2 all Y_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,X_2),X_2)),hAPP(X_a,X_a,times_times(X_a,Y_2),Y_2)))) <-> ti(X_a,Y_2) != zero_zero(X_a) | zero_zero(X_a) != ti(X_a,X_2))))) # label(fact_239_sum__squares__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 329 (all X_a (semiring(X_a) -> (all A_1 all E_1 all B all C hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),E_1)),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,B),E_1)),C)) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),E_1)),C)))) # label(fact_447_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 330 (all X_2 all W_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X_2)) | zero_zero(nat) = number_number_of(nat,W_1) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,power_power(nat,X_2),number_number_of(nat,W_1)))))) # label(fact_288_zero__less__power__nat__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 331 (all K_1 all M all N_1 hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,K_1),M)),hAPP(nat,nat,times_times(nat,K_1),N_1)) = hAPP(nat,nat,times_times(nat,K_1),hAPP(nat,nat,plus_plus(nat,M),N_1))) # label(fact_697_add__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 332 (all N all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Ma)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,times_times(nat,N),Ma)),Ma)) <-> one_one(nat) = N))) # label(fact_553_dvd__mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 333 (all X_a (semiring_div(X_a) -> (all A_1 all B all C hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),B)),C) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,div_mod(X_a,A_1),C)),B)),C)))) # label(fact_936_mod__mult__left__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 334 (all X_a (ordered_ring(X_a) -> (all A_3 all E all C_1 all B_2 all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,minus_minus(X_a,B_2),A_3)),E)),D))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,A_3),E)),C_1)),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,B_2),E)),D))))))) # label(fact_546_less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 335 (all A_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A_3)) <-> zero_zero(nat) = A_3)) # label(fact_868_gcd__lcm__complete__lattice__nat_Otop__unique) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 336 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(K)),pls)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),pls)))) # label(fact_92_rel__simps_I12_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 337 (all A_1 all N_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),hAPP(nat,int,power_power(int,A_1),N_1))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),A_1))))) # label(fact_220_zprime__zdvd__power) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 338 (all V_1 all V ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls)) -> hAPP(nat,nat,plus_plus(nat,number_number_of(nat,V)),number_number_of(nat,V_1)) = number_number_of(nat,V)) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls)) -> hAPP(nat,nat,plus_plus(nat,number_number_of(nat,V)),number_number_of(nat,V_1)) = number_number_of(nat,hAPP(int,int,plus_plus(int,V),V_1)))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls)) -> hAPP(nat,nat,plus_plus(nat,number_number_of(nat,V)),number_number_of(nat,V_1)) = number_number_of(nat,V_1)))) # label(fact_52_add__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 339 (all K all Ma all N (hAPP(nat,nat,times_times(nat,K),Ma) = hAPP(nat,nat,times_times(nat,K),N) <-> zero_zero(nat) = K | Ma = N)) # label(fact_651_mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 340 (all A_1 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1)) -> (exists X (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) & hAPP(nat,real,power_power(real,X),N_1) = A_1 & (all Y (hAPP(nat,real,power_power(real,Y),N_1) = A_1 & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y)) -> Y = X))))))) # label(fact_552_realpow__pos__nth__unique) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 341 (all X_a (comm_semiring_1(X_a) -> (all X_1 one_one(X_a) = hAPP(nat,X_a,power_power(X_a,X_1),zero_zero(nat))))) # label(fact_313_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 342 (all M hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),hAPP(nat,nat,times_times(nat,M),hAPP(nat,nat,times_times(nat,M),M))))) # label(fact_702_le__cube) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 343 (all M all N_1 (hAPP(nat,nat,minus_minus(nat,M),N_1) = zero_zero(nat) -> (hAPP(nat,nat,minus_minus(nat,N_1),M) = zero_zero(nat) -> M = N_1))) # label(fact_566_diffs0__imp__equal) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 344 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K)),bit0(L))))) # label(fact_44_rel__simps_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 345 (all X_a (ordered_ring(X_a) -> (all A_3 all E all C_1 all B_2 all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,A_3),E)),C_1)),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,B_2),E)),D))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,minus_minus(X_a,A_3),B_2)),E)),C_1)),D)))))) # label(fact_545_less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 346 (all B_1_1 all X_a (mult_zero(X_a) -> times_times(X_a,ti(X_a,B_1_1)) = times_times(X_a,B_1_1))) # label(tsy_c_Groups_Otimes__class_Otimes_4_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 347 (all N all K all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),Ma)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),N)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),N)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,minus_minus(nat,Ma),K)),hAPP(nat,nat,minus_minus(nat,N),K))))))) # label(fact_627_le__diff__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 348 (all A_3 all B_2 all Ma (hBOOL(hAPP(int,bool,zcong(hAPP(int,int,div_mod(int,A_3),Ma),hAPP(int,int,div_mod(int,B_2),Ma)),Ma)) <-> hBOOL(hAPP(int,bool,zcong(A_3,B_2),Ma)))) # label(fact_914_zcong__zmod) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 349 (all X_2 all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X_2)) | zero_zero(nat) = N <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,power_power(nat,X_2),N))))) # label(fact_287_zero__less__power__nat__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 350 (all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1)) -> M = hAPP(nat,nat,div_mod(nat,M),N_1))) # label(fact_984_mod__less) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 351 (all M all N_1 hAPP(nat,nat,times_times(nat,N_1),M) = hAPP(nat,nat,times_times(nat,M),N_1)) # label(fact_629_nat__mult__commute) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 352 (all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,minus_minus(int,P_1),one_one(int)))) -> zero_zero(int) != inv(P_1,A_1))))) # label(fact_827_inv__not__0) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 353 (all X_1 all Y_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) -> Y_1 = X_1 | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)))) # label(fact_596_dvd_Ole__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 354 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(K)),min)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),min)))) # label(fact_300_rel__simps_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 355 (all X_a (cancel_semigroup_add(X_a) -> (all B_2 all A_3 all C_1 (hAPP(X_a,X_a,plus_plus(X_a,B_2),A_3) = hAPP(X_a,X_a,plus_plus(X_a,C_1),A_3) <-> ti(X_a,B_2) = ti(X_a,C_1))))) # label(fact_760_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.59/1.97 356 (all X_a (monoid_mult(X_a) -> (all A_1 hAPP(nat,X_a,power_power(X_a,A_1),one_one(nat)) = ti(X_a,A_1)))) # label(fact_242_power__one__right) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 357 (all X_a (linordered_idom(X_a) -> (all X_2 all Y_2 (ti(X_a,Y_2) != zero_zero(X_a) | ti(X_a,X_2) != zero_zero(X_a) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,X_2),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,X_a,power_power(X_a,Y_2),number_number_of(nat,bit0(bit1(pls))))))))))) # label(fact_268_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 358 (all J_1 all K_1 all A_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_1)) -> (-hBOOL(hAPP(int,bool,zcong(K_1,zero_zero(int)),P_1)) -> (-hBOOL(hAPP(int,bool,zcong(J_1,zero_zero(int)),P_1)) -> (hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_1),multInv(P_1,J_1)),hAPP(int,int,times_times(int,A_1),multInv(P_1,K_1))),P_1)) -> hBOOL(hAPP(int,bool,zcong(J_1,K_1),P_1))))))))) # label(fact_836_MultInv__zcong__prop3) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 359 (all B_1_1 all X_a (power(X_a) -> power_power(X_a,B_1_1) = ti(fun(nat,X_a),power_power(X_a,B_1_1)))) # label(tsy_c_Power_Opower__class_Opower_res) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 360 (all X_a (comm_ring_1(X_a) -> (all Z_1 all X_1 all Y_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X_1),Y_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X_1),Z_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X_1),hAPP(X_a,X_a,minus_minus(X_a,Y_1),Z_1)))))))) # label(fact_461_dvd__diff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 361 (all X_a (linord20386208strict(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,B),A_1)),zero_zero(X_a)))))))) # label(fact_498_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 362 (all D_2 all C all A_1 all B all M (hBOOL(hAPP(int,bool,zcong(A_1,B),M)) -> (ti(int,B) = ti(int,C) -> (hBOOL(hAPP(int,bool,zcong(C,D_2),M)) -> hBOOL(hAPP(int,bool,zcong(A_1,D_2),M)))))) # label(fact_391_zcong__eq__trans) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 363 (all X_a (linordered_idom(X_a) -> (all X_1 all Y_1 -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,X_1),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,X_a,power_power(X_a,Y_1),number_number_of(nat,bit0(bit1(pls)))))),zero_zero(X_a)))))) # label(fact_267_not__sum__power2__lt__zero) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 364 (all M all N_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),hAPP(int,int,times_times(int,M),N_1))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),N_1)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),M))))) # label(fact_409_zprime__zdvd__zmult__better) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 365 (all X_1 all Y_1 all Z_1 hAPP(int,int,plus_plus(int,X_1),hAPP(int,int,plus_plus(int,Y_1),Z_1)) = hAPP(int,int,plus_plus(int,Y_1),hAPP(int,int,plus_plus(int,X_1),Z_1))) # label(fact_90_zadd__left__commute) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 366 (all T all A ti(T,ti(T,A)) = ti(T,A)) # label(help_ti_idem) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 367 (all X_b all X_c all X_a all B_1_1 all B_2_1 combb(X_b,X_c,X_a,B_1_1,B_2_1) = combb(X_b,X_c,X_a,ti(fun(X_b,X_c),B_1_1),B_2_1)) # label(tsy_c_COMBB_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 368 (all B_1_1 ti(fun(nat,nat),minus_minus(nat,B_1_1)) = minus_minus(nat,B_1_1)) # label(tsy_c_Groups_Ominus__class_Ominus_1_res) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 369 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),bit1(K))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),K)))) # label(fact_56_rel__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 370 (all N_1 zero_zero(nat) = hAPP(nat,nat,minus_minus(nat,zero_zero(nat)),N_1)) # label(fact_563_diff__0__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 371 (all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,plus_plus(int,one_one(int)),Z_1))))) # label(fact_250_le__imp__0__less) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 372 (all A_1 all P_1 ((hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_1)) -> legendre(A_1,P_1) = zero_zero(int)) & (-hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_1)) -> (-hBOOL(hAPP(int,bool,quadRes(P_1),A_1)) -> legendre(A_1,P_1) = number_number_of(int,min)) & (hBOOL(hAPP(int,bool,quadRes(P_1),A_1)) -> legendre(A_1,P_1) = one_one(int))))) # label(fact_418_Legendre__def) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 373 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B all C (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),B)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),C)))))) # label(fact_452_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 374 (all X_a (order(X_a) -> ord_less(X_a) = ti(fun(X_a,fun(X_a,bool)),ord_less(X_a)))) # label(tsy_c_Orderings_Oord__class_Oless_res) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 375 (all I_1 all K_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),J_1)) -> hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,minus_minus(nat,J_1),K_1)),I_1) = hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,plus_plus(nat,J_1),I_1)),K_1))) # label(fact_695_diff__add__assoc2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 376 (all Q all P (-hBOOL(P) | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,P),Q)) | -hBOOL(Q))) # label(help_fconj_1_1_U) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 377 (all P_3 sr(P_3) = collect(int,combs(int,bool,bool,combb(bool,fun(bool,bool),int,fconj,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int))),combc(int,int,bool,ord_less(int),P_3)))) # label(fact_883_SR__def) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 378 (all X_a (comm_semiring_1(X_a) -> (all Lx all Ly all Rx all Ry hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,times_times(X_a,Lx),Ly)),hAPP(X_a,X_a,times_times(X_a,Rx),Ry)) = hAPP(X_a,X_a,times_times(X_a,Lx),hAPP(X_a,X_a,times_times(X_a,Ly),hAPP(X_a,X_a,times_times(X_a,Rx),Ry)))))) # label(fact_68_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 379 (all B_3 all Q_4 all R_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B_3),Q_4)),R_3))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_3),B_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_3)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Q_4)))))) # label(fact_359_q__pos__lemma) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 380 (all X_a (linord219039673up_add(X_a) -> (all A_3 (hAPP(X_a,X_a,plus_plus(X_a,A_3),A_3) = zero_zero(X_a) <-> ti(X_a,A_3) = zero_zero(X_a))))) # label(fact_768_double__zero__sym) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 381 (all N_1 all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M)) -> hAPP(nat,nat,plus_plus(nat,N_1),hAPP(nat,nat,minus_minus(nat,M),N_1)) = M)) # label(fact_688_le__add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 382 (all B_1_1 d22set(B_1_1) = ti(fun(int,bool),d22set(B_1_1))) # label(tsy_c_IntFact_Od22set_res) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 383 (all X_c all X_a all B_1_1 all B_2_1 ti(X_c,hAPP(X_a,X_c,B_1_1,B_2_1)) = hAPP(X_a,X_c,B_1_1,B_2_1)) # label(tsy_c_hAPP_res) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 384 (all X_a (linordered_semiring(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,C),A_1)),hAPP(X_a,X_a,times_times(X_a,C),B))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))))))) # label(fact_536_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 385 (all K_1 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),N_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N_1))))) # label(fact_718_dvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 386 (all X_2 all Y_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),Y_2)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_2),Y_2)) | Y_2 = X_2)) # label(fact_429_less__eq__real__def) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 387 (all X_a (real_normed_algebra(X_a) -> (all Xa all X_1 all Y_1 hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,Xa),X_1)),hAPP(X_a,X_a,times_times(X_a,Xa),Y_1)) = hAPP(X_a,X_a,times_times(X_a,Xa),hAPP(X_a,X_a,plus_plus(X_a,X_1),Y_1))))) # label(fact_890_mult__right_Oadd) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 388 (all M all N_1 all K_1 hAPP(nat,nat,times_times(nat,hAPP(nat,nat,minus_minus(nat,M),N_1)),K_1) = hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,times_times(nat,M),K_1)),hAPP(nat,nat,times_times(nat,N_1),K_1))) # label(fact_630_diff__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 389 (all L (pls = bit0(L) <-> pls = ti(int,L))) # label(fact_121_rel__simps_I38_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 390 (all Z1 all Z2 all Z3 hAPP(int,int,plus_plus(int,Z1),hAPP(int,int,plus_plus(int,Z2),Z3)) = hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,Z1),Z2)),Z3)) # label(fact_89_zadd__assoc) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 391 (all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),P_3)) & (all M_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M_2),P_3)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M_2)) -> ti(int,M_2) = one_one(int) | ti(int,M_2) = ti(int,P_3))))) # label(fact_280_zprime__def) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 392 (all X_a (ordere216010020id_add(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),zero_zero(X_a)))))))) # label(fact_803_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 393 (all N_1 all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M),N_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N_1),M)) -> ti(int,M) = ti(int,N_1)))))) # label(fact_191_zdvd__antisym__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 394 (all U_1 all M all N_1 all J_1 all I_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_1),I_1)) -> hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,hAPP(nat,nat,minus_minus(nat,I_1),J_1)),U_1)),M)),N_1) = hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,I_1),U_1)),M)),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,J_1),U_1)),N_1)))) # label(fact_733_nat__diff__add__eq1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 395 (all X_a (idom(X_a) -> (all A_3 all C_1 all B_2 (zero_zero(X_a) = ti(X_a,C_1) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_3),B_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,times_times(X_a,A_3),C_1)),hAPP(X_a,X_a,times_times(X_a,B_2),C_1))))))) # label(fact_513_dvd__mult__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 396 (all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) -> wset(hAPP(int,int,minus_minus(int,P_3),number_number_of(int,bit0(bit1(pls)))),P_3) = d22set(hAPP(int,int,minus_minus(int,P_3),number_number_of(int,bit0(bit1(pls))))))) # label(fact_855_d22set__eq__wset) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 397 (all X_2 all Y_2 (zero_zero(int) = hAPP(int,int,div_mod(int,number_number_of(int,Y_2)),number_number_of(int,X_2)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),number_number_of(int,X_2)),number_number_of(int,Y_2))))) # label(fact_898_zdvd__iff__zmod__eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 398 (all X_a (ordere236663937imp_le(X_a) -> (all A_3 all C_1 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_3),C_1)),hAPP(X_a,X_a,plus_plus(X_a,B_2),C_1))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)))))) # label(fact_778_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 399 (all X_a (linord1278240602ring_1(X_a) -> (all V all U_1 all Y_1 all X_1 all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_1),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),U_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),V)) -> (one_one(X_a) = hAPP(X_a,X_a,plus_plus(X_a,U_1),V) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,U_1),X_1)),hAPP(X_a,X_a,times_times(X_a,V),Y_1))),A_1)))))))))) # label(fact_548_convex__bound__le) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 400 (all I_1 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),N_1)) -> I_1 = hAPP(nat,nat,minus_minus(nat,N_1),hAPP(nat,nat,minus_minus(nat,N_1),I_1)))) # label(fact_624_diff__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 401 (all W pls = hAPP(int,int,times_times(int,pls),W)) # label(fact_123_mult__Pls) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 402 (all B_1_1 all X_a (number(X_a) -> number_number_of(X_a,B_1_1) = number_number_of(X_a,ti(int,B_1_1)))) # label(tsy_c_Int_Onumber__class_Onumber__of_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 403 (all X_a (linordered_semidom(X_a) -> (all B all C all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),hAPP(X_a,X_a,plus_plus(X_a,A_1),C)))))))) # label(fact_505_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 404 (all B_1_1 all B_2_1 legendre(B_1_1,B_2_1) = legendre(B_1_1,ti(int,B_2_1))) # label(tsy_c_Residues_OLegendre_arg2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 405 (all X_1 all M hBOOL(hAPP(int,bool,zcong(X_1,standardRes(M,X_1)),M))) # label(fact_988_StandardRes__prop1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 406 (all Ma all N (zero_zero(nat) = hAPP(nat,nat,minus_minus(nat,Ma),N) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),N)))) # label(fact_647_diff__is__0__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 407 (all X_a (linordered_idom(X_a) -> (all X_2 all Y_2 (hAPP(X_a,X_a,plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,X_2),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,X_a,power_power(X_a,Y_2),number_number_of(nat,bit0(bit1(pls))))) = zero_zero(X_a) <-> ti(X_a,X_2) = zero_zero(X_a) & ti(X_a,Y_2) = zero_zero(X_a))))) # label(fact_258_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 408 (all X_a (linord20386208strict(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,times_times(X_a,A_1),B)))))))) # label(fact_500_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 409 (all A_1 all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,div_mod(int,A_1),B))))) # label(fact_955_pos__mod__sign) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 410 (all B_1_1 all B_2_1 product_Pair(int,int,B_1_1,B_2_1) = product_Pair(int,int,B_1_1,ti(int,B_2_1))) # label(tsy_c_Product__Type_OPair_arg2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 411 (all N_1 all M hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),hAPP(nat,nat,plus_plus(nat,M),N_1)))) # label(fact_674_le__add2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 412 (all X_1 all Y_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) -> -(hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1))))) # label(fact_584_dvd_Oless__imp__not__less) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 413 (all A_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A_1)) -> zero_zero(nat) = A_1)) # label(fact_867_gcd__lcm__complete__lattice__nat_Otop__le) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 414 (all X_a (comm_semiring_1(X_a) -> (all A_1 all C hAPP(X_a,X_a,plus_plus(X_a,C),A_1) = hAPP(X_a,X_a,plus_plus(X_a,A_1),C)))) # label(fact_81_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 415 (all X_a (number_ring(X_a) -> (all A_1 hAPP(X_a,X_a,plus_plus(X_a,A_1),number_number_of(X_a,pls)) = ti(X_a,A_1)))) # label(fact_144_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 416 (all Ma all N all K (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),N)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,times_times(nat,K),Ma)),hAPP(nat,nat,times_times(nat,K),N)))))) # label(fact_731_nat__mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 417 (all X_a (linord20386208strict(X_a) -> (all A_1 all C all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),C))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))))))) # label(fact_535_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 418 (all X_b (monoid_mult(X_b) & number(X_b) -> (all W hAPP(X_b,X_b,times_times(X_b,number_number_of(X_b,W)),number_number_of(X_b,W)) = hAPP(nat,X_b,power_power(X_b,number_number_of(X_b,W)),number_number_of(nat,bit0(bit1(pls))))))) # label(fact_10_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 419 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all Y_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),number_number_of(X_a,Y_2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(pls)),Y_2)))))) # label(fact_105_less__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 420 (all X_1 all Y_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),Y_1)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y_1),X_1)) | ti(int,X_1) = ti(int,Y_1))) # label(fact_23_zless__linear) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 421 (all X_a (monoid_mult(X_a) -> (all A_1 hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),A_1)),A_1) = hAPP(nat,X_a,power_power(X_a,A_1),number_number_of(nat,bit1(bit1(pls))))))) # label(fact_163_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 422 (all X_a (comm_semiring_1(X_a) -> (all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),zero_zero(X_a)),A_1)) -> ti(X_a,A_1) = zero_zero(X_a))))) # label(fact_449_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 423 (all X_a (comm_semiring_1(X_a) -> (all X_1 hAPP(X_a,X_a,times_times(X_a,X_1),X_1) = hAPP(nat,X_a,power_power(X_a,X_1),number_number_of(nat,bit0(bit1(pls))))))) # label(fact_13_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 424 (all C all A_1 all B (B = A_1 -> (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),C),B)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),C)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),C),A_1))))) # label(fact_589_dvd_Oord__eq__less__trans) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 425 (all W all Z_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W),Z_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,W),one_one(int))),Z_1)))) # label(fact_61_zless__imp__add1__zle) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 426 (all Ma all N (one_one(int) = ti(int,Ma) & ti(int,N) = one_one(int) | ti(int,N) = number_number_of(int,min) & ti(int,Ma) = number_number_of(int,min) <-> one_one(int) = hAPP(int,int,times_times(int,Ma),N))) # label(fact_320_zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 427 (all X_a (ring_div(X_a) -> (all B all B_3 all A_1 all C all A_4 (hAPP(X_a,X_a,div_mod(X_a,A_1),C) = hAPP(X_a,X_a,div_mod(X_a,A_4),C) -> (hAPP(X_a,X_a,div_mod(X_a,B_3),C) = hAPP(X_a,X_a,div_mod(X_a,B),C) -> hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,minus_minus(X_a,A_4),B_3)),C) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,minus_minus(X_a,A_1),B)),C)))))) # label(fact_947_mod__diff__cong) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 428 (all Ma all N (Ma != N <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),Ma)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),N)))) # label(fact_572_nat__neq__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 429 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),min)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K)),min)))) # label(fact_310_rel__simps_I28_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 430 (all X_a (linord20386208strict(X_a) -> (all A_1 all C all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),C))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))))))) # label(fact_538_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 431 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),bit1(K))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),K)))) # label(fact_97_rel__simps_I22_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 432 (all X_a (monoid_mult(X_a) -> (all A_1 all M all N_1 hAPP(nat,X_a,power_power(X_a,hAPP(nat,X_a,power_power(X_a,A_1),M)),N_1) = hAPP(nat,X_a,power_power(X_a,A_1),hAPP(nat,nat,times_times(nat,M),N_1))))) # label(fact_263_power__mult) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 433 (all K_1 number_number_of(int,K_1) = ti(int,K_1)) # label(fact_88_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 434 (all K_1 all L_1 all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),L_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,times_times(nat,I_1),K_1)),hAPP(nat,nat,times_times(nat,J_1),L_1)))))) # label(fact_699_mult__le__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 435 (all A_1 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1)) -> (exists R_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_2)) & hAPP(nat,real,power_power(real,R_2),N_1) = A_1))))) # label(fact_551_realpow__pos__nth) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 436 (all B_1_1 all X_a (ab_semigroup_add(X_a) -> plus_plus(X_a,ti(X_a,B_1_1)) = plus_plus(X_a,B_1_1))) # label(tsy_c_Groups_Oplus__class_Oplus_3_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 437 (all X_2 all Y_2 (Y_2 = X_2 <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_2),Y_2)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_2),X_2)))) # label(fact_603_dvd_Oeq__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 438 (all X_a (ordere216010020id_add(X_a) -> (all B all C all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),hAPP(X_a,X_a,plus_plus(X_a,A_1),C)))))))) # label(fact_814_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 439 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all X_2 all Y_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),number_number_of(X_a,X_2)),number_number_of(X_a,Y_2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X_2),Y_2)))))) # label(fact_31_le__number__of) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 440 (all X_a (ordere779506340up_add(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),C)),hAPP(X_a,X_a,plus_plus(X_a,B),C))))))) # label(fact_773_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 441 (all B_1_1 all B_2_1 inv(B_1_1,B_2_1) = inv(B_1_1,ti(int,B_2_1))) # label(tsy_c_WilsonRuss_Oinv_arg2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 442 (all A_3 all B_2 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Ma),hAPP(int,int,minus_minus(int,A_3),B_2))) <-> hBOOL(hAPP(int,bool,zcong(A_3,B_2),Ma)))) # label(fact_383_zcong__def) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 443 (all P_2 all I_2 all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_2),K)) -> (hBOOL(hAPP(int,bool,P_2,hAPP(int,int,minus_minus(int,K),one_one(int)))) -> ((all I (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I),K)) -> (hBOOL(hAPP(int,bool,P_2,I)) -> hBOOL(hAPP(int,bool,P_2,hAPP(int,int,minus_minus(int,I),one_one(int))))))) -> hBOOL(hAPP(int,bool,P_2,I_2)))))) # label(fact_850_int__less__induct) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 444 (all X_a (power(X_a) -> (all A_1 one_one(X_a) = hAPP(nat,X_a,power_power(X_a,A_1),zero_zero(nat))))) # label(fact_314_power__0) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 445 (all X_1 all Y_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) -> X_1 = Y_1))) # label(fact_591_dvd_Oantisym) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 446 (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) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,X_2),X_2)),hAPP(X_a,X_a,times_times(X_a,Y_2),Y_2))),zero_zero(X_a))))))) # label(fact_236_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 447 (all Z1 all Z2 all Z3 hAPP(real,real,times_times(real,Z1),hAPP(real,real,times_times(real,Z2),Z3)) = hAPP(real,real,times_times(real,hAPP(real,real,times_times(real,Z1),Z2)),Z3)) # label(fact_432_real__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 448 (all X_a all B_1_1 collect(X_a,B_1_1) = ti(fun(X_a,bool),collect(X_a,B_1_1))) # label(tsy_c_Set_OCollect_res) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 449 (all Q_1 all B all R_1 all C (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),R_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_1),zero_zero(int))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),hAPP(int,int,div_mod(int,Q_1),C))),R_1)),zero_zero(int))))))) # label(fact_967_zmult2__lemma__aux2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 450 (all X_a (semiring_div(X_a) -> (all A_1 all B all C hAPP(X_a,X_a,div_mod(X_a,A_1),B) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),hAPP(X_a,X_a,times_times(X_a,B),C))),B)))) # label(fact_903_mod__mult__self2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 451 (all K_1 all M all N_1 hAPP(nat,nat,times_times(nat,K_1),hAPP(nat,nat,div_mod(nat,M),N_1)) = hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,times_times(nat,K_1),M)),hAPP(nat,nat,times_times(nat,K_1),N_1))) # label(fact_983_mod__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 452 (all X_a (zero_neq_one(X_a) -> zero_zero(X_a) != one_one(X_a))) # label(fact_446_one__neq__zero) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 453 (all X_b all X_c all X_a all B_1_1 all B_2_1 combb(X_b,X_c,X_a,B_1_1,B_2_1) = combb(X_b,X_c,X_a,B_1_1,ti(fun(X_a,X_b),B_2_1))) # label(tsy_c_COMBB_arg2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 454 (all N_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N_1))) # label(fact_645_less__eq__nat_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 455 -(all T_4 hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,s),number_number_of(nat,bit0(bit1(pls))))),one_one(int)) != hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int))),T_4)) # label(fact_19__096_B_Bthesis_O_A_I_B_Bt_O_As_A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A_L_A1_) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 456 (all B_1_1 all X_a (group_add(X_a) -> minus_minus(X_a,ti(X_a,B_1_1)) = minus_minus(X_a,B_1_1))) # label(tsy_c_Groups_Ominus__class_Ominus_0_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 457 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all Y_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),Y_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),number_number_of(X_a,Y_2))))))) # label(fact_246_less__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 458 (all X_a (number_ring(X_a) -> (all A_1 hAPP(X_a,X_a,times_times(X_a,A_1),number_number_of(X_a,bit1(pls))) = ti(X_a,A_1)))) # label(fact_156_mult__numeral__1__right) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 459 (all X_a (semiring_div(X_a) -> (all A_1 all C all B hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),C)) = hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,div_mod(X_a,A_1),B)),C)))) # label(fact_939_mod__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 460 (all K_1 all M hBOOL(hAPP(int,bool,zcong(K_1,K_1),M))) # label(fact_327_zcong__refl) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 461 (all N_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),N_1))) # label(fact_616_le__refl) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 462 (all N all K all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),Ma)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),N)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,minus_minus(nat,Ma),K)),hAPP(nat,nat,minus_minus(nat,N),K))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),N)))))) # label(fact_669_less__diff__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 463 (all Ma all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),N)) & Ma != N <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),N)))) # label(fact_664_nat__less__le) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 464 (all W_1 all Z_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),Z_2)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W_1),hAPP(int,int,minus_minus(int,Z_2),one_one(int)))))) # label(fact_386_zle__diff1__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 465 (all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,minus_minus(int,P_1),one_one(int)))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),inv(P_1,A_1)),hAPP(int,int,minus_minus(int,P_1),one_one(int)))))))) # label(fact_825_inv__less__p__minus__1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 466 (all Z_1 all W hAPP(int,int,times_times(int,Z_1),W) = hAPP(int,int,times_times(int,W),Z_1)) # label(fact_87_zmult__commute) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 467 (all C all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,times_times(nat,A_1),C)),hAPP(nat,nat,times_times(nat,B),C))))) # label(fact_466_divides__mul__r) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 468 (all Z1 all Z2 all W hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,Z1),Z2)),W) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,Z1),W)),hAPP(int,int,times_times(int,Z2),W))) # label(fact_132_zadd__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 469 (all N_1 all M M = hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,plus_plus(nat,N_1),M)),N_1)) # label(fact_612_diff__add__inverse) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 470 (all X_a (comm_semiring_1(X_a) -> (all A_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),A_1))))) # label(fact_426_dvd__refl) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 471 (all Q_1 all B all R_1 all C (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_1),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),hAPP(int,int,div_mod(int,Q_1),C))),R_1)),hAPP(int,int,times_times(int,B),C))))))) # label(fact_965_zmult2__lemma__aux4) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 472 (all X_2 all Y_2 all Z_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Z_2)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),Y_2)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,Z_2),X_2)),hAPP(real,real,times_times(real,Z_2),Y_2)))))) # label(fact_521_real__mult__le__cancel__iff2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 473 (all X_a (linordered_semidom(X_a) -> (all A_1 all N_1 all B (hAPP(nat,X_a,power_power(X_a,A_1),N_1) = hAPP(nat,X_a,power_power(X_a,B),N_1) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) -> ti(X_a,B) = ti(X_a,A_1)))))))) # label(fact_189_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 474 (all X_a (number_semiring(X_a) -> (all V_1 all V (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_1)) -> hAPP(X_a,X_a,times_times(X_a,number_number_of(X_a,V)),number_number_of(X_a,V_1)) = number_number_of(X_a,hAPP(int,int,times_times(int,V),V_1))))))) # label(fact_135_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 475 (all X_a (monoid_add(X_a) -> (all A_1 hAPP(X_a,X_a,plus_plus(X_a,zero_zero(X_a)),A_1) = ti(X_a,A_1)))) # label(fact_766_add__0__left) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 476 (all X_2 all N all Y_2 (hAPP(int,int,div_mod(int,X_2),N) = hAPP(int,int,div_mod(int,Y_2),N) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N),hAPP(int,int,minus_minus(int,X_2),Y_2))))) # label(fact_911_zmod__eq__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 477 (all X_a (linordered_semidom(X_a) -> (all N_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N_1)),hAPP(nat,X_a,power_power(X_a,B),N_1))))))))) # label(fact_197_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 478 (all X_a (semiring_div(X_a) -> (all B all B_3 all A_1 all C all A_4 (hAPP(X_a,X_a,div_mod(X_a,A_1),C) = hAPP(X_a,X_a,div_mod(X_a,A_4),C) -> (hAPP(X_a,X_a,div_mod(X_a,B),C) = hAPP(X_a,X_a,div_mod(X_a,B_3),C) -> hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_4),B_3)),C) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),C)))))) # label(fact_934_mod__add__cong) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 479 (all C_1 all X_2 all Ta all A_3 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_3),D)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_3),hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,X_2),hAPP(int,int,times_times(int,C_1),D))),Ta))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_3),hAPP(int,int,plus_plus(int,X_2),Ta)))))) # label(fact_213_zdvd__period) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 480 (all X_a (semiring_div(X_a) -> (all A_1 all B all C hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,div_mod(X_a,A_1),C)),hAPP(X_a,X_a,div_mod(X_a,B),C))),C) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),B)),C)))) # label(fact_937_mod__mult__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 481 (all M all N_1 (N_1 = M -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)))) # label(fact_618_eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 482 (all M all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),hAPP(nat,nat,plus_plus(nat,M),J_1))))) # label(fact_657_trans__less__add2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 483 (all B all M all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),A_1)) -> -hBOOL(hAPP(int,bool,zcong(A_1,B),M))))))) # label(fact_347_zcong__not) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 484 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(K)),bit0(L))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L)))) # label(fact_58_rel__simps_I33_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 485 (all X_a (comm_ring(X_a) & dvd(X_a) -> (all Ta all D all D_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),D_3)) -> (all X all K_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,minus_minus(X_a,X),hAPP(X_a,X_a,times_times(X_a,K_2),D_3))),Ta))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),hAPP(X_a,X_a,plus_plus(X_a,X),Ta))))))))) # label(fact_746_inf__period_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 486 (all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> hBOOL(hAPP(int,bool,zcong(zfact(hAPP(int,int,minus_minus(int,P_1),one_one(int))),number_number_of(int,min)),P_1)))) # label(fact_753_Wilson__Russ) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 487 (all X_a (linord581940658strict(X_a) -> (all A_3 all C_1 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,A_3),C_1)),hAPP(X_a,X_a,times_times(X_a,B_2),C_1))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),A_3)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a))) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)))))) # label(fact_503_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 488 (all X_a (linord219039673up_add(X_a) -> (all A_3 (zero_zero(X_a) = ti(X_a,A_3) <-> zero_zero(X_a) = hAPP(X_a,X_a,plus_plus(X_a,A_3),A_3))))) # label(fact_203_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 489 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all X_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),number_number_of(X_a,X_2)),one_one(X_a))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_2),bit1(pls))))))) # label(fact_104_less__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 490 (all B_1_1 all X_a (power(X_a) -> times_times(X_a,ti(X_a,B_1_1)) = times_times(X_a,B_1_1))) # label(tsy_c_Groups_Otimes__class_Otimes_1_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 491 (all X_a (group_add(X_a) -> (all A_1 hAPP(X_a,X_a,minus_minus(X_a,A_1),zero_zero(X_a)) = ti(X_a,A_1)))) # label(fact_788_diff__0__right) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 492 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B all C hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),C)) = hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),C)))) # label(fact_110_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 493 (all X_a (linordered_semidom(X_a) -> (all X_1 all Y_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,X_1),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,X_a,power_power(X_a,Y_1),number_number_of(nat,bit0(bit1(pls)))))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1))))))) # label(fact_264_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 494 (all B all Q_1 all R_1 all B_3 all Q_4 all R_3 (hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B_3),Q_4)),R_3) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),Q_1)),R_1) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B_3),Q_4)),R_3))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_3),B_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_3),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_1),Q_4))))))))) # label(fact_356_zdiv__mono2__lemma) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 495 (all X_a (number_ring(X_a) -> (all V hAPP(X_a,X_a,plus_plus(X_a,number_number_of(X_a,V)),one_one(X_a)) = number_number_of(X_a,hAPP(int,int,plus_plus(int,V),bit1(pls)))))) # label(fact_17_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 496 (all X_a (number_ring(X_a) & ring_char_0(X_a) -> (all X_2 all Y_2 (ti(int,Y_2) = ti(int,X_2) <-> number_number_of(X_a,X_2) = number_number_of(X_a,Y_2))))) # label(fact_82_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 497 (all N all Ma (hAPP(nat,nat,times_times(nat,N),Ma) = one_one(nat) <-> N = one_one(nat) & one_one(nat) = Ma)) # label(fact_467_nat__mult__eq__one) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 498 (all X_a (semiri456707255roduct(X_a) -> (all C all D_2 all A_1 all B all R_1 (zero_zero(X_a) != ti(X_a,R_1) -> (ti(X_a,C) != ti(X_a,D_2) & ti(X_a,B) = ti(X_a,A_1) -> hAPP(X_a,X_a,plus_plus(X_a,A_1),hAPP(X_a,X_a,times_times(X_a,R_1),C)) != hAPP(X_a,X_a,plus_plus(X_a,B),hAPP(X_a,X_a,times_times(X_a,R_1),D_2))))))) # label(fact_219_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 499 (all X_a (ordere216010020id_add(X_a) -> (all B all A_1 all C (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),hAPP(X_a,X_a,plus_plus(X_a,A_1),C)))))))) # label(fact_802_add__increasing2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 500 (all A_3 all B_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),B_2)) -> -hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),d22set(A_3))))) # label(fact_858_d22set__le__swap) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 501 (all X_a (comm_semiring_1(X_a) -> (all X_1 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) | ti(X_a,X_1) = one_one(X_a) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X_1),hAPP(nat,X_a,power_power(X_a,X_1),N_1))))))) # label(fact_322_dvd__power) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 502 (all B all Q_4 all R_3 all Q_1 all R_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),Q_4)),R_3)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),Q_1)),R_1))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_3),B)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_1),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_4),Q_1))))))) # label(fact_357_unique__quotient__lemma) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 503 (all I_2 all J_2 all K (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,I_2),K)),J_2)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),hAPP(nat,nat,minus_minus(nat,J_2),K))))) # label(fact_663_less__diff__conv) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 504 (all Z_1 ti(int,Z_1) = hAPP(int,int,plus_plus(int,zero_zero(int)),Z_1)) # label(fact_206_zadd__0) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 505 (all K_1 bit0(K_1) != min) # label(fact_295_rel__simps_I45_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 506 (all X_a (ordered_ring(X_a) -> (all A_3 all E all C_1 all B_2 all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,A_3),E)),C_1)),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,B_2),E)),D))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,minus_minus(X_a,A_3),B_2)),E)),C_1)),D)))))) # label(fact_543_le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 507 (all X_a (semiring_div(X_a) -> (all B all A_1 hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,times_times(X_a,B),A_1)),B) = zero_zero(X_a)))) # label(fact_899_mod__mult__self1__is__0) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 508 (all N_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N_1))) # label(fact_555_le0) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 509 (all X_a (linordered_idom(X_a) -> (all A_1 all N_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,power_power(X_a,A_1),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),N_1))))))) # label(fact_269_zero__le__even__power_H) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 510 (all M hAPP(nat,nat,minus_minus(nat,M),M) = zero_zero(nat)) # label(fact_565_diff__self__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 511 (all N_1 all K_1 all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),M)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N_1)) -> hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,minus_minus(nat,M),K_1)),hAPP(nat,nat,minus_minus(nat,N_1),K_1)) = hAPP(nat,nat,minus_minus(nat,M),N_1)))) # label(fact_626_Nat_Odiff__diff__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 512 (all X_a (semiring_div(X_a) -> (all Ma all K all N (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K),N)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K),hAPP(X_a,X_a,div_mod(X_a,Ma),N))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K),Ma))))))) # label(fact_926_dvd__mod__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 513 (all X_a (linordered_semidom(X_a) -> (all A_1 all N_1 all N_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),N_3)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),one_one(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N_3)),hAPP(nat,X_a,power_power(X_a,A_1),N_1))))))))) # label(fact_216_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 514 (all V_2 (zero_zero(nat) = number_number_of(nat,V_2) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),V_2),pls)))) # label(fact_325_eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 515 (all X_a (dvd(X_a) -> ti(fun(X_a,fun(X_a,bool)),dvd_dvd(X_a)) = dvd_dvd(X_a))) # label(tsy_c_Rings_Odvd__class_Odvd_res) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 516 (all X_a all X_c all X_b all B_1_1 all B_2_1 ti(fun(X_a,X_c),combs(X_a,X_b,X_c,B_1_1,B_2_1)) = combs(X_a,X_b,X_c,B_1_1,B_2_1)) # label(tsy_c_COMBS_res) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 517 (all P_2 all Ma all N ((hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),N)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_2,N),Ma))) -> ((Ma = N -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_2,N),Ma))) -> ((hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),Ma)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_2,N),Ma))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_2,N),Ma)))))) # label(fact_567_nat__less__cases) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 518 (all K_1 all A_1 all B all M (hBOOL(hAPP(int,bool,zcong(A_1,B),M)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,K_1),A_1),hAPP(int,int,times_times(int,K_1),B)),M)))) # label(fact_338_zcong__scalar2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 519 (all M all K_1 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,M),K_1)),N_1)) -> -(hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)) -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N_1))))) # label(fact_684_add__leE) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 520 (all I_1 all J_1 all K_1 hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,minus_minus(nat,I_1),K_1)),J_1) = hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,minus_minus(nat,I_1),J_1)),K_1)) # label(fact_558_diff__commute) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 521 (all M all N_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,div_mod(nat,M),N_1)),M))) # label(fact_985_mod__less__eq__dividend) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 522 (all K all Ma all N ((hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),N))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,times_times(nat,K),Ma)),hAPP(nat,nat,times_times(nat,K),N))))) # label(fact_720_mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.98 523 (all X_a (number_ring(X_a) -> (all W number_number_of(X_a,bit0(W)) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,plus_plus(X_a,zero_zero(X_a)),number_number_of(X_a,W))),number_number_of(X_a,W))))) # label(fact_241_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 524 (all Ma all N (Ma = one_one(nat) & one_one(nat) = N <-> one_one(nat) = hAPP(nat,nat,times_times(nat,Ma),N))) # label(fact_706_nat__1__eq__mult__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 525 (all Z_1 hAPP(real,real,times_times(real,one_one(real)),Z_1) = Z_1) # label(fact_434_real__mult__1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 526 (all X_1 all Y_1 hAPP(nat,real,power_power(real,hAPP(real,real,plus_plus(real,X_1),Y_1)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(real,real,plus_plus(real,hAPP(real,real,plus_plus(real,hAPP(nat,real,power_power(real,X_1),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,real,power_power(real,Y_1),number_number_of(nat,bit0(bit1(pls)))))),hAPP(real,real,times_times(real,hAPP(real,real,times_times(real,number_number_of(real,bit0(bit1(pls)))),X_1)),Y_1))) # label(fact_174_real__sum__squared__expand) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 527 (all X_a (semiring_div(X_a) -> (all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_3),B_2)) <-> hAPP(X_a,X_a,div_mod(X_a,B_2),A_3) = zero_zero(X_a))))) # label(fact_904_dvd__eq__mod__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 528 (all X_a (semiring_div(X_a) -> (all B all B_3 all A_1 all C all A_4 (hAPP(X_a,X_a,div_mod(X_a,A_4),C) = hAPP(X_a,X_a,div_mod(X_a,A_1),C) -> (hAPP(X_a,X_a,div_mod(X_a,B_3),C) = hAPP(X_a,X_a,div_mod(X_a,B),C) -> hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,times_times(X_a,A_4),B_3)),C) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),B)),C)))))) # label(fact_941_mod__mult__cong) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 529 (all X_a (comm_semiring_1(X_a) -> (all Lx all Ly all Rx hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,times_times(X_a,Lx),Ly)),Rx) = hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,times_times(X_a,Lx),Rx)),Ly)))) # label(fact_69_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 530 (all N_1 all B all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,power_power(int,P_1),N_1)),hAPP(int,int,times_times(int,A_1),B))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,power_power(int,P_1),N_1)),B)))))) # label(fact_233_zprime__power__zdvd__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 531 (all M all K_1 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,plus_plus(nat,N_1),M)),K_1))))) # label(fact_687_le__add__diff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 532 (all X_a (linord581940658strict(X_a) -> (all C_1 all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a))) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B_2),A_3)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,C_1),A_3)),hAPP(X_a,X_a,times_times(X_a,C_1),B_2))))))) # label(fact_502_mult__less__cancel__left__disj) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 533 (all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) -> (A_1 != B -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),A_1))))) # label(fact_595_dvd_Ole__neq__trans) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 534 (all N_1 hAPP(nat,nat,times_times(nat,one_one(nat)),N_1) = N_1) # label(fact_707_nat__mult__1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 535 (all X_a (power(X_a) & no_zero_divisors(X_a) & zero_neq_one(X_a) & mult_zero(X_a) -> (all A_3 all N (zero_zero(X_a) = hAPP(nat,X_a,power_power(X_a,A_3),N) <-> N != zero_zero(nat) & zero_zero(X_a) = ti(X_a,A_3))))) # label(fact_185_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 536 (all B_1_1 all X_a (semiring_div(X_a) -> div_mod(X_a,B_1_1) = div_mod(X_a,ti(X_a,B_1_1)))) # label(tsy_c_Divides_Odiv__class_Omod_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 537 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B all C hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),C)),B) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),C)))) # label(fact_77_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 538 (all X_a (linord581940658strict(X_a) -> (all A_3 all B_2 all C_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),A_3)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,C_1),A_3)),hAPP(X_a,X_a,times_times(X_a,C_1),B_2)))))))) # label(fact_529_mult__le__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 539 (all X_a (number_ring(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,plus_plus(X_a,number_number_of(X_a,pls)),A_1)))) # label(fact_143_add__numeral__0) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 540 (all X_2 all P_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_3),X_2)) <-> hBOOL(hAPP(int,bool,zcong(X_2,zero_zero(int)),P_3)))) # label(fact_407_zcong__eq__zdvd__prop) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 541 (all U all Ma all N all J_2 all I_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_2),I_2)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,I_2),U)),Ma)),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,J_2),U)),N))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,hAPP(nat,nat,minus_minus(nat,I_2),J_2)),U)),Ma)),N))))) # label(fact_732_nat__le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 542 (all Z_1 all W (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_1),W)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W),Z_1)) -> ti(int,W) = ti(int,Z_1)))) # label(fact_25_zle__antisym) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 543 (all X_2 all Y_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_2),Y_2)) & X_2 != Y_2 <-> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_2),X_2)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_2),Y_2)))) # label(fact_601_dvd_Oless__le) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 544 (all X_a all X_c all B_1_1 all B_2_1 hAPP(X_a,X_c,B_1_1,ti(X_a,B_2_1)) = hAPP(X_a,X_c,B_1_1,B_2_1)) # label(tsy_c_hAPP_arg2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 545 (all X_a (linordered_idom(X_a) -> (all A_1 -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,A_1),number_number_of(nat,bit0(bit1(pls))))),zero_zero(X_a)))))) # label(fact_256_power2__less__0) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 546 (all M (hBOOL(hAPP(int,bool,zprime,hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),M)),one_one(int)))) -> legendre(number_number_of(int,min),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),M)),one_one(int))) = one_one(int))) # label(fact_417_Legendre__1mod4) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 547 (all B_3 all Q_4 all R_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B_3),Q_4)),R_3)),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_3)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_4),zero_zero(int))))))) # label(fact_358_q__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 548 (all X_a (ordered_ab_group_add(X_a) -> (all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,minus_minus(X_a,A_3),B_2)),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)))))) # label(fact_811_less__iff__diff__less__0) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 549 (all N_1 N_1 = hAPP(nat,nat,times_times(nat,N_1),one_one(nat))) # label(fact_705_nat__mult__1__right) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 550 (all B_1_1 twoSqu1929807760sum2sq(B_1_1) = ti(int,twoSqu1929807760sum2sq(B_1_1))) # label(tsy_c_TwoSquares__Mirabelle__lldtdwpyrp_Osum2sq_res) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 551 (all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M)))) # label(fact_617_nat__le__linear) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 552 (all X_a (comm_semiring_1(X_a) -> (all A_1 hAPP(X_a,X_a,times_times(X_a,A_1),zero_zero(X_a)) = zero_zero(X_a)))) # label(fact_199_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 553 (all K_1 all L_1 bit0(L_1) != bit1(K_1)) # label(fact_118_rel__simps_I50_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 554 (all K_1 all L_1 bit0(K_1) != bit1(L_1)) # label(fact_119_rel__simps_I49_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 555 (all X_1 all Y_1 all Z_1 hAPP(nat,nat,plus_plus(nat,Y_1),hAPP(nat,nat,plus_plus(nat,X_1),Z_1)) = hAPP(nat,nat,plus_plus(nat,X_1),hAPP(nat,nat,plus_plus(nat,Y_1),Z_1))) # label(fact_607_nat__add__left__commute) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 556 (all N_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,N_1))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,bit0(N_1)))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,bit1(N_1)))))) # label(fact_739_number__of1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 557 (all N_1 all J_1 all K_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J_1),K_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,minus_minus(nat,J_1),N_1)),K_1)))) # label(fact_575_less__imp__diff__less) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 558 (all K_1 all M all N_1 hAPP(nat,nat,minus_minus(nat,M),N_1) = hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,plus_plus(nat,K_1),M)),hAPP(nat,nat,plus_plus(nat,K_1),N_1))) # label(fact_614_Nat_Odiff__cancel) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 559 (all Z_1 all X_1 all Y_1 (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),Z_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z_1),Y_1)) -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z_1),X_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Z_1))))) # label(fact_577_dvd_Oless__trans) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 560 (all X_b (ring(X_b) & number(X_b) -> (all V all B all C hAPP(X_b,X_b,times_times(X_b,number_number_of(X_b,V)),hAPP(X_b,X_b,minus_minus(X_b,B),C)) = hAPP(X_b,X_b,minus_minus(X_b,hAPP(X_b,X_b,times_times(X_b,number_number_of(X_b,V)),B)),hAPP(X_b,X_b,times_times(X_b,number_number_of(X_b,V)),C))))) # label(fact_376_right__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 561 (all A_3 all P_3 (hBOOL(hAPP(int,bool,zcong(A_3,hAPP(int,int,minus_minus(int,P_3),one_one(int))),P_3)) <-> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_3),hAPP(int,int,minus_minus(int,P_3),one_one(int))),one_one(int)),P_3)))) # label(fact_390_inv__not__p__minus__1__aux) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 562 (all X_a (real_normed_algebra(X_a) -> (all Y_1 zero_zero(X_a) = hAPP(X_a,X_a,times_times(X_a,zero_zero(X_a)),Y_1)))) # label(fact_886_mult__left_Ozero) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 563 (all X_a (ring(X_a) -> (all X_1 all Y_1 all A_1 all B hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,X_1),hAPP(X_a,X_a,minus_minus(X_a,Y_1),B))),hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,minus_minus(X_a,X_1),A_1)),B)) = hAPP(X_a,X_a,minus_minus(X_a,hAPP(X_a,X_a,times_times(X_a,X_1),Y_1)),hAPP(X_a,X_a,times_times(X_a,A_1),B))))) # label(fact_516_mult__diff__mult) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 564 (all Ma all D ((exists Q_2 ti(int,Ma) = hAPP(int,int,times_times(int,D),Q_2)) <-> hAPP(int,int,div_mod(int,Ma),D) = zero_zero(int))) # label(fact_910_zmod__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 565 (all X_a (number_ring(X_a) -> (all V all W number_number_of(X_a,hAPP(int,int,times_times(int,V),W)) = hAPP(X_a,X_a,times_times(X_a,number_number_of(X_a,V)),number_number_of(X_a,W))))) # label(fact_146_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 566 (all V_1 all V ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls)) -> number_number_of(nat,hAPP(int,int,times_times(int,V),V_1)) = hAPP(nat,nat,times_times(nat,number_number_of(nat,V)),number_number_of(nat,V_1))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls)) -> hAPP(nat,nat,times_times(nat,number_number_of(nat,V)),number_number_of(nat,V_1)) = zero_zero(nat)))) # label(fact_331_mult__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 567 (all X_a (ring_1(X_a) -> (all X_1 hAPP(X_a,X_a,minus_minus(X_a,hAPP(X_a,X_a,times_times(X_a,X_1),X_1)),one_one(X_a)) = hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,plus_plus(X_a,X_1),one_one(X_a))),hAPP(X_a,X_a,minus_minus(X_a,X_1),one_one(X_a)))))) # label(fact_547_real__squared__diff__one__factored) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 568 (all K all Ma all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),N)) | zero_zero(nat) = K <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,times_times(nat,K),Ma)),hAPP(nat,nat,times_times(nat,K),N))))) # label(fact_729_nat__mult__dvd__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 569 (all M all N_1 ((hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1)) -> M = hAPP(nat,nat,div_mod(nat,M),N_1)) & (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1)) -> hAPP(nat,nat,div_mod(nat,M),N_1) = hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,minus_minus(nat,M),N_1)),N_1)))) # label(fact_978_mod__if) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 570 (all X_1 all Y_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) -> Y_1 != X_1)) # label(fact_583_dvd_Oless__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 571 (all X_2 (one_one(int) = hAPP(int,int,div_mod(int,X_2),number_number_of(int,bit0(bit1(pls)))) <-> zero_zero(int) != hAPP(int,int,div_mod(int,X_2),number_number_of(int,bit0(bit1(pls)))))) # label(fact_969_neq__one__mod__two) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 572 (all K all I_2 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_2)) -> (J_2 = hAPP(nat,nat,plus_plus(nat,K),I_2) <-> hAPP(nat,nat,minus_minus(nat,J_2),I_2) = K))) # label(fact_692_le__imp__diff__is__add) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 573 (all N_1 hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),hAPP(nat,real,power_power(real,number_number_of(real,bit0(bit1(pls)))),N_1)))) # label(fact_550_two__realpow__ge__one) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 574 (all K all N all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),hAPP(int,int,plus_plus(int,N),hAPP(int,int,times_times(int,K),Ma)))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),N)))) # label(fact_212_zdvd__reduce) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 575 (all X_a (ordere216010020id_add(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,A_1),B)))))))) # label(fact_816_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 576 (all V all W hAPP(int,int,plus_plus(int,number_number_of(int,V)),number_number_of(int,W)) = number_number_of(int,hAPP(int,int,plus_plus(int,V),W))) # label(fact_134_plus__numeral__code_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 577 (all X_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),P_1)) -> -hBOOL(hAPP(int,bool,zcong(multInv(P_1,X_1),zero_zero(int)),P_1)))))) # label(fact_831_MultInv__prop3) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 578 (all X_a (semiring_div(X_a) -> (all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B)) -> zero_zero(X_a) = hAPP(X_a,X_a,div_mod(X_a,B),A_1))))) # label(fact_905_dvd__imp__mod__0) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 579 (all Z1 all Z2 all W hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,Z1),W)),hAPP(real,real,times_times(real,Z2),W)) = hAPP(real,real,times_times(real,hAPP(real,real,plus_plus(real,Z1),Z2)),W)) # label(fact_468_real__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 580 (all Ma all X_2 (hBOOL(hAPP(int,bool,quadRes(Ma),X_2)) <-> (exists Y hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,Y),number_number_of(nat,bit0(bit1(pls)))),X_2),Ma))))) # label(fact_420_QuadRes__def) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 581 (all A_1 all B hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,A_1),B)),hAPP(int,int,minus_minus(int,A_1),B)) = hAPP(int,int,minus_minus(int,hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,B),number_number_of(nat,bit0(bit1(pls)))))) # label(fact_395_zspecial__product) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 582 (all J_1 all K_1 all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),M)) -> (hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,number_number_of(int,min)),J_1),hAPP(nat,int,power_power(int,number_number_of(int,min)),K_1)),M)) -> hAPP(nat,int,power_power(int,number_number_of(int,min)),K_1) = hAPP(nat,int,power_power(int,number_number_of(int,min)),J_1)))) # label(fact_363_neg__one__power__eq__mod__m) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 583 (all X_1 all Q_1 all N_1 all R_1 (hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,Q_1),N_1)),R_1) = X_1 -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),R_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),R_1),N_1)) -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N_1),X_1)))))) # label(fact_549_divides__div__not) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 584 (all Y_1 all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,div_mod(int,X_1),Y_1)))))) # label(fact_908_Divides_Otransfer__nat__int__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 585 (all V all W ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,W))) -> hAPP(int,int,div_mod(int,number_number_of(int,bit1(V))),number_number_of(int,bit0(W))) = hAPP(int,int,minus_minus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit1(pls)))),hAPP(int,int,div_mod(int,hAPP(int,int,plus_plus(int,number_number_of(int,V)),one_one(int))),number_number_of(int,W)))),one_one(int))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,W))) -> hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit1(pls)))),hAPP(int,int,div_mod(int,number_number_of(int,V)),number_number_of(int,W)))),one_one(int)) = hAPP(int,int,div_mod(int,number_number_of(int,bit1(V))),number_number_of(int,bit0(W)))))) # label(fact_897_zmod__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 586 (all L_1 bit1(hAPP(int,int,minus_minus(int,min),L_1)) = hAPP(int,int,minus_minus(int,min),bit0(L_1))) # label(fact_388_diff__bin__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 587 (all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,minus_minus(int,P_1),one_one(int)))) -> inv(P_1,A_1) != ti(int,A_1))))) # label(fact_821_inv__distinct) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 588 (all X_a (ordered_semiring(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),C)))))))) # label(fact_479_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 589 (all N_1 -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),N_1))) # label(fact_573_less__not__refl) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 590 (all K_1 all I_1 all J_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_1),J_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,I_1),K_1)),hAPP(int,int,plus_plus(int,J_1),K_1))))) # label(fact_50_zadd__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 591 (all K_1 all L_1 hAPP(int,int,plus_plus(int,bit0(hAPP(int,int,times_times(int,K_1),L_1))),L_1) = hAPP(int,int,times_times(int,bit1(K_1)),L_1)) # label(fact_161_mult__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 592 (all X_a (linord219039673up_add(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_3),A_3)),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),zero_zero(X_a))))))) # label(fact_805_double__add__less__zero__iff__single__add__less__zero) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 593 (all X_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),P_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),standardRes(P_1,X_1)),P_1)))) # label(fact_989_StandardRes__ubound) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 594 (all X_a (linord581940658strict(X_a) -> (all A_3 all B_2 all C_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,C_1),A_3)),hAPP(X_a,X_a,times_times(X_a,C_1),B_2)))))))) # label(fact_501_mult__less__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 595 (all B all D_2 all A_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_2),A_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_2),hAPP(nat,nat,plus_plus(nat,A_1),B))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_2),B))))) # label(fact_464_divides__add__revr) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 596 (all Z_1 hAPP(nat,nat,plus_plus(nat,Z_1),Z_1) = hAPP(nat,nat,times_times(nat,Z_1),number_number_of(nat,bit0(bit1(pls))))) # label(fact_36_nat__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 597 (all X_2 all Y_2 (Y_2 != X_2 & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),Y_2)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_2),Y_2)))) # label(fact_430_real__less__def) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 598 (all X_2 all Y_2 (zero_zero(real) = Y_2 & X_2 = zero_zero(real) <-> hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,X_2),X_2)),hAPP(real,real,times_times(real,Y_2),Y_2)) = zero_zero(real))) # label(fact_524_real__two__squares__add__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 599 (all M all X_1 standardRes(M,X_1) = hAPP(int,int,div_mod(int,X_1),M)) # label(fact_981_StandardRes__def) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 600 (all B_1_1 all X_a (power(X_a) -> ti(fun(X_a,X_a),times_times(X_a,B_1_1)) = times_times(X_a,B_1_1))) # label(tsy_c_Groups_Otimes__class_Otimes_1_res) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 601 (all N_1 all M hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),hAPP(nat,nat,plus_plus(nat,N_1),M)))) # label(fact_675_le__add1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 602 (all I_1 all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,power_power(nat,I_1),M)),hAPP(nat,nat,power_power(nat,I_1),N_1))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),I_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1))))) # label(fact_312_power__dvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 603 (all B_1_1 all X_a (semiring_div(X_a) -> ti(fun(X_a,X_a),div_mod(X_a,B_1_1)) = div_mod(X_a,B_1_1))) # label(tsy_c_Divides_Odiv__class_Omod_res) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 604 (all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),N_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))))) # label(fact_877_dvd__pos__nat) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 605 (all X_1 all Y_1 (X_1 != Y_1 -> (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_1),Y_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Y_1),X_1))))) # label(fact_571_linorder__neqE__nat) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 606 (all X_a (ordere453448008miring(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),B)),zero_zero(X_a)))))))) # label(fact_481_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 607 (all X_a (linordered_semidom(X_a) -> (all N_1 all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(X_a,X_a,times_times(X_a,A_1),hAPP(nat,X_a,power_power(X_a,A_1),N_1)))))))) # label(fact_177_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 608 (all X_a (ordere216010020id_add(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),zero_zero(X_a)))))))) # label(fact_812_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 609 (all X_a (number_ring(X_a) -> (all V all W hAPP(X_a,X_a,times_times(X_a,number_number_of(X_a,V)),number_number_of(X_a,W)) = number_number_of(X_a,hAPP(int,int,times_times(int,V),W))))) # label(fact_147_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 610 (all B_2 all A_3 all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,bit1(bit0(bit1(pls))))),P_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(int,int,minus_minus(int,P_3),one_one(int)))) -> (hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),wset(A_3,P_3))) -> hBOOL(hAPP(fun(int,bool),bool,member(int,inv(P_3,B_2)),wset(A_3,P_3)))))))) # label(fact_841_wset__mem__inv__mem) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 611 (all X_a all X_c all X_b all B_1_1 all B_2_1 combc(X_a,X_b,X_c,B_1_1,B_2_1) = ti(fun(X_a,X_c),combc(X_a,X_b,X_c,B_1_1,B_2_1))) # label(tsy_c_COMBC_res) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 612 (all B_1_1 all B_2_1 standardRes(B_1_1,B_2_1) = standardRes(B_1_1,ti(int,B_2_1))) # label(tsy_c_Residues_OStandardRes_arg2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 613 (all X_a (semiring_div(X_a) -> (all A_1 all C all B hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),C) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,div_mod(X_a,A_1),C)),B)),C)))) # label(fact_933_zmod__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 614 (all X_a (number(X_a) -> (all W_1 all X_2 (number_number_of(X_a,W_1) = ti(X_a,X_2) <-> ti(X_a,X_2) = number_number_of(X_a,W_1))))) # label(fact_83_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 615 (all X_a (semiring_div(X_a) -> (all N_1 all K_1 all M (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_1),M)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_1),N_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_1),hAPP(X_a,X_a,div_mod(X_a,M),N_1)))))))) # label(fact_924_dvd__mod) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 616 (all A_1 all B all P_1 all Q_1 twoSqu1929807760sum2sq(product_Pair(int,int,hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A_1),P_1)),hAPP(int,int,times_times(int,B),Q_1)),hAPP(int,int,minus_minus(int,hAPP(int,int,times_times(int,A_1),Q_1)),hAPP(int,int,times_times(int,B),P_1)))) = hAPP(int,int,times_times(int,twoSqu1929807760sum2sq(product_Pair(int,int,A_1,B))),twoSqu1929807760sum2sq(product_Pair(int,int,P_1,Q_1)))) # label(fact_392_mult__sum2sq) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 617 (all X_a (cancel146912293up_add(X_a) -> (all A_1 all B all C (hAPP(X_a,X_a,plus_plus(X_a,A_1),C) = hAPP(X_a,X_a,plus_plus(X_a,A_1),B) -> ti(X_a,C) = ti(X_a,B))))) # label(fact_762_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 618 (all B_1_1 all B_2_1 standardRes(ti(int,B_1_1),B_2_1) = standardRes(B_1_1,B_2_1)) # label(tsy_c_Residues_OStandardRes_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 619 (all Z_1 all X_1 all Y_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),Y_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,Z_1),X_1)),hAPP(real,real,plus_plus(real,Z_1),Y_1))))) # label(fact_435_real__add__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 620 (all X_a (comm_semiring_1(X_a) -> (all A_1 hAPP(X_a,X_a,times_times(X_a,one_one(X_a)),A_1) = ti(X_a,A_1)))) # label(fact_114_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 621 (all A_3 all B_2 all N (zero_zero(nat) != N -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_3),B_2)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,power_power(int,A_3),N)),hAPP(nat,int,power_power(int,B_2),N)))))) # label(fact_866_pow__divides__eq__int) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 622 (all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,div_mod(nat,M),N_1)),N_1)))) # label(fact_974_mod__le__divisor) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 623 (all X_a (linordered_idom(X_a) -> (all A_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,power_power(X_a,A_1),number_number_of(nat,bit0(bit1(pls))))))))) # label(fact_253_zero__le__power2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 624 (all W_1 all Z_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,W_1),one_one(int))),Z_2)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),Z_2)))) # label(fact_62_add1__zle__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 625 (all A_1 all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),X_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),A_1)) -> (ti(int,X_1) != hAPP(int,int,minus_minus(int,A_1),one_one(int)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),hAPP(int,int,minus_minus(int,A_1),one_one(int)))))))) # label(fact_385_Euler_Oaux1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 626 (all U all Ma all N all I_2 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_2)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,hAPP(nat,nat,minus_minus(nat,J_2),I_2)),U)),N))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,I_2),U)),Ma)),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,J_2),U)),N)))))) # label(fact_735_nat__le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 627 (all Y_1 all X_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),P_1)) -> (-hBOOL(hAPP(int,bool,zcong(Y_1,zero_zero(int)),P_1)) -> -hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,X_1),Y_1),zero_zero(int)),P_1)))))) # label(fact_413_zcong__zmult__prop3) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 628 (all X_a (comm_semiring_1(X_a) -> (all A_1 all C all D_2 hAPP(X_a,X_a,plus_plus(X_a,A_1),hAPP(X_a,X_a,plus_plus(X_a,C),D_2)) = hAPP(X_a,X_a,plus_plus(X_a,C),hAPP(X_a,X_a,plus_plus(X_a,A_1),D_2))))) # label(fact_80_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 629 (all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) -> zero_zero(nat) = B | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A_1),B)))) # label(fact_517_divides__ge) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 630 (all N_1 all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N_1),M)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),N_1),M)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),M),zero_zero(int))))) # label(fact_406_zdvd__bounds) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 631 (all X_a (semiring_div(X_a) -> (all A_1 all B all C hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,div_mod(X_a,A_1),C)),B)),C) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),C)))) # label(fact_930_mod__add__left__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 632 (all Z_1 all W (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Z_1),W)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),W),Z_1)))) # label(fact_560_real__le__linear) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 633 (all X_a (real_normed_algebra(X_a) -> (all A_1 zero_zero(X_a) = hAPP(X_a,X_a,times_times(X_a,A_1),zero_zero(X_a))))) # label(fact_885_mult_Ozero__right) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 634 (all X_2 (hBOOL(hAPP(int,bool,twoSqu658283162sum2sq,X_2)) <-> (exists A_5 exists B_4 twoSqu1929807760sum2sq(product_Pair(int,int,A_5,B_4)) = ti(int,X_2)))) # label(fact_740_is__sum2sq__def) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 635 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(K)),bit1(L))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L)))) # label(fact_41_rel__simps_I34_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 636 (all P all Q (hBOOL(Q) | -hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,P),Q)))) # label(help_fconj_3_1_U) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 637 (all X_a (linordered_semidom(X_a) -> (all N_1 all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),one_one(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),hAPP(nat,X_a,power_power(X_a,A_1),N_1))),hAPP(nat,X_a,power_power(X_a,A_1),N_1)))))))) # label(fact_232_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 638 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),pls)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K)),pls)))) # label(fact_95_rel__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 639 (all X_a (ordere223160158up_add(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,C),A_1)),hAPP(X_a,X_a,plus_plus(X_a,C),B))))))) # label(fact_781_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 640 (all N_1 all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M)) -> hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,minus_minus(nat,M),N_1)),N_1) = hAPP(nat,nat,div_mod(nat,M),N_1))) # label(fact_980_le__mod__geq) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 641 (all X_a (zero(X_a) -> ti(X_a,zero_zero(X_a)) = zero_zero(X_a))) # label(tsy_c_Groups_Ozero__class_Ozero_res) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 642 (all K_1 hAPP(int,int,plus_plus(int,pls),K_1) = ti(int,K_1)) # label(fact_126_add__Pls) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 643 (all B_2 all P_3 all A_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_3)) -> (hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),wset(hAPP(int,int,minus_minus(int,A_3),one_one(int)),P_3))) -> hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),wset(A_3,P_3)))))) # label(fact_844_wset__subset) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 644 (all K all L (ti(int,L) = ti(int,K) <-> bit0(L) = bit0(K))) # label(fact_85_rel__simps_I48_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 645 (all X_a (linord626643107strict(X_a) -> (all V all U_1 all Y_1 all X_1 all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_1),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),U_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),V)) -> (one_one(X_a) = hAPP(X_a,X_a,plus_plus(X_a,U_1),V) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,U_1),X_1)),hAPP(X_a,X_a,times_times(X_a,V),Y_1))),A_1)))))))))) # label(fact_422_convex__bound__lt) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 646 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K)),bit1(L))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L)))) # label(fact_99_rel__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 647 (all N_1 all A_1 all B all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),B)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,power_power(int,P_1),N_1)),hAPP(int,int,times_times(int,A_1),B))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,power_power(int,P_1),N_1)),A_1)))))) # label(fact_234_zprime__power__zdvd__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 648 (all N_1 (N_1 != zero_zero(nat) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)))) # label(fact_632_gr0I) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 649 (all X_a (linordered_semidom(X_a) -> (all N_1 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N_1)),hAPP(nat,X_a,power_power(X_a,B),N_1)))))))) # label(fact_209_power__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 650 (all X_2 all N (one_one(nat) = hAPP(nat,nat,power_power(nat,X_2),N) <-> N = zero_zero(nat) | one_one(nat) = X_2)) # label(fact_527_exp__eq__1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 651 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K)),bit0(L))))) # label(fact_47_rel__simps_I31_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 652 (all B_1_1 all X_a (ab_semigroup_mult(X_a) -> times_times(X_a,B_1_1) = times_times(X_a,ti(X_a,B_1_1)))) # label(tsy_c_Groups_Otimes__class_Otimes_0_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 653 (all X_1 all Y_1 all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),M)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,standardRes(M,X_1)),standardRes(M,Y_1)),hAPP(int,int,times_times(int,X_1),Y_1)),M)))) # label(fact_973_StandardRes__prop4) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 654 (all Q_1 all B all R_1 all C (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_1),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),hAPP(int,int,div_mod(int,Q_1),C))),R_1))))))) # label(fact_964_zmult2__lemma__aux3) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 655 (all X_a (linord20386208strict(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,times_times(X_a,B),A_1))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B))))))) # label(fact_496_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 656 (all N_1 all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),M),N_1)) -> -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N_1),M))))) # label(fact_190_zdvd__not__zless) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 657 (all Z_2 all W_1 (ti(int,W_1) != ti(int,Z_2) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_2),W_1)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_2),W_1)))) # label(fact_22_zless__le) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 658 (all X_a (linordered_idom(X_a) -> (all A_3 (ti(X_a,A_3) != zero_zero(X_a) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,power_power(X_a,A_3),number_number_of(nat,bit0(bit1(pls)))))))))) # label(fact_257_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 659 (all X_2 all Y_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_2),Y_2)) <-> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_2),X_2)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_2),Y_2)) | X_2 = Y_2)) # label(fact_602_dvd_Ole__less) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 660 (all X_a (linordered_semidom(X_a) -> (all A_1 all N_1 all N_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),N_3)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,A_1),N_1)),hAPP(nat,X_a,power_power(X_a,A_1),N_3)))))))) # label(fact_271_power__increasing) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 661 -(all S_1 -(hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),S_1),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int)))) & hBOOL(hAPP(int,bool,zcong(s1,S_1),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int)))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),S_1)))) # label(fact_183__096_B_Bthesis_O_A_I_B_Bs_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 662 (all X_a (linorder(X_a) & number(X_a) -> (all V_2 all W_1 (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),number_number_of(X_a,W_1)),number_number_of(X_a,V_2))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),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]. 1.59/1.99 663 (all Z_1 all W (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_1),W)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W),Z_1)))) # label(fact_21_zle__linear) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 664 (all A_3 all B_2 all Ma ((exists K_2 ti(int,B_2) = hAPP(int,int,plus_plus(int,A_3),hAPP(int,int,times_times(int,Ma),K_2))) <-> hBOOL(hAPP(int,bool,zcong(A_3,B_2),Ma)))) # label(fact_349_zcong__iff__lin) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 665 (all X_1 -(hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),X_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),X_1)))) # label(fact_604_dvd_Oless__irrefl) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 666 (all P_1 all M ((M = zero_zero(nat) -> hAPP(nat,nat,power_power(nat,P_1),M) = one_one(nat)) & (M != zero_zero(nat) -> hAPP(nat,nat,power_power(nat,P_1),M) = hAPP(nat,nat,times_times(nat,P_1),hAPP(nat,nat,power_power(nat,P_1),hAPP(nat,nat,minus_minus(nat,M),one_one(nat))))))) # label(fact_438_power__eq__if) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 667 (all X_a (comm_semiring_1(X_a) -> (all Lx all Ly all Rx all Ry hAPP(X_a,X_a,times_times(X_a,Rx),hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,times_times(X_a,Lx),Ly)),Ry)) = hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,times_times(X_a,Lx),Ly)),hAPP(X_a,X_a,times_times(X_a,Rx),Ry))))) # label(fact_67_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 668 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,minus_minus(int,K),L)),zero_zero(int))))) # label(fact_381_less__bin__lemma) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 669 (all X_a (semiri456707255roduct(X_a) -> (all B_2 all A_3 (ti(X_a,B_2) = hAPP(X_a,X_a,plus_plus(X_a,B_2),A_3) <-> zero_zero(X_a) = ti(X_a,A_3))))) # label(fact_202_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 670 (all K_1 all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),hAPP(nat,nat,minus_minus(nat,M),N_1))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),M)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_1),N_1)))))) # label(fact_672_dvd__diffD1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 671 (all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) <-> zero_zero(nat) != N)) # label(fact_635_neq0__conv) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 672 (all A_1 all B all C hAPP(int,int,div_mod(int,hAPP(int,int,times_times(int,A_1),B)),C) = hAPP(int,int,div_mod(int,hAPP(int,int,times_times(int,A_1),hAPP(int,int,div_mod(int,B),C))),C)) # label(fact_919_zmod__zmult1__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 673 (all I_1 all K_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),J_1)) -> hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,plus_plus(nat,J_1),I_1)),K_1) = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,minus_minus(nat,J_1),K_1)),I_1))) # label(fact_694_add__diff__assoc2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 674 (all A_1 all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M)) -> (exists X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),M)) & hBOOL(hAPP(int,bool,zcong(A_1,X),M)) & (all Y (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y),M)) & hBOOL(hAPP(int,bool,zcong(A_1,Y),M)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y)) -> ti(int,X) = ti(int,Y))))))) # label(fact_751_zcong__zless__unique) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 675 (all X_a (ordered_ab_group_add(X_a) -> (all A_3 all B_2 all C_1 all D (hAPP(X_a,X_a,minus_minus(X_a,A_3),B_2) = hAPP(X_a,X_a,minus_minus(X_a,C_1),D) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),D))))))) # label(fact_789_diff__eq__diff__less__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 676 (all X_a (ordere216010020id_add(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),zero_zero(X_a)))))))) # label(fact_813_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 677 (all B_1_1 all X_a (dvd(X_a) & linorder(X_a) & plus(X_a) -> plus_plus(X_a,B_1_1) = plus_plus(X_a,ti(X_a,B_1_1)))) # label(tsy_c_Groups_Oplus__class_Oplus_0_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 678 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all Y_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),number_number_of(X_a,Y_2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),Y_2)))))) # label(fact_107_le__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 679 (all X_a (linordered_idom(X_a) -> (all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_3),A_3)),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),zero_zero(X_a))))))) # label(fact_217_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 680 (all N_1 all M ((zero_zero(nat) != M -> hAPP(nat,nat,times_times(nat,M),N_1) = hAPP(nat,nat,plus_plus(nat,N_1),hAPP(nat,nat,times_times(nat,hAPP(nat,nat,minus_minus(nat,M),one_one(nat))),N_1))) & (zero_zero(nat) = M -> hAPP(nat,nat,times_times(nat,M),N_1) = zero_zero(nat)))) # label(fact_437_mult__eq__if) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 681 (all B_1_1 all X_a (number(X_a) -> number_number_of(X_a,B_1_1) = ti(X_a,number_number_of(X_a,B_1_1)))) # label(tsy_c_Int_Onumber__class_Onumber__of_res) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 682 (all L_1 all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,minus_minus(nat,L_1),N_1)),hAPP(nat,nat,minus_minus(nat,L_1),M))))) # label(fact_622_diff__le__mono2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 683 (all P_2 all I_2 all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_2),K)) -> (hBOOL(hAPP(int,bool,P_2,K)) -> ((all I (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I),K)) -> (hBOOL(hAPP(int,bool,P_2,I)) -> hBOOL(hAPP(int,bool,P_2,hAPP(int,int,minus_minus(int,I),one_one(int))))))) -> hBOOL(hAPP(int,bool,P_2,I_2)))))) # label(fact_849_int__le__induct) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 684 (all V_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),V_2)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),number_number_of(nat,V_2))))) # label(fact_323_less__0__number__of) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 685 (all X_a (linordered_ring(X_a) -> (all A_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,times_times(X_a,A_1),A_1)))))) # label(fact_487_zero__le__square) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 686 (all X_a (ordere216010020id_add(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),zero_zero(X_a)))))))) # label(fact_807_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 687 (all C all A_1 all B all M (hBOOL(hAPP(int,bool,zcong(A_1,B),M)) -> (hBOOL(hAPP(int,bool,zcong(B,C),M)) -> hBOOL(hAPP(int,bool,zcong(A_1,C),M))))) # label(fact_328_zcong__trans) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 688 (all X_a (comm_monoid_mult(X_a) -> (all A_1 hAPP(X_a,X_a,times_times(X_a,one_one(X_a)),A_1) = ti(X_a,A_1)))) # label(fact_792_mult__1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 689 (all X_a (number_ring(X_a) -> (all Z_1 hAPP(X_a,X_a,times_times(X_a,Z_1),number_number_of(X_a,bit0(bit1(pls)))) = hAPP(X_a,X_a,plus_plus(X_a,Z_1),Z_1)))) # label(fact_168_mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 690 (all K_1 all L_1 all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),L_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,I_1),K_1)),hAPP(nat,nat,plus_plus(nat,J_1),L_1)))))) # label(fact_681_add__le__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 691 (all X_a (linordered_semidom(X_a) -> (all A_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),hAPP(X_a,X_a,plus_plus(X_a,A_1),one_one(X_a))))))) # label(fact_511_less__add__one) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 692 (all Q_1 all R_1 all B all C (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),R_1),B)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,B),hAPP(nat,nat,div_mod(nat,Q_1),C))),R_1)),hAPP(nat,nat,times_times(nat,B),C)))))) # label(fact_992_mod__lemma) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 693 (all Z_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z_2)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),Z_2)))) # label(fact_243_int__one__le__iff__zero__less) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 694 (all X_a (ordere236663937imp_le(X_a) -> (all A_1 all C all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),C)),hAPP(X_a,X_a,plus_plus(X_a,B),C))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)))))) # label(fact_776_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 695 (all X_a (ordere216010020id_add(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,A_1),B)))))))) # label(fact_755_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 696 (all X_a (number_semiring(X_a) -> (all Z_1 hAPP(X_a,X_a,times_times(X_a,Z_1),number_number_of(X_a,bit0(bit1(pls)))) = hAPP(X_a,X_a,plus_plus(X_a,Z_1),Z_1)))) # label(fact_167_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 697 (all A_1 all B hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,A_1),number_number_of(nat,bit0(bit1(pls))))),hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit1(pls)))),A_1)),B))),hAPP(nat,int,power_power(int,B),number_number_of(nat,bit0(bit1(pls))))) = hAPP(nat,int,power_power(int,hAPP(int,int,plus_plus(int,A_1),B)),number_number_of(nat,bit0(bit1(pls))))) # label(fact_7_zadd__power2) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 698 (all B_1_1 all B_2_1 wset(B_1_1,B_2_1) = wset(ti(int,B_1_1),B_2_1)) # label(tsy_c_WilsonRuss_Owset_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 699 (all Z_1 ti(int,Z_1) = hAPP(int,int,plus_plus(int,Z_1),zero_zero(int))) # label(fact_207_zadd__0__right) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 700 (all V all W hAPP(int,int,times_times(int,number_number_of(int,V)),number_number_of(int,W)) = number_number_of(int,hAPP(int,int,times_times(int,V),W))) # label(fact_131_times__numeral__code_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 701 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,K)),number_number_of(int,L))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L)))) # label(fact_48_less__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 702 (all B_2 all A_3 all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(int,int,minus_minus(int,P_3),one_one(int)))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),B_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_2),A_3)) -> hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),wset(A_3,P_3)))))))) # label(fact_848_wset__mem) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 703 (all X_a (comm_semiring_1(X_a) -> (all A_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),zero_zero(X_a)))))) # label(fact_423_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 704 (all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,div_mod(nat,M),N_1)),N_1)))) # label(fact_975_mod__less__divisor) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 705 (all K_1 all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,I_1),K_1)),hAPP(nat,nat,plus_plus(nat,J_1),K_1))))) # label(fact_658_add__less__mono1) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 706 (all X_a (semiring_div(X_a) -> (all A_1 all B all C hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),hAPP(X_a,X_a,div_mod(X_a,B),C))),C) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),B)),C)))) # label(fact_935_mod__mult__right__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 707 (all X_a (semiring_div(X_a) -> (all A_1 all B all C hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),C) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),hAPP(X_a,X_a,div_mod(X_a,B),C))),C)))) # label(fact_929_mod__add__right__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 708 (all K_1 all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_1),K_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),K_1))))) # label(fact_619_le__trans) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 709 (all Z_1 zero_zero(int) != hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,one_one(int)),Z_1)),Z_1)) # label(fact_231_odd__nonzero) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 710 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K1)),bit1(K2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)))) # label(fact_98_less__eq__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 711 (all X_a (ordere216010020id_add(X_a) -> (all B all C all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),hAPP(X_a,X_a,plus_plus(X_a,A_1),C)))))))) # label(fact_801_add__increasing) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 712 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),K)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),bit0(K))))) # label(fact_317_rel__simps_I25_J) # label(axiom) # label(non_clause). [assumption]. 1.59/1.99 713 (all X_a (comm_semiring_1(X_a) -> (all X_1 ti(X_a,X_1) = hAPP(nat,X_a,power_power(X_a,X_1),one_one(nat))))) # label(fact_27_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 714 (all Z_1 hAPP(int,int,times_times(int,one_one(int)),Z_1) = ti(int,Z_1)) # label(fact_130_zmult__1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 715 (all X_a (plus(X_a) & dvd(X_a) & linorder(X_a) -> (all D all Sa exists Z all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),X)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),hAPP(X_a,X_a,plus_plus(X_a,X),Sa))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),hAPP(X_a,X_a,plus_plus(X_a,X),Sa)))))))) # label(fact_862_pinf_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 716 (all A_1 all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),hAPP(int,int,div_mod(int,A_1),B))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,div_mod(int,A_1),B)),zero_zero(int))))) # label(fact_959_neg__mod__conj) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 717 (all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),number_number_of(int,bit0(bit1(pls))))) -> ti(int,X_1) = one_one(int) | zero_zero(int) = ti(int,X_1)))) # label(fact_278_int__pos__lt__two__imp__zero__or__one) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 718 (all J_1 all A_1 all K_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,zcong(K_1,zero_zero(int)),P_1)) -> (hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,J_1),K_1),hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,A_1),multInv(P_1,K_1))),K_1)),P_1)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,J_1),K_1),A_1),P_1))))))) # label(fact_834_aux______2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 719 (all B_1_1 all B_2_1 ti(int,legendre(B_1_1,B_2_1)) = legendre(B_1_1,B_2_1)) # label(tsy_c_Residues_OLegendre_res) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 720 (all X_a (linord20386208strict(X_a) -> (all C all D_2 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_2)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),D_2)))))))))) # label(fact_532_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 721 (all B_1_1 all X_a (cancel_semigroup_add(X_a) -> plus_plus(X_a,B_1_1) = ti(fun(X_a,X_a),plus_plus(X_a,B_1_1)))) # label(tsy_c_Groups_Oplus__class_Oplus_2_res) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 722 (all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),N_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N_1),M)) -> M = N_1))) # label(fact_592_dvd__antisym) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 723 (all K all Ma all N (K = zero_zero(nat) | N = Ma <-> hAPP(nat,nat,times_times(nat,K),Ma) = hAPP(nat,nat,times_times(nat,K),N))) # label(fact_725_nat__mult__eq__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 724 (all K_1 all I_1 all J_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_1),J_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,K_1),I_1)),hAPP(int,int,plus_plus(int,K_1),J_1))))) # label(fact_51_zadd__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 725 (all X_2 all Y_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,minus_minus(real,X_2),Y_2)),zero_zero(real))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),Y_2)))) # label(fact_424_real__le__eq__diff) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 726 (all X_a (zero(X_a) -> (all X_2 (ti(X_a,X_2) = zero_zero(X_a) <-> ti(X_a,X_2) = zero_zero(X_a))))) # label(fact_756_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 727 (all Z_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_2),zero_zero(int))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,one_one(int)),Z_2)),Z_2)),zero_zero(int))))) # label(fact_245_odd__less__0) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 728 (all X_a all X_c all X_b all B_1_1 all B_2_1 ti(fun(X_a,X_c),combb(X_b,X_c,X_a,B_1_1,B_2_1)) = combb(X_b,X_c,X_a,B_1_1,B_2_1)) # label(tsy_c_COMBB_res) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 729 (all Ma all D ((exists Q_2 Ma = hAPP(nat,nat,times_times(nat,D),Q_2)) <-> zero_zero(nat) = hAPP(nat,nat,div_mod(nat,Ma),D))) # label(fact_976_mod__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 730 (all Z_1 all N_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Z_1),N_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_1),N_1))))) # label(fact_196_zdvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 731 (all Ma all K all N (zero_zero(nat) = K | Ma = N <-> hAPP(nat,nat,times_times(nat,N),K) = hAPP(nat,nat,times_times(nat,Ma),K))) # label(fact_652_mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 732 (all X_a (number_semiring(X_a) -> (all V_1 all V (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_1)) -> hAPP(X_a,X_a,plus_plus(X_a,number_number_of(X_a,V)),number_number_of(X_a,V_1)) = number_number_of(X_a,hAPP(int,int,plus_plus(int,V),V_1))))))) # label(fact_136_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 733 (all B_2 all A_3 all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(int,int,minus_minus(int,P_3),one_one(int)))) -> (hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),wset(A_3,P_3))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_2),hAPP(int,int,minus_minus(int,P_3),one_one(int)))))))) # label(fact_845_wset__less) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 734 (all A_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),zero_zero(nat)))) # label(fact_870_gcd__lcm__complete__lattice__nat_Otop__greatest) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 735 (all X_1 all Y_1 (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) -> X_1 != Y_1)) # label(fact_587_dvd_Oless__imp__neq) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 736 (all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,minus_minus(int,P_1),one_one(int)))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),inv(P_1,A_1))))))) # label(fact_824_inv__g__1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 737 (all A_1 all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,div_mod(int,A_1),B)),B)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,div_mod(int,A_1),B))))) # label(fact_956_pos__mod__conj) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 738 (all X_a (mult_zero(X_a) -> (all A_1 hAPP(X_a,X_a,times_times(X_a,A_1),zero_zero(X_a)) = zero_zero(X_a)))) # label(fact_443_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 739 (all A_3 all B_2 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma)) -> (hAPP(int,int,div_mod(int,A_3),Ma) = hAPP(int,int,div_mod(int,B_2),Ma) <-> hBOOL(hAPP(int,bool,zcong(A_3,B_2),Ma))))) # label(fact_954_zcong__zmod__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 740 (all M all N_1 all I_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),I_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,power_power(nat,I_1),M)),hAPP(nat,nat,power_power(nat,I_1),N_1))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1))))) # label(fact_289_nat__power__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 741 (all X_a (comm_semiring_1(X_a) -> (all X_1 all P_1 all Q_1 hAPP(nat,X_a,power_power(X_a,X_1),hAPP(nat,nat,plus_plus(nat,P_1),Q_1)) = hAPP(X_a,X_a,times_times(X_a,hAPP(nat,X_a,power_power(X_a,X_1),P_1)),hAPP(nat,X_a,power_power(X_a,X_1),Q_1))))) # label(fact_33_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 742 (all X_a (ordered_semiring(X_a) -> (all C all D_2 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_2)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),D_2)))))))))) # label(fact_474_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 743 (all X_a (number_semiring(X_a) -> (all X_1 all Y_1 hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,X_1),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,X_a,power_power(X_a,Y_1),number_number_of(nat,bit0(bit1(pls)))))),hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,times_times(X_a,number_number_of(X_a,bit0(bit1(pls)))),X_1)),Y_1)) = hAPP(nat,X_a,power_power(X_a,hAPP(X_a,X_a,plus_plus(X_a,X_1),Y_1)),number_number_of(nat,bit0(bit1(pls))))))) # label(fact_9_power2__sum) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 744 (all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,bit1(bit0(bit1(pls))))),P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),P_1)) -> inv(P_1,inv(P_1,A_1)) = ti(int,A_1)))))) # label(fact_817_inv__inv) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 745 (all M all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),hAPP(nat,nat,plus_plus(nat,J_1),M))))) # label(fact_678_trans__le__add1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 746 (all X_a (order(X_a) -> (all X_1 all Y_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1)) -> (ti(X_a,Y_1) != ti(X_a,X_1) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1))))))) # label(fact_332_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 747 (all X_a (comm_monoid_mult(X_a) -> (all A_1 all B all N_1 hAPP(X_a,X_a,times_times(X_a,hAPP(nat,X_a,power_power(X_a,A_1),N_1)),hAPP(nat,X_a,power_power(X_a,B),N_1)) = hAPP(nat,X_a,power_power(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),B)),N_1)))) # label(fact_260_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 748 (all K_1 all A_1 all J_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,zcong(J_1,zero_zero(int)),P_1)) -> (hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,multInv(P_1,J_1)),J_1)),K_1),hAPP(int,int,times_times(int,multInv(P_1,J_1)),A_1)),P_1)) -> hBOOL(hAPP(int,bool,zcong(K_1,hAPP(int,int,times_times(int,A_1),multInv(P_1,J_1))),P_1))))))) # label(fact_833_aux______4) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 749 (all Z_1 all W hAPP(int,int,plus_plus(int,Z_1),W) = hAPP(int,int,plus_plus(int,W),Z_1)) # label(fact_91_zadd__commute) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 750 (all X_a (comm_monoid_add(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,plus_plus(X_a,zero_zero(X_a)),A_1)))) # label(fact_767_add__0) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 751 (all X_a (comm_semiring_1(X_a) -> (all Lx all Rx all Ry hAPP(X_a,X_a,times_times(X_a,Lx),hAPP(X_a,X_a,times_times(X_a,Rx),Ry)) = hAPP(X_a,X_a,times_times(X_a,Rx),hAPP(X_a,X_a,times_times(X_a,Lx),Ry))))) # label(fact_72_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 752 (all Ma all K all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),N)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,times_times(nat,Ma),K)),hAPP(nat,nat,times_times(nat,N),K))))) # label(fact_714_mult__less__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 753 (all T_2 all T_1 (order(T_1) -> order(fun(T_2,T_1)))) # label(arity_fun___Orderings_Oorder) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 754 (all B_1_1 ti(fun(int,bool),quadRes(B_1_1)) = quadRes(B_1_1)) # label(tsy_c_Residues_OQuadRes_res) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 755 (all Q_1 all B all R_1 all C (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),R_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_1),zero_zero(int))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,B),C)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),hAPP(int,int,div_mod(int,Q_1),C))),R_1))))))) # label(fact_966_zmult2__lemma__aux1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 756 (all A_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_3),one_one(nat))) <-> A_3 = one_one(nat))) # label(fact_875_gcd__lcm__complete__lattice__nat_Obot__unique) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 757 (all X_a (linord581940658strict(X_a) -> (all A_3 all B_2 all C_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,C_1),A_3)),hAPP(X_a,X_a,times_times(X_a,C_1),B_2))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2))))))) # label(fact_528_mult__le__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 758 (all X_a (number_ring(X_a) -> (all Z_1 hAPP(X_a,X_a,times_times(X_a,number_number_of(X_a,bit0(bit1(pls)))),Z_1) = hAPP(X_a,X_a,plus_plus(X_a,Z_1),Z_1)))) # label(fact_166_mult__2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 759 (all X_1 all M all Y_1 (hAPP(int,int,div_mod(int,X_1),M) = hAPP(int,int,div_mod(int,Y_1),M) -> hBOOL(hAPP(int,bool,zcong(X_1,Y_1),M)))) # label(fact_912_Residues_Oaux) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 760 (all B_1_1 all X_a (semiring(X_a) -> ti(fun(X_a,X_a),times_times(X_a,B_1_1)) = times_times(X_a,B_1_1))) # label(tsy_c_Groups_Otimes__class_Otimes_3_res) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 761 (all X_a (ordere779506340up_add(X_a) -> (all C all D_2 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D_2)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),C)),hAPP(X_a,X_a,plus_plus(X_a,B),D_2)))))))) # label(fact_775_add__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 762 (all P_2 all N all K ((zero_zero(int) = ti(int,K) -> hBOOL(hAPP(int,bool,P_2,N))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),zero_zero(int))) -> (all I all J (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),J)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),J),zero_zero(int))) & ti(int,N) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,K),I)),J) -> hBOOL(hAPP(int,bool,P_2,J))))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K)) -> (all I all J (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),J),K)) & hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,K),I)),J) = ti(int,N) -> hBOOL(hAPP(int,bool,P_2,J))))) <-> hBOOL(hAPP(int,bool,P_2,hAPP(int,int,div_mod(int,N),K))))) # label(fact_963_split__zmod) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 763 (all X_a (linordered_ring(X_a) -> (all A_1 -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),A_1)),zero_zero(X_a)))))) # label(fact_504_not__square__less__zero) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 764 (all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M)) -> M = N_1))) # label(fact_620_le__antisym) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 765 (all X_a (linord20386208strict(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),C)))))))) # label(fact_492_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 766 (all Ma all N all K (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),N)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,times_times(nat,K),Ma)),hAPP(nat,nat,times_times(nat,K),N)))))) # label(fact_728_nat__mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 767 (all X_a (comm_semiring_1(X_a) -> (all M all A_1 hAPP(X_a,X_a,plus_plus(X_a,M),hAPP(X_a,X_a,times_times(X_a,A_1),M)) = hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),one_one(X_a))),M)))) # label(fact_141_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 768 (all X_a (power(X_a) & semiring_0(X_a) -> (all N_1 ((N_1 = zero_zero(nat) -> one_one(X_a) = hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),N_1)) & (zero_zero(nat) != N_1 -> hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),N_1) = zero_zero(X_a)))))) # label(fact_195_power__0__left) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 769 (all X_a (ordere1490568538miring(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,C),A_1)),hAPP(X_a,X_a,times_times(X_a,C),B)))))))) # label(fact_477_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 770 (all B_1_1 all B_2_1 zcong(B_1_1,B_2_1) = zcong(B_1_1,ti(int,B_2_1))) # label(tsy_c_IntPrimes_Ozcong_arg2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 771 (all A_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),one_one(nat)),A_1))) # label(fact_873_gcd__lcm__complete__lattice__nat_Obot__least) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 772 (all X_a (linord581940658strict(X_a) -> (all X_2 all Y_2 (zero_zero(X_a) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,X_2),X_2)),hAPP(X_a,X_a,times_times(X_a,Y_2),Y_2)) <-> zero_zero(X_a) = ti(X_a,X_2) & ti(X_a,Y_2) = zero_zero(X_a))))) # label(fact_218_sum__squares__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 773 (all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_1),A_1),one_one(int)),P_1)) -> hBOOL(hAPP(int,bool,zcong(A_1,hAPP(int,int,minus_minus(int,P_1),one_one(int))),P_1)) | hBOOL(hAPP(int,bool,zcong(A_1,one_one(int)),P_1)))))) # label(fact_393_zcong__square) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 774 (all U_1 all M all N_1 all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_1)) -> hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,I_1),U_1)),M)),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,J_1),U_1)),N_1)) = hAPP(nat,nat,minus_minus(nat,M),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,hAPP(nat,nat,minus_minus(nat,J_1),I_1)),U_1)),N_1)))) # label(fact_736_nat__diff__add__eq2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 775 (all X_a (linordered_semidom(X_a) -> (all Ma all N all A_3 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_3)) -> (Ma = N <-> hAPP(nat,X_a,power_power(X_a,A_3),N) = hAPP(nat,X_a,power_power(X_a,A_3),Ma)))))) # label(fact_272_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 776 (all U all Ma all N all J_2 all I_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_2),I_2)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,hAPP(nat,nat,minus_minus(nat,I_2),J_2)),U)),Ma)),N)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,I_2),U)),Ma)),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,J_2),U)),N)))))) # label(fact_738_nat__less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 777 (all X_a (ring(X_a) -> (all A_3 all E all C_1 all B_2 all D (hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,minus_minus(X_a,B_2),A_3)),E)),D) = ti(X_a,C_1) <-> hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,A_3),E)),C_1) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,B_2),E)),D))))) # label(fact_515_eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 778 (all X_1 all M all Y_1 hAPP(int,int,div_mod(int,hAPP(nat,int,power_power(int,hAPP(int,int,div_mod(int,X_1),M)),Y_1)),M) = hAPP(int,int,div_mod(int,hAPP(nat,int,power_power(int,X_1),Y_1)),M)) # label(fact_917_zpower__zmod) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 779 (all Y_1 all X_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),P_1)) -> (-hBOOL(hAPP(int,bool,zcong(Y_1,zero_zero(int)),P_1)) -> (hBOOL(hAPP(int,bool,zcong(multInv(P_1,X_1),multInv(P_1,Y_1)),P_1)) -> hBOOL(hAPP(int,bool,zcong(X_1,Y_1),P_1)))))))) # label(fact_829_MultInv__prop5) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 780 (all A_1 zero_zero(int) = hAPP(int,int,div_mod(int,A_1),number_number_of(int,min))) # label(fact_952_zmod__minus1__right) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 781 (all X_a (number_ring(X_a) -> zero_zero(X_a) = number_number_of(X_a,pls))) # label(fact_222_number__of__Pls) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 782 (all X_a (comm_semiring_1(X_a) -> (all A_1 hAPP(X_a,X_a,times_times(X_a,zero_zero(X_a)),A_1) = zero_zero(X_a)))) # label(fact_198_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 783 (all P_5 all P_2 all X_2 ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_2)) -> (hBOOL(P_2) <-> hBOOL(P_5))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_2)) & hBOOL(P_2) <-> hBOOL(P_5) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_2))))) # label(fact_742_conj__le__cong) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 784 (all Y_2 all X_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_2),X_2)) -> (Y_2 = X_2 <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_2),Y_2))))) # label(fact_597_dvd_Oantisym__conv) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 785 (all A_1 all J_1 all K_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zcong(J_1,K_1),P_1)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_1),multInv(P_1,J_1)),hAPP(int,int,times_times(int,A_1),multInv(P_1,K_1))),P_1))))) # label(fact_828_MultInv__zcong__prop1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 786 (all X_a (semiring_div(X_a) -> (all A_1 hAPP(X_a,X_a,div_mod(X_a,A_1),A_1) = zero_zero(X_a)))) # label(fact_942_mod__self) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 787 (all Ma all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),N)) <-> Ma = N | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),N)))) # label(fact_665_le__eq__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 788 (all C all A_1 all B all M (hBOOL(hAPP(int,bool,zcong(A_1,B),M)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,plus_plus(int,A_1),C),hAPP(int,int,plus_plus(int,B),C)),M)))) # label(fact_399_zcong__shift) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 789 (all K (bit0(K) = pls <-> pls = ti(int,K))) # label(fact_120_rel__simps_I44_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 790 (all Ma all K all F_1 ((all M_2 all N_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_2),N_2)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,F_1,M_2)),hAPP(nat,nat,F_1,N_2))))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,F_1,Ma)),K)),hAPP(nat,nat,F_1,hAPP(nat,nat,plus_plus(nat,Ma),K)))))) # label(fact_854_mono__nat__linear__lb) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 791 (all B_1_1 all X_a (dvd(X_a) -> times_times(X_a,ti(X_a,B_1_1)) = times_times(X_a,B_1_1))) # label(tsy_c_Groups_Otimes__class_Otimes_2_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 792 (all M M = hAPP(nat,nat,minus_minus(nat,M),zero_zero(nat))) # label(fact_564_minus__nat_Odiff__0) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 793 (all A_3 all B_2 (ti(int,B_2) = ti(int,A_3) <-> hBOOL(hAPP(int,bool,zcong(A_3,B_2),zero_zero(int))))) # label(fact_335_IntPrimes_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 794 (all X_a (group_add(X_a) -> (all A_1 zero_zero(X_a) = hAPP(X_a,X_a,minus_minus(X_a,A_1),A_1)))) # label(fact_787_diff__self) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 795 (all N_1 all M all K_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_1)) -> (hAPP(nat,nat,times_times(nat,K_1),N_1) = hAPP(nat,nat,times_times(nat,K_1),M) -> N_1 = M))) # label(fact_881_mult__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 796 (all X_1 all Y_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) -> -(-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1))))) # label(fact_586_dvd_Oless__not__sym) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 797 (all X_2 all P_3 (-hBOOL(hAPP(int,bool,zcong(X_2,zero_zero(int)),P_3)) <-> zero_zero(int) != standardRes(P_3,X_2))) # label(fact_987_StandardRes__prop3) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 798 (all A_1 zero_zero(int) = hAPP(int,int,div_mod(int,A_1),A_1)) # label(fact_921_zmod__self) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 799 -(all S1 -hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,S1),number_number_of(nat,bit0(bit1(pls)))),number_number_of(int,min)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int))))) # label(fact_284__096_B_Bthesis_O_A_I_B_Bs1_O_A_091s1_A_094_A2_A_061_A_N1_093_A_Imod_A4_) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 800 (all B_1_1 all B_2_1 product_Pair(int,int,B_1_1,B_2_1) = product_Pair(int,int,ti(int,B_1_1),B_2_1)) # label(tsy_c_Product__Type_OPair_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 801 (all X_a (number_ring(X_a) -> (all V all W all Z_1 hAPP(X_a,X_a,plus_plus(X_a,number_number_of(X_a,hAPP(int,int,plus_plus(int,V),W))),Z_1) = hAPP(X_a,X_a,plus_plus(X_a,number_number_of(X_a,V)),hAPP(X_a,X_a,plus_plus(X_a,number_number_of(X_a,W)),Z_1))))) # label(fact_148_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 802 (all X_a (number_ring(X_a) -> (all V all W hAPP(X_a,X_a,plus_plus(X_a,number_number_of(X_a,V)),number_number_of(X_a,W)) = number_number_of(X_a,hAPP(int,int,plus_plus(int,V),W))))) # label(fact_149_add__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 803 (all B_1_1 all B_2_1 product_Pair(int,int,B_1_1,B_2_1) = ti(product_prod(int,int),product_Pair(int,int,B_1_1,B_2_1))) # label(tsy_c_Product__Type_OPair_res) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 804 (all B_1_1 all X_a (semiring(X_a) -> times_times(X_a,B_1_1) = times_times(X_a,ti(X_a,B_1_1)))) # label(tsy_c_Groups_Otimes__class_Otimes_3_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 805 (all X_1 all Y_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zcong(X_1,Y_1),P_1)) -> hBOOL(hAPP(int,bool,zcong(multInv(P_1,X_1),multInv(P_1,Y_1)),P_1))))) # label(fact_826_MultInv__prop1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 806 (all X_a (ordered_ring(X_a) -> (all A_3 all E all C_1 all B_2 all D (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,minus_minus(X_a,B_2),A_3)),E)),D))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,A_3),E)),C_1)),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,B_2),E)),D))))))) # label(fact_544_le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 807 (all X_a (number_semiring(X_a) -> number_number_of(X_a,pls) = zero_zero(X_a))) # label(fact_223_semiring__numeral__0__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 808 (all Ma all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,plus_plus(nat,Ma),N))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Ma)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)))) # label(fact_709_add__gr__0) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 809 (all K_1 bit1(K_1) = hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,one_one(int)),K_1)),K_1)) # label(fact_153_Bit1__def) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 810 (all X_a (ring_div(X_a) -> (all A_1 all B all C hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,minus_minus(X_a,A_1),B)),C) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,minus_minus(X_a,hAPP(X_a,X_a,div_mod(X_a,A_1),C)),hAPP(X_a,X_a,div_mod(X_a,B),C))),C)))) # label(fact_948_mod__diff__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 811 (all X_a (ring_div(X_a) -> (all A_1 all B all C hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,minus_minus(X_a,hAPP(X_a,X_a,div_mod(X_a,A_1),C)),B)),C) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,minus_minus(X_a,A_1),B)),C)))) # label(fact_949_mod__diff__left__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 812 (all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)))) # label(fact_666_less__imp__le__nat) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 813 (all Ma all N all K (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),N)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,times_times(nat,K),Ma)),hAPP(nat,nat,times_times(nat,K),N)))))) # label(fact_730_nat__mult__dvd__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 814 (all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),one_one(nat))) <-> one_one(nat) = Ma)) # label(fact_698_nat__dvd__1__iff__1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 815 (all X_a (ordered_ab_group_add(X_a) -> (all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,minus_minus(X_a,A_3),B_2)),zero_zero(X_a))))))) # label(fact_810_le__iff__diff__le__0) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 816 (all X_a (monoid_mult(X_a) -> (all A_1 hAPP(X_a,X_a,times_times(X_a,A_1),A_1) = hAPP(nat,X_a,power_power(X_a,A_1),number_number_of(nat,bit0(bit1(pls))))))) # label(fact_14_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 817 (all Z_1 all X_1 all Y_1 all M (hBOOL(hAPP(int,bool,zcong(X_1,Y_1),M)) -> hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,X_1),Z_1),hAPP(nat,int,power_power(int,Y_1),Z_1)),M)))) # label(fact_400_zcong__zpower) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 818 (all K_1 all I_1 all J_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),I_1),J_1)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),J_1),K_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),I_1),K_1))))) # label(fact_561_real__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 819 (all X_a (real_normed_algebra(X_a) -> (all A_1 all B all B_3 hAPP(X_a,X_a,minus_minus(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),B)),hAPP(X_a,X_a,times_times(X_a,A_1),B_3)) = hAPP(X_a,X_a,times_times(X_a,A_1),hAPP(X_a,X_a,minus_minus(X_a,B),B_3))))) # label(fact_892_mult_Odiff__right) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 820 (all X_a (linordered_semidom(X_a) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,one_one(X_a)),one_one(X_a)))))) # label(fact_542_zero__less__two) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 821 (all K all Ma all N (hAPP(nat,nat,plus_plus(nat,K),Ma) = hAPP(nat,nat,plus_plus(nat,K),N) <-> N = Ma)) # label(fact_609_nat__add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 822 (all K all P_2 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D)) -> ((all X (hBOOL(hAPP(int,bool,P_2,X)) -> hBOOL(hAPP(int,bool,P_2,hAPP(int,int,plus_plus(int,X),D))))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) -> (all X (hBOOL(hAPP(int,bool,P_2,X)) -> hBOOL(hAPP(int,bool,P_2,hAPP(int,int,plus_plus(int,X),hAPP(int,int,times_times(int,K),D)))))))))) # label(fact_749_incr__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 823 (all X_a (linordered_idom(X_a) -> (all X_1 all Y_1 (ti(X_a,X_1) != ti(X_a,Y_1) -> (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_1),X_1))))))) # label(fact_425_linorder__neqE__linordered__idom) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 824 (all X_a (ab_group_add(X_a) -> (all A_3 all B_2 (hAPP(X_a,X_a,minus_minus(X_a,A_3),B_2) = zero_zero(X_a) <-> ti(X_a,B_2) = ti(X_a,A_3))))) # label(fact_786_eq__iff__diff__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 825 (all X_a (linord20386208strict(X_a) -> (all C all D_2 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_2)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),D_2)))))))))) # label(fact_531_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 826 (all K_1 hAPP(int,int,plus_plus(int,K_1),K_1) = bit0(K_1)) # label(fact_128_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 827 (all X_a (ring_div(X_a) -> (all A_1 all B all C hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,minus_minus(X_a,A_1),hAPP(X_a,X_a,div_mod(X_a,B),C))),C) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,minus_minus(X_a,A_1),B)),C)))) # label(fact_950_mod__diff__right__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 828 (all L_1 pls != bit1(L_1)) # label(fact_117_rel__simps_I39_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 829 (all W_1 all Z_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W_1),Z_2)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),hAPP(int,int,plus_plus(int,Z_2),one_one(int)))))) # label(fact_63_zle__add1__eq__le) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 830 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(K1)),bit0(K2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)))) # label(fact_57_less__eq__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 831 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all X_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),number_number_of(X_a,X_2)),zero_zero(X_a))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_2),pls)))))) # label(fact_247_less__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 832 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all X_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),number_number_of(X_a,X_2)),zero_zero(X_a))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X_2),pls)))))) # label(fact_249_le__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 833 (all X_a (semiring_div(X_a) -> (all A_1 all C all B hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),hAPP(X_a,X_a,times_times(X_a,C),B))),B) = hAPP(X_a,X_a,div_mod(X_a,A_1),B)))) # label(fact_902_mod__mult__self1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 834 (all X_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zcong(X_1,number_number_of(int,min)),P_1)) -> -hBOOL(hAPP(int,bool,zcong(X_1,one_one(int)),P_1))))) # label(fact_364_zcong__neg__1__impl__ne__1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 835 (all J_1 all I_1 -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,J_1),I_1)),I_1))) # label(fact_654_not__add__less2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 836 (all X_a (semiring_1(X_a) -> hAPP(nat,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]. 1.59/2.00 837 (all P_2 all A_3 all B_2 (-(hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_3),B_2)) & -hBOOL(hAPP(nat,bool,P_2,zero_zero(nat))) | (exists D_4 (-hBOOL(hAPP(nat,bool,P_2,D_4)) & hAPP(nat,nat,plus_plus(nat,B_2),D_4) = A_3))) <-> hBOOL(hAPP(nat,bool,P_2,hAPP(nat,nat,minus_minus(nat,A_3),B_2))))) # label(fact_711_nat__diff__split__asm) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 838 (all W all Z1 all Z2 hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,W),Z1)),hAPP(int,int,times_times(int,W),Z2)) = hAPP(int,int,times_times(int,W),hAPP(int,int,plus_plus(int,Z1),Z2))) # label(fact_133_zadd__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 839 (all B_1_1 (hBOOL(B_1_1) <-> hBOOL(ti(bool,B_1_1)))) # label(tsy_c_hBOOL_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 840 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),bit0(K))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),K)))) # label(fact_101_rel__simps_I21_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 841 (all X_a (number_ring(X_a) -> number_number_of(X_a,bit1(pls)) = one_one(X_a))) # label(fact_158_numeral__1__eq__1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 842 (all X_a (linordered_ring(X_a) -> (all X_1 all Y_1 -hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,X_1),X_1)),hAPP(X_a,X_a,times_times(X_a,Y_1),Y_1))),zero_zero(X_a)))))) # label(fact_238_not__sum__squares__lt__zero) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 843 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(K1)),bit1(K2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)))) # label(fact_40_less__eq__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 844 (all Z_1 all W (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Z_1),W)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),W),Z_1)) -> W = Z_1))) # label(fact_562_real__le__antisym) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 845 (all X_a (one(X_a) -> (all X_2 (one_one(X_a) = ti(X_a,X_2) <-> ti(X_a,X_2) = one_one(X_a))))) # label(fact_764_one__reorient) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 846 (all X_a (comm_semiring_1(X_a) -> (all A_1 hAPP(X_a,X_a,times_times(X_a,A_1),one_one(X_a)) = ti(X_a,A_1)))) # label(fact_113_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 847 (all Ma all N (Ma = zero_zero(nat) & N != zero_zero(nat) <-> zero_zero(nat) = hAPP(nat,nat,power_power(nat,Ma),N))) # label(fact_469_nat__power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 848 (all A_1 all J_1 all K_1 all P_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (hBOOL(hAPP(int,bool,zprime,P_1)) -> (-hBOOL(hAPP(int,bool,zcong(K_1,zero_zero(int)),P_1)) -> (-hBOOL(hAPP(int,bool,zcong(J_1,zero_zero(int)),P_1)) -> (hBOOL(hAPP(int,bool,zcong(J_1,hAPP(int,int,times_times(int,A_1),multInv(P_1,K_1))),P_1)) -> hBOOL(hAPP(int,bool,zcong(K_1,hAPP(int,int,times_times(int,A_1),multInv(P_1,J_1))),P_1)))))))) # label(fact_835_MultInv__zcong__prop2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 849 (all K_1 all N_1 all M hAPP(nat,nat,div_mod(nat,M),N_1) = hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,K_1),N_1)),M)),N_1)) # label(fact_979_mod__mult__self3) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 850 (all M all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),hAPP(nat,nat,plus_plus(nat,M),J_1))))) # label(fact_679_trans__le__add2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 851 (all X_a (ordere453448008miring(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),B)),zero_zero(X_a)))))))) # label(fact_483_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 852 (all N_1 -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),zero_zero(nat)))) # label(fact_557_less__zeroE) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 853 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,K)),number_number_of(int,L))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L)))) # label(fact_49_less__eq__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 854 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B all C all D_2 hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),C)),hAPP(X_a,X_a,plus_plus(X_a,B),D_2)) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),hAPP(X_a,X_a,plus_plus(X_a,C),D_2))))) # label(fact_74_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 855 (all P all Q (hBOOL(P) | -hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,P),Q)))) # label(help_fconj_2_1_U) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 856 (all K_1 all M all N_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,times_times(int,K_1),M)),hAPP(int,int,times_times(int,K_1),N_1))) -> (ti(int,K_1) != zero_zero(int) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M),N_1))))) # label(fact_192_zdvd__mult__cancel) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 857 (all X_a (number_ring(X_a) -> (all V all W hAPP(X_a,X_a,plus_plus(X_a,number_number_of(X_a,V)),number_number_of(X_a,W)) = number_number_of(X_a,hAPP(int,int,plus_plus(int,V),W))))) # label(fact_150_number__of__add) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 858 (all X_a (ordere236663937imp_le(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,C),A_1)),hAPP(X_a,X_a,plus_plus(X_a,C),B))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)))))) # label(fact_777_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 859 (all X_a (comm_semiring_1(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),hAPP(X_a,X_a,times_times(X_a,B),C))))))) # label(fact_456_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 860 (all X_a (real_normed_algebra(X_a) -> (all A_1 all B all B_3 hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),B)),hAPP(X_a,X_a,times_times(X_a,A_1),B_3)) = hAPP(X_a,X_a,times_times(X_a,A_1),hAPP(X_a,X_a,plus_plus(X_a,B),B_3))))) # label(fact_891_mult_Oadd__right) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 861 (all C_1 all D all A_3 all B_2 all Ma (hBOOL(hAPP(int,bool,zcong(A_3,B_2),Ma)) -> (hBOOL(hAPP(int,bool,zcong(C_1,hAPP(int,int,times_times(int,D),B_2)),Ma)) <-> hBOOL(hAPP(int,bool,zcong(C_1,hAPP(int,int,times_times(int,D),A_3)),Ma))))) # label(fact_398_zcong__zmult__prop2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 862 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),zero_zero(int))) -> hAPP(int,int,minus_minus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit1(pls)))),hAPP(int,int,div_mod(int,hAPP(int,int,plus_plus(int,B),one_one(int))),A_1))),one_one(int)) = hAPP(int,int,div_mod(int,hAPP(int,int,plus_plus(int,one_one(int)),hAPP(int,int,times_times(int,number_number_of(int,bit0(bit1(pls)))),B))),hAPP(int,int,times_times(int,number_number_of(int,bit0(bit1(pls)))),A_1)))) # label(fact_896_neg__zmod__mult__2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 863 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K)),min)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),min)))) # label(fact_318_rel__simps_I11_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 864 (all X_a (linord20386208strict(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),B)),zero_zero(X_a)))))))) # label(fact_499_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 865 (all X_1 all Y_1 (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)))) # label(fact_585_dvd_Oless__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 866 (all X_a (ordered_ring(X_a) -> (all C all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,C),A_1)),hAPP(X_a,X_a,times_times(X_a,C),B)))))))) # label(fact_475_mult__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 867 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,times_times(int,A_1),B))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))))) # label(fact_229_pos__zmult__pos) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 868 (all R_1 all Q_1 all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (ti(int,A_1) = hAPP(int,int,plus_plus(int,R_1),hAPP(int,int,times_times(int,A_1),Q_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_1),A_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),Q_1)))))) # label(fact_282_self__quotient__aux1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 869 (all K_1 all M all N_1 hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,times_times(nat,K_1),M)),hAPP(nat,nat,times_times(nat,K_1),N_1)) = hAPP(nat,nat,times_times(nat,K_1),hAPP(nat,nat,minus_minus(nat,M),N_1))) # label(fact_631_diff__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 870 (all B all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_1),B),zero_zero(int)),P_1)) -> hBOOL(hAPP(int,bool,zcong(B,zero_zero(int)),P_1)) | hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_1)))))) # label(fact_414_zcong__zprime__prod__zero) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 871 (all X_a (linordered_semidom(X_a) -> (all X_2 all Y_2 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),B_2)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,power_power(X_a,B_2),X_2)),hAPP(nat,X_a,power_power(X_a,B_2),Y_2))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_2),Y_2))))))) # label(fact_273_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 872 (all X_a (linordered_semidom(X_a) -> (all X_2 all Y_2 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),B_2)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,B_2),X_2)),hAPP(nat,X_a,power_power(X_a,B_2),Y_2))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X_2),Y_2))))))) # label(fact_179_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 873 (all B all Q_4 all R_3 all Q_1 all R_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),Q_4)),R_3)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),Q_1)),R_1))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_1),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),R_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),R_3)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_1),Q_4))))))) # label(fact_355_unique__quotient__lemma__neg) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 874 hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),t)) -> (exists X exists Y hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int)) = hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,X),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,Y),number_number_of(nat,bit0(bit1(pls)))))) # 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]. 1.59/2.00 875 (all B_1_1 all B_2_1 zcong(B_1_1,B_2_1) = zcong(ti(int,B_1_1),B_2_1)) # label(tsy_c_IntPrimes_Ozcong_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 876 (all X_a (dvd(X_a) & semiring_0(X_a) -> (all P_2 all L ((exists X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),L),hAPP(X_a,X_a,plus_plus(X_a,X),zero_zero(X_a)))) & hBOOL(hAPP(X_a,bool,P_2,X)))) <-> (exists X hBOOL(hAPP(X_a,bool,P_2,hAPP(X_a,X_a,times_times(X_a,L),X)))))))) # label(fact_744_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 1.59/2.00 877 (all X_a (ordere453448008miring(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a))) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),B)),zero_zero(X_a))))))) # label(fact_471_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 878 (all X_a (linordered_semidom(X_a) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),one_one(X_a))))) # label(fact_508_zero__less__one) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 879 (all X_a all X_c all X_b all P all Q all R hAPP(X_a,X_c,combc(X_a,X_b,X_c,P,Q),R) = hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),P,R),Q)) # label(help_COMBC_1_1_U) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 880 (all M all N_1 all K_1 hAPP(nat,nat,plus_plus(nat,M),hAPP(nat,nat,plus_plus(nat,N_1),K_1)) = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,plus_plus(nat,M),N_1)),K_1)) # label(fact_608_nat__add__assoc) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 881 (all X_a (ordere216010020id_add(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,A_1),B)))))))) # label(fact_806_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 882 (all N_1 all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),M)) -> M != N_1)) # label(fact_569_less__not__refl2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 883 (all B_1_1 zfact(B_1_1) = ti(int,zfact(B_1_1))) # label(tsy_c_IntFact_Ozfact_res) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 884 (all C all D_2 all A_1 all B all M (hBOOL(hAPP(int,bool,zcong(A_1,B),M)) -> (hBOOL(hAPP(int,bool,zcong(C,D_2),M)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,plus_plus(int,A_1),C),hAPP(int,int,plus_plus(int,B),D_2)),M))))) # label(fact_341_zcong__zadd) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 885 (all M hAPP(nat,nat,times_times(nat,M),zero_zero(nat)) = zero_zero(nat)) # label(fact_649_mult__0__right) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 886 (all X_a (number_semiring(X_a) -> number_number_of(X_a,bit1(pls)) = one_one(X_a))) # label(fact_157_semiring__numeral__1__eq__1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 887 (all L (bit1(L) = min <-> min = ti(int,L))) # label(fact_291_rel__simps_I43_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 888 (all B_1_1 sr(B_1_1) = sr(ti(int,B_1_1))) # label(tsy_c_Residues_OSR_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 889 (all X_a all P_2 ti(fun(X_a,bool),P_2) = collect(X_a,P_2)) # label(fact_76_Collect__def) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 890 (all X1 all X2 all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma)) -> (hBOOL(hAPP(int,bool,zcong(X1,X2),Ma)) <-> standardRes(Ma,X2) = standardRes(Ma,X1)))) # label(fact_994_StandardRes__prop2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 891 (all W_1 all Z_2 (ti(int,Z_2) = ti(int,W_1) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),Z_2)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),hAPP(int,int,plus_plus(int,Z_2),one_one(int)))))) # label(fact_102_zless__add1__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 892 (all A_3 (A_3 != one_one(nat) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),one_one(nat)),A_3)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_3),one_one(nat))))) # label(fact_874_gcd__lcm__complete__lattice__nat_Obot__less) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 893 (all X_a (semiring_div(X_a) -> (all K_1 all M all N_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_1),hAPP(X_a,X_a,div_mod(X_a,M),N_1))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_1),N_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_1),M))))))) # label(fact_923_dvd__mod__imp__dvd) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 894 (all P_2 all K all I_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),I_2)) -> (hBOOL(hAPP(int,bool,P_2,hAPP(int,int,plus_plus(int,K),one_one(int)))) -> ((all I (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),I)) -> (hBOOL(hAPP(int,bool,P_2,I)) -> hBOOL(hAPP(int,bool,P_2,hAPP(int,int,plus_plus(int,I),one_one(int))))))) -> hBOOL(hAPP(int,bool,P_2,I_2)))))) # label(fact_853_int__gr__induct) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 895 (all K_1 all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,times_times(nat,I_1),K_1)),hAPP(nat,nat,times_times(nat,J_1),K_1))))) # label(fact_701_mult__le__mono1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 896 (all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_1)) -> (exists K_2 (hAPP(nat,nat,plus_plus(nat,I_1),K_2) = J_1 & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_2)))))) # label(fact_861_less__imp__add__positive) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 897 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(K1)),bit1(K2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)))) # label(fact_38_less__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 898 (all U all Ma all N all I_2 all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),J_2)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,hAPP(nat,nat,minus_minus(nat,J_2),I_2)),U)),N))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,I_2),U)),Ma)),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,J_2),U)),N)))))) # label(fact_724_nat__less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 899 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K1)),bit1(K2))))) # label(fact_59_less__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 900 (all X_a (comm_semiring_1(X_a) -> (all A_1 all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(nat,X_a,power_power(X_a,A_1),M)),hAPP(nat,X_a,power_power(X_a,A_1),N_1))))))) # label(fact_186_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 901 (all X_a (linord20386208strict(X_a) -> (all C all D_2 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_2)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,A_1),C)),hAPP(X_a,X_a,times_times(X_a,B),D_2)))))))))) # label(fact_530_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 902 (all K_1 ti(int,K_1) = hAPP(int,int,minus_minus(int,K_1),pls)) # label(fact_370_diff__bin__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 903 (all A_1 -(-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),zero_zero(nat))) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A_1)))) # label(fact_871_gcd__lcm__complete__lattice__nat_Onot__top__less) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 904 (all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)) -> (M != N_1 -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1))))) # label(fact_667_le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 905 (all X_a (linordered_idom(X_a) -> (all A_1 all K_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,power_power(X_a,A_1),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),K_1))),zero_zero(X_a))) -> zero_zero(X_a) = ti(X_a,A_1))))) # label(fact_279_even__power__le__0__imp__0) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 906 (all A_1 all N_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,power_power(nat,A_1),N_1)),hAPP(nat,nat,power_power(nat,B),N_1))) -> (N_1 != zero_zero(nat) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))))) # label(fact_526_divides__rev) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 907 (all B_1_1 twoSqu1929807760sum2sq(ti(product_prod(int,int),B_1_1)) = twoSqu1929807760sum2sq(B_1_1)) # label(tsy_c_TwoSquares__Mirabelle__lldtdwpyrp_Osum2sq_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 908 (all X_a (plus(X_a) & dvd(X_a) & linorder(X_a) -> (all D all Sa exists Z all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Z)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),hAPP(X_a,X_a,plus_plus(X_a,X),Sa))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D),hAPP(X_a,X_a,plus_plus(X_a,X),Sa)))))))) # label(fact_865_minf_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 909 (all N all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma)) -> (one_one(int) = hAPP(int,int,times_times(int,Ma),N) <-> one_one(int) = ti(int,Ma) & one_one(int) = ti(int,N)))) # label(fact_244_pos__zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 910 (all W all Z1 all Z2 hAPP(int,int,minus_minus(int,hAPP(int,int,times_times(int,W),Z1)),hAPP(int,int,times_times(int,W),Z2)) = hAPP(int,int,times_times(int,W),hAPP(int,int,minus_minus(int,Z1),Z2))) # label(fact_372_zdiff__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 911 (all X_a (cancel_semigroup_add(X_a) -> (all A_1 all B all C (hAPP(X_a,X_a,plus_plus(X_a,A_1),B) = hAPP(X_a,X_a,plus_plus(X_a,A_1),C) -> ti(X_a,C) = ti(X_a,B))))) # label(fact_761_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 912 (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) != hAPP(X_a,X_a,times_times(X_a,A_1),B)))))) # label(fact_441_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 913 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(K)),bit1(L))))) # label(fact_39_rel__simps_I17_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 914 (all K (min = bit1(K) <-> min = ti(int,K))) # label(fact_290_rel__simps_I47_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 915 (all X_a (ordere216010020id_add(X_a) -> (all B all C all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),hAPP(X_a,X_a,plus_plus(X_a,A_1),C)))))))) # label(fact_815_add__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 916 (all C all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) -> (B = C -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C))))) # label(fact_593_dvd_Oord__le__eq__trans) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 917 (all B_1_1 bit1(ti(int,B_1_1)) = bit1(B_1_1)) # label(tsy_c_Int_OBit1_arg1) # label(hypothesis) # label(non_clause). [assumption]. 1.59/2.01 918 (all L_1 all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,minus_minus(nat,M),L_1)),hAPP(nat,nat,minus_minus(nat,N_1),L_1))))) # label(fact_623_diff__le__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 919 (all A_1 all B all C hAPP(int,int,div_mod(int,hAPP(int,int,times_times(int,A_1),B)),C) = hAPP(int,int,div_mod(int,hAPP(int,int,times_times(int,A_1),hAPP(int,int,div_mod(int,B),C))),C)) # label(fact_920_zmod__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 920 (all M hBOOL(hAPP(int,bool,zcong(M,zero_zero(int)),M))) # label(fact_396_zcong__id) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 921 (all Y_1 all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,times_times(int,X_1),Y_1)))))) # label(fact_345_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 922 (all A_1 all M all B hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_1),M),hAPP(int,int,times_times(int,B),M)),M))) # label(fact_340_zcong__zmult__self) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 923 (all X_a (ab_group_add(X_a) -> (all A_1 all C all B all D_2 hAPP(X_a,X_a,minus_minus(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),C)),hAPP(X_a,X_a,plus_plus(X_a,B),D_2)) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,minus_minus(X_a,A_1),B)),hAPP(X_a,X_a,minus_minus(X_a,C),D_2))))) # label(fact_450_add__diff__add) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 924 (all L_1 hAPP(int,int,minus_minus(int,min),bit1(L_1)) = bit0(hAPP(int,int,minus_minus(int,min),L_1))) # label(fact_389_diff__bin__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 925 (all M all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),X_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),M)) -> -hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),M))))) # label(fact_405_zcong__not__zero) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 926 (all M all N_1 all K_1 hAPP(nat,nat,times_times(nat,hAPP(nat,nat,div_mod(nat,M),N_1)),K_1) = hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,times_times(nat,M),K_1)),hAPP(nat,nat,times_times(nat,N_1),K_1))) # label(fact_982_mod__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 927 (all X_a (comm_semiring_1(X_a) -> (all M hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,plus_plus(X_a,one_one(X_a)),one_one(X_a))),M) = hAPP(X_a,X_a,plus_plus(X_a,M),M)))) # label(fact_142_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 928 (all Ma all X_2 (hBOOL(hAPP(int,bool,zcong(X_2,zero_zero(int)),Ma)) <-> standardRes(Ma,X_2) = zero_zero(int))) # label(fact_986_StandardRes__eq__zcong) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 929 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all X_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),number_number_of(X_a,X_2)),one_one(X_a))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X_2),bit1(pls))))))) # label(fact_106_le__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 930 (all V all W hAPP(int,int,div_mod(int,number_number_of(int,bit0(V))),number_number_of(int,bit0(W))) = hAPP(int,int,times_times(int,number_number_of(int,bit0(bit1(pls)))),hAPP(int,int,div_mod(int,number_number_of(int,V)),number_number_of(int,W)))) # label(fact_962_zmod__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 931 (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]. 1.59/2.01 932 (all K all Ma all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),N)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,K),Ma)),hAPP(nat,nat,plus_plus(nat,K),N))))) # label(fact_677_nat__add__left__cancel__le) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 933 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(K)),min)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),min)))) # label(fact_306_rel__simps_I30_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 934 (all X_a (ordere453448008miring(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,times_times(X_a,A_1),B)))))))) # label(fact_484_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 935 (all L_1 bit1(hAPP(int,int,minus_minus(int,min),L_1)) = hAPP(int,int,minus_minus(int,pls),bit1(L_1))) # label(fact_387_diff__bin__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 936 (all X_a (comm_monoid_add(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,plus_plus(X_a,A_1),zero_zero(X_a))))) # label(fact_770_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 937 (all J_1 all A_1 all P_1 all K_1 (hBOOL(hAPP(int,bool,zcong(J_1,hAPP(int,int,times_times(int,A_1),multInv(P_1,K_1))),P_1)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,J_1),K_1),hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,A_1),multInv(P_1,K_1))),K_1)),P_1)))) # label(fact_820_aux______1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 938 (all X_a (linord893533164strict(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,C),A_1)),hAPP(X_a,X_a,times_times(X_a,C),B)))))))) # label(fact_490_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 939 (all X_a (ordered_ring(X_a) -> (all B all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,times_times(X_a,A_1),B)))))))) # label(fact_480_mult__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 940 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),bit1(K))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),K)))) # label(fact_307_rel__simps_I26_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 941 (all Y_1 all N_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),hAPP(nat,int,power_power(int,Y_1),N_1))) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_1),Y_1)))))) # label(fact_416_zpower__zdvd__prop2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 942 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B all C hAPP(X_a,X_a,plus_plus(X_a,A_1),hAPP(X_a,X_a,plus_plus(X_a,B),C)) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),C)))) # label(fact_78_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 943 (all X_a (semiring_div(X_a) -> (all A_1 all B hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),B) = hAPP(X_a,X_a,div_mod(X_a,A_1),B)))) # label(fact_927_mod__add__self2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 944 (all X_a (number_semiring(X_a) -> (all Z_1 hAPP(X_a,X_a,times_times(X_a,number_number_of(X_a,bit0(bit1(pls)))),Z_1) = hAPP(X_a,X_a,plus_plus(X_a,Z_1),Z_1)))) # label(fact_165_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 945 (all X_1 all Y_1 all Z_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Z_1)) -> (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),Y_1)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,Z_1),X_1)),hAPP(real,real,times_times(real,Z_1),Y_1)))))) # label(fact_523_real__mult__less__mono2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 946 (all X_a all B_1_1 ti(fun(fun(X_a,bool),bool),member(X_a,B_1_1)) = member(X_a,B_1_1)) # label(tsy_c_member_res) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 947 (all X_a (monoid_mult(X_a) -> (all X_1 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) -> hAPP(X_a,X_a,times_times(X_a,hAPP(nat,X_a,power_power(X_a,X_1),hAPP(nat,nat,minus_minus(nat,N_1),one_one(nat)))),X_1) = hAPP(nat,X_a,power_power(X_a,X_1),N_1))))) # label(fact_436_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 948 (all A_1 all B (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,div_mod(int,A_1),B)),B)))) # label(fact_907_pos__mod__bound) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 949 (all Z_1 hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),Z_1) = hAPP(nat,nat,plus_plus(nat,Z_1),Z_1)) # label(fact_35_nat__mult__2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 950 (all Ma all N (hAPP(nat,nat,times_times(nat,Ma),N) = zero_zero(nat) <-> zero_zero(nat) = N | zero_zero(nat) = Ma)) # label(fact_650_mult__is__0) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 951 (all J_1 all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_1)) -> (-hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_1)) -> (-hBOOL(hAPP(int,bool,zcong(J_1,zero_zero(int)),P_1)) -> (-hBOOL(hAPP(int,bool,quadRes(P_1),A_1)) -> -hBOOL(hAPP(int,bool,zcong(J_1,hAPP(int,int,times_times(int,A_1),multInv(P_1,J_1))),P_1)))))))) # label(fact_818_MultInvPair__distinct) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 952 (all X_a (number_ring(X_a) -> (all W number_number_of(X_a,bit1(W)) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,plus_plus(X_a,one_one(X_a)),number_number_of(X_a,W))),number_number_of(X_a,W))))) # label(fact_154_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 953 (all X_a (ordere236663937imp_le(X_a) -> (all C_1 all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,C_1),A_3)),hAPP(X_a,X_a,plus_plus(X_a,C_1),B_2))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2)))))) # label(fact_772_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 954 (all X_1 all Y_1 (Y_1 = X_1 -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)))) # label(fact_598_dvd_Oeq__refl) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 955 (all X_a (ring_n68954251visors(X_a) -> (all A_3 all B_2 (zero_zero(X_a) = hAPP(X_a,X_a,times_times(X_a,A_3),B_2) <-> ti(X_a,B_2) = zero_zero(X_a) | ti(X_a,A_3) = zero_zero(X_a))))) # label(fact_442_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 956 (all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (ti(int,P_1) != number_number_of(int,bit0(bit1(pls))) -> (number_number_of(int,bit1(bit1(pls))) != ti(int,P_1) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,bit1(bit0(bit1(pls))))),P_1)))))) # label(fact_172_prime__g__5) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 957 (all L_1 bit0(L_1) != min) # label(fact_296_rel__simps_I42_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 958 (all X_a (linordered_idom(X_a) -> (all X_2 all Y_2 (ti(X_a,X_2) = zero_zero(X_a) & zero_zero(X_a) = ti(X_a,Y_2) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,X_2),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,X_a,power_power(X_a,Y_2),number_number_of(nat,bit0(bit1(pls)))))),zero_zero(X_a))))))) # label(fact_266_sum__power2__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 959 (all Ma all K all N (zero_zero(nat) = K | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),N)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,times_times(nat,Ma),K)),hAPP(nat,nat,times_times(nat,N),K))))) # label(fact_518_nat__mult__dvd__cancel__disj_H) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 960 (all X_a (monoid_mult(X_a) -> (all A_1 all N_1 hAPP(nat,X_a,power_power(X_a,A_1),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),N_1)) = hAPP(nat,X_a,power_power(X_a,hAPP(nat,X_a,power_power(X_a,A_1),N_1)),number_number_of(nat,bit0(bit1(pls))))))) # label(fact_103_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 961 (all X_a (power(X_a) -> (all M all N_1 ((zero_zero(nat) != N_1 -> hAPP(X_a,X_a,times_times(X_a,M),hAPP(nat,X_a,power_power(X_a,M),hAPP(nat,nat,minus_minus(nat,N_1),one_one(nat)))) = hAPP(nat,X_a,power_power(X_a,M),N_1)) & (N_1 = zero_zero(nat) -> hAPP(nat,X_a,power_power(X_a,M),N_1) = one_one(X_a)))))) # label(fact_722_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 962 (all N_1 -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),zero_zero(nat)))) # label(fact_636_not__less0) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 963 (all X_a (comm_monoid_mult(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,times_times(X_a,A_1),one_one(X_a))))) # label(fact_794_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 964 (all X_a all X_b all X_c all B_1_1 all B_2_1 combc(X_a,X_b,X_c,B_1_1,ti(X_b,B_2_1)) = combc(X_a,X_b,X_c,B_1_1,B_2_1)) # label(tsy_c_COMBC_arg2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 965 (all X_a (comm_semiring_1(X_a) -> (all A_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),one_one(X_a)),A_1))))) # label(fact_460_one__dvd) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 966 (all A_2 all B_1 all Q_3 all Y_1 (ti(int,A_2) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B_1),Q_3)),Y_1) -> ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_1),Y_1)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Y_1),zero_zero(int)))) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y_1)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y_1),B_1))) -> (ti(int,B_1) != zero_zero(int) -> ti(int,Y_1) = hAPP(int,int,div_mod(int,A_2),B_1))))) # label(fact_968_divmod__int__rel__mod__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 967 (all X_a (comm_semiring_1(X_a) -> (all N_1 all X_1 all Y_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X_1),Y_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(nat,X_a,power_power(X_a,X_1),N_1)),hAPP(nat,X_a,power_power(X_a,Y_1),N_1))))))) # label(fact_193_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 968 (all X_a (linordered_semidom(X_a) -> (all N_1 all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(nat,X_a,power_power(X_a,A_1),N_1)))))))) # label(fact_321_one__less__power) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 969 (all X_b (semiring(X_b) & number(X_b) -> (all V all B all C hAPP(X_b,X_b,times_times(X_b,number_number_of(X_b,V)),hAPP(X_b,X_b,plus_plus(X_b,B),C)) = hAPP(X_b,X_b,plus_plus(X_b,hAPP(X_b,X_b,times_times(X_b,number_number_of(X_b,V)),B)),hAPP(X_b,X_b,times_times(X_b,number_number_of(X_b,V)),C))))) # label(fact_139_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 970 (all X_a (semiring_div(X_a) -> (all C all A_1 all B hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,times_times(X_a,C),A_1)),hAPP(X_a,X_a,times_times(X_a,C),B)) = hAPP(X_a,X_a,times_times(X_a,C),hAPP(X_a,X_a,div_mod(X_a,A_1),B))))) # label(fact_938_mod__mult__mult1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 971 (all X_a (ring_11004092258visors(X_a) -> (all N_1 all A_1 (ti(X_a,A_1) != zero_zero(X_a) -> zero_zero(X_a) != hAPP(nat,X_a,power_power(X_a,A_1),N_1))))) # label(fact_194_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 972 (all M all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),M)) -> (hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),M)) -> ti(int,X_1) = zero_zero(int))))) # label(fact_410_Int2_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 973 (all C_1 all D all A_3 all B_2 all Ma (hBOOL(hAPP(int,bool,zcong(A_3,B_2),Ma)) -> (hBOOL(hAPP(int,bool,zcong(C_1,hAPP(int,int,times_times(int,B_2),D)),Ma)) <-> hBOOL(hAPP(int,bool,zcong(C_1,hAPP(int,int,times_times(int,A_3),D)),Ma))))) # label(fact_397_zcong__zmult__prop1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 974 (all B all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),zero_zero(int))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),A_1)) -> ti(int,A_1) = hAPP(int,int,div_mod(int,A_1),B)))) # label(fact_960_mod__neg__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 975 (all A_1 all B (B != A_1 -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),A_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))))) # label(fact_599_dvd_Oneq__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 976 (all X_a (linord20386208strict(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,times_times(X_a,C),A_1)),hAPP(X_a,X_a,times_times(X_a,C),B))) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))))))) # label(fact_537_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 977 (all X_a (ordere223160158up_add(X_a) -> (all C all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),C)),hAPP(X_a,X_a,plus_plus(X_a,B),C))))))) # label(fact_780_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 978 (all X_a (mult_zero(X_a) -> (all A_1 hAPP(X_a,X_a,times_times(X_a,zero_zero(X_a)),A_1) = zero_zero(X_a)))) # label(fact_444_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 979 (all X_a (semiring_div(X_a) -> (all A_1 hAPP(X_a,X_a,div_mod(X_a,A_1),one_one(X_a)) = zero_zero(X_a)))) # label(fact_901_mod__by__1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 980 (all Y_1 all X_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,plus_plus(int,X_1),Y_1)))))) # label(fact_346_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 981 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),hAPP(X_a,X_a,times_times(X_a,A_1),B)))))) # label(fact_458_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 982 (all X_a (no_zero_divisors(X_a) -> (all A_1 all B (zero_zero(X_a) = hAPP(X_a,X_a,times_times(X_a,A_1),B) -> zero_zero(X_a) = ti(X_a,A_1) | zero_zero(X_a) = ti(X_a,B))))) # label(fact_440_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 983 (all X_a (linordered_semidom(X_a) -> (all N_1 all A_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,power_power(X_a,A_1),N_1))))))) # label(fact_208_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 984 (all M all N_1 (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1)) -> M = hAPP(nat,nat,plus_plus(nat,N_1),hAPP(nat,nat,minus_minus(nat,M),N_1)))) # label(fact_662_add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 985 (all B_1_1 all X_a (monoid_add(X_a) -> plus_plus(X_a,B_1_1) = plus_plus(X_a,ti(X_a,B_1_1)))) # label(tsy_c_Groups_Oplus__class_Oplus_1_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 986 (all B_1_1 all X_a (mult_zero(X_a) -> ti(fun(X_a,X_a),times_times(X_a,B_1_1)) = times_times(X_a,B_1_1))) # label(tsy_c_Groups_Otimes__class_Otimes_4_res) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 987 (all X_a all X_b all X_c all B_1_1 all B_2_1 combs(X_a,X_b,X_c,B_1_1,B_2_1) = combs(X_a,X_b,X_c,B_1_1,ti(fun(X_a,X_b),B_2_1))) # label(tsy_c_COMBS_arg2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 988 (all X_a (cancel_semigroup_add(X_a) -> (all B all A_1 all C (hAPP(X_a,X_a,plus_plus(X_a,B),A_1) = hAPP(X_a,X_a,plus_plus(X_a,C),A_1) -> ti(X_a,C) = ti(X_a,B))))) # label(fact_763_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 989 (all I_2 all K all J_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),J_2)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),hAPP(nat,nat,minus_minus(nat,J_2),K))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,I_2),K)),J_2))))) # label(fact_690_le__diff__conv2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 990 (all X_a all B_1_1 member(X_a,ti(X_a,B_1_1)) = member(X_a,B_1_1)) # label(tsy_c_member_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 991 (all I_1 all J_1 -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,I_1),J_1)),I_1))) # label(fact_653_not__add__less1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 992 (all B_1_1 all X_a (monoid_add(X_a) -> ti(fun(X_a,X_a),plus_plus(X_a,B_1_1)) = plus_plus(X_a,B_1_1))) # label(tsy_c_Groups_Oplus__class_Oplus_1_res) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 993 (all Z_1 all X_1 all Y_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) -> (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z_1),Y_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),Z_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Z_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z_1),X_1))))) # label(fact_588_dvd_Ole__less__trans) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 994 (all W_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),zero_zero(int))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(W_1)),zero_zero(int))))) # label(fact_226_bin__less__0__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 995 (all X_a (linordered_idom(X_a) -> (all X_1 all Y_1 hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,X_1),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,X_a,power_power(X_a,Y_1),number_number_of(nat,bit0(bit1(pls)))))))))) # label(fact_265_sum__power2__ge__zero) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 996 (all I_1 all J_1 all K_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,I_1),J_1)),K_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),K_1)))) # label(fact_661_add__lessD1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 997 (all X_a (ab_group_add(X_a) -> (all A_3 all B_2 all C_1 all D (hAPP(X_a,X_a,minus_minus(X_a,A_3),B_2) = hAPP(X_a,X_a,minus_minus(X_a,C_1),D) -> (ti(X_a,C_1) = ti(X_a,D) <-> ti(X_a,B_2) = ti(X_a,A_3)))))) # label(fact_765_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 998 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),hAPP(X_a,X_a,times_times(X_a,B),A_1)))))) # label(fact_457_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 999 (all K_1 all L_1 bit0(hAPP(int,int,minus_minus(int,K_1),L_1)) = hAPP(int,int,minus_minus(int,bit0(K_1)),bit0(L_1))) # label(fact_371_diff__bin__simps_I7_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1000 (all A_1 all P_1 (hBOOL(hAPP(int,bool,zprime,P_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),P_1)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_1),inv(P_1,A_1)),one_one(int)),P_1)))))) # label(fact_832_inv__is__inv) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1001 (all X_a (ordere236663937imp_le(X_a) -> (all A_1 all C all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),C)),hAPP(X_a,X_a,plus_plus(X_a,B),C))) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)))))) # label(fact_783_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1002 (all X_a (real_normed_algebra(X_a) -> (all A_1 all A_4 all B hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),B)),hAPP(X_a,X_a,times_times(X_a,A_4),B)) = hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),A_4)),B)))) # label(fact_889_mult_Oadd__left) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1003 (all B_1_1 zfact(ti(int,B_1_1)) = zfact(B_1_1)) # label(tsy_c_IntFact_Ozfact_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1004 (all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),N_1)) -> N_1 = zero_zero(nat) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)))) # label(fact_880_divides__le) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1005 (all X_a (ring(X_a) -> (all A_3 all E all C_1 all B_2 all D (ti(X_a,D) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,minus_minus(X_a,A_3),B_2)),E)),C_1) <-> hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,A_3),E)),C_1) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,B_2),E)),D))))) # label(fact_514_eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1006 (all B_1_1 d22set(B_1_1) = d22set(ti(int,B_1_1))) # label(tsy_c_IntFact_Od22set_arg1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1007 (all Ma all N ((exists K_2 hAPP(nat,nat,plus_plus(nat,Ma),K_2) = N) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),N)))) # label(fact_676_le__iff__add) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1008 (all X_a (semiri456707255roduct(X_a) -> (all C_1 all D all A_3 all B_2 (ti(X_a,B_2) != ti(X_a,A_3) & ti(X_a,C_1) != ti(X_a,D) <-> hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,A_3),D)),hAPP(X_a,X_a,times_times(X_a,B_2),C_1)) != hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,A_3),C_1)),hAPP(X_a,X_a,times_times(X_a,B_2),D)))))) # label(fact_111_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1009 (all A_3 all B_2 all C_1 (C_1 != zero_zero(real) -> (hAPP(real,real,times_times(real,B_2),C_1) = hAPP(real,real,times_times(real,A_3),C_1) <-> A_3 = B_2))) # label(fact_462_real__mult__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1010 (all L_1 hAPP(int,int,minus_minus(int,pls),bit0(L_1)) = bit0(hAPP(int,int,minus_minus(int,pls),L_1))) # label(fact_380_diff__bin__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1011 (all X_a (linord581940658strict(X_a) -> (all A_3 all B_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,times_times(X_a,A_3),B_2)),zero_zero(X_a))) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B_2)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),zero_zero(X_a))) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_3)) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B_2),zero_zero(X_a))))))) # label(fact_485_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1012 (all M M = hAPP(nat,nat,plus_plus(nat,M),zero_zero(nat))) # label(fact_641_Nat_Oadd__0__right) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1013 (all X_1 all Y_1 (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) -> X_1 != Y_1)) # label(fact_582_dvd_Oless__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1014 (all X_a (semiring_div(X_a) -> (all A_1 all C all B hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,div_mod(X_a,A_1),C)),B)),C) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),B)),C)))) # label(fact_940_zmod__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1015 (all B_1_1 all B_2_1 ti(int,multInv(B_1_1,B_2_1)) = multInv(B_1_1,B_2_1)) # label(tsy_c_Int2_OMultInv_res) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1016 (all A_3 all Ma (hBOOL(hAPP(int,bool,zcong(A_3,zero_zero(int)),Ma)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Ma),A_3)))) # label(fact_408_zcong__zero__equiv__div) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1017 (all S all T_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),S),T_3)) -> S != T_3)) # label(fact_568_less__not__refl3) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1018 (all I_1 all K_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),J_1)) -> hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,plus_plus(nat,I_1),J_1)),K_1) = hAPP(nat,nat,plus_plus(nat,I_1),hAPP(nat,nat,minus_minus(nat,J_1),K_1)))) # label(fact_693_diff__add__assoc) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1019 (all B_1_1 all B_2_1 ti(int,inv(B_1_1,B_2_1)) = inv(B_1_1,B_2_1)) # label(tsy_c_WilsonRuss_Oinv_res) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1020 (all B_1_1 all X_a (group_add(X_a) -> minus_minus(X_a,B_1_1) = ti(fun(X_a,X_a),minus_minus(X_a,B_1_1)))) # label(tsy_c_Groups_Ominus__class_Ominus_0_res) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1021 (all P_5 all P_2 all X_2 ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_2)) -> (hBOOL(P_5) <-> hBOOL(P_2))) -> ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_2)) -> hBOOL(P_2)) <-> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_2)) -> hBOOL(P_5))))) # label(fact_741_imp__le__cong) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1022 (all P_2 all A_3 all B_2 (hBOOL(hAPP(nat,bool,P_2,hAPP(nat,nat,minus_minus(nat,A_3),B_2))) <-> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_3),B_2)) -> hBOOL(hAPP(nat,bool,P_2,zero_zero(nat)))) & (all D_4 (A_3 = hAPP(nat,nat,plus_plus(nat,B_2),D_4) -> hBOOL(hAPP(nat,bool,P_2,D_4)))))) # label(fact_710_nat__diff__split) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1023 (all K_1 all L_1 bit0(hAPP(int,int,minus_minus(int,K_1),L_1)) = hAPP(int,int,minus_minus(int,bit1(K_1)),bit1(L_1))) # label(fact_379_diff__bin__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1024 (all N (zero_zero(nat) = N <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),zero_zero(nat))))) # label(fact_644_le__0__eq) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1025 (all K all P_2 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D)) -> ((all X (hBOOL(hAPP(int,bool,P_2,X)) -> hBOOL(hAPP(int,bool,P_2,hAPP(int,int,minus_minus(int,X),D))))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) -> (all X (hBOOL(hAPP(int,bool,P_2,X)) -> hBOOL(hAPP(int,bool,P_2,hAPP(int,int,minus_minus(int,X),hAPP(int,int,times_times(int,K),D)))))))))) # label(fact_748_decr__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1026 (all X_a (real_normed_algebra(X_a) -> (all X_1 all Y_1 all Ya hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,minus_minus(X_a,X_1),Y_1)),Ya) = hAPP(X_a,X_a,minus_minus(X_a,hAPP(X_a,X_a,times_times(X_a,X_1),Ya)),hAPP(X_a,X_a,times_times(X_a,Y_1),Ya))))) # label(fact_895_mult__left_Odiff) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1027 (all X_a all X_2 all A_6 (hBOOL(hAPP(fun(X_a,bool),bool,member(X_a,X_2),A_6)) <-> hBOOL(hAPP(X_a,bool,A_6,X_2)))) # label(fact_75_mem__def) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1028 (all X_a (comm_semiring_1(X_a) -> (all X_1 all N_1 hAPP(nat,X_a,power_power(X_a,X_1),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),N_1)) = hAPP(X_a,X_a,times_times(X_a,hAPP(nat,X_a,power_power(X_a,X_1),N_1)),hAPP(nat,X_a,power_power(X_a,X_1),N_1))))) # label(fact_15_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1029 (all N_1 all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M)) -> M = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,minus_minus(nat,M),N_1)),N_1))) # label(fact_691_le__add__diff__inverse2) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1030 (all X_a (comm_semiring_1(X_a) -> (all B all A_1 all C (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),C)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),hAPP(X_a,X_a,times_times(X_a,B),C))))))) # label(fact_455_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1031 (all K_1 all M all N_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_1),hAPP(int,int,div_mod(int,M),N_1))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_1),N_1)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_1),M))))) # label(fact_915_zdvd__zmod__imp__zdvd) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1032 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K1)),bit0(K2))))) # label(fact_43_less__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1033 (all B_1_1 all X_a (dvd(X_a) -> times_times(X_a,B_1_1) = ti(fun(X_a,X_a),times_times(X_a,B_1_1)))) # label(tsy_c_Groups_Otimes__class_Otimes_2_res) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1034 (all X_a all X_c all X_b all P all Q all R hAPP(X_b,X_c,P,hAPP(X_a,X_b,Q,R)) = hAPP(X_a,X_c,combb(X_b,X_c,X_a,P,Q),R)) # label(help_COMBB_1_1_U) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1035 (all X_a (ordere223160158up_add(X_a) -> (all C all D_2 all A_1 all B (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D_2)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,plus_plus(X_a,A_1),C)),hAPP(X_a,X_a,plus_plus(X_a,B),D_2)))))))) # label(fact_809_add__le__less__mono) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1036 (all M all I_1 all J_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),hAPP(nat,nat,plus_plus(nat,J_1),M))))) # label(fact_656_trans__less__add1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1037 (all B_2 all A_3 (hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),d22set(A_3))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),B_2)))) # label(fact_859_d22set__g__1) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1038 (all X_a (cancel_semigroup_add(X_a) -> (all A_3 all B_2 all C_1 (hAPP(X_a,X_a,plus_plus(X_a,A_3),C_1) = hAPP(X_a,X_a,plus_plus(X_a,A_3),B_2) <-> ti(X_a,C_1) = ti(X_a,B_2))))) # label(fact_759_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1039 (all X_a (real_normed_algebra(X_a) -> (all A_1 all A_4 all B hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,minus_minus(X_a,A_1),A_4)),B) = hAPP(X_a,X_a,minus_minus(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),B)),hAPP(X_a,X_a,times_times(X_a,A_4),B))))) # label(fact_894_mult_Odiff__left) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1040 (all X_2 all Y_2 (zero_zero(real) = hAPP(real,real,plus_plus(real,hAPP(nat,real,power_power(real,X_2),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,real,power_power(real,Y_2),number_number_of(nat,bit0(bit1(pls))))) <-> zero_zero(real) = Y_2 & X_2 = zero_zero(real))) # label(fact_421_realpow__two__sum__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1041 (all X_a (number_ring(X_a) -> (all W number_number_of(X_a,hAPP(int,int,plus_plus(int,bit1(pls)),W)) = hAPP(X_a,X_a,plus_plus(X_a,one_one(X_a)),number_number_of(X_a,W))))) # label(fact_16_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1042 (all X_a (semiring_div(X_a) & number(X_a) -> (all X_2 all Y_2 (hAPP(X_a,X_a,div_mod(X_a,number_number_of(X_a,Y_2)),number_number_of(X_a,X_2)) = zero_zero(X_a) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),number_number_of(X_a,X_2)),number_number_of(X_a,Y_2))))))) # label(fact_951_dvd__eq__mod__eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 1.59/2.01 1043 -(exists X exists Y hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,X),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,Y),number_number_of(nat,bit0(bit1(pls))))) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int))) # label(conj_0) # label(negated_conjecture) # label(non_clause). [assumption]. 1.59/2.03 1.59/2.03 ============================== end of process non-clausal formulas === 1.59/2.03 1.59/2.03 ============================== PROCESS INITIAL CLAUSES =============== 1.59/2.03 1.59/2.03 ============================== PREDICATE ELIMINATION ================= 1.59/2.03 1044 comm_semiring_1(int) # label(arity_Int_Oint___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 1.59/2.03 1045 -comm_semiring_1(A) | hAPP(A,A,times_times(A,hAPP(A,A,times_times(A,B),C)),D) = hAPP(A,A,times_times(A,B),hAPP(A,A,times_times(A,C),D)) # label(fact_71_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom). [clausify(2)]. 1.59/2.03 1046 -comm_semiring_1(A) | hAPP(A,A,times_times(A,hAPP(A,A,times_times(A,B),C)),D) = hAPP(A,A,times_times(A,B),hAPP(A,A,times_times(A,C),D)) # label(fact_70_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom). [clausify(3)]. 1.59/2.03 1047 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),D),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(nat,A,power_power(A,B),D)),hAPP(nat,A,power_power(A,C),E))) # label(fact_187_dvd__power__le) # label(axiom). [clausify(14)]. 1.59/2.03 1048 -comm_semiring_1(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C)) = hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,B),D)),C) # label(fact_109_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom). [clausify(18)]. 1.59/2.03 1049 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),D),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),E))) # label(fact_454_mult__dvd__mono) # label(axiom). [clausify(55)]. 1.59/2.03 1050 -comm_semiring_1(A) | ti(A,B) = hAPP(A,A,plus_plus(A,B),zero_zero(A)) # label(fact_201_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom). [clausify(70)]. 1.59/2.03 Derived: hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,A),B)),C) = hAPP(int,int,times_times(int,A),hAPP(int,int,times_times(int,B),C)). [resolve(1044,a,1045,a)]. 1.59/2.03 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),D)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,power_power(int,A),C)),hAPP(nat,int,power_power(int,B),D))). [resolve(1044,a,1047,a)]. 1.59/2.03 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),C),D)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,times_times(int,A),C)),hAPP(int,int,times_times(int,B),D))). [resolve(1044,a,1049,a)]. 1.59/2.03 1051 -comm_semiring_1(A) | hAPP(A,A,times_times(A,B),C) = hAPP(A,A,times_times(A,C),B) # label(fact_73_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom). [clausify(78)]. 1.59/2.03 1052 -comm_semiring_1(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,plus_plus(A,B),C)),D) = hAPP(A,A,plus_plus(A,B),hAPP(A,A,plus_plus(A,C),D)) # label(fact_79_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom). [clausify(83)]. 1.59/2.03 1053 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,plus_plus(A,C),D))) # label(fact_459_dvd__add) # label(axiom). [clausify(90)]. 1.59/2.03 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(int,int,plus_plus(int,B),C))). [resolve(1053,a,1044,a)]. 1.59/2.03 1054 -comm_semiring_1(A) | hAPP(nat,A,power_power(A,hAPP(nat,A,power_power(A,B),C)),D) = hAPP(nat,A,power_power(A,B),hAPP(nat,nat,times_times(nat,C),D)) # label(fact_26_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom). [clausify(96)]. 1.59/2.03 1055 -comm_semiring_1(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),C) = hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,B),one_one(A))),C) # label(fact_140_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom). [clausify(106)]. 1.59/2.03 Derived: hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),B)),B) = hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,A),one_one(int))),B). [resolve(1055,a,1044,a)]. 1.59/2.03 1056 -comm_semiring_1(A) | ti(A,B) = hAPP(A,A,plus_plus(A,zero_zero(A)),B) # label(fact_200_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom). [clausify(112)]. 1.59/2.03 1057 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(A,A,times_times(A,B),C)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),D)) # label(fact_451_dvd__mult__right) # label(axiom). [clausify(116)]. 1.59/2.03 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,times_times(int,A),B)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B),C)). [resolve(1057,a,1044,a)]. 1.59/2.03 1058 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(nat,A,power_power(A,B),C)),D)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),E),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(nat,A,power_power(A,B),E)),D)) # label(fact_188_power__le__dvd) # label(axiom). [clausify(120)]. 1.59/2.03 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,power_power(int,A),B)),C)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),D),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,power_power(int,A),D)),C)). [resolve(1058,a,1044,a)]. 1.59/2.03 1059 -comm_semiring_1(A) | hAPP(nat,A,power_power(A,hAPP(A,A,times_times(A,B),C)),D) = hAPP(A,A,times_times(A,hAPP(nat,A,power_power(A,B),D)),hAPP(nat,A,power_power(A,C),D)) # label(fact_115_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom). [clausify(168)]. 1.59/2.03 Derived: hAPP(nat,int,power_power(int,hAPP(int,int,times_times(int,A),B)),C) = hAPP(int,int,times_times(int,hAPP(nat,int,power_power(int,A),C)),hAPP(nat,int,power_power(int,B),C)). [resolve(1059,a,1044,a)]. 1.59/2.03 1060 -comm_semiring_1(A) | hAPP(A,A,times_times(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),E)) = hAPP(A,A,times_times(A,hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),E)) # label(fact_66_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom). [clausify(169)]. 1.59/2.03 Derived: hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,A),B)),hAPP(int,int,times_times(int,C),D)) = hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,A),C)),hAPP(int,int,times_times(int,B),D)). [resolve(1060,a,1044,a)]. 1.59/2.03 1061 comm_semiring_1(real) # label(arity_RealDef_Oreal___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 1.59/2.03 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),D)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(nat,real,power_power(real,A),C)),hAPP(nat,real,power_power(real,B),D))). [resolve(1061,a,1047,a)]. 1.59/2.03 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),C),D)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(real,real,times_times(real,A),C)),hAPP(real,real,times_times(real,B),D))). [resolve(1061,a,1049,a)]. 1.59/2.03 Derived: ti(real,A) = hAPP(real,real,plus_plus(real,A),zero_zero(real)). [resolve(1061,a,1050,a)]. 1.59/2.03 Derived: hAPP(real,real,times_times(real,A),B) = hAPP(real,real,times_times(real,B),A). [resolve(1061,a,1051,a)]. 1.59/2.03 Derived: hAPP(real,real,plus_plus(real,hAPP(real,real,plus_plus(real,A),B)),C) = hAPP(real,real,plus_plus(real,A),hAPP(real,real,plus_plus(real,B),C)). [resolve(1061,a,1052,a)]. 1.59/2.03 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(real,real,plus_plus(real,B),C))). [resolve(1061,a,1053,a)]. 1.59/2.05 Derived: hAPP(nat,real,power_power(real,hAPP(nat,real,power_power(real,A),B)),C) = hAPP(nat,real,power_power(real,A),hAPP(nat,nat,times_times(nat,B),C)). [resolve(1061,a,1054,a)]. 1.59/2.05 Derived: hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),B)),B) = hAPP(real,real,times_times(real,hAPP(real,real,plus_plus(real,A),one_one(real))),B). [resolve(1061,a,1055,a)]. 1.59/2.05 Derived: ti(real,A) = hAPP(real,real,plus_plus(real,zero_zero(real)),A). [resolve(1061,a,1056,a)]. 1.59/2.05 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(real,real,times_times(real,A),B)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),B),C)). [resolve(1061,a,1057,a)]. 1.59/2.05 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(nat,real,power_power(real,A),B)),C)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),D),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(nat,real,power_power(real,A),D)),C)). [resolve(1061,a,1058,a)]. 1.59/2.05 Derived: hAPP(nat,real,power_power(real,hAPP(real,real,times_times(real,A),B)),C) = hAPP(real,real,times_times(real,hAPP(nat,real,power_power(real,A),C)),hAPP(nat,real,power_power(real,B),C)). [resolve(1061,a,1059,a)]. 1.59/2.05 Derived: hAPP(real,real,times_times(real,hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,C),D)) = hAPP(real,real,times_times(real,hAPP(real,real,times_times(real,A),C)),hAPP(real,real,times_times(real,B),D)). [resolve(1061,a,1060,a)]. 1.59/2.05 1062 -comm_semiring_1(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D)) = hAPP(A,A,times_times(A,B),hAPP(A,A,plus_plus(A,C),D)) # label(fact_112_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom). [clausify(294)]. 1.59/2.05 Derived: hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,A),C)) = hAPP(real,real,times_times(real,A),hAPP(real,real,plus_plus(real,B),C)). [resolve(1062,a,1061,a)]. 1.59/2.05 1063 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) # label(fact_427_dvd__trans) # label(axiom). [clausify(320)]. 1.59/2.05 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),C)). [resolve(1063,a,1044,a)]. 1.59/2.05 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),B),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),C)). [resolve(1063,a,1061,a)]. 1.59/2.05 1064 -comm_semiring_1(A) | one_one(A) = hAPP(nat,A,power_power(A,B),zero_zero(nat)) # label(fact_313_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom). [clausify(341)]. 1.59/2.05 Derived: one_one(int) = hAPP(nat,int,power_power(int,A),zero_zero(nat)). [resolve(1064,a,1044,a)]. 1.59/2.05 Derived: one_one(real) = hAPP(nat,real,power_power(real,A),zero_zero(nat)). [resolve(1064,a,1061,a)]. 1.59/2.05 1065 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(A,A,times_times(A,B),C)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) # label(fact_452_dvd__mult__left) # label(axiom). [clausify(373)]. 1.59/2.05 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,times_times(int,A),B)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),C)). [resolve(1065,a,1044,a)]. 1.59/2.05 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(real,real,times_times(real,A),B)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),C)). [resolve(1065,a,1061,a)]. 1.59/2.05 1066 -comm_semiring_1(A) | hAPP(A,A,times_times(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),E)) = hAPP(A,A,times_times(A,B),hAPP(A,A,times_times(A,C),hAPP(A,A,times_times(A,D),E))) # label(fact_68_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom). [clausify(378)]. 1.59/2.06 1067 -comm_semiring_1(A) | hAPP(A,A,plus_plus(A,B),C) = hAPP(A,A,plus_plus(A,C),B) # label(fact_81_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom). [clausify(414)]. 1.59/2.06 Derived: hAPP(real,real,plus_plus(real,A),B) = hAPP(real,real,plus_plus(real,B),A). [resolve(1067,a,1061,a)]. 1.59/2.06 1068 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),zero_zero(A)),B)) | zero_zero(A) = ti(A,B) # label(fact_449_dvd__0__left) # label(axiom). [clausify(422)]. 1.59/2.06 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),zero_zero(int)),A)) | zero_zero(int) = ti(int,A). [resolve(1068,a,1044,a)]. 1.59/2.06 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),zero_zero(real)),A)) | zero_zero(real) = ti(real,A). [resolve(1068,a,1061,a)]. 1.59/2.06 1069 -comm_semiring_1(A) | hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))) = hAPP(A,A,times_times(A,B),B) # label(fact_13_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom). [clausify(423)]. 1.59/2.07 Derived: hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls)))) = hAPP(int,int,times_times(int,A),A). [resolve(1069,a,1044,a)]. 1.59/2.07 Derived: hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))) = hAPP(real,real,times_times(real,A),A). [resolve(1069,a,1061,a)]. 1.59/2.07 1070 -comm_semiring_1(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),B)) # label(fact_426_dvd__refl) # label(axiom). [clausify(470)]. 1.59/2.07 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),A)). [resolve(1070,a,1044,a)]. 1.59/2.07 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),A)). [resolve(1070,a,1061,a)]. 1.59/2.07 1071 -comm_semiring_1(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C)) = hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,B),D)),C) # label(fact_110_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom). [clausify(492)]. 1.59/2.07 Derived: hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),B)),hAPP(int,int,times_times(int,C),B)) = hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,A),C)),B). [resolve(1071,a,1044,a)]. 1.59/2.07 1072 -comm_semiring_1(A) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),hAPP(nat,A,power_power(A,C),B))) # label(fact_322_dvd__power) # label(axiom). [clausify(501)]. 1.59/2.07 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B),hAPP(nat,int,power_power(int,B),A))). [resolve(1072,a,1044,a)]. 1.59/2.07 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),B),hAPP(nat,real,power_power(real,B),A))). [resolve(1072,a,1061,a)]. 1.59/2.07 1073 -comm_semiring_1(A) | one_one(A) != ti(A,B) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(nat,A,power_power(A,B),C))) # label(fact_322_dvd__power) # label(axiom). [clausify(501)]. 1.59/2.07 Derived: one_one(int) != ti(int,A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(nat,int,power_power(int,A),B))). [resolve(1073,a,1044,a)]. 1.59/2.07 Derived: one_one(real) != ti(real,A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(nat,real,power_power(real,A),B))). [resolve(1073,a,1061,a)]. 1.59/2.07 1074 -comm_semiring_1(A) | hAPP(A,A,times_times(A,hAPP(A,A,times_times(A,B),C)),D) = hAPP(A,A,times_times(A,hAPP(A,A,times_times(A,B),D)),C) # label(fact_69_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom). [clausify(529)]. 1.59/2.07 Derived: hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,A),B)),C) = hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,A),C)),B). [resolve(1074,a,1044,a)]. 1.59/2.07 Derived: hAPP(real,real,times_times(real,hAPP(real,real,times_times(real,A),B)),C) = hAPP(real,real,times_times(real,hAPP(real,real,times_times(real,A),C)),B). [resolve(1074,a,1061,a)]. 1.59/2.07 1075 -comm_semiring_1(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,plus_plus(A,B),C)),D) = hAPP(A,A,plus_plus(A,hAPP(A,A,plus_plus(A,B),D)),C) # label(fact_77_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom). [clausify(537)]. 1.59/2.09 Derived: hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,A),B)),C) = hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,A),C)),B). [resolve(1075,a,1044,a)]. 1.59/2.09 Derived: hAPP(real,real,plus_plus(real,hAPP(real,real,plus_plus(real,A),B)),C) = hAPP(real,real,plus_plus(real,hAPP(real,real,plus_plus(real,A),C)),B). [resolve(1075,a,1061,a)]. 1.59/2.09 1076 -comm_semiring_1(A) | zero_zero(A) = hAPP(A,A,times_times(A,B),zero_zero(A)) # label(fact_199_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom). [clausify(552)]. 1.59/2.09 Derived: zero_zero(int) = hAPP(int,int,times_times(int,A),zero_zero(int)). [resolve(1076,a,1044,a)]. 1.59/2.09 Derived: zero_zero(real) = hAPP(real,real,times_times(real,A),zero_zero(real)). [resolve(1076,a,1061,a)]. 1.59/2.09 1077 -comm_semiring_1(A) | ti(A,B) = hAPP(A,A,times_times(A,one_one(A)),B) # label(fact_114_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom). [clausify(620)]. 1.59/2.09 Derived: ti(real,A) = hAPP(real,real,times_times(real,one_one(real)),A). [resolve(1077,a,1061,a)]. 1.59/2.09 1078 -comm_semiring_1(A) | hAPP(A,A,plus_plus(A,B),hAPP(A,A,plus_plus(A,C),D)) = hAPP(A,A,plus_plus(A,C),hAPP(A,A,plus_plus(A,B),D)) # label(fact_80_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom). [clausify(628)]. 1.59/2.09 Derived: hAPP(int,int,plus_plus(int,A),hAPP(int,int,plus_plus(int,B),C)) = hAPP(int,int,plus_plus(int,B),hAPP(int,int,plus_plus(int,A),C)). [resolve(1078,a,1044,a)]. 1.59/2.09 Derived: hAPP(real,real,plus_plus(real,A),hAPP(real,real,plus_plus(real,B),C)) = hAPP(real,real,plus_plus(real,B),hAPP(real,real,plus_plus(real,A),C)). [resolve(1078,a,1061,a)]. 1.59/2.09 1079 -comm_semiring_1(A) | hAPP(A,A,times_times(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),E)) = hAPP(A,A,times_times(A,D),hAPP(A,A,times_times(A,hAPP(A,A,times_times(A,B),C)),E)) # label(fact_67_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom). [clausify(667)]. 1.59/2.09 Derived: hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,A),B)),hAPP(int,int,times_times(int,C),D)) = hAPP(int,int,times_times(int,C),hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,A),B)),D)). [resolve(1079,a,1044,a)]. 1.59/2.09 Derived: hAPP(real,real,times_times(real,hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,C),D)) = hAPP(real,real,times_times(real,C),hAPP(real,real,times_times(real,hAPP(real,real,times_times(real,A),B)),D)). [resolve(1079,a,1061,a)]. 1.59/2.09 1080 -comm_semiring_1(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),zero_zero(A))) # label(fact_423_dvd__0__right) # label(axiom). [clausify(703)]. 1.59/2.09 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),zero_zero(int))). [resolve(1080,a,1044,a)]. 1.59/2.09 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),zero_zero(real))). [resolve(1080,a,1061,a)]. 1.59/2.09 1081 -comm_semiring_1(A) | ti(A,B) = hAPP(nat,A,power_power(A,B),one_one(nat)) # label(fact_27_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom). [clausify(713)]. 1.59/2.09 Derived: ti(int,A) = hAPP(nat,int,power_power(int,A),one_one(nat)). [resolve(1081,a,1044,a)]. 1.59/2.09 Derived: ti(real,A) = hAPP(nat,real,power_power(real,A),one_one(nat)). [resolve(1081,a,1061,a)]. 1.59/2.09 1082 -comm_semiring_1(A) | hAPP(nat,A,power_power(A,B),hAPP(nat,nat,plus_plus(nat,C),D)) = hAPP(A,A,times_times(A,hAPP(nat,A,power_power(A,B),C)),hAPP(nat,A,power_power(A,B),D)) # label(fact_33_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom). [clausify(741)]. 1.59/2.09 Derived: hAPP(nat,int,power_power(int,A),hAPP(nat,nat,plus_plus(nat,B),C)) = hAPP(int,int,times_times(int,hAPP(nat,int,power_power(int,A),B)),hAPP(nat,int,power_power(int,A),C)). [resolve(1082,a,1044,a)]. 1.59/2.09 Derived: hAPP(nat,real,power_power(real,A),hAPP(nat,nat,plus_plus(nat,B),C)) = hAPP(real,real,times_times(real,hAPP(nat,real,power_power(real,A),B)),hAPP(nat,real,power_power(real,A),C)). [resolve(1082,a,1061,a)]. 1.59/2.11 1083 -comm_semiring_1(A) | hAPP(A,A,times_times(A,B),hAPP(A,A,times_times(A,C),D)) = hAPP(A,A,times_times(A,C),hAPP(A,A,times_times(A,B),D)) # label(fact_72_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom). [clausify(751)]. 1.59/2.11 Derived: hAPP(int,int,times_times(int,A),hAPP(int,int,times_times(int,B),C)) = hAPP(int,int,times_times(int,B),hAPP(int,int,times_times(int,A),C)). [resolve(1083,a,1044,a)]. 1.59/2.11 Derived: hAPP(real,real,times_times(real,A),hAPP(real,real,times_times(real,B),C)) = hAPP(real,real,times_times(real,B),hAPP(real,real,times_times(real,A),C)). [resolve(1083,a,1061,a)]. 1.59/2.11 1084 -comm_semiring_1(A) | hAPP(A,A,plus_plus(A,B),hAPP(A,A,times_times(A,C),B)) = hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,C),one_one(A))),B) # label(fact_141_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom). [clausify(767)]. 1.59/2.11 Derived: hAPP(int,int,plus_plus(int,A),hAPP(int,int,times_times(int,B),A)) = hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,B),one_one(int))),A). [resolve(1084,a,1044,a)]. 1.59/2.11 Derived: hAPP(real,real,plus_plus(real,A),hAPP(real,real,times_times(real,B),A)) = hAPP(real,real,times_times(real,hAPP(real,real,plus_plus(real,B),one_one(real))),A). [resolve(1084,a,1061,a)]. 1.59/2.11 1085 comm_semiring_1(nat) # label(arity_Nat_Onat___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 1.59/2.11 Derived: hAPP(nat,nat,times_times(nat,hAPP(nat,nat,times_times(nat,A),B)),C) = hAPP(nat,nat,times_times(nat,A),hAPP(nat,nat,times_times(nat,B),C)). [resolve(1085,a,1045,a)]. 1.59/2.11 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),D)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,power_power(nat,A),C)),hAPP(nat,nat,power_power(nat,B),D))). [resolve(1085,a,1047,a)]. 1.59/2.11 Derived: hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,A),B)),hAPP(nat,nat,times_times(nat,C),B)) = hAPP(nat,nat,times_times(nat,hAPP(nat,nat,plus_plus(nat,A),C)),B). [resolve(1085,a,1048,a)]. 1.59/2.11 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),C),D)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,times_times(nat,A),C)),hAPP(nat,nat,times_times(nat,B),D))). [resolve(1085,a,1049,a)]. 1.59/2.11 Derived: ti(nat,A) = hAPP(nat,nat,plus_plus(nat,A),zero_zero(nat)). [resolve(1085,a,1050,a)]. 1.59/2.11 Derived: hAPP(nat,nat,times_times(nat,A),B) = hAPP(nat,nat,times_times(nat,B),A). [resolve(1085,a,1051,a)]. 1.59/2.11 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,plus_plus(nat,B),C))). [resolve(1085,a,1053,a)]. 1.59/2.11 Derived: hAPP(nat,nat,power_power(nat,hAPP(nat,nat,power_power(nat,A),B)),C) = hAPP(nat,nat,power_power(nat,A),hAPP(nat,nat,times_times(nat,B),C)). [resolve(1085,a,1054,a)]. 1.59/2.11 Derived: hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,A),B)),B) = hAPP(nat,nat,times_times(nat,hAPP(nat,nat,plus_plus(nat,A),one_one(nat))),B). [resolve(1085,a,1055,a)]. 1.59/2.11 Derived: ti(nat,A) = hAPP(nat,nat,plus_plus(nat,zero_zero(nat)),A). [resolve(1085,a,1056,a)]. 1.59/2.11 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,times_times(nat,A),B)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),C)). [resolve(1085,a,1057,a)]. 1.59/2.11 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,power_power(nat,A),B)),C)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),D),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,power_power(nat,A),D)),C)). [resolve(1085,a,1058,a)]. 1.59/2.11 Derived: hAPP(nat,nat,power_power(nat,hAPP(nat,nat,times_times(nat,A),B)),C) = hAPP(nat,nat,times_times(nat,hAPP(nat,nat,power_power(nat,A),C)),hAPP(nat,nat,power_power(nat,B),C)). [resolve(1085,a,1059,a)]. 1.59/2.11 Derived: hAPP(nat,nat,times_times(nat,hAPP(nat,nat,times_times(nat,A),B)),hAPP(nat,nat,times_times(nat,C),D)) = hAPP(nat,nat,times_times(nat,hAPP(nat,nat,times_times(nat,A),C)),hAPP(nat,nat,times_times(nat,B),D)). [resolve(1085,a,1060,a)]. 1.59/2.12 Derived: hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,A),B)),hAPP(nat,nat,times_times(nat,A),C)) = hAPP(nat,nat,times_times(nat,A),hAPP(nat,nat,plus_plus(nat,B),C)). [resolve(1085,a,1062,a)]. 1.59/2.12 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),C)). [resolve(1085,a,1063,a)]. 1.59/2.12 Derived: one_one(nat) = hAPP(nat,nat,power_power(nat,A),zero_zero(nat)). [resolve(1085,a,1064,a)]. 1.59/2.12 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,times_times(nat,A),B)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),C)). [resolve(1085,a,1065,a)]. 1.59/2.12 Derived: hAPP(nat,nat,plus_plus(nat,A),B) = hAPP(nat,nat,plus_plus(nat,B),A). [resolve(1085,a,1067,a)]. 1.59/2.12 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A)) | zero_zero(nat) = ti(nat,A). [resolve(1085,a,1068,a)]. 1.59/2.12 Derived: hAPP(nat,nat,power_power(nat,A),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,nat,times_times(nat,A),A). [resolve(1085,a,1069,a)]. 1.59/2.12 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),A)). [resolve(1085,a,1070,a)]. 1.59/2.12 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),hAPP(nat,nat,power_power(nat,B),A))). [resolve(1085,a,1072,a)]. 1.59/2.12 Derived: one_one(nat) != ti(nat,A) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,power_power(nat,A),B))). [resolve(1085,a,1073,a)]. 1.59/2.12 Derived: hAPP(nat,nat,times_times(nat,hAPP(nat,nat,times_times(nat,A),B)),C) = hAPP(nat,nat,times_times(nat,hAPP(nat,nat,times_times(nat,A),C)),B). [resolve(1085,a,1074,a)]. 1.59/2.12 Derived: hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,plus_plus(nat,A),B)),C) = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,plus_plus(nat,A),C)),B). [resolve(1085,a,1075,a)]. 1.59/2.12 Derived: ti(nat,A) = hAPP(nat,nat,times_times(nat,one_one(nat)),A). [resolve(1085,a,1077,a)]. 1.59/2.12 Derived: hAPP(nat,nat,plus_plus(nat,A),hAPP(nat,nat,plus_plus(nat,B),C)) = hAPP(nat,nat,plus_plus(nat,B),hAPP(nat,nat,plus_plus(nat,A),C)). [resolve(1085,a,1078,a)]. 1.59/2.12 Derived: hAPP(nat,nat,times_times(nat,hAPP(nat,nat,times_times(nat,A),B)),hAPP(nat,nat,times_times(nat,C),D)) = hAPP(nat,nat,times_times(nat,C),hAPP(nat,nat,times_times(nat,hAPP(nat,nat,times_times(nat,A),B)),D)). [resolve(1085,a,1079,a)]. 1.59/2.12 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),zero_zero(nat))). [resolve(1085,a,1080,a)]. 1.59/2.12 Derived: ti(nat,A) = hAPP(nat,nat,power_power(nat,A),one_one(nat)). [resolve(1085,a,1081,a)]. 1.59/2.12 Derived: hAPP(nat,nat,power_power(nat,A),hAPP(nat,nat,plus_plus(nat,B),C)) = hAPP(nat,nat,times_times(nat,hAPP(nat,nat,power_power(nat,A),B)),hAPP(nat,nat,power_power(nat,A),C)). [resolve(1085,a,1082,a)]. 1.59/2.12 Derived: hAPP(nat,nat,times_times(nat,A),hAPP(nat,nat,times_times(nat,B),C)) = hAPP(nat,nat,times_times(nat,B),hAPP(nat,nat,times_times(nat,A),C)). [resolve(1085,a,1083,a)]. 1.59/2.12 Derived: hAPP(nat,nat,plus_plus(nat,A),hAPP(nat,nat,times_times(nat,B),A)) = hAPP(nat,nat,times_times(nat,hAPP(nat,nat,plus_plus(nat,B),one_one(nat))),A). [resolve(1085,a,1084,a)]. 1.59/2.12 1086 -comm_semiring_1(A) | zero_zero(A) = hAPP(A,A,times_times(A,zero_zero(A)),B) # label(fact_198_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom). [clausify(782)]. 1.59/2.12 Derived: zero_zero(int) = hAPP(int,int,times_times(int,zero_zero(int)),A). [resolve(1086,a,1044,a)]. 1.59/2.12 Derived: zero_zero(real) = hAPP(real,real,times_times(real,zero_zero(real)),A). [resolve(1086,a,1061,a)]. 1.59/2.12 Derived: zero_zero(nat) = hAPP(nat,nat,times_times(nat,zero_zero(nat)),A). [resolve(1086,a,1085,a)]. 1.59/2.12 1087 -comm_semiring_1(A) | ti(A,B) = hAPP(A,A,times_times(A,B),one_one(A)) # label(fact_113_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom). [clausify(846)]. 1.59/2.12 Derived: ti(int,A) = hAPP(int,int,times_times(int,A),one_one(int)). [resolve(1087,a,1044,a)]. 1.59/2.15 Derived: ti(real,A) = hAPP(real,real,times_times(real,A),one_one(real)). [resolve(1087,a,1061,a)]. 1.59/2.15 Derived: ti(nat,A) = hAPP(nat,nat,times_times(nat,A),one_one(nat)). [resolve(1087,a,1085,a)]. 1.59/2.15 1088 -comm_semiring_1(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,plus_plus(A,B),C)),hAPP(A,A,plus_plus(A,D),E)) = hAPP(A,A,plus_plus(A,hAPP(A,A,plus_plus(A,B),D)),hAPP(A,A,plus_plus(A,C),E)) # label(fact_74_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom). [clausify(854)]. 1.59/2.15 Derived: hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,A),B)),hAPP(int,int,plus_plus(int,C),D)) = hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,A),C)),hAPP(int,int,plus_plus(int,B),D)). [resolve(1088,a,1044,a)]. 1.59/2.15 Derived: hAPP(real,real,plus_plus(real,hAPP(real,real,plus_plus(real,A),B)),hAPP(real,real,plus_plus(real,C),D)) = hAPP(real,real,plus_plus(real,hAPP(real,real,plus_plus(real,A),C)),hAPP(real,real,plus_plus(real,B),D)). [resolve(1088,a,1061,a)]. 1.59/2.15 Derived: hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,plus_plus(nat,A),B)),hAPP(nat,nat,plus_plus(nat,C),D)) = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,plus_plus(nat,A),C)),hAPP(nat,nat,plus_plus(nat,B),D)). [resolve(1088,a,1085,a)]. 1.59/2.15 1089 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,times_times(A,C),D))) # label(fact_456_dvd__mult2) # label(axiom). [clausify(859)]. 1.59/2.15 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(int,int,times_times(int,B),C))). [resolve(1089,a,1044,a)]. 1.59/2.15 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(real,real,times_times(real,B),C))). [resolve(1089,a,1061,a)]. 1.59/2.15 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,times_times(nat,B),C))). [resolve(1089,a,1085,a)]. 1.59/2.15 1090 -comm_semiring_1(A) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(nat,A,power_power(A,D),B)),hAPP(nat,A,power_power(A,D),C))) # label(fact_186_le__imp__power__dvd) # label(axiom). [clausify(900)]. 1.59/2.15 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,power_power(int,C),A)),hAPP(nat,int,power_power(int,C),B))). [resolve(1090,a,1044,a)]. 1.59/2.15 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(nat,real,power_power(real,C),A)),hAPP(nat,real,power_power(real,C),B))). [resolve(1090,a,1061,a)]. 1.59/2.15 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,power_power(nat,C),A)),hAPP(nat,nat,power_power(nat,C),B))). [resolve(1090,a,1085,a)]. 1.59/2.15 1091 -comm_semiring_1(A) | hAPP(A,A,plus_plus(A,B),B) = hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,one_one(A)),one_one(A))),B) # label(fact_142_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom). [clausify(927)]. 1.59/2.15 Derived: hAPP(int,int,plus_plus(int,A),A) = hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,one_one(int)),one_one(int))),A). [resolve(1091,a,1044,a)]. 1.59/2.15 Derived: hAPP(real,real,plus_plus(real,A),A) = hAPP(real,real,times_times(real,hAPP(real,real,plus_plus(real,one_one(real)),one_one(real))),A). [resolve(1091,a,1061,a)]. 1.59/2.15 Derived: hAPP(nat,nat,plus_plus(nat,A),A) = hAPP(nat,nat,times_times(nat,hAPP(nat,nat,plus_plus(nat,one_one(nat)),one_one(nat))),A). [resolve(1091,a,1085,a)]. 1.59/2.15 1092 -comm_semiring_1(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,plus_plus(A,B),C)),D) = hAPP(A,A,plus_plus(A,B),hAPP(A,A,plus_plus(A,C),D)) # label(fact_78_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom). [clausify(942)]. 1.59/2.15 Derived: hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,A),B)),C) = hAPP(int,int,plus_plus(int,A),hAPP(int,int,plus_plus(int,B),C)). [resolve(1092,a,1044,a)]. 1.59/2.17 Derived: hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,plus_plus(nat,A),B)),C) = hAPP(nat,nat,plus_plus(nat,A),hAPP(nat,nat,plus_plus(nat,B),C)). [resolve(1092,a,1085,a)]. 1.59/2.17 1093 -comm_semiring_1(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),one_one(A)),B)) # label(fact_460_one__dvd) # label(axiom). [clausify(965)]. 1.59/2.17 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),one_one(int)),A)). [resolve(1093,a,1044,a)]. 1.59/2.17 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),one_one(real)),A)). [resolve(1093,a,1061,a)]. 1.59/2.17 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),one_one(nat)),A)). [resolve(1093,a,1085,a)]. 1.59/2.17 1094 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(nat,A,power_power(A,B),D)),hAPP(nat,A,power_power(A,C),D))) # label(fact_193_dvd__power__same) # label(axiom). [clausify(967)]. 1.59/2.17 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,power_power(int,A),C)),hAPP(nat,int,power_power(int,B),C))). [resolve(1094,a,1044,a)]. 1.59/2.17 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(nat,real,power_power(real,A),C)),hAPP(nat,real,power_power(real,B),C))). [resolve(1094,a,1061,a)]. 1.59/2.17 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,power_power(nat,A),C)),hAPP(nat,nat,power_power(nat,B),C))). [resolve(1094,a,1085,a)]. 1.59/2.17 1095 -comm_semiring_1(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,times_times(A,B),C))) # label(fact_458_dvd__triv__left) # label(axiom). [clausify(981)]. 1.59/2.17 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(int,int,times_times(int,A),B))). [resolve(1095,a,1044,a)]. 1.59/2.17 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(real,real,times_times(real,A),B))). [resolve(1095,a,1061,a)]. 1.59/2.17 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,times_times(nat,A),B))). [resolve(1095,a,1085,a)]. 1.59/2.17 1096 -comm_semiring_1(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,times_times(A,C),B))) # label(fact_457_dvd__triv__right) # label(axiom). [clausify(998)]. 1.59/2.17 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(int,int,times_times(int,B),A))). [resolve(1096,a,1044,a)]. 1.59/2.17 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(real,real,times_times(real,B),A))). [resolve(1096,a,1061,a)]. 1.59/2.17 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,times_times(nat,B),A))). [resolve(1096,a,1085,a)]. 1.59/2.17 1097 -comm_semiring_1(A) | hAPP(nat,A,power_power(A,B),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),C)) = hAPP(A,A,times_times(A,hAPP(nat,A,power_power(A,B),C)),hAPP(nat,A,power_power(A,B),C)) # label(fact_15_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom). [clausify(1028)]. 1.59/2.17 Derived: hAPP(nat,int,power_power(int,A),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),B)) = hAPP(int,int,times_times(int,hAPP(nat,int,power_power(int,A),B)),hAPP(nat,int,power_power(int,A),B)). [resolve(1097,a,1044,a)]. 1.59/2.17 Derived: hAPP(nat,real,power_power(real,A),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),B)) = hAPP(real,real,times_times(real,hAPP(nat,real,power_power(real,A),B)),hAPP(nat,real,power_power(real,A),B)). [resolve(1097,a,1061,a)]. 1.59/2.17 Derived: hAPP(nat,nat,power_power(nat,A),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),B)) = hAPP(nat,nat,times_times(nat,hAPP(nat,nat,power_power(nat,A),B)),hAPP(nat,nat,power_power(nat,A),B)). [resolve(1097,a,1085,a)]. 1.59/2.17 1098 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,times_times(A,D),C))) # label(fact_455_dvd__mult) # label(axiom). [clausify(1030)]. 1.59/2.28 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(int,int,times_times(int,C),B))). [resolve(1098,a,1044,a)]. 1.59/2.28 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(real,real,times_times(real,C),B))). [resolve(1098,a,1061,a)]. 1.59/2.28 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,times_times(nat,C),B))). [resolve(1098,a,1085,a)]. 1.59/2.28 1099 ordere223160158up_add(nat) # label(arity_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [assumption]. 1.59/2.28 1100 -ordere223160158up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,B),D)),hAPP(A,A,plus_plus(A,C),E))) # label(fact_782_add__strict__mono) # label(axiom). [clausify(8)]. 1.59/2.28 1101 -ordere223160158up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,B),D)),hAPP(A,A,plus_plus(A,C),E))) # label(fact_808_add__less__le__mono) # label(axiom). [clausify(215)]. 1.59/2.28 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),D)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,A),C)),hAPP(nat,nat,plus_plus(nat,B),D))). [resolve(1101,a,1099,a)]. 1.59/2.28 1102 -ordere223160158up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,D),B)),hAPP(A,A,plus_plus(A,D),C))) # label(fact_781_add__strict__left__mono) # label(axiom). [clausify(639)]. 1.59/2.28 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,C),A)),hAPP(nat,nat,plus_plus(nat,C),B))). [resolve(1102,a,1099,a)]. 1.59/2.28 1103 ordere223160158up_add(real) # label(arity_RealDef_Oreal___Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [assumption]. 1.59/2.28 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),D)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,A),C)),hAPP(real,real,plus_plus(real,B),D))). [resolve(1103,a,1100,a)]. 1.59/2.28 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),D)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,A),C)),hAPP(real,real,plus_plus(real,B),D))). [resolve(1103,a,1101,a)]. 1.59/2.28 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,C),A)),hAPP(real,real,plus_plus(real,C),B))). [resolve(1103,a,1102,a)]. 1.59/2.28 1104 ordere223160158up_add(int) # label(arity_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [assumption]. 1.59/2.28 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),D)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,A),C)),hAPP(int,int,plus_plus(int,B),D))). [resolve(1104,a,1100,a)]. 1.59/2.28 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),C),D)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,A),C)),hAPP(int,int,plus_plus(int,B),D))). [resolve(1104,a,1101,a)]. 1.59/2.28 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,C),A)),hAPP(int,int,plus_plus(int,C),B))). [resolve(1104,a,1102,a)]. 2.08/2.36 1105 -ordere223160158up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,B),D)),hAPP(A,A,plus_plus(A,C),D))) # label(fact_780_add__strict__right__mono) # label(axiom). [clausify(977)]. 2.08/2.36 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,A),C)),hAPP(nat,nat,plus_plus(nat,B),C))). [resolve(1105,a,1099,a)]. 2.08/2.36 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,A),C)),hAPP(real,real,plus_plus(real,B),C))). [resolve(1105,a,1103,a)]. 2.08/2.36 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,A),C)),hAPP(int,int,plus_plus(int,B),C))). [resolve(1105,a,1104,a)]. 2.08/2.36 1106 -ordere223160158up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,B),D)),hAPP(A,A,plus_plus(A,C),E))) # label(fact_809_add__le__less__mono) # label(axiom). [clausify(1035)]. 2.08/2.36 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),C),D)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,A),C)),hAPP(nat,nat,plus_plus(nat,B),D))). [resolve(1106,a,1099,a)]. 2.08/2.36 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),D)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,A),C)),hAPP(real,real,plus_plus(real,B),D))). [resolve(1106,a,1103,a)]. 2.08/2.36 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),D)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,A),C)),hAPP(int,int,plus_plus(int,B),D))). [resolve(1106,a,1104,a)]. 2.08/2.36 1107 linord581940658strict(int) # label(arity_Int_Oint___Rings_Olinordered__ring__strict) # label(axiom). [assumption]. 2.08/2.36 1108 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,C),D)),hAPP(A,A,times_times(A,B),D))) # label(fact_489_mult__strict__right__mono__neg) # label(axiom). [clausify(9)]. 2.08/2.36 1109 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,D),C)),hAPP(A,A,times_times(A,D),B))) # label(fact_488_mult__strict__left__mono__neg) # label(axiom). [clausify(160)]. 2.08/2.36 1110 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,B),C))) # label(fact_495_mult__less__cancel__left__neg) # label(axiom). [clausify(166)]. 2.08/2.36 1111 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,B),C))) # label(fact_495_mult__less__cancel__left__neg) # label(axiom). [clausify(166)]. 2.08/2.36 1112 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,times_times(A,B),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) # label(fact_486_zero__le__mult__iff) # label(axiom). [clausify(193)]. 2.08/2.36 1113 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,times_times(A,B),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) # label(fact_486_zero__le__mult__iff) # label(axiom). [clausify(193)]. 2.08/2.36 1114 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,times_times(A,B),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) # label(fact_486_zero__le__mult__iff) # label(axiom). [clausify(193)]. 2.08/2.36 1115 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,times_times(A,B),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) # label(fact_486_zero__le__mult__iff) # label(axiom). [clausify(193)]. 2.08/2.36 1116 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,times_times(A,B),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) # label(fact_486_zero__le__mult__iff) # label(axiom). [clausify(193)]. 2.08/2.36 1117 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,times_times(A,B),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) # label(fact_486_zero__le__mult__iff) # label(axiom). [clausify(193)]. 2.08/2.36 1118 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,times_times(A,B),C))) # label(fact_493_mult__neg__neg) # label(axiom). [clausify(234)]. 2.08/2.36 1119 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),B)),hAPP(A,A,times_times(A,C),C)))) | zero_zero(A) != ti(A,C) | zero_zero(A) != ti(A,B) # label(fact_239_sum__squares__gt__zero__iff) # label(axiom). [clausify(328)]. 2.08/2.36 1120 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),B)),hAPP(A,A,times_times(A,C),C)))) | zero_zero(A) = ti(A,C) # label(fact_239_sum__squares__gt__zero__iff) # label(axiom). [clausify(328)]. 2.08/2.36 1121 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),B)),hAPP(A,A,times_times(A,C),C)))) | zero_zero(A) = ti(A,B) # label(fact_239_sum__squares__gt__zero__iff) # label(axiom). [clausify(328)]. 2.08/2.36 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,B),C)),hAPP(int,int,times_times(int,A),C))). [resolve(1107,a,1108,a)]. 2.08/2.36 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,C),B)),hAPP(int,int,times_times(int,C),A))). [resolve(1107,a,1109,a)]. 2.08/2.36 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,A),C)),hAPP(int,int,times_times(int,A),B))). [resolve(1107,a,1111,a)]. 2.08/2.37 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,times_times(int,A),B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int))). [resolve(1107,a,1113,a)]. 2.08/2.37 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,times_times(int,A),B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),zero_zero(int))). [resolve(1107,a,1114,a)]. 2.08/2.37 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,times_times(int,A),B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int))). [resolve(1107,a,1117,a)]. 2.08/2.37 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,times_times(int,A),B))). [resolve(1107,a,1118,a)]. 2.08/2.37 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),A)),hAPP(int,int,times_times(int,B),B)))) | zero_zero(int) != ti(int,B) | zero_zero(int) != ti(int,A). [resolve(1107,a,1119,a)]. 2.08/2.37 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),A)),hAPP(int,int,times_times(int,B),B)))) | zero_zero(int) = ti(int,B). [resolve(1107,a,1120,a)]. 2.08/2.37 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),A)),hAPP(int,int,times_times(int,B),B)))) | zero_zero(int) = ti(int,A). [resolve(1107,a,1121,a)]. 2.08/2.37 1122 -linord581940658strict(A) | zero_zero(A) != ti(A,B) | zero_zero(A) != ti(A,C) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),B)),hAPP(A,A,times_times(A,C),C))),zero_zero(A))) # label(fact_236_sum__squares__le__zero__iff) # label(axiom). [clausify(446)]. 2.08/2.37 Derived: zero_zero(int) != ti(int,A) | zero_zero(int) != ti(int,B) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),A)),hAPP(int,int,times_times(int,B),B))),zero_zero(int))). [resolve(1122,a,1107,a)]. 2.08/2.37 1123 -linord581940658strict(A) | zero_zero(A) = ti(A,B) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),B)),hAPP(A,A,times_times(A,C),C))),zero_zero(A))) # label(fact_236_sum__squares__le__zero__iff) # label(axiom). [clausify(446)]. 2.08/2.37 Derived: zero_zero(int) = ti(int,A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),A)),hAPP(int,int,times_times(int,B),B))),zero_zero(int))). [resolve(1123,a,1107,a)]. 2.08/2.37 1124 -linord581940658strict(A) | zero_zero(A) = ti(A,B) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),C)),hAPP(A,A,times_times(A,B),B))),zero_zero(A))) # label(fact_236_sum__squares__le__zero__iff) # label(axiom). [clausify(446)]. 2.08/2.37 Derived: zero_zero(int) = ti(int,A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),B)),hAPP(int,int,times_times(int,A),A))),zero_zero(int))). [resolve(1124,a,1107,a)]. 2.08/2.37 1125 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),C)) # label(fact_503_mult__less__cancel__right__disj) # label(axiom). [clausify(487)]. 2.08/2.37 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,A),B)),hAPP(int,int,times_times(int,C),B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)). [resolve(1125,a,1107,a)]. 2.08/2.37 1126 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),D)) # label(fact_503_mult__less__cancel__right__disj) # label(axiom). [clausify(487)]. 2.08/2.37 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,A),B)),hAPP(int,int,times_times(int,C),B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),C)). [resolve(1126,a,1107,a)]. 2.08/2.37 1127 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),C)) # label(fact_503_mult__less__cancel__right__disj) # label(axiom). [clausify(487)]. 2.08/2.37 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,A),B)),hAPP(int,int,times_times(int,C),B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)). [resolve(1127,a,1107,a)]. 2.08/2.37 1128 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),D)) # label(fact_503_mult__less__cancel__right__disj) # label(axiom). [clausify(487)]. 2.08/2.37 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,A),B)),hAPP(int,int,times_times(int,C),B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),C)). [resolve(1128,a,1107,a)]. 2.08/2.37 1129 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),zero_zero(A))) # label(fact_503_mult__less__cancel__right__disj) # label(axiom). [clausify(487)]. 2.08/2.37 1130 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),D)) # label(fact_503_mult__less__cancel__right__disj) # label(axiom). [clausify(487)]. 2.08/2.37 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,A),B)),hAPP(int,int,times_times(int,C),B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),C)). [resolve(1130,a,1107,a)]. 2.08/2.37 1131 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D))) # label(fact_502_mult__less__cancel__left__disj) # label(axiom). [clausify(532)]. 2.08/2.37 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,A),B)),hAPP(int,int,times_times(int,A),C))). [resolve(1131,a,1107,a)]. 2.08/2.37 1132 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,B),C))) # label(fact_502_mult__less__cancel__left__disj) # label(axiom). [clausify(532)]. 2.08/2.38 1133 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D))) # label(fact_502_mult__less__cancel__left__disj) # label(axiom). [clausify(532)]. 2.08/2.38 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,A),B)),hAPP(int,int,times_times(int,A),C))). [resolve(1133,a,1107,a)]. 2.08/2.38 1134 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,B),C))) # label(fact_502_mult__less__cancel__left__disj) # label(axiom). [clausify(532)]. 2.08/2.38 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,A),C)),hAPP(int,int,times_times(int,A),B))). [resolve(1134,a,1107,a)]. 2.08/2.38 1135 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,D),B)),hAPP(A,A,times_times(A,D),C))) # label(fact_502_mult__less__cancel__left__disj) # label(axiom). [clausify(532)]. 2.08/2.38 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,C),A)),hAPP(int,int,times_times(int,C),B))). [resolve(1135,a,1107,a)]. 2.08/2.38 1136 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,D),B)),hAPP(A,A,times_times(A,D),C))) # label(fact_502_mult__less__cancel__left__disj) # label(axiom). [clausify(532)]. 2.08/2.38 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,C),A)),hAPP(int,int,times_times(int,C),B))). [resolve(1136,a,1107,a)]. 2.08/2.38 1137 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,B),C))) # label(fact_529_mult__le__cancel__left__neg) # label(axiom). [clausify(538)]. 2.08/2.38 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,times_times(int,A),C)),hAPP(int,int,times_times(int,A),B))). [resolve(1137,a,1107,a)]. 2.08/2.38 1138 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,B),C))) # label(fact_529_mult__le__cancel__left__neg) # label(axiom). [clausify(538)]. 2.08/2.38 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),C)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,times_times(int,A),C)),hAPP(int,int,times_times(int,A),B))). [resolve(1138,a,1107,a)]. 2.17/2.38 1139 linord581940658strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__ring__strict) # label(axiom). [assumption]. 2.17/2.38 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,B),C)),hAPP(real,real,times_times(real,A),C))). [resolve(1139,a,1108,a)]. 2.17/2.38 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,C),B)),hAPP(real,real,times_times(real,C),A))). [resolve(1139,a,1109,a)]. 2.17/2.38 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,A),C)),hAPP(real,real,times_times(real,A),B))). [resolve(1139,a,1111,a)]. 2.17/2.38 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,times_times(real,A),B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),zero_zero(real))). [resolve(1139,a,1113,a)]. 2.17/2.38 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,times_times(real,A),B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))). [resolve(1139,a,1114,a)]. 2.17/2.38 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,times_times(real,A),B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)). [resolve(1139,a,1116,a)]. 2.17/2.38 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,times_times(real,A),B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),zero_zero(real))). [resolve(1139,a,1117,a)]. 2.17/2.38 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,times_times(real,A),B))). [resolve(1139,a,1118,a)]. 2.17/2.38 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),A)),hAPP(real,real,times_times(real,B),B)))) | zero_zero(real) != ti(real,B) | zero_zero(real) != ti(real,A). [resolve(1139,a,1119,a)]. 2.17/2.38 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),A)),hAPP(real,real,times_times(real,B),B)))) | zero_zero(real) = ti(real,B). [resolve(1139,a,1120,a)]. 2.17/2.38 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),A)),hAPP(real,real,times_times(real,B),B)))) | zero_zero(real) = ti(real,A). [resolve(1139,a,1121,a)]. 2.17/2.38 Derived: zero_zero(real) != ti(real,A) | zero_zero(real) != ti(real,B) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),A)),hAPP(real,real,times_times(real,B),B))),zero_zero(real))). [resolve(1139,a,1122,a)]. 2.17/2.38 Derived: zero_zero(real) = ti(real,A) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),A)),hAPP(real,real,times_times(real,B),B))),zero_zero(real))). [resolve(1139,a,1123,a)]. 2.17/2.39 Derived: zero_zero(real) = ti(real,A) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,B),B)),hAPP(real,real,times_times(real,A),A))),zero_zero(real))). [resolve(1139,a,1124,a)]. 2.17/2.39 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,C),B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B)). [resolve(1139,a,1125,a)]. 2.17/2.39 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,C),B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),C)). [resolve(1139,a,1126,a)]. 2.17/2.39 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,C),B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B)). [resolve(1139,a,1127,a)]. 2.17/2.39 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,C),B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),C)). [resolve(1139,a,1128,a)]. 2.17/2.39 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,C),B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),C)). [resolve(1139,a,1130,a)]. 2.17/2.39 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,A),C))). [resolve(1139,a,1133,a)]. 2.17/2.39 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,A),C)),hAPP(real,real,times_times(real,A),B))). [resolve(1139,a,1134,a)]. 2.17/2.39 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,C),A)),hAPP(real,real,times_times(real,C),B))). [resolve(1139,a,1135,a)]. 2.17/2.39 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,C),A)),hAPP(real,real,times_times(real,C),B))). [resolve(1139,a,1136,a)]. 2.17/2.39 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,A),C)),hAPP(real,real,times_times(real,A),B))). [resolve(1139,a,1137,a)]. 2.17/2.39 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,A),C)),hAPP(real,real,times_times(real,A),B))). [resolve(1139,a,1138,a)]. 2.17/2.39 1140 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D))) # label(fact_501_mult__less__cancel__left__pos) # label(axiom). [clausify(594)]. 2.21/2.42 1141 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D))) # label(fact_501_mult__less__cancel__left__pos) # label(axiom). [clausify(594)]. 2.21/2.42 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,A),B)),hAPP(int,int,times_times(int,A),C))). [resolve(1141,a,1107,a)]. 2.21/2.42 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,A),C))). [resolve(1141,a,1139,a)]. 2.21/2.42 1142 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),D)) # label(fact_528_mult__le__cancel__left__pos) # label(axiom). [clausify(757)]. 2.21/2.42 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,times_times(int,A),B)),hAPP(int,int,times_times(int,A),C))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),C)). [resolve(1142,a,1107,a)]. 2.21/2.42 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,A),C))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),C)). [resolve(1142,a,1139,a)]. 2.21/2.42 1143 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),D)) # label(fact_528_mult__le__cancel__left__pos) # label(axiom). [clausify(757)]. 2.21/2.42 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,times_times(int,A),B)),hAPP(int,int,times_times(int,A),C))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),C)). [resolve(1143,a,1107,a)]. 2.21/2.42 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,A),C))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),C)). [resolve(1143,a,1139,a)]. 2.21/2.42 1144 -linord581940658strict(A) | zero_zero(A) != hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),B)),hAPP(A,A,times_times(A,C),C)) | zero_zero(A) = ti(A,B) # label(fact_218_sum__squares__eq__zero__iff) # label(axiom). [clausify(772)]. 2.21/2.42 Derived: zero_zero(int) != hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),A)),hAPP(int,int,times_times(int,B),B)) | zero_zero(int) = ti(int,A). [resolve(1144,a,1107,a)]. 2.21/2.42 Derived: zero_zero(real) != hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),A)),hAPP(real,real,times_times(real,B),B)) | zero_zero(real) = ti(real,A). [resolve(1144,a,1139,a)]. 2.21/2.42 1145 -linord581940658strict(A) | zero_zero(A) != hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),B)),hAPP(A,A,times_times(A,C),C)) | zero_zero(A) = ti(A,C) # label(fact_218_sum__squares__eq__zero__iff) # label(axiom). [clausify(772)]. 2.21/2.45 Derived: zero_zero(int) != hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),A)),hAPP(int,int,times_times(int,B),B)) | zero_zero(int) = ti(int,B). [resolve(1145,a,1107,a)]. 2.21/2.45 Derived: zero_zero(real) != hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),A)),hAPP(real,real,times_times(real,B),B)) | zero_zero(real) = ti(real,B). [resolve(1145,a,1139,a)]. 2.21/2.45 1146 -linord581940658strict(A) | zero_zero(A) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),B)),hAPP(A,A,times_times(A,C),C)) | zero_zero(A) != ti(A,B) | zero_zero(A) != ti(A,C) # label(fact_218_sum__squares__eq__zero__iff) # label(axiom). [clausify(772)]. 2.21/2.45 Derived: zero_zero(int) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),A)),hAPP(int,int,times_times(int,B),B)) | zero_zero(int) != ti(int,A) | zero_zero(int) != ti(int,B). [resolve(1146,a,1107,a)]. 2.21/2.45 Derived: zero_zero(real) = hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),A)),hAPP(real,real,times_times(real,B),B)) | zero_zero(real) != ti(real,A) | zero_zero(real) != ti(real,B). [resolve(1146,a,1139,a)]. 2.21/2.45 1147 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),C)),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) # label(fact_485_mult__le__0__iff) # label(axiom). [clausify(1011)]. 2.21/2.45 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,times_times(int,A),B)),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)). [resolve(1147,a,1107,a)]. 2.21/2.45 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,A),B)),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)). [resolve(1147,a,1139,a)]. 2.21/2.45 1148 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),C)),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) # label(fact_485_mult__le__0__iff) # label(axiom). [clausify(1011)]. 2.21/2.45 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,times_times(int,A),B)),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int))). [resolve(1148,a,1107,a)]. 2.21/2.45 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,A),B)),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),zero_zero(real))). [resolve(1148,a,1139,a)]. 2.21/2.45 1149 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),C)),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) # label(fact_485_mult__le__0__iff) # label(axiom). [clausify(1011)]. 2.21/2.45 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,times_times(int,A),B)),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)). [resolve(1149,a,1107,a)]. 2.21/2.45 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,A),B)),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)). [resolve(1149,a,1139,a)]. 2.21/2.50 1150 -linord581940658strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),C)),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) # label(fact_485_mult__le__0__iff) # label(axiom). [clausify(1011)]. 2.21/2.50 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,times_times(int,A),B)),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int))). [resolve(1150,a,1107,a)]. 2.21/2.50 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,A),B)),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),zero_zero(real))). [resolve(1150,a,1139,a)]. 2.21/2.50 1151 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),C)),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) # label(fact_485_mult__le__0__iff) # label(axiom). [clausify(1011)]. 2.21/2.50 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,times_times(int,A),B)),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),zero_zero(int))). [resolve(1151,a,1107,a)]. 2.21/2.50 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,A),B)),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))). [resolve(1151,a,1139,a)]. 2.21/2.50 1152 -linord581940658strict(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),C)),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) # label(fact_485_mult__le__0__iff) # label(axiom). [clausify(1011)]. 2.21/2.50 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,times_times(int,A),B)),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int))). [resolve(1152,a,1107,a)]. 2.21/2.50 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,A),B)),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),zero_zero(real))). [resolve(1152,a,1139,a)]. 2.21/2.50 1153 -linord219039673up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,plus_plus(A,B),B))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) # label(fact_797_zero__le__double__add__iff__zero__le__single__add) # label(axiom). [clausify(20)]. 2.21/2.50 1154 linord219039673up_add(real) # label(arity_RealDef_Oreal___Groups_Olinordered__ab__group__add) # label(axiom). [assumption]. 2.21/2.50 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,plus_plus(real,A),A))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)). [resolve(1153,a,1154,a)]. 2.21/2.50 1155 -linord219039673up_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,plus_plus(A,B),B))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) # label(fact_797_zero__le__double__add__iff__zero__le__single__add) # label(axiom). [clausify(20)]. 2.21/2.50 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,plus_plus(real,A),A))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)). [resolve(1155,a,1154,a)]. 2.30/2.51 1156 -linord219039673up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,plus_plus(A,B),B))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) # label(fact_804_zero__less__double__add__iff__zero__less__single__add) # label(axiom). [clausify(130)]. 2.30/2.51 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,plus_plus(real,A),A))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)). [resolve(1156,a,1154,a)]. 2.30/2.51 1157 -linord219039673up_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,plus_plus(A,B),B))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) # label(fact_804_zero__less__double__add__iff__zero__less__single__add) # label(axiom). [clausify(130)]. 2.30/2.51 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,plus_plus(real,A),A))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)). [resolve(1157,a,1154,a)]. 2.30/2.51 1158 -linord219039673up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,B),B)),zero_zero(A))) # label(fact_798_double__add__le__zero__iff__single__add__le__zero) # label(axiom). [clausify(271)]. 2.30/2.51 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,A),A)),zero_zero(real))). [resolve(1158,a,1154,a)]. 2.30/2.51 1159 -linord219039673up_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,B),B)),zero_zero(A))) # label(fact_798_double__add__le__zero__iff__single__add__le__zero) # label(axiom). [clausify(271)]. 2.30/2.51 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,A),A)),zero_zero(real))). [resolve(1159,a,1154,a)]. 2.30/2.51 1160 linord219039673up_add(int) # label(arity_Int_Oint___Groups_Olinordered__ab__group__add) # label(axiom). [assumption]. 2.30/2.51 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,plus_plus(int,A),A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)). [resolve(1160,a,1153,a)]. 2.30/2.51 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,plus_plus(int,A),A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)). [resolve(1160,a,1156,a)]. 2.30/2.51 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,plus_plus(int,A),A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)). [resolve(1160,a,1157,a)]. 2.30/2.51 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,A),A)),zero_zero(int))). [resolve(1160,a,1158,a)]. 2.30/2.51 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,A),A)),zero_zero(int))). [resolve(1160,a,1159,a)]. 2.30/2.51 1161 -linord219039673up_add(A) | zero_zero(A) != hAPP(A,A,plus_plus(A,B),B) | zero_zero(A) = ti(A,B) # label(fact_768_double__zero__sym) # label(axiom). [clausify(380)]. 2.30/2.51 Derived: zero_zero(real) != hAPP(real,real,plus_plus(real,A),A) | zero_zero(real) = ti(real,A). [resolve(1161,a,1154,a)]. 2.30/2.51 Derived: zero_zero(int) != hAPP(int,int,plus_plus(int,A),A) | zero_zero(int) = ti(int,A). [resolve(1161,a,1160,a)]. 2.30/2.51 1162 -linord219039673up_add(A) | zero_zero(A) = hAPP(A,A,plus_plus(A,B),B) | zero_zero(A) != ti(A,B) # label(fact_768_double__zero__sym) # label(axiom). [clausify(380)]. 2.39/2.64 Derived: zero_zero(real) = hAPP(real,real,plus_plus(real,A),A) | zero_zero(real) != ti(real,A). [resolve(1162,a,1154,a)]. 2.39/2.64 Derived: zero_zero(int) = hAPP(int,int,plus_plus(int,A),A) | zero_zero(int) != ti(int,A). [resolve(1162,a,1160,a)]. 2.39/2.64 1163 -linord219039673up_add(A) | zero_zero(A) != ti(A,B) | zero_zero(A) = hAPP(A,A,plus_plus(A,B),B) # label(fact_203_double__eq__0__iff) # label(axiom). [clausify(488)]. 2.39/2.64 1164 -linord219039673up_add(A) | zero_zero(A) = ti(A,B) | zero_zero(A) != hAPP(A,A,plus_plus(A,B),B) # label(fact_203_double__eq__0__iff) # label(axiom). [clausify(488)]. 2.39/2.64 1165 -linord219039673up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,B),B)),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) # label(fact_805_double__add__less__zero__iff__single__add__less__zero) # label(axiom). [clausify(592)]. 2.39/2.64 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,A),A)),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))). [resolve(1165,a,1154,a)]. 2.39/2.64 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,A),A)),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),zero_zero(int))). [resolve(1165,a,1160,a)]. 2.39/2.64 1166 -linord219039673up_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,B),B)),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) # label(fact_805_double__add__less__zero__iff__single__add__less__zero) # label(axiom). [clausify(592)]. 2.39/2.64 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,A),A)),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))). [resolve(1166,a,1154,a)]. 2.39/2.64 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,A),A)),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),zero_zero(int))). [resolve(1166,a,1160,a)]. 2.39/2.64 1167 number_ring(real) # label(arity_RealDef_Oreal___Int_Onumber__ring) # label(axiom). [assumption]. 2.39/2.64 1168 -number_ring(A) | hAPP(nat,A,power_power(A,hAPP(A,A,minus_minus(A,B),C)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(A,A,minus_minus(A,hAPP(A,A,plus_plus(A,hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,A,power_power(A,C),number_number_of(nat,bit0(bit1(pls)))))),hAPP(A,A,times_times(A,hAPP(A,A,times_times(A,number_number_of(A,bit0(bit1(pls)))),B)),C)) # label(fact_401_power2__diff) # label(axiom). [clausify(15)]. 2.39/2.64 1169 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),number_number_of(A,B))) # label(fact_248_le__special_I1_J) # label(axiom). [clausify(65)]. 2.39/2.64 1170 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),number_number_of(A,B))) # label(fact_248_le__special_I1_J) # label(axiom). [clausify(65)]. 2.39/2.64 1171 -number_ring(A) | number_number_of(A,bit0(bit1(pls))) = hAPP(A,A,plus_plus(A,one_one(A)),one_one(A)) # label(fact_18_one__add__one__is__two) # label(axiom). [clausify(95)]. 2.39/2.64 1172 -number_ring(A) | number_number_of(A,bit0(B)) = hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,one_one(A)),one_one(A))),number_number_of(A,B)) # label(fact_162_double__number__of__Bit0) # label(axiom). [clausify(151)]. 2.39/2.64 1173 -number_ring(A) | ti(A,B) = hAPP(A,A,times_times(A,number_number_of(A,bit1(pls))),B) # label(fact_155_mult__numeral__1) # label(axiom). [clausify(163)]. 2.39/2.64 1174 -number_ring(A) | one_one(A) = number_number_of(A,bit1(pls)) # label(fact_159_semiring__norm_I110_J) # label(axiom). [clausify(241)]. 2.39/2.64 1175 -number_ring(A) | zero_zero(A) = number_number_of(A,pls) # label(fact_221_semiring__norm_I112_J) # label(axiom). [clausify(266)]. 2.39/2.64 1176 -number_ring(A) | -linordered_idom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),number_number_of(A,B)),number_number_of(A,C))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C)) # label(fact_30_less__number__of) # label(axiom). [clausify(274)]. 2.39/2.64 1177 -number_ring(A) | -linordered_idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),number_number_of(A,B)),number_number_of(A,C))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C)) # label(fact_30_less__number__of) # label(axiom). [clausify(274)]. 2.39/2.64 1178 -number_ring(A) | hAPP(A,A,minus_minus(A,number_number_of(A,hAPP(int,int,plus_plus(int,B),C))),D) = hAPP(A,A,plus_plus(A,number_number_of(A,B)),hAPP(A,A,minus_minus(A,number_number_of(A,C)),D)) # label(fact_384_add__number__of__diff1) # label(axiom). [clausify(275)]. 2.39/2.64 1179 -number_ring(A) | hAPP(A,A,times_times(A,number_number_of(A,hAPP(int,int,times_times(int,B),C))),D) = hAPP(A,A,times_times(A,number_number_of(A,B)),hAPP(A,A,times_times(A,number_number_of(A,C)),D)) # label(fact_145_mult__number__of__left) # label(axiom). [clausify(284)]. 2.39/2.64 1180 -number_ring(A) | number_number_of(A,hAPP(int,int,minus_minus(int,B),C)) = hAPP(A,A,minus_minus(A,number_number_of(A,B)),number_number_of(A,C)) # label(fact_369_number__of__diff) # label(axiom). [clausify(310)]. 2.39/2.64 1181 -number_ring(A) | one_one(A) = hAPP(nat,A,power_power(A,number_number_of(A,min)),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),B)) # label(fact_342_power__m1__even) # label(axiom). [clausify(327)]. 2.39/2.64 1182 -number_ring(A) | ti(A,B) = hAPP(A,A,plus_plus(A,B),number_number_of(A,pls)) # label(fact_144_add__numeral__0__right) # label(axiom). [clausify(415)]. 2.39/2.64 1183 -number_ring(A) | -linordered_idom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),number_number_of(A,B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(pls)),B)) # label(fact_105_less__special_I2_J) # label(axiom). [clausify(419)]. 2.39/2.64 1184 -number_ring(A) | -linordered_idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),number_number_of(A,B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(pls)),B)) # label(fact_105_less__special_I2_J) # label(axiom). [clausify(419)]. 2.39/2.64 1185 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,B)),number_number_of(A,C))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),C)) # label(fact_31_le__number__of) # label(axiom). [clausify(439)]. 2.39/2.64 1186 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,B)),number_number_of(A,C))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),C)) # label(fact_31_le__number__of) # label(axiom). [clausify(439)]. 2.39/2.64 1187 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),number_number_of(A,B))) # label(fact_246_less__special_I1_J) # label(axiom). [clausify(457)]. 2.39/2.64 1188 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),number_number_of(A,B))) # label(fact_246_less__special_I1_J) # label(axiom). [clausify(457)]. 2.39/2.64 1189 -number_ring(A) | ti(A,B) = hAPP(A,A,times_times(A,B),number_number_of(A,bit1(pls))) # label(fact_156_mult__numeral__1__right) # label(axiom). [clausify(458)]. 2.39/2.64 1190 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),number_number_of(A,B)),one_one(A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),bit1(pls))) # label(fact_104_less__special_I4_J) # label(axiom). [clausify(489)]. 2.39/2.64 1191 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),number_number_of(A,B)),one_one(A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),bit1(pls))) # label(fact_104_less__special_I4_J) # label(axiom). [clausify(489)]. 2.39/2.64 1192 -number_ring(A) | number_number_of(A,hAPP(int,int,plus_plus(int,B),bit1(pls))) = hAPP(A,A,plus_plus(A,number_number_of(A,B)),one_one(A)) # label(fact_17_add__special_I3_J) # label(axiom). [clausify(495)]. 2.39/2.64 1193 -number_ring(A) | -ring_char_0(A) | ti(int,B) != ti(int,C) | number_number_of(A,B) = number_number_of(A,C) # label(fact_82_eq__number__of) # label(axiom). [clausify(496)]. 2.39/2.64 1194 -number_ring(A) | -ring_char_0(A) | ti(int,B) = ti(int,C) | number_number_of(A,B) != number_number_of(A,C) # label(fact_82_eq__number__of) # label(axiom). [clausify(496)]. 2.39/2.64 1195 -number_ring(A) | number_number_of(A,bit0(B)) = hAPP(A,A,plus_plus(A,hAPP(A,A,plus_plus(A,zero_zero(A)),number_number_of(A,B))),number_number_of(A,B)) # label(fact_241_number__of__Bit0) # label(axiom). [clausify(523)]. 2.39/2.64 1196 -number_ring(A) | ti(A,B) = hAPP(A,A,plus_plus(A,number_number_of(A,pls)),B) # label(fact_143_add__numeral__0) # label(axiom). [clausify(539)]. 2.39/2.64 1197 -number_ring(A) | number_number_of(A,hAPP(int,int,times_times(int,B),C)) = hAPP(A,A,times_times(A,number_number_of(A,B)),number_number_of(A,C)) # label(fact_146_arith__simps_I32_J) # label(axiom). [clausify(565)]. 2.39/2.64 Derived: hAPP(nat,real,power_power(real,hAPP(real,real,minus_minus(real,A),B)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(real,real,minus_minus(real,hAPP(real,real,plus_plus(real,hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,real,power_power(real,B),number_number_of(nat,bit0(bit1(pls)))))),hAPP(real,real,times_times(real,hAPP(real,real,times_times(real,number_number_of(real,bit0(bit1(pls)))),A)),B)). [resolve(1167,a,1168,a)]. 2.39/2.64 Derived: -linordered_idom(real) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),number_number_of(real,A))). [resolve(1167,a,1169,b)]. 2.39/2.64 Derived: -linordered_idom(real) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),number_number_of(real,A))). [resolve(1167,a,1170,b)]. 2.39/2.64 Derived: number_number_of(real,bit0(bit1(pls))) = hAPP(real,real,plus_plus(real,one_one(real)),one_one(real)). [resolve(1167,a,1171,a)]. 2.39/2.64 Derived: number_number_of(real,bit0(A)) = hAPP(real,real,times_times(real,hAPP(real,real,plus_plus(real,one_one(real)),one_one(real))),number_number_of(real,A)). [resolve(1167,a,1172,a)]. 2.39/2.64 Derived: ti(real,A) = hAPP(real,real,times_times(real,number_number_of(real,bit1(pls))),A). [resolve(1167,a,1173,a)]. 2.39/2.64 Derived: one_one(real) = number_number_of(real,bit1(pls)). [resolve(1167,a,1174,a)]. 2.39/2.64 Derived: zero_zero(real) = number_number_of(real,pls). [resolve(1167,a,1175,a)]. 2.39/2.64 Derived: -linordered_idom(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),number_number_of(real,A)),number_number_of(real,B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)). [resolve(1167,a,1176,a)]. 2.39/2.64 Derived: -linordered_idom(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),number_number_of(real,A)),number_number_of(real,B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)). [resolve(1167,a,1177,a)]. 2.39/2.64 Derived: hAPP(real,real,minus_minus(real,number_number_of(real,hAPP(int,int,plus_plus(int,A),B))),C) = hAPP(real,real,plus_plus(real,number_number_of(real,A)),hAPP(real,real,minus_minus(real,number_number_of(real,B)),C)). [resolve(1167,a,1178,a)]. 2.39/2.64 Derived: hAPP(real,real,times_times(real,number_number_of(real,hAPP(int,int,times_times(int,A),B))),C) = hAPP(real,real,times_times(real,number_number_of(real,A)),hAPP(real,real,times_times(real,number_number_of(real,B)),C)). [resolve(1167,a,1179,a)]. 2.39/2.64 Derived: number_number_of(real,hAPP(int,int,minus_minus(int,A),B)) = hAPP(real,real,minus_minus(real,number_number_of(real,A)),number_number_of(real,B)). [resolve(1167,a,1180,a)]. 2.39/2.64 Derived: one_one(real) = hAPP(nat,real,power_power(real,number_number_of(real,min)),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),A)). [resolve(1167,a,1181,a)]. 2.39/2.65 Derived: ti(real,A) = hAPP(real,real,plus_plus(real,A),number_number_of(real,pls)). [resolve(1167,a,1182,a)]. 2.39/2.65 Derived: -linordered_idom(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),number_number_of(real,A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(pls)),A)). [resolve(1167,a,1183,a)]. 2.39/2.65 Derived: -linordered_idom(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),number_number_of(real,A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(pls)),A)). [resolve(1167,a,1184,a)]. 2.39/2.65 Derived: -linordered_idom(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),number_number_of(real,B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(1167,a,1185,b)]. 2.39/2.65 Derived: -linordered_idom(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),number_number_of(real,B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(1167,a,1186,b)]. 2.39/2.65 Derived: -linordered_idom(real) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),number_number_of(real,A))). [resolve(1167,a,1187,b)]. 2.39/2.65 Derived: -linordered_idom(real) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),number_number_of(real,A))). [resolve(1167,a,1188,b)]. 2.39/2.65 Derived: ti(real,A) = hAPP(real,real,times_times(real,A),number_number_of(real,bit1(pls))). [resolve(1167,a,1189,a)]. 2.39/2.65 Derived: -linordered_idom(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),number_number_of(real,A)),one_one(real))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),bit1(pls))). [resolve(1167,a,1190,b)]. 2.39/2.65 Derived: -linordered_idom(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),number_number_of(real,A)),one_one(real))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),bit1(pls))). [resolve(1167,a,1191,b)]. 2.39/2.65 Derived: number_number_of(real,hAPP(int,int,plus_plus(int,A),bit1(pls))) = hAPP(real,real,plus_plus(real,number_number_of(real,A)),one_one(real)). [resolve(1167,a,1192,a)]. 2.39/2.65 Derived: -ring_char_0(real) | ti(int,A) != ti(int,B) | number_number_of(real,A) = number_number_of(real,B). [resolve(1167,a,1193,a)]. 2.39/2.65 Derived: -ring_char_0(real) | ti(int,A) = ti(int,B) | number_number_of(real,A) != number_number_of(real,B). [resolve(1167,a,1194,a)]. 2.39/2.65 Derived: number_number_of(real,bit0(A)) = hAPP(real,real,plus_plus(real,hAPP(real,real,plus_plus(real,zero_zero(real)),number_number_of(real,A))),number_number_of(real,A)). [resolve(1167,a,1195,a)]. 2.39/2.65 Derived: ti(real,A) = hAPP(real,real,plus_plus(real,number_number_of(real,pls)),A). [resolve(1167,a,1196,a)]. 2.39/2.65 Derived: number_number_of(real,hAPP(int,int,times_times(int,A),B)) = hAPP(real,real,times_times(real,number_number_of(real,A)),number_number_of(real,B)). [resolve(1167,a,1197,a)]. 2.39/2.65 1198 -number_ring(A) | number_number_of(A,hAPP(int,int,times_times(int,B),C)) = hAPP(A,A,times_times(A,number_number_of(A,B)),number_number_of(A,C)) # label(fact_147_number__of__mult) # label(axiom). [clausify(609)]. 2.39/2.65 1199 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),one_one(A)),number_number_of(A,B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),B)) # label(fact_107_le__special_I2_J) # label(axiom). [clausify(678)]. 2.39/2.65 Derived: -linordered_idom(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),number_number_of(real,A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),A)). [resolve(1199,b,1167,a)]. 2.39/2.65 1200 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),one_one(A)),number_number_of(A,B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),B)) # label(fact_107_le__special_I2_J) # label(axiom). [clausify(678)]. 2.48/2.68 Derived: -linordered_idom(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),number_number_of(real,A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),A)). [resolve(1200,b,1167,a)]. 2.48/2.68 1201 -number_ring(A) | hAPP(A,A,plus_plus(A,B),B) = hAPP(A,A,times_times(A,B),number_number_of(A,bit0(bit1(pls)))) # label(fact_168_mult__2__right) # label(axiom). [clausify(689)]. 2.48/2.68 Derived: hAPP(real,real,plus_plus(real,A),A) = hAPP(real,real,times_times(real,A),number_number_of(real,bit0(bit1(pls)))). [resolve(1201,a,1167,a)]. 2.48/2.68 1202 -number_ring(A) | hAPP(A,A,plus_plus(A,B),B) = hAPP(A,A,times_times(A,number_number_of(A,bit0(bit1(pls)))),B) # label(fact_166_mult__2) # label(axiom). [clausify(758)]. 2.48/2.68 Derived: hAPP(real,real,plus_plus(real,A),A) = hAPP(real,real,times_times(real,number_number_of(real,bit0(bit1(pls)))),A). [resolve(1202,a,1167,a)]. 2.48/2.68 1203 -number_ring(A) | zero_zero(A) = number_number_of(A,pls) # label(fact_222_number__of__Pls) # label(axiom). [clausify(781)]. 2.48/2.68 1204 -number_ring(A) | hAPP(A,A,plus_plus(A,number_number_of(A,hAPP(int,int,plus_plus(int,B),C))),D) = hAPP(A,A,plus_plus(A,number_number_of(A,B)),hAPP(A,A,plus_plus(A,number_number_of(A,C)),D)) # label(fact_148_add__number__of__left) # label(axiom). [clausify(801)]. 2.48/2.68 Derived: hAPP(real,real,plus_plus(real,number_number_of(real,hAPP(int,int,plus_plus(int,A),B))),C) = hAPP(real,real,plus_plus(real,number_number_of(real,A)),hAPP(real,real,plus_plus(real,number_number_of(real,B)),C)). [resolve(1204,a,1167,a)]. 2.48/2.68 1205 -number_ring(A) | number_number_of(A,hAPP(int,int,plus_plus(int,B),C)) = hAPP(A,A,plus_plus(A,number_number_of(A,B)),number_number_of(A,C)) # label(fact_149_add__number__of__eq) # label(axiom). [clausify(802)]. 2.48/2.68 Derived: number_number_of(real,hAPP(int,int,plus_plus(int,A),B)) = hAPP(real,real,plus_plus(real,number_number_of(real,A)),number_number_of(real,B)). [resolve(1205,a,1167,a)]. 2.48/2.68 1206 -number_ring(A) | -linordered_idom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),number_number_of(A,B)),zero_zero(A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),pls)) # label(fact_247_less__special_I3_J) # label(axiom). [clausify(831)]. 2.48/2.68 Derived: -linordered_idom(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),number_number_of(real,A)),zero_zero(real))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)). [resolve(1206,a,1167,a)]. 2.48/2.68 1207 -number_ring(A) | -linordered_idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),number_number_of(A,B)),zero_zero(A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),pls)) # label(fact_247_less__special_I3_J) # label(axiom). [clausify(831)]. 2.48/2.68 Derived: -linordered_idom(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),number_number_of(real,A)),zero_zero(real))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)). [resolve(1207,a,1167,a)]. 2.48/2.68 1208 -number_ring(A) | -linordered_idom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,B)),zero_zero(A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),pls)) # label(fact_249_le__special_I3_J) # label(axiom). [clausify(832)]. 2.48/2.68 Derived: -linordered_idom(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),zero_zero(real))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),pls)). [resolve(1208,a,1167,a)]. 2.48/2.68 1209 -number_ring(A) | -linordered_idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,B)),zero_zero(A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),pls)) # label(fact_249_le__special_I3_J) # label(axiom). [clausify(832)]. 2.48/2.68 Derived: -linordered_idom(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),zero_zero(real))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),pls)). [resolve(1209,a,1167,a)]. 2.48/2.68 1210 -number_ring(A) | one_one(A) = number_number_of(A,bit1(pls)) # label(fact_158_numeral__1__eq__1) # label(axiom). [clausify(841)]. 2.48/2.70 1211 -number_ring(A) | number_number_of(A,hAPP(int,int,plus_plus(int,B),C)) = hAPP(A,A,plus_plus(A,number_number_of(A,B)),number_number_of(A,C)) # label(fact_150_number__of__add) # label(axiom). [clausify(857)]. 2.48/2.70 1212 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,B)),one_one(A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),bit1(pls))) # label(fact_106_le__special_I4_J) # label(axiom). [clausify(929)]. 2.48/2.70 Derived: -linordered_idom(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),one_one(real))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),bit1(pls))). [resolve(1212,b,1167,a)]. 2.48/2.70 1213 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,B)),one_one(A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),bit1(pls))) # label(fact_106_le__special_I4_J) # label(axiom). [clausify(929)]. 2.48/2.70 Derived: -linordered_idom(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),one_one(real))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),bit1(pls))). [resolve(1213,b,1167,a)]. 2.48/2.70 1214 -number_ring(A) | number_number_of(A,bit1(B)) = hAPP(A,A,plus_plus(A,hAPP(A,A,plus_plus(A,one_one(A)),number_number_of(A,B))),number_number_of(A,B)) # label(fact_154_number__of__Bit1) # label(axiom). [clausify(952)]. 2.48/2.70 Derived: number_number_of(real,bit1(A)) = hAPP(real,real,plus_plus(real,hAPP(real,real,plus_plus(real,one_one(real)),number_number_of(real,A))),number_number_of(real,A)). [resolve(1214,a,1167,a)]. 2.48/2.70 1215 number_ring(int) # label(arity_Int_Oint___Int_Onumber__ring) # label(axiom). [assumption]. 2.48/2.70 Derived: hAPP(nat,int,power_power(int,hAPP(int,int,minus_minus(int,A),B)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(int,int,minus_minus(int,hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,B),number_number_of(nat,bit0(bit1(pls)))))),hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit1(pls)))),A)),B)). [resolve(1215,a,1168,a)]. 2.48/2.70 Derived: -linordered_idom(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,A))). [resolve(1215,a,1169,b)]. 2.48/2.70 Derived: -linordered_idom(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,A))). [resolve(1215,a,1170,b)]. 2.48/2.70 Derived: number_number_of(int,bit0(bit1(pls))) = hAPP(int,int,plus_plus(int,one_one(int)),one_one(int)). [resolve(1215,a,1171,a)]. 2.48/2.70 Derived: number_number_of(int,bit0(A)) = hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,one_one(int)),one_one(int))),number_number_of(int,A)). [resolve(1215,a,1172,a)]. 2.48/2.70 Derived: ti(int,A) = hAPP(int,int,times_times(int,number_number_of(int,bit1(pls))),A). [resolve(1215,a,1173,a)]. 2.48/2.70 Derived: one_one(int) = number_number_of(int,bit1(pls)). [resolve(1215,a,1174,a)]. 2.48/2.70 Derived: zero_zero(int) = number_number_of(int,pls). [resolve(1215,a,1175,a)]. 2.48/2.70 Derived: -linordered_idom(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,A)),number_number_of(int,B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)). [resolve(1215,a,1176,a)]. 2.48/2.70 Derived: -linordered_idom(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,A)),number_number_of(int,B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)). [resolve(1215,a,1177,a)]. 2.48/2.70 Derived: hAPP(int,int,minus_minus(int,number_number_of(int,hAPP(int,int,plus_plus(int,A),B))),C) = hAPP(int,int,plus_plus(int,number_number_of(int,A)),hAPP(int,int,minus_minus(int,number_number_of(int,B)),C)). [resolve(1215,a,1178,a)]. 2.48/2.70 Derived: hAPP(int,int,times_times(int,number_number_of(int,hAPP(int,int,times_times(int,A),B))),C) = hAPP(int,int,times_times(int,number_number_of(int,A)),hAPP(int,int,times_times(int,number_number_of(int,B)),C)). [resolve(1215,a,1179,a)]. 2.48/2.70 Derived: number_number_of(int,hAPP(int,int,minus_minus(int,A),B)) = hAPP(int,int,minus_minus(int,number_number_of(int,A)),number_number_of(int,B)). [resolve(1215,a,1180,a)]. 2.48/2.70 Derived: one_one(int) = hAPP(nat,int,power_power(int,number_number_of(int,min)),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),A)). [resolve(1215,a,1181,a)]. 2.48/2.70 Derived: ti(int,A) = hAPP(int,int,plus_plus(int,A),number_number_of(int,pls)). [resolve(1215,a,1182,a)]. 2.48/2.70 Derived: -linordered_idom(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),number_number_of(int,A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(pls)),A)). [resolve(1215,a,1183,a)]. 2.48/2.70 Derived: -linordered_idom(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),number_number_of(int,A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(pls)),A)). [resolve(1215,a,1184,a)]. 2.48/2.70 Derived: -linordered_idom(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,A)),number_number_of(int,B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(1215,a,1185,b)]. 2.48/2.70 Derived: -linordered_idom(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,A)),number_number_of(int,B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(1215,a,1186,b)]. 2.48/2.70 Derived: -linordered_idom(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),number_number_of(int,A))). [resolve(1215,a,1187,b)]. 2.48/2.70 Derived: -linordered_idom(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),number_number_of(int,A))). [resolve(1215,a,1188,b)]. 2.48/2.70 Derived: ti(int,A) = hAPP(int,int,times_times(int,A),number_number_of(int,bit1(pls))). [resolve(1215,a,1189,a)]. 2.48/2.70 Derived: -linordered_idom(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,A)),one_one(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),bit1(pls))). [resolve(1215,a,1190,b)]. 2.48/2.70 Derived: -linordered_idom(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,A)),one_one(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),bit1(pls))). [resolve(1215,a,1191,b)]. 2.48/2.70 Derived: number_number_of(int,hAPP(int,int,plus_plus(int,A),bit1(pls))) = hAPP(int,int,plus_plus(int,number_number_of(int,A)),one_one(int)). [resolve(1215,a,1192,a)]. 2.48/2.70 Derived: -ring_char_0(int) | ti(int,A) != ti(int,B) | number_number_of(int,A) = number_number_of(int,B). [resolve(1215,a,1193,a)]. 2.48/2.70 Derived: -ring_char_0(int) | ti(int,A) = ti(int,B) | number_number_of(int,A) != number_number_of(int,B). [resolve(1215,a,1194,a)]. 2.48/2.70 Derived: number_number_of(int,bit0(A)) = hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,zero_zero(int)),number_number_of(int,A))),number_number_of(int,A)). [resolve(1215,a,1195,a)]. 2.48/2.70 Derived: ti(int,A) = hAPP(int,int,plus_plus(int,number_number_of(int,pls)),A). [resolve(1215,a,1196,a)]. 2.48/2.70 Derived: number_number_of(int,hAPP(int,int,times_times(int,A),B)) = hAPP(int,int,times_times(int,number_number_of(int,A)),number_number_of(int,B)). [resolve(1215,a,1197,a)]. 2.48/2.70 Derived: -linordered_idom(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),number_number_of(int,A))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),A)). [resolve(1215,a,1199,b)]. 2.48/2.70 Derived: -linordered_idom(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),number_number_of(int,A))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),A)). [resolve(1215,a,1200,b)]. 2.48/2.70 Derived: hAPP(int,int,plus_plus(int,A),A) = hAPP(int,int,times_times(int,A),number_number_of(int,bit0(bit1(pls)))). [resolve(1215,a,1201,a)]. 2.52/2.80 Derived: hAPP(int,int,plus_plus(int,A),A) = hAPP(int,int,times_times(int,number_number_of(int,bit0(bit1(pls)))),A). [resolve(1215,a,1202,a)]. 2.52/2.80 Derived: hAPP(int,int,plus_plus(int,number_number_of(int,hAPP(int,int,plus_plus(int,A),B))),C) = hAPP(int,int,plus_plus(int,number_number_of(int,A)),hAPP(int,int,plus_plus(int,number_number_of(int,B)),C)). [resolve(1215,a,1204,a)]. 2.52/2.80 Derived: number_number_of(int,hAPP(int,int,plus_plus(int,A),B)) = hAPP(int,int,plus_plus(int,number_number_of(int,A)),number_number_of(int,B)). [resolve(1215,a,1205,a)]. 2.52/2.80 Derived: -linordered_idom(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,A)),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)). [resolve(1215,a,1206,a)]. 2.52/2.80 Derived: -linordered_idom(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,A)),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)). [resolve(1215,a,1207,a)]. 2.52/2.80 Derived: -linordered_idom(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,A)),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),pls)). [resolve(1215,a,1208,a)]. 2.52/2.80 Derived: -linordered_idom(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,A)),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),pls)). [resolve(1215,a,1209,a)]. 2.52/2.80 Derived: -linordered_idom(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,A)),one_one(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),bit1(pls))). [resolve(1215,a,1212,b)]. 2.52/2.80 Derived: -linordered_idom(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,A)),one_one(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),bit1(pls))). [resolve(1215,a,1213,b)]. 2.52/2.80 Derived: number_number_of(int,bit1(A)) = hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,one_one(int)),number_number_of(int,A))),number_number_of(int,A)). [resolve(1215,a,1214,a)]. 2.52/2.80 1216 -number_ring(A) | number_number_of(A,hAPP(int,int,plus_plus(int,bit1(pls)),B)) = hAPP(A,A,plus_plus(A,one_one(A)),number_number_of(A,B)) # label(fact_16_add__special_I2_J) # label(axiom). [clausify(1041)]. 2.52/2.80 Derived: number_number_of(real,hAPP(int,int,plus_plus(int,bit1(pls)),A)) = hAPP(real,real,plus_plus(real,one_one(real)),number_number_of(real,A)). [resolve(1216,a,1167,a)]. 2.52/2.80 Derived: number_number_of(int,hAPP(int,int,plus_plus(int,bit1(pls)),A)) = hAPP(int,int,plus_plus(int,one_one(int)),number_number_of(int,A)). [resolve(1216,a,1215,a)]. 2.52/2.80 1217 ordered_ring(real) # label(arity_RealDef_Oreal___Rings_Oordered__ring) # label(axiom). [assumption]. 2.52/2.80 1218 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,times_times(A,B),C))) # label(fact_472_split__mult__pos__le) # label(axiom). [clausify(16)]. 2.52/2.80 1219 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,times_times(A,C),B))) # label(fact_472_split__mult__pos__le) # label(axiom). [clausify(16)]. 2.52/2.80 1220 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,C),D)),hAPP(A,A,times_times(A,B),D))) # label(fact_476_mult__right__mono__neg) # label(axiom). [clausify(117)]. 2.52/2.80 1221 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,C),D)),E)),F))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,D),E)),B)),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),E)),F))) # label(fact_546_less__add__iff2) # label(axiom). [clausify(334)]. 2.52/2.81 1222 -ordered_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,C),D)),E)),F))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,D),E)),B)),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),E)),F))) # label(fact_546_less__add__iff2) # label(axiom). [clausify(334)]. 2.52/2.81 1223 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),D)),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,E),C)),F))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,B),E)),C)),D)),F)) # label(fact_545_less__add__iff1) # label(axiom). [clausify(345)]. 2.52/2.81 1224 -ordered_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),D)),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,E),C)),F))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,B),E)),C)),D)),F)) # label(fact_545_less__add__iff1) # label(axiom). [clausify(345)]. 2.52/2.81 1225 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),D)),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,E),C)),F))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,B),E)),C)),D)),F)) # label(fact_543_le__add__iff1) # label(axiom). [clausify(506)]. 2.52/2.81 1226 -ordered_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),D)),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,E),C)),F))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,B),E)),C)),D)),F)) # label(fact_543_le__add__iff1) # label(axiom). [clausify(506)]. 2.52/2.81 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,B),C)),hAPP(real,real,times_times(real,A),C))). [resolve(1217,a,1220,a)]. 2.52/2.81 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,hAPP(real,real,minus_minus(real,B),C)),D)),E))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,C),D)),A)),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,B),D)),E))). [resolve(1217,a,1221,a)]. 2.52/2.81 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,hAPP(real,real,minus_minus(real,B),C)),D)),E))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,C),D)),A)),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,B),D)),E))). [resolve(1217,a,1222,a)]. 2.52/2.81 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),B)),C)),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,D),B)),E))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,hAPP(real,real,minus_minus(real,A),D)),B)),C)),E)). [resolve(1217,a,1223,a)]. 2.52/2.81 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),B)),C)),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,D),B)),E))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,hAPP(real,real,minus_minus(real,A),D)),B)),C)),E)). [resolve(1217,a,1224,a)]. 2.61/2.83 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),B)),C)),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,D),B)),E))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,hAPP(real,real,minus_minus(real,A),D)),B)),C)),E)). [resolve(1217,a,1225,a)]. 2.61/2.83 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),B)),C)),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,D),B)),E))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,hAPP(real,real,minus_minus(real,A),D)),B)),C)),E)). [resolve(1217,a,1226,a)]. 2.61/2.83 1227 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,C),D)),E)),F))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,D),E)),B)),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),E)),F))) # label(fact_544_le__add__iff2) # label(axiom). [clausify(806)]. 2.61/2.83 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,hAPP(real,real,minus_minus(real,B),C)),D)),E))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,C),D)),A)),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,B),D)),E))). [resolve(1227,a,1217,a)]. 2.61/2.83 1228 -ordered_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,C),D)),E)),F))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,D),E)),B)),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),E)),F))) # label(fact_544_le__add__iff2) # label(axiom). [clausify(806)]. 2.61/2.83 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,hAPP(real,real,minus_minus(real,B),C)),D)),E))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,C),D)),A)),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,B),D)),E))). [resolve(1228,a,1217,a)]. 2.61/2.83 1229 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,D),C)),hAPP(A,A,times_times(A,D),B))) # label(fact_475_mult__left__mono__neg) # label(axiom). [clausify(866)]. 2.61/2.83 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,C),B)),hAPP(real,real,times_times(real,C),A))). [resolve(1229,a,1217,a)]. 2.61/2.83 1230 -ordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,times_times(A,B),C))) # label(fact_480_mult__nonpos__nonpos) # label(axiom). [clausify(939)]. 2.61/2.83 1231 ordered_ring(int) # label(arity_Int_Oint___Rings_Oordered__ring) # label(axiom). [assumption]. 2.61/2.83 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,times_times(int,A),B))). [resolve(1231,a,1218,a)]. 2.61/2.90 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),C),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,times_times(int,B),C)),hAPP(int,int,times_times(int,A),C))). [resolve(1231,a,1220,a)]. 2.61/2.90 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,B),C)),D)),E))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,C),D)),A)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),D)),E))). [resolve(1231,a,1221,a)]. 2.61/2.90 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,B),C)),D)),E))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,C),D)),A)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),D)),E))). [resolve(1231,a,1222,a)]. 2.61/2.90 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),B)),C)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,D),B)),E))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,A),D)),B)),C)),E)). [resolve(1231,a,1223,a)]. 2.61/2.90 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),B)),C)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,D),B)),E))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,A),D)),B)),C)),E)). [resolve(1231,a,1224,a)]. 2.61/2.90 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),B)),C)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,D),B)),E))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,A),D)),B)),C)),E)). [resolve(1231,a,1225,a)]. 2.61/2.90 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),B)),C)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,D),B)),E))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,A),D)),B)),C)),E)). [resolve(1231,a,1226,a)]. 2.61/2.90 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,B),C)),D)),E))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,C),D)),A)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),D)),E))). [resolve(1231,a,1227,a)]. 2.61/2.90 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,B),C)),D)),E))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,C),D)),A)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),D)),E))). [resolve(1231,a,1228,a)]. 2.61/2.90 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),C),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,times_times(int,C),B)),hAPP(int,int,times_times(int,C),A))). [resolve(1231,a,1229,a)]. 2.61/2.90 1232 -power(A) | -semiring_0(A) | zero_zero(nat) = number_number_of(nat,B) | zero_zero(A) = hAPP(nat,A,power_power(A,zero_zero(A)),number_number_of(nat,B)) # label(fact_350_power__0__left__number__of) # label(axiom). [clausify(251)]. 2.61/2.90 1233 semiring_0(int) # label(arity_Int_Oint___Rings_Osemiring__0) # label(axiom). [assumption]. 2.71/2.95 Derived: -power(int) | zero_zero(nat) = number_number_of(nat,A) | zero_zero(int) = hAPP(nat,int,power_power(int,zero_zero(int)),number_number_of(nat,A)). [resolve(1232,b,1233,a)]. 2.71/2.95 1234 -power(A) | -semiring_0(A) | zero_zero(nat) != number_number_of(nat,B) | one_one(A) = hAPP(nat,A,power_power(A,zero_zero(A)),number_number_of(nat,B)) # label(fact_350_power__0__left__number__of) # label(axiom). [clausify(251)]. 2.71/2.95 1235 semiring_0(nat) # label(arity_Nat_Onat___Rings_Osemiring__0) # label(axiom). [assumption]. 2.71/2.95 Derived: -power(nat) | zero_zero(nat) = number_number_of(nat,A) | zero_zero(nat) = hAPP(nat,nat,power_power(nat,zero_zero(nat)),number_number_of(nat,A)). [resolve(1235,a,1232,b)]. 2.71/2.95 1236 semiring_0(real) # label(arity_RealDef_Oreal___Rings_Osemiring__0) # label(axiom). [assumption]. 2.71/2.95 Derived: -power(real) | zero_zero(nat) = number_number_of(nat,A) | zero_zero(real) = hAPP(nat,real,power_power(real,zero_zero(real)),number_number_of(nat,A)). [resolve(1236,a,1232,b)]. 2.71/2.95 1237 -power(A) | -semiring_0(A) | zero_zero(nat) != B | one_one(A) = hAPP(nat,A,power_power(A,zero_zero(A)),B) # label(fact_195_power__0__left) # label(axiom). [clausify(768)]. 2.71/2.95 1238 -power(A) | -semiring_0(A) | zero_zero(nat) = B | zero_zero(A) = hAPP(nat,A,power_power(A,zero_zero(A)),B) # label(fact_195_power__0__left) # label(axiom). [clausify(768)]. 2.71/2.95 Derived: -power(int) | zero_zero(nat) = A | zero_zero(int) = hAPP(nat,int,power_power(int,zero_zero(int)),A). [resolve(1238,b,1233,a)]. 2.71/2.95 Derived: -power(nat) | zero_zero(nat) = A | zero_zero(nat) = hAPP(nat,nat,power_power(nat,zero_zero(nat)),A). [resolve(1238,b,1235,a)]. 2.71/2.95 Derived: -power(real) | zero_zero(nat) = A | zero_zero(real) = hAPP(nat,real,power_power(real,zero_zero(real)),A). [resolve(1238,b,1236,a)]. 2.71/2.95 1239 -dvd(A) | -semiring_0(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,plus_plus(A,C),zero_zero(A)))) | -hBOOL(hAPP(A,bool,D,C)) | hBOOL(hAPP(A,bool,D,hAPP(A,A,times_times(A,B),f32(A,D,B)))) # label(fact_744_unity__coeff__ex) # label(axiom). [clausify(876)]. 2.71/2.95 Derived: -dvd(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(int,int,plus_plus(int,B),zero_zero(int)))) | -hBOOL(hAPP(int,bool,C,B)) | hBOOL(hAPP(int,bool,C,hAPP(int,int,times_times(int,A),f32(int,C,A)))). [resolve(1239,b,1233,a)]. 2.71/2.95 Derived: -dvd(nat) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,plus_plus(nat,B),zero_zero(nat)))) | -hBOOL(hAPP(nat,bool,C,B)) | hBOOL(hAPP(nat,bool,C,hAPP(nat,nat,times_times(nat,A),f32(nat,C,A)))). [resolve(1239,b,1235,a)]. 2.71/2.95 Derived: -dvd(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(real,real,plus_plus(real,B),zero_zero(real)))) | -hBOOL(hAPP(real,bool,C,B)) | hBOOL(hAPP(real,bool,C,hAPP(real,real,times_times(real,A),f32(real,C,A)))). [resolve(1239,b,1236,a)]. 2.71/2.95 1240 -dvd(A) | -semiring_0(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,plus_plus(A,f33(A,C,B)),zero_zero(A)))) | -hBOOL(hAPP(A,bool,C,hAPP(A,A,times_times(A,B),D))) # label(fact_744_unity__coeff__ex) # label(axiom). [clausify(876)]. 2.71/2.95 Derived: -dvd(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(int,int,plus_plus(int,f33(int,B,A)),zero_zero(int)))) | -hBOOL(hAPP(int,bool,B,hAPP(int,int,times_times(int,A),C))). [resolve(1240,b,1233,a)]. 2.71/2.95 Derived: -dvd(nat) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,plus_plus(nat,f33(nat,B,A)),zero_zero(nat)))) | -hBOOL(hAPP(nat,bool,B,hAPP(nat,nat,times_times(nat,A),C))). [resolve(1240,b,1235,a)]. 2.71/2.95 Derived: -dvd(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(real,real,plus_plus(real,f33(real,B,A)),zero_zero(real)))) | -hBOOL(hAPP(real,bool,B,hAPP(real,real,times_times(real,A),C))). [resolve(1240,b,1236,a)]. 2.71/2.95 1241 -dvd(A) | -semiring_0(A) | hBOOL(hAPP(A,bool,B,f33(A,B,C))) | -hBOOL(hAPP(A,bool,B,hAPP(A,A,times_times(A,C),D))) # label(fact_744_unity__coeff__ex) # label(axiom). [clausify(876)]. 2.71/2.95 Derived: -dvd(int) | hBOOL(hAPP(int,bool,A,f33(int,A,B))) | -hBOOL(hAPP(int,bool,A,hAPP(int,int,times_times(int,B),C))). [resolve(1241,b,1233,a)]. 2.80/3.03 Derived: -dvd(nat) | hBOOL(hAPP(nat,bool,A,f33(nat,A,B))) | -hBOOL(hAPP(nat,bool,A,hAPP(nat,nat,times_times(nat,B),C))). [resolve(1241,b,1235,a)]. 2.80/3.03 Derived: -dvd(real) | hBOOL(hAPP(real,bool,A,f33(real,A,B))) | -hBOOL(hAPP(real,bool,A,hAPP(real,real,times_times(real,B),C))). [resolve(1241,b,1236,a)]. 2.80/3.03 1242 linordered_semidom(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semidom) # label(axiom). [assumption]. 2.80/3.03 1243 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),one_one(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),one_one(A)),hAPP(nat,A,power_power(A,B),C))) # label(fact_270_one__le__power) # label(axiom). [clausify(21)]. 2.80/3.03 1244 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,power_power(A,B),C)),hAPP(nat,A,power_power(A,D),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),D)) # label(fact_214_power__less__imp__less__base) # label(axiom). [clausify(22)]. 2.80/3.03 1245 -linordered_semidom(A) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),one_one(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(nat,A,power_power(A,D),C)),hAPP(nat,A,power_power(A,D),B))) # label(fact_215_power__decreasing) # label(axiom). [clausify(40)]. 2.80/3.03 1246 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(nat,A,power_power(A,B),C)),hAPP(nat,A,power_power(A,B),D))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),D)) # label(fact_178_power__le__imp__le__exp) # label(axiom). [clausify(76)]. 2.80/3.03 1247 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(nat,A,power_power(A,B),C))) # label(fact_210_zero__less__power) # label(axiom). [clausify(86)]. 2.80/3.03 1248 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,power_power(A,B),C)),hAPP(nat,A,power_power(A,B),D))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),C),D)) # label(fact_274_power__less__imp__less__exp) # label(axiom). [clausify(114)]. 2.80/3.03 1249 -linordered_semidom(A) | hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))) != hAPP(nat,A,power_power(A,C),number_number_of(nat,bit0(bit1(pls)))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | ti(A,B) = ti(A,C) # label(fact_255_power2__eq__imp__eq) # label(axiom). [clausify(156)]. 2.80/3.03 1250 -linordered_semidom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),one_one(A))) # label(fact_506_zero__le__one) # label(axiom). [clausify(158)]. 2.80/3.03 1251 -linordered_semidom(A) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,power_power(A,D),B)),hAPP(nat,A,power_power(A,D),C))) # label(fact_275_power__strict__increasing) # label(axiom). [clausify(180)]. 2.80/3.03 1252 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),zero_zero(A))) # label(fact_509_not__one__less__zero) # label(axiom). [clausify(191)]. 2.80/3.03 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),hAPP(nat,real,power_power(real,A),B))). [resolve(1242,a,1243,a)]. 2.80/3.03 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,power_power(real,A),B)),hAPP(nat,real,power_power(real,C),B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),C)). [resolve(1242,a,1244,a)]. 2.80/3.04 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),one_one(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,power_power(real,C),B)),hAPP(nat,real,power_power(real,C),A))). [resolve(1242,a,1245,a)]. 2.80/3.04 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,power_power(real,A),B)),hAPP(nat,real,power_power(real,A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)). [resolve(1242,a,1246,a)]. 2.80/3.04 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,power_power(real,A),B))). [resolve(1242,a,1247,a)]. 2.80/3.04 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,power_power(real,A),B)),hAPP(nat,real,power_power(real,A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)). [resolve(1242,a,1248,a)]. 2.80/3.04 Derived: hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))) != hAPP(nat,real,power_power(real,B),number_number_of(nat,bit0(bit1(pls)))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | ti(real,A) = ti(real,B). [resolve(1242,a,1249,a)]. 2.80/3.04 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),one_one(real))). [resolve(1242,a,1250,a)]. 2.80/3.04 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,power_power(real,C),A)),hAPP(nat,real,power_power(real,C),B))). [resolve(1242,a,1251,a)]. 2.80/3.04 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),zero_zero(real))). [resolve(1242,a,1252,a)]. 2.80/3.04 1253 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),hAPP(A,A,times_times(A,B),C))) # label(fact_510_less__1__mult) # label(axiom). [clausify(242)]. 2.80/3.04 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),hAPP(real,real,times_times(real,A),B))). [resolve(1253,a,1242,a)]. 2.80/3.04 1254 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),one_one(A)),zero_zero(A))) # label(fact_507_not__one__le__zero) # label(axiom). [clausify(267)]. 2.80/3.04 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),zero_zero(real))). [resolve(1254,a,1242,a)]. 2.80/3.04 1255 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,A,power_power(A,C),number_number_of(nat,bit0(bit1(pls)))))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) # label(fact_254_power2__le__imp__le) # label(axiom). [clausify(301)]. 2.80/3.04 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,real,power_power(real,B),number_number_of(nat,bit0(bit1(pls)))))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)). [resolve(1255,a,1242,a)]. 2.80/3.05 1256 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,power_power(A,B),C)),hAPP(A,A,times_times(A,B),hAPP(nat,A,power_power(A,B),C)))) # label(fact_176_power__less__power__Suc) # label(axiom). [clausify(305)]. 2.80/3.05 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,power_power(real,A),B)),hAPP(real,real,times_times(real,A),hAPP(nat,real,power_power(real,A),B)))). [resolve(1256,a,1242,a)]. 2.80/3.05 1257 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),hAPP(A,A,plus_plus(A,B),D))) # label(fact_505_pos__add__strict) # label(axiom). [clausify(403)]. 2.80/3.05 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),hAPP(real,real,plus_plus(real,A),C))). [resolve(1257,a,1242,a)]. 2.80/3.05 1258 -linordered_semidom(A) | hAPP(nat,A,power_power(A,B),C) != hAPP(nat,A,power_power(A,D),C) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C)) | ti(A,B) = ti(A,D) # label(fact_189_power__eq__imp__eq__base) # label(axiom). [clausify(473)]. 2.80/3.05 Derived: hAPP(nat,real,power_power(real,A),B) != hAPP(nat,real,power_power(real,C),B) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | ti(real,A) = ti(real,C). [resolve(1258,a,1242,a)]. 2.80/3.05 1259 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,power_power(A,B),D)),hAPP(nat,A,power_power(A,C),D))) # label(fact_197_power__strict__mono) # label(axiom). [clausify(477)]. 2.80/3.05 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,power_power(real,A),C)),hAPP(nat,real,power_power(real,B),C))). [resolve(1259,a,1242,a)]. 2.80/3.05 1260 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,A,power_power(A,C),number_number_of(nat,bit0(bit1(pls)))))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) # label(fact_264_power2__less__imp__less) # label(axiom). [clausify(493)]. 2.80/3.05 1261 -linordered_semidom(A) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),one_one(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,power_power(A,D),C)),hAPP(nat,A,power_power(A,D),B))) # label(fact_216_power__strict__decreasing) # label(axiom). [clausify(513)]. 2.80/3.05 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),one_one(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,power_power(real,C),B)),hAPP(nat,real,power_power(real,C),A))). [resolve(1261,a,1242,a)]. 2.80/3.07 1262 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),hAPP(A,A,times_times(A,B),hAPP(nat,A,power_power(A,B),C)))) # label(fact_177_power__gt1__lemma) # label(axiom). [clausify(607)]. 2.80/3.07 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),hAPP(real,real,times_times(real,A),hAPP(nat,real,power_power(real,A),B)))). [resolve(1262,a,1242,a)]. 2.80/3.07 1263 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),one_one(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),hAPP(nat,A,power_power(A,B),C))),hAPP(nat,A,power_power(A,B),C))) # label(fact_232_power__Suc__less) # label(axiom). [clausify(637)]. 2.80/3.07 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),one_one(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,A),hAPP(nat,real,power_power(real,A),B))),hAPP(nat,real,power_power(real,A),B))). [resolve(1263,a,1242,a)]. 2.80/3.07 1264 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(nat,A,power_power(A,B),D)),hAPP(nat,A,power_power(A,C),D))) # label(fact_209_power__mono) # label(axiom). [clausify(649)]. 2.80/3.07 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,power_power(real,A),C)),hAPP(nat,real,power_power(real,B),C))). [resolve(1264,a,1242,a)]. 2.80/3.07 1265 -linordered_semidom(A) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),one_one(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(nat,A,power_power(A,D),B)),hAPP(nat,A,power_power(A,D),C))) # label(fact_271_power__increasing) # label(axiom). [clausify(660)]. 2.80/3.07 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,power_power(real,C),A)),hAPP(nat,real,power_power(real,C),B))). [resolve(1265,a,1242,a)]. 2.80/3.07 1266 -linordered_semidom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),hAPP(A,A,plus_plus(A,B),one_one(A)))) # label(fact_511_less__add__one) # label(axiom). [clausify(691)]. 2.80/3.07 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),hAPP(real,real,plus_plus(real,A),one_one(real)))). [resolve(1266,a,1242,a)]. 2.80/3.07 1267 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | C != D | hAPP(nat,A,power_power(A,B),C) = hAPP(nat,A,power_power(A,B),D) # label(fact_272_power__inject__exp) # label(axiom). [clausify(775)]. 2.80/3.07 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),A)) | B != C | hAPP(nat,real,power_power(real,A),B) = hAPP(nat,real,power_power(real,A),C). [resolve(1267,a,1242,a)]. 2.80/3.07 1268 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | C = D | hAPP(nat,A,power_power(A,B),C) != hAPP(nat,A,power_power(A,B),D) # label(fact_272_power__inject__exp) # label(axiom). [clausify(775)]. 2.80/3.07 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),A)) | B = C | hAPP(nat,real,power_power(real,A),B) != hAPP(nat,real,power_power(real,A),C). [resolve(1268,a,1242,a)]. 2.80/3.07 1269 linordered_semidom(int) # label(arity_Int_Oint___Rings_Olinordered__semidom) # label(axiom). [assumption]. 2.80/3.08 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),hAPP(nat,int,power_power(int,A),B))). [resolve(1269,a,1243,a)]. 2.80/3.08 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,power_power(int,A),B)),hAPP(nat,int,power_power(int,C),B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),C)). [resolve(1269,a,1244,a)]. 2.80/3.08 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),C)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),C),one_one(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,power_power(int,C),B)),hAPP(nat,int,power_power(int,C),A))). [resolve(1269,a,1245,a)]. 2.80/3.08 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,power_power(int,A),B)),hAPP(nat,int,power_power(int,A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)). [resolve(1269,a,1246,a)]. 2.80/3.08 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(nat,int,power_power(int,A),B))). [resolve(1269,a,1247,a)]. 2.80/3.08 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,power_power(int,A),B)),hAPP(nat,int,power_power(int,A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)). [resolve(1269,a,1248,a)]. 2.80/3.08 Derived: hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls)))) != hAPP(nat,int,power_power(int,B),number_number_of(nat,bit0(bit1(pls)))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | ti(int,A) = ti(int,B). [resolve(1269,a,1249,a)]. 2.80/3.08 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),one_one(int))). [resolve(1269,a,1250,a)]. 2.80/3.08 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,power_power(int,C),A)),hAPP(nat,int,power_power(int,C),B))). [resolve(1269,a,1251,a)]. 2.80/3.08 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),zero_zero(int))). [resolve(1269,a,1252,a)]. 2.80/3.08 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(int,int,times_times(int,A),B))). [resolve(1269,a,1253,a)]. 2.80/3.08 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),zero_zero(int))). [resolve(1269,a,1254,a)]. 2.80/3.08 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,B),number_number_of(nat,bit0(bit1(pls)))))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(1269,a,1255,a)]. 2.80/3.08 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,power_power(int,A),B)),hAPP(int,int,times_times(int,A),hAPP(nat,int,power_power(int,A),B)))). [resolve(1269,a,1256,a)]. 2.80/3.08 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),hAPP(int,int,plus_plus(int,A),C))). [resolve(1269,a,1257,a)]. 2.80/3.09 Derived: hAPP(nat,int,power_power(int,A),B) != hAPP(nat,int,power_power(int,C),B) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),C)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | ti(int,A) = ti(int,C). [resolve(1269,a,1258,a)]. 2.80/3.09 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,power_power(int,A),C)),hAPP(nat,int,power_power(int,B),C))). [resolve(1269,a,1259,a)]. 2.80/3.09 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),one_one(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,power_power(int,C),B)),hAPP(nat,int,power_power(int,C),A))). [resolve(1269,a,1261,a)]. 2.80/3.09 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(int,int,times_times(int,A),hAPP(nat,int,power_power(int,A),B)))). [resolve(1269,a,1262,a)]. 2.80/3.09 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),one_one(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,A),hAPP(nat,int,power_power(int,A),B))),hAPP(nat,int,power_power(int,A),B))). [resolve(1269,a,1263,a)]. 2.80/3.09 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,power_power(int,A),C)),hAPP(nat,int,power_power(int,B),C))). [resolve(1269,a,1264,a)]. 2.80/3.09 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,power_power(int,C),A)),hAPP(nat,int,power_power(int,C),B))). [resolve(1269,a,1265,a)]. 2.80/3.09 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),hAPP(int,int,plus_plus(int,A),one_one(int)))). [resolve(1269,a,1266,a)]. 2.80/3.09 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A)) | B != C | hAPP(nat,int,power_power(int,A),B) = hAPP(nat,int,power_power(int,A),C). [resolve(1269,a,1267,a)]. 2.80/3.09 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A)) | B = C | hAPP(nat,int,power_power(int,A),B) != hAPP(nat,int,power_power(int,A),C). [resolve(1269,a,1268,a)]. 2.80/3.09 1270 -linordered_semidom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,plus_plus(A,one_one(A)),one_one(A)))) # label(fact_542_zero__less__two) # label(axiom). [clausify(820)]. 2.80/3.09 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,plus_plus(real,one_one(real)),one_one(real)))). [resolve(1270,a,1242,a)]. 2.80/3.09 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,plus_plus(int,one_one(int)),one_one(int)))). [resolve(1270,a,1269,a)]. 2.80/3.09 1271 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,power_power(A,B),C)),hAPP(nat,A,power_power(A,B),D))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),C),D)) # label(fact_273_power__strict__increasing__iff) # label(axiom). [clausify(871)]. 2.80/3.09 1272 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(nat,A,power_power(A,B),C)),hAPP(nat,A,power_power(A,B),D))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),C),D)) # label(fact_273_power__strict__increasing__iff) # label(axiom). [clausify(871)]. 2.80/3.09 1273 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(nat,A,power_power(A,B),C)),hAPP(nat,A,power_power(A,B),D))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),D)) # label(fact_179_power__increasing__iff) # label(axiom). [clausify(872)]. 2.80/3.09 1274 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(nat,A,power_power(A,B),C)),hAPP(nat,A,power_power(A,B),D))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),D)) # label(fact_179_power__increasing__iff) # label(axiom). [clausify(872)]. 2.80/3.09 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,power_power(real,A),B)),hAPP(nat,real,power_power(real,A),C))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)). [resolve(1274,a,1242,a)]. 2.80/3.09 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,power_power(int,A),B)),hAPP(nat,int,power_power(int,A),C))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)). [resolve(1274,a,1269,a)]. 2.80/3.09 1275 linordered_semidom(nat) # label(arity_Nat_Onat___Rings_Olinordered__semidom) # label(axiom). [assumption]. 2.80/3.09 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),hAPP(nat,nat,power_power(nat,A),B))). [resolve(1275,a,1243,a)]. 2.80/3.09 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,power_power(nat,A),B)),hAPP(nat,nat,power_power(nat,C),B))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),C)). [resolve(1275,a,1244,a)]. 2.80/3.09 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),C)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),one_one(nat))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,power_power(nat,C),B)),hAPP(nat,nat,power_power(nat,C),A))). [resolve(1275,a,1245,a)]. 2.80/3.09 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,power_power(nat,A),B)),hAPP(nat,nat,power_power(nat,A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)). [resolve(1275,a,1246,a)]. 2.80/3.09 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,power_power(nat,A),B))). [resolve(1275,a,1247,a)]. 2.80/3.09 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,power_power(nat,A),B)),hAPP(nat,nat,power_power(nat,A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)). [resolve(1275,a,1248,a)]. 2.80/3.09 Derived: hAPP(nat,nat,power_power(nat,A),number_number_of(nat,bit0(bit1(pls)))) != hAPP(nat,nat,power_power(nat,B),number_number_of(nat,bit0(bit1(pls)))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | ti(nat,A) = ti(nat,B). [resolve(1275,a,1249,a)]. 2.80/3.09 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),one_one(nat))). [resolve(1275,a,1250,a)]. 2.80/3.09 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,power_power(nat,C),A)),hAPP(nat,nat,power_power(nat,C),B))). [resolve(1275,a,1251,a)]. 2.80/3.09 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),hAPP(nat,nat,times_times(nat,A),B))). [resolve(1275,a,1253,a)]. 2.80/3.09 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),zero_zero(nat))). [resolve(1275,a,1254,a)]. 2.80/3.09 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,power_power(nat,A),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,nat,power_power(nat,B),number_number_of(nat,bit0(bit1(pls)))))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)). [resolve(1275,a,1255,a)]. 2.80/3.09 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,power_power(nat,A),B)),hAPP(nat,nat,times_times(nat,A),hAPP(nat,nat,power_power(nat,A),B)))). [resolve(1275,a,1256,a)]. 2.80/3.09 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),hAPP(nat,nat,plus_plus(nat,A),C))). [resolve(1275,a,1257,a)]. 2.80/3.09 Derived: hAPP(nat,nat,power_power(nat,A),B) != hAPP(nat,nat,power_power(nat,C),B) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),C)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | ti(nat,A) = ti(nat,C). [resolve(1275,a,1258,a)]. 2.80/3.09 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,power_power(nat,A),C)),hAPP(nat,nat,power_power(nat,B),C))). [resolve(1275,a,1259,a)]. 2.80/3.09 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),C),one_one(nat))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,power_power(nat,C),B)),hAPP(nat,nat,power_power(nat,C),A))). [resolve(1275,a,1261,a)]. 2.80/3.09 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),hAPP(nat,nat,times_times(nat,A),hAPP(nat,nat,power_power(nat,A),B)))). [resolve(1275,a,1262,a)]. 2.80/3.09 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),one_one(nat))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,times_times(nat,A),hAPP(nat,nat,power_power(nat,A),B))),hAPP(nat,nat,power_power(nat,A),B))). [resolve(1275,a,1263,a)]. 2.80/3.09 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,power_power(nat,A),C)),hAPP(nat,nat,power_power(nat,B),C))). [resolve(1275,a,1264,a)]. 2.80/3.09 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,power_power(nat,C),A)),hAPP(nat,nat,power_power(nat,C),B))). [resolve(1275,a,1265,a)]. 2.80/3.09 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),hAPP(nat,nat,plus_plus(nat,A),one_one(nat)))). [resolve(1275,a,1266,a)]. 2.90/3.18 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),A)) | B != C | hAPP(nat,nat,power_power(nat,A),B) = hAPP(nat,nat,power_power(nat,A),C). [resolve(1275,a,1267,a)]. 2.90/3.18 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),A)) | B = C | hAPP(nat,nat,power_power(nat,A),B) != hAPP(nat,nat,power_power(nat,A),C). [resolve(1275,a,1268,a)]. 2.90/3.18 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,plus_plus(nat,one_one(nat)),one_one(nat)))). [resolve(1275,a,1270,a)]. 2.90/3.18 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,power_power(nat,A),B)),hAPP(nat,nat,power_power(nat,A),C))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)). [resolve(1275,a,1274,a)]. 2.90/3.18 1276 -linordered_semidom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),one_one(A))) # label(fact_508_zero__less__one) # label(axiom). [clausify(878)]. 2.90/3.18 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),one_one(real))). [resolve(1276,a,1242,a)]. 2.90/3.18 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),one_one(int))). [resolve(1276,a,1269,a)]. 2.90/3.18 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),one_one(nat))). [resolve(1276,a,1275,a)]. 2.90/3.18 1277 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),hAPP(nat,A,power_power(A,B),C))) # label(fact_321_one__less__power) # label(axiom). [clausify(968)]. 2.90/3.18 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),hAPP(nat,real,power_power(real,A),B))). [resolve(1277,a,1242,a)]. 2.90/3.18 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(nat,int,power_power(int,A),B))). [resolve(1277,a,1269,a)]. 2.90/3.18 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),hAPP(nat,nat,power_power(nat,A),B))). [resolve(1277,a,1275,a)]. 2.90/3.18 1278 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(nat,A,power_power(A,B),C))) # label(fact_208_zero__le__power) # label(axiom). [clausify(983)]. 2.90/3.18 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,power_power(real,A),B))). [resolve(1278,a,1242,a)]. 2.90/3.18 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,power_power(int,A),B))). [resolve(1278,a,1269,a)]. 2.90/3.18 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),hAPP(nat,nat,power_power(nat,A),B))). [resolve(1278,a,1275,a)]. 2.90/3.18 1279 no_zero_divisors(real) # label(arity_RealDef_Oreal___Rings_Ono__zero__divisors) # label(axiom). [assumption]. 2.90/3.18 1280 -no_zero_divisors(A) | ti(fun(A,A),times_times(A,B)) = times_times(A,B) # label(tsy_c_Groups_Otimes__class_Otimes_5_res) # label(axiom). [clausify(25)]. 2.90/3.18 Derived: ti(fun(real,real),times_times(real,A)) = times_times(real,A). [resolve(1279,a,1280,a)]. 2.99/3.20 1281 no_zero_divisors(int) # label(arity_Int_Oint___Rings_Ono__zero__divisors) # label(axiom). [assumption]. 2.99/3.20 Derived: ti(fun(int,int),times_times(int,A)) = times_times(int,A). [resolve(1281,a,1280,a)]. 2.99/3.20 1282 -power(A) | -zero_neq_one(A) | -no_zero_divisors(A) | -mult_zero(A) | zero_zero(nat) = number_number_of(nat,B) | zero_zero(A) != ti(A,C) | zero_zero(A) = hAPP(nat,A,power_power(A,C),number_number_of(nat,B)) # label(fact_343_power__eq__0__iff__number__of) # label(axiom). [clausify(283)]. 2.99/3.20 Derived: -power(real) | -zero_neq_one(real) | -mult_zero(real) | zero_zero(nat) = number_number_of(nat,A) | zero_zero(real) != ti(real,B) | zero_zero(real) = hAPP(nat,real,power_power(real,B),number_number_of(nat,A)). [resolve(1282,c,1279,a)]. 2.99/3.20 Derived: -power(int) | -zero_neq_one(int) | -mult_zero(int) | zero_zero(nat) = number_number_of(nat,A) | zero_zero(int) != ti(int,B) | zero_zero(int) = hAPP(nat,int,power_power(int,B),number_number_of(nat,A)). [resolve(1282,c,1281,a)]. 2.99/3.20 1283 -power(A) | -zero_neq_one(A) | -no_zero_divisors(A) | -mult_zero(A) | zero_zero(nat) != number_number_of(nat,B) | zero_zero(A) != hAPP(nat,A,power_power(A,C),number_number_of(nat,B)) # label(fact_343_power__eq__0__iff__number__of) # label(axiom). [clausify(283)]. 2.99/3.20 Derived: -power(real) | -zero_neq_one(real) | -mult_zero(real) | zero_zero(nat) != number_number_of(nat,A) | zero_zero(real) != hAPP(nat,real,power_power(real,B),number_number_of(nat,A)). [resolve(1283,c,1279,a)]. 2.99/3.20 Derived: -power(int) | -zero_neq_one(int) | -mult_zero(int) | zero_zero(nat) != number_number_of(nat,A) | zero_zero(int) != hAPP(nat,int,power_power(int,B),number_number_of(nat,A)). [resolve(1283,c,1281,a)]. 2.99/3.20 1284 -power(A) | -zero_neq_one(A) | -no_zero_divisors(A) | -mult_zero(A) | zero_zero(A) = ti(A,B) | zero_zero(A) != hAPP(nat,A,power_power(A,B),number_number_of(nat,C)) # label(fact_343_power__eq__0__iff__number__of) # label(axiom). [clausify(283)]. 2.99/3.20 Derived: -power(real) | -zero_neq_one(real) | -mult_zero(real) | zero_zero(real) = ti(real,A) | zero_zero(real) != hAPP(nat,real,power_power(real,A),number_number_of(nat,B)). [resolve(1284,c,1279,a)]. 2.99/3.20 Derived: -power(int) | -zero_neq_one(int) | -mult_zero(int) | zero_zero(int) = ti(int,A) | zero_zero(int) != hAPP(nat,int,power_power(int,A),number_number_of(nat,B)). [resolve(1284,c,1281,a)]. 2.99/3.20 1285 -no_zero_divisors(A) | times_times(A,ti(A,B)) = times_times(A,B) # label(tsy_c_Groups_Otimes__class_Otimes_5_arg1) # label(axiom). [clausify(299)]. 2.99/3.20 Derived: times_times(real,ti(real,A)) = times_times(real,A). [resolve(1285,a,1279,a)]. 2.99/3.20 Derived: times_times(int,ti(int,A)) = times_times(int,A). [resolve(1285,a,1281,a)]. 2.99/3.20 1286 -power(A) | -no_zero_divisors(A) | -zero_neq_one(A) | -mult_zero(A) | zero_zero(A) != hAPP(nat,A,power_power(A,B),C) | zero_zero(nat) != C # label(fact_185_power__eq__0__iff) # label(axiom). [clausify(535)]. 2.99/3.20 Derived: -power(real) | -zero_neq_one(real) | -mult_zero(real) | zero_zero(real) != hAPP(nat,real,power_power(real,A),B) | zero_zero(nat) != B. [resolve(1286,b,1279,a)]. 2.99/3.20 Derived: -power(int) | -zero_neq_one(int) | -mult_zero(int) | zero_zero(int) != hAPP(nat,int,power_power(int,A),B) | zero_zero(nat) != B. [resolve(1286,b,1281,a)]. 2.99/3.20 1287 -power(A) | -no_zero_divisors(A) | -zero_neq_one(A) | -mult_zero(A) | zero_zero(A) != hAPP(nat,A,power_power(A,B),C) | zero_zero(A) = ti(A,B) # label(fact_185_power__eq__0__iff) # label(axiom). [clausify(535)]. 2.99/3.20 Derived: -power(real) | -zero_neq_one(real) | -mult_zero(real) | zero_zero(real) != hAPP(nat,real,power_power(real,A),B) | zero_zero(real) = ti(real,A). [resolve(1287,b,1279,a)]. 2.99/3.20 Derived: -power(int) | -zero_neq_one(int) | -mult_zero(int) | zero_zero(int) != hAPP(nat,int,power_power(int,A),B) | zero_zero(int) = ti(int,A). [resolve(1287,b,1281,a)]. 2.99/3.20 1288 -power(A) | -no_zero_divisors(A) | -zero_neq_one(A) | -mult_zero(A) | zero_zero(A) = hAPP(nat,A,power_power(A,B),C) | zero_zero(nat) = C | zero_zero(A) != ti(A,B) # label(fact_185_power__eq__0__iff) # label(axiom). [clausify(535)]. 2.99/3.20 Derived: -power(real) | -zero_neq_one(real) | -mult_zero(real) | zero_zero(real) = hAPP(nat,real,power_power(real,A),B) | zero_zero(nat) = B | zero_zero(real) != ti(real,A). [resolve(1288,b,1279,a)]. 3.10/3.39 Derived: -power(int) | -zero_neq_one(int) | -mult_zero(int) | zero_zero(int) = hAPP(nat,int,power_power(int,A),B) | zero_zero(nat) = B | zero_zero(int) != ti(int,A). [resolve(1288,b,1281,a)]. 3.10/3.39 1289 no_zero_divisors(nat) # label(arity_Nat_Onat___Rings_Ono__zero__divisors) # label(axiom). [assumption]. 3.10/3.39 Derived: ti(fun(nat,nat),times_times(nat,A)) = times_times(nat,A). [resolve(1289,a,1280,a)]. 3.10/3.39 Derived: -power(nat) | -zero_neq_one(nat) | -mult_zero(nat) | zero_zero(nat) = number_number_of(nat,A) | zero_zero(nat) != ti(nat,B) | zero_zero(nat) = hAPP(nat,nat,power_power(nat,B),number_number_of(nat,A)). [resolve(1289,a,1282,c)]. 3.10/3.39 Derived: -power(nat) | -zero_neq_one(nat) | -mult_zero(nat) | zero_zero(nat) = ti(nat,A) | zero_zero(nat) != hAPP(nat,nat,power_power(nat,A),number_number_of(nat,B)). [resolve(1289,a,1284,c)]. 3.10/3.39 Derived: times_times(nat,ti(nat,A)) = times_times(nat,A). [resolve(1289,a,1285,a)]. 3.10/3.39 Derived: -power(nat) | -zero_neq_one(nat) | -mult_zero(nat) | zero_zero(nat) != hAPP(nat,nat,power_power(nat,A),B) | zero_zero(nat) = ti(nat,A). [resolve(1289,a,1287,b)]. 3.10/3.39 Derived: -power(nat) | -zero_neq_one(nat) | -mult_zero(nat) | zero_zero(nat) = hAPP(nat,nat,power_power(nat,A),B) | zero_zero(nat) = B | zero_zero(nat) != ti(nat,A). [resolve(1289,a,1288,b)]. 3.10/3.39 1290 -no_zero_divisors(A) | zero_zero(A) = ti(A,B) | zero_zero(A) = ti(A,C) | zero_zero(A) != hAPP(A,A,times_times(A,B),C) # label(fact_441_no__zero__divisors) # label(axiom). [clausify(912)]. 3.10/3.39 Derived: zero_zero(real) = ti(real,A) | zero_zero(real) = ti(real,B) | zero_zero(real) != hAPP(real,real,times_times(real,A),B). [resolve(1290,a,1279,a)]. 3.10/3.39 Derived: zero_zero(int) = ti(int,A) | zero_zero(int) = ti(int,B) | zero_zero(int) != hAPP(int,int,times_times(int,A),B). [resolve(1290,a,1281,a)]. 3.10/3.39 Derived: zero_zero(nat) = ti(nat,A) | zero_zero(nat) = ti(nat,B) | zero_zero(nat) != hAPP(nat,nat,times_times(nat,A),B). [resolve(1290,a,1289,a)]. 3.10/3.39 1291 -no_zero_divisors(A) | zero_zero(A) != hAPP(A,A,times_times(A,B),C) | zero_zero(A) = ti(A,B) | zero_zero(A) = ti(A,C) # label(fact_440_divisors__zero) # label(axiom). [clausify(982)]. 3.10/3.39 1292 ordere779506340up_add(int) # label(arity_Int_Oint___Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 3.10/3.39 1293 -ordere779506340up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,D),B)),hAPP(A,A,plus_plus(A,D),C))) # label(fact_774_add__left__mono) # label(axiom). [clausify(26)]. 3.10/3.39 1294 -ordere779506340up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,B),D)),hAPP(A,A,plus_plus(A,C),D))) # label(fact_773_add__right__mono) # label(axiom). [clausify(440)]. 3.10/3.39 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,A),C)),hAPP(int,int,plus_plus(int,B),C))). [resolve(1294,a,1292,a)]. 3.10/3.39 1295 -ordere779506340up_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,B),D)),hAPP(A,A,plus_plus(A,C),E))) # label(fact_775_add__mono) # label(axiom). [clausify(761)]. 3.10/3.39 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),C),D)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,A),C)),hAPP(int,int,plus_plus(int,B),D))). [resolve(1295,a,1292,a)]. 3.10/3.39 1296 ordere779506340up_add(real) # label(arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 3.10/3.39 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,C),A)),hAPP(real,real,plus_plus(real,C),B))). [resolve(1296,a,1293,a)]. 3.20/3.47 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,A),C)),hAPP(real,real,plus_plus(real,B),C))). [resolve(1296,a,1294,a)]. 3.20/3.47 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),D)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,A),C)),hAPP(real,real,plus_plus(real,B),D))). [resolve(1296,a,1295,a)]. 3.20/3.47 1297 ordere779506340up_add(nat) # label(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 3.20/3.47 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,C),A)),hAPP(nat,nat,plus_plus(nat,C),B))). [resolve(1297,a,1293,a)]. 3.20/3.47 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,A),C)),hAPP(nat,nat,plus_plus(nat,B),C))). [resolve(1297,a,1294,a)]. 3.20/3.47 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),D)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,A),C)),hAPP(nat,nat,plus_plus(nat,B),D))). [resolve(1297,a,1295,a)]. 3.20/3.47 1298 semiring_div(nat) # label(arity_Nat_Onat___Divides_Osemiring__div) # label(axiom). [assumption]. 3.20/3.47 1299 -semiring_div(A) | ti(A,B) = hAPP(A,A,div_mod(A,B),zero_zero(A)) # label(fact_943_mod__by__0) # label(axiom). [clausify(33)]. 3.20/3.47 1300 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,B),C)),B) = hAPP(A,A,div_mod(A,C),B) # label(fact_928_mod__add__self1) # label(axiom). [clausify(68)]. 3.20/3.47 Derived: ti(nat,A) = hAPP(nat,nat,div_mod(nat,A),zero_zero(nat)). [resolve(1298,a,1299,a)]. 3.20/3.47 Derived: hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,plus_plus(nat,A),B)),A) = hAPP(nat,nat,div_mod(nat,B),A). [resolve(1298,a,1300,a)]. 3.20/3.47 1301 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | hAPP(A,A,div_mod(A,hAPP(A,A,div_mod(A,D),C)),B) = hAPP(A,A,div_mod(A,D),B) # label(fact_925_mod__mod__cancel) # label(axiom). [clausify(171)]. 3.20/3.47 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,div_mod(nat,C),B)),A) = hAPP(nat,nat,div_mod(nat,C),A). [resolve(1301,a,1298,a)]. 3.20/3.47 1302 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,B),hAPP(A,A,div_mod(A,C),D))),D) = hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,B),C)),D) # label(fact_932_zmod__simps_I2_J) # label(axiom). [clausify(209)]. 3.20/3.47 Derived: hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,plus_plus(nat,A),hAPP(nat,nat,div_mod(nat,B),C))),C) = hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,plus_plus(nat,A),B)),C). [resolve(1302,a,1298,a)]. 3.20/3.47 1303 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,hAPP(A,A,div_mod(A,B),C)),hAPP(A,A,div_mod(A,D),C))),C) = hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,B),D)),C) # label(fact_931_mod__add__eq) # label(axiom). [clausify(217)]. 3.20/3.47 Derived: hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,div_mod(nat,A),B)),hAPP(nat,nat,div_mod(nat,C),B))),B) = hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,plus_plus(nat,A),C)),B). [resolve(1303,a,1298,a)]. 3.20/3.47 1304 -semiring_div(A) | zero_zero(A) = hAPP(A,A,div_mod(A,zero_zero(A)),B) # label(fact_944_mod__0) # label(axiom). [clausify(291)]. 3.20/3.47 Derived: zero_zero(nat) = hAPP(nat,nat,div_mod(nat,zero_zero(nat)),A). [resolve(1304,a,1298,a)]. 3.20/3.47 1305 -semiring_div(A) | zero_zero(A) = hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,B),C)),C) # label(fact_900_mod__mult__self2__is__0) # label(axiom). [clausify(304)]. 3.20/3.47 Derived: zero_zero(nat) = hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,times_times(nat,A),B)),B). [resolve(1305,a,1298,a)]. 3.20/3.47 1306 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,div_mod(A,B),C)),C) = hAPP(A,A,div_mod(A,B),C) # label(fact_918_mod__mod__trivial) # label(axiom). [clausify(309)]. 3.20/3.48 Derived: hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,div_mod(nat,A),B)),B) = hAPP(nat,nat,div_mod(nat,A),B). [resolve(1306,a,1298,a)]. 3.20/3.48 1307 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,hAPP(A,A,div_mod(A,B),C)),D)),C) = hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,B),D)),C) # label(fact_936_mod__mult__left__eq) # label(axiom). [clausify(333)]. 3.20/3.48 Derived: hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,times_times(nat,hAPP(nat,nat,div_mod(nat,A),B)),C)),B) = hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,times_times(nat,A),C)),B). [resolve(1307,a,1298,a)]. 3.20/3.48 1308 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,B),hAPP(A,A,times_times(A,C),D))),C) = hAPP(A,A,div_mod(A,B),C) # label(fact_903_mod__mult__self2) # label(axiom). [clausify(450)]. 3.20/3.48 Derived: hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,plus_plus(nat,A),hAPP(nat,nat,times_times(nat,B),C))),B) = hAPP(nat,nat,div_mod(nat,A),B). [resolve(1308,a,1298,a)]. 3.20/3.48 1309 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C)) = hAPP(A,A,times_times(A,hAPP(A,A,div_mod(A,B),D)),C) # label(fact_939_mod__mult__mult2) # label(axiom). [clausify(459)]. 3.20/3.48 1310 -semiring_div(A) | hAPP(A,A,div_mod(A,B),C) != hAPP(A,A,div_mod(A,D),C) | hAPP(A,A,div_mod(A,E),C) != hAPP(A,A,div_mod(A,F),C) | hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,B),E)),C) = hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,D),F)),C) # label(fact_934_mod__add__cong) # label(axiom). [clausify(478)]. 3.20/3.48 Derived: hAPP(nat,nat,div_mod(nat,A),B) != hAPP(nat,nat,div_mod(nat,C),B) | hAPP(nat,nat,div_mod(nat,D),B) != hAPP(nat,nat,div_mod(nat,E),B) | hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,plus_plus(nat,A),D)),B) = hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,plus_plus(nat,C),E)),B). [resolve(1310,a,1298,a)]. 3.20/3.48 1311 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,hAPP(A,A,div_mod(A,B),C)),hAPP(A,A,div_mod(A,D),C))),C) = hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,B),D)),C) # label(fact_937_mod__mult__eq) # label(axiom). [clausify(480)]. 3.20/3.48 Derived: hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,times_times(nat,hAPP(nat,nat,div_mod(nat,A),B)),hAPP(nat,nat,div_mod(nat,C),B))),B) = hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,times_times(nat,A),C)),B). [resolve(1311,a,1298,a)]. 3.20/3.48 1312 -semiring_div(A) | zero_zero(A) = hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,B),C)),B) # label(fact_899_mod__mult__self1__is__0) # label(axiom). [clausify(507)]. 3.20/3.48 Derived: zero_zero(nat) = hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,times_times(nat,A),B)),A). [resolve(1312,a,1298,a)]. 3.20/3.48 1313 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,div_mod(A,D),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) # label(fact_926_dvd__mod__iff) # label(axiom). [clausify(512)]. 3.20/3.48 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,div_mod(nat,C),B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),C)). [resolve(1313,a,1298,a)]. 3.20/3.48 1314 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,div_mod(A,D),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) # label(fact_926_dvd__mod__iff) # label(axiom). [clausify(512)]. 3.20/3.48 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,div_mod(nat,C),B))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),C)). [resolve(1314,a,1298,a)]. 3.20/3.48 1315 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | zero_zero(A) = hAPP(A,A,div_mod(A,C),B) # label(fact_904_dvd__eq__mod__eq__0) # label(axiom). [clausify(527)]. 3.20/3.48 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | zero_zero(nat) = hAPP(nat,nat,div_mod(nat,B),A). [resolve(1315,a,1298,a)]. 3.20/3.48 1316 -semiring_div(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | zero_zero(A) != hAPP(A,A,div_mod(A,C),B) # label(fact_904_dvd__eq__mod__eq__0) # label(axiom). [clausify(527)]. 3.20/3.49 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),B)) | zero_zero(nat) != hAPP(nat,nat,div_mod(nat,B),A). [resolve(1316,a,1298,a)]. 3.20/3.49 1317 -semiring_div(A) | hAPP(A,A,div_mod(A,B),C) != hAPP(A,A,div_mod(A,D),C) | hAPP(A,A,div_mod(A,E),C) != hAPP(A,A,div_mod(A,F),C) | hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,B),E)),C) = hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,D),F)),C) # label(fact_941_mod__mult__cong) # label(axiom). [clausify(528)]. 3.20/3.49 Derived: hAPP(nat,nat,div_mod(nat,A),B) != hAPP(nat,nat,div_mod(nat,C),B) | hAPP(nat,nat,div_mod(nat,D),B) != hAPP(nat,nat,div_mod(nat,E),B) | hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,times_times(nat,A),D)),B) = hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,times_times(nat,C),E)),B). [resolve(1317,a,1298,a)]. 3.20/3.49 1318 -semiring_div(A) | div_mod(A,ti(A,B)) = div_mod(A,B) # label(tsy_c_Divides_Odiv__class_Omod_arg1) # label(axiom). [clausify(536)]. 3.20/3.49 Derived: div_mod(nat,ti(nat,A)) = div_mod(nat,A). [resolve(1318,a,1298,a)]. 3.20/3.49 1319 semiring_div(int) # label(arity_Int_Oint___Divides_Osemiring__div) # label(axiom). [assumption]. 3.20/3.49 Derived: ti(int,A) = hAPP(int,int,div_mod(int,A),zero_zero(int)). [resolve(1319,a,1299,a)]. 3.20/3.49 Derived: hAPP(int,int,div_mod(int,hAPP(int,int,plus_plus(int,A),B)),A) = hAPP(int,int,div_mod(int,B),A). [resolve(1319,a,1300,a)]. 3.20/3.49 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | hAPP(int,int,div_mod(int,hAPP(int,int,div_mod(int,C),B)),A) = hAPP(int,int,div_mod(int,C),A). [resolve(1319,a,1301,a)]. 3.20/3.49 Derived: hAPP(int,int,div_mod(int,hAPP(int,int,plus_plus(int,A),hAPP(int,int,div_mod(int,B),C))),C) = hAPP(int,int,div_mod(int,hAPP(int,int,plus_plus(int,A),B)),C). [resolve(1319,a,1302,a)]. 3.20/3.49 Derived: hAPP(int,int,div_mod(int,hAPP(int,int,plus_plus(int,hAPP(int,int,div_mod(int,A),B)),hAPP(int,int,div_mod(int,C),B))),B) = hAPP(int,int,div_mod(int,hAPP(int,int,plus_plus(int,A),C)),B). [resolve(1319,a,1303,a)]. 3.20/3.49 Derived: zero_zero(int) = hAPP(int,int,div_mod(int,zero_zero(int)),A). [resolve(1319,a,1304,a)]. 3.20/3.49 Derived: zero_zero(int) = hAPP(int,int,div_mod(int,hAPP(int,int,times_times(int,A),B)),B). [resolve(1319,a,1305,a)]. 3.20/3.49 Derived: hAPP(int,int,div_mod(int,hAPP(int,int,div_mod(int,A),B)),B) = hAPP(int,int,div_mod(int,A),B). [resolve(1319,a,1306,a)]. 3.20/3.49 Derived: hAPP(int,int,div_mod(int,hAPP(int,int,times_times(int,hAPP(int,int,div_mod(int,A),B)),C)),B) = hAPP(int,int,div_mod(int,hAPP(int,int,times_times(int,A),C)),B). [resolve(1319,a,1307,a)]. 3.20/3.49 Derived: hAPP(int,int,div_mod(int,hAPP(int,int,plus_plus(int,A),hAPP(int,int,times_times(int,B),C))),B) = hAPP(int,int,div_mod(int,A),B). [resolve(1319,a,1308,a)]. 3.20/3.49 Derived: hAPP(int,int,div_mod(int,hAPP(int,int,times_times(int,A),B)),hAPP(int,int,times_times(int,C),B)) = hAPP(int,int,times_times(int,hAPP(int,int,div_mod(int,A),C)),B). [resolve(1319,a,1309,a)]. 3.20/3.49 Derived: hAPP(int,int,div_mod(int,A),B) != hAPP(int,int,div_mod(int,C),B) | hAPP(int,int,div_mod(int,D),B) != hAPP(int,int,div_mod(int,E),B) | hAPP(int,int,div_mod(int,hAPP(int,int,plus_plus(int,A),D)),B) = hAPP(int,int,div_mod(int,hAPP(int,int,plus_plus(int,C),E)),B). [resolve(1319,a,1310,a)]. 3.20/3.49 Derived: hAPP(int,int,div_mod(int,hAPP(int,int,times_times(int,hAPP(int,int,div_mod(int,A),B)),hAPP(int,int,div_mod(int,C),B))),B) = hAPP(int,int,div_mod(int,hAPP(int,int,times_times(int,A),C)),B). [resolve(1319,a,1311,a)]. 3.20/3.49 Derived: zero_zero(int) = hAPP(int,int,div_mod(int,hAPP(int,int,times_times(int,A),B)),A). [resolve(1319,a,1312,a)]. 3.20/3.49 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(int,int,div_mod(int,C),B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),C)). [resolve(1319,a,1314,a)]. 3.20/3.49 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | zero_zero(int) = hAPP(int,int,div_mod(int,B),A). [resolve(1319,a,1315,a)]. 3.20/3.49 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | zero_zero(int) != hAPP(int,int,div_mod(int,B),A). [resolve(1319,a,1316,a)]. 3.31/3.53 Derived: hAPP(int,int,div_mod(int,A),B) != hAPP(int,int,div_mod(int,C),B) | hAPP(int,int,div_mod(int,D),B) != hAPP(int,int,div_mod(int,E),B) | hAPP(int,int,div_mod(int,hAPP(int,int,times_times(int,A),D)),B) = hAPP(int,int,div_mod(int,hAPP(int,int,times_times(int,C),E)),B). [resolve(1319,a,1317,a)]. 3.31/3.53 Derived: div_mod(int,ti(int,A)) = div_mod(int,A). [resolve(1319,a,1318,a)]. 3.31/3.53 1320 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | zero_zero(A) = hAPP(A,A,div_mod(A,C),B) # label(fact_905_dvd__imp__mod__0) # label(axiom). [clausify(578)]. 3.31/3.53 1321 -semiring_div(A) | div_mod(A,B) = ti(fun(A,A),div_mod(A,B)) # label(tsy_c_Divides_Odiv__class_Omod_res) # label(axiom). [clausify(603)]. 3.31/3.53 Derived: div_mod(nat,A) = ti(fun(nat,nat),div_mod(nat,A)). [resolve(1321,a,1298,a)]. 3.31/3.53 Derived: div_mod(int,A) = ti(fun(int,int),div_mod(int,A)). [resolve(1321,a,1319,a)]. 3.31/3.53 1322 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,hAPP(A,A,div_mod(A,B),C)),D)),C) = hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,B),D)),C) # label(fact_933_zmod__simps_I1_J) # label(axiom). [clausify(613)]. 3.31/3.53 Derived: hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,div_mod(nat,A),B)),C)),B) = hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,plus_plus(nat,A),C)),B). [resolve(1322,a,1298,a)]. 3.31/3.53 Derived: hAPP(int,int,div_mod(int,hAPP(int,int,plus_plus(int,hAPP(int,int,div_mod(int,A),B)),C)),B) = hAPP(int,int,div_mod(int,hAPP(int,int,plus_plus(int,A),C)),B). [resolve(1322,a,1319,a)]. 3.31/3.53 1323 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,div_mod(A,C),D))) # label(fact_924_dvd__mod) # label(axiom). [clausify(615)]. 3.31/3.53 1324 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,hAPP(A,A,div_mod(A,B),C)),D)),C) = hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,B),D)),C) # label(fact_930_mod__add__left__eq) # label(axiom). [clausify(631)]. 3.31/3.53 1325 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,B),hAPP(A,A,div_mod(A,C),D))),D) = hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,B),C)),D) # label(fact_935_mod__mult__right__eq) # label(axiom). [clausify(706)]. 3.31/3.53 Derived: hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,times_times(nat,A),hAPP(nat,nat,div_mod(nat,B),C))),C) = hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,times_times(nat,A),B)),C). [resolve(1325,a,1298,a)]. 3.31/3.53 1326 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,B),hAPP(A,A,div_mod(A,C),D))),D) = hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,B),C)),D) # label(fact_929_mod__add__right__eq) # label(axiom). [clausify(707)]. 3.31/3.53 1327 -semiring_div(A) | zero_zero(A) = hAPP(A,A,div_mod(A,B),B) # label(fact_942_mod__self) # label(axiom). [clausify(786)]. 3.31/3.53 Derived: zero_zero(nat) = hAPP(nat,nat,div_mod(nat,A),A). [resolve(1327,a,1298,a)]. 3.31/3.53 1328 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,B),hAPP(A,A,times_times(A,C),D))),D) = hAPP(A,A,div_mod(A,B),D) # label(fact_902_mod__mult__self1) # label(axiom). [clausify(833)]. 3.31/3.53 Derived: hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,plus_plus(nat,A),hAPP(nat,nat,times_times(nat,B),C))),C) = hAPP(nat,nat,div_mod(nat,A),C). [resolve(1328,a,1298,a)]. 3.31/3.53 Derived: hAPP(int,int,div_mod(int,hAPP(int,int,plus_plus(int,A),hAPP(int,int,times_times(int,B),C))),C) = hAPP(int,int,div_mod(int,A),C). [resolve(1328,a,1319,a)]. 3.31/3.53 1329 -semiring_div(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,div_mod(A,C),D))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) # label(fact_923_dvd__mod__imp__dvd) # label(axiom). [clausify(893)]. 3.31/3.53 1330 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,plus_plus(A,B),C)),C) = hAPP(A,A,div_mod(A,B),C) # label(fact_927_mod__add__self2) # label(axiom). [clausify(943)]. 3.31/3.53 Derived: hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,plus_plus(nat,A),B)),B) = hAPP(nat,nat,div_mod(nat,A),B). [resolve(1330,a,1298,a)]. 3.31/3.53 Derived: hAPP(int,int,div_mod(int,hAPP(int,int,plus_plus(int,A),B)),B) = hAPP(int,int,div_mod(int,A),B). [resolve(1330,a,1319,a)]. 3.42/3.63 1331 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D)) = hAPP(A,A,times_times(A,B),hAPP(A,A,div_mod(A,C),D)) # label(fact_938_mod__mult__mult1) # label(axiom). [clausify(970)]. 3.42/3.63 Derived: hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,times_times(nat,A),B)),hAPP(nat,nat,times_times(nat,A),C)) = hAPP(nat,nat,times_times(nat,A),hAPP(nat,nat,div_mod(nat,B),C)). [resolve(1331,a,1298,a)]. 3.42/3.63 Derived: hAPP(int,int,div_mod(int,hAPP(int,int,times_times(int,A),B)),hAPP(int,int,times_times(int,A),C)) = hAPP(int,int,times_times(int,A),hAPP(int,int,div_mod(int,B),C)). [resolve(1331,a,1319,a)]. 3.42/3.63 1332 -semiring_div(A) | zero_zero(A) = hAPP(A,A,div_mod(A,B),one_one(A)) # label(fact_901_mod__by__1) # label(axiom). [clausify(979)]. 3.42/3.63 Derived: zero_zero(nat) = hAPP(nat,nat,div_mod(nat,A),one_one(nat)). [resolve(1332,a,1298,a)]. 3.42/3.63 Derived: zero_zero(int) = hAPP(int,int,div_mod(int,A),one_one(int)). [resolve(1332,a,1319,a)]. 3.42/3.63 1333 -semiring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,hAPP(A,A,div_mod(A,B),C)),D)),C) = hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,B),D)),C) # label(fact_940_zmod__simps_I4_J) # label(axiom). [clausify(1014)]. 3.42/3.63 1334 -semiring_div(A) | -number(A) | zero_zero(A) != hAPP(A,A,div_mod(A,number_number_of(A,B)),number_number_of(A,C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),number_number_of(A,C)),number_number_of(A,B))) # label(fact_951_dvd__eq__mod__eq__0__number__of) # label(axiom). [clausify(1042)]. 3.42/3.63 1335 -semiring_div(A) | -number(A) | zero_zero(A) = hAPP(A,A,div_mod(A,number_number_of(A,B)),number_number_of(A,C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),number_number_of(A,C)),number_number_of(A,B))) # label(fact_951_dvd__eq__mod__eq__0__number__of) # label(axiom). [clausify(1042)]. 3.42/3.63 1336 comm_ring(int) # label(arity_Int_Oint___Rings_Ocomm__ring) # label(axiom). [assumption]. 3.42/3.63 1337 -dvd(A) | -comm_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,plus_plus(A,hAPP(A,A,minus_minus(A,D),hAPP(A,A,times_times(A,E),C))),F))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,plus_plus(A,D),F))) # label(fact_745_inf__period_I4_J) # label(axiom). [clausify(36)]. 3.42/3.63 1338 -dvd(A) | -comm_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,plus_plus(A,hAPP(A,A,minus_minus(A,D),hAPP(A,A,times_times(A,E),C))),F))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,plus_plus(A,D),F))) # label(fact_745_inf__period_I4_J) # label(axiom). [clausify(36)]. 3.42/3.63 Derived: -dvd(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(int,int,plus_plus(int,hAPP(int,int,minus_minus(int,C),hAPP(int,int,times_times(int,D),B))),E))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(int,int,plus_plus(int,C),E))). [resolve(1336,a,1337,b)]. 3.42/3.63 Derived: -dvd(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(int,int,plus_plus(int,hAPP(int,int,minus_minus(int,C),hAPP(int,int,times_times(int,D),B))),E))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(int,int,plus_plus(int,C),E))). [resolve(1336,a,1338,b)]. 3.42/3.63 1339 -comm_ring(A) | -dvd(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,plus_plus(A,hAPP(A,A,minus_minus(A,D),hAPP(A,A,times_times(A,E),C))),F))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,plus_plus(A,D),F))) # label(fact_746_inf__period_I3_J) # label(axiom). [clausify(485)]. 3.42/3.63 1340 -comm_ring(A) | -dvd(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,plus_plus(A,hAPP(A,A,minus_minus(A,D),hAPP(A,A,times_times(A,E),C))),F))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,plus_plus(A,D),F))) # label(fact_746_inf__period_I3_J) # label(axiom). [clausify(485)]. 3.70/3.90 1341 comm_ring(real) # label(arity_RealDef_Oreal___Rings_Ocomm__ring) # label(axiom). [assumption]. 3.70/3.90 Derived: -dvd(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(real,real,plus_plus(real,hAPP(real,real,minus_minus(real,C),hAPP(real,real,times_times(real,D),B))),E))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(real,real,plus_plus(real,C),E))). [resolve(1341,a,1337,b)]. 3.70/3.90 Derived: -dvd(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(real,real,plus_plus(real,hAPP(real,real,minus_minus(real,C),hAPP(real,real,times_times(real,D),B))),E))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(real,real,plus_plus(real,C),E))). [resolve(1341,a,1338,b)]. 3.70/3.90 1342 semiring(real) # label(arity_RealDef_Oreal___Rings_Osemiring) # label(axiom). [assumption]. 3.70/3.90 1343 -semiring(A) | -number(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),number_number_of(A,C))),hAPP(A,A,times_times(A,D),number_number_of(A,C))) = hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,B),D)),number_number_of(A,C)) # label(fact_138_left__distrib__number__of) # label(axiom). [clausify(38)]. 3.70/3.90 1344 -semiring(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,B),C)),D)),E) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),D)),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),D)),E)) # label(fact_447_combine__common__factor) # label(axiom). [clausify(329)]. 3.70/3.90 Derived: hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,hAPP(real,real,plus_plus(real,A),B)),C)),D) = hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),C)),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,B),C)),D)). [resolve(1342,a,1344,a)]. 3.70/3.90 1345 -semiring(A) | ti(fun(A,A),times_times(A,B)) = times_times(A,B) # label(tsy_c_Groups_Otimes__class_Otimes_3_res) # label(axiom). [clausify(760)]. 3.70/3.90 1346 -semiring(A) | times_times(A,ti(A,B)) = times_times(A,B) # label(tsy_c_Groups_Otimes__class_Otimes_3_arg1) # label(axiom). [clausify(804)]. 3.70/3.90 1347 semiring(int) # label(arity_Int_Oint___Rings_Osemiring) # label(axiom). [assumption]. 3.70/3.90 Derived: hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,A),B)),C)),D) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),C)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),C)),D)). [resolve(1347,a,1344,a)]. 3.70/3.90 1348 -semiring(A) | -number(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,number_number_of(A,B)),C)),hAPP(A,A,times_times(A,number_number_of(A,B)),D)) = hAPP(A,A,times_times(A,number_number_of(A,B)),hAPP(A,A,plus_plus(A,C),D)) # label(fact_139_right__distrib__number__of) # label(axiom). [clausify(969)]. 3.70/3.90 Derived: -number(int) | hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,A)),B)),hAPP(int,int,times_times(int,number_number_of(int,A)),C)) = hAPP(int,int,times_times(int,number_number_of(int,A)),hAPP(int,int,plus_plus(int,B),C)). [resolve(1348,a,1347,a)]. 3.70/3.90 1349 semiring(nat) # label(arity_Nat_Onat___Rings_Osemiring) # label(axiom). [assumption]. 3.70/3.90 Derived: hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,hAPP(nat,nat,plus_plus(nat,A),B)),C)),D) = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,A),C)),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,B),C)),D)). [resolve(1349,a,1344,a)]. 3.70/3.90 1350 -plus(A) | -dvd(A) | -linorder(A) | ti(fun(A,A),plus_plus(A,B)) = plus_plus(A,B) # label(tsy_c_Groups_Oplus__class_Oplus_0_res) # label(axiom). [clausify(113)]. 3.70/3.90 1351 linorder(real) # label(arity_RealDef_Oreal___Orderings_Olinorder) # label(axiom). [assumption]. 3.70/3.90 Derived: -plus(real) | -dvd(real) | ti(fun(real,real),plus_plus(real,A)) = plus_plus(real,A). [resolve(1350,c,1351,a)]. 3.70/3.90 1352 linorder(nat) # label(arity_Nat_Onat___Orderings_Olinorder) # label(axiom). [assumption]. 3.70/3.90 Derived: -plus(nat) | -dvd(nat) | ti(fun(nat,nat),plus_plus(nat,A)) = plus_plus(nat,A). [resolve(1352,a,1350,c)]. 3.81/4.04 1353 linorder(int) # label(arity_Int_Oint___Orderings_Olinorder) # label(axiom). [assumption]. 3.81/4.04 Derived: -plus(int) | -dvd(int) | ti(fun(int,int),plus_plus(int,A)) = plus_plus(int,A). [resolve(1353,a,1350,c)]. 3.81/4.04 1354 -linorder(A) | -number(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),number_number_of(A,B)),number_number_of(A,C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),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(662)]. 3.81/4.04 Derived: -number(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),number_number_of(real,A)),number_number_of(real,B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,B)),number_number_of(real,A))). [resolve(1354,a,1351,a)]. 3.81/4.04 Derived: -number(nat) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),number_number_of(nat,A)),number_number_of(nat,B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),number_number_of(nat,B)),number_number_of(nat,A))). [resolve(1354,a,1352,a)]. 3.81/4.04 Derived: -number(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,A)),number_number_of(int,B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,B)),number_number_of(int,A))). [resolve(1354,a,1353,a)]. 3.81/4.04 1355 -linorder(A) | -number(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),number_number_of(A,B)),number_number_of(A,C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),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(662)]. 3.81/4.04 Derived: -number(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),number_number_of(real,A)),number_number_of(real,B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,B)),number_number_of(real,A))). [resolve(1355,a,1351,a)]. 3.81/4.04 Derived: -number(nat) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),number_number_of(nat,A)),number_number_of(nat,B))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),number_number_of(nat,B)),number_number_of(nat,A))). [resolve(1355,a,1352,a)]. 3.81/4.04 Derived: -number(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,A)),number_number_of(int,B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,B)),number_number_of(int,A))). [resolve(1355,a,1353,a)]. 3.81/4.04 1356 -dvd(A) | -linorder(A) | -plus(A) | plus_plus(A,ti(A,B)) = plus_plus(A,B) # label(tsy_c_Groups_Oplus__class_Oplus_0_arg1) # label(axiom). [clausify(677)]. 3.81/4.04 Derived: -dvd(real) | -plus(real) | plus_plus(real,ti(real,A)) = plus_plus(real,A). [resolve(1356,b,1351,a)]. 3.81/4.04 Derived: -dvd(nat) | -plus(nat) | plus_plus(nat,ti(nat,A)) = plus_plus(nat,A). [resolve(1356,b,1352,a)]. 3.81/4.04 Derived: -dvd(int) | -plus(int) | plus_plus(int,ti(int,A)) = plus_plus(int,A). [resolve(1356,b,1353,a)]. 3.81/4.04 1357 -ordere453448008miring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,C),B)),zero_zero(A))) # label(fact_482_mult__nonneg__nonpos2) # label(axiom). [clausify(280)]. 3.81/4.04 1358 ordere453448008miring(nat) # label(arity_Nat_Onat___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 3.81/4.04 1359 ordere453448008miring(int) # label(arity_Int_Oint___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 3.81/4.04 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),zero_zero(nat))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,times_times(nat,B),A)),zero_zero(nat))). [resolve(1357,a,1358,a)]. 3.81/4.04 1360 -ordere453448008miring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),C)),zero_zero(A))) # label(fact_481_mult__nonpos__nonneg) # label(axiom). [clausify(606)]. 3.89/4.16 1361 ordere453448008miring(real) # label(arity_RealDef_Oreal___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 3.89/4.16 1362 -ordere453448008miring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),C)),zero_zero(A))) # label(fact_483_mult__nonneg__nonpos) # label(axiom). [clausify(851)]. 3.89/4.16 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),zero_zero(nat))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,times_times(nat,A),B)),zero_zero(nat))). [resolve(1362,a,1358,a)]. 3.89/4.16 1363 -ordere453448008miring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,C),B)),zero_zero(A))) # label(fact_471_split__mult__neg__le) # label(axiom). [clausify(877)]. 3.89/4.16 1364 -ordere453448008miring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),C)),zero_zero(A))) # label(fact_471_split__mult__neg__le) # label(axiom). [clausify(877)]. 3.89/4.16 1365 -ordere453448008miring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,times_times(A,B),C))) # label(fact_484_mult__nonneg__nonneg) # label(axiom). [clausify(934)]. 3.89/4.16 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),hAPP(nat,nat,times_times(nat,A),B))). [resolve(1365,a,1358,a)]. 3.89/4.16 1366 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),D)) # label(fact_539_mult__left__le__imp__le) # label(axiom). [clausify(87)]. 3.89/4.16 1367 linord20386208strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 3.89/4.16 1368 linord20386208strict(int) # label(arity_Int_Oint___Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 3.89/4.16 1369 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),C)),zero_zero(A))) # label(fact_494_mult__neg__pos) # label(axiom). [clausify(142)]. 3.89/4.16 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,A),B)),zero_zero(real))). [resolve(1369,a,1367,a)]. 3.89/4.16 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,A),B)),zero_zero(int))). [resolve(1369,a,1368,a)]. 3.89/4.16 1370 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,D),B)),hAPP(A,A,times_times(A,D),C))) # label(fact_491_mult__strict__left__mono) # label(axiom). [clausify(223)]. 3.89/4.17 1371 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,times_times(A,B),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),C)) # label(fact_497_zero__less__mult__pos) # label(axiom). [clausify(281)]. 3.89/4.17 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,times_times(real,A),B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B)). [resolve(1371,a,1367,a)]. 3.89/4.17 1372 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),E)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),E))) # label(fact_533_mult__le__less__imp__less) # label(axiom). [clausify(290)]. 3.89/4.17 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),D)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,A),C)),hAPP(real,real,times_times(real,B),D))). [resolve(1372,a,1367,a)]. 3.89/4.17 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),D)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,A),C)),hAPP(int,int,times_times(int,B),D))). [resolve(1372,a,1368,a)]. 3.89/4.17 1373 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,C),B)),zero_zero(A))) # label(fact_498_mult__pos__neg2) # label(axiom). [clausify(361)]. 3.89/4.17 1374 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,times_times(A,B),C))) # label(fact_500_mult__pos__pos) # label(axiom). [clausify(408)]. 3.89/4.17 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,times_times(real,A),B))). [resolve(1374,a,1367,a)]. 3.89/4.17 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,times_times(int,A),B))). [resolve(1374,a,1368,a)]. 3.89/4.17 1375 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),D)) # label(fact_535_mult__less__imp__less__right) # label(axiom). [clausify(417)]. 3.89/4.17 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,C),B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),C)). [resolve(1375,a,1367,a)]. 3.89/4.18 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,A),B)),hAPP(int,int,times_times(int,C),B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),C)). [resolve(1375,a,1368,a)]. 3.89/4.18 1376 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),D)) # label(fact_538_mult__right__le__imp__le) # label(axiom). [clausify(430)]. 3.89/4.18 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,C),B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),C)). [resolve(1376,a,1367,a)]. 3.89/4.18 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,times_times(int,A),B)),hAPP(int,int,times_times(int,C),B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),C)). [resolve(1376,a,1368,a)]. 3.89/4.18 1377 linord20386208strict(nat) # label(arity_Nat_Onat___Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 3.89/4.18 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,times_times(nat,A),B)),hAPP(nat,nat,times_times(nat,A),C))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)). [resolve(1377,a,1366,a)]. 3.89/4.18 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,times_times(nat,A),B))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)). [resolve(1377,a,1371,a)]. 3.89/4.18 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),C),D)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,times_times(nat,A),C)),hAPP(nat,nat,times_times(nat,B),D))). [resolve(1377,a,1372,a)]. 3.89/4.18 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,times_times(nat,A),B))). [resolve(1377,a,1374,a)]. 3.89/4.18 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,times_times(nat,A),B)),hAPP(nat,nat,times_times(nat,C),B))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),C)). [resolve(1377,a,1376,a)]. 3.89/4.18 1378 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,times_times(A,B),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) # label(fact_496_zero__less__mult__pos2) # label(axiom). [clausify(655)]. 3.89/4.18 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,times_times(real,A),B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)). [resolve(1378,a,1367,a)]. 3.89/4.18 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,times_times(int,A),B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)). [resolve(1378,a,1368,a)]. 4.01/4.20 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,times_times(nat,A),B))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)). [resolve(1378,a,1377,a)]. 4.01/4.20 1379 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),E)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),E))) # label(fact_532_mult__less__le__imp__less) # label(axiom). [clausify(720)]. 4.01/4.20 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),D)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,A),C)),hAPP(real,real,times_times(real,B),D))). [resolve(1379,a,1367,a)]. 4.01/4.20 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),C),D)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,A),C)),hAPP(int,int,times_times(int,B),D))). [resolve(1379,a,1368,a)]. 4.01/4.20 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),D)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,times_times(nat,A),C)),hAPP(nat,nat,times_times(nat,B),D))). [resolve(1379,a,1377,a)]. 4.01/4.20 1380 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),D))) # label(fact_492_mult__strict__right__mono) # label(axiom). [clausify(765)]. 4.01/4.20 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,times_times(nat,A),C)),hAPP(nat,nat,times_times(nat,B),C))). [resolve(1380,a,1377,a)]. 4.01/4.20 1381 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),E)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),E))) # label(fact_531_mult__strict__mono_H) # label(axiom). [clausify(825)]. 4.01/4.20 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),D)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,A),C)),hAPP(real,real,times_times(real,B),D))). [resolve(1381,a,1367,a)]. 4.02/4.21 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),D)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,A),C)),hAPP(int,int,times_times(int,B),D))). [resolve(1381,a,1368,a)]. 4.02/4.21 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),C),D)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,times_times(nat,A),C)),hAPP(nat,nat,times_times(nat,B),D))). [resolve(1381,a,1377,a)]. 4.02/4.21 1382 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),C)),zero_zero(A))) # label(fact_499_mult__pos__neg) # label(axiom). [clausify(864)]. 4.02/4.21 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,A),B)),zero_zero(real))). [resolve(1382,a,1367,a)]. 4.02/4.21 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,A),B)),zero_zero(int))). [resolve(1382,a,1368,a)]. 4.02/4.21 1383 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),E)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),E))) # label(fact_530_mult__strict__mono) # label(axiom). [clausify(901)]. 4.02/4.21 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),D)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,A),C)),hAPP(real,real,times_times(real,B),D))). [resolve(1383,a,1367,a)]. 4.02/4.21 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),D)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,A),C)),hAPP(int,int,times_times(int,B),D))). [resolve(1383,a,1368,a)]. 4.02/4.21 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),C),D)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,times_times(nat,A),C)),hAPP(nat,nat,times_times(nat,B),D))). [resolve(1383,a,1377,a)]. 4.02/4.21 1384 -linord20386208strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) # label(fact_537_mult__less__imp__less__left) # label(axiom). [clausify(976)]. 4.12/4.33 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,A),C))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),C)). [resolve(1384,a,1367,a)]. 4.12/4.33 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,A),B)),hAPP(int,int,times_times(int,A),C))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C)). [resolve(1384,a,1368,a)]. 4.12/4.33 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,times_times(nat,A),B)),hAPP(nat,nat,times_times(nat,A),C))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)). [resolve(1384,a,1377,a)]. 4.12/4.33 1385 real_normed_algebra(real) # label(arity_RealDef_Oreal___RealVector_Oreal__normed__algebra) # label(axiom). [assumption]. 4.12/4.33 1386 -real_normed_algebra(A) | zero_zero(A) = hAPP(A,A,times_times(A,B),zero_zero(A)) # label(fact_884_mult__right_Ozero) # label(axiom). [clausify(50)]. 4.12/4.33 1387 -real_normed_algebra(A) | hAPP(A,A,minus_minus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),E)) = hAPP(A,A,plus_plus(A,hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,B),D)),hAPP(A,A,minus_minus(A,C),E))),hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,B),D)),E))),hAPP(A,A,times_times(A,D),hAPP(A,A,minus_minus(A,C),E))) # label(fact_882_mult_Oprod__diff__prod) # label(axiom). [clausify(107)]. 4.12/4.33 1388 -real_normed_algebra(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C)) = hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,B),D)),C) # label(fact_888_mult__left_Oadd) # label(axiom). [clausify(149)]. 4.12/4.33 1389 -real_normed_algebra(A) | hAPP(A,A,minus_minus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D)) = hAPP(A,A,times_times(A,B),hAPP(A,A,minus_minus(A,C),D)) # label(fact_893_mult__right_Odiff) # label(axiom). [clausify(237)]. 4.12/4.33 1390 -real_normed_algebra(A) | zero_zero(A) = hAPP(A,A,times_times(A,zero_zero(A)),B) # label(fact_887_mult_Ozero__left) # label(axiom). [clausify(318)]. 4.12/4.33 1391 -real_normed_algebra(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D)) = hAPP(A,A,times_times(A,B),hAPP(A,A,plus_plus(A,C),D)) # label(fact_890_mult__right_Oadd) # label(axiom). [clausify(387)]. 4.12/4.33 Derived: hAPP(real,real,minus_minus(real,hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,C),D)) = hAPP(real,real,plus_plus(real,hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,hAPP(real,real,minus_minus(real,A),C)),hAPP(real,real,minus_minus(real,B),D))),hAPP(real,real,times_times(real,hAPP(real,real,minus_minus(real,A),C)),D))),hAPP(real,real,times_times(real,C),hAPP(real,real,minus_minus(real,B),D))). [resolve(1385,a,1387,a)]. 4.12/4.33 Derived: hAPP(real,real,minus_minus(real,hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,A),C)) = hAPP(real,real,times_times(real,A),hAPP(real,real,minus_minus(real,B),C)). [resolve(1385,a,1389,a)]. 4.12/4.33 1392 -real_normed_algebra(A) | zero_zero(A) = hAPP(A,A,times_times(A,zero_zero(A)),B) # label(fact_886_mult__left_Ozero) # label(axiom). [clausify(562)]. 4.12/4.33 1393 -real_normed_algebra(A) | zero_zero(A) = hAPP(A,A,times_times(A,B),zero_zero(A)) # label(fact_885_mult_Ozero__right) # label(axiom). [clausify(633)]. 4.12/4.33 1394 -real_normed_algebra(A) | hAPP(A,A,minus_minus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D)) = hAPP(A,A,times_times(A,B),hAPP(A,A,minus_minus(A,C),D)) # label(fact_892_mult_Odiff__right) # label(axiom). [clausify(819)]. 4.12/4.33 1395 -real_normed_algebra(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D)) = hAPP(A,A,times_times(A,B),hAPP(A,A,plus_plus(A,C),D)) # label(fact_891_mult_Oadd__right) # label(axiom). [clausify(860)]. 4.19/4.45 1396 -real_normed_algebra(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C)) = hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,B),D)),C) # label(fact_889_mult_Oadd__left) # label(axiom). [clausify(1002)]. 4.19/4.45 Derived: hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,C),B)) = hAPP(real,real,times_times(real,hAPP(real,real,plus_plus(real,A),C)),B). [resolve(1396,a,1385,a)]. 4.19/4.45 1397 -real_normed_algebra(A) | hAPP(A,A,minus_minus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C)) = hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,B),D)),C) # label(fact_895_mult__left_Odiff) # label(axiom). [clausify(1026)]. 4.19/4.45 Derived: hAPP(real,real,minus_minus(real,hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,C),B)) = hAPP(real,real,times_times(real,hAPP(real,real,minus_minus(real,A),C)),B). [resolve(1397,a,1385,a)]. 4.19/4.45 1398 -real_normed_algebra(A) | hAPP(A,A,minus_minus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C)) = hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,B),D)),C) # label(fact_894_mult_Odiff__left) # label(axiom). [clausify(1039)]. 4.19/4.45 1399 ordere236663937imp_le(nat) # label(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 4.19/4.45 1400 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,D),B)),hAPP(A,A,plus_plus(A,D),C))) # label(fact_779_add__less__cancel__left) # label(axiom). [clausify(67)]. 4.19/4.45 1401 -ordere236663937imp_le(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,D),B)),hAPP(A,A,plus_plus(A,D),C))) # label(fact_779_add__less__cancel__left) # label(axiom). [clausify(67)]. 4.19/4.45 1402 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,B),C)),hAPP(A,A,plus_plus(A,D),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),D)) # label(fact_771_add__le__cancel__right) # label(axiom). [clausify(111)]. 4.19/4.45 1403 -ordere236663937imp_le(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,B),C)),hAPP(A,A,plus_plus(A,D),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),D)) # label(fact_771_add__le__cancel__right) # label(axiom). [clausify(111)]. 4.19/4.45 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,A),B)),hAPP(nat,nat,plus_plus(nat,C),B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),C)). [resolve(1399,a,1402,a)]. 4.19/4.45 1404 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,B),C)),hAPP(A,A,plus_plus(A,B),D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) # label(fact_784_add__less__imp__less__left) # label(axiom). [clausify(324)]. 4.19/4.45 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,A),B)),hAPP(nat,nat,plus_plus(nat,A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)). [resolve(1404,a,1399,a)]. 4.19/4.45 1405 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,B),C)),hAPP(A,A,plus_plus(A,D),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),D)) # label(fact_778_add__less__cancel__right) # label(axiom). [clausify(398)]. 4.19/4.45 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,A),B)),hAPP(nat,nat,plus_plus(nat,C),B))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),C)). [resolve(1405,a,1399,a)]. 4.19/4.45 1406 -ordere236663937imp_le(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,B),C)),hAPP(A,A,plus_plus(A,D),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),D)) # label(fact_778_add__less__cancel__right) # label(axiom). [clausify(398)]. 4.19/4.45 1407 ordere236663937imp_le(int) # label(arity_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 4.19/4.45 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,C),A)),hAPP(int,int,plus_plus(int,C),B))). [resolve(1407,a,1401,a)]. 4.34/4.59 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,A),B)),hAPP(int,int,plus_plus(int,C),B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),C)). [resolve(1407,a,1402,a)]. 4.34/4.59 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,A),B)),hAPP(int,int,plus_plus(int,C),B))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),C)). [resolve(1407,a,1405,a)]. 4.34/4.59 1408 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,B),C)),hAPP(A,A,plus_plus(A,D),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),D)) # label(fact_776_add__le__imp__le__right) # label(axiom). [clausify(694)]. 4.34/4.59 1409 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,B),C)),hAPP(A,A,plus_plus(A,B),D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),D)) # label(fact_777_add__le__imp__le__left) # label(axiom). [clausify(858)]. 4.34/4.59 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,A),B)),hAPP(int,int,plus_plus(int,A),C))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),C)). [resolve(1409,a,1407,a)]. 4.34/4.59 1410 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,B),C)),hAPP(A,A,plus_plus(A,B),D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),D)) # label(fact_772_add__le__cancel__left) # label(axiom). [clausify(953)]. 4.34/4.59 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,A),B)),hAPP(nat,nat,plus_plus(nat,A),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)). [resolve(1410,a,1399,a)]. 4.34/4.59 1411 -ordere236663937imp_le(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,B),C)),hAPP(A,A,plus_plus(A,B),D))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),D)) # label(fact_772_add__le__cancel__left) # label(axiom). [clausify(953)]. 4.34/4.59 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,A),B)),hAPP(int,int,plus_plus(int,A),C))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),C)). [resolve(1411,a,1407,a)]. 4.34/4.59 1412 ordere236663937imp_le(real) # label(arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 4.34/4.59 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,C),A)),hAPP(real,real,plus_plus(real,C),B))). [resolve(1412,a,1401,a)]. 4.34/4.59 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,A),B)),hAPP(real,real,plus_plus(real,C),B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),C)). [resolve(1412,a,1402,a)]. 4.34/4.59 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,A),B)),hAPP(real,real,plus_plus(real,C),B))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),C)). [resolve(1412,a,1405,a)]. 4.34/4.59 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,A),B)),hAPP(real,real,plus_plus(real,A),C))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),C)). [resolve(1412,a,1409,a)]. 4.34/4.59 1413 -ordere236663937imp_le(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,B),C)),hAPP(A,A,plus_plus(A,D),C))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),D)) # label(fact_783_add__less__imp__less__right) # label(axiom). [clausify(1001)]. 4.34/4.59 1414 -ordere1490568538miring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,D),B)),hAPP(A,A,times_times(A,D),C))) # label(fact_477_comm__mult__left__mono) # label(axiom). [clausify(769)]. 4.49/4.73 1415 ordere1490568538miring(real) # label(arity_RealDef_Oreal___Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 4.49/4.73 1416 ordere1490568538miring(int) # label(arity_Int_Oint___Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 4.49/4.73 1417 ordere1490568538miring(nat) # label(arity_Nat_Onat___Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 4.49/4.73 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,C),A)),hAPP(real,real,times_times(real,C),B))). [resolve(1414,a,1415,a)]. 4.49/4.73 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,times_times(int,C),A)),hAPP(int,int,times_times(int,C),B))). [resolve(1414,a,1416,a)]. 4.49/4.73 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,times_times(nat,C),A)),hAPP(nat,nat,times_times(nat,C),B))). [resolve(1414,a,1417,a)]. 4.49/4.73 1418 monoid_mult(real) # label(arity_RealDef_Oreal___Groups_Omonoid__mult) # label(axiom). [assumption]. 4.49/4.73 1419 -monoid_mult(A) | hAPP(nat,A,power_power(A,B),hAPP(nat,nat,plus_plus(nat,C),D)) = hAPP(A,A,times_times(A,hAPP(nat,A,power_power(A,B),C)),hAPP(nat,A,power_power(A,B),D)) # label(fact_261_power__add) # label(axiom). [clausify(82)]. 4.49/4.73 1420 -monoid_mult(A) | one_one(A) = hAPP(nat,A,power_power(A,one_one(A)),B) # label(fact_262_power__one) # label(axiom). [clausify(131)]. 4.49/4.73 1421 -monoid_mult(A) | hAPP(A,A,times_times(A,hAPP(nat,A,power_power(A,B),C)),B) = hAPP(A,A,times_times(A,B),hAPP(nat,A,power_power(A,B),C)) # label(fact_259_power__commutes) # label(axiom). [clausify(162)]. 4.49/4.73 1422 -monoid_mult(A) | ti(A,B) = hAPP(A,A,times_times(A,B),one_one(A)) # label(fact_793_mult__1__right) # label(axiom). [clausify(232)]. 4.49/4.73 1423 -monoid_mult(A) | ti(A,B) = hAPP(A,A,times_times(A,one_one(A)),B) # label(fact_791_mult__1__left) # label(axiom). [clausify(277)]. 4.49/4.73 1424 -monoid_mult(A) | ti(A,B) = hAPP(nat,A,power_power(A,B),one_one(nat)) # label(fact_242_power__one__right) # label(axiom). [clausify(356)]. 4.49/4.73 1425 -monoid_mult(A) | -number(A) | hAPP(A,A,times_times(A,number_number_of(A,B)),number_number_of(A,B)) = hAPP(nat,A,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(418)]. 4.49/4.73 1426 -monoid_mult(A) | hAPP(nat,A,power_power(A,B),number_number_of(nat,bit1(bit1(pls)))) = hAPP(A,A,times_times(A,hAPP(A,A,times_times(A,B),B)),B) # label(fact_163_power3__eq__cube) # label(axiom). [clausify(421)]. 4.49/4.73 1427 -monoid_mult(A) | hAPP(nat,A,power_power(A,hAPP(nat,A,power_power(A,B),C)),D) = hAPP(nat,A,power_power(A,B),hAPP(nat,nat,times_times(nat,C),D)) # label(fact_263_power__mult) # label(axiom). [clausify(432)]. 4.49/4.73 Derived: one_one(real) = hAPP(nat,real,power_power(real,one_one(real)),A). [resolve(1418,a,1420,a)]. 4.49/4.73 Derived: -number(real) | hAPP(real,real,times_times(real,number_number_of(real,A)),number_number_of(real,A)) = hAPP(nat,real,power_power(real,number_number_of(real,A)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1418,a,1425,a)]. 4.49/4.73 Derived: hAPP(nat,real,power_power(real,A),number_number_of(nat,bit1(bit1(pls)))) = hAPP(real,real,times_times(real,hAPP(real,real,times_times(real,A),A)),A). [resolve(1418,a,1426,a)]. 4.49/4.73 1428 -monoid_mult(A) | hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))) = hAPP(A,A,times_times(A,B),B) # label(fact_14_power2__eq__square) # label(axiom). [clausify(816)]. 4.49/4.73 1429 monoid_mult(nat) # label(arity_Nat_Onat___Groups_Omonoid__mult) # label(axiom). [assumption]. 4.64/4.84 Derived: one_one(nat) = hAPP(nat,nat,power_power(nat,one_one(nat)),A). [resolve(1429,a,1420,a)]. 4.64/4.84 Derived: -number(nat) | hAPP(nat,nat,times_times(nat,number_number_of(nat,A)),number_number_of(nat,A)) = hAPP(nat,nat,power_power(nat,number_number_of(nat,A)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1429,a,1425,a)]. 4.64/4.84 Derived: hAPP(nat,nat,power_power(nat,A),number_number_of(nat,bit1(bit1(pls)))) = hAPP(nat,nat,times_times(nat,hAPP(nat,nat,times_times(nat,A),A)),A). [resolve(1429,a,1426,a)]. 4.64/4.84 1430 -monoid_mult(A) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B)) | hAPP(nat,A,power_power(A,C),B) = hAPP(A,A,times_times(A,hAPP(nat,A,power_power(A,C),hAPP(nat,nat,minus_minus(nat,B),one_one(nat)))),C) # label(fact_436_realpow__minus__mult) # label(axiom). [clausify(947)]. 4.64/4.84 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hAPP(nat,real,power_power(real,B),A) = hAPP(real,real,times_times(real,hAPP(nat,real,power_power(real,B),hAPP(nat,nat,minus_minus(nat,A),one_one(nat)))),B). [resolve(1430,a,1418,a)]. 4.64/4.84 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hAPP(nat,nat,power_power(nat,B),A) = hAPP(nat,nat,times_times(nat,hAPP(nat,nat,power_power(nat,B),hAPP(nat,nat,minus_minus(nat,A),one_one(nat)))),B). [resolve(1430,a,1429,a)]. 4.64/4.84 1431 monoid_mult(int) # label(arity_Int_Oint___Groups_Omonoid__mult) # label(axiom). [assumption]. 4.64/4.84 Derived: one_one(int) = hAPP(nat,int,power_power(int,one_one(int)),A). [resolve(1431,a,1420,a)]. 4.64/4.84 Derived: hAPP(int,int,times_times(int,hAPP(nat,int,power_power(int,A),B)),A) = hAPP(int,int,times_times(int,A),hAPP(nat,int,power_power(int,A),B)). [resolve(1431,a,1421,a)]. 4.64/4.84 Derived: ti(int,A) = hAPP(int,int,times_times(int,one_one(int)),A). [resolve(1431,a,1423,a)]. 4.64/4.84 Derived: -number(int) | hAPP(int,int,times_times(int,number_number_of(int,A)),number_number_of(int,A)) = hAPP(nat,int,power_power(int,number_number_of(int,A)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1431,a,1425,a)]. 4.64/4.84 Derived: hAPP(nat,int,power_power(int,A),number_number_of(nat,bit1(bit1(pls)))) = hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,A),A)),A). [resolve(1431,a,1426,a)]. 4.64/4.84 Derived: hAPP(nat,int,power_power(int,hAPP(nat,int,power_power(int,A),B)),C) = hAPP(nat,int,power_power(int,A),hAPP(nat,nat,times_times(nat,B),C)). [resolve(1431,a,1427,a)]. 4.64/4.84 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | hAPP(nat,int,power_power(int,B),A) = hAPP(int,int,times_times(int,hAPP(nat,int,power_power(int,B),hAPP(nat,nat,minus_minus(nat,A),one_one(nat)))),B). [resolve(1431,a,1430,a)]. 4.64/4.84 1432 -monoid_mult(A) | hAPP(nat,A,power_power(A,hAPP(nat,A,power_power(A,B),C)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,A,power_power(A,B),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),C)) # label(fact_103_power__even__eq) # label(axiom). [clausify(960)]. 4.64/4.84 Derived: hAPP(nat,real,power_power(real,hAPP(nat,real,power_power(real,A),B)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,real,power_power(real,A),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),B)). [resolve(1432,a,1418,a)]. 4.64/4.84 Derived: hAPP(nat,nat,power_power(nat,hAPP(nat,nat,power_power(nat,A),B)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,nat,power_power(nat,A),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),B)). [resolve(1432,a,1429,a)]. 4.64/4.84 Derived: hAPP(nat,int,power_power(int,hAPP(nat,int,power_power(int,A),B)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,int,power_power(int,A),hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),B)). [resolve(1432,a,1431,a)]. 4.64/4.84 1433 -zero(A) | zero_zero(A) = ti(A,zero_zero(A)) # label(tsy_c_Groups_Ozero__class_Ozero_res) # label(axiom). [clausify(641)]. 4.64/4.84 1434 zero(int) # label(arity_Int_Oint___Groups_Ozero) # label(axiom). [assumption]. 4.64/4.84 Derived: zero_zero(int) = ti(int,zero_zero(int)). [resolve(1433,a,1434,a)]. 4.64/4.84 1435 zero(nat) # label(arity_Nat_Onat___Groups_Ozero) # label(axiom). [assumption]. 4.81/5.01 Derived: zero_zero(nat) = ti(nat,zero_zero(nat)). [resolve(1435,a,1433,a)]. 4.81/5.01 1436 zero(real) # label(arity_RealDef_Oreal___Groups_Ozero) # label(axiom). [assumption]. 4.81/5.01 Derived: zero_zero(real) = ti(real,zero_zero(real)). [resolve(1436,a,1433,a)]. 4.81/5.01 1437 cancel_semigroup_add(real) # label(arity_RealDef_Oreal___Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 4.81/5.01 1438 -cancel_semigroup_add(A) | plus_plus(A,ti(A,B)) = plus_plus(A,B) # label(tsy_c_Groups_Oplus__class_Oplus_2_arg1) # label(axiom). [clausify(92)]. 4.81/5.01 Derived: plus_plus(real,ti(real,A)) = plus_plus(real,A). [resolve(1437,a,1438,a)]. 4.81/5.01 1439 -cancel_semigroup_add(A) | hAPP(A,A,plus_plus(A,B),C) != hAPP(A,A,plus_plus(A,D),C) | ti(A,B) = ti(A,D) # label(fact_760_add__right__cancel) # label(axiom). [clausify(355)]. 4.81/5.01 Derived: hAPP(real,real,plus_plus(real,A),B) != hAPP(real,real,plus_plus(real,C),B) | ti(real,A) = ti(real,C). [resolve(1439,a,1437,a)]. 4.81/5.01 1440 -cancel_semigroup_add(A) | hAPP(A,A,plus_plus(A,B),C) = hAPP(A,A,plus_plus(A,D),C) | ti(A,B) != ti(A,D) # label(fact_760_add__right__cancel) # label(axiom). [clausify(355)]. 4.81/5.01 Derived: hAPP(real,real,plus_plus(real,A),B) = hAPP(real,real,plus_plus(real,C),B) | ti(real,A) != ti(real,C). [resolve(1440,a,1437,a)]. 4.81/5.01 1441 cancel_semigroup_add(int) # label(arity_Int_Oint___Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 4.81/5.01 Derived: plus_plus(int,ti(int,A)) = plus_plus(int,A). [resolve(1441,a,1438,a)]. 4.81/5.01 Derived: hAPP(int,int,plus_plus(int,A),B) != hAPP(int,int,plus_plus(int,C),B) | ti(int,A) = ti(int,C). [resolve(1441,a,1439,a)]. 4.81/5.01 Derived: hAPP(int,int,plus_plus(int,A),B) = hAPP(int,int,plus_plus(int,C),B) | ti(int,A) != ti(int,C). [resolve(1441,a,1440,a)]. 4.81/5.01 1442 -cancel_semigroup_add(A) | ti(fun(A,A),plus_plus(A,B)) = plus_plus(A,B) # label(tsy_c_Groups_Oplus__class_Oplus_2_res) # label(axiom). [clausify(721)]. 4.81/5.01 Derived: ti(fun(real,real),plus_plus(real,A)) = plus_plus(real,A). [resolve(1442,a,1437,a)]. 4.81/5.01 Derived: ti(fun(int,int),plus_plus(int,A)) = plus_plus(int,A). [resolve(1442,a,1441,a)]. 4.81/5.01 1443 -cancel_semigroup_add(A) | hAPP(A,A,plus_plus(A,B),C) != hAPP(A,A,plus_plus(A,B),D) | ti(A,C) = ti(A,D) # label(fact_761_add__left__imp__eq) # label(axiom). [clausify(911)]. 4.81/5.01 Derived: hAPP(real,real,plus_plus(real,A),B) != hAPP(real,real,plus_plus(real,A),C) | ti(real,B) = ti(real,C). [resolve(1443,a,1437,a)]. 4.81/5.01 Derived: hAPP(int,int,plus_plus(int,A),B) != hAPP(int,int,plus_plus(int,A),C) | ti(int,B) = ti(int,C). [resolve(1443,a,1441,a)]. 4.81/5.01 1444 -cancel_semigroup_add(A) | hAPP(A,A,plus_plus(A,B),C) != hAPP(A,A,plus_plus(A,D),C) | ti(A,B) = ti(A,D) # label(fact_763_add__right__imp__eq) # label(axiom). [clausify(988)]. 4.81/5.01 1445 cancel_semigroup_add(nat) # label(arity_Nat_Onat___Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 4.81/5.01 Derived: plus_plus(nat,ti(nat,A)) = plus_plus(nat,A). [resolve(1445,a,1438,a)]. 4.81/5.01 Derived: hAPP(nat,nat,plus_plus(nat,A),B) != hAPP(nat,nat,plus_plus(nat,C),B) | ti(nat,A) = ti(nat,C). [resolve(1445,a,1439,a)]. 4.81/5.01 Derived: hAPP(nat,nat,plus_plus(nat,A),B) = hAPP(nat,nat,plus_plus(nat,C),B) | ti(nat,A) != ti(nat,C). [resolve(1445,a,1440,a)]. 4.81/5.01 Derived: ti(fun(nat,nat),plus_plus(nat,A)) = plus_plus(nat,A). [resolve(1445,a,1442,a)]. 4.81/5.01 Derived: hAPP(nat,nat,plus_plus(nat,A),B) != hAPP(nat,nat,plus_plus(nat,A),C) | ti(nat,B) = ti(nat,C). [resolve(1445,a,1443,a)]. 4.81/5.01 1446 -cancel_semigroup_add(A) | hAPP(A,A,plus_plus(A,B),C) != hAPP(A,A,plus_plus(A,B),D) | ti(A,C) = ti(A,D) # label(fact_759_add__left__cancel) # label(axiom). [clausify(1038)]. 4.81/5.01 1447 -cancel_semigroup_add(A) | hAPP(A,A,plus_plus(A,B),C) = hAPP(A,A,plus_plus(A,B),D) | ti(A,C) != ti(A,D) # label(fact_759_add__left__cancel) # label(axiom). [clausify(1038)]. 4.81/5.01 Derived: hAPP(real,real,plus_plus(real,A),B) = hAPP(real,real,plus_plus(real,A),C) | ti(real,B) != ti(real,C). [resolve(1447,a,1437,a)]. 4.81/5.01 Derived: hAPP(int,int,plus_plus(int,A),B) = hAPP(int,int,plus_plus(int,A),C) | ti(int,B) != ti(int,C). [resolve(1447,a,1441,a)]. 4.81/5.01 Derived: hAPP(nat,nat,plus_plus(nat,A),B) = hAPP(nat,nat,plus_plus(nat,A),C) | ti(nat,B) != ti(nat,C). [resolve(1447,a,1445,a)]. 5.04/5.26 1448 -linordered_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),B)),hAPP(A,A,times_times(A,C),C)))) # label(fact_235_sum__squares__ge__zero) # label(axiom). [clausify(249)]. 5.04/5.26 1449 linordered_ring(real) # label(arity_RealDef_Oreal___Rings_Olinordered__ring) # label(axiom). [assumption]. 5.04/5.26 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),A)),hAPP(real,real,times_times(real,B),B)))). [resolve(1448,a,1449,a)]. 5.04/5.26 1450 -linordered_ring(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,times_times(A,B),B))) # label(fact_487_zero__le__square) # label(axiom). [clausify(685)]. 5.04/5.26 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,times_times(real,A),A))). [resolve(1450,a,1449,a)]. 5.04/5.26 1451 -linordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),B)),zero_zero(A))) # label(fact_504_not__square__less__zero) # label(axiom). [clausify(763)]. 5.04/5.26 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,A),A)),zero_zero(real))). [resolve(1451,a,1449,a)]. 5.04/5.26 1452 linordered_ring(int) # label(arity_Int_Oint___Rings_Olinordered__ring) # label(axiom). [assumption]. 5.04/5.26 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),A)),hAPP(int,int,times_times(int,B),B)))). [resolve(1452,a,1448,a)]. 5.04/5.26 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,times_times(int,A),A))). [resolve(1452,a,1450,a)]. 5.04/5.26 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,A),A)),zero_zero(int))). [resolve(1452,a,1451,a)]. 5.04/5.26 1453 -linordered_ring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),B)),hAPP(A,A,times_times(A,C),C))),zero_zero(A))) # label(fact_238_not__sum__squares__lt__zero) # label(axiom). [clausify(842)]. 5.04/5.26 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),A)),hAPP(real,real,times_times(real,B),B))),zero_zero(real))). [resolve(1453,a,1449,a)]. 5.04/5.26 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),A)),hAPP(int,int,times_times(int,B),B))),zero_zero(int))). [resolve(1453,a,1452,a)]. 5.04/5.26 1454 ring_11004092258visors(real) # label(arity_RealDef_Oreal___Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 5.04/5.26 1455 -ring_11004092258visors(A) | zero_zero(A) != hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))) | zero_zero(A) = ti(A,B) # label(fact_252_zero__eq__power2) # label(axiom). [clausify(100)]. 5.04/5.26 1456 -ring_11004092258visors(A) | zero_zero(A) = hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))) | zero_zero(A) != ti(A,B) # label(fact_252_zero__eq__power2) # label(axiom). [clausify(100)]. 5.04/5.26 Derived: zero_zero(real) != hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))) | zero_zero(real) = ti(real,A). [resolve(1454,a,1455,a)]. 5.04/5.26 Derived: zero_zero(real) = hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))) | zero_zero(real) != ti(real,A). [resolve(1454,a,1456,a)]. 5.04/5.26 1457 ring_11004092258visors(int) # label(arity_Int_Oint___Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 5.04/5.26 Derived: zero_zero(int) != hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls)))) | zero_zero(int) = ti(int,A). [resolve(1457,a,1455,a)]. 5.04/5.26 Derived: zero_zero(int) = hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls)))) | zero_zero(int) != ti(int,A). [resolve(1457,a,1456,a)]. 5.04/5.26 1458 -ring_11004092258visors(A) | zero_zero(A) = ti(A,B) | zero_zero(A) != hAPP(nat,A,power_power(A,B),C) # label(fact_194_field__power__not__zero) # label(axiom). [clausify(971)]. 5.11/5.35 Derived: zero_zero(real) = ti(real,A) | zero_zero(real) != hAPP(nat,real,power_power(real,A),B). [resolve(1458,a,1454,a)]. 5.11/5.35 Derived: zero_zero(int) = ti(int,A) | zero_zero(int) != hAPP(nat,int,power_power(int,A),B). [resolve(1458,a,1457,a)]. 5.11/5.35 1459 ordere216010020id_add(int) # label(arity_Int_Oint___Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 5.11/5.35 1460 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | zero_zero(A) != hAPP(A,A,plus_plus(A,B),C) | zero_zero(A) = ti(A,C) # label(fact_800_add__nonneg__eq__0__iff) # label(axiom). [clausify(109)]. 5.11/5.35 1461 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | zero_zero(A) != hAPP(A,A,plus_plus(A,B),C) | zero_zero(A) = ti(A,B) # label(fact_800_add__nonneg__eq__0__iff) # label(axiom). [clausify(109)]. 5.11/5.35 1462 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | zero_zero(A) = hAPP(A,A,plus_plus(A,B),C) | zero_zero(A) != ti(A,C) | zero_zero(A) != ti(A,B) # label(fact_800_add__nonneg__eq__0__iff) # label(axiom). [clausify(109)]. 5.11/5.35 1463 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),hAPP(A,A,plus_plus(A,B),C))) # label(fact_799_add__nonneg__nonneg) # label(axiom). [clausify(179)]. 5.11/5.35 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | zero_zero(int) != hAPP(int,int,plus_plus(int,A),B) | zero_zero(int) = ti(int,B). [resolve(1459,a,1460,a)]. 5.11/5.35 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | zero_zero(int) != hAPP(int,int,plus_plus(int,A),B) | zero_zero(int) = ti(int,A). [resolve(1459,a,1461,a)]. 5.11/5.35 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | zero_zero(int) = hAPP(int,int,plus_plus(int,A),B) | zero_zero(int) != ti(int,B) | zero_zero(int) != ti(int,A). [resolve(1459,a,1462,a)]. 5.11/5.35 1464 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,B),C)),zero_zero(A))) # label(fact_803_add__nonpos__nonpos) # label(axiom). [clausify(392)]. 5.11/5.35 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,A),B)),zero_zero(int))). [resolve(1464,a,1459,a)]. 5.11/5.35 1465 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),hAPP(A,A,plus_plus(A,B),D))) # label(fact_814_add__strict__increasing2) # label(axiom). [clausify(438)]. 5.11/5.35 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),hAPP(int,int,plus_plus(int,A),C))). [resolve(1465,a,1459,a)]. 5.11/5.35 1466 ordere216010020id_add(real) # label(arity_RealDef_Oreal___Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 5.11/5.35 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | zero_zero(real) != hAPP(real,real,plus_plus(real,A),B) | zero_zero(real) = ti(real,B). [resolve(1466,a,1460,a)]. 5.11/5.36 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | zero_zero(real) != hAPP(real,real,plus_plus(real,A),B) | zero_zero(real) = ti(real,A). [resolve(1466,a,1461,a)]. 5.11/5.36 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | zero_zero(real) = hAPP(real,real,plus_plus(real,A),B) | zero_zero(real) != ti(real,B) | zero_zero(real) != ti(real,A). [resolve(1466,a,1462,a)]. 5.11/5.36 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,plus_plus(real,A),B))). [resolve(1466,a,1463,a)]. 5.11/5.36 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,A),B)),zero_zero(real))). [resolve(1466,a,1464,a)]. 5.11/5.36 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),hAPP(real,real,plus_plus(real,A),C))). [resolve(1466,a,1465,a)]. 5.11/5.36 1467 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),hAPP(A,A,plus_plus(A,D),B))) # label(fact_802_add__increasing2) # label(axiom). [clausify(499)]. 5.11/5.36 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),hAPP(int,int,plus_plus(int,C),A))). [resolve(1467,a,1459,a)]. 5.11/5.36 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),hAPP(real,real,plus_plus(real,C),A))). [resolve(1467,a,1466,a)]. 5.11/5.36 1468 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,plus_plus(A,B),C))) # label(fact_816_add__nonneg__pos) # label(axiom). [clausify(575)]. 5.11/5.36 1469 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,B),C)),zero_zero(A))) # label(fact_812_add__nonpos__neg) # label(axiom). [clausify(608)]. 5.11/5.36 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,A),B)),zero_zero(int))). [resolve(1469,a,1459,a)]. 5.11/5.36 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,A),B)),zero_zero(real))). [resolve(1469,a,1466,a)]. 5.11/5.36 1470 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,B),C)),zero_zero(A))) # label(fact_813_add__neg__nonpos) # label(axiom). [clausify(676)]. 5.19/5.38 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,A),B)),zero_zero(int))). [resolve(1470,a,1459,a)]. 5.19/5.38 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,A),B)),zero_zero(real))). [resolve(1470,a,1466,a)]. 5.19/5.38 1471 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,B),C)),zero_zero(A))) # label(fact_807_add__neg__neg) # label(axiom). [clausify(686)]. 5.19/5.38 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,A),B)),zero_zero(int))). [resolve(1471,a,1459,a)]. 5.19/5.38 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,A),B)),zero_zero(real))). [resolve(1471,a,1466,a)]. 5.19/5.38 1472 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,plus_plus(A,B),C))) # label(fact_755_add__pos__nonneg) # label(axiom). [clausify(695)]. 5.19/5.38 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,plus_plus(int,A),B))). [resolve(1472,a,1459,a)]. 5.19/5.38 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,plus_plus(real,A),B))). [resolve(1472,a,1466,a)]. 5.19/5.38 1473 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),hAPP(A,A,plus_plus(A,B),D))) # label(fact_801_add__increasing) # label(axiom). [clausify(711)]. 5.19/5.38 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),hAPP(int,int,plus_plus(int,A),C))). [resolve(1473,a,1459,a)]. 5.19/5.38 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),hAPP(real,real,plus_plus(real,A),C))). [resolve(1473,a,1466,a)]. 5.19/5.38 1474 ordere216010020id_add(nat) # label(arity_Nat_Onat___Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 5.19/5.38 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),B)) | zero_zero(nat) != hAPP(nat,nat,plus_plus(nat,A),B) | zero_zero(nat) = ti(nat,B). [resolve(1474,a,1460,a)]. 5.24/5.50 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),B)) | zero_zero(nat) != hAPP(nat,nat,plus_plus(nat,A),B) | zero_zero(nat) = ti(nat,A). [resolve(1474,a,1461,a)]. 5.24/5.50 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),B)) | zero_zero(nat) = hAPP(nat,nat,plus_plus(nat,A),B) | zero_zero(nat) != ti(nat,B) | zero_zero(nat) != ti(nat,A). [resolve(1474,a,1462,a)]. 5.24/5.50 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),hAPP(nat,nat,plus_plus(nat,A),B))). [resolve(1474,a,1463,a)]. 5.24/5.50 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),zero_zero(nat))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),zero_zero(nat))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,plus_plus(nat,A),B)),zero_zero(nat))). [resolve(1474,a,1464,a)]. 5.24/5.50 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),hAPP(nat,nat,plus_plus(nat,A),C))). [resolve(1474,a,1465,a)]. 5.24/5.50 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),hAPP(nat,nat,plus_plus(nat,C),A))). [resolve(1474,a,1467,a)]. 5.24/5.50 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),hAPP(nat,nat,plus_plus(nat,A),C))). [resolve(1474,a,1473,a)]. 5.24/5.50 1475 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),hAPP(A,A,plus_plus(A,B),C))) # label(fact_806_add__pos__pos) # label(axiom). [clausify(881)]. 5.24/5.50 1476 -ordere216010020id_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),C),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),hAPP(A,A,plus_plus(A,B),D))) # label(fact_815_add__strict__increasing) # label(axiom). [clausify(915)]. 5.24/5.50 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),hAPP(int,int,plus_plus(int,A),C))). [resolve(1476,a,1459,a)]. 5.24/5.50 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),B),hAPP(real,real,plus_plus(real,A),C))). [resolve(1476,a,1466,a)]. 5.24/5.50 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),B),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),B),hAPP(nat,nat,plus_plus(nat,A),C))). [resolve(1476,a,1474,a)]. 5.24/5.50 1477 -monoid_add(A) | ti(A,B) = hAPP(A,A,plus_plus(A,B),zero_zero(A)) # label(fact_769_add__0__right) # label(axiom). [clausify(265)]. 5.24/5.50 1478 monoid_add(int) # label(arity_Int_Oint___Groups_Omonoid__add) # label(axiom). [assumption]. 5.24/5.50 1479 monoid_add(nat) # label(arity_Nat_Onat___Groups_Omonoid__add) # label(axiom). [assumption]. 5.24/5.50 1480 -monoid_add(A) | ti(A,B) = hAPP(A,A,plus_plus(A,zero_zero(A)),B) # label(fact_766_add__0__left) # label(axiom). [clausify(475)]. 5.39/5.63 1481 monoid_add(real) # label(arity_RealDef_Oreal___Groups_Omonoid__add) # label(axiom). [assumption]. 5.39/5.63 1482 -monoid_add(A) | plus_plus(A,ti(A,B)) = plus_plus(A,B) # label(tsy_c_Groups_Oplus__class_Oplus_1_arg1) # label(axiom). [clausify(985)]. 5.39/5.63 1483 -monoid_add(A) | ti(fun(A,A),plus_plus(A,B)) = plus_plus(A,B) # label(tsy_c_Groups_Oplus__class_Oplus_1_res) # label(axiom). [clausify(992)]. 5.39/5.63 1484 -semiri456707255roduct(A) | ti(A,B) != ti(A,C) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,D),B)),hAPP(A,A,times_times(A,E),C)) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,D),C)),hAPP(A,A,times_times(A,E),B)) # label(fact_108_crossproduct__eq) # label(axiom). [clausify(140)]. 5.39/5.63 1485 semiri456707255roduct(int) # label(arity_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc) # label(axiom). [assumption]. 5.39/5.63 Derived: ti(int,A) != ti(int,B) | hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,C),A)),hAPP(int,int,times_times(int,D),B)) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,C),B)),hAPP(int,int,times_times(int,D),A)). [resolve(1484,a,1485,a)]. 5.39/5.63 1486 -semiri456707255roduct(A) | ti(A,B) != ti(A,C) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),E)) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),E)),hAPP(A,A,times_times(A,C),D)) # label(fact_108_crossproduct__eq) # label(axiom). [clausify(140)]. 5.39/5.63 Derived: ti(int,A) != ti(int,B) | hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),C)),hAPP(int,int,times_times(int,B),D)) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),D)),hAPP(int,int,times_times(int,B),C)). [resolve(1486,a,1485,a)]. 5.39/5.63 1487 -semiri456707255roduct(A) | ti(A,B) = ti(A,C) | ti(A,D) = ti(A,E) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,D),B)),hAPP(A,A,times_times(A,E),C)) != hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,D),C)),hAPP(A,A,times_times(A,E),B)) # label(fact_108_crossproduct__eq) # label(axiom). [clausify(140)]. 5.39/5.63 Derived: ti(int,A) = ti(int,B) | ti(int,C) = ti(int,D) | hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,C),A)),hAPP(int,int,times_times(int,D),B)) != hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,C),B)),hAPP(int,int,times_times(int,D),A)). [resolve(1487,a,1485,a)]. 5.39/5.63 1488 semiri456707255roduct(nat) # label(arity_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc) # label(axiom). [assumption]. 5.39/5.63 Derived: ti(nat,A) != ti(nat,B) | hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,C),A)),hAPP(nat,nat,times_times(nat,D),B)) = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,C),B)),hAPP(nat,nat,times_times(nat,D),A)). [resolve(1488,a,1484,a)]. 5.39/5.63 Derived: ti(nat,A) != ti(nat,B) | hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,A),C)),hAPP(nat,nat,times_times(nat,B),D)) = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,A),D)),hAPP(nat,nat,times_times(nat,B),C)). [resolve(1488,a,1486,a)]. 5.39/5.63 Derived: ti(nat,A) = ti(nat,B) | ti(nat,C) = ti(nat,D) | hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,C),A)),hAPP(nat,nat,times_times(nat,D),B)) != hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,C),B)),hAPP(nat,nat,times_times(nat,D),A)). [resolve(1488,a,1487,a)]. 5.39/5.63 1489 semiri456707255roduct(real) # label(arity_RealDef_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossp) # label(axiom). [assumption]. 5.39/5.63 Derived: ti(real,A) != ti(real,B) | hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,C),A)),hAPP(real,real,times_times(real,D),B)) = hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,C),B)),hAPP(real,real,times_times(real,D),A)). [resolve(1489,a,1484,a)]. 5.39/5.63 Derived: ti(real,A) != ti(real,B) | hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),C)),hAPP(real,real,times_times(real,B),D)) = hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),D)),hAPP(real,real,times_times(real,B),C)). [resolve(1489,a,1486,a)]. 5.39/5.63 Derived: ti(real,A) = ti(real,B) | ti(real,C) = ti(real,D) | hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,C),A)),hAPP(real,real,times_times(real,D),B)) != hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,C),B)),hAPP(real,real,times_times(real,D),A)). [resolve(1489,a,1487,a)]. 5.47/5.66 1490 -semiri456707255roduct(A) | zero_zero(A) = ti(A,B) | ti(A,C) = ti(A,D) | ti(A,E) != ti(A,F) | hAPP(A,A,plus_plus(A,E),hAPP(A,A,times_times(A,B),C)) != hAPP(A,A,plus_plus(A,F),hAPP(A,A,times_times(A,B),D)) # label(fact_219_add__scale__eq__noteq) # label(axiom). [clausify(498)]. 5.47/5.66 Derived: zero_zero(int) = ti(int,A) | ti(int,B) = ti(int,C) | ti(int,D) != ti(int,E) | hAPP(int,int,plus_plus(int,D),hAPP(int,int,times_times(int,A),B)) != hAPP(int,int,plus_plus(int,E),hAPP(int,int,times_times(int,A),C)). [resolve(1490,a,1485,a)]. 5.47/5.66 Derived: zero_zero(nat) = ti(nat,A) | ti(nat,B) = ti(nat,C) | ti(nat,D) != ti(nat,E) | hAPP(nat,nat,plus_plus(nat,D),hAPP(nat,nat,times_times(nat,A),B)) != hAPP(nat,nat,plus_plus(nat,E),hAPP(nat,nat,times_times(nat,A),C)). [resolve(1490,a,1488,a)]. 5.47/5.66 Derived: zero_zero(real) = ti(real,A) | ti(real,B) = ti(real,C) | ti(real,D) != ti(real,E) | hAPP(real,real,plus_plus(real,D),hAPP(real,real,times_times(real,A),B)) != hAPP(real,real,plus_plus(real,E),hAPP(real,real,times_times(real,A),C)). [resolve(1490,a,1489,a)]. 5.47/5.66 1491 -semiri456707255roduct(A) | ti(A,B) != hAPP(A,A,plus_plus(A,B),C) | zero_zero(A) = ti(A,C) # label(fact_202_add__0__iff) # label(axiom). [clausify(669)]. 5.47/5.66 Derived: ti(int,A) != hAPP(int,int,plus_plus(int,A),B) | zero_zero(int) = ti(int,B). [resolve(1491,a,1485,a)]. 5.47/5.66 Derived: ti(nat,A) != hAPP(nat,nat,plus_plus(nat,A),B) | zero_zero(nat) = ti(nat,B). [resolve(1491,a,1488,a)]. 5.47/5.66 Derived: ti(real,A) != hAPP(real,real,plus_plus(real,A),B) | zero_zero(real) = ti(real,B). [resolve(1491,a,1489,a)]. 5.47/5.66 1492 -semiri456707255roduct(A) | ti(A,B) = hAPP(A,A,plus_plus(A,B),C) | zero_zero(A) != ti(A,C) # label(fact_202_add__0__iff) # label(axiom). [clausify(669)]. 5.47/5.66 Derived: ti(int,A) = hAPP(int,int,plus_plus(int,A),B) | zero_zero(int) != ti(int,B). [resolve(1492,a,1485,a)]. 5.47/5.66 Derived: ti(nat,A) = hAPP(nat,nat,plus_plus(nat,A),B) | zero_zero(nat) != ti(nat,B). [resolve(1492,a,1488,a)]. 5.47/5.66 Derived: ti(real,A) = hAPP(real,real,plus_plus(real,A),B) | zero_zero(real) != ti(real,B). [resolve(1492,a,1489,a)]. 5.47/5.66 1493 -semiri456707255roduct(A) | ti(A,B) = ti(A,C) | ti(A,D) = ti(A,E) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),D)),hAPP(A,A,times_times(A,B),E)) != hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),E)),hAPP(A,A,times_times(A,B),D)) # label(fact_111_crossproduct__noteq) # label(axiom). [clausify(1008)]. 5.47/5.66 Derived: ti(int,A) = ti(int,B) | ti(int,C) = ti(int,D) | hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),C)),hAPP(int,int,times_times(int,A),D)) != hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),D)),hAPP(int,int,times_times(int,A),C)). [resolve(1493,a,1485,a)]. 5.47/5.66 Derived: ti(nat,A) = ti(nat,B) | ti(nat,C) = ti(nat,D) | hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,B),C)),hAPP(nat,nat,times_times(nat,A),D)) != hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,B),D)),hAPP(nat,nat,times_times(nat,A),C)). [resolve(1493,a,1488,a)]. 5.47/5.66 Derived: ti(real,A) = ti(real,B) | ti(real,C) = ti(real,D) | hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,B),C)),hAPP(real,real,times_times(real,A),D)) != hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,B),D)),hAPP(real,real,times_times(real,A),C)). [resolve(1493,a,1489,a)]. 5.47/5.66 1494 -semiri456707255roduct(A) | ti(A,B) != ti(A,C) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),D)),hAPP(A,A,times_times(A,B),E)) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),E)),hAPP(A,A,times_times(A,B),D)) # label(fact_111_crossproduct__noteq) # label(axiom). [clausify(1008)]. 5.47/5.66 Derived: ti(int,A) != ti(int,B) | hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),C)),hAPP(int,int,times_times(int,A),D)) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),D)),hAPP(int,int,times_times(int,A),C)). [resolve(1494,a,1485,a)]. 5.47/5.66 Derived: ti(nat,A) != ti(nat,B) | hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,B),C)),hAPP(nat,nat,times_times(nat,A),D)) = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,B),D)),hAPP(nat,nat,times_times(nat,A),C)). [resolve(1494,a,1488,a)]. 5.92/6.15 Derived: ti(real,A) != ti(real,B) | hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,B),C)),hAPP(real,real,times_times(real,A),D)) = hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,B),D)),hAPP(real,real,times_times(real,A),C)). [resolve(1494,a,1489,a)]. 5.92/6.15 1495 -semiri456707255roduct(A) | ti(A,B) != ti(A,C) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,D),B)),hAPP(A,A,times_times(A,E),C)) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,D),C)),hAPP(A,A,times_times(A,E),B)) # label(fact_111_crossproduct__noteq) # label(axiom). [clausify(1008)]. 5.92/6.15 1496 linordered_semiring(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semiring) # label(axiom). [assumption]. 5.92/6.15 1497 -linordered_semiring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),D)) # label(fact_534_mult__right__less__imp__less) # label(axiom). [clausify(125)]. 5.92/6.15 1498 -linordered_semiring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),C),D)) # label(fact_536_mult__left__less__imp__less) # label(axiom). [clausify(384)]. 5.92/6.15 1499 linordered_semiring(nat) # label(arity_Nat_Onat___Rings_Olinordered__semiring) # label(axiom). [assumption]. 5.92/6.15 1500 linordered_semiring(int) # label(arity_Int_Oint___Rings_Olinordered__semiring) # label(axiom). [assumption]. 5.92/6.15 1501 -semiring_1(A) | zero_zero(A) = hAPP(nat,A,power_power(A,zero_zero(A)),number_number_of(nat,bit0(bit1(pls)))) # label(fact_251_zero__power2) # label(axiom). [clausify(170)]. 5.92/6.15 1502 semiring_1(real) # label(arity_RealDef_Oreal___Rings_Osemiring__1) # label(axiom). [assumption]. 5.92/6.15 Derived: zero_zero(real) = hAPP(nat,real,power_power(real,zero_zero(real)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1501,a,1502,a)]. 5.92/6.15 1503 semiring_1(nat) # label(arity_Nat_Onat___Rings_Osemiring__1) # label(axiom). [assumption]. 5.92/6.15 Derived: zero_zero(nat) = hAPP(nat,nat,power_power(nat,zero_zero(nat)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1503,a,1501,a)]. 5.92/6.15 1504 semiring_1(int) # label(arity_Int_Oint___Rings_Osemiring__1) # label(axiom). [assumption]. 5.92/6.15 Derived: zero_zero(int) = hAPP(nat,int,power_power(int,zero_zero(int)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1504,a,1501,a)]. 5.92/6.15 1505 -semiring_1(A) | one_one(A) = hAPP(nat,A,power_power(A,one_one(A)),number_number_of(nat,bit0(bit1(pls)))) # label(fact_12_one__power2) # label(axiom). [clausify(836)]. 5.92/6.15 1506 one(nat) # label(arity_Nat_Onat___Groups_Oone) # label(axiom). [assumption]. 5.92/6.15 1507 -one(A) | one_one(A) = ti(A,one_one(A)) # label(tsy_c_Groups_Oone__class_Oone_res) # label(axiom). [clausify(133)]. 5.92/6.15 Derived: one_one(nat) = ti(nat,one_one(nat)). [resolve(1506,a,1507,a)]. 5.92/6.15 1508 one(int) # label(arity_Int_Oint___Groups_Oone) # label(axiom). [assumption]. 5.92/6.15 Derived: one_one(int) = ti(int,one_one(int)). [resolve(1508,a,1507,a)]. 5.92/6.15 1509 one(real) # label(arity_RealDef_Oreal___Groups_Oone) # label(axiom). [assumption]. 5.92/6.15 Derived: one_one(real) = ti(real,one_one(real)). [resolve(1509,a,1507,a)]. 5.92/6.15 1510 ab_semigroup_mult(int) # label(arity_Int_Oint___Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 5.92/6.15 1511 -ab_semigroup_mult(A) | ti(fun(A,A),times_times(A,B)) = times_times(A,B) # label(tsy_c_Groups_Otimes__class_Otimes_0_res) # label(axiom). [clausify(136)]. 5.92/6.15 1512 -ab_semigroup_mult(A) | hAPP(A,A,times_times(A,hAPP(A,A,times_times(A,B),C)),D) = hAPP(A,A,times_times(A,B),hAPP(A,A,times_times(A,C),D)) # label(fact_757_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom). [clausify(323)]. 5.92/6.15 1513 -ab_semigroup_mult(A) | times_times(A,ti(A,B)) = times_times(A,B) # label(tsy_c_Groups_Otimes__class_Otimes_0_arg1) # label(axiom). [clausify(652)]. 6.33/6.54 1514 ab_semigroup_mult(real) # label(arity_RealDef_Oreal___Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 6.33/6.54 Derived: hAPP(real,real,times_times(real,hAPP(real,real,times_times(real,A),B)),C) = hAPP(real,real,times_times(real,A),hAPP(real,real,times_times(real,B),C)). [resolve(1514,a,1512,a)]. 6.33/6.54 1515 ab_semigroup_mult(nat) # label(arity_Nat_Onat___Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 6.33/6.54 1516 -cancel146912293up_add(A) | hAPP(A,A,plus_plus(A,B),C) != hAPP(A,A,plus_plus(A,B),D) | ti(A,C) = ti(A,D) # label(fact_762_add__imp__eq) # label(axiom). [clausify(617)]. 6.33/6.54 1517 cancel146912293up_add(int) # label(arity_Int_Oint___Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 6.33/6.54 1518 cancel146912293up_add(real) # label(arity_RealDef_Oreal___Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 6.33/6.54 1519 cancel146912293up_add(nat) # label(arity_Nat_Onat___Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 6.33/6.54 1520 ab_semigroup_add(nat) # label(arity_Nat_Onat___Groups_Oab__semigroup__add) # label(axiom). [assumption]. 6.33/6.54 1521 -ab_semigroup_add(A) | ti(fun(A,A),plus_plus(A,B)) = plus_plus(A,B) # label(tsy_c_Groups_Oplus__class_Oplus_3_res) # label(axiom). [clausify(138)]. 6.33/6.54 1522 -ab_semigroup_add(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,plus_plus(A,B),C)),D) = hAPP(A,A,plus_plus(A,B),hAPP(A,A,plus_plus(A,C),D)) # label(fact_758_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom). [clausify(240)]. 6.33/6.54 1523 -ab_semigroup_add(A) | plus_plus(A,ti(A,B)) = plus_plus(A,B) # label(tsy_c_Groups_Oplus__class_Oplus_3_arg1) # label(axiom). [clausify(436)]. 6.33/6.54 1524 ab_semigroup_add(real) # label(arity_RealDef_Oreal___Groups_Oab__semigroup__add) # label(axiom). [assumption]. 6.33/6.54 1525 ab_semigroup_add(int) # label(arity_Int_Oint___Groups_Oab__semigroup__add) # label(axiom). [assumption]. 6.33/6.54 1526 ordered_semiring(nat) # label(arity_Nat_Onat___Rings_Oordered__semiring) # label(axiom). [assumption]. 6.33/6.54 1527 -ordered_semiring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),E)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),E))) # label(fact_473_mult__mono) # label(axiom). [clausify(183)]. 6.33/6.54 1528 -ordered_semiring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,D),B)),hAPP(A,A,times_times(A,D),C))) # label(fact_478_mult__left__mono) # label(axiom). [clausify(313)]. 6.33/6.54 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),D)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,times_times(nat,A),C)),hAPP(nat,nat,times_times(nat,B),D))). [resolve(1526,a,1527,a)]. 6.33/6.54 1529 ordered_semiring(real) # label(arity_RealDef_Oreal___Rings_Oordered__semiring) # label(axiom). [assumption]. 6.33/6.54 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),D)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,A),C)),hAPP(real,real,times_times(real,B),D))). [resolve(1529,a,1527,a)]. 6.33/6.54 1530 -ordered_semiring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),D))) # label(fact_479_mult__right__mono) # label(axiom). [clausify(588)]. 6.42/6.66 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,A),C)),hAPP(real,real,times_times(real,B),C))). [resolve(1530,a,1529,a)]. 6.42/6.66 1531 -ordered_semiring(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),E)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),E))) # label(fact_474_mult__mono_H) # label(axiom). [clausify(742)]. 6.42/6.66 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),D)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,times_times(nat,A),C)),hAPP(nat,nat,times_times(nat,B),D))). [resolve(1531,a,1526,a)]. 6.42/6.66 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),D)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,A),C)),hAPP(real,real,times_times(real,B),D))). [resolve(1531,a,1529,a)]. 6.42/6.66 1532 ordered_semiring(int) # label(arity_Int_Oint___Rings_Oordered__semiring) # label(axiom). [assumption]. 6.42/6.66 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),C),D)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,times_times(int,A),C)),hAPP(int,int,times_times(int,B),D))). [resolve(1532,a,1527,a)]. 6.42/6.66 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,times_times(int,A),C)),hAPP(int,int,times_times(int,B),C))). [resolve(1532,a,1530,a)]. 6.42/6.66 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),C),D)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,times_times(int,A),C)),hAPP(int,int,times_times(int,B),D))). [resolve(1532,a,1531,a)]. 6.42/6.66 1533 -ring_1(A) | hAPP(A,A,minus_minus(A,hAPP(A,A,times_times(A,B),B)),one_one(A)) = hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,B),one_one(A))),hAPP(A,A,minus_minus(A,B),one_one(A))) # label(fact_547_real__squared__diff__one__factored) # label(axiom). [clausify(567)]. 6.42/6.66 1534 ring_1(int) # label(arity_Int_Oint___Rings_Oring__1) # label(axiom). [assumption]. 6.42/6.66 1535 ring_1(real) # label(arity_RealDef_Oreal___Rings_Oring__1) # label(axiom). [assumption]. 6.42/6.66 Derived: hAPP(int,int,minus_minus(int,hAPP(int,int,times_times(int,A),A)),one_one(int)) = hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,A),one_one(int))),hAPP(int,int,minus_minus(int,A),one_one(int))). [resolve(1533,a,1534,a)]. 6.42/6.66 Derived: hAPP(real,real,minus_minus(real,hAPP(real,real,times_times(real,A),A)),one_one(real)) = hAPP(real,real,times_times(real,hAPP(real,real,plus_plus(real,A),one_one(real))),hAPP(real,real,minus_minus(real,A),one_one(real))). [resolve(1533,a,1535,a)]. 6.52/6.79 1536 idom(int) # label(arity_Int_Oint___Rings_Oidom) # label(axiom). [assumption]. 6.52/6.79 1537 -idom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D))) | zero_zero(A) = ti(A,B) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),D)) # label(fact_512_dvd__mult__cancel__left) # label(axiom). [clausify(189)]. 6.52/6.79 1538 -idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D))) | zero_zero(A) != ti(A,B) # label(fact_512_dvd__mult__cancel__left) # label(axiom). [clausify(189)]. 6.52/6.79 1539 -idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),D)) # label(fact_512_dvd__mult__cancel__left) # label(axiom). [clausify(189)]. 6.52/6.79 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,times_times(int,A),B)),hAPP(int,int,times_times(int,A),C))) | zero_zero(int) != ti(int,A). [resolve(1536,a,1538,a)]. 6.52/6.79 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,times_times(int,A),B)),hAPP(int,int,times_times(int,A),C))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B),C)). [resolve(1536,a,1539,a)]. 6.52/6.79 1540 idom(real) # label(arity_RealDef_Oreal___Rings_Oidom) # label(axiom). [assumption]. 6.52/6.79 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,A),C))) | zero_zero(real) = ti(real,A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),B),C)). [resolve(1540,a,1537,a)]. 6.52/6.79 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,A),C))) | zero_zero(real) != ti(real,A). [resolve(1540,a,1538,a)]. 6.52/6.79 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,A),C))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),B),C)). [resolve(1540,a,1539,a)]. 6.52/6.79 1541 -idom(A) | zero_zero(A) != ti(A,B) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(A,A,times_times(A,C),B)),hAPP(A,A,times_times(A,D),B))) # label(fact_513_dvd__mult__cancel__right) # label(axiom). [clausify(395)]. 6.52/6.79 Derived: zero_zero(int) != ti(int,A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,times_times(int,B),A)),hAPP(int,int,times_times(int,C),A))). [resolve(1541,a,1536,a)]. 6.52/6.79 Derived: zero_zero(real) != ti(real,A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(real,real,times_times(real,B),A)),hAPP(real,real,times_times(real,C),A))). [resolve(1541,a,1540,a)]. 6.52/6.79 1542 -idom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),D))) # label(fact_513_dvd__mult__cancel__right) # label(axiom). [clausify(395)]. 6.52/6.79 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,times_times(int,A),C)),hAPP(int,int,times_times(int,B),C))). [resolve(1542,a,1536,a)]. 6.52/6.79 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(real,real,times_times(real,A),C)),hAPP(real,real,times_times(real,B),C))). [resolve(1542,a,1540,a)]. 6.52/6.79 1543 -idom(A) | zero_zero(A) = ti(A,B) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),D)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(A,A,times_times(A,C),B)),hAPP(A,A,times_times(A,D),B))) # label(fact_513_dvd__mult__cancel__right) # label(axiom). [clausify(395)]. 6.52/6.79 Derived: zero_zero(int) = ti(int,A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B),C)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,times_times(int,B),A)),hAPP(int,int,times_times(int,C),A))). [resolve(1543,a,1536,a)]. 6.72/6.94 Derived: zero_zero(real) = ti(real,A) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),B),C)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(real,real,times_times(real,B),A)),hAPP(real,real,times_times(real,C),A))). [resolve(1543,a,1540,a)]. 6.72/6.94 1544 number_semiring(nat) # label(arity_Nat_Onat___Int_Onumber__semiring) # label(axiom). [assumption]. 6.72/6.94 1545 -number_semiring(A) | number_number_of(A,bit0(bit1(pls))) = hAPP(A,A,plus_plus(A,one_one(A)),one_one(A)) # label(fact_169_semiring__one__add__one__is__two) # label(axiom). [clausify(196)]. 6.72/6.94 Derived: number_number_of(nat,bit0(bit1(pls))) = hAPP(nat,nat,plus_plus(nat,one_one(nat)),one_one(nat)). [resolve(1544,a,1545,a)]. 6.72/6.94 1546 -number_semiring(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),C)) | number_number_of(A,hAPP(int,int,times_times(int,B),C)) = hAPP(A,A,times_times(A,number_number_of(A,B)),number_number_of(A,C)) # label(fact_135_semiring__mult__number__of) # label(axiom). [clausify(474)]. 6.72/6.94 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) | number_number_of(nat,hAPP(int,int,times_times(int,A),B)) = hAPP(nat,nat,times_times(nat,number_number_of(nat,A)),number_number_of(nat,B)). [resolve(1546,a,1544,a)]. 6.72/6.94 1547 -number_semiring(A) | hAPP(A,A,plus_plus(A,B),B) = hAPP(A,A,times_times(A,B),number_number_of(A,bit0(bit1(pls)))) # label(fact_167_semiring__mult__2__right) # label(axiom). [clausify(696)]. 6.72/6.94 Derived: hAPP(nat,nat,plus_plus(nat,A),A) = hAPP(nat,nat,times_times(nat,A),number_number_of(nat,bit0(bit1(pls)))). [resolve(1547,a,1544,a)]. 6.72/6.94 1548 -number_semiring(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),C)) | number_number_of(A,hAPP(int,int,plus_plus(int,B),C)) = hAPP(A,A,plus_plus(A,number_number_of(A,B)),number_number_of(A,C)) # label(fact_136_semiring__add__number__of) # label(axiom). [clausify(732)]. 6.72/6.94 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),B)) | number_number_of(nat,hAPP(int,int,plus_plus(int,A),B)) = hAPP(nat,nat,plus_plus(nat,number_number_of(nat,A)),number_number_of(nat,B)). [resolve(1548,a,1544,a)]. 6.72/6.94 1549 -number_semiring(A) | hAPP(nat,A,power_power(A,hAPP(A,A,plus_plus(A,B),C)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(A,A,plus_plus(A,hAPP(A,A,plus_plus(A,hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,A,power_power(A,C),number_number_of(nat,bit0(bit1(pls)))))),hAPP(A,A,times_times(A,hAPP(A,A,times_times(A,number_number_of(A,bit0(bit1(pls)))),B)),C)) # label(fact_9_power2__sum) # label(axiom). [clausify(743)]. 6.72/6.94 Derived: hAPP(nat,nat,power_power(nat,hAPP(nat,nat,plus_plus(nat,A),B)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,power_power(nat,A),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,nat,power_power(nat,B),number_number_of(nat,bit0(bit1(pls)))))),hAPP(nat,nat,times_times(nat,hAPP(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),A)),B)). [resolve(1549,a,1544,a)]. 6.72/6.94 1550 -number_semiring(A) | zero_zero(A) = number_number_of(A,pls) # label(fact_223_semiring__numeral__0__eq__0) # label(axiom). [clausify(807)]. 6.72/6.94 Derived: zero_zero(nat) = number_number_of(nat,pls). [resolve(1550,a,1544,a)]. 6.72/6.94 1551 -number_semiring(A) | one_one(A) = number_number_of(A,bit1(pls)) # label(fact_157_semiring__numeral__1__eq__1) # label(axiom). [clausify(886)]. 6.72/6.94 1552 number_semiring(real) # label(arity_RealDef_Oreal___Int_Onumber__semiring) # label(axiom). [assumption]. 6.72/6.94 Derived: hAPP(nat,real,power_power(real,hAPP(real,real,plus_plus(real,A),B)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(real,real,plus_plus(real,hAPP(real,real,plus_plus(real,hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,real,power_power(real,B),number_number_of(nat,bit0(bit1(pls)))))),hAPP(real,real,times_times(real,hAPP(real,real,times_times(real,number_number_of(real,bit0(bit1(pls)))),A)),B)). [resolve(1552,a,1549,a)]. 6.82/7.06 1553 -number_semiring(A) | hAPP(A,A,plus_plus(A,B),B) = hAPP(A,A,times_times(A,number_number_of(A,bit0(bit1(pls)))),B) # label(fact_165_semiring__mult__2) # label(axiom). [clausify(944)]. 6.82/7.06 1554 number_semiring(int) # label(arity_Int_Oint___Int_Onumber__semiring) # label(axiom). [assumption]. 6.82/7.06 Derived: hAPP(nat,int,power_power(int,hAPP(int,int,plus_plus(int,A),B)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,B),number_number_of(nat,bit0(bit1(pls)))))),hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit1(pls)))),A)),B)). [resolve(1554,a,1549,a)]. 6.82/7.06 1555 ordered_ab_group_add(real) # label(arity_RealDef_Oreal___Groups_Oordered__ab__group__add) # label(axiom). [assumption]. 6.82/7.06 1556 -ordered_ab_group_add(A) | hAPP(A,A,minus_minus(A,B),C) != hAPP(A,A,minus_minus(A,D),E) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) # label(fact_790_diff__eq__diff__less) # label(axiom). [clausify(216)]. 6.82/7.06 1557 -ordered_ab_group_add(A) | hAPP(A,A,minus_minus(A,B),C) != hAPP(A,A,minus_minus(A,D),E) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),E)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) # label(fact_790_diff__eq__diff__less) # label(axiom). [clausify(216)]. 6.82/7.06 Derived: hAPP(real,real,minus_minus(real,A),B) != hAPP(real,real,minus_minus(real,C),D) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),D)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)). [resolve(1555,a,1556,a)]. 6.82/7.06 Derived: hAPP(real,real,minus_minus(real,A),B) != hAPP(real,real,minus_minus(real,C),D) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),D)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)). [resolve(1555,a,1557,a)]. 6.82/7.06 1558 ordered_ab_group_add(int) # label(arity_Int_Oint___Groups_Oordered__ab__group__add) # label(axiom). [assumption]. 6.82/7.06 Derived: hAPP(int,int,minus_minus(int,A),B) != hAPP(int,int,minus_minus(int,C),D) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),D)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)). [resolve(1558,a,1556,a)]. 6.82/7.06 Derived: hAPP(int,int,minus_minus(int,A),B) != hAPP(int,int,minus_minus(int,C),D) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),D)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)). [resolve(1558,a,1557,a)]. 6.82/7.06 1559 -ordered_ab_group_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,minus_minus(A,B),C)),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) # label(fact_811_less__iff__diff__less__0) # label(axiom). [clausify(548)]. 6.82/7.06 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,minus_minus(real,A),B)),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)). [resolve(1559,a,1555,a)]. 6.82/7.06 1560 -ordered_ab_group_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,minus_minus(A,B),C)),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) # label(fact_811_less__iff__diff__less__0) # label(axiom). [clausify(548)]. 6.82/7.06 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,minus_minus(real,A),B)),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)). [resolve(1560,a,1555,a)]. 6.82/7.06 1561 -ordered_ab_group_add(A) | hAPP(A,A,minus_minus(A,B),C) != hAPP(A,A,minus_minus(A,D),E) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),E)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) # label(fact_789_diff__eq__diff__less__eq) # label(axiom). [clausify(675)]. 6.82/7.06 Derived: hAPP(real,real,minus_minus(real,A),B) != hAPP(real,real,minus_minus(real,C),D) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),D)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)). [resolve(1561,a,1555,a)]. 7.03/7.30 Derived: hAPP(int,int,minus_minus(int,A),B) != hAPP(int,int,minus_minus(int,C),D) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),C),D)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(1561,a,1558,a)]. 7.03/7.30 1562 -ordered_ab_group_add(A) | hAPP(A,A,minus_minus(A,B),C) != hAPP(A,A,minus_minus(A,D),E) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),E)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) # label(fact_789_diff__eq__diff__less__eq) # label(axiom). [clausify(675)]. 7.03/7.30 Derived: hAPP(real,real,minus_minus(real,A),B) != hAPP(real,real,minus_minus(real,C),D) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),D)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)). [resolve(1562,a,1555,a)]. 7.03/7.30 Derived: hAPP(int,int,minus_minus(int,A),B) != hAPP(int,int,minus_minus(int,C),D) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),C),D)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)). [resolve(1562,a,1558,a)]. 7.03/7.30 1563 -ordered_ab_group_add(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,minus_minus(A,B),C)),zero_zero(A))) # label(fact_810_le__iff__diff__le__0) # label(axiom). [clausify(815)]. 7.03/7.30 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,minus_minus(real,A),B)),zero_zero(real))). [resolve(1563,a,1555,a)]. 7.03/7.30 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,minus_minus(int,A),B)),zero_zero(int))). [resolve(1563,a,1558,a)]. 7.03/7.30 1564 -ordered_ab_group_add(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,minus_minus(A,B),C)),zero_zero(A))) # label(fact_810_le__iff__diff__le__0) # label(axiom). [clausify(815)]. 7.03/7.30 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,minus_minus(real,A),B)),zero_zero(real))). [resolve(1564,a,1555,a)]. 7.03/7.30 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,minus_minus(int,A),B)),zero_zero(int))). [resolve(1564,a,1558,a)]. 7.03/7.30 1565 -ring_n68954251visors(A) | zero_zero(A) != hAPP(A,A,times_times(A,B),C) | zero_zero(A) = ti(A,C) | zero_zero(A) = ti(A,B) # label(fact_442_mult__eq__0__iff) # label(axiom). [clausify(955)]. 7.03/7.30 1566 ring_n68954251visors(real) # label(arity_RealDef_Oreal___Rings_Oring__no__zero__divisors) # label(axiom). [assumption]. 7.03/7.30 1567 ring_n68954251visors(int) # label(arity_Int_Oint___Rings_Oring__no__zero__divisors) # label(axiom). [assumption]. 7.03/7.30 1568 -ring_n68954251visors(A) | zero_zero(A) = hAPP(A,A,times_times(A,B),C) | zero_zero(A) != ti(A,C) # label(fact_442_mult__eq__0__iff) # label(axiom). [clausify(955)]. 7.03/7.30 Derived: zero_zero(real) = hAPP(real,real,times_times(real,A),B) | zero_zero(real) != ti(real,B). [resolve(1568,a,1566,a)]. 7.03/7.30 Derived: zero_zero(int) = hAPP(int,int,times_times(int,A),B) | zero_zero(int) != ti(int,B). [resolve(1568,a,1567,a)]. 7.03/7.30 1569 -ring_n68954251visors(A) | zero_zero(A) = hAPP(A,A,times_times(A,B),C) | zero_zero(A) != ti(A,B) # label(fact_442_mult__eq__0__iff) # label(axiom). [clausify(955)]. 7.03/7.30 Derived: zero_zero(real) = hAPP(real,real,times_times(real,A),B) | zero_zero(real) != ti(real,A). [resolve(1569,a,1566,a)]. 7.03/7.30 Derived: zero_zero(int) = hAPP(int,int,times_times(int,A),B) | zero_zero(int) != ti(int,A). [resolve(1569,a,1567,a)]. 7.03/7.30 1570 ring(int) # label(arity_Int_Oint___Rings_Oring) # label(axiom). [assumption]. 7.03/7.30 1571 -ring(A) | -number(A) | hAPP(A,A,minus_minus(A,hAPP(A,A,times_times(A,B),number_number_of(A,C))),hAPP(A,A,times_times(A,D),number_number_of(A,C))) = hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,B),D)),number_number_of(A,C)) # label(fact_377_left__diff__distrib__number__of) # label(axiom). [clausify(227)]. 7.12/7.33 Derived: -number(int) | hAPP(int,int,minus_minus(int,hAPP(int,int,times_times(int,A),number_number_of(int,B))),hAPP(int,int,times_times(int,C),number_number_of(int,B))) = hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,A),C)),number_number_of(int,B)). [resolve(1570,a,1571,a)]. 7.12/7.33 1572 -ring(A) | -number(A) | hAPP(A,A,minus_minus(A,hAPP(A,A,times_times(A,number_number_of(A,B)),C)),hAPP(A,A,times_times(A,number_number_of(A,B)),D)) = hAPP(A,A,times_times(A,number_number_of(A,B)),hAPP(A,A,minus_minus(A,C),D)) # label(fact_376_right__diff__distrib__number__of) # label(axiom). [clausify(560)]. 7.12/7.33 1573 -ring(A) | hAPP(A,A,minus_minus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),E)) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),hAPP(A,A,minus_minus(A,C),E))),hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,B),D)),E)) # label(fact_516_mult__diff__mult) # label(axiom). [clausify(563)]. 7.12/7.33 Derived: hAPP(int,int,minus_minus(int,hAPP(int,int,times_times(int,A),B)),hAPP(int,int,times_times(int,C),D)) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),hAPP(int,int,minus_minus(int,B),D))),hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,A),C)),D)). [resolve(1573,a,1570,a)]. 7.12/7.33 1574 ring(real) # label(arity_RealDef_Oreal___Rings_Oring) # label(axiom). [assumption]. 7.12/7.33 Derived: hAPP(real,real,minus_minus(real,hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,C),D)) = hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),hAPP(real,real,minus_minus(real,B),D))),hAPP(real,real,times_times(real,hAPP(real,real,minus_minus(real,A),C)),D)). [resolve(1574,a,1573,a)]. 7.12/7.33 1575 -ring(A) | ti(A,B) != hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,C),D)),E)),F) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),E)),F) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,D),E)),B) # label(fact_515_eq__add__iff2) # label(axiom). [clausify(777)]. 7.12/7.33 Derived: ti(int,A) != hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,B),C)),D)),E) | hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),D)),E) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,C),D)),A). [resolve(1575,a,1570,a)]. 7.12/7.33 Derived: ti(real,A) != hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,hAPP(real,real,minus_minus(real,B),C)),D)),E) | hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,B),D)),E) = hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,C),D)),A). [resolve(1575,a,1574,a)]. 7.12/7.33 1576 -ring(A) | ti(A,B) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,C),D)),E)),F) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),E)),F) != hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,D),E)),B) # label(fact_515_eq__add__iff2) # label(axiom). [clausify(777)]. 7.12/7.33 Derived: ti(int,A) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,B),C)),D)),E) | hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),D)),E) != hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,C),D)),A). [resolve(1576,a,1570,a)]. 7.12/7.33 Derived: ti(real,A) = hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,hAPP(real,real,minus_minus(real,B),C)),D)),E) | hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,B),D)),E) != hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,C),D)),A). [resolve(1576,a,1574,a)]. 7.12/7.33 1577 -ring(A) | ti(A,B) != hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,C),D)),E)),F) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,D),E)),B) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),E)),F) # label(fact_514_eq__add__iff1) # label(axiom). [clausify(1005)]. 7.12/7.33 Derived: ti(int,A) != hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,B),C)),D)),E) | hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,C),D)),A) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),D)),E). [resolve(1577,a,1570,a)]. 7.20/7.46 Derived: ti(real,A) != hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,hAPP(real,real,minus_minus(real,B),C)),D)),E) | hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,C),D)),A) = hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,B),D)),E). [resolve(1577,a,1574,a)]. 7.20/7.46 1578 -ring(A) | ti(A,B) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,hAPP(A,A,minus_minus(A,C),D)),E)),F) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,D),E)),B) != hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),E)),F) # label(fact_514_eq__add__iff1) # label(axiom). [clausify(1005)]. 7.20/7.46 Derived: ti(int,A) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,B),C)),D)),E) | hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,C),D)),A) != hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),D)),E). [resolve(1578,a,1570,a)]. 7.20/7.46 Derived: ti(real,A) = hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,hAPP(real,real,minus_minus(real,B),C)),D)),E) | hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,C),D)),A) != hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,B),D)),E). [resolve(1578,a,1574,a)]. 7.20/7.46 1579 group_add(real) # label(arity_RealDef_Oreal___Groups_Ogroup__add) # label(axiom). [assumption]. 7.20/7.46 1580 -group_add(A) | zero_zero(A) != hAPP(A,A,minus_minus(A,B),C) | ti(A,C) = ti(A,B) # label(fact_785_right__minus__eq) # label(axiom). [clausify(263)]. 7.20/7.46 1581 -group_add(A) | zero_zero(A) = hAPP(A,A,minus_minus(A,B),C) | ti(A,C) != ti(A,B) # label(fact_785_right__minus__eq) # label(axiom). [clausify(263)]. 7.20/7.46 1582 -group_add(A) | ti(A,B) = hAPP(A,A,plus_plus(A,hAPP(A,A,minus_minus(A,B),C)),C) # label(fact_796_diff__add__cancel) # label(axiom). [clausify(272)]. 7.20/7.46 1583 -group_add(A) | ti(A,B) = hAPP(A,A,minus_minus(A,hAPP(A,A,plus_plus(A,B),C)),C) # label(fact_795_add__diff__cancel) # label(axiom). [clausify(303)]. 7.20/7.46 1584 -group_add(A) | minus_minus(A,ti(A,B)) = minus_minus(A,B) # label(tsy_c_Groups_Ominus__class_Ominus_0_arg1) # label(axiom). [clausify(456)]. 7.20/7.46 1585 -group_add(A) | ti(A,B) = hAPP(A,A,minus_minus(A,B),zero_zero(A)) # label(fact_788_diff__0__right) # label(axiom). [clausify(491)]. 7.20/7.46 1586 -group_add(A) | zero_zero(A) = hAPP(A,A,minus_minus(A,B),B) # label(fact_787_diff__self) # label(axiom). [clausify(794)]. 7.20/7.46 Derived: zero_zero(real) != hAPP(real,real,minus_minus(real,A),B) | ti(real,B) = ti(real,A). [resolve(1579,a,1580,a)]. 7.20/7.46 Derived: zero_zero(real) = hAPP(real,real,minus_minus(real,A),B) | ti(real,B) != ti(real,A). [resolve(1579,a,1581,a)]. 7.20/7.46 Derived: ti(real,A) = hAPP(real,real,plus_plus(real,hAPP(real,real,minus_minus(real,A),B)),B). [resolve(1579,a,1582,a)]. 7.20/7.46 Derived: ti(real,A) = hAPP(real,real,minus_minus(real,hAPP(real,real,plus_plus(real,A),B)),B). [resolve(1579,a,1583,a)]. 7.20/7.46 Derived: minus_minus(real,ti(real,A)) = minus_minus(real,A). [resolve(1579,a,1584,a)]. 7.20/7.46 Derived: ti(real,A) = hAPP(real,real,minus_minus(real,A),zero_zero(real)). [resolve(1579,a,1585,a)]. 7.20/7.46 Derived: zero_zero(real) = hAPP(real,real,minus_minus(real,A),A). [resolve(1579,a,1586,a)]. 7.20/7.46 1587 group_add(int) # label(arity_Int_Oint___Groups_Ogroup__add) # label(axiom). [assumption]. 7.20/7.46 Derived: zero_zero(int) != hAPP(int,int,minus_minus(int,A),B) | ti(int,B) = ti(int,A). [resolve(1587,a,1580,a)]. 7.20/7.46 Derived: zero_zero(int) = hAPP(int,int,minus_minus(int,A),B) | ti(int,B) != ti(int,A). [resolve(1587,a,1581,a)]. 7.20/7.46 Derived: ti(int,A) = hAPP(int,int,plus_plus(int,hAPP(int,int,minus_minus(int,A),B)),B). [resolve(1587,a,1582,a)]. 7.20/7.46 Derived: ti(int,A) = hAPP(int,int,minus_minus(int,hAPP(int,int,plus_plus(int,A),B)),B). [resolve(1587,a,1583,a)]. 7.20/7.46 Derived: minus_minus(int,ti(int,A)) = minus_minus(int,A). [resolve(1587,a,1584,a)]. 7.20/7.46 Derived: ti(int,A) = hAPP(int,int,minus_minus(int,A),zero_zero(int)). [resolve(1587,a,1585,a)]. 7.20/7.46 Derived: zero_zero(int) = hAPP(int,int,minus_minus(int,A),A). [resolve(1587,a,1586,a)]. 7.20/7.46 1588 -group_add(A) | minus_minus(A,B) = ti(fun(A,A),minus_minus(A,B)) # label(tsy_c_Groups_Ominus__class_Ominus_0_res) # label(axiom). [clausify(1020)]. 7.73/7.96 Derived: minus_minus(real,A) = ti(fun(real,real),minus_minus(real,A)). [resolve(1588,a,1579,a)]. 7.73/7.96 Derived: minus_minus(int,A) = ti(fun(int,int),minus_minus(int,A)). [resolve(1588,a,1587,a)]. 7.73/7.96 1589 comm_semiring(int) # label(arity_Int_Oint___Rings_Ocomm__semiring) # label(axiom). [assumption]. 7.73/7.96 1590 -comm_semiring(A) | hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,D),C)) = hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,B),D)),C) # label(fact_448_comm__semiring__class_Odistrib) # label(axiom). [clausify(288)]. 7.73/7.96 1591 comm_semiring(real) # label(arity_RealDef_Oreal___Rings_Ocomm__semiring) # label(axiom). [assumption]. 7.73/7.96 1592 comm_semiring(nat) # label(arity_Nat_Onat___Rings_Ocomm__semiring) # label(axiom). [assumption]. 7.73/7.96 1593 comm_ring_1(int) # label(arity_Int_Oint___Rings_Ocomm__ring__1) # label(axiom). [assumption]. 7.73/7.96 1594 -comm_ring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(A,A,minus_minus(A,C),D))) # label(fact_461_dvd__diff) # label(axiom). [clausify(360)]. 7.73/7.96 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(int,int,minus_minus(int,B),C))). [resolve(1593,a,1594,a)]. 7.73/7.96 1595 comm_ring_1(real) # label(arity_RealDef_Oreal___Rings_Ocomm__ring__1) # label(axiom). [assumption]. 7.73/7.96 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(real,real,minus_minus(real,B),C))). [resolve(1595,a,1594,a)]. 7.73/7.96 1596 linord1278240602ring_1(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semiring__1) # label(axiom). [assumption]. 7.73/7.96 1597 -linord1278240602ring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),D),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),E)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),F)) | one_one(A) != hAPP(A,A,plus_plus(A,E),F) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,E),B)),hAPP(A,A,times_times(A,F),D))),C)) # label(fact_548_convex__bound__le) # label(axiom). [clausify(399)]. 7.73/7.96 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),D)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),E)) | one_one(real) != hAPP(real,real,plus_plus(real,D),E) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,D),A)),hAPP(real,real,times_times(real,E),C))),B)). [resolve(1596,a,1597,a)]. 7.73/7.96 1598 linord1278240602ring_1(int) # label(arity_Int_Oint___Rings_Olinordered__semiring__1) # label(axiom). [assumption]. 7.73/7.96 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),C),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),D)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),E)) | one_one(int) != hAPP(int,int,plus_plus(int,D),E) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,D),A)),hAPP(int,int,times_times(int,E),C))),B)). [resolve(1598,a,1597,a)]. 7.73/7.96 1599 -linord893533164strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),D)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,times_times(A,D),B)),hAPP(A,A,times_times(A,D),C))) # label(fact_490_comm__mult__strict__left__mono) # label(axiom). [clausify(938)]. 8.10/8.33 1600 linord893533164strict(nat) # label(arity_Nat_Onat___Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 8.10/8.33 1601 linord893533164strict(int) # label(arity_Int_Oint___Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 8.10/8.33 1602 linord893533164strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 8.10/8.33 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A),B)) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,times_times(nat,C),A)),hAPP(nat,nat,times_times(nat,C),B))). [resolve(1599,a,1600,a)]. 8.10/8.33 1603 ring_div(int) # label(arity_Int_Oint___Divides_Oring__div) # label(axiom). [assumption]. 8.10/8.33 1604 -ring_div(A) | hAPP(A,A,div_mod(A,B),C) != hAPP(A,A,div_mod(A,D),C) | hAPP(A,A,div_mod(A,E),C) != hAPP(A,A,div_mod(A,F),C) | hAPP(A,A,div_mod(A,hAPP(A,A,minus_minus(A,B),E)),C) = hAPP(A,A,div_mod(A,hAPP(A,A,minus_minus(A,D),F)),C) # label(fact_947_mod__diff__cong) # label(axiom). [clausify(427)]. 8.10/8.33 Derived: hAPP(int,int,div_mod(int,A),B) != hAPP(int,int,div_mod(int,C),B) | hAPP(int,int,div_mod(int,D),B) != hAPP(int,int,div_mod(int,E),B) | hAPP(int,int,div_mod(int,hAPP(int,int,minus_minus(int,A),D)),B) = hAPP(int,int,div_mod(int,hAPP(int,int,minus_minus(int,C),E)),B). [resolve(1603,a,1604,a)]. 8.10/8.33 1605 -ring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,minus_minus(A,hAPP(A,A,div_mod(A,B),C)),hAPP(A,A,div_mod(A,D),C))),C) = hAPP(A,A,div_mod(A,hAPP(A,A,minus_minus(A,B),D)),C) # label(fact_948_mod__diff__eq) # label(axiom). [clausify(810)]. 8.10/8.33 Derived: hAPP(int,int,div_mod(int,hAPP(int,int,minus_minus(int,hAPP(int,int,div_mod(int,A),B)),hAPP(int,int,div_mod(int,C),B))),B) = hAPP(int,int,div_mod(int,hAPP(int,int,minus_minus(int,A),C)),B). [resolve(1605,a,1603,a)]. 8.10/8.33 1606 -ring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,minus_minus(A,hAPP(A,A,div_mod(A,B),C)),D)),C) = hAPP(A,A,div_mod(A,hAPP(A,A,minus_minus(A,B),D)),C) # label(fact_949_mod__diff__left__eq) # label(axiom). [clausify(811)]. 8.10/8.33 Derived: hAPP(int,int,div_mod(int,hAPP(int,int,minus_minus(int,hAPP(int,int,div_mod(int,A),B)),C)),B) = hAPP(int,int,div_mod(int,hAPP(int,int,minus_minus(int,A),C)),B). [resolve(1606,a,1603,a)]. 8.10/8.33 1607 -ring_div(A) | hAPP(A,A,div_mod(A,hAPP(A,A,minus_minus(A,B),hAPP(A,A,div_mod(A,C),D))),D) = hAPP(A,A,div_mod(A,hAPP(A,A,minus_minus(A,B),C)),D) # label(fact_950_mod__diff__right__eq) # label(axiom). [clausify(827)]. 8.10/8.33 Derived: hAPP(int,int,div_mod(int,hAPP(int,int,minus_minus(int,A),hAPP(int,int,div_mod(int,B),C))),C) = hAPP(int,int,div_mod(int,hAPP(int,int,minus_minus(int,A),B)),C). [resolve(1607,a,1603,a)]. 8.10/8.33 1608 -comm_monoid_mult(A) | ti(A,B) = hAPP(A,A,times_times(A,one_one(A)),B) # label(fact_792_mult__1) # label(axiom). [clausify(688)]. 8.10/8.33 1609 comm_monoid_mult(int) # label(arity_Int_Oint___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 8.10/8.33 1610 comm_monoid_mult(nat) # label(arity_Nat_Onat___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 8.10/8.33 1611 comm_monoid_mult(real) # label(arity_RealDef_Oreal___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 8.10/8.33 1612 -comm_monoid_mult(A) | hAPP(nat,A,power_power(A,hAPP(A,A,times_times(A,B),C)),D) = hAPP(A,A,times_times(A,hAPP(nat,A,power_power(A,B),D)),hAPP(nat,A,power_power(A,C),D)) # label(fact_260_power__mult__distrib) # label(axiom). [clausify(747)]. 8.10/8.33 1613 -comm_monoid_mult(A) | ti(A,B) = hAPP(A,A,times_times(A,B),one_one(A)) # label(fact_794_mult_Ocomm__neutral) # label(axiom). [clausify(963)]. 8.10/8.33 1614 -ab_group_add(A) | zero_zero(A) != hAPP(A,A,minus_minus(A,B),C) | ti(A,C) = ti(A,B) # label(fact_786_eq__iff__diff__eq__0) # label(axiom). [clausify(824)]. 8.10/8.33 1615 ab_group_add(real) # label(arity_RealDef_Oreal___Groups_Oab__group__add) # label(axiom). [assumption]. 8.10/8.33 1616 -ab_group_add(A) | zero_zero(A) = hAPP(A,A,minus_minus(A,B),C) | ti(A,C) != ti(A,B) # label(fact_786_eq__iff__diff__eq__0) # label(axiom). [clausify(824)]. 8.35/8.59 1617 ab_group_add(int) # label(arity_Int_Oint___Groups_Oab__group__add) # label(axiom). [assumption]. 8.35/8.59 1618 -ab_group_add(A) | hAPP(A,A,minus_minus(A,hAPP(A,A,plus_plus(A,B),C)),hAPP(A,A,plus_plus(A,D),E)) = hAPP(A,A,plus_plus(A,hAPP(A,A,minus_minus(A,B),D)),hAPP(A,A,minus_minus(A,C),E)) # label(fact_450_add__diff__add) # label(axiom). [clausify(923)]. 8.35/8.59 Derived: hAPP(real,real,minus_minus(real,hAPP(real,real,plus_plus(real,A),B)),hAPP(real,real,plus_plus(real,C),D)) = hAPP(real,real,plus_plus(real,hAPP(real,real,minus_minus(real,A),C)),hAPP(real,real,minus_minus(real,B),D)). [resolve(1618,a,1615,a)]. 8.35/8.59 Derived: hAPP(int,int,minus_minus(int,hAPP(int,int,plus_plus(int,A),B)),hAPP(int,int,plus_plus(int,C),D)) = hAPP(int,int,plus_plus(int,hAPP(int,int,minus_minus(int,A),C)),hAPP(int,int,minus_minus(int,B),D)). [resolve(1618,a,1617,a)]. 8.35/8.59 1619 -ab_group_add(A) | hAPP(A,A,minus_minus(A,B),C) != hAPP(A,A,minus_minus(A,D),E) | ti(A,B) != ti(A,C) | ti(A,E) = ti(A,D) # label(fact_765_diff__eq__diff__eq) # label(axiom). [clausify(997)]. 8.35/8.59 Derived: hAPP(real,real,minus_minus(real,A),B) != hAPP(real,real,minus_minus(real,C),D) | ti(real,A) != ti(real,B) | ti(real,D) = ti(real,C). [resolve(1619,a,1615,a)]. 8.35/8.59 Derived: hAPP(int,int,minus_minus(int,A),B) != hAPP(int,int,minus_minus(int,C),D) | ti(int,A) != ti(int,B) | ti(int,D) = ti(int,C). [resolve(1619,a,1617,a)]. 8.35/8.59 1620 -ab_group_add(A) | hAPP(A,A,minus_minus(A,B),C) != hAPP(A,A,minus_minus(A,D),E) | ti(A,B) = ti(A,C) | ti(A,E) != ti(A,D) # label(fact_765_diff__eq__diff__eq) # label(axiom). [clausify(997)]. 8.35/8.59 Derived: hAPP(real,real,minus_minus(real,A),B) != hAPP(real,real,minus_minus(real,C),D) | ti(real,A) = ti(real,B) | ti(real,D) != ti(real,C). [resolve(1620,a,1615,a)]. 8.35/8.59 Derived: hAPP(int,int,minus_minus(int,A),B) != hAPP(int,int,minus_minus(int,C),D) | ti(int,A) = ti(int,B) | ti(int,D) != ti(int,C). [resolve(1620,a,1617,a)]. 8.35/8.59 1621 -comm_monoid_add(A) | ti(A,B) = hAPP(A,A,plus_plus(A,zero_zero(A)),B) # label(fact_767_add__0) # label(axiom). [clausify(750)]. 8.41/8.59 1622 comm_monoid_add(int) # label(arity_Int_Oint___Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 8.41/8.59 Derived: ti(int,A) = hAPP(int,int,plus_plus(int,zero_zero(int)),A). [resolve(1621,a,1622,a)]. 8.41/8.59 1623 comm_monoid_add(nat) # label(arity_Nat_Onat___Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 8.41/8.59 1624 comm_monoid_add(real) # label(arity_RealDef_Oreal___Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 8.41/8.59 1625 -comm_monoid_add(A) | ti(A,B) = hAPP(A,A,plus_plus(A,B),zero_zero(A)) # label(fact_770_add_Ocomm__neutral) # label(axiom). [clausify(936)]. 8.41/8.59 Derived: ti(int,A) = hAPP(int,int,plus_plus(int,A),zero_zero(int)). [resolve(1625,a,1622,a)]. 8.41/8.59 1626 linord626643107strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semiring__1__strict) # label(axiom). [assumption]. 8.41/8.59 1627 -linord626643107strict(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),D),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),E)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),F)) | one_one(A) != hAPP(A,A,plus_plus(A,E),F) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,E),B)),hAPP(A,A,times_times(A,F),D))),C)) # label(fact_422_convex__bound__lt) # label(axiom). [clausify(645)]. 8.41/8.59 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),D)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),E)) | one_one(real) != hAPP(real,real,plus_plus(real,D),E) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,D),A)),hAPP(real,real,times_times(real,E),C))),B)). [resolve(1626,a,1627,a)]. 8.41/8.59 1628 linord626643107strict(int) # label(arity_Int_Oint___Rings_Olinordered__semiring__1__strict) # label(axiom). [assumption]. 9.15/9.37 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),B)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),D)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),E)) | one_one(int) != hAPP(int,int,plus_plus(int,D),E) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,D),A)),hAPP(int,int,times_times(int,E),C))),B)). [resolve(1628,a,1627,a)]. 9.15/9.37 9.15/9.37 ============================== end predicate elimination ============= 9.15/9.37 9.15/9.37 Auto_denials: (non-Horn, no changes). 9.15/9.37 9.15/9.37 Term ordering decisions: 9.15/9.37 Function symbol KB weights: int=1. nat=1. bool=1. real=1. pls=1. min=1. zprime=1. m=1. s=1. t=1. s1=1. twoSqu658283162sum2sq=1. fconj=1. c1=1. c2=1. c3=1. c4=1. c5=1. c6=1. c7=1. c8=1. fun=1. times_times=1. plus_plus=1. ti=1. number_number_of=1. power_power=1. minus_minus=1. div_mod=1. zcong=1. multInv=1. member=1. standardRes=1. wset=1. inv=1. legendre=1. collect=1. product_prod=1. f1=1. f5=1. f7=1. f9=1. f11=1. f13=1. f15=1. f16=1. f20=1. f23=1. f35=1. f37=1. zero_zero=1. ord_less=1. ord_less_eq=1. one_one=1. dvd_dvd=1. bit1=1. bit0=1. quadRes=1. twoSqu1929807760sum2sq=1. d22set=1. zfact=1. sr=1. undefined=1. f10=1. f12=1. f17=1. f18=1. f2=1. f3=1. f4=1. f14=1. f19=1. f21=1. f24=1. f25=1. f26=1. f27=1. f28=1. f29=1. f30=1. f31=1. f32=1. f33=1. f34=1. f38=1. f39=1. hAPP=1. product_Pair=1. combb=1. combc=1. combs=1. 9.15/9.37 9.15/9.37 ============================== end of process initial clauses ======== 9.15/9.37 9.15/9.37 ============================== CLAUSES FOR SEARCH ==================== 9.15/9.37 9.15/9.37 ============================== end of clauses for search ============= 9.15/9.37 9.15/9.37 ============================== SEARCH ================================ 9.15/9.37 9.15/9.37 % Starting search at 7.17 seconds. 9.15/9.37 9.15/9.37 ============================== PROOF ================================= 9.15/9.37 % SZS status Theorem 9.15/9.37 % SZS output start Refutation 9.15/9.37 9.15/9.37 % Proof 1 at 7.57 (+ 0.01) seconds. 9.15/9.37 % Length of proof is 51. 9.15/9.37 % Level of proof is 7. 9.15/9.37 % Maximum clause weight is 90.000. 9.15/9.37 % Given clauses 415. 9.15/9.37 9.15/9.37 228 t = one_one(int) -> (exists X exists Y hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,X),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,Y),number_number_of(nat,bit0(bit1(pls))))) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int))) # label(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]. 9.15/9.37 241 (all X_a (number_ring(X_a) -> one_one(X_a) = number_number_of(X_a,bit1(pls)))) # label(fact_159_semiring__norm_I110_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.37 366 (all T all A ti(T,ti(T,A)) = ti(T,A)) # label(help_ti_idem) # label(axiom) # label(non_clause). [assumption]. 9.15/9.37 383 (all X_c all X_a all B_1_1 all B_2_1 ti(X_c,hAPP(X_a,X_c,B_1_1,B_2_1)) = hAPP(X_a,X_c,B_1_1,B_2_1)) # label(tsy_c_hAPP_res) # label(axiom) # label(non_clause). [assumption]. 9.15/9.37 390 (all Z1 all Z2 all Z3 hAPP(int,int,plus_plus(int,Z1),hAPP(int,int,plus_plus(int,Z2),Z3)) = hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,Z1),Z2)),Z3)) # label(fact_89_zadd__assoc) # label(axiom) # label(non_clause). [assumption]. 9.15/9.37 433 (all K_1 number_number_of(int,K_1) = ti(int,K_1)) # label(fact_88_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 9.15/9.37 544 (all X_a all X_c all B_1_1 all B_2_1 hAPP(X_a,X_c,B_1_1,ti(X_a,B_2_1)) = hAPP(X_a,X_c,B_1_1,B_2_1)) # label(tsy_c_hAPP_arg2) # label(axiom) # label(non_clause). [assumption]. 9.15/9.37 576 (all V all W hAPP(int,int,plus_plus(int,number_number_of(int,V)),number_number_of(int,W)) = number_number_of(int,hAPP(int,int,plus_plus(int,V),W))) # label(fact_134_plus__numeral__code_I9_J) # label(axiom) # label(non_clause). [assumption]. 9.15/9.37 657 (all Z_2 all W_1 (ti(int,W_1) != ti(int,Z_2) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_2),W_1)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_2),W_1)))) # label(fact_22_zless__le) # label(axiom) # label(non_clause). [assumption]. 9.15/9.37 699 (all Z_1 ti(int,Z_1) = hAPP(int,int,plus_plus(int,Z_1),zero_zero(int))) # label(fact_207_zadd__0__right) # label(axiom) # label(non_clause). [assumption]. 9.15/9.37 809 (all K_1 bit1(K_1) = hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,one_one(int)),K_1)),K_1)) # label(fact_153_Bit1__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.37 826 (all K_1 hAPP(int,int,plus_plus(int,K_1),K_1) = bit0(K_1)) # label(fact_128_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 9.15/9.37 874 hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),t)) -> (exists X exists Y hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int)) = hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,X),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,Y),number_number_of(nat,bit0(bit1(pls)))))) # 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]. 9.15/9.37 1043 -(exists X exists Y hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,X),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,Y),number_number_of(nat,bit0(bit1(pls))))) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int))) # label(conj_0) # label(negated_conjecture) # label(non_clause). [assumption]. 9.15/9.37 1174 -number_ring(A) | one_one(A) = number_number_of(A,bit1(pls)) # label(fact_159_semiring__norm_I110_J) # label(axiom). [clausify(241)]. 9.15/9.37 1215 number_ring(int) # label(arity_Int_Oint___Int_Onumber__ring) # label(axiom). [assumption]. 9.15/9.37 1876 one_one(int) != t | hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,c1),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,c2),number_number_of(nat,bit0(bit1(pls))))) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int)) # label(fact_1__096t_A_061_A1_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06) # label(axiom). [clausify(228)]. 9.15/9.37 1974 ti(int,t) = t # label(tsy_v_t_____res) # label(axiom). [assumption]. 9.15/9.37 2036 zero_zero(int) = pls # label(fact_204_Pls__def) # label(axiom). [assumption]. 9.15/9.37 2037 pls = zero_zero(int). [copy(2036),flip(a)]. 9.15/9.37 2045 ti(A,ti(A,B)) = ti(A,B) # label(help_ti_idem) # label(axiom). [clausify(366)]. 9.15/9.37 2072 ti(A,hAPP(B,A,C,D)) = hAPP(B,A,C,D) # label(tsy_c_hAPP_res) # label(axiom). [clausify(383)]. 9.15/9.37 2077 hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),t)) # label(fact_0_tpos) # label(axiom). [assumption]. 9.15/9.37 2083 hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,A),B)),C) = hAPP(int,int,plus_plus(int,A),hAPP(int,int,plus_plus(int,B),C)) # label(fact_89_zadd__assoc) # label(axiom). [clausify(390)]. 9.15/9.37 2149 number_number_of(int,A) = ti(int,A) # label(fact_88_number__of__is__id) # label(axiom). [clausify(433)]. 9.15/9.37 2247 zero_zero(int) = number_number_of(int,pls) # label(fact_227_zero__is__num__zero) # label(axiom). [assumption]. 9.15/9.37 2248 ti(int,zero_zero(int)) = zero_zero(int). [copy(2247),rewrite([2037(4),2149(6)]),flip(a)]. 9.15/9.37 2254 hAPP(A,B,C,ti(A,D)) = hAPP(A,B,C,D) # label(tsy_c_hAPP_arg2) # label(axiom). [clausify(544)]. 9.15/9.37 2298 number_number_of(int,hAPP(int,int,plus_plus(int,A),B)) = hAPP(int,int,plus_plus(int,number_number_of(int,A)),number_number_of(int,B)) # label(fact_134_plus__numeral__code_I9_J) # label(axiom). [clausify(576)]. 9.15/9.37 2299 hAPP(int,int,plus_plus(int,ti(int,A)),B) = hAPP(int,int,plus_plus(int,A),B). [copy(2298),rewrite([2149(7),2072(7),2149(10),2149(13),2254(14)]),flip(a)]. 9.15/9.37 2396 ti(int,A) = ti(int,B) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),A)) # label(fact_22_zless__le) # label(axiom). [clausify(657)]. 9.15/9.37 2451 ti(int,A) = hAPP(int,int,plus_plus(int,A),zero_zero(int)) # label(fact_207_zadd__0__right) # label(axiom). [clausify(699)]. 9.15/9.37 2452 hAPP(int,int,plus_plus(int,A),zero_zero(int)) = ti(int,A). [copy(2451),flip(a)]. 9.15/9.37 2518 hAPP(nat,nat,plus_plus(nat,one_one(nat)),one_one(nat)) = number_number_of(nat,bit0(bit1(pls))) # label(fact_37_nat__1__add__1) # label(axiom). [assumption]. 9.15/9.37 2519 hAPP(nat,nat,plus_plus(nat,one_one(nat)),one_one(nat)) = number_number_of(nat,bit0(bit1(zero_zero(int)))). [copy(2518),rewrite([2037(11)])]. 9.15/9.37 2649 bit1(A) = hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,one_one(int)),A)),A) # label(fact_153_Bit1__def) # label(axiom). [clausify(809)]. 9.15/9.37 2650 bit1(A) = hAPP(int,int,plus_plus(int,one_one(int)),hAPP(int,int,plus_plus(int,A),A)). [copy(2649),rewrite([2083(13)])]. 9.15/9.37 2658 hAPP(int,int,plus_plus(int,A),A) = bit0(A) # label(fact_128_Bit0__def) # label(axiom). [clausify(826)]. 9.15/9.37 2659 bit0(A) = hAPP(int,int,plus_plus(int,A),A). [copy(2658),flip(a)]. 9.15/9.37 2721 -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),t)) | hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,c7),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,c8),number_number_of(nat,bit0(bit1(pls))))) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int)) # label(fact_2__0961_A_060_At_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_06) # label(axiom). [clausify(874)]. 9.15/9.37 2722 -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),t)) | hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,c7),number_number_of(nat,hAPP(int,int,plus_plus(int,one_one(int)),one_one(int))))),hAPP(nat,int,power_power(int,c8),number_number_of(nat,hAPP(int,int,plus_plus(int,one_one(int)),one_one(int))))) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,one_one(int)),hAPP(int,int,plus_plus(int,one_one(int)),hAPP(int,int,plus_plus(int,one_one(int)),one_one(int))))),m)),one_one(int)). [copy(2721),rewrite([2037(24),2650(26),2452(38),2248(33),2452(32),2659(28),2254(36),2299(34),2037(42),2650(44),2452(56),2248(51),2452(50),2659(46),2254(54),2299(52),2037(61),2650(63),2452(75),2248(70),2452(69),2659(65),2254(73),2299(71),2659(70),2083(83),2149(84),2072(84)])]. 9.15/9.37 2925 hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,int,power_power(int,B),number_number_of(nat,bit0(bit1(pls))))) != hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls))))),m)),one_one(int)) # label(conj_0) # label(negated_conjecture). [clausify(1043)]. 9.15/9.37 2926 hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,A),number_number_of(nat,hAPP(int,int,plus_plus(int,one_one(int)),one_one(int))))),hAPP(nat,int,power_power(int,B),number_number_of(nat,hAPP(int,int,plus_plus(int,one_one(int)),one_one(int))))) != hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,one_one(int)),hAPP(int,int,plus_plus(int,one_one(int)),hAPP(int,int,plus_plus(int,one_one(int)),one_one(int))))),m)),one_one(int)). [copy(2925),rewrite([2037(9),2650(11),2452(23),2248(18),2452(17),2659(13),2254(21),2299(19),2037(26),2650(28),2452(40),2248(35),2452(34),2659(30),2254(38),2299(36),2037(45),2650(47),2452(59),2248(54),2452(53),2659(49),2254(57),2299(55),2659(54),2083(67),2149(68),2072(68)])]. 9.15/9.37 3299 one_one(int) = number_number_of(int,bit1(pls)). [resolve(1215,a,1174,a)]. 9.15/9.37 3300 ti(int,one_one(int)) = one_one(int). [copy(3299),rewrite([2037(4),2650(6),2452(18),2248(13),2452(12),2149(8),2045(8)]),flip(a)]. 9.15/9.37 4041 one_one(int) != t. [back_rewrite(1876),rewrite([2037(14),2650(16),2452(28),2248(23),2452(22),3300(17),2659(16),2037(32),2650(34),2452(46),2248(41),2452(40),3300(35),2659(34),2037(51),2650(53),2452(65),2248(60),2452(59),3300(54),2659(53),2659(60),2083(73),2149(74),2072(74)]),unit_del(b,2926)]. 9.15/9.37 4107 number_number_of(nat,hAPP(int,int,plus_plus(int,one_one(int)),one_one(int))) = hAPP(nat,nat,plus_plus(nat,one_one(nat)),one_one(nat)). [back_rewrite(2519),rewrite([2650(13),2452(25),2248(20),2452(19),3300(14),2659(13)]),flip(a)]. 9.15/9.37 4197 -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),t)) | hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,one_one(int)),hAPP(int,int,plus_plus(int,one_one(int)),hAPP(int,int,plus_plus(int,one_one(int)),one_one(int))))),m)),one_one(int)) = hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,c7),hAPP(nat,nat,plus_plus(nat,one_one(nat)),one_one(nat)))),hAPP(nat,int,power_power(int,c8),hAPP(nat,nat,plus_plus(nat,one_one(nat)),one_one(nat)))). [back_unit_del(2722),rewrite([4107(33),4107(49)]),flip(b)]. 9.15/9.37 4391 hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,A),hAPP(nat,nat,plus_plus(nat,one_one(nat)),one_one(nat)))),hAPP(nat,int,power_power(int,B),hAPP(nat,nat,plus_plus(nat,one_one(nat)),one_one(nat)))) != hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,one_one(int)),hAPP(int,int,plus_plus(int,one_one(int)),hAPP(int,int,plus_plus(int,one_one(int)),one_one(int))))),m)),one_one(int)). [back_rewrite(2926),rewrite([4107(18),4107(33)])]. 9.15/9.37 4420 -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),t)). [back_unit_del(4197),unit_del(b(flip),4391)]. 9.15/9.37 6275 $F. [resolve(2396,b,2077,a),rewrite([1974(3),3300(5)]),flip(a),unit_del(a,4041),unit_del(b,4420)]. 9.15/9.37 9.15/9.37 % SZS output end Refutation 9.15/9.37 ============================== end of proof ========================== 9.15/9.37 9.15/9.37 ============================== STATISTICS ============================ 9.15/9.37 9.15/9.37 Given=415. Generated=8375. Kept=3869. proofs=1. 9.15/9.37 Usable=415. Sos=3049. Demods=383. Limbo=0, Disabled=2908. Hints=0. 9.15/9.37 Megabytes=22.25. 9.15/9.37 User_CPU=7.57, System_CPU=0.01, Wall_clock=8. 9.15/9.37 9.15/9.37 ============================== end of statistics ===================== 9.15/9.37 9.15/9.37 ============================== end of search ========================= 9.15/9.37 9.15/9.37 THEOREM PROVED 9.15/9.37 % SZS status Theorem 9.15/9.37 9.15/9.37 Exiting with 1 proof. 9.15/9.37 9.15/9.37 Process 3775 exit (max_proofs) Tue Aug 9 04:29:44 2022 9.15/9.37 Prover9 interrupted 9.15/9.38 EOF