0.00/0.04 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.00/0.04 % Command : tptp2X_and_run_prover9 %d %s 0.03/0.23 % Computer : n009.star.cs.uiowa.edu 0.03/0.23 % Model : x86_64 x86_64 0.03/0.23 % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz 0.03/0.23 % Memory : 32218.625MB 0.03/0.23 % OS : Linux 3.10.0-693.2.2.el7.x86_64 0.03/0.23 % CPULimit : 300 0.03/0.23 % DateTime : Sat Jul 14 04:49:26 CDT 2018 0.03/0.24 % CPUTime : 1.53/1.77 ============================== Prover9 =============================== 1.53/1.77 Prover9 (32) version 2009-11A, November 2009. 1.53/1.77 Process 37457 was started by sandbox2 on n009.star.cs.uiowa.edu, 1.53/1.77 Sat Jul 14 04:49:28 2018 1.53/1.77 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 300 -f /tmp/Prover9_37425_n009.star.cs.uiowa.edu". 1.53/1.77 ============================== end of head =========================== 1.53/1.77 1.53/1.77 ============================== INPUT ================================= 1.53/1.77 1.53/1.77 % Reading from file /tmp/Prover9_37425_n009.star.cs.uiowa.edu 1.53/1.77 1.53/1.77 set(prolog_style_variables). 1.53/1.77 set(auto2). 1.53/1.77 % set(auto2) -> set(auto). 1.53/1.77 % set(auto) -> set(auto_inference). 1.53/1.77 % set(auto) -> set(auto_setup). 1.53/1.77 % set(auto_setup) -> set(predicate_elim). 1.53/1.77 % set(auto_setup) -> assign(eq_defs, unfold). 1.53/1.77 % set(auto) -> set(auto_limits). 1.53/1.77 % set(auto_limits) -> assign(max_weight, "100.000"). 1.53/1.77 % set(auto_limits) -> assign(sos_limit, 20000). 1.53/1.77 % set(auto) -> set(auto_denials). 1.53/1.77 % set(auto) -> set(auto_process). 1.53/1.77 % set(auto2) -> assign(new_constants, 1). 1.53/1.77 % set(auto2) -> assign(fold_denial_max, 3). 1.53/1.77 % set(auto2) -> assign(max_weight, "200.000"). 1.53/1.77 % set(auto2) -> assign(max_hours, 1). 1.53/1.77 % assign(max_hours, 1) -> assign(max_seconds, 3600). 1.53/1.77 % set(auto2) -> assign(max_seconds, 0). 1.53/1.77 % set(auto2) -> assign(max_minutes, 5). 1.53/1.77 % assign(max_minutes, 5) -> assign(max_seconds, 300). 1.53/1.77 % set(auto2) -> set(sort_initial_sos). 1.53/1.77 % set(auto2) -> assign(sos_limit, -1). 1.53/1.77 % set(auto2) -> assign(lrs_ticks, 3000). 1.53/1.77 % set(auto2) -> assign(max_megs, 400). 1.53/1.77 % set(auto2) -> assign(stats, some). 1.53/1.77 % set(auto2) -> clear(echo_input). 1.53/1.77 % set(auto2) -> set(quiet). 1.53/1.77 % set(auto2) -> clear(print_initial_clauses). 1.53/1.77 % set(auto2) -> clear(print_given). 1.53/1.77 assign(lrs_ticks,-1). 1.53/1.77 assign(sos_limit,10000). 1.53/1.77 assign(order,kbo). 1.53/1.77 set(lex_order_vars). 1.53/1.77 clear(print_given). 1.53/1.77 1.53/1.77 % formulas(sos). % not echoed (1257 formulas) 1.53/1.77 1.53/1.77 ============================== end of input ========================== 1.53/1.77 1.53/1.77 % From the command line: assign(max_seconds, 300). 1.53/1.77 1.53/1.77 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 1.53/1.77 1.53/1.77 % Formulas that are not ordinary clauses: 1.53/1.77 1 (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_174_odd__less__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 2 (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_768_add__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 3 (all X_1 all Y all N_1 (N_1 != zero_zero(nat) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,power_power(nat,X_1),N_1)),Y)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y))))) # label(fact_524_divides__exp2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 4 (all X_a (plus(X_a) & linorder(X_a) & dvd(X_a) -> (all D all Sa exists Z all X_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),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_2),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_2),Sa)))))))) # label(fact_866_minf_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 5 (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_714_mult__le__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 6 (all B_1_1 all B_2_1 multInv(B_1_1,ti(int,B_2_1)) = multInv(B_1_1,B_2_1)) # label(tsy_c_Int2_OMultInv_arg2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 7 (all X_a (number_ring(X_a) & ring_char_0(X_a) -> (all X all Y_1 (number_number_of(X_a,Y_1) = number_number_of(X_a,X) <-> ti(int,X) = ti(int,Y_1))))) # label(fact_88_eq__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 8 (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),A_3),B_2)) | ti(X_a,C_1) = zero_zero(X_a) <-> 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))))))) # label(fact_405_dvd__mult__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 9 (all B_1_1 all X_a (dvd(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_2_res) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 10 (all K_1 all M all 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)) = hAPP(nat,nat,times_times(nat,K_1),hAPP(nat,nat,div_mod(nat,M),N_1))) # label(fact_977_mod__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 11 (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),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_223_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 12 (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_423_mult__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 13 (all X_a (real_normed_algebra(X_a) -> (all B hAPP(X_a,X_a,times_times(X_a,zero_zero(X_a)),B) = zero_zero(X_a)))) # label(fact_886_mult_Ozero__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 14 (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_360_power__m1__even) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 15 (all K all Ma all 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))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),N)))) # label(fact_661_nat__add__left__cancel__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 16 (all X all Y_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y_1)) <-> Y_1 != X & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),Y_1)))) # label(fact_542_real__less__def) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 17 (all X_a (comm_semiring_1(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_237_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 18 (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_38_zadd__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 19 (all X_1 all P_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_3)) -> (hBOOL(hAPP(int,bool,zprime,P_3)) -> (-hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),P_3)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,X_1),multInv(P_3,X_1)),one_one(int)),P_3)))))) # label(fact_851_MultInv__prop2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 20 (all N_1 (zero_zero(nat) != N_1 -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)))) # label(fact_618_gr0I) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 21 (all N all P_1 (-hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) -> (hBOOL(hAPP(nat,bool,P_1,N)) -> (exists K_2 (hBOOL(hAPP(nat,bool,P_1,K_2)) & (all I (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),K_2)) -> -hBOOL(hAPP(nat,bool,P_1,I)))) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_2),N))))))) # label(fact_862_ex__least__nat__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 22 (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]. 1.53/1.77 23 (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]. 1.53/1.77 24 (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_772_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 25 (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),B)),C) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),C)),B)))) # label(fact_224_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 26 (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_490_less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 27 (all D_2 all C all A_1 all B all M (hBOOL(hAPP(int,bool,zcong(A_1,B),M)) -> (ti(int,C) = ti(int,B) -> (hBOOL(hAPP(int,bool,zcong(C,D_2),M)) -> hBOOL(hAPP(int,bool,zcong(A_1,D_2),M)))))) # label(fact_311_zcong__eq__trans) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 28 (all B_2 all P_2 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_2))) -> (hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),wset(A_3,P_2))) -> inv(P_2,A_3) = ti(int,B_2) | ti(int,B_2) = ti(int,A_3))))) # label(fact_833_wset__mem__imp__or) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 29 (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]. 1.53/1.77 30 (all A_3 (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_3),one_one(nat))) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),one_one(nat)),A_3)) <-> one_one(nat) != A_3)) # label(fact_877_gcd__lcm__complete__lattice__nat_Obot__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 31 (all X_a (group_add(X_a) -> (all A_1 all B ti(X_a,A_1) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,minus_minus(X_a,A_1),B)),B)))) # label(fact_797_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 32 (all Ma all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,times_times(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_698_nat__0__less__mult__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 33 (all B_1_1 ti(fun(int,bool),d22set(B_1_1)) = d22set(B_1_1)) # label(tsy_c_IntFact_Od22set_res) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 34 (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)),bit1(K2))))) # label(fact_52_less__eq__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 35 (all X_a (cancel_semigroup_add(X_a) -> (all A_3 all B_2 all C_1 (ti(X_a,C_1) = ti(X_a,B_2) <-> 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))))) # label(fact_760_add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 36 (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)),bit0(L))))) # label(fact_155_rel__simps_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 37 (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(X_a),C_1),D)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_3),B_2))))))) # label(fact_791_diff__eq__diff__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 38 (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_939_neg__mod__bound) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 39 (all X_a (semiring_div(X_a) -> (all A_3 all B_2 (zero_zero(X_a) = hAPP(X_a,X_a,div_mod(X_a,B_2),A_3) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_3),B_2)))))) # label(fact_940_dvd__eq__mod__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 40 (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_483_diff__bin__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 41 (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_916_zdvd__zmod__imp__zdvd) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 42 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all Y_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),Y_1)) <-> 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_1))))))) # label(fact_176_less__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 43 (all X_a (number_semiring(X_a) -> one_one(X_a) = number_number_of(X_a,bit1(pls)))) # label(fact_171_semiring__numeral__1__eq__1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 44 (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,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,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))) # label(fact_734_nat__diff__add__eq1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 45 (all Z_1 hAPP(int,int,plus_plus(int,Z_1),zero_zero(int)) = ti(int,Z_1)) # label(fact_133_zadd__0__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 46 (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_566_dvd_Oord__eq__less__trans) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 47 (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_63_mult__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 48 -(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_195__096_B_Bthesis_O_A_I_B_Bs_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 49 (all M all N_1 all 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)) = hAPP(nat,nat,times_times(nat,hAPP(nat,nat,div_mod(nat,M),N_1)),K_1)) # label(fact_978_mod__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 50 (all B_1_1 sr(B_1_1) = ti(fun(int,bool),sr(B_1_1))) # label(tsy_c_Residues_OSR_res) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 51 (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_637_not__add__less1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 52 (all X_1 all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X_1)) -> X_1 != Y)) # label(fact_560_dvd_Oless__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 53 (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_469_number__of__diff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 54 (all X_a (number_semiring(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_177_semiring__one__add__one__is__two) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 55 (all K_1 pls != bit1(K_1)) # label(fact_115_rel__simps_I46_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 56 (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_363_mult__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 57 (all Y 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)) -> 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)))))) # label(fact_258_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 58 (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_957_pos__mod__sign) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 59 (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_994_mod__lemma) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 60 (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_291_rel__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 61 (all X_1 all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X_1)) -> Y != X_1)) # label(fact_564_dvd_Oless__imp__neq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 62 (all X_a (number_ring(X_a) -> (all W hAPP(X_a,X_a,plus_plus(X_a,one_one(X_a)),number_number_of(X_a,W)) = number_number_of(X_a,hAPP(int,int,plus_plus(int,bit1(pls)),W))))) # label(fact_15_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 63 (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]. 1.53/1.77 64 (all N_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N_1))) # label(fact_626_less__eq__nat_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 65 (all X_a (semiring_div(X_a) & number(X_a) -> (all X all Y_1 (hAPP(X_a,X_a,div_mod(X_a,number_number_of(X_a,Y_1)),number_number_of(X_a,X)) = 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)),number_number_of(X_a,Y_1))))))) # label(fact_900_dvd__eq__mod__eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 66 (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_976_mod__le__divisor) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 67 (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_809_add__less__le__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 68 (all X_1 all P_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_3)) -> (hBOOL(hAPP(int,bool,zprime,P_3)) -> (-hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),P_3)) -> hBOOL(hAPP(int,bool,zcong(multInv(P_3,multInv(P_3,X_1)),X_1),P_3)))))) # label(fact_842_MultInv__prop4) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 69 (all W all Z1 all Z2 hAPP(int,int,times_times(int,W),hAPP(int,int,minus_minus(int,Z1),Z2)) = hAPP(int,int,minus_minus(int,hAPP(int,int,times_times(int,W),Z1)),hAPP(int,int,times_times(int,W),Z2))) # label(fact_474_zdiff__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 70 (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_85_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 71 (all X all Y_1 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,Z_2),X)),hAPP(real,real,times_times(real,Z_2),Y_1))) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),Y_1))))) # label(fact_704_real__mult__le__cancel__iff2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 72 (all X_a (dvd(X_a) & linorder(X_a) & plus(X_a) -> (all D all Sa exists Z all X_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),X_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,X_2),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_2),Sa)))))))) # label(fact_864_pinf_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 73 (all X_a (mult_zero(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_318_mult__zero__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 74 (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_123_less__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 75 (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_2 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_2)) & (all Y_2 (hAPP(nat,real,power_power(real,Y_2),N_1) = A_1 & hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y_2)) -> X_2 = Y_2)) & A_1 = hAPP(nat,real,power_power(real,X_2),N_1)))))) # label(fact_529_realpow__pos__nth__unique) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 76 (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_611_diff__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 77 (all X_a (comm_semiring_1(X_a) -> (all X_1 all Y all Z_1 hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,X_1),Y)),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),Z_1))))) # label(fact_239_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 78 (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_956_zcong__zmod__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 79 (all B_1_1 all B_2_1 standardRes(B_1_1,B_2_1) = ti(int,standardRes(B_1_1,B_2_1))) # label(tsy_c_Residues_OStandardRes_res) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 80 (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_763_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 81 (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,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_924_zmod__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 82 (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_267_unique__quotient__lemma__neg) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 83 (all A_1 all J_1 all K_1 all P_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_3)) -> (hBOOL(hAPP(int,bool,zprime,P_3)) -> (-hBOOL(hAPP(int,bool,zcong(K_1,zero_zero(int)),P_3)) -> (-hBOOL(hAPP(int,bool,zcong(J_1,zero_zero(int)),P_3)) -> (hBOOL(hAPP(int,bool,zcong(J_1,hAPP(int,int,times_times(int,A_1),multInv(P_3,K_1))),P_3)) -> hBOOL(hAPP(int,bool,zcong(K_1,hAPP(int,int,times_times(int,A_1),multInv(P_3,J_1))),P_3)))))))) # label(fact_846_MultInv__zcong__prop2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 84 (all X all Y_1 (zero_zero(real) = X & zero_zero(real) = Y_1 <-> zero_zero(real) = hAPP(real,real,plus_plus(real,hAPP(nat,real,power_power(real,X),number_number_of(nat,bit0(bit1(pls))))),hAPP(nat,real,power_power(real,Y_1),number_number_of(nat,bit0(bit1(pls))))))) # label(fact_528_realpow__two__sum__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.77 85 (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,B) = ti(X_a,C))))) # label(fact_762_add__left__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 86 (all Y_1 all X (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y_1)) <-> X = Y_1))) # label(fact_575_dvd_Oantisym__conv) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 87 (all J_2 all K all I_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,minus_minus(nat,J_2),K)),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))))) # label(fact_670_le__diff__conv) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 88 (all X_a all B_1_1 member(X_a,B_1_1) = ti(fun(fun(X_a,bool),bool),member(X_a,B_1_1))) # label(tsy_c_member_res) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 89 (all X_1 all Y all M hAPP(int,int,div_mod(int,hAPP(int,int,minus_minus(int,X_1),hAPP(int,int,div_mod(int,Y),M))),M) = hAPP(int,int,div_mod(int,hAPP(int,int,minus_minus(int,X_1),Y)),M)) # label(fact_949_zdiff__zmod__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 90 (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_371_mult__nonpos__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 91 (all X_a (comm_semiring_1(X_a) -> (all N_1 all X_1 all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X_1),Y)) -> 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),N_1))))))) # label(fact_339_dvd__power__same) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 92 (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_395_pos__add__strict) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 93 (all X_a (ab_group_add(X_a) -> (all A_1 all C all B all 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)) = 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))))) # label(fact_613_add__diff__add) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 94 (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_599_nat__le__linear) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 95 (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_138_semiring__mult__2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 96 (all X all Y_1 (Y_1 = X <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X)))) # label(fact_512_divides__antisym) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 97 (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_735_nat__eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 98 (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_157_rel__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 99 (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_583_nat__add__commute) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 100 (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,plus_plus(X_a,A_1),A_4)),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))))) # label(fact_892_mult_Oadd__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 101 (all X_a (linordered_idom(X_a) -> (all X all Y_1 (ti(X_a,X) = zero_zero(X_a) & zero_zero(X_a) = ti(X_a,Y_1) <-> hAPP(X_a,X_a,plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,X),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_9_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 102 (all N (N = zero_zero(nat) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),zero_zero(nat))))) # label(fact_627_le__0__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 103 (all B all A_1 all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1)) -> (-hBOOL(hAPP(int,bool,zcong(B,zero_zero(int)),P_3)) & -hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_3)) -> -hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_1),B),zero_zero(int)),P_3)))))) # label(fact_460_zcong__zprime__prod__zero__contra) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 104 (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_eq(int),bit1(K)),bit0(L))))) # label(fact_73_rel__simps_I33_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 105 (all P_1 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_1,K)) -> ((all I (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),I)) -> (hBOOL(hAPP(int,bool,P_1,I)) -> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,plus_plus(int,I),one_one(int))))))) -> hBOOL(hAPP(int,bool,P_1,I_2)))))) # label(fact_854_int__ge__induct) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 106 (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]. 1.53/1.78 107 (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_499_power__dvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 108 (all K all P_1 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D)) -> ((all X_2 (hBOOL(hAPP(int,bool,P_1,X_2)) -> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,plus_plus(int,X_2),D))))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) -> (all X_2 (hBOOL(hAPP(int,bool,P_1,X_2)) -> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,plus_plus(int,X_2),hAPP(int,int,times_times(int,K),D)))))))))) # label(fact_749_incr__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 109 (all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)) -> (N_1 != M -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1))))) # label(fact_651_le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 110 (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_124_rel__simps_I17_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 111 (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_26_less__eq__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 112 (all B_1_1 all X_a (number(X_a) -> number_number_of(X_a,ti(int,B_1_1)) = number_number_of(X_a,B_1_1))) # label(tsy_c_Int_Onumber__class_Onumber__of_arg1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 113 (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_667_add__leD1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 114 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),min)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(K)),min)))) # label(fact_290_rel__simps_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 115 (all K_1 hAPP(int,int,plus_plus(int,K_1),K_1) = bit0(K_1)) # label(fact_132_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 116 (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_335_dvd__mult__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 117 (all B_1_1 all B_2_1 inv(ti(int,B_1_1),B_2_1) = inv(B_1_1,B_2_1)) # label(tsy_c_WilsonRuss_Oinv_arg1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 118 (all P_1 all A_3 all B_2 (-((exists D_4 (hAPP(nat,nat,plus_plus(nat,B_2),D_4) = A_3 & -hBOOL(hAPP(nat,bool,P_1,D_4)))) | -hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_3),B_2))) <-> hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,minus_minus(nat,A_3),B_2))))) # label(fact_696_nat__diff__split__asm) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 119 (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_381_mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 120 (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,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),E_1)),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))))) # label(fact_322_combine__common__factor) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 121 (all M all N_1 (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1)) -> hAPP(nat,nat,plus_plus(nat,N_1),hAPP(nat,nat,minus_minus(nat,M),N_1)) = M)) # label(fact_646_add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 122 (all Ma all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),N)) <-> hAPP(nat,nat,minus_minus(nat,Ma),N) = zero_zero(nat))) # label(fact_628_diff__is__0__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 123 (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_162_add__Bit0__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 124 (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_244_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 125 (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),B),A_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))))) # label(fact_573_dvd_Ole__neq__trans) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 126 (all X_1 all Y all P_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_3)) -> (hBOOL(hAPP(int,bool,zcong(X_1,Y),P_3)) -> hBOOL(hAPP(int,bool,zcong(multInv(P_3,X_1),multInv(P_3,Y)),P_3))))) # label(fact_839_MultInv__prop1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 127 (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_494_zle__diff1__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 128 (all Z_1 all X_1 all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y)) -> (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z_1),Y)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),Z_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_565_dvd_Ole__less__trans) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 129 (all X_b (ring(X_b) & number(X_b) -> (all A_1 all B all 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)) = 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)))))) # label(fact_481_left__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 130 (all B_1_1 all B_2_1 zcong(B_1_1,ti(int,B_2_1)) = zcong(B_1_1,B_2_1)) # label(tsy_c_IntPrimes_Ozcong_arg2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 131 (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_982_mod__mult__self3) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 132 (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),B_2),zero_zero(X_a))) & hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),zero_zero(X_a))))))) # label(fact_376_zero__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 133 (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),A_1),B)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),A_1))))) # label(fact_346_Euler_Oaux2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 134 (all X_a (linordered_semidom(X_a) -> (all X all Y_1 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)),hAPP(nat,X_a,power_power(X_a,B_2),Y_1))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),Y_1))))))) # label(fact_445_power__strict__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 135 (all Y all X_1 all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) -> (-hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),P_3)) -> (-hBOOL(hAPP(int,bool,zcong(Y,zero_zero(int)),P_3)) -> -hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,X_1),Y),zero_zero(int)),P_3)))))) # label(fact_452_zcong__zmult__prop3) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 136 (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_908_zmod__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 137 (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_402_less__1__mult) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 138 (all Ma all N (hAPP(nat,nat,times_times(nat,Ma),N) = one_one(nat) <-> one_one(nat) = Ma & N = one_one(nat))) # label(fact_689_nat__1__eq__mult__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 139 (all X all P_2 (-hBOOL(hAPP(int,bool,zcong(X,zero_zero(int)),P_2)) <-> standardRes(P_2,X) != zero_zero(int))) # label(fact_989_StandardRes__prop3) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 140 (all X_a (number_ring(X_a) -> (all A_1 hAPP(X_a,X_a,times_times(X_a,number_number_of(X_a,bit1(pls))),A_1) = ti(X_a,A_1)))) # label(fact_109_mult__numeral__1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 141 (all Y 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)) -> 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)))))) # label(fact_936_Divides_Otransfer__nat__int__function__closures_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 142 (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) -> M = N_1))) # label(fact_883_mult__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 143 (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_64_less__number__of__int__code) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 144 (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_678_add__diff__assoc2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 145 (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_351_zcong__zmult__self) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 146 (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_514_divides__mul__r) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 147 (all A_1 all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) -> (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_3)) -> hBOOL(hAPP(int,bool,zcong(A_1,one_one(int)),P_3)) | hBOOL(hAPP(int,bool,zcong(A_1,hAPP(int,int,minus_minus(int,P_3),one_one(int))),P_3)))))) # label(fact_501_zcong__square) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 148 (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_645_add__lessD1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 149 (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_903_mod__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 150 (all B_2 all A_3 all P_2 (hBOOL(hAPP(int,bool,zprime,P_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(int,int,minus_minus(int,P_2),one_one(int)))) -> (hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),wset(A_3,P_2))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_2),hAPP(int,int,minus_minus(int,P_2),one_one(int)))))))) # label(fact_832_wset__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 151 (all P_1 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_1,K)) -> ((all I (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I),K)) -> (hBOOL(hAPP(int,bool,P_1,I)) -> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,minus_minus(int,I),one_one(int))))))) -> hBOOL(hAPP(int,bool,P_1,I_2)))))) # label(fact_836_int__le__induct) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 152 (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_191_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 153 (all X_a (monoid_mult(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(nat,X_a,power_power(X_a,A_1),one_one(nat))))) # label(fact_347_power__one__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 154 (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_668_add__leE) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 155 (all X all Y_1 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),Y_1)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,times_times(real,X),Z_2)),hAPP(real,real,times_times(real,Y_1),Z_2)))))) # label(fact_703_real__mult__le__cancel__iff1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 156 (all P_1 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_1,N),Ma))) -> ((Ma = N -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,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_1,N),Ma))) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,N),Ma)))))) # label(fact_550_nat__less__cases) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 157 (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_657_dvd__diffD1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 158 (all X_a (group_add(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,minus_minus(X_a,A_1),zero_zero(X_a))))) # label(fact_789_diff__0__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 159 (all N_1 N_1 = hAPP(nat,nat,plus_plus(nat,zero_zero(nat)),N_1)) # label(fact_621_plus__nat_Oadd__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 160 (all A_1 all B all C (ti(int,C) = hAPP(int,int,minus_minus(int,A_1),B) -> hAPP(int,int,plus_plus(int,C),B) = ti(int,A_1))) # label(fact_475_Int2_Oaux1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 161 (all A_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),one_one(nat)),A_1))) # label(fact_878_gcd__lcm__complete__lattice__nat_Obot__least) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 162 (all X all Y_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),Y_1)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y_1)) | Y_1 = X)) # label(fact_543_less__eq__real__def) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 163 (all X_1 all P_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_3)) -> (hBOOL(hAPP(int,bool,zprime,P_3)) -> (-hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),P_3)) -> -hBOOL(hAPP(int,bool,zcong(multInv(P_3,X_1),zero_zero(int)),P_3)))))) # label(fact_843_MultInv__prop3) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 164 (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_725_real__le__antisym) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 165 (all B_2 all P_2 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_2))) -> hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),wset(A_3,P_2)))))) # label(fact_830_wset__subset) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 166 (all J_1 all A_1 all K_1 all P_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_3)) -> (hBOOL(hAPP(int,bool,zprime,P_3)) -> (-hBOOL(hAPP(int,bool,zcong(K_1,zero_zero(int)),P_3)) -> (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_3,K_1))),K_1)),P_3)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,J_1),K_1),A_1),P_3))))))) # label(fact_845_aux______2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 167 (all X_1 all Y 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)) -> 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)))))) # label(fact_705_real__mult__less__mono2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 168 (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_444_power__less__imp__less__exp) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 169 (all A_1 zero_zero(int) = hAPP(int,int,div_mod(int,A_1),number_number_of(int,min))) # label(fact_954_zmod__minus1__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 170 (all X_1 hAPP(nat,int,power_power(int,X_1),number_number_of(nat,bit0(bit0(bit1(pls))))) = 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))))) # label(fact_269_quartic__square__square) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 171 (all B_1_1 all B_2_1 inv(B_1_1,B_2_1) = ti(int,inv(B_1_1,B_2_1))) # label(tsy_c_WilsonRuss_Oinv_res) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 172 (all N_1 -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),zero_zero(nat)))) # label(fact_533_less__zeroE) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 173 (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_356_zcong__zmult) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 174 (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_79_int__one__le__iff__zero__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 175 (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_513_divides__add__revr) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 176 (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_245_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 177 (all Z_1 all W hAPP(int,int,plus_plus(int,W),Z_1) = hAPP(int,int,plus_plus(int,Z_1),W)) # label(fact_98_zadd__commute) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 178 (all K_1 all M all 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)) = hAPP(nat,nat,minus_minus(nat,M),N_1)) # label(fact_591_Nat_Odiff__cancel) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 179 (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_922_dvd__mod__imp__dvd) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 180 (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_671_le__add__diff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 181 (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_802_add__increasing) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 182 (all X_a (comm_semiring_1(X_a) -> (all N_1 all M all X_1 all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X_1),Y)) -> (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),M)))))))) # label(fact_341_dvd__power__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 183 (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]. 1.53/1.78 184 (all N all P_1 (-hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) -> (hBOOL(hAPP(nat,bool,P_1,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_1,I)))) & hBOOL(hAPP(nat,bool,P_1,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_719_ex__least__nat__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 185 -(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_279__096_B_Bthesis_O_A_I_B_Bs1_O_A_091s1_A_094_A2_A_061_A_N1_093_A_Imod_A4_) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 186 (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_256_pos__zmult__pos) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 187 (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_355_zcong__scalar2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 188 (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_666_add__leD2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 189 (all X_a (monoid_mult(X_a) -> (all A_1 all M all 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)) = hAPP(nat,X_a,power_power(X_a,A_1),hAPP(nat,nat,plus_plus(nat,M),N_1))))) # label(fact_412_power__add) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 190 (all Y all X_1 all P_3 (-hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),P_3)) -> (hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,Y),number_number_of(nat,bit0(bit1(pls)))),X_1),P_3)) -> -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_3),Y))))) # label(fact_271_Euler_Oaux____1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 191 (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_374_mult__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 192 (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_334_dvd__mult__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 193 (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_810_add__le__less__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 194 (all T all A ti(T,A) = ti(T,ti(T,A))) # label(help_ti_idem) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 195 (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,hAPP(X_a,X_a,plus_plus(X_a,A_1),C)),D_2)))) # label(fact_222_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 196 (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,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)) = 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_737_nat__diff__add__eq2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 197 (all X_a (power(X_a) -> (all M all N_1 ((zero_zero(nat) = N_1 -> one_one(X_a) = hAPP(nat,X_a,power_power(X_a,M),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)))))) # label(fact_718_realpow__num__eq__if) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 198 (all X_b (number(X_b) & semiring(X_b) -> (all V all B all 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)) = 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))))) # label(fact_95_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 199 (all K all L (bit1(K) = bit1(L) <-> ti(int,L) = ti(int,K))) # label(fact_90_rel__simps_I51_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 200 (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_489_le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 201 (all X_a (linordered_idom(X_a) -> (all Y 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)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),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),X_1)),X_1)))))))) # label(fact_433_mult__left__le__one__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 202 (all X_a (number_ring(X_a) -> (all A_1 ti(X_a,A_1) = hAPP(X_a,X_a,plus_plus(X_a,A_1),number_number_of(X_a,pls))))) # label(fact_148_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 203 (all L (pls = ti(int,L) <-> bit0(L) = pls)) # label(fact_120_rel__simps_I38_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 204 (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_610_diff__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 205 (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_358_zcong__shift) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 206 (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_387_zero__less__mult__pos) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 207 (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]. 1.53/1.78 208 (all X_a (one(X_a) -> (all X (one_one(X_a) = ti(X_a,X) <-> one_one(X_a) = ti(X_a,X))))) # label(fact_765_one__reorient) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 209 (all A_1 all 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))))) = hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,A_1),B)),hAPP(int,int,minus_minus(int,A_1),B))) # label(fact_502_zspecial__product) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 210 (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),Rx)),Ly) = hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,times_times(X_a,Lx),Ly)),Rx)))) # label(fact_215_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 211 (all V_3 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),number_number_of(nat,V_3))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),V_3)))) # label(fact_49_less__0__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 212 (all X_a (no_zero_divisors(X_a) -> (all A_1 all B (hAPP(X_a,X_a,times_times(X_a,A_1),B) = zero_zero(X_a) -> zero_zero(X_a) = ti(X_a,B) | zero_zero(X_a) = ti(X_a,A_1))))) # label(fact_315_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 213 (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_eq(int),bit1(K)),pls)))) # label(fact_70_rel__simps_I29_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 214 (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_888_mult_Ozero__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 215 (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_235_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 216 (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_436_power__less__imp__less__base) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 217 (all B all A_1 all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) -> (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_3)) -> hBOOL(hAPP(int,bool,zcong(B,zero_zero(int)),P_3)) | hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_3)))))) # label(fact_459_zcong__zprime__prod__zero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 218 (all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)) -> zero_zero(nat) = hAPP(nat,nat,minus_minus(nat,M),N_1))) # label(fact_629_diff__is__0__eq_H) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 219 (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_186_cube__square) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 220 (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_675_le__add__diff__inverse2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 221 (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)),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_906_mod__mult__left__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 222 (all B_1_1 bit0(ti(int,B_1_1)) = bit0(B_1_1)) # label(tsy_c_Int_OBit0_arg1) # label(hypothesis) # label(non_clause). [assumption]. 1.53/1.78 223 (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_114_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 224 (all X_a (zero_neq_one(X_a) -> one_one(X_a) != zero_zero(X_a))) # label(fact_320_zero__neq__one) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 225 (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_43_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 226 (all B_1_1 all X_a (cancel_semigroup_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_2_arg1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 227 (all X_a (linorder(X_a) & number(X_a) -> (all V_3 all W_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),number_number_of(X_a,V_3)),number_number_of(X_a,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_3))))))) # label(fact_50_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 228 (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_87_le__imp__0__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 229 (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_2 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,X_2),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,hAPP(X_a,X_a,minus_minus(X_a,X_2),hAPP(X_a,X_a,times_times(X_a,K_2),D_3))),Ta))))))))) # label(fact_745_inf__period_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 230 (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]. 1.53/1.78 231 (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_158_rel__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 232 (all X_a (linordered_idom(X_a) -> (all X_1 all Y -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),number_number_of(nat,bit0(bit1(pls)))))),zero_zero(X_a)))))) # label(fact_7_not__sum__power2__lt__zero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 233 (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_808_add__neg__neg) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 234 (all L (ti(int,L) = min <-> min = bit1(L))) # label(fact_282_rel__simps_I43_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 235 (all X_a (number_semiring(X_a) -> (all X_1 all Y 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),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)) = hAPP(nat,X_a,power_power(X_a,hAPP(X_a,X_a,plus_plus(X_a,X_1),Y)),number_number_of(nat,bit0(bit1(pls))))))) # label(fact_168_power2__sum) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 236 (all A_1 zero_zero(int) = hAPP(int,int,div_mod(int,A_1),A_1)) # label(fact_917_zmod__self) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 237 (all X_a (linordered_idom(X_a) -> (all X_1 all Y (ti(X_a,X_1) != ti(X_a,Y) -> (-hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X_1))))))) # label(fact_306_linorder__neqE__linordered__idom) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 238 (all X_a (number(X_a) -> (all W_1 all X (ti(X_a,X) = number_number_of(X_a,W_1) <-> number_number_of(X_a,W_1) = ti(X_a,X))))) # label(fact_89_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 239 (all B_1_1 bit0(B_1_1) = ti(int,bit0(B_1_1))) # label(tsy_c_Int_OBit0_res) # label(hypothesis) # label(non_clause). [assumption]. 1.53/1.78 240 (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_400_power__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 241 (all X_a (linordered_semidom(X_a) -> (all X all Y_1 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(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X),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,B_2),X)),hAPP(nat,X_a,power_power(X_a,B_2),Y_1)))))))) # label(fact_439_power__increasing__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 242 (all X all W_1 (zero_zero(nat) = number_number_of(nat,W_1) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,power_power(nat,X),number_number_of(nat,W_1)))))) # label(fact_227_zero__less__power__nat__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 243 (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_388_mult__pos__neg2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 244 (all X_a (linord581940658strict(X_a) -> (all X all Y_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,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,X),X)),hAPP(X_a,X_a,times_times(X_a,Y_1),Y_1)))) <-> zero_zero(X_a) != ti(X_a,X) | ti(X_a,Y_1) != zero_zero(X_a))))) # label(fact_107_sum__squares__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 245 (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_264_q__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 246 (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_327_power__commutes) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 247 (all X_a (idom(X_a) -> (all A_3 all C_1 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,A_3),C_1)),hAPP(X_a,X_a,times_times(X_a,B_2),C_1))) <-> ti(X_a,C_1) = zero_zero(X_a) | hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_3),B_2)))))) # label(fact_406_dvd__mult__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 248 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),K)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),bit0(K))))) # label(fact_55_rel__simps_I21_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 249 (all N_1 all A_1 all B all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_3),B)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,power_power(int,P_3),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_3),N_1)),A_1)))))) # label(fact_200_zprime__power__zdvd__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 250 (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_403_zero__less__power) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 251 (all P_4 all P_1 all X ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)) -> (hBOOL(P_4) <-> hBOOL(P_1))) -> ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)) -> hBOOL(P_1)) <-> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)) -> hBOOL(P_4))))) # label(fact_742_imp__le__cong) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 252 (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_687_mult__le__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 253 (all B_1_1 all X_a (power(X_a) -> power_power(X_a,B_1_1) = power_power(X_a,ti(X_a,B_1_1)))) # label(tsy_c_Power_Opower__class_Opower_arg1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 254 (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,B_1_1,ti(X_a,B_2_1))) # label(tsy_c_hAPP_arg2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 255 (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)) -> (Ma = N <-> hAPP(nat,nat,minus_minus(nat,Ma),K) = hAPP(nat,nat,minus_minus(nat,N),K))))) # label(fact_603_eq__diff__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 256 (all A_1 -(hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),zero_zero(nat))))) # label(fact_874_gcd__lcm__complete__lattice__nat_Onot__top__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 257 (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_386_zero__less__mult__pos2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 258 (all K_1 ti(int,K_1) = hAPP(int,int,plus_plus(int,pls),K_1)) # label(fact_130_add__Pls) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 259 (all X_a (comm_semiring_1(X_a) -> (all X_1 all Y 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),Q_1)) = hAPP(nat,X_a,power_power(X_a,hAPP(X_a,X_a,times_times(X_a,X_1),Y)),Q_1)))) # label(fact_246_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 260 (all A_3 all B_2 all N (N != zero_zero(nat) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_3),B_2)) <-> 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)))))) # label(fact_881_pow__divides__eq__nat) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 261 (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_756_add__nonpos__neg) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 262 (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_896_mult_Odiff__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 263 (all N_1 all M (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N_1),M)) -> M = zero_zero(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)))) # label(fact_511_divides__cases) # label(axiom) # label(non_clause). [assumption]. 1.53/1.78 264 (all A_1 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)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,minus_minus(int,P_3),one_one(int)))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),inv(P_3,A_1)),hAPP(int,int,minus_minus(int,P_3),one_one(int)))))))) # label(fact_820_inv__less__p__minus__1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 265 (all X_a (real_normed_algebra(X_a) -> (all Y zero_zero(X_a) = hAPP(X_a,X_a,times_times(X_a,zero_zero(X_a)),Y)))) # label(fact_887_mult__left_Ozero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 266 (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_656_dvd__diffD) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 267 (all X_1 all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X_1)) -> -(hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y))))) # label(fact_553_dvd_Oless__asym) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 268 (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_694_nat__dvd__not__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 269 (all X_a (number_ring(X_a) -> (all V all W number_number_of(X_a,hAPP(int,int,plus_plus(int,V),W)) = hAPP(X_a,X_a,plus_plus(X_a,number_number_of(X_a,V)),number_number_of(X_a,W))))) # label(fact_152_number__of__add) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 270 (all X_a (mult_zero(X_a) -> (all A_1 zero_zero(X_a) = hAPP(X_a,X_a,times_times(X_a,zero_zero(X_a)),A_1)))) # label(fact_319_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 271 (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_112_mult__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 272 (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_597_le__trans) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 273 (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]. 1.53/1.79 274 (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_449_zcong__zless__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 275 (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_617_gr__implies__not0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 276 (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_945_mod__mult__self2__is__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 277 (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_228_nat__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 278 (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_549_less__not__refl3) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 279 (all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) -> hBOOL(hAPP(int,bool,zcong(zfact(hAPP(int,int,minus_minus(int,P_3),one_one(int))),number_number_of(int,min)),P_3)))) # label(fact_755_Wilson__Russ) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 280 (all V all W number_number_of(int,hAPP(int,int,plus_plus(int,V),W)) = hAPP(int,int,plus_plus(int,number_number_of(int,V)),number_number_of(int,W))) # label(fact_67_plus__numeral__code_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 281 (all M all N_1 (hAPP(int,int,times_times(int,M),N_1) = one_one(int) -> ti(int,M) = one_one(int) | number_number_of(int,min) = ti(int,M))) # label(fact_305_pos__zmult__eq__1__iff__lemma) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 282 (all Y 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)) -> 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)))))) # label(fact_259_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 283 (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_135_zadd__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 284 (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_328_dvd__triv__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 285 (all X_a (comm_semiring_1(X_a) -> (all M hAPP(X_a,X_a,plus_plus(X_a,M),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)))) # label(fact_252_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 286 (all A_1 all J_1 all K_1 all P_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_3)) -> (hBOOL(hAPP(int,bool,zcong(J_1,K_1),P_3)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_1),multInv(P_3,J_1)),hAPP(int,int,times_times(int,A_1),multInv(P_3,K_1))),P_3))))) # label(fact_840_MultInv__zcong__prop1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 287 (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_777_add__le__imp__le__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 288 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all X all Y_1 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),number_number_of(X_a,X)),number_number_of(X_a,Y_1))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),Y_1)))))) # label(fact_149_less__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 289 (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]. 1.53/1.79 290 (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_10_power2__less__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 291 (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_338_one__dvd) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 292 (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]. 1.53/1.79 293 (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,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))) & (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)))) # label(fact_231_nat__number__of__mult__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 294 (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]. 1.53/1.79 295 (all X_a (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,A_3) = ti(X_a,B_2))))) # label(fact_786_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 296 (all X_a (monoid_mult(X_a) -> (all A_1 hAPP(nat,X_a,power_power(X_a,A_1),number_number_of(nat,bit0(bit1(pls)))) = hAPP(X_a,X_a,times_times(X_a,A_1),A_1)))) # label(fact_142_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 297 (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_192_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 298 (all X all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,power_power(nat,X),N))) <-> zero_zero(nat) = N | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X)))) # label(fact_197_zero__less__power__nat__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 299 (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_572_dvd_Oord__eq__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 300 (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),ord_less_eq(nat),M),N_1)) | N_1 = zero_zero(nat))) # label(fact_882_divides__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 301 (all X all Y_1 (X != Y_1 & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y_1)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X)))) # label(fact_579_dvd_Oless__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 302 (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_83_add1__zle__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 303 (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_664_add__le__mono1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 304 (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_582_dvd_Oless__irrefl) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 305 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),bit0(K))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),K)))) # label(fact_303_rel__simps_I25_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 306 (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),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))))) # label(fact_728_nat__mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 307 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all X (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),pls)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),number_number_of(X_a,X)),zero_zero(X_a))))))) # label(fact_175_less__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 308 (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_309_zcong__sym) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 309 (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_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,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]. 1.53/1.79 310 (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_771_add_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 311 (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_265_unique__quotient__lemma) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 312 (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_602_Nat_Odiff__diff__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 313 (all X_a (linord581940658strict(X_a) -> (all X all Y_1 (ti(X_a,Y_1) = zero_zero(X_a) & ti(X_a,X) = zero_zero(X_a) <-> hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,X),X)),hAPP(X_a,X_a,times_times(X_a,Y_1),Y_1)) = zero_zero(X_a))))) # label(fact_93_sum__squares__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 314 (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_783_add__strict__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 315 (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]. 1.53/1.79 316 (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_29_zle__trans) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 317 (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_333_dvdI) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 318 (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_769_double__zero__sym) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 319 (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_437_power__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 320 (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_448_zcong__zless__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 321 (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]. 1.53/1.79 322 (all X_b (semiring(X_b) & number(X_b) -> (all A_1 all B all 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)) = 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)))))) # label(fact_94_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 323 (all B_2 all A_3 all P_2 (hBOOL(hAPP(int,bool,zprime,P_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(int,int,minus_minus(int,P_2),one_one(int)))) -> (hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),wset(A_3,P_2))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),B_2)))))) # label(fact_831_wset__g__1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 324 (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_30_zle__antisym) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 325 (all X_a (no_zero_divisors(X_a) -> (all B all A_1 (ti(X_a,A_1) != zero_zero(X_a) -> (ti(X_a,B) != zero_zero(X_a) -> hAPP(X_a,X_a,times_times(X_a,A_1),B) != zero_zero(X_a)))))) # label(fact_316_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 326 (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_465_one__not__neg__one__mod__m) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 327 (all X_a (comm_semiring_1(X_a) -> (all X_1 hAPP(nat,X_a,power_power(X_a,X_1),one_one(nat)) = ti(X_a,X_1)))) # label(fact_226_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 328 (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,times_times(nat,M),N_1)) = hAPP(nat,X_a,power_power(X_a,hAPP(nat,X_a,power_power(X_a,A_1),M)),N_1)))) # label(fact_345_power__mult) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 329 (all X_a (comm_monoid_mult(X_a) -> (all A_1 all B all N_1 hAPP(nat,X_a,power_power(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),B)),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))))) # label(fact_326_power__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 330 (all M zero_zero(nat) = hAPP(nat,nat,times_times(nat,M),zero_zero(nat))) # label(fact_631_mult__0__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 331 (all X_1 all M all Y hAPP(int,int,div_mod(int,hAPP(nat,int,power_power(int,X_1),Y)),M) = hAPP(int,int,div_mod(int,hAPP(nat,int,power_power(int,hAPP(int,int,div_mod(int,X_1),M)),Y)),M)) # label(fact_911_zpower__zmod) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 332 (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_110_mult__numeral__1__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 333 (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_209_zdvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 334 (all B_1_1 all X_a (no_zero_divisors(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_5_arg1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 335 (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_139_mult__2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 336 (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_606_diff__le__mono2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 337 (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_781_add__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 338 (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_869_pow__divides__pow__int) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 339 (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_796_add__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 340 (all Ma all N (one_one(int) = ti(int,Ma) & ti(int,N) = one_one(int) | number_number_of(int,min) = ti(int,N) & number_number_of(int,min) = ti(int,Ma) <-> one_one(int) = hAPP(int,int,times_times(int,Ma),N))) # label(fact_304_zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 341 (all X_a (number_ring(X_a) -> (all X_1 all Y hAPP(nat,X_a,power_power(X_a,hAPP(X_a,X_a,minus_minus(X_a,X_1),Y)),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),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))))) # label(fact_503_power2__diff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 342 (all Ma all K all N (K = zero_zero(nat) | N = Ma <-> hAPP(nat,nat,times_times(nat,N),K) = hAPP(nat,nat,times_times(nat,Ma),K))) # label(fact_634_mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 343 (all Ma all X ((exists Y_2 hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,Y_2),number_number_of(nat,bit0(bit1(pls)))),X),Ma))) <-> hBOOL(hAPP(int,bool,quadRes(Ma),X)))) # label(fact_509_QuadRes__def) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 344 (all A_2 all B_1 all Q_3 all Y (hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B_1),Q_3)),Y) = ti(int,A_2) -> ((-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),Y),zero_zero(int))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_1),Y))) & (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),Y),B_1)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))) -> (zero_zero(int) != ti(int,B_1) -> hAPP(int,int,div_mod(int,A_2),B_1) = ti(int,Y))))) # label(fact_965_divmod__int__rel__mod__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 345 (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_890_mult_Oadd__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 346 (all X_a (number_ring(X_a) -> (all V number_number_of(X_a,hAPP(int,int,plus_plus(int,V),bit1(pls))) = hAPP(X_a,X_a,plus_plus(X_a,number_number_of(X_a,V)),one_one(X_a))))) # label(fact_16_add__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 347 (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,B_1_1,ti(X_b,B_2_1))) # label(tsy_c_COMBC_arg2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 348 (all X_a (zero_neq_one(X_a) -> one_one(X_a) != zero_zero(X_a))) # label(fact_321_one__neq__zero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 349 (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_312_zcong__trans) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 350 (all Q all P (-hBOOL(P) | -hBOOL(Q) | hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,P),Q)))) # label(help_fconj_1_1_U) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 351 (all X_a (ordere216010020id_add(X_a) -> (all Y_1 all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y_1)) -> (zero_zero(X_a) = ti(X_a,Y_1) & ti(X_a,X) = zero_zero(X_a) <-> hAPP(X_a,X_a,plus_plus(X_a,X),Y_1) = zero_zero(X_a))))))) # label(fact_801_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 352 (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_151_add__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 353 (all X_a (number_ring(X_a) -> one_one(X_a) = number_number_of(X_a,bit1(pls)))) # label(fact_173_semiring__norm_I110_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 354 (all X_1 all M all Y (hAPP(int,int,div_mod(int,X_1),M) = hAPP(int,int,div_mod(int,Y),M) -> hBOOL(hAPP(int,bool,zcong(X_1,Y),M)))) # label(fact_914_Residues_Oaux) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 355 (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_325_dvd__0__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 356 (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_397_not__one__le__zero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 357 (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),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(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)))))) # label(fact_392_mult__less__cancel__left__disj) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 358 (all W hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W),W))) # label(fact_21_zle__refl) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 359 (all L_1 bit0(hAPP(int,int,minus_minus(int,pls),L_1)) = hAPP(int,int,minus_minus(int,pls),bit0(L_1))) # label(fact_484_diff__bin__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 360 (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_958_pos__mod__conj) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 361 (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_188_self__quotient__aux1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 362 (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_456_power__eq__imp__eq__base) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 363 (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),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_eq(nat),Ma),N))))) # label(fact_732_nat__mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 364 (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_352_zcong__zmult__prop1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 365 (all X_a (comm_semiring_1(X_a) -> (all Lx all Ly all Rx hAPP(X_a,X_a,times_times(X_a,Lx),hAPP(X_a,X_a,times_times(X_a,Ly),Rx)) = hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,times_times(X_a,Lx),Ly)),Rx)))) # label(fact_216_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 366 (all P_3 all M ((zero_zero(nat) != M -> hAPP(nat,nat,power_power(nat,P_3),M) = hAPP(nat,nat,times_times(nat,P_3),hAPP(nat,nat,power_power(nat,P_3),hAPP(nat,nat,minus_minus(nat,M),one_one(nat))))) & (zero_zero(nat) = M -> hAPP(nat,nat,power_power(nat,P_3),M) = one_one(nat)))) # label(fact_518_power__eq__if) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 367 (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_710_real__squared__diff__one__factored) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 368 (all X_a all X all A_6 (hBOOL(hAPP(fun(X_a,bool),bool,member(X_a,X),A_6)) <-> hBOOL(hAPP(X_a,bool,A_6,X)))) # label(fact_75_mem__def) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 369 (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_329_dvd__triv__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 370 (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_422_mult__right__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 371 (all X_a (real_normed_algebra(X_a) -> (all X_1 all Y all 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),Ya)) = hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,minus_minus(X_a,X_1),Y)),Ya)))) # label(fact_897_mult__left_Odiff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 372 (all B_1_1 minus_minus(nat,B_1_1) = minus_minus(nat,ti(nat,B_1_1))) # label(tsy_c_Groups_Ominus__class_Ominus_1_arg1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 373 (all V_3 all V_2 ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_3),V_2)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),V_2))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_3),V_2)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),number_number_of(nat,V_3)),number_number_of(nat,V_2))))) # label(fact_78_less__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 374 (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_942_mod__mult__self2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 375 (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_778_add__le__imp__le__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 376 (all X_a (number_ring(X_a) -> number_number_of(X_a,bit1(pls)) = one_one(X_a))) # label(fact_172_numeral__1__eq__1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 377 (all K_1 all L_1 bit0(L_1) != bit1(K_1)) # label(fact_117_rel__simps_I50_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 378 (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)) -> N_1 = M))) # label(fact_570_dvd__antisym) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 379 (all X_1 all P_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_3)) -> (hBOOL(hAPP(int,bool,zcong(X_1,number_number_of(int,min)),P_3)) -> -hBOOL(hAPP(int,bool,zcong(X_1,one_one(int)),P_3))))) # label(fact_313_zcong__neg__1__impl__ne__1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 380 (all Z_1 all W (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W),Z_1)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_1),W)))) # label(fact_24_zle__linear) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 381 (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]. 1.53/1.79 382 (all A_3 all B_2 all Ma ((exists K_2 hAPP(int,int,plus_plus(int,A_3),hAPP(int,int,times_times(int,Ma),K_2)) = ti(int,B_2)) <-> hBOOL(hAPP(int,bool,zcong(A_3,B_2),Ma)))) # label(fact_417_zcong__iff__lin) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 383 (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_42_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 384 (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_396_zero__le__one) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 385 (all X_a (plus(X_a) & linorder(X_a) & dvd(X_a) -> (all D all Sa exists Z all X_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),X_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,X_2),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_2),Sa)))))))) # label(fact_867_pinf_I10_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 386 (all X all Y_1 (X = zero_zero(real) & Y_1 = zero_zero(real) <-> hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,X),X)),hAPP(real,real,times_times(real,Y_1),Y_1)) = zero_zero(real))) # label(fact_709_real__two__squares__add__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 387 (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]. 1.53/1.79 388 (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)) -> (hAPP(nat,nat,plus_plus(nat,K),I_2) = J_2 <-> hAPP(nat,nat,minus_minus(nat,J_2),I_2) = K))) # label(fact_676_le__imp__diff__is__add) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 389 (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_754_norR__mem__unique__aux) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 390 (all M all N_1 (M = hAPP(nat,nat,times_times(nat,M),N_1) -> zero_zero(nat) = M | one_one(nat) = N_1)) # label(fact_708_mult__eq__self__implies__10) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 391 (all X_a (linordered_ring(X_a) -> (all X_1 all Y 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),Y))))))) # label(fact_46_sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 392 (all X_a all X_c all X_b all B_1_1 all B_2_1 combb(X_b,X_c,X_a,B_1_1,B_2_1) = ti(fun(X_a,X_c),combb(X_b,X_c,X_a,B_1_1,B_2_1))) # label(tsy_c_COMBB_res) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 393 (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),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,times_times(X_a,C_1),A_3)),hAPP(X_a,X_a,times_times(X_a,C_1),B_2)))))))) # label(fact_432_mult__le__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 394 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K)),bit0(L))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L)))) # label(fact_36_rel__simps_I31_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 395 (all N_1 (hAPP(nat,int,power_power(int,number_number_of(int,min)),N_1) = number_number_of(int,min) | hAPP(nat,int,power_power(int,number_number_of(int,min)),N_1) = one_one(int))) # label(fact_506_neg__one__power) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 396 (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_178_one__add__one__is__two) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 397 (all X_a (semiring_0(X_a) & power(X_a) -> (all 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)) & (N_1 = zero_zero(nat) -> hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),N_1) = one_one(X_a)))))) # label(fact_442_power__0__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 398 (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_782_add__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 399 (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_879_gcd__lcm__complete__lattice__nat_Onot__less__bot) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 400 (all X_a (ab_semigroup_mult(X_a) -> (all A_1 all B all C hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),B)),C) = hAPP(X_a,X_a,times_times(X_a,A_1),hAPP(X_a,X_a,times_times(X_a,B),C))))) # label(fact_758_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 401 (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)) -> (C = B -> 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_557_dvd_Oord__less__eq__trans) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 402 (all B_1_1 sr(ti(int,B_1_1)) = sr(B_1_1)) # label(tsy_c_Residues_OSR_arg1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 403 (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(int),bit0(K)),min)))) # label(fact_302_rel__simps_I11_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 404 (all X_1 all P_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),P_3)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),standardRes(P_3,X_1)),P_3)))) # label(fact_991_StandardRes__ubound) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 405 (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_655_dvd__reduce) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 406 (all B_1_1 all X_a (power(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_1_res) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 407 (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),M),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),N_1),M)))) # label(fact_416_zdvd__bounds) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 408 (all A_1 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)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,minus_minus(int,P_3),one_one(int)))) -> inv(P_3,A_1) != one_one(int))))) # label(fact_823_inv__not__1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 409 (all K_1 ti(int,K_1) = hAPP(int,int,minus_minus(int,K_1),pls)) # label(fact_471_diff__bin__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 410 (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_961_neg__mod__conj) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 411 (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_342_le__imp__power__dvd) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 412 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K1)),bit0(K2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)))) # label(fact_35_less__eq__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 413 (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_204_zdvd__antisym__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 414 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),number_number_of(X_a,X)),one_one(X_a))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),bit1(pls))))))) # label(fact_179_less__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 415 (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_353_zcong__zmult__prop2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 416 (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_160_bin__less__0__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 417 (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_701_mult__less__mono1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 418 -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))) -> 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))) # label(fact_467__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]. 1.53/1.79 419 (all N_1 -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),N_1))) # label(fact_547_less__irrefl__nat) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 420 (all X_a (linordered_ring(X_a) -> (all X_1 all Y -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),Y))),zero_zero(X_a)))))) # label(fact_108_not__sum__squares__lt__zero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 421 (all M all N_1 (M = N_1 -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)))) # label(fact_598_eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 422 (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_817_add__neg__nonpos) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 423 (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_733_nat__le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 424 (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_804_add__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 425 (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_147_add__numeral__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 426 (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_330_dvd__mult2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 427 (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_933_mod__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 428 (all B_1_1 all X_a (group_add(X_a) -> ti(fun(X_a,X_a),minus_minus(X_a,B_1_1)) = minus_minus(X_a,B_1_1))) # label(tsy_c_Groups_Ominus__class_Ominus_0_res) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 429 (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_425_mult__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 430 (all X_a all X_c all X_b all P all Q all R hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),P,R),Q) = hAPP(X_a,X_c,combc(X_a,X_b,X_c,P,Q),R)) # label(help_COMBC_1_1_U) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 431 (all X all N (hAPP(nat,nat,power_power(nat,X),N) = one_one(nat) <-> X = one_one(nat) | zero_zero(nat) = N)) # label(fact_526_exp__eq__1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 432 (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,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_4),B_3)),C)))))) # label(fact_923_mod__add__cong) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 433 (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_712_dvd__mult__cancel) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 434 (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]. 1.53/1.79 435 (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]. 1.53/1.79 436 (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]. 1.53/1.79 437 (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_816_add__strict__increasing2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 438 (all X all N all Y_1 (hAPP(int,int,div_mod(int,Y_1),N) = hAPP(int,int,div_mod(int,X),N) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N),hAPP(int,int,minus_minus(int,X),Y_1))))) # label(fact_934_zmod__eq__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 439 (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_727_left__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 440 (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_383_mult__neg__neg) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 441 (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_567_dvd__diff__nat) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 442 (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]. 1.53/1.79 443 (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,A_1),hAPP(X_a,X_a,div_mod(X_a,B),C))),C)))) # label(fact_952_mod__diff__right__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 444 (all 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))) <-> zero_zero(nat) != A_3)) # label(fact_872_gcd__lcm__complete__lattice__nat_Oless__top) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 445 (all M all N_1 all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_3),hAPP(int,int,times_times(int,M),N_1))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_3),N_1)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_3),M))))) # label(fact_420_zprime__zdvd__zmult__better) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 446 (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_876_gcd__lcm__complete__lattice__nat_Obot__unique) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 447 (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_807_add__pos__pos) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 448 (all Ma all D (hAPP(int,int,div_mod(int,Ma),D) = zero_zero(int) <-> (exists Q_2 hAPP(int,int,times_times(int,D),Q_2) = ti(int,Ma)))) # label(fact_935_zmod__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 449 (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_925_zmod__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 450 (all A_1 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)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,minus_minus(int,P_3),one_one(int)))) -> inv(P_3,A_1) != zero_zero(int))))) # label(fact_825_inv__not__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.79 451 (all A_1 all R_1 all B all M all C all D_2 all 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)))) = 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))) # label(fact_486_xzgcda__linear__aux1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 452 (all Ma all N (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)),Ma)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,plus_plus(nat,Ma),N))))) # label(fact_695_add__gr__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 453 (all X_a all P_1 ti(fun(X_a,bool),P_1) = collect(X_a,P_1)) # label(fact_76_Collect__def) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 454 (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_479_eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 455 (all M hAPP(nat,nat,minus_minus(nat,M),M) = zero_zero(nat)) # label(fact_538_diff__self__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 456 (all N all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ma)) -> (one_one(int) = ti(int,N) & ti(int,Ma) = one_one(int) <-> one_one(int) = hAPP(int,int,times_times(int,Ma),N)))) # label(fact_113_pos__zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 457 (all A_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1)) -> (exists P_5 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),A_1)) & hBOOL(hAPP(int,bool,zprime,P_5)))))) # label(fact_740_zprime__factor__exists) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 458 (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_681_add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 459 (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_275_power__strict__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 460 (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_487_zcong__def) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 461 (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_967_zmult2__lemma__aux1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 462 (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_373_mult__nonneg__nonpos) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 463 (all X (-hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,times_times(real,X),X))) <-> X = zero_zero(real))) # label(fact_468_not__real__square__gt__zero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 464 (all N_1 hAPP(nat,nat,times_times(nat,one_one(nat)),N_1) = N_1) # label(fact_688_nat__mult__1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 465 (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_421_mult__left__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 466 (all V_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),V_3),pls)) <-> zero_zero(nat) = number_number_of(nat,V_3))) # label(fact_40_eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 467 (all X_a (number_ring(X_a) -> (all 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)) = number_number_of(X_a,bit1(W))))) # label(fact_170_number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 468 (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_480_right__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 469 (all X_a (number_ring(X_a) -> zero_zero(X_a) = number_number_of(X_a,pls))) # label(fact_146_semiring__norm_I112_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 470 (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_126_less__int__code_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 471 (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_987_mod__less__eq__dividend) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 472 (all X all P_2 (hBOOL(hAPP(fun(int,bool),bool,member(int,X),sr(P_2))) -> standardRes(P_2,X) = ti(int,X))) # label(fact_992_StandardRes__SR__prop) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 473 (all N_1 zero_zero(nat) = hAPP(nat,nat,minus_minus(nat,zero_zero(nat)),N_1)) # label(fact_536_diff__0__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 474 (all J_1 all A_1 all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_3)) -> (-hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_3)) -> (-hBOOL(hAPP(int,bool,zcong(J_1,zero_zero(int)),P_3)) -> (-hBOOL(hAPP(int,bool,quadRes(P_3),A_1)) -> -hBOOL(hAPP(int,bool,zcong(J_1,hAPP(int,int,times_times(int,A_1),multInv(P_3,J_1))),P_3)))))))) # label(fact_835_MultInvPair__distinct) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 475 (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_261_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 476 (all N_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N_1))) # label(fact_534_le0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 477 (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_250_zero__le__even__power_H) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 478 (all V all 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))) = hAPP(int,int,div_mod(int,number_number_of(int,bit0(V))),number_number_of(int,bit0(W)))) # label(fact_964_zmod__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 479 (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_364_mult__mono_H) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 480 (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_983_mod__geq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 481 (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_919_dvd__mod__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 482 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(K1)),bit0(K2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)))) # label(fact_154_less__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 483 (all B_2 all A_3 all P_2 (hBOOL(hAPP(int,bool,zprime,P_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(int,int,minus_minus(int,P_2),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_2)))))))) # label(fact_834_wset__mem) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 484 (all X_a (ring_n68954251visors(X_a) -> (all A_3 all B_2 (zero_zero(X_a) = ti(X_a,A_3) | zero_zero(X_a) = ti(X_a,B_2) <-> zero_zero(X_a) = hAPP(X_a,X_a,times_times(X_a,A_3),B_2))))) # label(fact_317_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 485 (all B_1_1 all X_a (ab_semigroup_mult(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_0_arg1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 486 (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_2 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_2),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_2),Ta))))))))) # label(fact_746_inf__period_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 487 (all X_a (cancel_semigroup_add(X_a) -> (all B 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,B),A_1) -> ti(X_a,C) = ti(X_a,B))))) # label(fact_764_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 488 (all P_2 (hBOOL(hAPP(int,bool,zprime,P_2)) -> wset(hAPP(int,int,minus_minus(int,P_2),number_number_of(int,bit0(bit1(pls)))),P_2) = d22set(hAPP(int,int,minus_minus(int,P_2),number_number_of(int,bit0(bit1(pls))))))) # label(fact_857_d22set__eq__wset) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 489 (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)),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))) | 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))))))) # label(fact_375_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 490 (all B_1_1 all B_2_1 ti(product_prod(int,int),product_Pair(int,int,B_1_1,B_2_1)) = product_Pair(int,int,B_1_1,B_2_1)) # label(tsy_c_Product__Type_OPair_res) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 491 (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_856_mono__nat__linear__lb) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 492 (all X_1 all P_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_3)) -> (hBOOL(hAPP(int,bool,zprime,P_3)) -> (-hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),P_3)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,X_1),multInv(P_3,X_1))),multInv(P_3,multInv(P_3,X_1))),X_1),P_3)))))) # label(fact_848_Int2_Oaux____2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 493 (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),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),B_2),A_3))))))) # label(fact_431_mult__le__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 494 (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_69_zadd__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 495 (all K all Ma all N (Ma = N <-> hAPP(nat,nat,plus_plus(nat,K),N) = hAPP(nat,nat,plus_plus(nat,K),Ma))) # label(fact_586_nat__add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 496 (all X_a (linordered_idom(X_a) -> (all Y 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)) -> (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),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)),X_1)))))))) # label(fact_434_mult__right__le__one__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 497 (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_477_zdvd__zdiffD) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 498 (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),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_eq(nat),Ma),N)))))) # label(fact_601_le__diff__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 499 (all X_a (comm_semiring_1(X_a) -> (all M all A_1 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) = hAPP(X_a,X_a,plus_plus(X_a,M),hAPP(X_a,X_a,times_times(X_a,A_1),M))))) # label(fact_253_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 500 (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_56_zadd__zless__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 501 (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_812_less__iff__diff__less__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 502 (all X_a (semiring_div(X_a) -> (all A_1 zero_zero(X_a) = hAPP(X_a,X_a,div_mod(X_a,A_1),one_one(X_a))))) # label(fact_944_mod__by__1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 503 (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),zero_zero(X_a)),hAPP(nat,X_a,power_power(X_a,A_3),number_number_of(nat,bit0(bit1(pls)))))) <-> ti(X_a,A_3) != zero_zero(X_a))))) # label(fact_11_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 504 (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)) -> zero_zero(nat) = hAPP(nat,nat,times_times(nat,number_number_of(nat,V)),number_number_of(nat,V_1))))) # label(fact_230_mult__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 505 (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_683_le__square) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 506 (all X_a (linord626643107strict(X_a) -> (all V all U_1 all Y 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),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)) -> (hAPP(X_a,X_a,plus_plus(X_a,U_1),V) = 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,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))),A_1)))))))))) # label(fact_276_convex__bound__lt) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 507 (all X all Y_1 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),X),Y_1)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,times_times(real,X),Z_2)),hAPP(real,real,times_times(real,Y_1),Z_2)))))) # label(fact_707_real__mult__less__iff1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 508 (all X all P_1 ((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_1,hAPP(int,int,minus_minus(int,A_5),one_one(int))))) -> hBOOL(hAPP(int,bool,P_1,A_5)))) -> hBOOL(hAPP(int,bool,P_1,X)))) # label(fact_852_d22set__induct__old) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 509 (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_548_less__not__refl2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 510 (all A_1 all B all P_3 all Q_1 twoSqu1929807760sum2sq(product_Pair(int,int,hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A_1),P_3)),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_3)))) = hAPP(int,int,times_times(int,twoSqu1929807760sum2sq(product_Pair(int,int,A_1,B))),twoSqu1929807760sum2sq(product_Pair(int,int,P_3,Q_1)))) # label(fact_752_mult__sum2sq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 511 (all X_a (ab_semigroup_add(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_759_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 512 (all A_3 all B_2 (hBOOL(hAPP(int,bool,zcong(A_3,B_2),zero_zero(int))) <-> ti(int,A_3) = ti(int,B_2))) # label(fact_349_IntPrimes_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 513 (all M all N_1 (hAPP(nat,nat,minus_minus(nat,M),N_1) = zero_zero(nat) -> (zero_zero(nat) = hAPP(nat,nat,minus_minus(nat,N_1),M) -> N_1 = M))) # label(fact_539_diffs0__imp__equal) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 514 (all B_1_1 all B_2_1 wset(B_1_1,B_2_1) = ti(fun(int,bool),wset(B_1_1,B_2_1))) # label(tsy_c_WilsonRuss_Owset_res) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 515 (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,hAPP(X_a,X_a,times_times(X_a,Lx),Rx)),hAPP(X_a,X_a,times_times(X_a,Ly),Ry))))) # label(fact_212_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 516 (all X all P_2 (hBOOL(hAPP(int,bool,zcong(X,zero_zero(int)),P_2)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_2),X)))) # label(fact_418_zcong__eq__zdvd__prop) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 517 (all X_a (linordered_idom(X_a) -> (all X_1 all Y 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),number_number_of(nat,bit0(bit1(pls)))))))))) # label(fact_106_sum__power2__ge__zero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 518 (all Ma all Ta all K (ti(int,K) != zero_zero(int) -> (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))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Ma),Ta))))) # label(fact_743_zdvd__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 519 (all B_1_1 all X_a (ab_semigroup_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_3_arg1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 520 (all X_a all X_c all B_1_1 all B_2_1 hAPP(X_a,X_c,ti(fun(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_arg1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 521 (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_297_rel__simps_I30_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 522 (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_189_self__quotient__aux2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 523 (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),ord_less_eq(nat),Ma),N)) & Ma != N)) # label(fact_648_nat__less__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 524 (all X all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X)) | N = zero_zero(nat) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,power_power(nat,X),N))))) # label(fact_344_nat__zero__less__power__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 525 (all B_1_1 quadRes(B_1_1) = quadRes(ti(int,B_1_1))) # label(tsy_c_Residues_OQuadRes_arg1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 526 (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_377_zero__le__square) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 527 (all K_1 all L_1 bit0(K_1) != bit1(L_1)) # label(fact_118_rel__simps_I49_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 528 (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,hAPP(X_a,X_a,minus_minus(X_a,A_3),B_2)),E)),C_1)),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_488_le__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 529 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),number_number_of(X_a,X)),one_one(X_a))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X),bit1(pls))))))) # label(fact_59_le__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 530 (all B_1_1 all X_a (cancel_semigroup_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_2_res) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 531 (all K 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),ord_less(nat),hAPP(nat,nat,plus_plus(nat,K),Ma)),hAPP(nat,nat,plus_plus(nat,K),N))))) # label(fact_639_nat__add__left__cancel__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 532 (all Y all N_1 all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_3),hAPP(nat,int,power_power(int,Y),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_3),Y)))))) # label(fact_277_zpower__zdvd__prop2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 533 (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))))) -> one_one(int) = ti(int,X_1) | ti(int,X_1) = zero_zero(int)))) # label(fact_184_int__pos__lt__two__imp__zero__or__one) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 534 (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,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)) = 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))))) # label(fact_973_pos__zmod__mult__2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 535 (all Z_1 all X_1 all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X_1)) -> (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z_1),Y)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),Z_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_554_dvd_Oless__trans) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 536 (all N (zero_zero(nat) != N <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)))) # label(fact_615_neq0__conv) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 537 (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_658_le__add2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 538 (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,M),N_1) = hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,minus_minus(nat,M),N_1)),N_1))) # label(fact_981_le__mod__geq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 539 (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]. 1.53/1.80 540 (all A_3 all B_2 all C_1 (C_1 != zero_zero(real) -> (B_2 = A_3 <-> hAPP(real,real,times_times(real,A_3),C_1) = hAPP(real,real,times_times(real,B_2),C_1)))) # label(fact_636_real__mult__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 541 (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_428_mult__less__le__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 542 (all C_1 all X 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,X),Ta))) <-> 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),hAPP(int,int,times_times(int,C_1),D))),Ta)))))) # label(fact_207_zdvd__period) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 543 (all Ma all N (zero_zero(nat) = hAPP(nat,nat,plus_plus(nat,Ma),N) <-> Ma = zero_zero(nat) & N = zero_zero(nat))) # label(fact_623_add__is__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 544 (all X_a (monoid_mult(X_a) -> (all A_1 hAPP(nat,X_a,power_power(X_a,A_1),number_number_of(nat,bit1(bit1(pls)))) = hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),A_1)),A_1)))) # label(fact_137_power3__eq__cube) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 545 (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_398_zero__less__one) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 546 (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_404_less__add__one) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 547 (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]. 1.53/1.80 548 (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]. 1.53/1.80 549 (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_96_zadd__assoc) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 550 (all X all Y_1 (X = Y_1 <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y_1)))) # label(fact_581_dvd_Oeq__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 551 (all X_a (semiri456707255roduct(X_a) -> (all W_1 all Y_1 all X all 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),Y_1)) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,W_1),Y_1)),hAPP(X_a,X_a,times_times(X_a,X),Z_2)) <-> ti(X_a,Y_1) = ti(X_a,Z_2) | ti(X_a,W_1) = ti(X_a,X))))) # label(fact_243_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 552 (all Z_1 ti(int,Z_1) = hAPP(int,int,plus_plus(int,zero_zero(int)),Z_1)) # label(fact_134_zadd__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 553 (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_457_one__less__power) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 554 (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_711_dvd__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 555 (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_493_Euler_Oaux1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 556 (all Ma all X (hBOOL(hAPP(int,bool,zcong(X,zero_zero(int)),Ma)) <-> standardRes(Ma,X) = zero_zero(int))) # label(fact_988_StandardRes__eq__zcong) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 557 (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_640_trans__less__add1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 558 (all Ma (Ma = one_one(nat) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),one_one(nat))))) # label(fact_682_nat__dvd__1__iff__1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 559 (all X_a (number_semiring(X_a) -> zero_zero(X_a) = number_number_of(X_a,pls))) # label(fact_144_semiring__numeral__0__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 560 (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),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_eq(X_a),A_3),zero_zero(X_a))))))) # label(fact_799_double__add__le__zero__iff__single__add__le__zero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 561 (all B_1_1 (hBOOL(ti(bool,B_1_1)) <-> hBOOL(B_1_1))) # label(tsy_c_hBOOL_arg1) # label(hypothesis) # label(non_clause). [assumption]. 1.53/1.80 562 (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_700_mult__less__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 563 (all M all Y 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)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M)) -> (hBOOL(hAPP(int,bool,zcong(X_1,Y),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),M)) -> ti(int,Y) = ti(int,X_1)))))))) # label(fact_415_zcong__less__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 564 (all L_1 bit0(L_1) != min) # label(fact_287_rel__simps_I42_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 565 (all A_3 all B_2 all C_1 (C_1 != zero_zero(real) -> (hAPP(real,real,times_times(real,C_1),A_3) = hAPP(real,real,times_times(real,C_1),B_2) <-> A_3 = B_2))) # label(fact_635_real__mult__left__cancel) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 566 (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_921_dvd__mod) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 567 (all X_a (linordered_semidom(X_a) -> (all X_1 all Y (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),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)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y))))))) # label(fact_102_power2__le__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 568 (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_713_mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 569 (all Z_1 all X_1 all Y (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),Y)) -> 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))))) # label(fact_612_real__add__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 570 (all B hAPP(int,int,div_mod(int,zero_zero(int)),B) = zero_zero(int)) # label(fact_918_zmod__zero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 571 (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]. 1.53/1.80 572 (all X_a (semiring_1(X_a) -> zero_zero(X_a) = hAPP(nat,X_a,power_power(X_a,zero_zero(X_a)),number_number_of(nat,bit0(bit1(pls)))))) # label(fact_13_zero__power2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 573 (all X_a all X_c all X_b all P all Q all R hAPP(X_a,X_c,combb(X_b,X_c,X_a,P,Q),R) = hAPP(X_b,X_c,P,hAPP(X_a,X_b,Q,R))) # label(help_COMBB_1_1_U) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 574 (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_454_power__Suc__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 575 (all X_1 all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y)))) # label(fact_562_dvd_Oless__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 576 (all P_4 all P_1 all X ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)) -> (hBOOL(P_1) <-> hBOOL(P_4))) -> (hBOOL(P_1) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X)) <-> hBOOL(P_4) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))))) # label(fact_741_conj__le__cong) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 577 (all N_1 -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),N_1))) # label(fact_544_less__not__refl) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 578 (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_187_power2__ge__self) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 579 (all X_a (semiring_div(X_a) -> (all A_1 all C all B 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,C),B))),B)))) # label(fact_943_mod__mult__self1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 580 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all X all Y_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X),Y_1)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),number_number_of(X_a,X)),number_number_of(X_a,Y_1))))))) # label(fact_27_le__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 581 (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,M),L_1) = hAPP(nat,nat,plus_plus(nat,K_1),N_1) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1))))) # label(fact_644_less__add__eq__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 582 (all X_1 all M all Y hAPP(int,int,div_mod(int,hAPP(int,int,minus_minus(int,X_1),Y)),M) = hAPP(int,int,div_mod(int,hAPP(int,int,minus_minus(int,hAPP(int,int,div_mod(int,X_1),M)),Y)),M)) # label(fact_948_zdiff__zmod__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 583 (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_609_nat__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 584 (all X_a (comm_semiring_1(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_241_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 585 (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_100_zmult__zless__mono2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 586 (all A_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),zero_zero(nat)))) # label(fact_873_gcd__lcm__complete__lattice__nat_Otop__greatest) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 587 (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_131_add__Bit0__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 588 (all X_a (real_normed_algebra(X_a) -> (all X_1 all Y all Ya hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,plus_plus(X_a,X_1),Y)),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),Ya))))) # label(fact_893_mult__left_Oadd) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 589 (all X_a (ring_11004092258visors(X_a) -> (all A_3 (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) = zero_zero(X_a))))) # label(fact_14_zero__eq__power2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 590 (all Z1 all Z2 all W hAPP(int,int,minus_minus(int,hAPP(int,int,times_times(int,Z1),W)),hAPP(int,int,times_times(int,Z2),W)) = hAPP(int,int,times_times(int,hAPP(int,int,minus_minus(int,Z1),Z2)),W)) # label(fact_473_zdiff__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 591 (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_813_add__pos__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 592 (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_300_rel__simps_I28_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 593 (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)) -> hAPP(int,int,div_mod(int,A_1),B) = ti(int,A_1)))) # label(fact_962_mod__neg__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 594 (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]. 1.53/1.80 595 (all A_1 all B all C hAPP(int,int,div_mod(int,hAPP(int,int,times_times(int,A_1),hAPP(int,int,div_mod(int,B),C))),C) = hAPP(int,int,div_mod(int,hAPP(int,int,times_times(int,A_1),B)),C)) # label(fact_909_zmod__zmult1__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 596 (all X_a (power(X_a) & zero_neq_one(X_a) & no_zero_divisors(X_a) & mult_zero(X_a) -> (all A_3 all N (hAPP(nat,X_a,power_power(X_a,A_3),N) = zero_zero(X_a) <-> zero_zero(X_a) = ti(X_a,A_3) & zero_zero(nat) != N)))) # label(fact_410_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 597 (all X all Y_1 (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),Y_1)) <-> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,minus_minus(real,X),Y_1)),zero_zero(real))))) # label(fact_541_real__le__eq__diff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 598 (all X all Y_1 (hAPP(int,int,div_mod(int,number_number_of(int,Y_1)),number_number_of(int,X)) = zero_zero(int) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),number_number_of(int,X)),number_number_of(int,Y_1))))) # label(fact_953_zdvd__iff__zmod__eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 599 (all M all N_1 (M = hAPP(nat,nat,plus_plus(nat,M),N_1) -> N_1 = zero_zero(nat))) # label(fact_624_add__eq__self__zero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 600 (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_225_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 601 (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_232_power__even__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 602 (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_530_realpow__pos__nth) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 603 (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_221_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 604 (all M all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N_1)) | M = N_1 -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N_1)))) # label(fact_652_less__or__eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 605 (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),C_1),D)) <-> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_3),B_2))))))) # label(fact_790_diff__eq__diff__less__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 606 (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_476_zcong__zdiff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 607 (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_446_four__x__squared) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 608 (all N all Ma (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Ma)) -> (one_one(nat) = N <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,times_times(nat,N),Ma)),Ma))))) # label(fact_531_dvd__mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 609 (all X_a (comm_semiring_1(X_a) -> (all A_1 all B hAPP(X_a,X_a,times_times(X_a,B),A_1) = hAPP(X_a,X_a,times_times(X_a,A_1),B)))) # label(fact_219_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 610 (all K all P_1 all D (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D)) -> ((all X_2 (hBOOL(hAPP(int,bool,P_1,X_2)) -> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,minus_minus(int,X_2),D))))) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) -> (all X_2 (hBOOL(hAPP(int,bool,P_1,X_2)) -> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,minus_minus(int,X_2),hAPP(int,int,times_times(int,K),D)))))))))) # label(fact_748_decr__mult__lemma) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 611 (all B_1_1 all B_2_1 wset(ti(int,B_1_1),B_2_1) = wset(B_1_1,B_2_1)) # label(tsy_c_WilsonRuss_Owset_arg1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 612 (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_937_zmod__le__nonneg__dividend) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 613 (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_294_rel__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 614 (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_72_less__eq__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 615 (all X_a (comm_semiring_1(X_a) -> (all X_1 hAPP(nat,X_a,power_power(X_a,X_1),zero_zero(nat)) = one_one(X_a)))) # label(fact_248_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 616 (all X_1 all Y (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),Y)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y),X_1)) | ti(int,Y) = ti(int,X_1))) # label(fact_92_zless__linear) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 617 (all C all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) -> (C = B -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C))))) # label(fact_571_dvd_Oord__le__eq__trans) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 618 (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_912_mod__mod__trivial) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 619 (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_685_mult__le__mono1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.80 620 (all B_2 all A_3 all P_2 (hBOOL(hAPP(int,bool,zprime,P_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,bit1(bit0(bit1(pls))))),P_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(int,int,minus_minus(int,P_2),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_2),one_one(int)))) -> (hBOOL(hAPP(fun(int,bool),bool,member(int,inv(P_2,B_2)),wset(A_3,P_2))) -> hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),wset(A_3,P_2)))))))))) # label(fact_828_wset__inv__mem__mem) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 621 (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_440_power__less__power__Suc) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 622 (all X_a all X_b all X_c all B_1_1 all 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) = combc(X_a,X_b,X_c,B_1_1,B_2_1)) # label(tsy_c_COMBC_arg1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 623 (all X_a (semiri456707255roduct(X_a) -> (all C all D_2 all A_1 all B all R_1 (ti(X_a,R_1) != zero_zero(X_a) -> (ti(X_a,C) != ti(X_a,D_2) & ti(X_a,A_1) = ti(X_a,B) -> hAPP(X_a,X_a,plus_plus(X_a,B),hAPP(X_a,X_a,times_times(X_a,R_1),D_2)) != hAPP(X_a,X_a,plus_plus(X_a,A_1),hAPP(X_a,X_a,times_times(X_a,R_1),C))))))) # label(fact_251_add__scale__eq__noteq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 624 (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]. 1.53/1.81 625 (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_23_zmult__commute) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 626 (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_859_d22set__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 627 (all J_1 all K_1 all A_1 all P_3 (hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,J_1),K_1),A_1),P_3)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,multInv(P_3,J_1)),J_1)),K_1),hAPP(int,int,times_times(int,multInv(P_3,J_1)),A_1)),P_3)))) # label(fact_838_aux______3) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 628 (all N_1 all B all A_1 all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) -> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_3),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,power_power(int,P_3),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_3),N_1)),B)))))) # label(fact_201_zprime__power__zdvd__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 629 (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)))) # label(fact_871_gcd__lcm__complete__lattice__nat_Otop__unique) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 630 (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]. 1.53/1.81 631 (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_776_add__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 632 (all N_1 zero_zero(nat) = hAPP(nat,nat,times_times(nat,zero_zero(nat)),N_1)) # label(fact_630_mult__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 633 (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_969_zmult2__lemma__aux3) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 634 (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_354_zcong__scalar) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 635 (all X_a (real_normed_algebra(X_a) -> (all X_1 zero_zero(X_a) = hAPP(X_a,X_a,times_times(X_a,X_1),zero_zero(X_a))))) # label(fact_889_mult__right_Ozero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 636 (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_674_le__diff__conv2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 637 (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,ti(fun(X_a,X_b),B_2_1)) = combb(X_b,X_c,X_a,B_1_1,B_2_1)) # label(tsy_c_COMBB_arg2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 638 (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_44_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 639 (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_620_zero__less__diff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 640 (all N_1 ((-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),N_1),zero_zero(int))) -> hAPP(int,int,times_times(int,N_1),zfact(hAPP(int,int,minus_minus(int,N_1),one_one(int)))) = zfact(N_1)) & (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),N_1),zero_zero(int))) -> zfact(N_1) = one_one(int)))) # label(fact_818_zfact_Osimps) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 641 (all X_a (one(X_a) -> one_one(X_a) = ti(X_a,one_one(X_a)))) # label(tsy_c_Groups_Oone__class_Oone_res) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 642 (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),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,A_3),C_1)),hAPP(X_a,X_a,plus_plus(X_a,B_2),C_1))))))) # label(fact_779_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 643 (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,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),K_1)),J_1))) # label(fact_669_diff__diff__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 644 (all X_1 all Y 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),number_number_of(nat,bit0(bit1(pls))))) = hAPP(nat,nat,times_times(nat,hAPP(nat,nat,plus_plus(nat,X_1),Y)),hAPP(nat,nat,minus_minus(nat,X_1),Y))) # label(fact_519_diff__square) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 645 (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,hAPP(nat,nat,minus_minus(nat,I_1),K_1)),J_1)) # label(fact_535_diff__commute) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 646 (all K all Ma 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,K),Ma)),hAPP(nat,nat,times_times(nat,K),N))))) # label(fact_730_nat__mult__dvd__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 647 (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_800_add__nonneg__nonneg) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 648 (all X_a (real_normed_algebra(X_a) -> (all X_1 all Y all A_1 all B hAPP(X_a,X_a,minus_minus(X_a,hAPP(X_a,X_a,times_times(X_a,X_1),Y)),hAPP(X_a,X_a,times_times(X_a,A_1),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),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),B)))))) # label(fact_884_mult_Oprod__diff__prod) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 649 (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_788_diff__self) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 650 (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_938_pos__mod__bound) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 651 (all N_1 all M ((zero_zero(nat) = M -> zero_zero(nat) = hAPP(nat,nat,times_times(nat,M),N_1)) & (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))))) # label(fact_517_mult__eq__if) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 652 (all P_3 all Y 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_3),Y)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_3),hAPP(nat,int,power_power(int,Y),N_1)))))) # label(fact_450_zpower__zdvd__prop1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 653 (all K all N all Ma (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),N)) <-> 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)))))) # label(fact_206_zdvd__reduce) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 654 (all Z1 all Z2 all Z3 hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,Z1),Z2)),Z3) = hAPP(int,int,times_times(int,Z1),hAPP(int,int,times_times(int,Z2),Z3))) # label(fact_28_zmult__assoc) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 655 (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_972_zmod__minus1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 656 (all X_a (real_normed_algebra(X_a) -> (all Xa all X_1 all Y hAPP(X_a,X_a,times_times(X_a,Xa),hAPP(X_a,X_a,minus_minus(X_a,X_1),Y)) = 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))))) # label(fact_895_mult__right_Odiff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 657 (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),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_928_mod__add__right__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 658 (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_659_le__add1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 659 (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_236_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 660 (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_296_rel__simps_I26_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 661 (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_774_add__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 662 (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_379_mult__strict__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 663 (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,hAPP(X_a,X_a,div_mod(X_a,A_1),B)),C) = hAPP(X_a,X_a,div_mod(X_a,A_1),C))))) # label(fact_920_mod__mod__cancel) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 664 (all X_c all X_a all B_1_1 all B_2_1 hAPP(X_a,X_c,B_1_1,B_2_1) = ti(X_c,hAPP(X_a,X_c,B_1_1,B_2_1))) # label(tsy_c_hAPP_res) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 665 (all X_a (linordered_idom(X_a) -> (all X all Y_1 (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),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))) <-> zero_zero(X_a) = ti(X_a,Y_1) & zero_zero(X_a) = ti(X_a,X))))) # label(fact_105_sum__power2__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 666 (all Z1 all Z2 all W hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,Z1),W)),hAPP(int,int,times_times(int,Z2),W)) = hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,Z1),Z2)),W)) # label(fact_68_zadd__zmult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 667 (all P_1 all N all 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_1,J)))))) & (zero_zero(nat) = K -> hBOOL(hAPP(nat,bool,P_1,N))) <-> hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,div_mod(nat,N),K))))) # label(fact_993_split__mod) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 668 (all Z_1 all X_1 all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),Z_1)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Z_1))))) # label(fact_568_dvd_Oorder__trans) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 669 (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_811_le__iff__diff__le__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 670 (all X_1 all Y 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)),hAPP(int,int,times_times(int,X_1),Y)),M)))) # label(fact_975_StandardRes__prop4) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 671 (all Z_1 all W hAPP(real,real,times_times(real,W),Z_1) = hAPP(real,real,times_times(real,Z_1),W)) # label(fact_594_real__mult__commute) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 672 (all B_1_1 all B_2_1 multInv(ti(int,B_1_1),B_2_1) = multInv(B_1_1,B_2_1)) # label(tsy_c_Int2_OMultInv_arg1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 673 (all Ma all N (hAPP(nat,nat,times_times(nat,Ma),N) = one_one(nat) <-> N = one_one(nat) & Ma = one_one(nat))) # label(fact_691_nat__mult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 674 (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]. 1.53/1.81 675 (all Z1 all Z2 all Z3 hAPP(real,real,times_times(real,hAPP(real,real,times_times(real,Z1),Z2)),Z3) = hAPP(real,real,times_times(real,Z1),hAPP(real,real,times_times(real,Z2),Z3))) # label(fact_593_real__mult__assoc) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 676 (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_660_le__iff__add) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 677 (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_427_mult__le__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 678 (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_343_nat__power__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 679 (all X_a (linord1278240602ring_1(X_a) -> (all V all U_1 all Y 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),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)) -> (hAPP(X_a,X_a,plus_plus(X_a,U_1),V) = 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,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))),A_1)))))))))) # label(fact_453_convex__bound__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 680 (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),bit1(K)),bit1(L))))) # label(fact_33_rel__simps_I34_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 681 (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_686_mult__le__mono2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 682 (all X_1 all Y hAPP(nat,real,power_power(real,hAPP(real,real,plus_plus(real,X_1),Y)),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),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))) # label(fact_510_real__sum__squared__expand) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 683 (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_380_comm__mult__strict__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 684 (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_951_mod__diff__left__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 685 (all X_a (ab_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_787_eq__iff__diff__eq__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 686 (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_585_nat__add__assoc) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 687 (all X_1 all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X_1)) -> Y = X_1))) # label(fact_569_dvd_Oantisym) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 688 (all X_a (comm_ring_1(X_a) -> (all Z_1 all X_1 all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X_1),Y)) -> (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),Z_1)))))))) # label(fact_470_dvd__diff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 689 (all A_3 all P_2 (hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_3),hAPP(int,int,minus_minus(int,P_2),one_one(int))),one_one(int)),P_2)) <-> hBOOL(hAPP(int,bool,zcong(A_3,hAPP(int,int,minus_minus(int,P_2),one_one(int))),P_2)))) # label(fact_498_inv__not__p__minus__1__aux) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 690 (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_82_zless__imp__add1__zle) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 691 (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_784_add__less__imp__less__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 692 (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_863_less__imp__add__positive) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 693 (all X_a (real_normed_algebra(X_a) -> (all Xa all X_1 all Y hAPP(X_a,X_a,times_times(X_a,Xa),hAPP(X_a,X_a,plus_plus(X_a,X_1),Y)) = 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))))) # label(fact_891_mult__right_Oadd) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 694 (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_941_dvd__imp__mod__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 695 (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_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_814_add__nonneg__pos) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 696 (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_724_real__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 697 (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)),bit1(L))))) # label(fact_53_rel__simps_I32_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 698 (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_775_add__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 699 (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]. 1.53/1.81 700 (all X_1 all P_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_3)) -> (hBOOL(hAPP(int,bool,zprime,P_3)) -> (-hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),P_3)) -> hBOOL(hAPP(int,bool,zcong(multInv(P_3,multInv(P_3,X_1)),hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,X_1),multInv(P_3,X_1))),multInv(P_3,multInv(P_3,X_1)))),P_3)))))) # label(fact_849_Int2_Oaux____1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 701 (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_673_add__diff__assoc) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 702 (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)) -> -(-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_555_dvd_Oless__asym_H) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 703 (all K (ti(int,K) = min <-> bit1(K) = min)) # label(fact_281_rel__simps_I47_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 704 (all Ma all K all N (N = Ma <-> hAPP(nat,nat,plus_plus(nat,N),K) = hAPP(nat,nat,plus_plus(nat,Ma),K))) # label(fact_587_nat__add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 705 (all X_a (plus(X_a) & dvd(X_a) & linorder(X_a) -> (all D all Sa exists Z all X_2 (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),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_2),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_2),Sa)))))))) # label(fact_865_minf_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 706 (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)),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))))))) # label(fact_805_zero__less__double__add__iff__zero__less__single__add) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 707 (all X_a (ring(X_a) -> (all X_1 all Y 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),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)),hAPP(X_a,X_a,times_times(X_a,A_1),B))))) # label(fact_693_mult__diff__mult) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 708 (all X_1 all Q_1 all N_1 all R_1 (X_1 = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,Q_1),N_1)),R_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_527_divides__div__not) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 709 (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]. 1.53/1.81 710 (all M hAPP(nat,nat,minus_minus(nat,M),zero_zero(nat)) = M) # label(fact_537_minus__nat_Odiff__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 711 (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_429_mult__strict__mono_H) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 712 (all Z_1 all X_1 all Y all P_3 (hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,X_1),Y),one_one(int)),P_3)) -> hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,X_1),hAPP(nat,nat,times_times(nat,Y),Z_1)),one_one(int)),P_3)))) # label(fact_451_zcong__zpower__zmult) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 713 (all X_1 all Y (Y != X_1 -> (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_1),Y)) -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Y),X_1))))) # label(fact_546_linorder__neqE__nat) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 714 (all X_1 all M hBOOL(hAPP(int,bool,zcong(X_1,standardRes(M,X_1)),M))) # label(fact_990_StandardRes__prop1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 715 (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_679_diff__add__assoc2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 716 (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_340_power__le__dvd) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 717 (all A_1 all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) -> (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_3)) -> (hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_1),A_1),one_one(int)),P_3)) -> ti(int,A_1) = one_one(int) | hAPP(int,int,minus_minus(int,P_3),one_one(int)) = ti(int,A_1)))))) # label(fact_500_zcong__square__zless) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 718 (all X_a (order(X_a) -> (all X_1 all Y (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y)) -> (ti(X_a,X_1) != ti(X_a,Y) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y))))))) # label(fact_314_order__le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 719 (all X_1 all P_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_3)) -> (hBOOL(hAPP(int,bool,zprime,P_3)) -> (-hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),P_3)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,multInv(P_3,X_1)),X_1),one_one(int)),P_3)))))) # label(fact_850_MultInv__prop2a) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 720 (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_721_nat__less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 721 (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_399_not__one__less__zero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 722 (all V_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),V_3),pls)) <-> number_number_of(nat,V_3) = zero_zero(nat))) # label(fact_39_eq__number__of__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 723 (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]. 1.53/1.81 724 (all X all Y_1 (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y_1)) | X = Y_1 <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y_1)))) # label(fact_580_dvd_Ole__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 725 (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]. 1.53/1.81 726 (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]. 1.53/1.81 727 (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_731_nat__mult__dvd__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 728 (all X_a (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__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 729 (all X_a (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__1__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 730 (all X_a (comm_semiring_1(X_a) -> (all A_1 zero_zero(X_a) = hAPP(X_a,X_a,times_times(X_a,zero_zero(X_a)),A_1)))) # label(fact_234_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 731 (all W pls = hAPP(int,int,times_times(int,pls),W)) # label(fact_62_mult__Pls) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 732 (all B_1_1 all X_a (dvd(X_a) & linorder(X_a) & plus(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_0_res) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 733 (all W hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),W),W))) # label(fact_722_real__le__refl) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 734 (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_369_mult__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 735 (all N_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),N_1))) # label(fact_600_le__refl) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 736 (all N_1 all M zero_zero(nat) = hAPP(nat,nat,minus_minus(nat,N_1),hAPP(nat,nat,plus_plus(nat,N_1),M))) # label(fact_625_diff__add__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 737 (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_447_Int2_Ozcong__zero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 738 -(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_18__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]. 1.53/1.81 739 (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_702_mult__less__mono2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 740 (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)),hAPP(X_a,X_a,div_mod(X_a,B),C))),C)))) # label(fact_905_mod__mult__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 741 (all Ma all N (hAPP(nat,nat,times_times(nat,Ma),N) = zero_zero(nat) <-> N = zero_zero(nat) | zero_zero(nat) = Ma)) # label(fact_632_mult__is__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 742 (all Ma all K all 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))) <-> zero_zero(nat) = K | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Ma),N)))) # label(fact_521_nat__mult__dvd__cancel__disj_H) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 743 (all A_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),one_one(nat))) -> A_1 = one_one(nat))) # label(fact_875_gcd__lcm__complete__lattice__nat_Ole__bot) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 744 (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_552_diff__less__mono2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 745 (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_384_mult__neg__pos) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 746 (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_391_mult__less__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 747 (all X_a (zero_neq_one(X_a) & no_zero_divisors(X_a) & mult_zero(X_a) & power(X_a) -> (all A_3 all W_1 (zero_zero(X_a) = ti(X_a,A_3) & zero_zero(nat) != number_number_of(nat,W_1) <-> hAPP(nat,X_a,power_power(X_a,A_3),number_number_of(nat,W_1)) = zero_zero(X_a))))) # label(fact_255_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 748 (all X_b (monoid_mult(X_b) & number(X_b) -> (all W hAPP(nat,X_b,power_power(X_b,number_number_of(X_b,W)),number_number_of(nat,bit0(bit1(pls)))) = hAPP(X_b,X_b,times_times(X_b,number_number_of(X_b,W)),number_number_of(X_b,W))))) # label(fact_167_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 749 (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_907_mod__mult__right__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 750 (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]. 1.53/1.81 751 (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_153_rel__simps_I12_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 752 (all L_1 bit0(hAPP(int,int,minus_minus(int,min),L_1)) = hAPP(int,int,minus_minus(int,min),bit1(L_1))) # label(fact_497_diff__bin__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 753 (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),A_3),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_3),A_3)),zero_zero(X_a))))))) # label(fact_806_double__add__less__zero__iff__single__add__less__zero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 754 (all X (one_one(int) = hAPP(int,int,div_mod(int,X),number_number_of(int,bit0(bit1(pls)))) <-> zero_zero(int) != hAPP(int,int,div_mod(int,X),number_number_of(int,bit0(bit1(pls)))))) # label(fact_971_neq__one__mod__two) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 755 (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_472_diff__bin__simps_I7_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 756 (all X_a (comm_semiring_1(X_a) -> (all Lx all Rx all Ry hAPP(X_a,X_a,times_times(X_a,Rx),hAPP(X_a,X_a,times_times(X_a,Lx),Ry)) = hAPP(X_a,X_a,times_times(X_a,Lx),hAPP(X_a,X_a,times_times(X_a,Rx),Ry))))) # label(fact_218_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 757 (all K all L (ti(int,K) = ti(int,L) <-> bit0(L) = bit0(K))) # label(fact_91_rel__simps_I48_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 758 (all X_a (comm_semiring_1(X_a) -> (all A_1 all M all B hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,plus_plus(X_a,A_1),B)),M) = 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))))) # label(fact_242_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 759 (all K_1 hAPP(int,int,plus_plus(int,K_1),pls) = ti(int,K_1)) # label(fact_129_add__Pls__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 760 (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_684_le__cube) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 761 (all A_1 all B hAPP(nat,int,power_power(int,hAPP(int,int,minus_minus(int,A_1),B)),number_number_of(nat,bit0(bit1(pls)))) = 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)))))) # label(fact_504_zdiff__power2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 762 (all P_2 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_2))))) # label(fact_829_wset__mem__mem) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 763 (all X_a all X_c all X_b all P all Q all R hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),P,R),hAPP(X_a,X_b,Q,R)) = hAPP(X_a,X_c,combs(X_a,X_b,X_c,P,Q),R)) # label(help_COMBS_1_1_U) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 764 (all X_1 all Y all Z_1 hAPP(nat,nat,plus_plus(nat,X_1),hAPP(nat,nat,plus_plus(nat,Y),Z_1)) = hAPP(nat,nat,plus_plus(nat,Y),hAPP(nat,nat,plus_plus(nat,X_1),Z_1))) # label(fact_584_nat__add__left__commute) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 765 (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_205_zdvd__mult__cancel) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 766 (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_260_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 767 (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_411_power__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 768 (all X_a (linordered_idom(X_a) & number_ring(X_a) -> (all Y_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),Y_1)) <-> 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_1))))))) # label(fact_60_le__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 769 (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_966_zmult2__lemma__aux2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 770 (all J_1 all K_1 all A_1 all P_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_3)) -> (hBOOL(hAPP(int,bool,zprime,P_3)) -> (-hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_3)) -> (-hBOOL(hAPP(int,bool,zcong(K_1,zero_zero(int)),P_3)) -> (-hBOOL(hAPP(int,bool,zcong(J_1,zero_zero(int)),P_3)) -> (hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_1),multInv(P_3,J_1)),hAPP(int,int,times_times(int,A_1),multInv(P_3,K_1))),P_3)) -> hBOOL(hAPP(int,bool,zcong(J_1,K_1),P_3))))))))) # label(fact_847_MultInv__zcong__prop3) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 771 (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_736_nat__le__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 772 (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_455_power__strict__decreasing) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 773 (all P_2 ((all M_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M_2)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M_2),P_2)) -> ti(int,M_2) = ti(int,P_2) | one_one(int) = ti(int,M_2))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),P_2)) <-> hBOOL(hAPP(int,bool,zprime,P_2)))) # label(fact_272_zprime__def) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 774 (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,C_1),D) = hAPP(X_a,X_a,minus_minus(X_a,A_3),B_2) -> (ti(X_a,C_1) = ti(X_a,D) <-> ti(X_a,A_3) = ti(X_a,B_2)))))) # label(fact_766_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 775 (all Ma all D (zero_zero(nat) = hAPP(nat,nat,div_mod(nat,Ma),D) <-> (exists Q_2 Ma = hAPP(nat,nat,times_times(nat,D),Q_2)))) # label(fact_985_mod__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 776 (all K 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),ord_less(nat),zero_zero(nat)),K)) <-> 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_699_mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 777 (all B_1_1 all X_a (power(X_a) -> ti(fun(nat,X_a),power_power(X_a,B_1_1)) = power_power(X_a,B_1_1))) # label(tsy_c_Power_Opower__class_Opower_res) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 778 (all K all Ma all N (K = zero_zero(nat) | Ma = N <-> hAPP(nat,nat,times_times(nat,K),Ma) = hAPP(nat,nat,times_times(nat,K),N))) # label(fact_633_mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 779 (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_430_mult__strict__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 780 (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_357_zcong__zadd) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 781 (all A_1 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),zero_zero(int)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),P_3)) -> inv(P_3,inv(P_3,A_1)) = ti(int,A_1)))))) # label(fact_819_inv__inv) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 782 (all X_1 all Y all Z_1 hAPP(int,int,plus_plus(int,Y),hAPP(int,int,plus_plus(int,X_1),Z_1)) = hAPP(int,int,plus_plus(int,X_1),hAPP(int,int,plus_plus(int,Y),Z_1))) # label(fact_97_zadd__left__commute) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 783 (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_716_two__realpow__ge__one) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 784 (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_435_zero__less__two) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 785 (all X_1 hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),X_1))) # label(fact_532_dvd_Oorder__refl) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 786 (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_589_diff__add__inverse) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 787 (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_361_split__mult__neg__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 788 (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_482_diff__bin__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 789 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all Y_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),Y_1)) <-> 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_1))))))) # label(fact_48_le__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 790 (all I_2 all J_2 all K (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),hAPP(nat,nat,minus_minus(nat,J_2),K))) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,plus_plus(nat,I_2),K)),J_2)))) # label(fact_647_less__diff__conv) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 791 (all K_1 hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,one_one(int)),K_1)),K_1) = bit1(K_1)) # label(fact_164_Bit1__def) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 792 (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_551_less__imp__diff__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 793 (all M_1 all D_1 (zero_zero(int) = hAPP(int,int,div_mod(int,M_1),D_1) -> (exists Q_2 ti(int,M_1) = hAPP(int,int,times_times(int,D_1),Q_2)))) # label(fact_974_zmod__eq__0D) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 794 (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_927_mod__add__left__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 795 (all W_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(W_1)),zero_zero(int))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),zero_zero(int))))) # label(fact_156_bin__less__0__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 796 (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) = M)) # label(fact_980_mod__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 797 (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_77_rel__simps_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 798 (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_372_mult__nonneg__nonpos2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.81 799 (all A_1 all N_1 all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_3),hAPP(nat,int,power_power(int,A_1),N_1))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_3),A_1))))) # label(fact_198_zprime__zdvd__power) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 800 (all X_a ti(X_a,undefined(X_a)) = undefined(X_a)) # label(tsy_c_HOL_Oundefined_res) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 801 (all X_a (ring_11004092258visors(X_a) -> (all N_1 all A_1 (zero_zero(X_a) != ti(X_a,A_1) -> hAPP(nat,X_a,power_power(X_a,A_1),N_1) != zero_zero(X_a))))) # label(fact_324_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 802 (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_183_zadd__power2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 803 (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_368_mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 804 (all X_a (linordered_idom(X_a) -> (all X all Y_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,plus_plus(X_a,hAPP(nat,X_a,power_power(X_a,X),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) != ti(X_a,X) | zero_zero(X_a) != ti(X_a,Y_1))))) # label(fact_8_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 805 (all A_3 all B_2 all Ma (hBOOL(hAPP(int,bool,zcong(A_3,B_2),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)))) # label(fact_910_zcong__zmod) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 806 (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_169_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 807 (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_332_mult__dvd__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 808 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all Y_1 (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_1))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit1(pls)),Y_1)))))) # label(fact_180_less__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 809 (all Y 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)) -> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,times_times(real,X_1),Y)))))) # label(fact_706_real__mult__order) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 810 (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_653_diff__less__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 811 (all K_1 all M hBOOL(hAPP(int,bool,zcong(K_1,K_1),M))) # label(fact_310_zcong__refl) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 812 (all X_a (comm_semiring_1(X_a) -> (all A_1 all 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) = hAPP(X_a,X_a,plus_plus(X_a,hAPP(X_a,X_a,times_times(X_a,A_1),M)),M)))) # label(fact_254_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 813 (all X_a (comm_semiring_1(X_a) -> (all X_1 all P_3 all Q_1 hAPP(nat,X_a,power_power(X_a,hAPP(nat,X_a,power_power(X_a,X_1),P_3)),Q_1) = hAPP(nat,X_a,power_power(X_a,X_1),hAPP(nat,nat,times_times(nat,P_3),Q_1))))) # label(fact_199_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 814 (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_331_dvd__mult) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 815 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),K)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),bit1(K))))) # label(fact_51_rel__simps_I22_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 816 (all X_a (comm_semiring_1(X_a) -> (all X_1 all P_3 all Q_1 hAPP(X_a,X_a,times_times(X_a,hAPP(nat,X_a,power_power(X_a,X_1),P_3)),hAPP(nat,X_a,power_power(X_a,X_1),Q_1)) = hAPP(nat,X_a,power_power(X_a,X_1),hAPP(nat,nat,plus_plus(nat,P_3),Q_1))))) # label(fact_247_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 817 (all Z_1 ti(int,Z_1) = hAPP(int,int,times_times(int,one_one(int)),Z_1)) # label(fact_66_zmult__1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 818 (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_950_mod__diff__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 819 (all A_1 all P_3 ((hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_3)) -> legendre(A_1,P_3) = zero_zero(int)) & (-hBOOL(hAPP(int,bool,zcong(A_1,zero_zero(int)),P_3)) -> (-hBOOL(hAPP(int,bool,quadRes(P_3),A_1)) -> number_number_of(int,min) = legendre(A_1,P_3)) & (hBOOL(hAPP(int,bool,quadRes(P_3),A_1)) -> one_one(int) = legendre(A_1,P_3))))) # label(fact_508_Legendre__def) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 820 (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_915_zdvd__zmod) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 821 (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_904_mod__mult__mult1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 822 (all X_b all X_c all X_a all B_1_1 all B_2_1 combb(X_b,X_c,X_a,ti(fun(X_b,X_c),B_1_1),B_2_1) = combb(X_b,X_c,X_a,B_1_1,B_2_1)) # label(tsy_c_COMBB_arg1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 823 (all Ma all N (N != 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),N),Ma)))) # label(fact_545_nat__neq__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 824 (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_861_d22set__mem) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 825 (all A_1 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)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),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)),inv(P_3,A_1))))))) # label(fact_821_inv__g__1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 826 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),K)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),bit1(K))))) # label(fact_71_rel__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 827 (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_103_zero__le__power2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 828 (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_605_diff__le__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 829 (all L_1 hAPP(int,int,minus_minus(int,pls),bit1(L_1)) = bit1(hAPP(int,int,minus_minus(int,min),L_1))) # label(fact_495_diff__bin__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 830 (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]. 1.53/1.82 831 (all B_1_1 all X_a (power(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_1_arg1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 832 (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)))) -> 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))))) # label(fact_507_Legendre__1mod4) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 833 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all X (hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),number_number_of(X_a,X)),zero_zero(X_a))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X),pls)))))) # label(fact_47_le__special_I3_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 834 (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_607_Nat_Odiff__le__self) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 835 (all X_1 all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y)) -> -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y)) | Y = X_1)) # label(fact_574_dvd_Ole__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 836 (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]. 1.53/1.82 837 (all Z_2 all W_1 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_2),W_1)) & ti(int,Z_2) != ti(int,W_1) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_2),W_1)))) # label(fact_37_zless__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 838 (all B_1_1 all X_a (group_add(X_a) -> minus_minus(X_a,B_1_1) = minus_minus(X_a,ti(X_a,B_1_1)))) # label(tsy_c_Groups_Ominus__class_Ominus_0_arg1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 839 (all M all X_1 hAPP(int,int,div_mod(int,X_1),M) = standardRes(M,X_1)) # label(fact_979_StandardRes__def) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 840 (all X_1 all Y (X_1 = Y -> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y)))) # label(fact_576_dvd_Oeq__refl) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 841 (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_438_power__le__imp__le__exp) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 842 (all Y all X_1 all P_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_3)) -> (hBOOL(hAPP(int,bool,zprime,P_3)) -> (-hBOOL(hAPP(int,bool,zcong(X_1,zero_zero(int)),P_3)) -> (-hBOOL(hAPP(int,bool,zcong(Y,zero_zero(int)),P_3)) -> (hBOOL(hAPP(int,bool,zcong(multInv(P_3,X_1),multInv(P_3,Y)),P_3)) -> hBOOL(hAPP(int,bool,zcong(X_1,Y),P_3)))))))) # label(fact_841_MultInv__prop5) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 843 (all X_a (real_normed_algebra(X_a) -> (all A_1 all B all 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)) = 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))))) # label(fact_894_mult_Odiff__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 844 (all B_1_1 all B_2_1 legendre(B_1_1,B_2_1) = ti(int,legendre(B_1_1,B_2_1))) # label(tsy_c_Residues_OLegendre_res) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 845 (all X_a (semiring_div(X_a) -> (all A_1 hAPP(X_a,X_a,div_mod(X_a,A_1),zero_zero(X_a)) = ti(X_a,A_1)))) # label(fact_932_mod__by__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 846 (all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) -> (number_number_of(int,bit0(bit1(pls))) != ti(int,P_3) -> (ti(int,P_3) != number_number_of(int,bit1(bit1(pls))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,bit1(bit0(bit1(pls))))),P_3)))))) # label(fact_273_prime__g__5) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 847 (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,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_901_mod__mult__cong) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 848 (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,V)),hAPP(X_a,X_a,plus_plus(X_a,number_number_of(X_a,W)),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)))) # label(fact_150_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 849 (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),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,plus_plus(X_a,C_1),A_3)),hAPP(X_a,X_a,plus_plus(X_a,C_1),B_2))))))) # label(fact_773_add__le__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 850 (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_478_eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 851 (all N_1 all X_1 all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y)) -> 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),N_1))))) # label(fact_516_divides__exp) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 852 (all I_1 all N_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),N_1)) -> hAPP(nat,nat,minus_minus(nat,N_1),hAPP(nat,nat,minus_minus(nat,N_1),I_1)) = I_1)) # label(fact_604_diff__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 853 (all P_2 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_2))) = sr(P_2)) # label(fact_885_SR__def) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 854 t = zero_zero(int) -> hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,s),number_number_of(nat,bit0(bit1(pls))))),one_one(int)) = zero_zero(int) # label(fact_4_calculation_I2_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 855 (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),bit1(K1)),bit1(K2))))) # label(fact_32_less__eq__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 856 (all X_a (linordered_semidom(X_a) -> (all X_1 all Y (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),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)) -> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y))))))) # label(fact_104_power2__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 857 (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,hAPP(nat,nat,minus_minus(nat,M),N_1)),N_1) = hAPP(nat,nat,div_mod(nat,M),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__if) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 858 (all X_a (zero(X_a) -> (all X (zero_zero(X_a) = ti(X_a,X) <-> zero_zero(X_a) = ti(X_a,X))))) # label(fact_757_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 859 (all K (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit0(K)),pls)) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),pls)))) # label(fact_54_rel__simps_I27_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 860 (all J_1 all A_1 all P_3 all K_1 (hBOOL(hAPP(int,bool,zcong(J_1,hAPP(int,int,times_times(int,A_1),multInv(P_3,K_1))),P_3)) -> 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_3,K_1))),K_1)),P_3)))) # label(fact_837_aux______1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 861 (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]. 1.53/1.82 862 (all X_1 all Y (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y)) -> -(-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X_1))))) # label(fact_561_dvd_Oless__imp__not__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 863 (all A_1 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)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,minus_minus(int,P_3),one_one(int)))) -> hAPP(int,int,minus_minus(int,P_3),one_one(int)) != inv(P_3,A_1))))) # label(fact_822_inv__not__p__minus__1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 864 (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]. 1.53/1.82 865 (all X (hBOOL(hAPP(int,bool,twoSqu658283162sum2sq,X)) <-> (exists A_5 exists B_4 ti(int,X) = twoSqu1929807760sum2sq(product_Pair(int,int,A_5,B_4))))) # label(fact_753_is__sum2sq__def) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 866 (all B_2 all A_3 all P_2 (hBOOL(hAPP(int,bool,zprime,P_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),number_number_of(int,bit1(bit0(bit1(pls))))),P_2)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),hAPP(int,int,minus_minus(int,P_2),one_one(int)))) -> (hBOOL(hAPP(fun(int,bool),bool,member(int,B_2),wset(A_3,P_2))) -> hBOOL(hAPP(fun(int,bool),bool,member(int,inv(P_2,B_2)),wset(A_3,P_2)))))))) # label(fact_827_wset__mem__inv__mem) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 867 (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_229_nat__mult__2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 868 (all M hBOOL(hAPP(int,bool,zcong(M,zero_zero(int)),M))) # label(fact_348_zcong__id) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 869 (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,Lx),hAPP(X_a,X_a,times_times(X_a,Ly),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),Ly)),hAPP(X_a,X_a,times_times(X_a,Rx),Ry))))) # label(fact_214_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 870 (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_394_not__square__less__zero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 871 (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_443_power__strict__increasing) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 872 (all A_1 all M (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M)) -> (exists X_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_2)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_2),M)) & (all Y_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y_2)) & hBOOL(hAPP(int,bool,zcong(A_1,Y_2),M)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y_2),M)) -> ti(int,Y_2) = ti(int,X_2))) & hBOOL(hAPP(int,bool,zcong(A_1,X_2),M)))))) # label(fact_750_zcong__zless__unique) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 873 (all Z_1 all W (hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),W),Z_1)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Z_1),W)))) # label(fact_723_real__le__linear) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 874 (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_780_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 875 (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_323_comm__semiring__class_Odistrib) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 876 (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_960_neg__mod__sign) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 877 (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_577_dvd_Oneq__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 878 (all X_a (semiring_div(X_a) -> (all B all A_1 zero_zero(X_a) = hAPP(X_a,X_a,div_mod(X_a,hAPP(X_a,X_a,times_times(X_a,B),A_1)),B)))) # label(fact_946_mod__mult__self1__is__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 879 (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_389_mult__pos__neg) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 880 (all Z_1 hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,one_one(int)),Z_1)),Z_1) != zero_zero(int)) # label(fact_165_odd__nonzero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 881 (all N_1 all P_3 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_3)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_3),hAPP(int,int,times_times(int,M),N_1))) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_3),M)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_3),N_1)))))) # label(fact_461_zprime__zdvd__zmult) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 882 (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_926_mod__add__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 883 (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_263_q__pos__lemma) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 884 (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_419_zcong__zero__equiv__div) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 885 (all Z_1 all X_1 all Y all M (hBOOL(hAPP(int,bool,zcong(X_1,Y),M)) -> hBOOL(hAPP(int,bool,zcong(hAPP(nat,int,power_power(int,X_1),Z_1),hAPP(nat,int,power_power(int,Y),Z_1)),M)))) # label(fact_359_zcong__zpower) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 886 (all A_1 all B hBOOL(hAPP(int,bool,zcong(A_1,B),one_one(int)))) # label(fact_350_zcong__1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 887 (all L_1 pls != bit1(L_1)) # label(fact_116_rel__simps_I39_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 888 (all A_1 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)),A_1)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,minus_minus(int,P_3),one_one(int)))) -> ti(int,A_1) != inv(P_3,A_1))))) # label(fact_824_inv__distinct) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 889 (all X all Y_1 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X)) <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X)))) # label(fact_578_dvd_Oless__le__not__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 890 (all X_a (monoid_mult(X_a) -> (all N_1 one_one(X_a) = hAPP(nat,X_a,power_power(X_a,one_one(X_a)),N_1)))) # label(fact_337_power__one) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 891 (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_233_even__power__le__0__imp__0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 892 (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_650_less__imp__le__nat) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 893 (all X_a (comm_semiring_1(X_a) -> (all Lx all Rx all Ry hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,times_times(X_a,Lx),Rx)),Ry) = hAPP(X_a,X_a,times_times(X_a,Lx),hAPP(X_a,X_a,times_times(X_a,Rx),Ry))))) # label(fact_217_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 894 (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_959_mod__pos__pos__trivial) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 895 (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_641_trans__less__add2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 896 (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)) -> N_1 = M))) # label(fact_596_le__antisym) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 897 (all P all Q (-hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,P),Q)) | hBOOL(Q))) # label(help_fconj_3_1_U) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 898 (all V_3 all V_2 (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),number_number_of(nat,V_3)),number_number_of(nat,V_2))) <-> (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),V_3),V_2)) -> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),V_3),pls))))) # label(fact_211_le__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 899 (all A_1 all P_3 (hBOOL(hAPP(int,bool,zprime,P_3)) -> (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_3)) -> hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,A_1),inv(P_3,A_1)),one_one(int)),P_3)))))) # label(fact_826_inv__is__inv) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 900 (all X_1 all Y all Z_1 hAPP(nat,int,power_power(int,hAPP(nat,int,power_power(int,X_1),Y)),Z_1) = hAPP(nat,int,power_power(int,X_1),hAPP(nat,nat,times_times(nat,Y),Z_1))) # label(fact_202_zpower__zpower) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 901 (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_407_power__increasing) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 902 (exists X_2 (hBOOL(hAPP(int,bool,zcong(s1,X_2),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_2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y_2)) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y_2),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_2),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_2) = ti(int,X_2))) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_2),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)),X_2)))) # label(fact_194__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]. 1.53/1.82 903 (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,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)))) & (-hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),number_number_of(int,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)) = hAPP(int,int,div_mod(int,number_number_of(int,bit1(V))),number_number_of(int,bit0(W)))))) # label(fact_898_zmod__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 904 (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)),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_362_split__mult__pos__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 905 (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_401_zero__le__power) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 906 (all B_1_1 all X_a (linorder(X_a) & dvd(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]. 1.53/1.82 907 (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_336_dvd__add) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 908 (all Z_1 hAPP(int,int,times_times(int,Z_1),one_one(int)) = ti(int,Z_1)) # label(fact_65_zmult__1__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 909 (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_638_not__add__less2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 910 (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_968_zmult2__lemma__aux4) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 911 (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_662_trans__le__add1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 912 (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]. 1.53/1.82 913 (all P_1 all N all K ((zero_zero(int) = ti(int,K) -> hBOOL(hAPP(int,bool,P_1,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_1,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(int),J),K)) & ti(int,N) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,K),I)),J) & hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J)) -> hBOOL(hAPP(int,bool,P_1,J))))) <-> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,div_mod(int,N),K))))) # label(fact_970_split__zmod) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 914 (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_720_nat__less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 915 (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]. 1.53/1.82 916 (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]. 1.53/1.82 917 (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_715_realpow__minus__mult) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 918 (all Z_1 hAPP(real,real,times_times(real,one_one(real)),Z_1) = Z_1) # label(fact_595_real__mult__1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 919 (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_792_mult__1__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 920 (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_717_dvd__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 921 (all X_a (cancel_semigroup_add(X_a) -> (all B_2 all A_3 all C_1 (ti(X_a,C_1) = ti(X_a,B_2) <-> hAPP(X_a,X_a,plus_plus(X_a,C_1),A_3) = hAPP(X_a,X_a,plus_plus(X_a,B_2),A_3))))) # label(fact_761_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 922 (all X_a (semiring_div(X_a) -> (all A_1 all B 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),B)),B)))) # label(fact_930_mod__add__self2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 923 (all B_1_1 d22set(ti(int,B_1_1)) = d22set(B_1_1)) # label(tsy_c_IntFact_Od22set_arg1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 924 (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_25_times__numeral__code_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 925 (all A_1 all B 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))))) = hAPP(nat,int,power_power(int,hAPP(int,int,minus_minus(int,A_1),B)),number_number_of(nat,bit1(bit1(pls))))) # label(fact_505_zdiff__power3) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 926 (all M all K_1 all 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)) = hAPP(nat,nat,minus_minus(nat,M),N_1)) # label(fact_592_diff__cancel2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 927 (all P_1 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_1,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_1,I)) -> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,minus_minus(int,I),one_one(int))))))) -> hBOOL(hAPP(int,bool,P_1,I_2)))))) # label(fact_853_int__less__induct) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 928 (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]. 1.53/1.82 929 (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_408_one__le__power) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 930 (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_785_add__less__imp__less__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 931 (all X_a (number_ring(X_a) -> (all Z_1 hAPP(X_a,X_a,plus_plus(X_a,Z_1),Z_1) = hAPP(X_a,X_a,times_times(X_a,Z_1),number_number_of(X_a,bit0(bit1(pls))))))) # label(fact_141_mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 932 (all X_a (semiri456707255roduct(X_a) -> (all B_2 all A_3 (zero_zero(X_a) = ti(X_a,A_3) <-> ti(X_a,B_2) = hAPP(X_a,X_a,plus_plus(X_a,B_2),A_3))))) # label(fact_238_add__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 933 (all K_1 bit0(K_1) != min) # label(fact_286_rel__simps_I45_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 934 (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_390_mult__pos__pos) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 935 (all X_a (semiring_div(X_a) -> (all A_1 zero_zero(X_a) = hAPP(X_a,X_a,div_mod(X_a,A_1),A_1)))) # label(fact_931_mod__self) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 936 (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_515_divides__mul__l) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 937 (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_84_zle__add1__eq__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 938 (all X_1 all Y (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X_1)) -> -(hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X_1)) & -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y))))) # label(fact_563_dvd_Oless__not__sym) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 939 (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_414_zcong__not__zero) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 940 (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_525_divides__rev) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 941 (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),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_902_zmod__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 942 (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,hAPP(X_a,X_a,minus_minus(X_a,A_3),B_2)),E)),C_1)),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_491_less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 943 (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)),J_1) = hAPP(nat,int,power_power(int,number_number_of(int,min)),K_1)))) # label(fact_464_neg__one__power__eq__mod__m) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 944 (all V_1 all V ((hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls)) -> number_number_of(nat,V_1) = hAPP(nat,nat,plus_plus(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)) -> (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),pls)) -> number_number_of(nat,V) = hAPP(nat,nat,plus_plus(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_1),pls)) -> number_number_of(nat,hAPP(int,int,plus_plus(int,V),V_1)) = hAPP(nat,nat,plus_plus(nat,number_number_of(nat,V)),number_number_of(nat,V_1)))))) # label(fact_86_add__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 945 (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_963_mod__pos__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 946 (all X_a (number_ring(X_a) -> zero_zero(X_a) = number_number_of(X_a,pls))) # label(fact_145_number__of__Pls) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 947 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K)),bit0(L))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L)))) # label(fact_127_rel__simps_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 948 (all B_1_1 all X_a (semiring_div(X_a) -> div_mod(X_a,ti(X_a,B_1_1)) = div_mod(X_a,B_1_1))) # label(tsy_c_Divides_Odiv__class_Omod_arg1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 949 (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_308_dvd__trans) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 950 (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_608_nat__mult__commute) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 951 (all Ma all N (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Ma),N)) | Ma = N <-> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Ma),N)))) # label(fact_649_le__eq__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 952 (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_986_mod__less__divisor) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 953 (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_795_mult_Ocomm__neutral) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 954 (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_860_d22set__g__1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 955 (all N_1 -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),zero_zero(nat)))) # label(fact_616_less__nat__zero__code) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 956 (all Z_1 all X_1 all Y (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y)) -> (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),Z_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_556_dvd_Oless__le__trans) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 957 (all K1 all K2 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),bit0(K1)),bit1(K2))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)))) # label(fact_74_less__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 958 (all N_1 N_1 = hAPP(nat,nat,times_times(nat,N_1),one_one(nat))) # label(fact_690_nat__mult__1__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 959 (all X_a (semiri456707255roduct(X_a) -> (all C_1 all D all A_3 all B_2 (ti(X_a,C_1) != ti(X_a,D) & ti(X_a,A_3) != ti(X_a,B_2) <-> 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)) != 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)))))) # label(fact_240_crossproduct__noteq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 960 (all A_3 all B_2 all N (zero_zero(nat) != N -> (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))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_3),B_2))))) # label(fact_868_pow__divides__eq__int) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 961 (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_672_le__add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 962 (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_692_real__add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 963 (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_642_add__less__mono1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 964 (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_619_diff__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 965 (all K_1 all M all N_1 hAPP(nat,nat,times_times(nat,K_1),hAPP(nat,nat,plus_plus(nat,M),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))) # label(fact_680_add__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 966 (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_182_zadd__power3) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 967 (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_413_zcong__not) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 968 (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_677_diff__add__assoc) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 969 (all B_1_1 all B_2_1 zcong(B_1_1,B_2_1) = ti(fun(int,bool),zcong(B_1_1,B_2_1))) # label(tsy_c_IntPrimes_Ozcong_res) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 970 (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_367_comm__mult__left__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 971 (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_726_nat__mult__eq__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 972 (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_161_add__Bit1__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 973 (all P_1 all A_3 all B_2 (hBOOL(hAPP(nat,bool,P_1,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_1,zero_zero(nat)))) & (all D_4 (A_3 = hAPP(nat,nat,plus_plus(nat,B_2),D_4) -> hBOOL(hAPP(nat,bool,P_1,D_4)))))) # label(fact_697_nat__diff__split) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 974 (all B_1_1 all X_a (monoid_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_1_arg1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.82 975 (all P_1 all X all Y_1 (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y_1)) -> (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X)) -> hBOOL(P_1)))) # label(fact_558_dvd_Oless__imp__triv) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 976 (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,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)) = 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)))) # label(fact_899_neg__zmod__mult__2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 977 (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,Rx),hAPP(X_a,X_a,times_times(X_a,hAPP(X_a,X_a,times_times(X_a,Lx),Ly)),Ry))))) # label(fact_213_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 978 (all N_1 -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),zero_zero(nat)))) # label(fact_614_not__less0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 979 (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_220_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 980 (all K_1 ti(int,K_1) = number_number_of(int,K_1)) # label(fact_22_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 981 (all K_1 all A_1 all J_1 all P_3 (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,bit0(bit1(pls)))),P_3)) -> (hBOOL(hAPP(int,bool,zprime,P_3)) -> (-hBOOL(hAPP(int,bool,zcong(J_1,zero_zero(int)),P_3)) -> (hBOOL(hAPP(int,bool,zcong(hAPP(int,int,times_times(int,hAPP(int,int,times_times(int,multInv(P_3,J_1)),J_1)),K_1),hAPP(int,int,times_times(int,multInv(P_3,J_1)),A_1)),P_3)) -> hBOOL(hAPP(int,bool,zcong(K_1,hAPP(int,int,times_times(int,A_1),multInv(P_3,J_1))),P_3))))))) # label(fact_844_aux______4) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 982 (all K (ti(int,K) = pls <-> bit0(K) = pls)) # label(fact_119_rel__simps_I44_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 983 (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)) -> (hAPP(nat,X_a,power_power(X_a,A_3),Ma) = hAPP(nat,X_a,power_power(X_a,A_3),N) <-> Ma = N))))) # label(fact_409_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 984 (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) <-> hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,hAPP(nat,nat,minus_minus(nat,J_2),I_2)),U)),N) = Ma))) # label(fact_738_nat__eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 985 (all B_1_1 all B_2_1 multInv(B_1_1,B_2_1) = ti(int,multInv(B_1_1,B_2_1))) # label(tsy_c_Int2_OMultInv_res) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 986 (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_385_mult__less__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 987 (all X_1 all Y (-hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X_1)) & hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y)) -> Y != X_1)) # label(fact_559_dvd_Oless__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 988 (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_365_mult__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 989 (all Ma all N (zero_zero(nat) = hAPP(nat,nat,power_power(nat,Ma),N) <-> zero_zero(nat) != N & zero_zero(nat) = Ma)) # label(fact_523_nat__power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 990 (all N all Ma (N = one_one(nat) & one_one(nat) = Ma <-> hAPP(nat,nat,times_times(nat,N),Ma) = one_one(nat))) # label(fact_522_nat__mult__eq__one) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 991 (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_166_zless__add1__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 992 (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_203_zdvd__not__zless) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 993 (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_278_dvd__0__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 994 (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_307_dvd__refl) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 995 (all X_a (linordered_semidom(X_a) -> (all X_1 all Y (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),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)) -> ti(X_a,Y) = ti(X_a,X_1))))))) # label(fact_101_power2__eq__imp__eq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 996 (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_268_zdiv__mono2__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 997 (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_426_mult__right__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 998 (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_590_diff__diff__left) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 999 (all X_a (number_semiring(X_a) -> (all Z_1 hAPP(X_a,X_a,plus_plus(X_a,Z_1),Z_1) = hAPP(X_a,X_a,times_times(X_a,Z_1),number_number_of(X_a,bit0(bit1(pls))))))) # label(fact_140_semiring__mult__2__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1000 (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_665_add__le__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1001 (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_955_zmod__zdvd__zmod) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1002 (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_654_less__diff__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1003 (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),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),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_393_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1004 (all A_1 all B (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) -> B = zero_zero(nat) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A_1),B)))) # label(fact_520_divides__ge) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1005 (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_378_mult__strict__left__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1006 (all L_1 hAPP(int,int,minus_minus(int,min),bit0(L_1)) = bit1(hAPP(int,int,minus_minus(int,min),L_1))) # label(fact_496_diff__bin__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1007 (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_929_mod__add__self1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1008 (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_136_double__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1009 (all P_1 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_1,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_1,I)) -> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,plus_plus(int,I),one_one(int))))))) -> hBOOL(hAPP(int,bool,P_1,I_2)))))) # label(fact_855_int__gr__induct) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1010 (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]. 1.53/1.83 1011 (all X_a (dvd(X_a) & semiring_0(X_a) -> (all P_1 all L ((exists X_2 hBOOL(hAPP(X_a,bool,P_1,hAPP(X_a,X_a,times_times(X_a,L),X_2)))) <-> (exists X_2 (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_2),zero_zero(X_a)))) & hBOOL(hAPP(X_a,bool,P_1,X_2)))))))) # label(fact_744_unity__coeff__ex) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1012 (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),Q_1)),R_1) = 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_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_266_zdiv__mono2__lemma) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1013 (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_793_mult__1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1014 (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_370_mult__nonpos__nonpos) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1015 (all M hAPP(nat,nat,plus_plus(nat,M),zero_zero(nat)) = M) # label(fact_622_Nat_Oadd__0__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1016 (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]. 1.53/1.83 1017 (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_382_mult__strict__right__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1018 (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_441_power__gt1__lemma) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1019 (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_41_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1020 (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]. 1.53/1.83 1021 (all X_a (number_ring(X_a) -> (all V all W all C hAPP(X_a,X_a,minus_minus(X_a,number_number_of(X_a,hAPP(int,int,plus_plus(int,V),W))),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))))) # label(fact_492_add__number__of__diff1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1022 (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_870_gcd__lcm__complete__lattice__nat_Otop__le) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1023 (all K all L (hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,minus_minus(int,K),L)),zero_zero(int))) <-> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L)))) # label(fact_485_less__bin__lemma) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1024 (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]. 1.53/1.83 1025 (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_880_dvd__pos__nat) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1026 (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)),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))))))) # label(fact_798_zero__le__double__add__iff__zero__le__single__add) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1027 (all X_a (linord581940658strict(X_a) -> (all X all Y_1 (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),X)),hAPP(X_a,X_a,times_times(X_a,Y_1),Y_1))),zero_zero(X_a))) <-> zero_zero(X_a) = ti(X_a,X) & zero_zero(X_a) = ti(X_a,Y_1))))) # label(fact_45_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1028 (all X_a (comm_semiring_1(X_a) -> (all X_1 all N_1 (one_one(X_a) = ti(X_a,X_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),dvd_dvd(X_a),X_1),hAPP(nat,X_a,power_power(X_a,X_1),N_1))))))) # label(fact_458_dvd__power) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1029 (all X_1 all Y all Z_1 hAPP(int,int,times_times(int,hAPP(nat,int,power_power(int,X_1),Y)),hAPP(nat,int,power_power(int,X_1),Z_1)) = hAPP(nat,int,power_power(int,X_1),hAPP(nat,nat,plus_plus(nat,Y),Z_1))) # label(fact_31_zpower__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1030 (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_663_trans__le__add2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1031 (all Y all X_1 (hBOOL(hAPP(int,bool,twoSqu658283162sum2sq,X_1)) -> (hBOOL(hAPP(int,bool,twoSqu658283162sum2sq,Y)) -> hBOOL(hAPP(int,bool,twoSqu658283162sum2sq,hAPP(int,int,times_times(int,X_1),Y)))))) # label(fact_210_is__mult__sum2sq) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1032 (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_143_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1033 (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_424_mult__left__less__imp__less) # label(axiom) # label(non_clause). [assumption]. 1.53/1.83 1034 (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_643_add__less__mono) # label(axiom) # label(non_clause). [assumption]. 1.53/1.98 1035 (all Ma all N all K (hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K)) -> (Ma = N <-> hAPP(nat,nat,times_times(nat,K),Ma) = hAPP(nat,nat,times_times(nat,K),N)))) # label(fact_729_nat__mult__eq__cancel1) # label(axiom) # label(non_clause). [assumption]. 1.53/1.98 1036 (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_366_mult__right__mono__neg) # label(axiom) # label(non_clause). [assumption]. 1.53/1.98 1037 (all M all N_1 hAPP(nat,nat,minus_minus(nat,hAPP(nat,nat,plus_plus(nat,M),N_1)),N_1) = M) # label(fact_588_diff__add__inverse2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.98 1038 (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_803_add__increasing2) # label(axiom) # label(non_clause). [assumption]. 1.53/1.98 1039 (all X_a (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__0__right) # label(axiom) # label(non_clause). [assumption]. 1.53/1.98 1.53/1.98 ============================== end of process non-clausal formulas === 1.53/1.98 1.53/1.98 ============================== PROCESS INITIAL CLAUSES =============== 1.53/1.98 1.53/1.98 ============================== PREDICATE ELIMINATION ================= 1.53/1.98 1040 comm_monoid_add(nat) # label(arity_Nat_Onat___Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 1.53/1.98 1041 -comm_monoid_add(A) | ti(A,B) = hAPP(A,A,plus_plus(A,zero_zero(A)),B) # label(fact_768_add__0) # label(axiom). [clausify(2)]. 1.53/1.98 1042 -comm_monoid_add(A) | ti(A,B) = hAPP(A,A,plus_plus(A,B),zero_zero(A)) # label(fact_771_add_Ocomm__neutral) # label(axiom). [clausify(310)]. 1.53/1.98 Derived: ti(nat,A) = hAPP(nat,nat,plus_plus(nat,zero_zero(nat)),A). [resolve(1040,a,1041,a)]. 1.53/1.98 Derived: ti(nat,A) = hAPP(nat,nat,plus_plus(nat,A),zero_zero(nat)). [resolve(1040,a,1042,a)]. 1.53/1.98 1043 comm_monoid_add(int) # label(arity_Int_Oint___Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 1.53/1.98 Derived: ti(int,A) = hAPP(int,int,plus_plus(int,zero_zero(int)),A). [resolve(1043,a,1041,a)]. 1.53/1.98 Derived: ti(int,A) = hAPP(int,int,plus_plus(int,A),zero_zero(int)). [resolve(1043,a,1042,a)]. 1.53/1.98 1044 comm_monoid_add(real) # label(arity_RealDef_Oreal___Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 1.53/1.98 Derived: ti(real,A) = hAPP(real,real,plus_plus(real,zero_zero(real)),A). [resolve(1044,a,1041,a)]. 1.53/1.98 Derived: ti(real,A) = hAPP(real,real,plus_plus(real,A),zero_zero(real)). [resolve(1044,a,1042,a)]. 1.53/1.98 1045 -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(309)]. 1.53/1.98 1046 ring_div(int) # label(arity_Int_Oint___Divides_Oring__div) # label(axiom). [assumption]. 1.53/1.98 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(1045,a,1046,a)]. 1.53/1.98 1047 -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_952_mod__diff__right__eq) # label(axiom). [clausify(443)]. 1.53/2.10 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(1047,a,1046,a)]. 1.53/2.10 1048 -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_951_mod__diff__left__eq) # label(axiom). [clausify(684)]. 1.53/2.10 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(1048,a,1046,a)]. 1.53/2.10 1049 -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_950_mod__diff__eq) # label(axiom). [clausify(818)]. 1.53/2.10 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(1049,a,1046,a)]. 1.53/2.10 1050 -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_802_add__increasing) # label(axiom). [clausify(181)]. 1.53/2.10 1051 ordere216010020id_add(nat) # label(arity_Nat_Onat___Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 1.53/2.10 1052 ordere216010020id_add(real) # label(arity_RealDef_Oreal___Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 1.53/2.10 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(1050,a,1052,a)]. 1.53/2.10 1053 -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_808_add__neg__neg) # label(axiom). [clausify(233)]. 1.53/2.10 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(1053,a,1052,a)]. 1.53/2.10 1054 -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_756_add__nonpos__neg) # label(axiom). [clausify(261)]. 1.53/2.10 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(1054,a,1052,a)]. 1.53/2.10 1055 ordere216010020id_add(int) # label(arity_Int_Oint___Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 1.53/2.10 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(1055,a,1050,a)]. 1.53/2.10 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(1055,a,1053,a)]. 1.53/2.10 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(1055,a,1054,a)]. 1.53/2.11 1056 -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)) | ti(A,C) != zero_zero(A) | ti(A,B) != zero_zero(A) | zero_zero(A) = hAPP(A,A,plus_plus(A,B),C) # label(fact_801_add__nonneg__eq__0__iff) # label(axiom). [clausify(351)]. 1.53/2.11 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)) | ti(nat,B) != zero_zero(nat) | ti(nat,A) != zero_zero(nat) | zero_zero(nat) = hAPP(nat,nat,plus_plus(nat,A),B). [resolve(1056,a,1051,a)]. 1.53/2.11 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)) | ti(real,B) != zero_zero(real) | ti(real,A) != zero_zero(real) | zero_zero(real) = hAPP(real,real,plus_plus(real,A),B). [resolve(1056,a,1052,a)]. 1.53/2.11 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)) | ti(int,B) != zero_zero(int) | ti(int,A) != zero_zero(int) | zero_zero(int) = hAPP(int,int,plus_plus(int,A),B). [resolve(1056,a,1055,a)]. 1.53/2.11 1057 -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)) | ti(A,C) = zero_zero(A) | zero_zero(A) != hAPP(A,A,plus_plus(A,B),C) # label(fact_801_add__nonneg__eq__0__iff) # label(axiom). [clausify(351)]. 1.53/2.11 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)) | ti(nat,B) = zero_zero(nat) | zero_zero(nat) != hAPP(nat,nat,plus_plus(nat,A),B). [resolve(1057,a,1051,a)]. 1.53/2.11 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)) | ti(real,B) = zero_zero(real) | zero_zero(real) != hAPP(real,real,plus_plus(real,A),B). [resolve(1057,a,1052,a)]. 1.53/2.11 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)) | ti(int,B) = zero_zero(int) | zero_zero(int) != hAPP(int,int,plus_plus(int,A),B). [resolve(1057,a,1055,a)]. 1.53/2.11 1058 -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)) | ti(A,B) = zero_zero(A) | zero_zero(A) != hAPP(A,A,plus_plus(A,B),C) # label(fact_801_add__nonneg__eq__0__iff) # label(axiom). [clausify(351)]. 1.53/2.11 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)) | ti(nat,A) = zero_zero(nat) | zero_zero(nat) != hAPP(nat,nat,plus_plus(nat,A),B). [resolve(1058,a,1051,a)]. 1.53/2.11 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)) | ti(real,A) = zero_zero(real) | zero_zero(real) != hAPP(real,real,plus_plus(real,A),B). [resolve(1058,a,1052,a)]. 1.53/2.11 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)) | ti(int,A) = zero_zero(int) | zero_zero(int) != hAPP(int,int,plus_plus(int,A),B). [resolve(1058,a,1055,a)]. 1.53/2.11 1059 -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_817_add__neg__nonpos) # label(axiom). [clausify(422)]. 1.53/2.13 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(1059,a,1052,a)]. 1.53/2.13 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(1059,a,1055,a)]. 1.53/2.13 1060 -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_804_add__nonpos__nonpos) # label(axiom). [clausify(424)]. 1.53/2.13 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(1060,a,1051,a)]. 1.53/2.13 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(1060,a,1052,a)]. 1.53/2.13 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(1060,a,1055,a)]. 1.53/2.13 1061 -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_816_add__strict__increasing2) # label(axiom). [clausify(437)]. 1.53/2.13 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(1061,a,1052,a)]. 1.53/2.13 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(1061,a,1055,a)]. 1.53/2.13 1062 -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_807_add__pos__pos) # label(axiom). [clausify(447)]. 1.53/2.13 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,plus_plus(real,A),B))). [resolve(1062,a,1052,a)]. 1.53/2.13 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,plus_plus(int,A),B))). [resolve(1062,a,1055,a)]. 1.53/2.13 1063 -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_813_add__pos__nonneg) # label(axiom). [clausify(591)]. 2.03/2.22 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),zero_zero(nat)),B)) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,plus_plus(nat,A),B))). [resolve(1063,a,1051,a)]. 2.03/2.22 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(1063,a,1052,a)]. 2.03/2.22 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(1063,a,1055,a)]. 2.03/2.22 1064 -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_800_add__nonneg__nonneg) # label(axiom). [clausify(647)]. 2.03/2.22 1065 -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_814_add__nonneg__pos) # label(axiom). [clausify(695)]. 2.03/2.22 1066 -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(1024)]. 2.03/2.22 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(1066,a,1051,a)]. 2.03/2.22 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(1066,a,1052,a)]. 2.03/2.22 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(1066,a,1055,a)]. 2.03/2.22 1067 -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_803_add__increasing2) # label(axiom). [clausify(1038)]. 2.03/2.22 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(1067,a,1051,a)]. 2.03/2.22 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(1067,a,1052,a)]. 2.03/2.22 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(1067,a,1055,a)]. 2.12/2.29 1068 number_ring(int) # label(arity_Int_Oint___Int_Onumber__ring) # label(axiom). [assumption]. 2.12/2.29 1069 -number_ring(A) | -ring_char_0(A) | number_number_of(A,B) != number_number_of(A,C) | ti(int,B) = ti(int,C) # label(fact_88_eq__number__of) # label(axiom). [clausify(7)]. 2.12/2.29 1070 -number_ring(A) | -ring_char_0(A) | number_number_of(A,B) = number_number_of(A,C) | ti(int,B) != ti(int,C) # label(fact_88_eq__number__of) # label(axiom). [clausify(7)]. 2.12/2.29 1071 -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_360_power__m1__even) # label(axiom). [clausify(14)]. 2.12/2.29 1072 -number_ring(A) | -linordered_idom(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_176_less__special_I1_J) # label(axiom). [clausify(42)]. 2.12/2.29 1073 -number_ring(A) | -linordered_idom(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_176_less__special_I1_J) # label(axiom). [clausify(42)]. 2.12/2.29 1074 -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_469_number__of__diff) # label(axiom). [clausify(53)]. 2.12/2.29 1075 -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_15_add__special_I2_J) # label(axiom). [clausify(62)]. 2.12/2.29 1076 -number_ring(A) | ti(A,B) = hAPP(A,A,times_times(A,number_number_of(A,bit1(pls))),B) # label(fact_109_mult__numeral__1) # label(axiom). [clausify(140)]. 2.12/2.29 1077 -number_ring(A) | ti(A,B) = hAPP(A,A,plus_plus(A,B),number_number_of(A,pls)) # label(fact_148_add__numeral__0__right) # label(axiom). [clausify(202)]. 2.12/2.29 1078 -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_43_arith__simps_I32_J) # label(axiom). [clausify(225)]. 2.12/2.29 1079 -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_152_number__of__add) # label(axiom). [clausify(269)]. 2.12/2.29 1080 -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_149_less__number__of) # label(axiom). [clausify(288)]. 2.12/2.29 1081 -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_149_less__number__of) # label(axiom). [clausify(288)]. 2.12/2.29 1082 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),pls)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),number_number_of(A,B)),zero_zero(A))) # label(fact_175_less__special_I3_J) # label(axiom). [clausify(307)]. 2.12/2.29 1083 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),pls)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),number_number_of(A,B)),zero_zero(A))) # label(fact_175_less__special_I3_J) # label(axiom). [clausify(307)]. 2.12/2.29 1084 -number_ring(A) | ti(A,B) = hAPP(A,A,times_times(A,B),number_number_of(A,bit1(pls))) # label(fact_110_mult__numeral__1__right) # label(axiom). [clausify(332)]. 2.12/2.29 1085 -number_ring(A) | hAPP(A,A,times_times(A,number_number_of(A,bit0(bit1(pls)))),B) = hAPP(A,A,plus_plus(A,B),B) # label(fact_139_mult__2) # label(axiom). [clausify(335)]. 2.12/2.29 1086 -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_503_power2__diff) # label(axiom). [clausify(341)]. 2.12/2.29 1087 -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_16_add__special_I3_J) # label(axiom). [clausify(346)]. 2.12/2.29 1088 -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_151_add__number__of__eq) # label(axiom). [clausify(352)]. 2.12/2.29 1089 -number_ring(A) | number_number_of(A,bit1(pls)) = one_one(A) # label(fact_173_semiring__norm_I110_J) # label(axiom). [clausify(353)]. 2.12/2.29 1090 -number_ring(A) | number_number_of(A,bit1(pls)) = one_one(A) # label(fact_172_numeral__1__eq__1) # label(axiom). [clausify(376)]. 2.12/2.29 1091 -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_42_mult__number__of__left) # label(axiom). [clausify(383)]. 2.12/2.29 1092 -number_ring(A) | number_number_of(A,bit0(bit1(pls))) = hAPP(A,A,plus_plus(A,one_one(A)),one_one(A)) # label(fact_178_one__add__one__is__two) # label(axiom). [clausify(396)]. 2.12/2.29 1093 -number_ring(A) | -linordered_idom(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_179_less__special_I4_J) # label(axiom). [clausify(414)]. 2.12/2.29 1094 -number_ring(A) | -linordered_idom(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_179_less__special_I4_J) # label(axiom). [clausify(414)]. 2.12/2.29 1095 -number_ring(A) | ti(A,B) = hAPP(A,A,plus_plus(A,number_number_of(A,pls)),B) # label(fact_147_add__numeral__0) # label(axiom). [clausify(425)]. 2.12/2.29 1096 -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_170_number__of__Bit1) # label(axiom). [clausify(467)]. 2.12/2.29 1097 -number_ring(A) | number_number_of(A,pls) = zero_zero(A) # label(fact_146_semiring__norm_I112_J) # label(axiom). [clausify(469)]. 2.12/2.29 1098 -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_59_le__special_I4_J) # label(axiom). [clausify(529)]. 2.12/2.29 1099 -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_59_le__special_I4_J) # label(axiom). [clausify(529)]. 2.12/2.29 1100 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),C)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,B)),number_number_of(A,C))) # label(fact_27_le__number__of) # label(axiom). [clausify(580)]. 2.12/2.29 1101 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),C)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),number_number_of(A,B)),number_number_of(A,C))) # label(fact_27_le__number__of) # label(axiom). [clausify(580)]. 2.12/2.29 1102 -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_44_number__of__mult) # label(axiom). [clausify(638)]. 2.12/2.29 Derived: -ring_char_0(int) | number_number_of(int,A) != number_number_of(int,B) | ti(int,A) = ti(int,B). [resolve(1068,a,1069,a)]. 2.12/2.29 Derived: -ring_char_0(int) | number_number_of(int,A) = number_number_of(int,B) | ti(int,A) != ti(int,B). [resolve(1068,a,1070,a)]. 2.12/2.29 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(1068,a,1071,a)]. 2.12/2.29 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(1068,a,1072,a)]. 2.12/2.29 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(1068,a,1073,a)]. 2.12/2.29 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(1068,a,1074,a)]. 2.12/2.29 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(1068,a,1075,a)]. 2.12/2.29 Derived: ti(int,A) = hAPP(int,int,times_times(int,number_number_of(int,bit1(pls))),A). [resolve(1068,a,1076,a)]. 2.12/2.29 Derived: ti(int,A) = hAPP(int,int,plus_plus(int,A),number_number_of(int,pls)). [resolve(1068,a,1077,a)]. 2.12/2.29 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(1068,a,1079,a)]. 2.12/2.29 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(1068,a,1080,a)]. 2.12/2.29 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(1068,a,1081,a)]. 2.12/2.29 Derived: -linordered_idom(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,A)),zero_zero(int))). [resolve(1068,a,1082,b)]. 2.12/2.29 Derived: -linordered_idom(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),number_number_of(int,A)),zero_zero(int))). [resolve(1068,a,1083,b)]. 2.12/2.29 Derived: ti(int,A) = hAPP(int,int,times_times(int,A),number_number_of(int,bit1(pls))). [resolve(1068,a,1084,a)]. 2.12/2.29 Derived: hAPP(int,int,times_times(int,number_number_of(int,bit0(bit1(pls)))),A) = hAPP(int,int,plus_plus(int,A),A). [resolve(1068,a,1085,a)]. 2.12/2.29 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(1068,a,1086,a)]. 2.12/2.29 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(1068,a,1087,a)]. 2.12/2.29 Derived: number_number_of(int,bit1(pls)) = one_one(int). [resolve(1068,a,1089,a)]. 2.12/2.29 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(1068,a,1091,a)]. 2.12/2.29 Derived: number_number_of(int,bit0(bit1(pls))) = hAPP(int,int,plus_plus(int,one_one(int)),one_one(int)). [resolve(1068,a,1092,a)]. 2.12/2.29 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(1068,a,1093,a)]. 2.12/2.29 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(1068,a,1094,a)]. 2.15/2.31 Derived: ti(int,A) = hAPP(int,int,plus_plus(int,number_number_of(int,pls)),A). [resolve(1068,a,1095,a)]. 2.15/2.31 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(1068,a,1096,a)]. 2.15/2.31 Derived: number_number_of(int,pls) = zero_zero(int). [resolve(1068,a,1097,a)]. 2.15/2.31 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(1068,a,1098,b)]. 2.15/2.31 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(1068,a,1099,b)]. 2.15/2.31 Derived: -linordered_idom(int) | -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),number_number_of(int,A)),number_number_of(int,B))). [resolve(1068,a,1100,b)]. 2.15/2.31 Derived: -linordered_idom(int) | 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),number_number_of(int,A)),number_number_of(int,B))). [resolve(1068,a,1101,b)]. 2.15/2.31 1103 -linordered_idom(A) | -number_ring(A) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),B)) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),one_one(A)),number_number_of(A,B))) # label(fact_60_le__special_I2_J) # label(axiom). [clausify(768)]. 2.15/2.31 Derived: -linordered_idom(int) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),number_number_of(int,A))). [resolve(1103,b,1068,a)]. 2.15/2.31 1104 -linordered_idom(A) | -number_ring(A) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),B)) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),one_one(A)),number_number_of(A,B))) # label(fact_60_le__special_I2_J) # label(axiom). [clausify(768)]. 2.15/2.31 Derived: -linordered_idom(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),A)) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),number_number_of(int,A))). [resolve(1104,b,1068,a)]. 2.15/2.31 1105 -number_ring(A) | -linordered_idom(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_48_le__special_I1_J) # label(axiom). [clausify(789)]. 2.15/2.31 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(1105,a,1068,a)]. 2.15/2.31 1106 -number_ring(A) | -linordered_idom(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_48_le__special_I1_J) # label(axiom). [clausify(789)]. 2.15/2.31 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(1106,a,1068,a)]. 2.15/2.31 1107 -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_169_number__of__Bit0) # label(axiom). [clausify(806)]. 2.15/2.31 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(1107,a,1068,a)]. 2.15/2.31 1108 -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_180_less__special_I2_J) # label(axiom). [clausify(808)]. 2.15/2.32 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(1108,a,1068,a)]. 2.15/2.32 1109 -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_180_less__special_I2_J) # label(axiom). [clausify(808)]. 2.15/2.32 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(1109,a,1068,a)]. 2.15/2.32 1110 -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_47_le__special_I3_J) # label(axiom). [clausify(833)]. 2.15/2.32 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(1110,a,1068,a)]. 2.15/2.32 1111 -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_47_le__special_I3_J) # label(axiom). [clausify(833)]. 2.15/2.32 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(1111,a,1068,a)]. 2.15/2.32 1112 -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_150_add__number__of__left) # label(axiom). [clausify(848)]. 2.15/2.32 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(1112,a,1068,a)]. 2.15/2.32 1113 number_ring(real) # label(arity_RealDef_Oreal___Int_Onumber__ring) # label(axiom). [assumption]. 2.15/2.32 Derived: -ring_char_0(real) | number_number_of(real,A) != number_number_of(real,B) | ti(int,A) = ti(int,B). [resolve(1113,a,1069,a)]. 2.15/2.32 Derived: -ring_char_0(real) | number_number_of(real,A) = number_number_of(real,B) | ti(int,A) != ti(int,B). [resolve(1113,a,1070,a)]. 2.15/2.32 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(1113,a,1071,a)]. 2.15/2.32 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(1113,a,1072,a)]. 2.15/2.32 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(1113,a,1073,a)]. 2.15/2.32 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(1113,a,1074,a)]. 2.15/2.32 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(1113,a,1075,a)]. 2.15/2.32 Derived: ti(real,A) = hAPP(real,real,times_times(real,number_number_of(real,bit1(pls))),A). [resolve(1113,a,1076,a)]. 2.15/2.32 Derived: ti(real,A) = hAPP(real,real,plus_plus(real,A),number_number_of(real,pls)). [resolve(1113,a,1077,a)]. 2.15/2.32 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(1113,a,1078,a)]. 2.15/2.32 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(1113,a,1079,a)]. 2.15/2.32 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(1113,a,1080,a)]. 2.15/2.32 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(1113,a,1081,a)]. 2.15/2.32 Derived: -linordered_idom(real) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),number_number_of(real,A)),zero_zero(real))). [resolve(1113,a,1082,b)]. 2.15/2.32 Derived: -linordered_idom(real) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),pls)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),number_number_of(real,A)),zero_zero(real))). [resolve(1113,a,1083,b)]. 2.15/2.32 Derived: ti(real,A) = hAPP(real,real,times_times(real,A),number_number_of(real,bit1(pls))). [resolve(1113,a,1084,a)]. 2.15/2.32 Derived: hAPP(real,real,times_times(real,number_number_of(real,bit0(bit1(pls)))),A) = hAPP(real,real,plus_plus(real,A),A). [resolve(1113,a,1085,a)]. 2.15/2.32 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(1113,a,1086,a)]. 2.15/2.32 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(1113,a,1087,a)]. 2.15/2.32 Derived: number_number_of(real,bit1(pls)) = one_one(real). [resolve(1113,a,1089,a)]. 2.15/2.32 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(1113,a,1091,a)]. 2.15/2.32 Derived: number_number_of(real,bit0(bit1(pls))) = hAPP(real,real,plus_plus(real,one_one(real)),one_one(real)). [resolve(1113,a,1092,a)]. 2.15/2.32 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(1113,a,1093,a)]. 2.15/2.32 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(1113,a,1094,a)]. 2.15/2.32 Derived: ti(real,A) = hAPP(real,real,plus_plus(real,number_number_of(real,pls)),A). [resolve(1113,a,1095,a)]. 2.15/2.32 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(1113,a,1096,a)]. 2.15/2.32 Derived: number_number_of(real,pls) = zero_zero(real). [resolve(1113,a,1097,a)]. 2.15/2.32 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(1113,a,1098,b)]. 2.15/2.32 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(1113,a,1099,b)]. 2.15/2.32 Derived: -linordered_idom(real) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),number_number_of(real,B))). [resolve(1113,a,1100,b)]. 2.17/2.35 Derived: -linordered_idom(real) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),B)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),number_number_of(real,A)),number_number_of(real,B))). [resolve(1113,a,1101,b)]. 2.17/2.35 Derived: -linordered_idom(real) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),number_number_of(real,A))). [resolve(1113,a,1103,b)]. 2.17/2.35 Derived: -linordered_idom(real) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),bit1(pls)),A)) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),number_number_of(real,A))). [resolve(1113,a,1104,b)]. 2.17/2.35 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(1113,a,1105,a)]. 2.17/2.35 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(1113,a,1106,a)]. 2.17/2.35 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(1113,a,1107,a)]. 2.17/2.35 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(1113,a,1108,a)]. 2.17/2.35 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(1113,a,1109,a)]. 2.17/2.35 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(1113,a,1110,a)]. 2.17/2.35 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(1113,a,1111,a)]. 2.17/2.35 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(1113,a,1112,a)]. 2.17/2.35 1114 -number_ring(A) | hAPP(A,A,times_times(A,B),number_number_of(A,bit0(bit1(pls)))) = hAPP(A,A,plus_plus(A,B),B) # label(fact_141_mult__2__right) # label(axiom). [clausify(931)]. 2.17/2.35 Derived: hAPP(int,int,times_times(int,A),number_number_of(int,bit0(bit1(pls)))) = hAPP(int,int,plus_plus(int,A),A). [resolve(1114,a,1068,a)]. 2.17/2.35 Derived: hAPP(real,real,times_times(real,A),number_number_of(real,bit0(bit1(pls)))) = hAPP(real,real,plus_plus(real,A),A). [resolve(1114,a,1113,a)]. 2.17/2.35 1115 -number_ring(A) | number_number_of(A,pls) = zero_zero(A) # label(fact_145_number__of__Pls) # label(axiom). [clausify(946)]. 2.17/2.35 1116 -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_136_double__number__of__Bit0) # label(axiom). [clausify(1008)]. 2.17/2.35 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(1116,a,1068,a)]. 2.17/2.35 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(1116,a,1113,a)]. 2.17/2.35 1117 -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_492_add__number__of__diff1) # label(axiom). [clausify(1021)]. 2.28/2.45 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(1117,a,1068,a)]. 2.28/2.45 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(1117,a,1113,a)]. 2.28/2.45 1118 idom(int) # label(arity_Int_Oint___Rings_Oidom) # label(axiom). [assumption]. 2.28/2.45 1119 -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,D),B)),hAPP(A,A,times_times(A,D),C))) # label(fact_405_dvd__mult__cancel__left) # label(axiom). [clausify(8)]. 2.28/2.45 1120 -idom(A) | ti(A,B) != zero_zero(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))) # label(fact_405_dvd__mult__cancel__left) # label(axiom). [clausify(8)]. 2.28/2.45 1121 -idom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),C)) | ti(A,D) = zero_zero(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),hAPP(A,A,times_times(A,D),B)),hAPP(A,A,times_times(A,D),C))) # label(fact_405_dvd__mult__cancel__left) # label(axiom). [clausify(8)]. 2.28/2.45 1122 -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,D),C))) | ti(A,C) = zero_zero(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) # label(fact_406_dvd__mult__cancel__right) # label(axiom). [clausify(247)]. 2.28/2.45 1123 -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,D),C))) | ti(A,C) != zero_zero(A) # label(fact_406_dvd__mult__cancel__right) # label(axiom). [clausify(247)]. 2.28/2.45 1124 -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,D),C))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),D)) # label(fact_406_dvd__mult__cancel__right) # label(axiom). [clausify(247)]. 2.28/2.45 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,C),A)),hAPP(int,int,times_times(int,C),B))). [resolve(1118,a,1119,a)]. 2.28/2.45 Derived: ti(int,A) != zero_zero(int) | 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))). [resolve(1118,a,1120,a)]. 2.28/2.45 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,C),B))) | ti(int,B) = zero_zero(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),C)). [resolve(1118,a,1122,a)]. 2.28/2.45 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,C),B))) | ti(int,B) != zero_zero(int). [resolve(1118,a,1123,a)]. 2.28/2.45 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,C),B))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),C)). [resolve(1118,a,1124,a)]. 2.28/2.45 1125 idom(real) # label(arity_RealDef_Oreal___Rings_Oidom) # label(axiom). [assumption]. 2.28/2.45 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,C),A)),hAPP(real,real,times_times(real,C),B))). [resolve(1125,a,1119,a)]. 2.28/2.45 Derived: ti(real,A) != zero_zero(real) | 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))). [resolve(1125,a,1120,a)]. 2.28/2.45 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),B)) | ti(real,C) = zero_zero(real) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),hAPP(real,real,times_times(real,C),A)),hAPP(real,real,times_times(real,C),B))). [resolve(1125,a,1121,a)]. 2.28/2.45 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,C),B))) | ti(real,B) = zero_zero(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),C)). [resolve(1125,a,1122,a)]. 2.35/2.57 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,C),B))) | ti(real,B) != zero_zero(real). [resolve(1125,a,1123,a)]. 2.35/2.57 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,C),B))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),C)). [resolve(1125,a,1124,a)]. 2.35/2.57 1126 dvd(real) # label(arity_RealDef_Oreal___Rings_Odvd) # label(axiom). [assumption]. 2.35/2.57 1127 -dvd(A) | times_times(A,B) = ti(fun(A,A),times_times(A,B)) # label(tsy_c_Groups_Otimes__class_Otimes_2_res) # label(axiom). [clausify(9)]. 2.35/2.57 1128 -dvd(A) | dvd_dvd(A) = ti(fun(A,fun(A,bool)),dvd_dvd(A)) # label(tsy_c_Rings_Odvd__class_Odvd_res) # label(axiom). [clausify(207)]. 2.35/2.57 1129 -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,D),E))) | 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,F),C))),E))) # label(fact_745_inf__period_I3_J) # label(axiom). [clausify(229)]. 2.35/2.57 1130 -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,D),E))) | -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,F),C))),E))) # label(fact_745_inf__period_I3_J) # label(axiom). [clausify(229)]. 2.35/2.57 Derived: times_times(real,A) = ti(fun(real,real),times_times(real,A)). [resolve(1126,a,1127,a)]. 2.35/2.57 Derived: dvd_dvd(real) = ti(fun(real,fun(real,bool)),dvd_dvd(real)). [resolve(1126,a,1128,a)]. 2.35/2.57 Derived: -comm_ring(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,C),D))) | 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,E),B))),D))). [resolve(1126,a,1129,b)]. 2.35/2.57 Derived: -comm_ring(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,C),D))) | -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,E),B))),D))). [resolve(1126,a,1130,b)]. 2.35/2.57 1131 -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_333_dvdI) # label(axiom). [clausify(317)]. 2.35/2.57 Derived: hAPP(real,real,times_times(real,A),B) != C | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),C)). [resolve(1131,a,1126,a)]. 2.35/2.57 1132 -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_746_inf__period_I4_J) # label(axiom). [clausify(486)]. 2.35/2.57 1133 -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_746_inf__period_I4_J) # label(axiom). [clausify(486)]. 2.35/2.57 1134 -dvd(A) | times_times(A,ti(A,B)) = times_times(A,B) # label(tsy_c_Groups_Otimes__class_Otimes_2_arg1) # label(axiom). [clausify(630)]. 2.35/2.57 Derived: times_times(real,ti(real,A)) = times_times(real,A). [resolve(1134,a,1126,a)]. 2.48/2.63 1135 -dvd(A) | -linorder(A) | -plus(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(732)]. 2.48/2.63 Derived: -linorder(real) | -plus(real) | ti(fun(real,real),plus_plus(real,A)) = plus_plus(real,A). [resolve(1135,a,1126,a)]. 2.48/2.63 1136 dvd(int) # label(arity_Int_Oint___Rings_Odvd) # label(axiom). [assumption]. 2.48/2.63 Derived: times_times(int,A) = ti(fun(int,int),times_times(int,A)). [resolve(1136,a,1127,a)]. 2.48/2.63 Derived: dvd_dvd(int) = ti(fun(int,fun(int,bool)),dvd_dvd(int)). [resolve(1136,a,1128,a)]. 2.48/2.63 Derived: -comm_ring(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,C),D))) | 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,E),B))),D))). [resolve(1136,a,1129,b)]. 2.48/2.63 Derived: -comm_ring(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,C),D))) | -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,E),B))),D))). [resolve(1136,a,1130,b)]. 2.48/2.63 Derived: hAPP(int,int,times_times(int,A),B) != C | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),C)). [resolve(1136,a,1131,a)]. 2.48/2.63 Derived: times_times(int,ti(int,A)) = times_times(int,A). [resolve(1136,a,1134,a)]. 2.48/2.63 Derived: -linorder(int) | -plus(int) | ti(fun(int,int),plus_plus(int,A)) = plus_plus(int,A). [resolve(1136,a,1135,a)]. 2.48/2.63 1137 -linorder(A) | -dvd(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(906)]. 2.48/2.63 Derived: -linorder(real) | -plus(real) | plus_plus(real,ti(real,A)) = plus_plus(real,A). [resolve(1137,b,1126,a)]. 2.48/2.63 Derived: -linorder(int) | -plus(int) | plus_plus(int,ti(int,A)) = plus_plus(int,A). [resolve(1137,b,1136,a)]. 2.48/2.63 1138 dvd(nat) # label(arity_Nat_Onat___Rings_Odvd) # label(axiom). [assumption]. 2.48/2.63 Derived: times_times(nat,A) = ti(fun(nat,nat),times_times(nat,A)). [resolve(1138,a,1127,a)]. 2.48/2.63 Derived: dvd_dvd(nat) = ti(fun(nat,fun(nat,bool)),dvd_dvd(nat)). [resolve(1138,a,1128,a)]. 2.48/2.63 Derived: -comm_ring(nat) | -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,plus_plus(nat,C),D))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,minus_minus(nat,C),hAPP(nat,nat,times_times(nat,E),B))),D))). [resolve(1138,a,1129,b)]. 2.48/2.63 Derived: -comm_ring(nat) | -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,plus_plus(nat,C),D))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,minus_minus(nat,C),hAPP(nat,nat,times_times(nat,E),B))),D))). [resolve(1138,a,1130,b)]. 2.48/2.63 Derived: hAPP(nat,nat,times_times(nat,A),B) != C | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),C)). [resolve(1138,a,1131,a)]. 2.48/2.63 Derived: times_times(nat,ti(nat,A)) = times_times(nat,A). [resolve(1138,a,1134,a)]. 2.48/2.63 Derived: -linorder(nat) | -plus(nat) | ti(fun(nat,nat),plus_plus(nat,A)) = plus_plus(nat,A). [resolve(1138,a,1135,a)]. 2.48/2.63 Derived: -linorder(nat) | -plus(nat) | plus_plus(nat,ti(nat,A)) = plus_plus(nat,A). [resolve(1138,a,1137,b)]. 2.48/2.63 1139 -dvd(A) | -semiring_0(A) | -hBOOL(hAPP(A,bool,B,hAPP(A,A,times_times(A,C),D))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),hAPP(A,A,plus_plus(A,f38(A,B,C)),zero_zero(A)))) # label(fact_744_unity__coeff__ex) # label(axiom). [clausify(1011)]. 2.48/2.63 Derived: -semiring_0(real) | -hBOOL(hAPP(real,bool,A,hAPP(real,real,times_times(real,B),C))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),B),hAPP(real,real,plus_plus(real,f38(real,A,B)),zero_zero(real)))). [resolve(1139,a,1126,a)]. 2.48/2.63 Derived: -semiring_0(int) | -hBOOL(hAPP(int,bool,A,hAPP(int,int,times_times(int,B),C))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B),hAPP(int,int,plus_plus(int,f38(int,A,B)),zero_zero(int)))). [resolve(1139,a,1136,a)]. 2.48/2.69 Derived: -semiring_0(nat) | -hBOOL(hAPP(nat,bool,A,hAPP(nat,nat,times_times(nat,B),C))) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),hAPP(nat,nat,plus_plus(nat,f38(nat,A,B)),zero_zero(nat)))). [resolve(1139,a,1138,a)]. 2.48/2.69 1140 -dvd(A) | -semiring_0(A) | -hBOOL(hAPP(A,bool,B,hAPP(A,A,times_times(A,C),D))) | hBOOL(hAPP(A,bool,B,f38(A,B,C))) # label(fact_744_unity__coeff__ex) # label(axiom). [clausify(1011)]. 2.48/2.69 Derived: -semiring_0(real) | -hBOOL(hAPP(real,bool,A,hAPP(real,real,times_times(real,B),C))) | hBOOL(hAPP(real,bool,A,f38(real,A,B))). [resolve(1140,a,1126,a)]. 2.48/2.69 Derived: -semiring_0(int) | -hBOOL(hAPP(int,bool,A,hAPP(int,int,times_times(int,B),C))) | hBOOL(hAPP(int,bool,A,f38(int,A,B))). [resolve(1140,a,1136,a)]. 2.48/2.69 Derived: -semiring_0(nat) | -hBOOL(hAPP(nat,bool,A,hAPP(nat,nat,times_times(nat,B),C))) | hBOOL(hAPP(nat,bool,A,f38(nat,A,B))). [resolve(1140,a,1138,a)]. 2.48/2.69 1141 -dvd(A) | -semiring_0(A) | hBOOL(hAPP(A,bool,B,hAPP(A,A,times_times(A,C),f39(A,B,C)))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),hAPP(A,A,plus_plus(A,D),zero_zero(A)))) | -hBOOL(hAPP(A,bool,B,D)) # label(fact_744_unity__coeff__ex) # label(axiom). [clausify(1011)]. 2.48/2.69 Derived: -semiring_0(real) | hBOOL(hAPP(real,bool,A,hAPP(real,real,times_times(real,B),f39(real,A,B)))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),B),hAPP(real,real,plus_plus(real,C),zero_zero(real)))) | -hBOOL(hAPP(real,bool,A,C)). [resolve(1141,a,1126,a)]. 2.48/2.69 Derived: -semiring_0(int) | hBOOL(hAPP(int,bool,A,hAPP(int,int,times_times(int,B),f39(int,A,B)))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B),hAPP(int,int,plus_plus(int,C),zero_zero(int)))) | -hBOOL(hAPP(int,bool,A,C)). [resolve(1141,a,1136,a)]. 2.48/2.69 Derived: -semiring_0(nat) | hBOOL(hAPP(nat,bool,A,hAPP(nat,nat,times_times(nat,B),f39(nat,A,B)))) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),hAPP(nat,nat,plus_plus(nat,C),zero_zero(nat)))) | -hBOOL(hAPP(nat,bool,A,C)). [resolve(1141,a,1138,a)]. 2.48/2.69 1142 comm_semiring_1(int) # label(arity_Int_Oint___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 2.48/2.69 1143 -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_223_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom). [clausify(11)]. 2.48/2.69 1144 -comm_semiring_1(A) | ti(A,B) = hAPP(A,A,plus_plus(A,B),zero_zero(A)) # label(fact_237_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J) # label(axiom). [clausify(17)]. 2.48/2.69 1145 -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_224_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J) # label(axiom). [clausify(25)]. 2.48/2.69 1146 -comm_semiring_1(A) | hAPP(A,A,times_times(A,B),hAPP(A,A,plus_plus(A,C),D)) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D)) # label(fact_239_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J) # label(axiom). [clausify(77)]. 2.48/2.69 1147 -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_339_dvd__power__same) # label(axiom). [clausify(91)]. 2.48/2.69 1148 -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_335_dvd__mult__right) # label(axiom). [clausify(116)]. 2.48/2.69 1149 -comm_semiring_1(A) | ti(A,B) = hAPP(A,A,times_times(A,one_one(A)),B) # label(fact_244_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J) # label(axiom). [clausify(124)]. 2.48/2.69 1150 -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_191_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J) # label(axiom). [clausify(152)]. 2.48/2.69 1151 -comm_semiring_1(A) | ti(A,B) = hAPP(A,A,times_times(A,B),one_one(A)) # label(fact_245_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom). [clausify(176)]. 2.48/2.69 1152 -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_341_dvd__power__le) # label(axiom). [clausify(182)]. 2.48/2.69 1153 -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_334_dvd__mult__left) # label(axiom). [clausify(192)]. 2.48/2.69 1154 -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_222_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J) # label(axiom). [clausify(195)]. 2.48/2.69 1155 -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_215_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom). [clausify(210)]. 2.48/2.69 1156 -comm_semiring_1(A) | zero_zero(A) = hAPP(A,A,times_times(A,B),zero_zero(A)) # label(fact_235_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom). [clausify(215)]. 2.48/2.69 1157 -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_246_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J) # label(axiom). [clausify(259)]. 2.48/2.69 1158 -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_328_dvd__triv__left) # label(axiom). [clausify(284)]. 2.48/2.70 1159 -comm_semiring_1(A) | hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,one_one(A)),one_one(A))),B) = hAPP(A,A,plus_plus(A,B),B) # label(fact_252_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J) # label(axiom). [clausify(285)]. 2.48/2.70 1160 -comm_semiring_1(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),one_one(A)),B)) # label(fact_338_one__dvd) # label(axiom). [clausify(291)]. 2.48/2.70 1161 -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_192_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom). [clausify(297)]. 2.48/2.70 1162 -comm_semiring_1(A) | ti(A,B) = hAPP(nat,A,power_power(A,B),one_one(nat)) # label(fact_226_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom). [clausify(327)]. 2.48/2.70 1163 -comm_semiring_1(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),zero_zero(A)),B)) | ti(A,B) = zero_zero(A) # label(fact_325_dvd__0__left) # label(axiom). [clausify(355)]. 2.48/2.70 1164 -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_216_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom). [clausify(365)]. 2.48/2.70 1165 -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_329_dvd__triv__right) # label(axiom). [clausify(369)]. 2.48/2.70 1166 -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_342_le__imp__power__dvd) # label(axiom). [clausify(411)]. 2.48/2.70 1167 -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_330_dvd__mult2) # label(axiom). [clausify(426)]. 2.48/2.70 1168 -comm_semiring_1(A) | hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,B),one_one(A))),C) = hAPP(A,A,plus_plus(A,C),hAPP(A,A,times_times(A,B),C)) # label(fact_253_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom). [clausify(499)]. 2.48/2.70 1169 -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_212_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J) # label(axiom). [clausify(515)]. 2.48/2.70 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(1142,a,1145,a)]. 2.48/2.70 Derived: hAPP(int,int,times_times(int,A),hAPP(int,int,plus_plus(int,B),C)) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),B)),hAPP(int,int,times_times(int,A),C)). [resolve(1142,a,1146,a)]. 2.48/2.70 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(1142,a,1147,a)]. 2.48/2.70 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(1142,a,1148,a)]. 2.48/2.70 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(1142,a,1150,a)]. 2.48/2.70 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(1142,a,1152,a)]. 2.48/2.70 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(1142,a,1153,a)]. 2.48/2.70 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(1142,a,1155,a)]. 2.48/2.70 Derived: zero_zero(int) = hAPP(int,int,times_times(int,A),zero_zero(int)). [resolve(1142,a,1156,a)]. 2.48/2.70 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(1142,a,1157,a)]. 2.48/2.70 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(int,int,times_times(int,A),B))). [resolve(1142,a,1158,a)]. 2.48/2.70 Derived: hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,one_one(int)),one_one(int))),A) = hAPP(int,int,plus_plus(int,A),A). [resolve(1142,a,1159,a)]. 2.48/2.70 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),one_one(int)),A)). [resolve(1142,a,1160,a)]. 2.48/2.70 Derived: hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls)))) = hAPP(int,int,times_times(int,A),A). [resolve(1142,a,1161,a)]. 2.48/2.70 Derived: ti(int,A) = hAPP(nat,int,power_power(int,A),one_one(nat)). [resolve(1142,a,1162,a)]. 2.48/2.70 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),zero_zero(int)),A)) | ti(int,A) = zero_zero(int). [resolve(1142,a,1163,a)]. 2.48/2.70 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(int,int,times_times(int,B),A))). [resolve(1142,a,1165,a)]. 2.48/2.70 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(1142,a,1166,a)]. 2.48/2.70 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(1142,a,1167,a)]. 2.48/2.70 Derived: hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,A),one_one(int))),B) = hAPP(int,int,plus_plus(int,B),hAPP(int,int,times_times(int,A),B)). [resolve(1142,a,1168,a)]. 2.48/2.70 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(1142,a,1169,a)]. 2.58/2.72 1170 -comm_semiring_1(A) | hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,B),C)),D) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),D)) # label(fact_241_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom). [clausify(584)]. 2.58/2.72 1171 -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_225_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J) # label(axiom). [clausify(600)]. 2.58/2.72 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(1171,a,1142,a)]. 2.58/2.72 1172 -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_221_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J) # label(axiom). [clausify(603)]. 2.58/2.72 1173 -comm_semiring_1(A) | hAPP(A,A,times_times(A,B),C) = hAPP(A,A,times_times(A,C),B) # label(fact_219_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) # label(axiom). [clausify(609)]. 2.58/2.72 1174 -comm_semiring_1(A) | one_one(A) = hAPP(nat,A,power_power(A,B),zero_zero(nat)) # label(fact_248_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom). [clausify(615)]. 2.58/2.72 Derived: one_one(int) = hAPP(nat,int,power_power(int,A),zero_zero(nat)). [resolve(1174,a,1142,a)]. 2.58/2.72 1175 -comm_semiring_1(A) | ti(A,B) = hAPP(A,A,plus_plus(A,zero_zero(A)),B) # label(fact_236_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J) # label(axiom). [clausify(659)]. 2.58/2.72 1176 comm_semiring_1(nat) # label(arity_Nat_Onat___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 2.58/2.72 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(1176,a,1145,a)]. 2.58/2.72 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(1176,a,1148,a)]. 2.58/2.72 Derived: ti(nat,A) = hAPP(nat,nat,times_times(nat,one_one(nat)),A). [resolve(1176,a,1149,a)]. 2.58/2.72 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(1176,a,1150,a)]. 2.58/2.72 Derived: ti(nat,A) = hAPP(nat,nat,times_times(nat,A),one_one(nat)). [resolve(1176,a,1151,a)]. 2.58/2.72 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(1176,a,1152,a)]. 2.58/2.72 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(1176,a,1153,a)]. 2.58/2.72 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(1176,a,1155,a)]. 2.58/2.72 Derived: zero_zero(nat) = hAPP(nat,nat,times_times(nat,A),zero_zero(nat)). [resolve(1176,a,1156,a)]. 2.58/2.72 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(1176,a,1157,a)]. 2.58/2.72 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,times_times(nat,A),B))). [resolve(1176,a,1158,a)]. 2.58/2.72 Derived: hAPP(nat,nat,times_times(nat,hAPP(nat,nat,plus_plus(nat,one_one(nat)),one_one(nat))),A) = hAPP(nat,nat,plus_plus(nat,A),A). [resolve(1176,a,1159,a)]. 2.58/2.73 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),one_one(nat)),A)). [resolve(1176,a,1160,a)]. 2.58/2.73 Derived: hAPP(nat,nat,power_power(nat,A),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,nat,times_times(nat,A),A). [resolve(1176,a,1161,a)]. 2.58/2.73 Derived: ti(nat,A) = hAPP(nat,nat,power_power(nat,A),one_one(nat)). [resolve(1176,a,1162,a)]. 2.58/2.73 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A)) | ti(nat,A) = zero_zero(nat). [resolve(1176,a,1163,a)]. 2.58/2.73 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(1176,a,1164,a)]. 2.58/2.73 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,times_times(nat,B),A))). [resolve(1176,a,1165,a)]. 2.58/2.73 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),dvd_dvd(nat),hAPP(nat,nat,power_power(nat,C),A)),hAPP(nat,nat,power_power(nat,C),B))). [resolve(1176,a,1166,a)]. 2.58/2.73 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(1176,a,1167,a)]. 2.58/2.73 Derived: hAPP(nat,nat,times_times(nat,hAPP(nat,nat,plus_plus(nat,A),one_one(nat))),B) = hAPP(nat,nat,plus_plus(nat,B),hAPP(nat,nat,times_times(nat,A),B)). [resolve(1176,a,1168,a)]. 2.58/2.73 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(1176,a,1169,a)]. 2.58/2.73 Derived: hAPP(nat,nat,times_times(nat,hAPP(nat,nat,plus_plus(nat,A),B)),C) = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,A),C)),hAPP(nat,nat,times_times(nat,B),C)). [resolve(1176,a,1170,a)]. 2.58/2.73 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(1176,a,1171,a)]. 2.58/2.74 Derived: one_one(nat) = hAPP(nat,nat,power_power(nat,A),zero_zero(nat)). [resolve(1176,a,1174,a)]. 2.58/2.74 1177 -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_340_power__le__dvd) # label(axiom). [clausify(716)]. 2.58/2.74 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(1177,a,1142,a)]. 2.58/2.74 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(1177,a,1176,a)]. 2.58/2.74 1178 -comm_semiring_1(A) | zero_zero(A) = hAPP(A,A,times_times(A,zero_zero(A)),B) # label(fact_234_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J) # label(axiom). [clausify(730)]. 2.58/2.74 Derived: zero_zero(int) = hAPP(int,int,times_times(int,zero_zero(int)),A). [resolve(1178,a,1142,a)]. 2.58/2.74 Derived: zero_zero(nat) = hAPP(nat,nat,times_times(nat,zero_zero(nat)),A). [resolve(1178,a,1176,a)]. 2.58/2.74 1179 comm_semiring_1(real) # label(arity_RealDef_Oreal___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 2.58/2.74 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(1179,a,1143,a)]. 2.58/2.74 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(1179,a,1145,a)]. 2.58/2.74 Derived: hAPP(real,real,times_times(real,A),hAPP(real,real,plus_plus(real,B),C)) = hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,A),C)). [resolve(1179,a,1146,a)]. 2.58/2.74 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(1179,a,1147,a)]. 2.58/2.74 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(1179,a,1148,a)]. 2.58/2.74 Derived: ti(real,A) = hAPP(real,real,times_times(real,one_one(real)),A). [resolve(1179,a,1149,a)]. 2.58/2.74 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(1179,a,1150,a)]. 2.58/2.74 Derived: ti(real,A) = hAPP(real,real,times_times(real,A),one_one(real)). [resolve(1179,a,1151,a)]. 2.58/2.74 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(1179,a,1152,a)]. 2.58/2.74 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(1179,a,1153,a)]. 2.58/2.74 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(1179,a,1155,a)]. 2.58/2.74 Derived: zero_zero(real) = hAPP(real,real,times_times(real,A),zero_zero(real)). [resolve(1179,a,1156,a)]. 2.58/2.74 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(1179,a,1157,a)]. 2.58/2.74 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(real,real,times_times(real,A),B))). [resolve(1179,a,1158,a)]. 2.58/2.74 Derived: hAPP(real,real,times_times(real,hAPP(real,real,plus_plus(real,one_one(real)),one_one(real))),A) = hAPP(real,real,plus_plus(real,A),A). [resolve(1179,a,1159,a)]. 2.58/2.74 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),one_one(real)),A)). [resolve(1179,a,1160,a)]. 2.58/2.74 Derived: hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))) = hAPP(real,real,times_times(real,A),A). [resolve(1179,a,1161,a)]. 2.58/2.74 Derived: ti(real,A) = hAPP(nat,real,power_power(real,A),one_one(nat)). [resolve(1179,a,1162,a)]. 2.58/2.74 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),zero_zero(real)),A)) | ti(real,A) = zero_zero(real). [resolve(1179,a,1163,a)]. 2.58/2.74 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(1179,a,1164,a)]. 2.58/2.74 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(real,real,times_times(real,B),A))). [resolve(1179,a,1165,a)]. 2.58/2.74 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(1179,a,1166,a)]. 2.58/2.74 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(1179,a,1167,a)]. 2.58/2.74 Derived: hAPP(real,real,times_times(real,hAPP(real,real,plus_plus(real,A),one_one(real))),B) = hAPP(real,real,plus_plus(real,B),hAPP(real,real,times_times(real,A),B)). [resolve(1179,a,1168,a)]. 2.58/2.74 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(1179,a,1169,a)]. 2.58/2.75 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(1179,a,1171,a)]. 2.58/2.75 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(1179,a,1172,a)]. 2.58/2.75 Derived: hAPP(real,real,times_times(real,A),B) = hAPP(real,real,times_times(real,B),A). [resolve(1179,a,1173,a)]. 2.58/2.75 Derived: one_one(real) = hAPP(nat,real,power_power(real,A),zero_zero(nat)). [resolve(1179,a,1174,a)]. 2.58/2.75 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(1179,a,1177,a)]. 2.58/2.75 Derived: zero_zero(real) = hAPP(real,real,times_times(real,zero_zero(real)),A). [resolve(1179,a,1178,a)]. 2.58/2.75 1180 -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_218_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom). [clausify(756)]. 2.58/2.75 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(1180,a,1142,a)]. 2.58/2.75 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(1180,a,1176,a)]. 2.58/2.75 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(1180,a,1179,a)]. 2.58/2.75 1181 -comm_semiring_1(A) | hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,B),C)),D) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),D)) # label(fact_242_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J) # label(axiom). [clausify(758)]. 2.58/2.75 Derived: hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,A),B)),C) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),C)),hAPP(int,int,times_times(int,B),C)). [resolve(1181,a,1142,a)]. 2.58/2.75 1182 -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_332_mult__dvd__mono) # label(axiom). [clausify(807)]. 2.58/2.75 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(1182,a,1142,a)]. 2.58/2.75 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(1182,a,1176,a)]. 2.58/2.75 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(1182,a,1179,a)]. 2.58/2.75 1183 -comm_semiring_1(A) | hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,B),one_one(A))),C) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),C) # label(fact_254_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom). [clausify(812)]. 2.58/2.75 Derived: hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,A),one_one(int))),B) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,A),B)),B). [resolve(1183,a,1142,a)]. 2.58/2.75 Derived: hAPP(nat,nat,times_times(nat,hAPP(nat,nat,plus_plus(nat,A),one_one(nat))),B) = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,A),B)),B). [resolve(1183,a,1176,a)]. 2.58/2.77 Derived: hAPP(real,real,times_times(real,hAPP(real,real,plus_plus(real,A),one_one(real))),B) = hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),B)),B). [resolve(1183,a,1179,a)]. 2.58/2.77 1184 -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_199_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J) # label(axiom). [clausify(813)]. 2.58/2.77 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(1184,a,1176,a)]. 2.58/2.77 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(1184,a,1179,a)]. 2.58/2.77 1185 -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_331_dvd__mult) # label(axiom). [clausify(814)]. 2.58/2.77 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(1185,a,1142,a)]. 2.58/2.77 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(1185,a,1176,a)]. 2.58/2.77 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(1185,a,1179,a)]. 2.58/2.77 1186 -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_247_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom). [clausify(816)]. 2.58/2.77 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(1186,a,1176,a)]. 2.58/2.77 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(1186,a,1179,a)]. 2.58/2.77 1187 -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_214_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J) # label(axiom). [clausify(869)]. 2.58/2.77 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,A),hAPP(int,int,times_times(int,B),hAPP(int,int,times_times(int,C),D))). [resolve(1187,a,1142,a)]. 2.58/2.77 1188 -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_217_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J) # label(axiom). [clausify(893)]. 2.58/2.77 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(1188,a,1142,a)]. 2.58/2.77 1189 -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_336_dvd__add) # label(axiom). [clausify(907)]. 2.58/2.77 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(1189,a,1142,a)]. 2.58/2.77 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(1189,a,1176,a)]. 2.66/2.80 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(1189,a,1179,a)]. 2.66/2.80 1190 -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_308_dvd__trans) # label(axiom). [clausify(949)]. 2.66/2.80 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(1190,a,1142,a)]. 2.66/2.80 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(1190,a,1176,a)]. 2.66/2.80 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(1190,a,1179,a)]. 2.66/2.80 1191 -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_213_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom). [clausify(977)]. 2.66/2.80 1192 -comm_semiring_1(A) | hAPP(A,A,plus_plus(A,B),C) = hAPP(A,A,plus_plus(A,C),B) # label(fact_220_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom). [clausify(979)]. 2.66/2.80 Derived: hAPP(int,int,plus_plus(int,A),B) = hAPP(int,int,plus_plus(int,B),A). [resolve(1192,a,1142,a)]. 2.66/2.80 Derived: hAPP(nat,nat,plus_plus(nat,A),B) = hAPP(nat,nat,plus_plus(nat,B),A). [resolve(1192,a,1176,a)]. 2.66/2.80 Derived: hAPP(real,real,plus_plus(real,A),B) = hAPP(real,real,plus_plus(real,B),A). [resolve(1192,a,1179,a)]. 2.66/2.80 1193 -comm_semiring_1(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),zero_zero(A))) # label(fact_278_dvd__0__right) # label(axiom). [clausify(993)]. 2.66/2.80 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),zero_zero(int))). [resolve(1193,a,1142,a)]. 2.66/2.80 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),zero_zero(nat))). [resolve(1193,a,1176,a)]. 2.66/2.80 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),zero_zero(real))). [resolve(1193,a,1179,a)]. 2.66/2.80 1194 -comm_semiring_1(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),B)) # label(fact_307_dvd__refl) # label(axiom). [clausify(994)]. 2.66/2.80 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),A)). [resolve(1194,a,1142,a)]. 2.66/2.80 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),A)). [resolve(1194,a,1176,a)]. 2.66/2.80 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),A)). [resolve(1194,a,1179,a)]. 2.66/2.80 1195 -comm_semiring_1(A) | ti(A,B) != one_one(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),B),hAPP(nat,A,power_power(A,B),C))) # label(fact_458_dvd__power) # label(axiom). [clausify(1028)]. 2.66/2.80 Derived: ti(int,A) != one_one(int) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(nat,int,power_power(int,A),B))). [resolve(1195,a,1142,a)]. 2.66/2.80 Derived: ti(nat,A) != one_one(nat) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,power_power(nat,A),B))). [resolve(1195,a,1176,a)]. 2.66/2.80 Derived: ti(real,A) != one_one(real) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),dvd_dvd(real),A),hAPP(nat,real,power_power(real,A),B))). [resolve(1195,a,1179,a)]. 2.66/2.80 1196 -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_458_dvd__power) # label(axiom). [clausify(1028)]. 2.66/2.87 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(1196,a,1142,a)]. 2.66/2.87 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(1196,a,1176,a)]. 2.66/2.87 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(1196,a,1179,a)]. 2.66/2.87 1197 linord20386208strict(nat) # label(arity_Nat_Onat___Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 2.66/2.87 1198 -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_423_mult__less__imp__less__left) # label(axiom). [clausify(12)]. 2.66/2.87 1199 -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_381_mult__strict__left__mono) # label(axiom). [clausify(119)]. 2.66/2.87 1200 -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_387_zero__less__mult__pos) # label(axiom). [clausify(206)]. 2.66/2.87 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(1200,a,1197,a)]. 2.66/2.87 1201 -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_388_mult__pos__neg2) # label(axiom). [clausify(243)]. 2.66/2.87 1202 -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_386_zero__less__mult__pos2) # label(axiom). [clausify(257)]. 2.66/2.87 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(1202,a,1197,a)]. 2.66/2.87 1203 linord20386208strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 2.66/2.87 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(1203,a,1198,a)]. 2.66/2.87 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),zero_zero(real)),C)) | 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(1203,a,1199,a)]. 2.66/2.87 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(1203,a,1200,a)]. 2.74/2.89 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,B),A)),zero_zero(real))). [resolve(1203,a,1201,a)]. 2.74/2.89 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(1203,a,1202,a)]. 2.74/2.89 1204 -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_422_mult__right__le__imp__le) # label(axiom). [clausify(370)]. 2.74/2.89 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(1204,a,1197,a)]. 2.74/2.89 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(1204,a,1203,a)]. 2.74/2.89 1205 -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_425_mult__less__imp__less__right) # label(axiom). [clausify(429)]. 2.74/2.89 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(1205,a,1203,a)]. 2.74/2.89 1206 -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_421_mult__left__le__imp__le) # label(axiom). [clausify(465)]. 2.74/2.89 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,A),C))) | -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)). [resolve(1206,a,1203,a)]. 2.74/2.89 1207 -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_428_mult__less__le__imp__less) # label(axiom). [clausify(541)]. 2.74/2.89 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(1207,a,1197,a)]. 2.76/2.90 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(1207,a,1203,a)]. 2.76/2.90 1208 linord20386208strict(int) # label(arity_Int_Oint___Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 2.76/2.90 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(1208,a,1198,a)]. 2.76/2.90 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),zero_zero(int)),C)) | 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(1208,a,1199,a)]. 2.76/2.90 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)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)). [resolve(1208,a,1200,a)]. 2.76/2.90 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,B),A)),zero_zero(int))). [resolve(1208,a,1201,a)]. 2.76/2.90 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(1208,a,1202,a)]. 2.76/2.90 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(1208,a,1204,a)]. 2.76/2.90 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(1208,a,1205,a)]. 2.76/2.90 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,A),C))) | -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)). [resolve(1208,a,1206,a)]. 2.76/2.90 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(1208,a,1207,a)]. 2.76/2.90 1209 -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_427_mult__le__less__imp__less) # label(axiom). [clausify(677)]. 2.76/2.91 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(1209,a,1197,a)]. 2.76/2.91 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(1209,a,1203,a)]. 2.76/2.91 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(1209,a,1208,a)]. 2.76/2.91 1210 -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_429_mult__strict__mono_H) # label(axiom). [clausify(711)]. 2.76/2.91 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(1210,a,1197,a)]. 2.76/2.91 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(1210,a,1203,a)]. 2.76/2.91 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(1210,a,1208,a)]. 2.76/2.91 1211 -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_384_mult__neg__pos) # label(axiom). [clausify(745)]. 2.76/2.91 1212 -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_430_mult__strict__mono) # label(axiom). [clausify(779)]. 2.77/2.95 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(1212,a,1197,a)]. 2.77/2.95 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(1212,a,1203,a)]. 2.77/2.95 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(1212,a,1208,a)]. 2.77/2.95 1213 -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_389_mult__pos__neg) # label(axiom). [clausify(879)]. 2.77/2.95 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(1213,a,1203,a)]. 2.77/2.95 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(1213,a,1208,a)]. 2.77/2.95 1214 -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_390_mult__pos__pos) # label(axiom). [clausify(934)]. 2.77/2.95 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(1214,a,1197,a)]. 2.77/2.95 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(1214,a,1203,a)]. 2.77/2.95 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(1214,a,1208,a)]. 2.77/2.95 1215 -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_382_mult__strict__right__mono) # label(axiom). [clausify(1017)]. 3.07/3.22 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(1215,a,1197,a)]. 3.07/3.22 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),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),C))). [resolve(1215,a,1203,a)]. 3.07/3.22 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),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),C))). [resolve(1215,a,1208,a)]. 3.07/3.22 1216 -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_763_add__imp__eq) # label(axiom). [clausify(80)]. 3.07/3.22 1217 cancel146912293up_add(real) # label(arity_RealDef_Oreal___Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 3.07/3.22 Derived: hAPP(real,real,plus_plus(real,A),B) != hAPP(real,real,plus_plus(real,A),C) | ti(real,B) = ti(real,C). [resolve(1216,a,1217,a)]. 3.07/3.22 1218 cancel146912293up_add(nat) # label(arity_Nat_Onat___Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 3.07/3.22 Derived: hAPP(nat,nat,plus_plus(nat,A),B) != hAPP(nat,nat,plus_plus(nat,A),C) | ti(nat,B) = ti(nat,C). [resolve(1218,a,1216,a)]. 3.07/3.22 1219 cancel146912293up_add(int) # label(arity_Int_Oint___Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 3.07/3.22 Derived: hAPP(int,int,plus_plus(int,A),B) != hAPP(int,int,plus_plus(int,A),C) | ti(int,B) = ti(int,C). [resolve(1219,a,1216,a)]. 3.07/3.22 1220 real_normed_algebra(real) # label(arity_RealDef_Oreal___RealVector_Oreal__normed__algebra) # label(axiom). [assumption]. 3.07/3.22 1221 -real_normed_algebra(A) | zero_zero(A) = hAPP(A,A,times_times(A,zero_zero(A)),B) # label(fact_886_mult_Ozero__left) # label(axiom). [clausify(13)]. 3.07/3.22 1222 -real_normed_algebra(A) | hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,B),C)),D) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),D)) # label(fact_892_mult_Oadd__left) # label(axiom). [clausify(100)]. 3.07/3.22 1223 -real_normed_algebra(A) | zero_zero(A) = hAPP(A,A,times_times(A,B),zero_zero(A)) # label(fact_888_mult_Ozero__right) # label(axiom). [clausify(214)]. 3.07/3.22 1224 -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_896_mult_Odiff__left) # label(axiom). [clausify(262)]. 3.07/3.22 1225 -real_normed_algebra(A) | zero_zero(A) = hAPP(A,A,times_times(A,zero_zero(A)),B) # label(fact_887_mult__left_Ozero) # label(axiom). [clausify(265)]. 3.07/3.22 1226 -real_normed_algebra(A) | hAPP(A,A,times_times(A,B),hAPP(A,A,plus_plus(A,C),D)) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D)) # label(fact_890_mult_Oadd__right) # label(axiom). [clausify(345)]. 3.07/3.22 1227 -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_897_mult__left_Odiff) # label(axiom). [clausify(371)]. 3.07/3.22 1228 -real_normed_algebra(A) | hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,B),C)),D) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),D)) # label(fact_893_mult__left_Oadd) # label(axiom). [clausify(588)]. 3.07/3.22 1229 -real_normed_algebra(A) | zero_zero(A) = hAPP(A,A,times_times(A,B),zero_zero(A)) # label(fact_889_mult__right_Ozero) # label(axiom). [clausify(635)]. 3.07/3.22 1230 -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_884_mult_Oprod__diff__prod) # label(axiom). [clausify(648)]. 3.34/3.53 1231 -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_895_mult__right_Odiff) # label(axiom). [clausify(656)]. 3.34/3.53 1232 -real_normed_algebra(A) | hAPP(A,A,times_times(A,B),hAPP(A,A,plus_plus(A,C),D)) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),C)),hAPP(A,A,times_times(A,B),D)) # label(fact_891_mult__right_Oadd) # label(axiom). [clausify(693)]. 3.34/3.53 1233 -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_894_mult_Odiff__right) # label(axiom). [clausify(843)]. 3.34/3.53 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(1220,a,1224,a)]. 3.34/3.53 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(1220,a,1230,a)]. 3.34/3.53 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(1220,a,1231,a)]. 3.34/3.53 1234 -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(434)]. 3.34/3.53 1235 ab_semigroup_add(nat) # label(arity_Nat_Onat___Groups_Oab__semigroup__add) # label(axiom). [assumption]. 3.34/3.53 1236 ab_semigroup_add(real) # label(arity_RealDef_Oreal___Groups_Oab__semigroup__add) # label(axiom). [assumption]. 3.34/3.53 Derived: ti(fun(nat,nat),plus_plus(nat,A)) = plus_plus(nat,A). [resolve(1234,a,1235,a)]. 3.34/3.53 Derived: ti(fun(real,real),plus_plus(real,A)) = plus_plus(real,A). [resolve(1234,a,1236,a)]. 3.34/3.53 1237 -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_759_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom). [clausify(511)]. 3.34/3.53 1238 -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(519)]. 3.34/3.53 Derived: plus_plus(nat,ti(nat,A)) = plus_plus(nat,A). [resolve(1238,a,1235,a)]. 3.34/3.53 Derived: plus_plus(real,ti(real,A)) = plus_plus(real,A). [resolve(1238,a,1236,a)]. 3.34/3.53 1239 ab_semigroup_add(int) # label(arity_Int_Oint___Groups_Oab__semigroup__add) # label(axiom). [assumption]. 3.34/3.53 Derived: ti(fun(int,int),plus_plus(int,A)) = plus_plus(int,A). [resolve(1239,a,1234,a)]. 3.34/3.53 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(1239,a,1237,a)]. 3.34/3.53 Derived: plus_plus(int,ti(int,A)) = plus_plus(int,A). [resolve(1239,a,1238,a)]. 3.34/3.53 1240 monoid_add(real) # label(arity_RealDef_Oreal___Groups_Omonoid__add) # label(axiom). [assumption]. 3.34/3.53 1241 -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(23)]. 3.34/3.53 1242 -monoid_add(A) | ti(A,B) = hAPP(A,A,plus_plus(A,zero_zero(A)),B) # label(fact_767_add__0__left) # label(axiom). [clausify(728)]. 3.34/3.53 1243 monoid_add(nat) # label(arity_Nat_Onat___Groups_Omonoid__add) # label(axiom). [assumption]. 3.34/3.53 1244 monoid_add(int) # label(arity_Int_Oint___Groups_Omonoid__add) # label(axiom). [assumption]. 3.34/3.53 1245 -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(974)]. 3.47/3.63 1246 -monoid_add(A) | ti(A,B) = hAPP(A,A,plus_plus(A,B),zero_zero(A)) # label(fact_770_add__0__right) # label(axiom). [clausify(1039)]. 3.47/3.63 1247 ordere236663937imp_le(real) # label(arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 3.47/3.63 1248 -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_772_add__le__cancel__right) # label(axiom). [clausify(24)]. 3.47/3.63 1249 -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_772_add__le__cancel__right) # label(axiom). [clausify(24)]. 3.47/3.63 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(1247,a,1248,a)]. 3.47/3.63 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(1247,a,1249,a)]. 3.47/3.63 1250 ordere236663937imp_le(nat) # label(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 3.47/3.63 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(1250,a,1248,a)]. 3.47/3.63 1251 -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_777_add__le__imp__le__right) # label(axiom). [clausify(287)]. 3.47/3.63 1252 ordere236663937imp_le(int) # label(arity_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 3.47/3.63 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(1252,a,1248,a)]. 3.47/3.63 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(1252,a,1249,a)]. 3.47/3.63 1253 -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_778_add__le__imp__le__left) # label(axiom). [clausify(375)]. 3.47/3.63 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(1253,a,1247,a)]. 3.47/3.63 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(1253,a,1250,a)]. 3.47/3.63 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(1253,a,1252,a)]. 3.47/3.63 1254 -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,B),D)),hAPP(A,A,plus_plus(A,C),D))) # label(fact_779_add__less__cancel__right) # label(axiom). [clausify(642)]. 3.47/3.63 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(1254,a,1247,a)]. 3.47/3.66 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(1254,a,1252,a)]. 3.47/3.66 1255 -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,B),D)),hAPP(A,A,plus_plus(A,C),D))) # label(fact_779_add__less__cancel__right) # label(axiom). [clausify(642)]. 3.47/3.66 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(1255,a,1247,a)]. 3.47/3.66 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(1255,a,1250,a)]. 3.47/3.66 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(1255,a,1252,a)]. 3.47/3.66 1256 -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_784_add__less__imp__less__right) # label(axiom). [clausify(691)]. 3.47/3.66 1257 -ordere236663937imp_le(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_773_add__le__cancel__left) # label(axiom). [clausify(849)]. 3.47/3.66 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),hAPP(real,real,plus_plus(real,C),A)),hAPP(real,real,plus_plus(real,C),B))). [resolve(1257,a,1247,a)]. 3.47/3.66 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),hAPP(nat,nat,plus_plus(nat,C),A)),hAPP(nat,nat,plus_plus(nat,C),B))). [resolve(1257,a,1250,a)]. 3.47/3.66 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),hAPP(int,int,plus_plus(int,C),A)),hAPP(int,int,plus_plus(int,C),B))). [resolve(1257,a,1252,a)]. 3.47/3.66 1258 -ordere236663937imp_le(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_773_add__le__cancel__left) # label(axiom). [clausify(849)]. 3.47/3.66 1259 -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_780_add__less__cancel__left) # label(axiom). [clausify(874)]. 3.47/3.66 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(1259,a,1247,a)]. 3.47/3.66 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(1259,a,1250,a)]. 3.47/3.66 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(1259,a,1252,a)]. 3.47/3.66 1260 -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_780_add__less__cancel__left) # label(axiom). [clausify(874)]. 3.64/3.80 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(1260,a,1247,a)]. 3.64/3.80 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(1260,a,1250,a)]. 3.64/3.80 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(1260,a,1252,a)]. 3.64/3.80 1261 -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_785_add__less__imp__less__left) # label(axiom). [clausify(930)]. 3.64/3.80 1262 -no_zero_divisors(A) | zero_zero(A) != hAPP(A,A,times_times(A,B),C) | ti(A,C) = zero_zero(A) | ti(A,B) = zero_zero(A) # label(fact_315_divisors__zero) # label(axiom). [clausify(212)]. 3.64/3.80 1263 no_zero_divisors(real) # label(arity_RealDef_Oreal___Rings_Ono__zero__divisors) # label(axiom). [assumption]. 3.64/3.80 Derived: zero_zero(real) != hAPP(real,real,times_times(real,A),B) | ti(real,B) = zero_zero(real) | ti(real,A) = zero_zero(real). [resolve(1262,a,1263,a)]. 3.64/3.80 1264 -no_zero_divisors(A) | ti(A,B) = zero_zero(A) | ti(A,C) = zero_zero(A) | zero_zero(A) != hAPP(A,A,times_times(A,B),C) # label(fact_316_no__zero__divisors) # label(axiom). [clausify(325)]. 3.64/3.80 1265 -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(334)]. 3.64/3.80 1266 no_zero_divisors(nat) # label(arity_Nat_Onat___Rings_Ono__zero__divisors) # label(axiom). [assumption]. 3.64/3.80 Derived: zero_zero(nat) != hAPP(nat,nat,times_times(nat,A),B) | ti(nat,B) = zero_zero(nat) | ti(nat,A) = zero_zero(nat). [resolve(1266,a,1262,a)]. 3.64/3.80 1267 -power(A) | -zero_neq_one(A) | -no_zero_divisors(A) | -mult_zero(A) | zero_zero(A) != hAPP(nat,A,power_power(A,B),C) | ti(A,B) = zero_zero(A) # label(fact_410_power__eq__0__iff) # label(axiom). [clausify(596)]. 3.64/3.80 Derived: -power(real) | -zero_neq_one(real) | -mult_zero(real) | zero_zero(real) != hAPP(nat,real,power_power(real,A),B) | ti(real,A) = zero_zero(real). [resolve(1267,c,1263,a)]. 3.64/3.80 Derived: -power(nat) | -zero_neq_one(nat) | -mult_zero(nat) | zero_zero(nat) != hAPP(nat,nat,power_power(nat,A),B) | ti(nat,A) = zero_zero(nat). [resolve(1267,c,1266,a)]. 3.64/3.80 1268 -power(A) | -zero_neq_one(A) | -no_zero_divisors(A) | -mult_zero(A) | zero_zero(A) != hAPP(nat,A,power_power(A,B),C) | zero_zero(nat) != C # label(fact_410_power__eq__0__iff) # label(axiom). [clausify(596)]. 3.64/3.80 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(1268,c,1263,a)]. 3.64/3.80 1269 -power(A) | -zero_neq_one(A) | -no_zero_divisors(A) | -mult_zero(A) | zero_zero(A) = hAPP(nat,A,power_power(A,B),C) | ti(A,B) != zero_zero(A) | zero_zero(nat) = C # label(fact_410_power__eq__0__iff) # label(axiom). [clausify(596)]. 3.64/3.80 Derived: -power(real) | -zero_neq_one(real) | -mult_zero(real) | zero_zero(real) = hAPP(nat,real,power_power(real,A),B) | ti(real,A) != zero_zero(real) | zero_zero(nat) = B. [resolve(1269,c,1263,a)]. 3.64/3.80 Derived: -power(nat) | -zero_neq_one(nat) | -mult_zero(nat) | zero_zero(nat) = hAPP(nat,nat,power_power(nat,A),B) | ti(nat,A) != zero_zero(nat) | zero_zero(nat) = B. [resolve(1269,c,1266,a)]. 3.64/3.80 1270 -zero_neq_one(A) | -no_zero_divisors(A) | -mult_zero(A) | -power(A) | ti(A,B) != zero_zero(A) | number_number_of(nat,C) = zero_zero(nat) | zero_zero(A) = hAPP(nat,A,power_power(A,B),number_number_of(nat,C)) # label(fact_255_power__eq__0__iff__number__of) # label(axiom). [clausify(747)]. 3.64/3.80 Derived: -zero_neq_one(real) | -mult_zero(real) | -power(real) | ti(real,A) != zero_zero(real) | number_number_of(nat,B) = zero_zero(nat) | zero_zero(real) = hAPP(nat,real,power_power(real,A),number_number_of(nat,B)). [resolve(1270,b,1263,a)]. 3.78/3.93 Derived: -zero_neq_one(nat) | -mult_zero(nat) | -power(nat) | ti(nat,A) != zero_zero(nat) | number_number_of(nat,B) = zero_zero(nat) | zero_zero(nat) = hAPP(nat,nat,power_power(nat,A),number_number_of(nat,B)). [resolve(1270,b,1266,a)]. 3.78/3.93 1271 -zero_neq_one(A) | -no_zero_divisors(A) | -mult_zero(A) | -power(A) | ti(A,B) = zero_zero(A) | zero_zero(A) != hAPP(nat,A,power_power(A,B),number_number_of(nat,C)) # label(fact_255_power__eq__0__iff__number__of) # label(axiom). [clausify(747)]. 3.78/3.93 1272 -zero_neq_one(A) | -no_zero_divisors(A) | -mult_zero(A) | -power(A) | number_number_of(nat,B) != zero_zero(nat) | zero_zero(A) != hAPP(nat,A,power_power(A,C),number_number_of(nat,B)) # label(fact_255_power__eq__0__iff__number__of) # label(axiom). [clausify(747)]. 3.78/3.93 Derived: -zero_neq_one(real) | -mult_zero(real) | -power(real) | number_number_of(nat,A) != zero_zero(nat) | zero_zero(real) != hAPP(nat,real,power_power(real,B),number_number_of(nat,A)). [resolve(1272,b,1263,a)]. 3.78/3.93 Derived: -zero_neq_one(nat) | -mult_zero(nat) | -power(nat) | number_number_of(nat,A) != zero_zero(nat) | zero_zero(nat) != hAPP(nat,nat,power_power(nat,B),number_number_of(nat,A)). [resolve(1272,b,1266,a)]. 3.78/3.93 1273 no_zero_divisors(int) # label(arity_Int_Oint___Rings_Ono__zero__divisors) # label(axiom). [assumption]. 3.78/3.93 Derived: zero_zero(int) != hAPP(int,int,times_times(int,A),B) | ti(int,B) = zero_zero(int) | ti(int,A) = zero_zero(int). [resolve(1273,a,1262,a)]. 3.78/3.93 Derived: -power(int) | -zero_neq_one(int) | -mult_zero(int) | zero_zero(int) != hAPP(nat,int,power_power(int,A),B) | ti(int,A) = zero_zero(int). [resolve(1273,a,1267,c)]. 3.78/3.93 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(1273,a,1268,c)]. 3.78/3.93 Derived: -power(int) | -zero_neq_one(int) | -mult_zero(int) | zero_zero(int) = hAPP(nat,int,power_power(int,A),B) | ti(int,A) != zero_zero(int) | zero_zero(nat) = B. [resolve(1273,a,1269,c)]. 3.78/3.93 Derived: -zero_neq_one(int) | -mult_zero(int) | -power(int) | ti(int,A) != zero_zero(int) | number_number_of(nat,B) = zero_zero(nat) | zero_zero(int) = hAPP(nat,int,power_power(int,A),number_number_of(nat,B)). [resolve(1273,a,1270,b)]. 3.78/3.93 Derived: -zero_neq_one(int) | -mult_zero(int) | -power(int) | number_number_of(nat,A) != zero_zero(nat) | zero_zero(int) != hAPP(nat,int,power_power(int,B),number_number_of(nat,A)). [resolve(1273,a,1272,b)]. 3.78/3.93 1274 -no_zero_divisors(A) | times_times(A,B) = ti(fun(A,A),times_times(A,B)) # label(tsy_c_Groups_Otimes__class_Otimes_5_res) # label(axiom). [clausify(928)]. 3.78/3.93 1275 ordered_ring(int) # label(arity_Int_Oint___Rings_Oordered__ring) # label(axiom). [assumption]. 3.78/3.93 1276 -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_490_less__add__iff2) # label(axiom). [clausify(26)]. 3.78/3.93 1277 -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_490_less__add__iff2) # label(axiom). [clausify(26)]. 3.78/3.93 1278 -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_489_le__add__iff2) # label(axiom). [clausify(200)]. 3.78/3.93 1279 -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_489_le__add__iff2) # label(axiom). [clausify(200)]. 3.78/3.94 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(1275,a,1276,a)]. 3.78/3.94 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(1275,a,1277,a)]. 3.78/3.94 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(1275,a,1278,a)]. 3.78/3.94 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(1275,a,1279,a)]. 3.78/3.94 1280 -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,hAPP(A,A,minus_minus(A,B),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,B),D)),E)),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),D)),F))) # label(fact_488_le__add__iff1) # label(axiom). [clausify(528)]. 3.78/3.94 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,hAPP(int,int,minus_minus(int,A),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,A),C)),D)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),C)),E))). [resolve(1280,a,1275,a)]. 3.78/3.94 1281 -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,hAPP(A,A,minus_minus(A,B),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,B),D)),E)),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),D)),F))) # label(fact_488_le__add__iff1) # label(axiom). [clausify(528)]. 3.78/3.94 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,hAPP(int,int,minus_minus(int,A),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,A),C)),D)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),C)),E))). [resolve(1281,a,1275,a)]. 3.78/3.94 1282 ordered_ring(real) # label(arity_RealDef_Oreal___Rings_Oordered__ring) # label(axiom). [assumption]. 3.78/3.94 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(1282,a,1276,a)]. 3.78/3.94 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(1282,a,1277,a)]. 3.78/3.98 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(1282,a,1278,a)]. 3.78/3.98 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(1282,a,1279,a)]. 3.78/3.98 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,hAPP(real,real,minus_minus(real,A),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,A),C)),D)),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,B),C)),E))). [resolve(1282,a,1280,a)]. 3.78/3.98 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,hAPP(real,real,minus_minus(real,A),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,A),C)),D)),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,B),C)),E))). [resolve(1282,a,1281,a)]. 3.78/3.98 1283 -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_362_split__mult__pos__le) # label(axiom). [clausify(904)]. 3.78/3.98 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),zero_zero(int)),hAPP(int,int,times_times(int,A),B))). [resolve(1283,a,1275,a)]. 3.78/3.98 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),zero_zero(real)),hAPP(real,real,times_times(real,A),B))). [resolve(1283,a,1282,a)]. 3.78/3.98 1284 -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_362_split__mult__pos__le) # label(axiom). [clausify(904)]. 3.78/3.98 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(1284,a,1275,a)]. 3.78/3.98 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,times_times(real,A),B))). [resolve(1284,a,1282,a)]. 3.78/3.98 1285 -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,hAPP(A,A,minus_minus(A,B),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,B),D)),E)),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),D)),F))) # label(fact_491_less__add__iff1) # label(axiom). [clausify(942)]. 3.78/4.00 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,hAPP(int,int,minus_minus(int,A),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,A),C)),D)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),C)),E))). [resolve(1285,a,1275,a)]. 3.78/4.00 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,hAPP(real,real,minus_minus(real,A),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,A),C)),D)),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,B),C)),E))). [resolve(1285,a,1282,a)]. 3.78/4.00 1286 -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,hAPP(A,A,minus_minus(A,B),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,B),D)),E)),hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),D)),F))) # label(fact_491_less__add__iff1) # label(axiom). [clausify(942)]. 3.78/4.00 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,hAPP(int,int,minus_minus(int,A),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,A),C)),D)),hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),C)),E))). [resolve(1286,a,1275,a)]. 3.78/4.00 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,hAPP(real,real,minus_minus(real,A),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,A),C)),D)),hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,B),C)),E))). [resolve(1286,a,1282,a)]. 3.78/4.00 1287 -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_365_mult__left__mono__neg) # label(axiom). [clausify(988)]. 3.78/4.00 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(1287,a,1275,a)]. 3.78/4.00 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(1287,a,1282,a)]. 3.78/4.00 1288 -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_370_mult__nonpos__nonpos) # label(axiom). [clausify(1014)]. 3.78/4.00 1289 -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_366_mult__right__mono__neg) # label(axiom). [clausify(1036)]. 3.78/4.00 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(1289,a,1275,a)]. 4.08/4.26 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(1289,a,1282,a)]. 4.08/4.26 1290 semiring_1(real) # label(arity_RealDef_Oreal___Rings_Osemiring__1) # label(axiom). [assumption]. 4.08/4.26 1291 -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(29)]. 4.08/4.26 Derived: one_one(real) = hAPP(nat,real,power_power(real,one_one(real)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1290,a,1291,a)]. 4.08/4.26 1292 -semiring_1(A) | zero_zero(A) = hAPP(nat,A,power_power(A,zero_zero(A)),number_number_of(nat,bit0(bit1(pls)))) # label(fact_13_zero__power2) # label(axiom). [clausify(572)]. 4.08/4.26 Derived: zero_zero(real) = hAPP(nat,real,power_power(real,zero_zero(real)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1292,a,1290,a)]. 4.08/4.26 1293 semiring_1(nat) # label(arity_Nat_Onat___Rings_Osemiring__1) # label(axiom). [assumption]. 4.08/4.26 Derived: one_one(nat) = hAPP(nat,nat,power_power(nat,one_one(nat)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1293,a,1291,a)]. 4.08/4.26 Derived: zero_zero(nat) = hAPP(nat,nat,power_power(nat,zero_zero(nat)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1293,a,1292,a)]. 4.08/4.26 1294 semiring_1(int) # label(arity_Int_Oint___Rings_Osemiring__1) # label(axiom). [assumption]. 4.08/4.26 Derived: one_one(int) = hAPP(nat,int,power_power(int,one_one(int)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1294,a,1291,a)]. 4.08/4.26 Derived: zero_zero(int) = hAPP(nat,int,power_power(int,zero_zero(int)),number_number_of(nat,bit0(bit1(pls)))). [resolve(1294,a,1292,a)]. 4.08/4.26 1295 group_add(int) # label(arity_Int_Oint___Groups_Ogroup__add) # label(axiom). [assumption]. 4.08/4.26 1296 -group_add(A) | ti(A,B) = hAPP(A,A,plus_plus(A,hAPP(A,A,minus_minus(A,B),C)),C) # label(fact_797_diff__add__cancel) # label(axiom). [clausify(31)]. 4.08/4.26 1297 -group_add(A) | ti(A,B) = hAPP(A,A,minus_minus(A,B),zero_zero(A)) # label(fact_789_diff__0__right) # label(axiom). [clausify(158)]. 4.08/4.26 1298 -group_add(A) | zero_zero(A) != hAPP(A,A,minus_minus(A,B),C) | ti(A,C) = ti(A,B) # label(fact_786_right__minus__eq) # label(axiom). [clausify(295)]. 4.08/4.26 1299 -group_add(A) | zero_zero(A) = hAPP(A,A,minus_minus(A,B),C) | ti(A,C) != ti(A,B) # label(fact_786_right__minus__eq) # label(axiom). [clausify(295)]. 4.08/4.26 1300 -group_add(A) | ti(A,B) = hAPP(A,A,minus_minus(A,hAPP(A,A,plus_plus(A,B),C)),C) # label(fact_796_add__diff__cancel) # label(axiom). [clausify(339)]. 4.08/4.26 1301 -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(428)]. 4.08/4.26 Derived: ti(int,A) = hAPP(int,int,plus_plus(int,hAPP(int,int,minus_minus(int,A),B)),B). [resolve(1295,a,1296,a)]. 4.08/4.26 Derived: ti(int,A) = hAPP(int,int,minus_minus(int,A),zero_zero(int)). [resolve(1295,a,1297,a)]. 4.08/4.26 Derived: zero_zero(int) != hAPP(int,int,minus_minus(int,A),B) | ti(int,B) = ti(int,A). [resolve(1295,a,1298,a)]. 4.08/4.26 Derived: zero_zero(int) = hAPP(int,int,minus_minus(int,A),B) | ti(int,B) != ti(int,A). [resolve(1295,a,1299,a)]. 4.08/4.26 Derived: ti(int,A) = hAPP(int,int,minus_minus(int,hAPP(int,int,plus_plus(int,A),B)),B). [resolve(1295,a,1300,a)]. 4.08/4.26 Derived: minus_minus(int,A) = ti(fun(int,int),minus_minus(int,A)). [resolve(1295,a,1301,a)]. 4.08/4.26 1302 -group_add(A) | zero_zero(A) = hAPP(A,A,minus_minus(A,B),B) # label(fact_788_diff__self) # label(axiom). [clausify(649)]. 4.08/4.26 Derived: zero_zero(int) = hAPP(int,int,minus_minus(int,A),A). [resolve(1302,a,1295,a)]. 4.08/4.26 1303 -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(838)]. 4.08/4.26 Derived: minus_minus(int,ti(int,A)) = minus_minus(int,A). [resolve(1303,a,1295,a)]. 4.39/4.55 1304 group_add(real) # label(arity_RealDef_Oreal___Groups_Ogroup__add) # label(axiom). [assumption]. 4.39/4.55 Derived: ti(real,A) = hAPP(real,real,plus_plus(real,hAPP(real,real,minus_minus(real,A),B)),B). [resolve(1304,a,1296,a)]. 4.39/4.55 Derived: ti(real,A) = hAPP(real,real,minus_minus(real,A),zero_zero(real)). [resolve(1304,a,1297,a)]. 4.39/4.55 Derived: zero_zero(real) != hAPP(real,real,minus_minus(real,A),B) | ti(real,B) = ti(real,A). [resolve(1304,a,1298,a)]. 4.39/4.55 Derived: zero_zero(real) = hAPP(real,real,minus_minus(real,A),B) | ti(real,B) != ti(real,A). [resolve(1304,a,1299,a)]. 4.39/4.55 Derived: ti(real,A) = hAPP(real,real,minus_minus(real,hAPP(real,real,plus_plus(real,A),B)),B). [resolve(1304,a,1300,a)]. 4.39/4.55 Derived: minus_minus(real,A) = ti(fun(real,real),minus_minus(real,A)). [resolve(1304,a,1301,a)]. 4.39/4.55 Derived: zero_zero(real) = hAPP(real,real,minus_minus(real,A),A). [resolve(1304,a,1302,a)]. 4.39/4.55 Derived: minus_minus(real,ti(real,A)) = minus_minus(real,A). [resolve(1304,a,1303,a)]. 4.39/4.55 1305 cancel_semigroup_add(real) # label(arity_RealDef_Oreal___Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 4.39/4.55 1306 -cancel_semigroup_add(A) | ti(A,B) != ti(A,C) | hAPP(A,A,plus_plus(A,D),B) = hAPP(A,A,plus_plus(A,D),C) # label(fact_760_add__left__cancel) # label(axiom). [clausify(35)]. 4.39/4.55 1307 -cancel_semigroup_add(A) | ti(A,B) = ti(A,C) | hAPP(A,A,plus_plus(A,D),B) != hAPP(A,A,plus_plus(A,D),C) # label(fact_760_add__left__cancel) # label(axiom). [clausify(35)]. 4.39/4.55 Derived: ti(real,A) != ti(real,B) | hAPP(real,real,plus_plus(real,C),A) = hAPP(real,real,plus_plus(real,C),B). [resolve(1305,a,1306,a)]. 4.39/4.55 1308 -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_762_add__left__imp__eq) # label(axiom). [clausify(85)]. 4.39/4.55 1309 -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(226)]. 4.39/4.55 1310 -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_764_add__right__imp__eq) # label(axiom). [clausify(487)]. 4.39/4.55 Derived: hAPP(real,real,plus_plus(real,A),B) != hAPP(real,real,plus_plus(real,C),B) | ti(real,A) = ti(real,C). [resolve(1310,a,1305,a)]. 4.39/4.55 1311 -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(530)]. 4.39/4.55 1312 cancel_semigroup_add(int) # label(arity_Int_Oint___Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 4.39/4.55 Derived: ti(int,A) != ti(int,B) | hAPP(int,int,plus_plus(int,C),A) = hAPP(int,int,plus_plus(int,C),B). [resolve(1312,a,1306,a)]. 4.39/4.55 Derived: hAPP(int,int,plus_plus(int,A),B) != hAPP(int,int,plus_plus(int,C),B) | ti(int,A) = ti(int,C). [resolve(1312,a,1310,a)]. 4.39/4.55 1313 -cancel_semigroup_add(A) | ti(A,B) != ti(A,C) | hAPP(A,A,plus_plus(A,B),D) = hAPP(A,A,plus_plus(A,C),D) # label(fact_761_add__right__cancel) # label(axiom). [clausify(921)]. 4.39/4.55 Derived: ti(real,A) != ti(real,B) | hAPP(real,real,plus_plus(real,A),C) = hAPP(real,real,plus_plus(real,B),C). [resolve(1313,a,1305,a)]. 4.39/4.55 Derived: ti(int,A) != ti(int,B) | hAPP(int,int,plus_plus(int,A),C) = hAPP(int,int,plus_plus(int,B),C). [resolve(1313,a,1312,a)]. 4.39/4.55 1314 -cancel_semigroup_add(A) | ti(A,B) = ti(A,C) | hAPP(A,A,plus_plus(A,B),D) != hAPP(A,A,plus_plus(A,C),D) # label(fact_761_add__right__cancel) # label(axiom). [clausify(921)]. 4.39/4.55 1315 cancel_semigroup_add(nat) # label(arity_Nat_Onat___Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 4.39/4.55 Derived: ti(nat,A) != ti(nat,B) | hAPP(nat,nat,plus_plus(nat,C),A) = hAPP(nat,nat,plus_plus(nat,C),B). [resolve(1315,a,1306,a)]. 4.39/4.55 Derived: hAPP(nat,nat,plus_plus(nat,A),B) != hAPP(nat,nat,plus_plus(nat,C),B) | ti(nat,A) = ti(nat,C). [resolve(1315,a,1310,a)]. 4.39/4.55 Derived: ti(nat,A) != ti(nat,B) | hAPP(nat,nat,plus_plus(nat,A),C) = hAPP(nat,nat,plus_plus(nat,B),C). [resolve(1315,a,1313,a)]. 4.39/4.55 1316 ordered_ab_group_add(real) # label(arity_RealDef_Oreal___Groups_Oordered__ab__group__add) # label(axiom). [assumption]. 4.39/4.55 1317 -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_791_diff__eq__diff__less) # label(axiom). [clausify(37)]. 4.39/4.56 1318 -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_791_diff__eq__diff__less) # label(axiom). [clausify(37)]. 4.39/4.56 1319 -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_812_less__iff__diff__less__0) # label(axiom). [clausify(501)]. 4.39/4.56 1320 -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_812_less__iff__diff__less__0) # label(axiom). [clausify(501)]. 4.39/4.56 1321 -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_790_diff__eq__diff__less__eq) # label(axiom). [clausify(605)]. 4.39/4.56 1322 -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_790_diff__eq__diff__less__eq) # label(axiom). [clausify(605)]. 4.39/4.56 1323 -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_811_le__iff__diff__le__0) # label(axiom). [clausify(669)]. 4.39/4.56 1324 -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_811_le__iff__diff__le__0) # label(axiom). [clausify(669)]. 4.39/4.56 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(1316,a,1317,a)]. 4.39/4.56 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(1316,a,1318,a)]. 4.39/4.56 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(1316,a,1319,a)]. 4.39/4.56 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(1316,a,1320,a)]. 4.39/4.56 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(1316,a,1321,a)]. 4.39/4.56 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(1316,a,1322,a)]. 4.39/4.56 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(1316,a,1323,a)]. 4.39/4.56 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(1316,a,1324,a)]. 4.47/4.67 1325 ordered_ab_group_add(int) # label(arity_Int_Oint___Groups_Oordered__ab__group__add) # label(axiom). [assumption]. 4.47/4.67 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(1325,a,1317,a)]. 4.47/4.67 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(1325,a,1318,a)]. 4.47/4.67 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(1325,a,1321,a)]. 4.47/4.67 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(1325,a,1322,a)]. 4.47/4.67 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(1325,a,1323,a)]. 4.47/4.67 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(1325,a,1324,a)]. 4.47/4.67 1326 semiring_div(int) # label(arity_Int_Oint___Divides_Osemiring__div) # label(axiom). [assumption]. 4.47/4.67 1327 -semiring_div(A) | zero_zero(A) != hAPP(A,A,div_mod(A,B),C) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),B)) # label(fact_940_dvd__eq__mod__eq__0) # label(axiom). [clausify(39)]. 4.47/4.67 1328 -semiring_div(A) | zero_zero(A) = hAPP(A,A,div_mod(A,B),C) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),dvd_dvd(A),C),B)) # label(fact_940_dvd__eq__mod__eq__0) # label(axiom). [clausify(39)]. 4.47/4.67 1329 -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_900_dvd__eq__mod__eq__0__number__of) # label(axiom). [clausify(65)]. 4.47/4.67 1330 -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_900_dvd__eq__mod__eq__0__number__of) # label(axiom). [clausify(65)]. 4.47/4.67 Derived: zero_zero(int) != hAPP(int,int,div_mod(int,A),B) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B),A)). [resolve(1326,a,1327,a)]. 4.47/4.67 Derived: zero_zero(int) = hAPP(int,int,div_mod(int,A),B) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B),A)). [resolve(1326,a,1328,a)]. 4.47/4.67 1331 -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_924_zmod__simps_I1_J) # label(axiom). [clausify(81)]. 4.47/4.67 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(1331,a,1326,a)]. 4.47/4.67 1332 -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_903_mod__mult__mult2) # label(axiom). [clausify(149)]. 4.47/4.67 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(1332,a,1326,a)]. 4.47/4.67 1333 -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_922_dvd__mod__imp__dvd) # label(axiom). [clausify(179)]. 4.47/4.67 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A),hAPP(int,int,div_mod(int,B),C))) | -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),B)). [resolve(1333,a,1326,a)]. 4.53/4.69 1334 -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_906_mod__mult__left__eq) # label(axiom). [clausify(221)]. 4.53/4.69 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(1334,a,1326,a)]. 4.53/4.69 1335 -semiring_div(A) | zero_zero(A) = hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,B),C)),C) # label(fact_945_mod__mult__self2__is__0) # label(axiom). [clausify(276)]. 4.53/4.69 Derived: zero_zero(int) = hAPP(int,int,div_mod(int,hAPP(int,int,times_times(int,A),B)),B). [resolve(1335,a,1326,a)]. 4.53/4.69 1336 -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_942_mod__mult__self2) # label(axiom). [clausify(374)]. 4.53/4.69 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(1336,a,1326,a)]. 4.53/4.69 1337 -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(381)]. 4.53/4.69 Derived: div_mod(int,A) = ti(fun(int,int),div_mod(int,A)). [resolve(1337,a,1326,a)]. 4.53/4.69 1338 -semiring_div(A) | zero_zero(A) = hAPP(A,A,div_mod(A,zero_zero(A)),B) # label(fact_933_mod__0) # label(axiom). [clausify(427)]. 4.53/4.69 1339 -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_923_mod__add__cong) # label(axiom). [clausify(432)]. 4.53/4.69 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(1339,a,1326,a)]. 4.53/4.69 1340 -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_925_zmod__simps_I2_J) # label(axiom). [clausify(449)]. 4.53/4.69 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(1340,a,1326,a)]. 4.53/4.69 1341 -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_919_dvd__mod__iff) # label(axiom). [clausify(481)]. 4.53/4.69 1342 -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_919_dvd__mod__iff) # label(axiom). [clausify(481)]. 4.53/4.69 1343 -semiring_div(A) | zero_zero(A) = hAPP(A,A,div_mod(A,B),one_one(A)) # label(fact_944_mod__by__1) # label(axiom). [clausify(502)]. 4.53/4.69 Derived: zero_zero(int) = hAPP(int,int,div_mod(int,A),one_one(int)). [resolve(1343,a,1326,a)]. 4.53/4.69 1344 semiring_div(nat) # label(arity_Nat_Onat___Divides_Osemiring__div) # label(axiom). [assumption]. 4.53/4.69 Derived: zero_zero(nat) != hAPP(nat,nat,div_mod(nat,A),B) | hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),A)). [resolve(1344,a,1327,a)]. 4.53/4.69 Derived: zero_zero(nat) = hAPP(nat,nat,div_mod(nat,A),B) | -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),A)). [resolve(1344,a,1328,a)]. 4.53/4.69 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(1344,a,1331,a)]. 4.53/4.69 Derived: hAPP(nat,nat,div_mod(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,div_mod(nat,A),C)),B). [resolve(1344,a,1332,a)]. 4.53/4.71 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A),hAPP(nat,nat,div_mod(nat,B),C))) | -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),B)). [resolve(1344,a,1333,a)]. 4.53/4.71 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(1344,a,1334,a)]. 4.53/4.71 Derived: zero_zero(nat) = hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,times_times(nat,A),B)),B). [resolve(1344,a,1335,a)]. 4.53/4.71 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(1344,a,1336,a)]. 4.53/4.71 Derived: div_mod(nat,A) = ti(fun(nat,nat),div_mod(nat,A)). [resolve(1344,a,1337,a)]. 4.53/4.71 Derived: zero_zero(nat) = hAPP(nat,nat,div_mod(nat,zero_zero(nat)),A). [resolve(1344,a,1338,a)]. 4.53/4.71 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(1344,a,1339,a)]. 4.53/4.71 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(1344,a,1340,a)]. 4.53/4.71 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(1344,a,1342,a)]. 4.53/4.71 Derived: zero_zero(nat) = hAPP(nat,nat,div_mod(nat,A),one_one(nat)). [resolve(1344,a,1343,a)]. 4.53/4.71 1345 -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_921_dvd__mod) # label(axiom). [clausify(566)]. 4.53/4.71 1346 -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_943_mod__mult__self1) # label(axiom). [clausify(579)]. 4.53/4.71 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(1346,a,1326,a)]. 4.53/4.71 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(1346,a,1344,a)]. 4.53/4.71 1347 -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_912_mod__mod__trivial) # label(axiom). [clausify(618)]. 4.53/4.71 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(1347,a,1326,a)]. 4.53/4.71 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(1347,a,1344,a)]. 4.53/4.71 1348 -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_928_mod__add__right__eq) # label(axiom). [clausify(657)]. 4.53/4.71 1349 -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_920_mod__mod__cancel) # label(axiom). [clausify(663)]. 4.53/4.71 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(1349,a,1326,a)]. 4.53/4.71 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(1349,a,1344,a)]. 4.53/4.71 1350 -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_941_dvd__imp__mod__0) # label(axiom). [clausify(694)]. 4.53/4.73 1351 -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_905_mod__mult__eq) # label(axiom). [clausify(740)]. 4.53/4.73 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(1351,a,1326,a)]. 4.53/4.73 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(1351,a,1344,a)]. 4.53/4.73 1352 -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_907_mod__mult__right__eq) # label(axiom). [clausify(749)]. 4.53/4.73 Derived: hAPP(int,int,div_mod(int,hAPP(int,int,times_times(int,A),hAPP(int,int,div_mod(int,B),C))),C) = hAPP(int,int,div_mod(int,hAPP(int,int,times_times(int,A),B)),C). [resolve(1352,a,1326,a)]. 4.53/4.73 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(1352,a,1344,a)]. 4.53/4.73 1353 -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_927_mod__add__left__eq) # label(axiom). [clausify(794)]. 4.53/4.73 1354 -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_904_mod__mult__mult1) # label(axiom). [clausify(821)]. 4.53/4.73 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(1354,a,1326,a)]. 4.53/4.73 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(1354,a,1344,a)]. 4.53/4.73 1355 -semiring_div(A) | ti(A,B) = hAPP(A,A,div_mod(A,B),zero_zero(A)) # label(fact_932_mod__by__0) # label(axiom). [clausify(845)]. 4.53/4.73 Derived: ti(int,A) = hAPP(int,int,div_mod(int,A),zero_zero(int)). [resolve(1355,a,1326,a)]. 4.53/4.73 Derived: ti(nat,A) = hAPP(nat,nat,div_mod(nat,A),zero_zero(nat)). [resolve(1355,a,1344,a)]. 4.53/4.73 1356 -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_901_mod__mult__cong) # label(axiom). [clausify(847)]. 4.53/4.73 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(1356,a,1326,a)]. 4.53/4.73 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(1356,a,1344,a)]. 4.53/4.73 1357 -semiring_div(A) | zero_zero(A) = hAPP(A,A,div_mod(A,hAPP(A,A,times_times(A,B),C)),B) # label(fact_946_mod__mult__self1__is__0) # label(axiom). [clausify(878)]. 4.53/4.73 Derived: zero_zero(int) = hAPP(int,int,div_mod(int,hAPP(int,int,times_times(int,A),B)),A). [resolve(1357,a,1326,a)]. 4.53/4.73 Derived: zero_zero(nat) = hAPP(nat,nat,div_mod(nat,hAPP(nat,nat,times_times(nat,A),B)),A). [resolve(1357,a,1344,a)]. 4.53/4.73 1358 -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_926_mod__add__eq) # label(axiom). [clausify(882)]. 4.69/4.84 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(1358,a,1326,a)]. 4.69/4.84 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(1358,a,1344,a)]. 4.69/4.84 1359 -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_930_mod__add__self2) # label(axiom). [clausify(922)]. 4.69/4.84 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(1359,a,1326,a)]. 4.69/4.84 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(1359,a,1344,a)]. 4.69/4.84 1360 -semiring_div(A) | zero_zero(A) = hAPP(A,A,div_mod(A,B),B) # label(fact_931_mod__self) # label(axiom). [clausify(935)]. 4.69/4.84 Derived: zero_zero(int) = hAPP(int,int,div_mod(int,A),A). [resolve(1360,a,1326,a)]. 4.69/4.84 Derived: zero_zero(nat) = hAPP(nat,nat,div_mod(nat,A),A). [resolve(1360,a,1344,a)]. 4.69/4.84 1361 -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_902_zmod__simps_I4_J) # label(axiom). [clausify(941)]. 4.69/4.84 1362 -semiring_div(A) | div_mod(A,ti(A,B)) = div_mod(A,B) # label(tsy_c_Divides_Odiv__class_Omod_arg1) # label(axiom). [clausify(948)]. 4.69/4.84 Derived: div_mod(int,ti(int,A)) = div_mod(int,A). [resolve(1362,a,1326,a)]. 4.69/4.84 Derived: div_mod(nat,ti(nat,A)) = div_mod(nat,A). [resolve(1362,a,1344,a)]. 4.69/4.84 1363 -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_929_mod__add__self1) # label(axiom). [clausify(1007)]. 4.69/4.84 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(1363,a,1326,a)]. 4.69/4.84 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(1363,a,1344,a)]. 4.69/4.84 1364 number_semiring(real) # label(arity_RealDef_Oreal___Int_Onumber__semiring) # label(axiom). [assumption]. 4.69/4.84 1365 -number_semiring(A) | number_number_of(A,bit1(pls)) = one_one(A) # label(fact_171_semiring__numeral__1__eq__1) # label(axiom). [clausify(43)]. 4.69/4.84 1366 -number_semiring(A) | number_number_of(A,bit0(bit1(pls))) = hAPP(A,A,plus_plus(A,one_one(A)),one_one(A)) # label(fact_177_semiring__one__add__one__is__two) # label(axiom). [clausify(54)]. 4.69/4.84 1367 -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_85_semiring__add__number__of) # label(axiom). [clausify(70)]. 4.69/4.84 1368 -number_semiring(A) | hAPP(A,A,times_times(A,number_number_of(A,bit0(bit1(pls)))),B) = hAPP(A,A,plus_plus(A,B),B) # label(fact_138_semiring__mult__2) # label(axiom). [clausify(95)]. 4.69/4.84 1369 -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_168_power2__sum) # label(axiom). [clausify(235)]. 4.69/4.84 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(1369,a,1364,a)]. 4.99/5.14 1370 -number_semiring(A) | number_number_of(A,pls) = zero_zero(A) # label(fact_144_semiring__numeral__0__eq__0) # label(axiom). [clausify(559)]. 4.99/5.14 1371 number_semiring(int) # label(arity_Int_Oint___Int_Onumber__semiring) # label(axiom). [assumption]. 4.99/5.14 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(1371,a,1369,a)]. 4.99/5.14 1372 number_semiring(nat) # label(arity_Nat_Onat___Int_Onumber__semiring) # label(axiom). [assumption]. 4.99/5.14 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(1372,a,1367,a)]. 4.99/5.14 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(1372,a,1369,a)]. 4.99/5.14 Derived: number_number_of(nat,pls) = zero_zero(nat). [resolve(1372,a,1370,a)]. 4.99/5.14 1373 -number_semiring(A) | hAPP(A,A,times_times(A,B),number_number_of(A,bit0(bit1(pls)))) = hAPP(A,A,plus_plus(A,B),B) # label(fact_140_semiring__mult__2__right) # label(axiom). [clausify(999)]. 4.99/5.14 Derived: hAPP(nat,nat,times_times(nat,A),number_number_of(nat,bit0(bit1(pls)))) = hAPP(nat,nat,plus_plus(nat,A),A). [resolve(1373,a,1372,a)]. 4.99/5.14 1374 -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_41_semiring__mult__number__of) # label(axiom). [clausify(1019)]. 4.99/5.14 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(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(1374,a,1371,a)]. 4.99/5.14 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(1374,a,1372,a)]. 4.99/5.14 1375 -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_380_comm__mult__strict__left__mono) # label(axiom). [clausify(683)]. 4.99/5.14 1376 linord893533164strict(nat) # label(arity_Nat_Onat___Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 4.99/5.14 1377 linord893533164strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 4.99/5.14 1378 linord893533164strict(int) # label(arity_Int_Oint___Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 4.99/5.14 1379 -monoid_mult(A) | ti(A,B) = hAPP(nat,A,power_power(A,B),one_one(nat)) # label(fact_347_power__one__right) # label(axiom). [clausify(153)]. 4.99/5.14 1380 monoid_mult(nat) # label(arity_Nat_Onat___Groups_Omonoid__mult) # label(axiom). [assumption]. 4.99/5.14 1381 -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_412_power__add) # label(axiom). [clausify(189)]. 4.99/5.18 1382 -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_327_power__commutes) # label(axiom). [clausify(246)]. 4.99/5.18 1383 monoid_mult(real) # label(arity_RealDef_Oreal___Groups_Omonoid__mult) # label(axiom). [assumption]. 4.99/5.18 1384 -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_142_power2__eq__square) # label(axiom). [clausify(296)]. 4.99/5.18 1385 -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_345_power__mult) # label(axiom). [clausify(328)]. 4.99/5.18 1386 monoid_mult(int) # label(arity_Int_Oint___Groups_Omonoid__mult) # label(axiom). [assumption]. 4.99/5.18 1387 -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_137_power3__eq__cube) # label(axiom). [clausify(544)]. 4.99/5.18 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(1387,a,1380,a)]. 4.99/5.18 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(1387,a,1383,a)]. 4.99/5.18 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(1387,a,1386,a)]. 4.99/5.18 1388 -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_232_power__even__eq) # label(axiom). [clausify(601)]. 4.99/5.18 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(1388,a,1380,a)]. 4.99/5.18 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(1388,a,1383,a)]. 4.99/5.18 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(1388,a,1386,a)]. 4.99/5.18 1389 -monoid_mult(A) | ti(A,B) = hAPP(A,A,times_times(A,B),one_one(A)) # label(fact_794_mult__1__right) # label(axiom). [clausify(729)]. 4.99/5.18 1390 -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_167_power2__eq__square__number__of) # label(axiom). [clausify(748)]. 4.99/5.18 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(1390,a,1380,a)]. 4.99/5.18 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(1390,a,1383,a)]. 4.99/5.18 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(1390,a,1386,a)]. 4.99/5.18 1391 -monoid_mult(A) | one_one(A) = hAPP(nat,A,power_power(A,one_one(A)),B) # label(fact_337_power__one) # label(axiom). [clausify(890)]. 4.99/5.18 Derived: one_one(nat) = hAPP(nat,nat,power_power(nat,one_one(nat)),A). [resolve(1391,a,1380,a)]. 5.12/5.29 Derived: one_one(real) = hAPP(nat,real,power_power(real,one_one(real)),A). [resolve(1391,a,1383,a)]. 5.12/5.29 Derived: one_one(int) = hAPP(nat,int,power_power(int,one_one(int)),A). [resolve(1391,a,1386,a)]. 5.12/5.29 1392 -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_715_realpow__minus__mult) # label(axiom). [clausify(917)]. 5.12/5.29 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(1392,a,1380,a)]. 5.12/5.29 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(1392,a,1383,a)]. 5.12/5.29 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(1392,a,1386,a)]. 5.12/5.29 1393 -monoid_mult(A) | ti(A,B) = hAPP(A,A,times_times(A,one_one(A)),B) # label(fact_792_mult__1__left) # label(axiom). [clausify(919)]. 5.12/5.29 Derived: ti(int,A) = hAPP(int,int,times_times(int,one_one(int)),A). [resolve(1393,a,1386,a)]. 5.12/5.29 1394 -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_376_zero__le__mult__iff) # label(axiom). [clausify(132)]. 5.12/5.29 1395 linord581940658strict(int) # label(arity_Int_Oint___Rings_Olinordered__ring__strict) # label(axiom). [assumption]. 5.12/5.29 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(1394,a,1395,a)]. 5.12/5.29 1396 -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_376_zero__le__mult__iff) # label(axiom). [clausify(132)]. 5.12/5.29 1397 -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_376_zero__le__mult__iff) # label(axiom). [clausify(132)]. 5.12/5.29 1398 -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_376_zero__le__mult__iff) # label(axiom). [clausify(132)]. 5.12/5.29 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(1398,a,1395,a)]. 5.12/5.29 1399 -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_376_zero__le__mult__iff) # label(axiom). [clausify(132)]. 5.12/5.29 1400 -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),C),zero_zero(A))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) # label(fact_376_zero__le__mult__iff) # label(axiom). [clausify(132)]. 5.12/5.30 1401 -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)))) | ti(A,B) != zero_zero(A) | ti(A,C) != zero_zero(A) # label(fact_107_sum__squares__gt__zero__iff) # label(axiom). [clausify(244)]. 5.12/5.30 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)))) | ti(int,A) != zero_zero(int) | ti(int,B) != zero_zero(int). [resolve(1401,a,1395,a)]. 5.12/5.30 1402 -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)))) | ti(A,B) = zero_zero(A) # label(fact_107_sum__squares__gt__zero__iff) # label(axiom). [clausify(244)]. 5.12/5.30 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)))) | ti(int,A) = zero_zero(int). [resolve(1402,a,1395,a)]. 5.12/5.30 1403 -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)))) | ti(A,C) = zero_zero(A) # label(fact_107_sum__squares__gt__zero__iff) # label(axiom). [clausify(244)]. 5.12/5.30 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)))) | ti(int,B) = zero_zero(int). [resolve(1403,a,1395,a)]. 5.12/5.30 1404 -linord581940658strict(A) | ti(A,B) != zero_zero(A) | ti(A,C) != zero_zero(A) | zero_zero(A) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),C)),hAPP(A,A,times_times(A,B),B)) # label(fact_93_sum__squares__eq__zero__iff) # label(axiom). [clausify(313)]. 5.12/5.30 Derived: ti(int,A) != zero_zero(int) | ti(int,B) != zero_zero(int) | zero_zero(int) = hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),B)),hAPP(int,int,times_times(int,A),A)). [resolve(1404,a,1395,a)]. 5.12/5.30 1405 -linord581940658strict(A) | ti(A,B) = zero_zero(A) | zero_zero(A) != hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,C),C)),hAPP(A,A,times_times(A,B),B)) # label(fact_93_sum__squares__eq__zero__iff) # label(axiom). [clausify(313)]. 5.12/5.30 Derived: ti(int,A) = zero_zero(int) | zero_zero(int) != hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,B),B)),hAPP(int,int,times_times(int,A),A)). [resolve(1405,a,1395,a)]. 5.12/5.30 1406 -linord581940658strict(A) | ti(A,B) = zero_zero(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_93_sum__squares__eq__zero__iff) # label(axiom). [clausify(313)]. 5.12/5.30 Derived: ti(int,A) = zero_zero(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(1406,a,1395,a)]. 5.12/5.30 1407 -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,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(A),B),zero_zero(A))) # label(fact_392_mult__less__cancel__left__disj) # label(axiom). [clausify(357)]. 5.12/5.30 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(int),zero_zero(int)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),zero_zero(int))). [resolve(1407,a,1395,a)]. 5.12/5.30 1408 -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,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(A),D),C)) # label(fact_392_mult__less__cancel__left__disj) # label(axiom). [clausify(357)]. 5.12/5.31 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(int),zero_zero(int)),A)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),B)). [resolve(1408,a,1395,a)]. 5.12/5.31 1409 -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,B),D))) | 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),B),zero_zero(A))) # label(fact_392_mult__less__cancel__left__disj) # label(axiom). [clausify(357)]. 5.12/5.31 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(int),B),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A),zero_zero(int))). [resolve(1409,a,1395,a)]. 5.12/5.31 1410 -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,B),D))) | 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),D),C)) # label(fact_392_mult__less__cancel__left__disj) # label(axiom). [clausify(357)]. 5.12/5.31 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(int),B),C)) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),B)). [resolve(1410,a,1395,a)]. 5.12/5.31 1411 -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,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(A),C),D)) # label(fact_392_mult__less__cancel__left__disj) # label(axiom). [clausify(357)]. 5.12/5.31 1412 -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,B),D))) | -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),D),C)) # label(fact_392_mult__less__cancel__left__disj) # label(axiom). [clausify(357)]. 5.12/5.31 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(int),A),zero_zero(int))) | -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),C),B)). [resolve(1412,a,1395,a)]. 5.12/5.31 1413 -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),C),D)) | 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))) # label(fact_432_mult__le__cancel__left__pos) # label(axiom). [clausify(393)]. 5.12/5.31 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_eq(int),hAPP(int,int,times_times(int,A),B)),hAPP(int,int,times_times(int,A),C))). [resolve(1413,a,1395,a)]. 5.12/5.31 1414 -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),C),D)) | -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))) # label(fact_432_mult__le__cancel__left__pos) # label(axiom). [clausify(393)]. 5.12/5.31 1415 -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_383_mult__neg__neg) # label(axiom). [clausify(440)]. 5.12/5.31 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(1415,a,1395,a)]. 5.12/5.31 1416 -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),zero_zero(A)),C)) # label(fact_375_mult__le__0__iff) # label(axiom). [clausify(489)]. 5.12/5.31 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),zero_zero(int)),B)). [resolve(1416,a,1395,a)]. 5.12/5.31 1417 -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),B),zero_zero(A))) # label(fact_375_mult__le__0__iff) # label(axiom). [clausify(489)]. 5.12/5.31 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),A),zero_zero(int))). [resolve(1417,a,1395,a)]. 5.12/5.31 1418 -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),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C)) # label(fact_375_mult__le__0__iff) # label(axiom). [clausify(489)]. 5.12/5.31 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),B),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B)). [resolve(1418,a,1395,a)]. 5.12/5.31 1419 -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),C),zero_zero(A))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) # label(fact_375_mult__le__0__iff) # label(axiom). [clausify(489)]. 5.12/5.31 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),B),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),zero_zero(int))). [resolve(1419,a,1395,a)]. 5.12/5.31 1420 -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_375_mult__le__0__iff) # label(axiom). [clausify(489)]. 5.12/5.31 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(1420,a,1395,a)]. 5.12/5.31 1421 -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_375_mult__le__0__iff) # label(axiom). [clausify(489)]. 5.12/5.31 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(1421,a,1395,a)]. 5.12/5.32 1422 -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),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),D),C)) # label(fact_431_mult__le__cancel__left__neg) # label(axiom). [clausify(493)]. 5.12/5.32 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),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),C),B)). [resolve(1422,a,1395,a)]. 5.12/5.32 1423 -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),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),D),C)) # label(fact_431_mult__le__cancel__left__neg) # label(axiom). [clausify(493)]. 5.12/5.32 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),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),C),B)). [resolve(1423,a,1395,a)]. 5.12/5.32 1424 linord581940658strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__ring__strict) # label(axiom). [assumption]. 5.12/5.32 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(1424,a,1394,a)]. 5.12/5.32 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(1424,a,1398,a)]. 5.12/5.32 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)))) | ti(real,A) != zero_zero(real) | ti(real,B) != zero_zero(real). [resolve(1424,a,1401,a)]. 5.12/5.32 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)))) | ti(real,A) = zero_zero(real). [resolve(1424,a,1402,a)]. 5.12/5.32 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)))) | ti(real,B) = zero_zero(real). [resolve(1424,a,1403,a)]. 5.12/5.32 Derived: ti(real,A) != zero_zero(real) | ti(real,B) != zero_zero(real) | zero_zero(real) = hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,B),B)),hAPP(real,real,times_times(real,A),A)). [resolve(1424,a,1404,a)]. 5.12/5.32 Derived: ti(real,A) = zero_zero(real) | zero_zero(real) != hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,B),B)),hAPP(real,real,times_times(real,A),A)). [resolve(1424,a,1405,a)]. 5.12/5.32 Derived: ti(real,A) = zero_zero(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(1424,a,1406,a)]. 5.12/5.32 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(real),zero_zero(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))). [resolve(1424,a,1407,a)]. 5.12/5.33 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(real),zero_zero(real)),A)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),B)). [resolve(1424,a,1408,a)]. 5.12/5.33 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(real),B),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A),zero_zero(real))). [resolve(1424,a,1409,a)]. 5.12/5.33 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(real),B),C)) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),B)). [resolve(1424,a,1410,a)]. 5.12/5.33 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(real),A),zero_zero(real))) | -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C),B)). [resolve(1424,a,1412,a)]. 5.12/5.33 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_eq(real),hAPP(real,real,times_times(real,A),B)),hAPP(real,real,times_times(real,A),C))). [resolve(1424,a,1413,a)]. 5.12/5.33 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(1424,a,1415,a)]. 5.12/5.33 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),zero_zero(real)),B)). [resolve(1424,a,1416,a)]. 5.12/5.33 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),B),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))). [resolve(1424,a,1419,a)]. 5.12/5.33 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(1424,a,1420,a)]. 5.12/5.33 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(1424,a,1421,a)]. 5.12/5.33 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),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),C),B)). [resolve(1424,a,1422,a)]. 5.12/5.33 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),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),C),B)). [resolve(1424,a,1423,a)]. 5.12/5.33 1425 -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_379_mult__strict__right__mono__neg) # label(axiom). [clausify(662)]. 5.22/5.37 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),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(1425,a,1395,a)]. 5.22/5.37 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(1425,a,1424,a)]. 5.22/5.37 1426 -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_391_mult__less__cancel__left__pos) # label(axiom). [clausify(746)]. 5.22/5.37 1427 -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_391_mult__less__cancel__left__pos) # label(axiom). [clausify(746)]. 5.22/5.37 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(1427,a,1395,a)]. 5.22/5.37 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(1427,a,1424,a)]. 5.22/5.37 1428 -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_385_mult__less__cancel__left__neg) # label(axiom). [clausify(986)]. 5.22/5.37 1429 -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_385_mult__less__cancel__left__neg) # label(axiom). [clausify(986)]. 5.22/5.37 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),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(1429,a,1395,a)]. 5.22/5.37 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(1429,a,1424,a)]. 5.22/5.37 1430 -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_393_mult__less__cancel__right__disj) # label(axiom). [clausify(1003)]. 5.22/5.37 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(1430,a,1395,a)]. 5.22/5.37 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(1430,a,1424,a)]. 5.22/5.37 1431 -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_393_mult__less__cancel__right__disj) # label(axiom). [clausify(1003)]. 5.22/5.37 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(1431,a,1395,a)]. 5.22/5.37 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(1431,a,1424,a)]. 5.22/5.37 1432 -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_393_mult__less__cancel__right__disj) # label(axiom). [clausify(1003)]. 5.22/5.37 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(1432,a,1395,a)]. 5.22/5.37 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(1432,a,1424,a)]. 5.22/5.37 1433 -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_393_mult__less__cancel__right__disj) # label(axiom). [clausify(1003)]. 5.22/5.37 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(1433,a,1395,a)]. 5.22/5.37 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(1433,a,1424,a)]. 5.22/5.37 1434 -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),D),B)) # label(fact_393_mult__less__cancel__right__disj) # label(axiom). [clausify(1003)]. 5.22/5.37 1435 -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_393_mult__less__cancel__right__disj) # label(axiom). [clausify(1003)]. 5.29/5.49 1436 -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_378_mult__strict__left__mono__neg) # label(axiom). [clausify(1005)]. 5.29/5.49 1437 -linord581940658strict(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),B)),hAPP(A,A,times_times(A,C),C))),zero_zero(A))) | ti(A,B) = zero_zero(A) # label(fact_45_sum__squares__le__zero__iff) # label(axiom). [clausify(1027)]. 5.29/5.49 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),A)),hAPP(int,int,times_times(int,B),B))),zero_zero(int))) | ti(int,A) = zero_zero(int). [resolve(1437,a,1395,a)]. 5.29/5.49 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),A)),hAPP(real,real,times_times(real,B),B))),zero_zero(real))) | ti(real,A) = zero_zero(real). [resolve(1437,a,1424,a)]. 5.29/5.49 1438 -linord581940658strict(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),B)),hAPP(A,A,times_times(A,C),C))),zero_zero(A))) | ti(A,C) = zero_zero(A) # label(fact_45_sum__squares__le__zero__iff) # label(axiom). [clausify(1027)]. 5.29/5.49 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),A)),hAPP(int,int,times_times(int,B),B))),zero_zero(int))) | ti(int,B) = zero_zero(int). [resolve(1438,a,1395,a)]. 5.29/5.49 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),A)),hAPP(real,real,times_times(real,B),B))),zero_zero(real))) | ti(real,B) = zero_zero(real). [resolve(1438,a,1424,a)]. 5.29/5.49 1439 -linord581940658strict(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),B)),hAPP(A,A,times_times(A,C),C))),zero_zero(A))) | ti(A,B) != zero_zero(A) | ti(A,C) != zero_zero(A) # label(fact_45_sum__squares__le__zero__iff) # label(axiom). [clausify(1027)]. 5.29/5.49 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),A)),hAPP(int,int,times_times(int,B),B))),zero_zero(int))) | ti(int,A) != zero_zero(int) | ti(int,B) != zero_zero(int). [resolve(1439,a,1395,a)]. 5.29/5.49 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),A)),hAPP(real,real,times_times(real,B),B))),zero_zero(real))) | ti(real,A) != zero_zero(real) | ti(real,B) != zero_zero(real). [resolve(1439,a,1424,a)]. 5.29/5.49 1440 ordered_semiring(real) # label(arity_RealDef_Oreal___Rings_Oordered__semiring) # label(axiom). [assumption]. 5.29/5.49 1441 -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_363_mult__mono) # label(axiom). [clausify(56)]. 5.29/5.49 1442 -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_364_mult__mono_H) # label(axiom). [clausify(479)]. 5.29/5.50 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(1440,a,1441,a)]. 5.29/5.50 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(1440,a,1442,a)]. 5.29/5.50 1443 -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_369_mult__right__mono) # label(axiom). [clausify(734)]. 5.29/5.50 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(1443,a,1440,a)]. 5.29/5.50 1444 ordered_semiring(int) # label(arity_Int_Oint___Rings_Oordered__semiring) # label(axiom). [assumption]. 5.29/5.50 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(1444,a,1441,a)]. 5.29/5.50 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(1444,a,1442,a)]. 5.29/5.50 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(1444,a,1443,a)]. 5.29/5.50 1445 -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_368_mult__left__mono) # label(axiom). [clausify(803)]. 5.29/5.50 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(1445,a,1440,a)]. 5.29/5.50 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(1445,a,1444,a)]. 5.58/5.78 1446 ordered_semiring(nat) # label(arity_Nat_Onat___Rings_Oordered__semiring) # label(axiom). [assumption]. 5.58/5.78 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(1446,a,1441,a)]. 5.58/5.78 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(1446,a,1442,a)]. 5.58/5.78 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,A),C)),hAPP(nat,nat,times_times(nat,B),C))). [resolve(1446,a,1443,a)]. 5.58/5.78 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(1446,a,1445,a)]. 5.58/5.78 1447 -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_322_combine__common__factor) # label(axiom). [clausify(120)]. 5.58/5.78 1448 semiring(int) # label(arity_Int_Oint___Rings_Osemiring) # label(axiom). [assumption]. 5.58/5.78 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(1447,a,1448,a)]. 5.58/5.78 1449 semiring(nat) # label(arity_Nat_Onat___Rings_Osemiring) # label(axiom). [assumption]. 5.58/5.78 1450 -number(A) | -semiring(A) | hAPP(A,A,times_times(A,number_number_of(A,B)),hAPP(A,A,plus_plus(A,C),D)) = 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)) # label(fact_95_right__distrib__number__of) # label(axiom). [clausify(198)]. 5.58/5.78 1451 -semiring(A) | times_times(A,ti(A,B)) = times_times(A,B) # label(tsy_c_Groups_Otimes__class_Otimes_3_arg1) # label(axiom). [clausify(315)]. 5.58/5.78 1452 -semiring(A) | -number(A) | hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,B),C)),number_number_of(A,D)) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),number_number_of(A,D))),hAPP(A,A,times_times(A,C),number_number_of(A,D))) # label(fact_94_left__distrib__number__of) # label(axiom). [clausify(322)]. 5.58/5.78 1453 -semiring(A) | times_times(A,B) = ti(fun(A,A),times_times(A,B)) # label(tsy_c_Groups_Otimes__class_Otimes_3_res) # label(axiom). [clausify(750)]. 5.58/5.78 1454 semiring(real) # label(arity_RealDef_Oreal___Rings_Osemiring) # label(axiom). [assumption]. 5.58/5.78 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(1454,a,1447,a)]. 5.58/5.78 Derived: -number(real) | hAPP(real,real,times_times(real,hAPP(real,real,plus_plus(real,A),B)),number_number_of(real,C)) = hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),number_number_of(real,C))),hAPP(real,real,times_times(real,B),number_number_of(real,C))). [resolve(1454,a,1452,a)]. 5.58/5.78 1455 -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_453_convex__bound__le) # label(axiom). [clausify(679)]. 5.88/6.04 1456 linord1278240602ring_1(int) # label(arity_Int_Oint___Rings_Olinordered__semiring__1) # label(axiom). [assumption]. 5.88/6.04 1457 linord1278240602ring_1(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semiring__1) # label(axiom). [assumption]. 5.88/6.04 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(1455,a,1456,a)]. 5.88/6.04 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(1455,a,1457,a)]. 5.88/6.04 1458 ab_semigroup_mult(real) # label(arity_RealDef_Oreal___Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 5.88/6.04 1459 -ab_semigroup_mult(A) | times_times(A,B) = ti(fun(A,A),times_times(A,B)) # label(tsy_c_Groups_Otimes__class_Otimes_0_res) # label(axiom). [clausify(63)]. 5.88/6.04 1460 ab_semigroup_mult(nat) # label(arity_Nat_Onat___Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 5.88/6.04 1461 -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_758_ab__semigroup__mult__class_Omult__ac_I1_J) # label(axiom). [clausify(400)]. 5.88/6.04 1462 ab_semigroup_mult(int) # label(arity_Int_Oint___Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 5.88/6.04 1463 -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(485)]. 5.88/6.04 1464 ordere223160158up_add(nat) # label(arity_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [assumption]. 5.88/6.04 1465 -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_809_add__less__le__mono) # label(axiom). [clausify(67)]. 5.88/6.04 1466 -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_810_add__le__less__mono) # label(axiom). [clausify(193)]. 5.88/6.04 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(1464,a,1465,a)]. 5.88/6.04 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(1464,a,1466,a)]. 6.04/6.20 1467 -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_783_add__strict__mono) # label(axiom). [clausify(314)]. 6.04/6.20 1468 -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_781_add__strict__right__mono) # label(axiom). [clausify(337)]. 6.04/6.20 1469 -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_782_add__strict__left__mono) # label(axiom). [clausify(398)]. 6.04/6.20 1470 ordere223160158up_add(real) # label(arity_RealDef_Oreal___Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [assumption]. 6.04/6.20 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(real),hAPP(real,real,plus_plus(real,A),C)),hAPP(real,real,plus_plus(real,B),D))). [resolve(1470,a,1465,a)]. 6.04/6.20 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(1470,a,1466,a)]. 6.04/6.20 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(real),hAPP(real,real,plus_plus(real,A),C)),hAPP(real,real,plus_plus(real,B),D))). [resolve(1470,a,1467,a)]. 6.04/6.20 1471 ordere223160158up_add(int) # label(arity_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add) # label(axiom). [assumption]. 6.04/6.20 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(int),hAPP(int,int,plus_plus(int,A),C)),hAPP(int,int,plus_plus(int,B),D))). [resolve(1471,a,1465,a)]. 6.04/6.20 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(1471,a,1466,a)]. 6.04/6.20 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(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(1471,a,1467,a)]. 6.04/6.20 1472 ordere453448008miring(real) # label(arity_RealDef_Oreal___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 6.04/6.20 1473 -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_371_mult__nonpos__nonneg) # label(axiom). [clausify(90)]. 6.04/6.20 1474 -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_374_mult__nonneg__nonneg) # label(axiom). [clausify(191)]. 6.04/6.20 1475 -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_373_mult__nonneg__nonpos) # label(axiom). [clausify(462)]. 6.18/6.33 1476 ordere453448008miring(nat) # label(arity_Nat_Onat___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 6.18/6.33 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),zero_zero(nat)),B)) | 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(1476,a,1473,a)]. 6.18/6.33 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(1476,a,1474,a)]. 6.18/6.33 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(1476,a,1475,a)]. 6.18/6.33 1477 ordere453448008miring(int) # label(arity_Int_Oint___Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 6.18/6.33 1478 -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_361_split__mult__neg__le) # label(axiom). [clausify(787)]. 6.18/6.33 1479 -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_361_split__mult__neg__le) # label(axiom). [clausify(787)]. 6.18/6.33 1480 -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_372_mult__nonneg__nonpos2) # label(axiom). [clausify(798)]. 6.18/6.33 1481 linordered_semidom(nat) # label(arity_Nat_Onat___Rings_Olinordered__semidom) # label(axiom). [assumption]. 6.18/6.33 1482 -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_395_pos__add__strict) # label(axiom). [clausify(92)]. 6.18/6.33 1483 -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_445_power__strict__increasing__iff) # label(axiom). [clausify(134)]. 6.18/6.33 1484 -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_445_power__strict__increasing__iff) # label(axiom). [clausify(134)]. 6.18/6.33 1485 -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_402_less__1__mult) # label(axiom). [clausify(137)]. 6.18/6.33 1486 -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_444_power__less__imp__less__exp) # label(axiom). [clausify(168)]. 6.18/6.34 1487 -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_436_power__less__imp__less__base) # label(axiom). [clausify(216)]. 6.18/6.34 1488 -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_400_power__mono) # label(axiom). [clausify(240)]. 6.18/6.34 1489 -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_eq(nat),C),D)) | 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))) # label(fact_439_power__increasing__iff) # label(axiom). [clausify(241)]. 6.18/6.34 1490 -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_eq(nat),C),D)) | -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))) # label(fact_439_power__increasing__iff) # label(axiom). [clausify(241)]. 6.18/6.34 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(1481,a,1483,a)]. 6.18/6.34 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(1481,a,1484,a)]. 6.18/6.34 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(1481,a,1485,a)]. 6.18/6.34 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(1481,a,1487,a)]. 6.18/6.34 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(1481,a,1488,a)]. 6.18/6.34 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),B),C)) | 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))). [resolve(1481,a,1489,a)]. 6.18/6.34 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),B),C)) | -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))). [resolve(1481,a,1490,a)]. 6.18/6.34 1491 -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_403_zero__less__power) # label(axiom). [clausify(250)]. 6.18/6.34 1492 -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_437_power__decreasing) # label(axiom). [clausify(319)]. 6.18/6.35 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(1492,a,1481,a)]. 6.18/6.35 1493 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),one_one(A)),zero_zero(A))) # label(fact_397_not__one__le__zero) # label(axiom). [clausify(356)]. 6.18/6.35 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),zero_zero(nat))). [resolve(1493,a,1481,a)]. 6.18/6.35 1494 -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_456_power__eq__imp__eq__base) # label(axiom). [clausify(362)]. 6.18/6.35 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(1494,a,1481,a)]. 6.18/6.35 1495 -linordered_semidom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),one_one(A))) # label(fact_396_zero__le__one) # label(axiom). [clausify(384)]. 6.18/6.35 1496 -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_275_power__strict__mono) # label(axiom). [clausify(459)]. 6.18/6.35 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(1496,a,1481,a)]. 6.18/6.35 1497 -linordered_semidom(A) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),zero_zero(A)),one_one(A))) # label(fact_398_zero__less__one) # label(axiom). [clausify(545)]. 6.18/6.35 Derived: hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),one_one(nat))). [resolve(1497,a,1481,a)]. 6.18/6.35 1498 -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_404_less__add__one) # label(axiom). [clausify(546)]. 6.18/6.35 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(1498,a,1481,a)]. 6.18/6.35 1499 -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_457_one__less__power) # label(axiom). [clausify(553)]. 6.18/6.35 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(1499,a,1481,a)]. 6.18/6.37 1500 -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_102_power2__le__imp__le) # label(axiom). [clausify(567)]. 6.18/6.37 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(1500,a,1481,a)]. 6.18/6.37 1501 -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_454_power__Suc__less) # label(axiom). [clausify(574)]. 6.18/6.37 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(1501,a,1481,a)]. 6.18/6.37 1502 -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_440_power__less__power__Suc) # label(axiom). [clausify(621)]. 6.18/6.37 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(1502,a,1481,a)]. 6.18/6.37 1503 linordered_semidom(int) # label(arity_Int_Oint___Rings_Olinordered__semidom) # label(axiom). [assumption]. 6.18/6.37 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),B),hAPP(int,int,plus_plus(int,A),C))). [resolve(1503,a,1482,a)]. 6.18/6.37 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(1503,a,1483,a)]. 6.18/6.37 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(1503,a,1484,a)]. 6.18/6.37 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(1503,a,1485,a)]. 6.18/6.37 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(1503,a,1487,a)]. 6.18/6.37 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(1503,a,1488,a)]. 6.18/6.37 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_eq(nat),B),C)) | 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))). [resolve(1503,a,1489,a)]. 6.18/6.37 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_eq(nat),B),C)) | -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))). [resolve(1503,a,1490,a)]. 6.18/6.37 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),zero_zero(int)),hAPP(nat,int,power_power(int,A),B))). [resolve(1503,a,1491,a)]. 6.18/6.37 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(1503,a,1492,a)]. 6.18/6.37 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),zero_zero(int))). [resolve(1503,a,1493,a)]. 6.18/6.37 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(1503,a,1494,a)]. 6.18/6.37 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),one_one(int))). [resolve(1503,a,1495,a)]. 6.18/6.37 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_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(1503,a,1496,a)]. 6.18/6.37 Derived: hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),one_one(int))). [resolve(1503,a,1497,a)]. 6.18/6.37 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(1503,a,1498,a)]. 6.18/6.37 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(1503,a,1499,a)]. 6.18/6.37 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(1503,a,1500,a)]. 6.18/6.37 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),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(1503,a,1501,a)]. 6.18/6.37 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(1503,a,1502,a)]. 6.18/6.37 1504 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),zero_zero(A))) # label(fact_399_not__one__less__zero) # label(axiom). [clausify(721)]. 6.18/6.39 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),zero_zero(int))). [resolve(1504,a,1503,a)]. 6.18/6.39 1505 -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_455_power__strict__decreasing) # label(axiom). [clausify(772)]. 6.18/6.39 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(1505,a,1481,a)]. 6.18/6.39 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(1505,a,1503,a)]. 6.18/6.39 1506 -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_435_zero__less__two) # label(axiom). [clausify(784)]. 6.18/6.39 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(1506,a,1481,a)]. 6.18/6.39 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(1506,a,1503,a)]. 6.18/6.39 1507 -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_438_power__le__imp__le__exp) # label(axiom). [clausify(841)]. 6.18/6.39 1508 -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_104_power2__less__imp__less) # label(axiom). [clausify(856)]. 6.18/6.39 1509 linordered_semidom(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semidom) # label(axiom). [assumption]. 6.18/6.39 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),B),hAPP(real,real,plus_plus(real,A),C))). [resolve(1509,a,1482,a)]. 6.18/6.39 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(1509,a,1483,a)]. 6.18/6.39 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(1509,a,1484,a)]. 6.18/6.39 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(1509,a,1485,a)]. 6.18/6.39 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(1509,a,1487,a)]. 6.18/6.39 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),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(1509,a,1488,a)]. 6.18/6.39 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_eq(nat),B),C)) | 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))). [resolve(1509,a,1489,a)]. 6.18/6.39 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_eq(nat),B),C)) | -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))). [resolve(1509,a,1490,a)]. 6.18/6.39 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),zero_zero(real)),hAPP(nat,real,power_power(real,A),B))). [resolve(1509,a,1491,a)]. 6.18/6.39 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(1509,a,1492,a)]. 6.18/6.39 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),zero_zero(real))). [resolve(1509,a,1493,a)]. 6.18/6.39 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(1509,a,1494,a)]. 6.18/6.39 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),one_one(real))). [resolve(1509,a,1495,a)]. 6.18/6.39 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_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(1509,a,1496,a)]. 6.18/6.39 Derived: hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),one_one(real))). [resolve(1509,a,1497,a)]. 6.18/6.39 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(1509,a,1498,a)]. 6.18/6.39 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(1509,a,1499,a)]. 6.18/6.39 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(1509,a,1500,a)]. 6.18/6.39 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),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(1509,a,1501,a)]. 6.26/6.41 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(1509,a,1502,a)]. 6.26/6.41 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),zero_zero(real))). [resolve(1509,a,1504,a)]. 6.26/6.41 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(1509,a,1505,a)]. 6.26/6.41 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(1509,a,1506,a)]. 6.26/6.41 1510 -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_443_power__strict__increasing) # label(axiom). [clausify(871)]. 6.26/6.41 1511 -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_407_power__increasing) # label(axiom). [clausify(901)]. 6.26/6.41 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(1511,a,1481,a)]. 6.26/6.41 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(1511,a,1503,a)]. 6.26/6.41 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(1511,a,1509,a)]. 6.26/6.41 1512 -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_401_zero__le__power) # label(axiom). [clausify(905)]. 6.26/6.41 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(1512,a,1481,a)]. 6.26/6.41 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(1512,a,1503,a)]. 6.26/6.41 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(1512,a,1509,a)]. 6.26/6.41 1513 -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_408_one__le__power) # label(axiom). [clausify(929)]. 6.26/6.42 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(1513,a,1481,a)]. 6.26/6.42 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(1513,a,1503,a)]. 6.26/6.42 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(1513,a,1509,a)]. 6.26/6.42 1514 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | hAPP(nat,A,power_power(A,B),C) != hAPP(nat,A,power_power(A,B),D) | C = D # label(fact_409_power__inject__exp) # label(axiom). [clausify(983)]. 6.26/6.42 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),A)) | hAPP(nat,nat,power_power(nat,A),B) != hAPP(nat,nat,power_power(nat,A),C) | B = C. [resolve(1514,a,1481,a)]. 6.26/6.42 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A)) | hAPP(nat,int,power_power(int,A),B) != hAPP(nat,int,power_power(int,A),C) | B = C. [resolve(1514,a,1503,a)]. 6.26/6.42 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),A)) | hAPP(nat,real,power_power(real,A),B) != hAPP(nat,real,power_power(real,A),C) | B = C. [resolve(1514,a,1509,a)]. 6.26/6.42 1515 -linordered_semidom(A) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less(A),one_one(A)),B)) | hAPP(nat,A,power_power(A,B),C) = hAPP(nat,A,power_power(A,B),D) | C != D # label(fact_409_power__inject__exp) # label(axiom). [clausify(983)]. 6.26/6.42 Derived: -hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),A)) | hAPP(nat,nat,power_power(nat,A),B) = hAPP(nat,nat,power_power(nat,A),C) | B != C. [resolve(1515,a,1481,a)]. 6.26/6.42 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A)) | hAPP(nat,int,power_power(int,A),B) = hAPP(nat,int,power_power(int,A),C) | B != C. [resolve(1515,a,1503,a)]. 6.26/6.42 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),A)) | hAPP(nat,real,power_power(real,A),B) = hAPP(nat,real,power_power(real,A),C) | B != C. [resolve(1515,a,1509,a)]. 6.26/6.42 1516 -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_101_power2__eq__imp__eq) # label(axiom). [clausify(995)]. 6.26/6.42 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(1516,a,1481,a)]. 6.26/6.42 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(1516,a,1503,a)]. 6.26/6.42 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(1516,a,1509,a)]. 6.49/6.68 1517 -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_441_power__gt1__lemma) # label(axiom). [clausify(1018)]. 6.49/6.68 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(1517,a,1481,a)]. 6.49/6.68 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(1517,a,1503,a)]. 6.49/6.68 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(1517,a,1509,a)]. 6.49/6.68 1518 ab_group_add(int) # label(arity_Int_Oint___Groups_Oab__group__add) # label(axiom). [assumption]. 6.49/6.68 1519 -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_613_add__diff__add) # label(axiom). [clausify(93)]. 6.49/6.68 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(1518,a,1519,a)]. 6.49/6.68 1520 ab_group_add(real) # label(arity_RealDef_Oreal___Groups_Oab__group__add) # label(axiom). [assumption]. 6.49/6.68 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(1520,a,1519,a)]. 6.49/6.68 1521 -ab_group_add(A) | zero_zero(A) != hAPP(A,A,minus_minus(A,B),C) | ti(A,C) = ti(A,B) # label(fact_787_eq__iff__diff__eq__0) # label(axiom). [clausify(685)]. 6.49/6.68 1522 -ab_group_add(A) | zero_zero(A) = hAPP(A,A,minus_minus(A,B),C) | ti(A,C) != ti(A,B) # label(fact_787_eq__iff__diff__eq__0) # label(axiom). [clausify(685)]. 6.49/6.68 1523 -ab_group_add(A) | hAPP(A,A,minus_minus(A,B),C) != hAPP(A,A,minus_minus(A,D),E) | ti(A,D) != ti(A,E) | ti(A,C) = ti(A,B) # label(fact_766_diff__eq__diff__eq) # label(axiom). [clausify(774)]. 6.49/6.68 Derived: hAPP(int,int,minus_minus(int,A),B) != hAPP(int,int,minus_minus(int,C),D) | ti(int,C) != ti(int,D) | ti(int,B) = ti(int,A). [resolve(1523,a,1518,a)]. 6.49/6.68 Derived: hAPP(real,real,minus_minus(real,A),B) != hAPP(real,real,minus_minus(real,C),D) | ti(real,C) != ti(real,D) | ti(real,B) = ti(real,A). [resolve(1523,a,1520,a)]. 6.49/6.68 1524 -ab_group_add(A) | hAPP(A,A,minus_minus(A,B),C) != hAPP(A,A,minus_minus(A,D),E) | ti(A,D) = ti(A,E) | ti(A,C) != ti(A,B) # label(fact_766_diff__eq__diff__eq) # label(axiom). [clausify(774)]. 6.49/6.68 Derived: hAPP(int,int,minus_minus(int,A),B) != hAPP(int,int,minus_minus(int,C),D) | ti(int,C) = ti(int,D) | ti(int,B) != ti(int,A). [resolve(1524,a,1518,a)]. 6.49/6.68 Derived: hAPP(real,real,minus_minus(real,A),B) != hAPP(real,real,minus_minus(real,C),D) | ti(real,C) = ti(real,D) | ti(real,B) != ti(real,A). [resolve(1524,a,1520,a)]. 6.49/6.68 1525 -semiri456707255roduct(A) | hAPP(A,A,plus_plus(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),E)),hAPP(A,A,times_times(A,D),C)) | ti(A,C) = ti(A,E) | ti(A,B) = ti(A,D) # label(fact_243_crossproduct__eq) # label(axiom). [clausify(551)]. 6.49/6.68 1526 semiri456707255roduct(nat) # label(arity_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc) # label(axiom). [assumption]. 6.49/6.68 1527 semiri456707255roduct(real) # label(arity_RealDef_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossp) # label(axiom). [assumption]. 6.49/6.68 Derived: hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,A),B)),hAPP(nat,nat,times_times(nat,C),D)) != hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,A),D)),hAPP(nat,nat,times_times(nat,C),B)) | ti(nat,B) = ti(nat,D) | ti(nat,A) = ti(nat,C). [resolve(1525,a,1526,a)]. 6.49/6.69 Derived: hAPP(real,real,plus_plus(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),D)),hAPP(real,real,times_times(real,C),B)) | ti(real,B) = ti(real,D) | ti(real,A) = ti(real,C). [resolve(1525,a,1527,a)]. 6.49/6.69 1528 -semiri456707255roduct(A) | hAPP(A,A,plus_plus(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),E)),hAPP(A,A,times_times(A,D),C)) | ti(A,C) != ti(A,E) # label(fact_243_crossproduct__eq) # label(axiom). [clausify(551)]. 6.49/6.69 Derived: hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,A),B)),hAPP(nat,nat,times_times(nat,C),D)) = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,A),D)),hAPP(nat,nat,times_times(nat,C),B)) | ti(nat,B) != ti(nat,D). [resolve(1528,a,1526,a)]. 6.49/6.69 Derived: hAPP(real,real,plus_plus(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),D)),hAPP(real,real,times_times(real,C),B)) | ti(real,B) != ti(real,D). [resolve(1528,a,1527,a)]. 6.49/6.69 1529 -semiri456707255roduct(A) | hAPP(A,A,plus_plus(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),E)),hAPP(A,A,times_times(A,D),C)) | ti(A,B) != ti(A,D) # label(fact_243_crossproduct__eq) # label(axiom). [clausify(551)]. 6.49/6.69 Derived: hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,A),B)),hAPP(nat,nat,times_times(nat,C),D)) = hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,A),D)),hAPP(nat,nat,times_times(nat,C),B)) | ti(nat,A) != ti(nat,C). [resolve(1529,a,1526,a)]. 6.49/6.69 Derived: hAPP(real,real,plus_plus(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),D)),hAPP(real,real,times_times(real,C),B)) | ti(real,A) != ti(real,C). [resolve(1529,a,1527,a)]. 6.49/6.69 1530 -semiri456707255roduct(A) | ti(A,B) = zero_zero(A) | 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_251_add__scale__eq__noteq) # label(axiom). [clausify(623)]. 6.49/6.69 Derived: ti(nat,A) = zero_zero(nat) | 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(1530,a,1526,a)]. 6.49/6.69 Derived: ti(real,A) = zero_zero(real) | 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(1530,a,1527,a)]. 6.49/6.69 1531 semiri456707255roduct(int) # label(arity_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc) # label(axiom). [assumption]. 6.49/6.69 Derived: hAPP(int,int,plus_plus(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),D)),hAPP(int,int,times_times(int,C),B)) | ti(int,B) = ti(int,D) | ti(int,A) = ti(int,C). [resolve(1531,a,1525,a)]. 6.49/6.69 Derived: hAPP(int,int,plus_plus(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),D)),hAPP(int,int,times_times(int,C),B)) | ti(int,B) != ti(int,D). [resolve(1531,a,1528,a)]. 6.49/6.69 Derived: hAPP(int,int,plus_plus(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),D)),hAPP(int,int,times_times(int,C),B)) | ti(int,A) != ti(int,C). [resolve(1531,a,1529,a)]. 6.49/6.69 Derived: ti(int,A) = zero_zero(int) | 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(1531,a,1530,a)]. 6.65/6.83 1532 -semiri456707255roduct(A) | ti(A,B) != zero_zero(A) | ti(A,C) = hAPP(A,A,plus_plus(A,C),B) # label(fact_238_add__0__iff) # label(axiom). [clausify(932)]. 6.65/6.83 Derived: ti(nat,A) != zero_zero(nat) | ti(nat,B) = hAPP(nat,nat,plus_plus(nat,B),A). [resolve(1532,a,1526,a)]. 6.65/6.83 Derived: ti(real,A) != zero_zero(real) | ti(real,B) = hAPP(real,real,plus_plus(real,B),A). [resolve(1532,a,1527,a)]. 6.65/6.83 Derived: ti(int,A) != zero_zero(int) | ti(int,B) = hAPP(int,int,plus_plus(int,B),A). [resolve(1532,a,1531,a)]. 6.65/6.83 1533 -semiri456707255roduct(A) | ti(A,B) = zero_zero(A) | ti(A,C) != hAPP(A,A,plus_plus(A,C),B) # label(fact_238_add__0__iff) # label(axiom). [clausify(932)]. 6.65/6.83 Derived: ti(nat,A) = zero_zero(nat) | ti(nat,B) != hAPP(nat,nat,plus_plus(nat,B),A). [resolve(1533,a,1526,a)]. 6.65/6.83 Derived: ti(real,A) = zero_zero(real) | ti(real,B) != hAPP(real,real,plus_plus(real,B),A). [resolve(1533,a,1527,a)]. 6.65/6.83 Derived: ti(int,A) = zero_zero(int) | ti(int,B) != hAPP(int,int,plus_plus(int,B),A). [resolve(1533,a,1531,a)]. 6.65/6.83 1534 -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,E),B)),hAPP(A,A,times_times(A,D),C)) != hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,E),C)),hAPP(A,A,times_times(A,D),B)) # label(fact_240_crossproduct__noteq) # label(axiom). [clausify(959)]. 6.65/6.83 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,D),A)),hAPP(nat,nat,times_times(nat,C),B)) != hAPP(nat,nat,plus_plus(nat,hAPP(nat,nat,times_times(nat,D),B)),hAPP(nat,nat,times_times(nat,C),A)). [resolve(1534,a,1526,a)]. 6.65/6.83 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,D),A)),hAPP(real,real,times_times(real,C),B)) != hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,D),B)),hAPP(real,real,times_times(real,C),A)). [resolve(1534,a,1527,a)]. 6.65/6.83 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,D),A)),hAPP(int,int,times_times(int,C),B)) != hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,D),B)),hAPP(int,int,times_times(int,C),A)). [resolve(1534,a,1531,a)]. 6.65/6.83 1535 -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_240_crossproduct__noteq) # label(axiom). [clausify(959)]. 6.65/6.83 1536 -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_240_crossproduct__noteq) # label(axiom). [clausify(959)]. 6.65/6.83 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(1536,a,1526,a)]. 6.65/6.83 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(1536,a,1527,a)]. 6.65/6.83 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(1536,a,1531,a)]. 6.65/6.83 1537 ring(real) # label(arity_RealDef_Oreal___Rings_Oring) # label(axiom). [assumption]. 6.65/6.83 1538 -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_481_left__diff__distrib__number__of) # label(axiom). [clausify(129)]. 6.65/6.86 1539 -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_479_eq__add__iff1) # label(axiom). [clausify(454)]. 6.65/6.86 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(1539,a,1537,a)]. 6.65/6.86 1540 -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_479_eq__add__iff1) # label(axiom). [clausify(454)]. 6.65/6.86 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(1540,a,1537,a)]. 6.65/6.86 1541 -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_480_right__diff__distrib__number__of) # label(axiom). [clausify(468)]. 6.65/6.86 1542 ring(int) # label(arity_Int_Oint___Rings_Oring) # label(axiom). [assumption]. 6.65/6.86 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(1542,a,1539,a)]. 6.65/6.86 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(1542,a,1540,a)]. 6.65/6.86 Derived: -number(int) | hAPP(int,int,minus_minus(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,minus_minus(int,B),C)). [resolve(1542,a,1541,a)]. 6.65/6.86 1543 -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_693_mult__diff__mult) # label(axiom). [clausify(707)]. 6.65/6.86 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(1543,a,1537,a)]. 6.65/6.86 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(1543,a,1542,a)]. 6.65/6.86 1544 -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_478_eq__add__iff2) # label(axiom). [clausify(850)]. 6.65/6.86 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(1544,a,1537,a)]. 6.65/6.86 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(1544,a,1542,a)]. 6.78/6.99 1545 -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_478_eq__add__iff2) # label(axiom). [clausify(850)]. 6.78/6.99 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(1545,a,1537,a)]. 6.78/6.99 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(1545,a,1542,a)]. 6.78/6.99 1546 -linord219039673up_add(A) | zero_zero(A) != hAPP(A,A,plus_plus(A,B),B) | ti(A,B) = zero_zero(A) # label(fact_114_double__eq__0__iff) # label(axiom). [clausify(223)]. 6.78/6.99 1547 linord219039673up_add(int) # label(arity_Int_Oint___Groups_Olinordered__ab__group__add) # label(axiom). [assumption]. 6.78/6.99 Derived: zero_zero(int) != hAPP(int,int,plus_plus(int,A),A) | ti(int,A) = zero_zero(int). [resolve(1546,a,1547,a)]. 6.78/6.99 1548 -linord219039673up_add(A) | zero_zero(A) = hAPP(A,A,plus_plus(A,B),B) | ti(A,B) != zero_zero(A) # label(fact_114_double__eq__0__iff) # label(axiom). [clausify(223)]. 6.78/6.99 Derived: zero_zero(int) = hAPP(int,int,plus_plus(int,A),A) | ti(int,A) != zero_zero(int). [resolve(1548,a,1547,a)]. 6.78/6.99 1549 -linord219039673up_add(A) | zero_zero(A) != hAPP(A,A,plus_plus(A,B),B) | ti(A,B) = zero_zero(A) # label(fact_769_double__zero__sym) # label(axiom). [clausify(318)]. 6.78/6.99 1550 -linord219039673up_add(A) | zero_zero(A) = hAPP(A,A,plus_plus(A,B),B) | ti(A,B) != zero_zero(A) # label(fact_769_double__zero__sym) # label(axiom). [clausify(318)]. 6.78/6.99 1551 linord219039673up_add(real) # label(arity_RealDef_Oreal___Groups_Olinordered__ab__group__add) # label(axiom). [assumption]. 6.78/6.99 Derived: zero_zero(real) != hAPP(real,real,plus_plus(real,A),A) | ti(real,A) = zero_zero(real). [resolve(1551,a,1546,a)]. 6.78/6.99 Derived: zero_zero(real) = hAPP(real,real,plus_plus(real,A),A) | ti(real,A) != zero_zero(real). [resolve(1551,a,1548,a)]. 6.78/6.99 1552 -linord219039673up_add(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))) | hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) # label(fact_799_double__add__le__zero__iff__single__add__le__zero) # label(axiom). [clausify(560)]. 6.78/6.99 Derived: -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,plus_plus(int,A),A)),zero_zero(int))) | hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A),zero_zero(int))). [resolve(1552,a,1547,a)]. 6.78/6.99 Derived: -hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,plus_plus(real,A),A)),zero_zero(real))) | hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A),zero_zero(real))). [resolve(1552,a,1551,a)]. 6.78/6.99 1553 -linord219039673up_add(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))) | -hBOOL(hAPP(A,bool,hAPP(A,fun(A,bool),ord_less_eq(A),B),zero_zero(A))) # label(fact_799_double__add__le__zero__iff__single__add__le__zero) # label(axiom). [clausify(560)]. 6.78/6.99 1554 -linord219039673up_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)),hAPP(A,A,plus_plus(A,B),B))) # label(fact_805_zero__less__double__add__iff__zero__less__single__add) # label(axiom). [clausify(706)]. 6.78/6.99 1555 -linord219039673up_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)),hAPP(A,A,plus_plus(A,B),B))) # label(fact_805_zero__less__double__add__iff__zero__less__single__add) # label(axiom). [clausify(706)]. 7.09/7.28 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(int,int,plus_plus(int,A),A))). [resolve(1555,a,1547,a)]. 7.09/7.28 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(real,real,plus_plus(real,A),A))). [resolve(1555,a,1551,a)]. 7.09/7.28 1556 -linord219039673up_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),hAPP(A,A,plus_plus(A,B),B)),zero_zero(A))) # label(fact_806_double__add__less__zero__iff__single__add__less__zero) # label(axiom). [clausify(753)]. 7.09/7.28 1557 -linord219039673up_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),hAPP(A,A,plus_plus(A,B),B)),zero_zero(A))) # label(fact_806_double__add__less__zero__iff__single__add__less__zero) # label(axiom). [clausify(753)]. 7.09/7.28 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),hAPP(int,int,plus_plus(int,A),A)),zero_zero(int))). [resolve(1557,a,1547,a)]. 7.09/7.28 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),hAPP(real,real,plus_plus(real,A),A)),zero_zero(real))). [resolve(1557,a,1551,a)]. 7.09/7.28 1558 -linord219039673up_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)),hAPP(A,A,plus_plus(A,B),B))) # label(fact_798_zero__le__double__add__iff__zero__le__single__add) # label(axiom). [clausify(1026)]. 7.09/7.28 1559 -linord219039673up_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)),hAPP(A,A,plus_plus(A,B),B))) # label(fact_798_zero__le__double__add__iff__zero__le__single__add) # label(axiom). [clausify(1026)]. 7.09/7.28 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(int,int,plus_plus(int,A),A))). [resolve(1559,a,1547,a)]. 7.09/7.28 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(real,real,plus_plus(real,A),A))). [resolve(1559,a,1551,a)]. 7.09/7.28 1560 -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_326_power__mult__distrib) # label(axiom). [clausify(329)]. 7.09/7.28 1561 comm_monoid_mult(int) # label(arity_Int_Oint___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 7.09/7.28 1562 comm_monoid_mult(real) # label(arity_RealDef_Oreal___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 7.09/7.28 1563 comm_monoid_mult(nat) # label(arity_Nat_Onat___Groups_Ocomm__monoid__mult) # label(axiom). [assumption]. 7.09/7.28 1564 -comm_monoid_mult(A) | ti(A,B) = hAPP(A,A,times_times(A,B),one_one(A)) # label(fact_795_mult_Ocomm__neutral) # label(axiom). [clausify(953)]. 7.09/7.28 Derived: ti(int,A) = hAPP(int,int,times_times(int,A),one_one(int)). [resolve(1564,a,1561,a)]. 7.09/7.28 1565 -comm_monoid_mult(A) | ti(A,B) = hAPP(A,A,times_times(A,one_one(A)),B) # label(fact_793_mult__1) # label(axiom). [clausify(1013)]. 7.09/7.28 1566 -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_470_dvd__diff) # label(axiom). [clausify(688)]. 7.09/7.28 1567 comm_ring_1(int) # label(arity_Int_Oint___Rings_Ocomm__ring__1) # label(axiom). [assumption]. 7.09/7.28 1568 comm_ring_1(real) # label(arity_RealDef_Oreal___Rings_Ocomm__ring__1) # label(axiom). [assumption]. 7.56/7.74 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(1566,a,1567,a)]. 7.56/7.74 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(1566,a,1568,a)]. 7.56/7.74 1569 -ring_n68954251visors(A) | ti(A,B) != zero_zero(A) | zero_zero(A) = hAPP(A,A,times_times(A,B),C) # label(fact_317_mult__eq__0__iff) # label(axiom). [clausify(484)]. 7.56/7.74 1570 ring_n68954251visors(int) # label(arity_Int_Oint___Rings_Oring__no__zero__divisors) # label(axiom). [assumption]. 7.56/7.74 Derived: ti(int,A) != zero_zero(int) | zero_zero(int) = hAPP(int,int,times_times(int,A),B). [resolve(1569,a,1570,a)]. 7.56/7.74 1571 -ring_n68954251visors(A) | ti(A,B) != zero_zero(A) | zero_zero(A) = hAPP(A,A,times_times(A,C),B) # label(fact_317_mult__eq__0__iff) # label(axiom). [clausify(484)]. 7.56/7.74 Derived: ti(int,A) != zero_zero(int) | zero_zero(int) = hAPP(int,int,times_times(int,B),A). [resolve(1571,a,1570,a)]. 7.56/7.74 1572 -ring_n68954251visors(A) | ti(A,B) = zero_zero(A) | ti(A,C) = zero_zero(A) | zero_zero(A) != hAPP(A,A,times_times(A,B),C) # label(fact_317_mult__eq__0__iff) # label(axiom). [clausify(484)]. 7.56/7.74 1573 ring_n68954251visors(real) # label(arity_RealDef_Oreal___Rings_Oring__no__zero__divisors) # label(axiom). [assumption]. 7.56/7.74 Derived: ti(real,A) != zero_zero(real) | zero_zero(real) = hAPP(real,real,times_times(real,A),B). [resolve(1573,a,1569,a)]. 7.56/7.74 Derived: ti(real,A) != zero_zero(real) | zero_zero(real) = hAPP(real,real,times_times(real,B),A). [resolve(1573,a,1571,a)]. 7.56/7.74 1574 -one(A) | ti(A,one_one(A)) = one_one(A) # label(tsy_c_Groups_Oone__class_Oone_res) # label(axiom). [clausify(641)]. 7.56/7.74 1575 one(real) # label(arity_RealDef_Oreal___Groups_Oone) # label(axiom). [assumption]. 7.56/7.74 1576 one(int) # label(arity_Int_Oint___Groups_Oone) # label(axiom). [assumption]. 7.56/7.74 Derived: ti(real,one_one(real)) = one_one(real). [resolve(1574,a,1575,a)]. 7.56/7.74 Derived: ti(int,one_one(int)) = one_one(int). [resolve(1574,a,1576,a)]. 7.56/7.74 1577 one(nat) # label(arity_Nat_Onat___Groups_Oone) # label(axiom). [assumption]. 7.56/7.74 Derived: ti(nat,one_one(nat)) = one_one(nat). [resolve(1577,a,1574,a)]. 7.56/7.74 1578 -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_426_mult__right__less__imp__less) # label(axiom). [clausify(997)]. 7.56/7.74 1579 linordered_semiring(int) # label(arity_Int_Oint___Rings_Olinordered__semiring) # label(axiom). [assumption]. 7.56/7.74 1580 linordered_semiring(nat) # label(arity_Nat_Onat___Rings_Olinordered__semiring) # label(axiom). [assumption]. 7.56/7.74 1581 linordered_semiring(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semiring) # label(axiom). [assumption]. 7.56/7.74 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,C),B))) | -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(nat),A),C)). [resolve(1578,a,1580,a)]. 7.56/7.74 1582 -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_424_mult__left__less__imp__less) # label(axiom). [clausify(1033)]. 7.56/7.74 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(1582,a,1580,a)]. 8.15/8.29 1583 -zero(A) | ti(A,zero_zero(A)) = zero_zero(A) # label(tsy_c_Groups_Ozero__class_Ozero_res) # label(axiom). [clausify(674)]. 8.15/8.29 1584 zero(real) # label(arity_RealDef_Oreal___Groups_Ozero) # label(axiom). [assumption]. 8.15/8.29 Derived: ti(real,zero_zero(real)) = zero_zero(real). [resolve(1583,a,1584,a)]. 8.15/8.29 1585 zero(nat) # label(arity_Nat_Onat___Groups_Ozero) # label(axiom). [assumption]. 8.15/8.29 Derived: ti(nat,zero_zero(nat)) = zero_zero(nat). [resolve(1585,a,1583,a)]. 8.15/8.29 1586 zero(int) # label(arity_Int_Oint___Groups_Ozero) # label(axiom). [assumption]. 8.15/8.29 Derived: ti(int,zero_zero(int)) = zero_zero(int). [resolve(1586,a,1583,a)]. 8.15/8.29 1587 -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_367_comm__mult__left__mono) # label(axiom). [clausify(970)]. 8.15/8.29 1588 ordere1490568538miring(int) # label(arity_Int_Oint___Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 8.15/8.29 1589 ordere1490568538miring(real) # label(arity_RealDef_Oreal___Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 8.15/8.29 1590 ordere1490568538miring(nat) # label(arity_Nat_Onat___Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 8.15/8.29 1591 ring_1(int) # label(arity_Int_Oint___Rings_Oring__1) # label(axiom). [assumption]. 8.15/8.29 1592 -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_710_real__squared__diff__one__factored) # label(axiom). [clausify(367)]. 8.15/8.29 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(1591,a,1592,a)]. 8.15/8.29 1593 ring_1(real) # label(arity_RealDef_Oreal___Rings_Oring__1) # label(axiom). [assumption]. 8.15/8.29 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(1593,a,1592,a)]. 8.15/8.29 1594 -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_776_add__mono) # label(axiom). [clausify(631)]. 8.15/8.29 1595 ordere779506340up_add(int) # label(arity_Int_Oint___Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 8.15/8.29 1596 ordere779506340up_add(nat) # label(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 8.15/8.29 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(1594,a,1595,a)]. 8.15/8.29 1597 -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_774_add__right__mono) # label(axiom). [clausify(661)]. 8.15/8.29 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(1597,a,1596,a)]. 8.15/8.29 1598 ordere779506340up_add(real) # label(arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add) # label(axiom). [assumption]. 8.15/8.29 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(1598,a,1594,a)]. 8.56/8.71 1599 -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_775_add__left__mono) # label(axiom). [clausify(698)]. 8.56/8.71 1600 linordered_ring(int) # label(arity_Int_Oint___Rings_Olinordered__ring) # label(axiom). [assumption]. 8.56/8.71 1601 -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_46_sum__squares__ge__zero) # label(axiom). [clausify(391)]. 8.56/8.71 1602 -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_108_not__sum__squares__lt__zero) # label(axiom). [clausify(420)]. 8.56/8.71 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(1600,a,1601,a)]. 8.56/8.71 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(1600,a,1602,a)]. 8.56/8.71 1603 -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_377_zero__le__square) # label(axiom). [clausify(526)]. 8.56/8.71 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(1603,a,1600,a)]. 8.56/8.71 1604 -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_394_not__square__less__zero) # label(axiom). [clausify(870)]. 8.56/8.71 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(1604,a,1600,a)]. 8.56/8.71 1605 linordered_ring(real) # label(arity_RealDef_Oreal___Rings_Olinordered__ring) # label(axiom). [assumption]. 8.56/8.71 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(1605,a,1601,a)]. 8.56/8.71 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(1605,a,1602,a)]. 8.56/8.71 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(1605,a,1603,a)]. 8.56/8.71 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(1605,a,1604,a)]. 8.56/8.71 1606 -comm_semiring(A) | hAPP(A,A,times_times(A,hAPP(A,A,plus_plus(A,B),C)),D) = hAPP(A,A,plus_plus(A,hAPP(A,A,times_times(A,B),D)),hAPP(A,A,times_times(A,C),D)) # label(fact_323_comm__semiring__class_Odistrib) # label(axiom). [clausify(875)]. 8.56/8.71 1607 comm_semiring(int) # label(arity_Int_Oint___Rings_Ocomm__semiring) # label(axiom). [assumption]. 8.56/8.71 1608 comm_semiring(real) # label(arity_RealDef_Oreal___Rings_Ocomm__semiring) # label(axiom). [assumption]. 8.56/8.71 Derived: hAPP(real,real,times_times(real,hAPP(real,real,plus_plus(real,A),B)),C) = hAPP(real,real,plus_plus(real,hAPP(real,real,times_times(real,A),C)),hAPP(real,real,times_times(real,B),C)). [resolve(1608,a,1606,a)]. 8.56/8.71 1609 comm_semiring(nat) # label(arity_Nat_Onat___Rings_Ocomm__semiring) # label(axiom). [assumption]. 8.56/8.71 1610 linord626643107strict(real) # label(arity_RealDef_Oreal___Rings_Olinordered__semiring__1__strict) # label(axiom). [assumption]. 8.56/8.71 1611 -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_276_convex__bound__lt) # label(axiom). [clausify(506)]. 9.21/9.33 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(1610,a,1611,a)]. 9.21/9.33 1612 linord626643107strict(int) # label(arity_Int_Oint___Rings_Olinordered__semiring__1__strict) # label(axiom). [assumption]. 9.21/9.33 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(1612,a,1611,a)]. 9.21/9.33 1613 ring_11004092258visors(int) # label(arity_Int_Oint___Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 9.21/9.33 1614 -ring_11004092258visors(A) | zero_zero(A) != hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))) | ti(A,B) = zero_zero(A) # label(fact_14_zero__eq__power2) # label(axiom). [clausify(589)]. 9.21/9.33 1615 -ring_11004092258visors(A) | zero_zero(A) = hAPP(nat,A,power_power(A,B),number_number_of(nat,bit0(bit1(pls)))) | ti(A,B) != zero_zero(A) # label(fact_14_zero__eq__power2) # label(axiom). [clausify(589)]. 9.21/9.33 1616 -ring_11004092258visors(A) | ti(A,B) = zero_zero(A) | zero_zero(A) != hAPP(nat,A,power_power(A,B),C) # label(fact_324_field__power__not__zero) # label(axiom). [clausify(801)]. 9.21/9.33 Derived: zero_zero(int) != hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls)))) | ti(int,A) = zero_zero(int). [resolve(1613,a,1614,a)]. 9.21/9.33 Derived: zero_zero(int) = hAPP(nat,int,power_power(int,A),number_number_of(nat,bit0(bit1(pls)))) | ti(int,A) != zero_zero(int). [resolve(1613,a,1615,a)]. 9.21/9.33 Derived: ti(int,A) = zero_zero(int) | zero_zero(int) != hAPP(nat,int,power_power(int,A),B). [resolve(1613,a,1616,a)]. 9.21/9.33 1617 ring_11004092258visors(real) # label(arity_RealDef_Oreal___Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 9.21/9.33 Derived: zero_zero(real) != hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))) | ti(real,A) = zero_zero(real). [resolve(1617,a,1614,a)]. 9.21/9.33 Derived: zero_zero(real) = hAPP(nat,real,power_power(real,A),number_number_of(nat,bit0(bit1(pls)))) | ti(real,A) != zero_zero(real). [resolve(1617,a,1615,a)]. 9.21/9.33 Derived: ti(real,A) = zero_zero(real) | zero_zero(real) != hAPP(nat,real,power_power(real,A),B). [resolve(1617,a,1616,a)]. 9.21/9.33 9.21/9.33 ============================== end predicate elimination ============= 9.21/9.33 9.21/9.33 Auto_denials: (non-Horn, no changes). 9.21/9.33 9.21/9.33 Term ordering decisions: 9.21/9.33 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. 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. wset=1. standardRes=1. inv=1. legendre=1. collect=1. product_prod=1. f2=1. f4=1. f9=1. f10=1. f13=1. f17=1. f18=1. f22=1. f23=1. f26=1. f27=1. f30=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. f14=1. f25=1. f28=1. f29=1. f5=1. f6=1. f7=1. f8=1. f11=1. f15=1. f16=1. f19=1. f20=1. f21=1. f31=1. f32=1. f33=1. f34=1. f35=1. f36=1. f37=1. f38=1. f39=1. hAPP=1. product_Pair=1. combb=1. combc=1. combs=1. 9.21/9.33 9.21/9.33 ============================== PROOF ================================= 9.21/9.33 % SZS status Theorem 9.21/9.33 % SZS output start Refutation 9.21/9.33 9.21/9.33 % Proof 1 at 7.64 (+ 0.01) seconds. 9.21/9.33 % Length of proof is 50. 9.21/9.33 % Level of proof is 6. 9.21/9.33 % Maximum clause weight is 100.000. 9.21/9.33 % Given clauses 0. 9.21/9.33 9.21/9.33 45 (all Z_1 hAPP(int,int,plus_plus(int,Z_1),zero_zero(int)) = ti(int,Z_1)) # label(fact_133_zadd__0__right) # label(axiom) # label(non_clause). [assumption]. 9.21/9.33 115 (all K_1 hAPP(int,int,plus_plus(int,K_1),K_1) = bit0(K_1)) # label(fact_132_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 9.21/9.33 194 (all T all A ti(T,A) = ti(T,ti(T,A))) # label(help_ti_idem) # label(axiom) # label(non_clause). [assumption]. 9.21/9.33 215 (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_235_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom) # label(non_clause). [assumption]. 9.21/9.33 254 (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,B_1_1,ti(X_a,B_2_1))) # label(tsy_c_hAPP_arg2) # label(axiom) # label(non_clause). [assumption]. 9.21/9.33 353 (all X_a (number_ring(X_a) -> one_one(X_a) = number_number_of(X_a,bit1(pls)))) # label(fact_173_semiring__norm_I110_J) # label(axiom) # label(non_clause). [assumption]. 9.21/9.33 469 (all X_a (number_ring(X_a) -> zero_zero(X_a) = number_number_of(X_a,pls))) # label(fact_146_semiring__norm_I112_J) # label(axiom) # label(non_clause). [assumption]. 9.21/9.33 549 (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_96_zadd__assoc) # label(axiom) # label(non_clause). [assumption]. 9.21/9.33 664 (all X_c all X_a all B_1_1 all B_2_1 hAPP(X_a,X_c,B_1_1,B_2_1) = ti(X_c,hAPP(X_a,X_c,B_1_1,B_2_1))) # label(tsy_c_hAPP_res) # label(axiom) # label(non_clause). [assumption]. 9.21/9.33 759 (all K_1 hAPP(int,int,plus_plus(int,K_1),pls) = ti(int,K_1)) # label(fact_129_add__Pls__right) # label(axiom) # label(non_clause). [assumption]. 9.21/9.33 791 (all K_1 hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,one_one(int)),K_1)),K_1) = bit1(K_1)) # label(fact_164_Bit1__def) # label(axiom) # label(non_clause). [assumption]. 9.21/9.33 980 (all K_1 ti(int,K_1) = number_number_of(int,K_1)) # label(fact_22_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 9.21/9.33 1068 number_ring(int) # label(arity_Int_Oint___Int_Onumber__ring) # label(axiom). [assumption]. 9.21/9.33 1089 -number_ring(A) | number_number_of(A,bit1(pls)) = one_one(A) # label(fact_173_semiring__norm_I110_J) # label(axiom). [clausify(353)]. 9.21/9.33 1097 -number_ring(A) | number_number_of(A,pls) = zero_zero(A) # label(fact_146_semiring__norm_I112_J) # label(axiom). [clausify(469)]. 9.21/9.33 1142 comm_semiring_1(int) # label(arity_Int_Oint___Rings_Ocomm__semiring__1) # label(axiom). [assumption]. 9.21/9.33 1156 -comm_semiring_1(A) | zero_zero(A) = hAPP(A,A,times_times(A,B),zero_zero(A)) # label(fact_235_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J) # label(axiom). [clausify(215)]. 9.21/9.33 1662 ti(int,A) = hAPP(int,int,plus_plus(int,A),zero_zero(int)) # label(fact_133_zadd__0__right) # label(axiom). [clausify(45)]. 9.21/9.33 1663 hAPP(int,int,plus_plus(int,A),zero_zero(int)) = ti(int,A). [copy(1662),flip(a)]. 9.21/9.33 1756 bit0(A) = hAPP(int,int,plus_plus(int,A),A) # label(fact_132_Bit0__def) # label(axiom). [clausify(115)]. 9.21/9.33 1854 ti(A,ti(A,B)) = ti(A,B) # label(help_ti_idem) # label(axiom). [clausify(194)]. 9.21/9.33 1919 hAPP(A,B,C,ti(A,D)) = hAPP(A,B,C,D) # label(tsy_c_hAPP_arg2) # label(axiom). [clausify(254)]. 9.21/9.33 2009 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) = hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,s),number_number_of(nat,bit0(bit1(pls))))),one_one(int)) # label(fact_3_t) # label(axiom). [assumption]. 9.21/9.33 2010 hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,bit1(pls)),bit1(pls))),hAPP(int,int,plus_plus(int,bit1(pls)),bit1(pls))))),m)),one_one(int))),t) = hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,s),number_number_of(nat,hAPP(int,int,plus_plus(int,bit1(pls)),bit1(pls))))),one_one(int)). [copy(2009),rewrite([1756(13),1756(20),1756(56)])]. 9.21/9.33 2268 pls = ti(int,pls) # label(tsy_c_Int_OPls_res) # label(hypothesis). [assumption]. 9.21/9.33 2269 ti(int,pls) = pls. [copy(2268),flip(a)]. 9.21/9.33 2290 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_96_zadd__assoc) # label(axiom). [clausify(549)]. 9.21/9.33 2383 hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(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)),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))),zero_zero(int)))) # label(fact_2__096_I4_A_K_Am_A_L_A1_J_A_K_At_A_060_A_I4_A_K_Am_A_L_A1_J_A_K_A0_096) # label(axiom). [assumption]. 9.21/9.33 2384 hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,hAPP(int,int,plus_plus(int,bit1(pls)),hAPP(int,int,plus_plus(int,bit1(pls)),hAPP(int,int,plus_plus(int,bit1(pls)),bit1(pls)))))),m)),one_one(int))),t)),hAPP(int,int,times_times(int,hAPP(int,int,plus_plus(int,hAPP(int,int,times_times(int,number_number_of(int,hAPP(int,int,plus_plus(int,bit1(pls)),hAPP(int,int,plus_plus(int,bit1(pls)),hAPP(int,int,plus_plus(int,bit1(pls)),bit1(pls)))))),m)),one_one(int))),zero_zero(int)))). [copy(2383),rewrite([1756(21),1756(28),2290(41),1756(66),1756(73),2290(86)])]. 9.21/9.33 2414 ti(A,hAPP(B,A,C,D)) = hAPP(B,A,C,D) # label(tsy_c_hAPP_res) # label(axiom). [clausify(664)]. 9.21/9.33 2515 ti(int,A) = hAPP(int,int,plus_plus(int,A),pls) # label(fact_129_add__Pls__right) # label(axiom). [clausify(759)]. 9.21/9.33 2516 hAPP(int,int,plus_plus(int,A),pls) = ti(int,A). [copy(2515),flip(a)]. 9.21/9.33 2567 bit1(A) = hAPP(int,int,plus_plus(int,hAPP(int,int,plus_plus(int,one_one(int)),A)),A) # label(fact_164_Bit1__def) # label(axiom). [clausify(791)]. 9.21/9.33 2568 bit1(A) = hAPP(int,int,plus_plus(int,one_one(int)),hAPP(int,int,plus_plus(int,A),A)). [copy(2567),rewrite([2290(13)])]. 9.21/9.33 2855 number_number_of(int,A) = ti(int,A) # label(fact_22_number__of__is__id) # label(axiom). [clausify(980)]. 9.21/9.33 2864 number_number_of(nat,bit0(bit1(pls))) = hAPP(nat,nat,plus_plus(nat,one_one(nat)),one_one(nat)) # label(fact_61_nat__1__add__1) # label(axiom). [assumption]. 9.21/9.33 2865 number_number_of(nat,hAPP(int,int,plus_plus(int,ti(int,one_one(int))),one_one(int))) = hAPP(nat,nat,plus_plus(nat,one_one(nat)),one_one(nat)). [copy(2864),rewrite([2568(3),2516(14),2269(10),2516(9),1756(6),1919(14)])]. 9.21/9.33 2880 pls = zero_zero(int) # label(fact_122_Pls__def) # label(axiom). [assumption]. 9.21/9.33 2909 -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,s),number_number_of(nat,bit0(bit1(pls))))),one_one(int))),zero_zero(int))) # label(conj_0) # label(negated_conjecture). [assumption]. 9.21/9.33 2910 -hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,s),hAPP(nat,nat,plus_plus(nat,one_one(nat)),one_one(nat)))),one_one(int))),zero_zero(int))). [copy(2909),rewrite([2880(18),2568(20),1663(32),1919(28),1663(26),1756(22),1919(30),2865(29)])]. 9.21/9.33 2988 number_number_of(int,bit1(pls)) = one_one(int). [resolve(1068,a,1089,a)]. 9.21/9.33 2989 ti(int,one_one(int)) = one_one(int). [copy(2988),rewrite([2880(2),2568(4),1663(16),1919(12),1663(10),2855(6),1854(6)])]. 9.21/9.33 2997 number_number_of(int,pls) = zero_zero(int). [resolve(1068,a,1097,a)]. 9.21/9.33 2998 ti(int,zero_zero(int)) = zero_zero(int). [copy(2997),rewrite([2880(2),2855(4)])]. 9.21/9.33 3184 zero_zero(int) = hAPP(int,int,times_times(int,A),zero_zero(int)). [resolve(1142,a,1156,a)]. 9.21/9.33 3185 hAPP(int,int,times_times(int,A),zero_zero(int)) = zero_zero(int). [copy(3184),flip(a)]. 9.21/9.33 4021 hAPP(int,int,times_times(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))),t) = hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,s),number_number_of(nat,hAPP(int,int,plus_plus(int,one_one(int)),one_one(int))))),one_one(int)). [back_rewrite(2010),rewrite([2880(17),2568(19),1663(31),2998(26),1663(25),2989(20),2880(20),2568(22),1663(34),2998(29),1663(28),2989(23),2880(27),2568(29),1663(41),2998(36),1663(35),2989(30),2880(30),2568(32),1663(44),2998(39),1663(38),2989(33),2290(33),2855(34),2414(34),2880(55),2568(57),1663(69),2998(64),1663(63),2989(58),2880(58),2568(60),1663(72),2998(67),1663(66),2989(61)])]. 9.21/9.33 4045 hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,plus_plus(int,hAPP(nat,int,power_power(int,s),number_number_of(nat,hAPP(int,int,plus_plus(int,one_one(int)),one_one(int))))),one_one(int))),zero_zero(int))). [back_rewrite(2384),rewrite([2880(22),2568(24),1663(36),2998(31),1663(30),2989(25),2880(28),2568(30),1663(42),2998(37),1663(36),2989(31),2880(34),2568(36),1663(48),2998(43),1663(42),2989(37),2880(37),2568(39),1663(51),2998(46),1663(45),2989(40),2855(42),2414(42),4021(50),2880(47),2568(49),1663(61),2998(56),1663(55),2989(50),2880(53),2568(55),1663(67),2998(62),1663(61),2989(56),2880(59),2568(61),1663(73),2998(68),1663(67),2989(62),2880(62),2568(64),1663(76),2998(71),1663(70),2989(65),2855(67),2414(67),3185(76)])]. 9.21/9.33 4212 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(2865),rewrite([2989(8)])]. 9.21/9.33 4378 $F. [back_rewrite(4045),rewrite([4212(27)]),unit_del(a,2910)]. 9.21/9.33 9.21/9.33 % SZS output end Refutation 9.21/9.33 ============================== end of proof ========================== 9.21/9.33 9.21/9.33 ============================== STATISTICS ============================ 9.21/9.33 9.21/9.33 Given=0. Generated=2629. Kept=2042. proofs=1. 9.21/9.33 Usable=0. Sos=1380. Demods=242. Limbo=302, Disabled=2838. Hints=0. 9.21/9.33 Megabytes=12.36. 9.21/9.33 User_CPU=7.64, System_CPU=0.01, Wall_clock=8. 9.21/9.33 9.21/9.33 ============================== end of statistics ===================== 9.21/9.33 9.21/9.33 ============================== end of search ========================= 9.21/9.33 9.21/9.33 THEOREM PROVED 9.21/9.33 % SZS status Theorem 9.21/9.33 9.21/9.33 Exiting with 1 proof. 9.21/9.33 9.21/9.33 Process 37457 exit (max_proofs) Sat Jul 14 04:49:36 2018 9.21/9.33 Prover9 interrupted 9.21/9.33 EOF